H ANDBOOK OF M ATHEMATICAL F LUID DYNAMICS Volume I
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H ANDBOOK OF M ATHEMATICAL F LUID DYNAMICS Volume I Edited by
S. FRIEDLANDER University of Illinois-Chicago, Chicago, Illinois, USA
D. SERRE Ecole Normale Supérieure de Lyon, Lyon, France
2002 ELSEVIER Amsterdam • Boston • London • New York • Oxford • Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo
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ISBN: 0-444-50330-7 First edition 2002 Library of Congress Cataloging-in-Publication Data Handbook of mathematical fluid dynamics/edited by S.J. Friedlander, D. Serre. p. cm. Includes bibliographical references and index. ISBN 0-444-50330-7 1. Fluid dynamics. I. Friedlander, S. J. II. Serre, D. (Denis) QA911 .H34 2002 532’.05–dc21 2002019382 British Library Cataloguing in Publication Data Handbook of mathematical fluid dynamics Vol. 1 edited by S.J. Friedlander, D. Serre 1. Fluid dynamics – Mathematics I. Friedlander, Susan, 1946 – II. Serre, D. (Denis) III. Mathematical fluid dynamics 532’.05’0151 ISBN: 0444503307 ∞ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper).
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Preface
The motion of fluids has intrigued scientists since antiquity and we may say that the field of mathematical fluid dynamics originated more than two centuries ago. In 1755 Euler [2] gave a mathematical formulation of the principle of conservation of mass in terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear and angular momentum by PDEs. Material symmetry and frame invariance were used by Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of internal frictional forces were modeled mathematically by Navier [8], Poisson [9], SaintVenant [11] and Stokes [12]. In the 19th century no sharp distinction was drawn between mathematicians and physicists as we sometime see in more recent times. The formulation of the equations of fluid motion could be considered as either mathematics or physics. The first work in fluid dynamics that has a “modern” mathematical flavor may have been done by Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous problem. Riemann recognized the mathematical nature of the entropy. This notion led him to his duality method for solving the non-characteristic Cauchy problem for linear hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his contemporaries. What we now call the Cauchy problem for a PDE and the search for its solution did not have the significance that it is accorded nowadays. Even Poincaré did not raise that kind of question in his Théorie des tourbillons. For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28. Then in 1933 Wolibner [13] proves their persistence. Last, Leray’s fundamental analysis of the Navier–Stokes equations for an incompressible fluid is published in 1934 [3]. As much as Riemann, Leray developed new mathematical tools which proved to have independent interest: e.g., weak solutions (that we now call Leray’s solutions in this context) and topological degree (a joint work with Schauder [4]). Since the 1930s, the interest that mathematicians devote to fluid dynamics has unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on the Navier–Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic displays such a variety of models and questions that thousands of scientists, among them many mathematicians, focus their research on fluid dynamics. v
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Preface
Because of the intense activity and the rapid increase of our knowledge, it appeared desirable to set up a landmark. Named “The Handbook of Mathematical Fluid Dynamics”, it is a collection of refereed review articles written by some of the very best specialists in their discipline. The authors were also chosen for the high quality of their expository style. We, the editors, are much indebted to our colleagues who enthusiastically accepted this challenge, and who made great efforts to write for a wide audience. We also thank the referees who worked hard to ensure the excellent quality of the articles. Of course, the length of these articles varies considerably since each topic can be narrow or wide. A few of them have the appearance of a small book. Their authors deserve special thanks, for the immense work that they achieved and for their generosity in choosing to publish their work in this Handbook. At the begining of our editorial work, we decided to restrict the contents to mathematical aspects of fluid dynamics, avoiding to a large extent the physical and the numerical aspects. We highly respect these facets of fluid dynamics and we encouraged the authors to describe the physical meaning of their mathematical results and questions. But we considered that the physics and the numerics were extremely well developed in other collections of a similar breadth (see, for instance, several articles in the Handbook of Numerical Analysis, Elsevier, edited by P. Ciarlet and J.-L. Lions). Furthermore, if we had made a wider choice, our editing work would have been an endless task! This has been our only restriction. We have tried to cover many kinds of fluid models, including ones that are rarefied, compressible, incompressible, viscous or inviscid, heat conducting, capillary, perfect or real, coupled with solid mechanics or with electromagnetism. We have also included many kinds of questions: the Cauchy problem, steady flows, boundary value problems, stability issues, turbulence, etc. These lists are by no mean exhaustive. We were only limited in some places by the lack, at present, of mathematical theories. Our first volume is more or less specialized to compressible issues. There might be valid mathematical, historical or physical reasons to explain such a choice, arguing, for instance, for the priority of Riemann’s work, or that kinetic models are at the very source of almost all other fluid models under various limiting regimes. The truth is more fortuitous, namely that the authors writing on compressible issues were the most prompt in delivering their articles in final form. The second and third volumes will be primarily devoted to problems arising in incompressible flows. Last, but not least, we thank the Editors at Elsevier, who gave us the opportunity of making available a collection of articles that we hope will be useful to many mathematicians and those beyond the mathematical community. We are also happy to thank Sylvie Benzoni-Gavage for her invaluable assistance. Chicago, Lyon September 2001 Susan Friedlander and Denis Serre
[email protected] [email protected]
Preface
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References [1] A.-L. Cauchy, Bull. Soc. Philomathique (1823), 9–13; Exercices de Mathématiques 2 (1827), 42–56, 108– 111; 4 (1829), 293–319. [2] L. Euler, Mém. Acad. Sci. Berlin 11 (1755), 274–315; 15 (1759), 210–240. [3] J. Leray, J. Math. Pures Appl. 12 (1933), 1–82; 13 (1934), 331–418; Acta Math. 63 (1934), 193–248. [4] J. Leray and J. Schauder, Ann. Sci. Ecole Norm. Sup. (3) 51 (1934), 45–78. [5] L. Lichtenstein, Math. Z. 23 (1925), 89–154; 26 (1926), 196–323; 28 (1928), 387–415; 32 (1930), 608–725. [6] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vols. 1, 2, Oxford Univ. Press (1998). [7] J. Nash, Bull. Soc. Math. France 90 (1962), 487–497. [8] C.L.M.H. Navier, Mém. Acad. Sci. Inst. France 6 (1822), 375–394. [9] S.D. Poisson, J. Ecole Polytechnique 13 (1831), 1–174. [10] B. Riemann, Gött. Abh. Math. Cl. 8 (1860), 43–65. [11] B. de Saint-Venant, C. R. Acad. Sci. Paris 17 (1843). [12] G.G. Stokes, Trans. Cambridge Philos. Soc. 8 (1849), 207–319. [13] W. Wolibner, Math. Z. 37 (1933), 698–726.
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List of Contributors Blokhin, A., Sobolev Institute of Mathematics, Novosibirsk, Russia (Ch. 6) Cercignani, C., Politecnico di Milano, Milano, Italy (Ch. 1) Chen, G.-Q., Northwestern University, Evanston, IL (Ch. 5) Fan, H., Georgetown University, Washington DC (Ch. 4) ˇ Praha, Czech Republic (Ch. 3) Feireisl, E., Institute of Mathematics AV CR, Galdi, G.P., University of Pittsburgh, Pittsburgh, PA (Ch. 7) Slemrod, M., University of Wisconsin-Madison, Madison, WI (Ch. 4) Trakhinin, Yu., Sobolev Institute of Mathematics, Novosibirsk, Russia (Ch. 6) Villani, C., UMPA, ENS Lyon, Lyon, France (Ch. 2) Wang, D., University of Pittsburgh, Pittsburgh, PA (Ch. 5)
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Contents Preface List of Contributors
v ix
1. The Boltzmann equation and fluid dynamics C. Cercignani 2. A review of mathematical topics in collisional kinetic theory C. Villani 3. Viscous and/or heat conducting compressible fluids E. Feireisl 4. Dynamic flows with liquid/vapor phase transitions H. Fan and M. Slemrod 5. The Cauchy problem for the Euler equations for compressible fluids G.-Q. Chen and D. Wang 6. Stability of strong discontinuities in fluids and MHD A. Blokhin and Y. Trakhinin 7. On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications G.P. Galdi
1
Author Index Subject Index
71 307 373 421 545
653
793 807
xi
CHAPTER 1
The Boltzmann Equation and Fluid Dynamics C. Cercignani Dipartimento di Matematica, Politecnico di Milano, Milano, Italy
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2. The basic molecular model . . . . . . . . . . . . . . . . 3. The Boltzmann equation . . . . . . . . . . . . . . . . . . 4. Molecules different from hard spheres . . . . . . . . . . 5. Collision invariants . . . . . . . . . . . . . . . . . . . . . 6. The Boltzmann inequality and the Maxwell distributions 7. The macroscopic balance equations . . . . . . . . . . . . 8. The H-theorem . . . . . . . . . . . . . . . . . . . . . . . 9. Model equations . . . . . . . . . . . . . . . . . . . . . . 10. The linearized collision operator . . . . . . . . . . . . . 11. Boundary conditions . . . . . . . . . . . . . . . . . . . . 12. The continuum limit . . . . . . . . . . . . . . . . . . . . 13. Free-molecule and nearly free-molecule flows . . . . . . 14. Perturbations of equilibria . . . . . . . . . . . . . . . . . 15. Approximate methods for linearized problems . . . . . . 16. Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . 17. Polyatomic gases . . . . . . . . . . . . . . . . . . . . . . 18. Chemistry and radiation . . . . . . . . . . . . . . . . . . 19. The DSMC method . . . . . . . . . . . . . . . . . . . . 20. Some applications of the DSMC method . . . . . . . . . 21. Concluding remarks . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The Boltzmann equation and fluid dynamics
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1. Introduction We say that a gas flow is rarefied when the so-called mean free-path of the gas molecules, i.e., the average distance covered by a molecule between to subsequent collisions, is not completely negligible with respect to a typical geometric length (the radius of curvature of the nose of a flying vehicle, the radius of a pipe, etc.). The most remarkable feature of rarefied flows is that the Navier–Stokes equations do not apply. One must then resort to the concepts of kinetic theory of gases and the Navier–Stokes equations must be replaced by the Boltzmann equation [43]. Thus the Boltzmann equation became a practical tool for the aerospace engineers, when they started to remark that flight in the upper atmosphere must face the problem of a decrease in the ambient density with increasing height. This density reduction would alleviate the aerodynamic forces and heat fluxes that a flying vehicle would have to withstand. However, for virtually all missions, the increase of altitude is accompanied by an increase in speed; thus it is not uncommon for spacecraft to experience its peak heating at considerable altitudes, such as, e.g., 70 km. When the density of a gas decreases, there is, of course, a reduction of the number of molecules in a given volume and, what is more important, an increase in the distance between two subsequent collisions of a given molecule, till one may well question the validity of the Euler and Navier–Stokes equations, which are usually introduced on the basis of a continuum model which does not take into account the molecular nature of a gas. It is to be remarked that, as we shall see, the use of those equations can also be based on the kinetic theory of gases, which justifies them as asymptotically useful models when the mean free path is negligible. In the area of environmental problems, the Boltzmann equation is also required. Understanding and controlling the formation, motion, reactions and evolution of particles of varying composition and shapes, ranging from a diameter of the order of 0.001 µm to 50 µm, as well as their space-time distribution under gradients of concentration, pressure, temperature and the action of radiation, has grown in importance, because of the increasing awareness of the local and global problems related to the emission of particles from electric power plants, chemical plants, vehicles as well as of the role played by small particles in the formation of fog and clouds, in the release of radioactivity from nuclear reactor accidents, and in the problems arising from the exhaust streams of aerosol reactors, such as those used to produce optical fibers, catalysts, ceramics, silicon chips and carbon whiskers. One cubic centimeter of atmospheric air at ground level contains approximately 2.5 × 1019 molecules. About a thousand of them may be charged (ions). A typical molecular diameter is 3 × 10−10 m (3×10−4 µm) and the average distance between the molecules is about ten times as much. The mean free path is of the order of 10−8 m, or 10−2 µm. In addition to molecules and ions one cubic centimeter of air also contains a significant number of particles varying in size, as indicated above. In relatively clean air, the number of these particles can be 105 or more, including pollen, bacteria, dust, and industrial emissions. They can be both beneficial and detrimental, and arise from a number of natural sources as well as from the activities of all living organisms, especially humans. The particles can have complex chemical compositions and shapes, and may even be toxic or radioactive.
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C. Cercignani
A suspension of particles in a gas is known as an aerosol. Atmospheric aerosols are of global interest and have important impact on our lives. Aerosols are also of great interest in numerous scientific and engineering applications [175]. A third area of application of rarefied gas dynamics has emerged in the last quarter of the twentieth century. Small size machines, called micromachines, are being designed and built. Their typical sizes range from a few microns to a few millimiters. Rarefied flow phenomena that are more or less laboratory curiosities in machines of more usual size can form the basis of important systems in the micromechanical domain. A further area of interest occurs in the vacuum industry. Although this area existed for a long time, the expense of the early computations with kinetic theory precluded applications of numerical methods. The latter could develop only in the context of the aerospace industry, because the big budgets required till recently were available only there. The basic parameter measuring the degree of rarefaction of a gas is the Knudsen number (Kn), the ratio between the mean free path λ and another typical length. Of course, one can consider several Knudsen numbers, based on different characteristic lengths, exactly as one does for the Reynolds number. Thus, in the flow past a body, there are two important macroscopic lengths: the local radius of curvature and the thickness of the viscous boundary layer δ, and one can consider Knudsen numbers based on either length. Usually the second one (Knδ = λ/δ), gives the most severe restriction to the use of Navier– Stokes equations in aerospace applications. When Kn is larger than (say) 0.01, the presence of a thin layer near the wall, of thickness of the order λ (Knudsen layer), influences the viscous profile in a significant way. This and other effects are of interest in both high altitude flight and aerosol science; in particular they are all met by a shuttle when returning to Earth. Another phenomenon of importance is the formation of shock waves, which are not discontinuity surfaces, but thin layers (the thickness is zero only if the Euler model is adopted). When the mean free path increases, one witnesses a thickening of the shock waves, whose thickness is of the order of 6λ. The bow shock in front of a body merges with the viscous boundary layer; that is why this regime is sometimes called the merged layer regime by aerodynamicists. We shall use the other frequently used name of transition regime. When Kn is large (few collisions), phenomena related to gas-surface interaction play an important role. They enter the theory in the form of boundary conditions for the Boltzmann equation. One distinguishes between free-molecule and nearly free-molecule regimes. In the first case the molecular collisions are completely negligible, while in the second they can be treated as a perturbation.
2. The basic molecular model According to kinetic theory, a gas in normal conditions (no chemical reactions, no ionization phenomena, etc.) is formed of elastic molecules rushing hither and thither at high speed, colliding and rebounding according to the laws of elementary mechanics. Monatomic molecules of a gas are frequently assumed to be hard, elastic, and perfectly smooth spheres. One can also consider these molecules to be centers of forces that move
The Boltzmann equation and fluid dynamics
5
according to the laws of classical mechanics. More complex models are needed to describe polyatomic molecules. The rules generating the dynamics of many spheres are easy to describe: thus, e.g., if no body forces, such as gravity, are assumed to act on the molecules, each of them will move in a straight line unless it happens to strike another molecule or a solid wall. The phenomena associated with this dynamics are not so simple, especially when the number of spheres is large. It turns out that this complication is always present when dealing with a gas, because the number of molecules usually considered is extremely large: there are about 2.7 · 1019 in a cubic centimeter of a gas at atmospheric pressure and a temperature of 0 ◦ C. Given the vast number of particles to be considered, it would of course be a hopeless task to attempt to describe the state of the gas by specifying the so-called microscopic state, i.e., the position and velocity of every individual sphere; we must have recourse to statistics. A description of this kind is made possible because in practice all that our typical observations can detect are changes in the macroscopic state of the gas, described by quantities such as density, bulk velocity, temperature, stresses, heat-flow, which are related to some suitable averages of quantities depending on the microscopic state.
3. The Boltzmann equation The exact dynamics of N particles is a useful conceptual tool, but cannot in any way be used in practical calculations because it requires a huge number of real variables (of the order of 1020). The basic tool is the one-particle probability density, or distribution function P (1) (x, ξ , t). The latter is a function of seven variables, i.e., the components of the two vectors x and ξ and time t. Let us consider the meaning of P (1) (x, ξ , t); it gives the probability density of finding one fixed particle (say, the one labelled by 1) at a certain point (x, ξ ) of the six-dimensional reduced phase space associated with the position and velocity of that molecule. In order to simplify the treatment, we shall for the moment assume that the molecules are hard spheres, whose center has position x. When the molecules collide, momentum and kinetic energy must be conserved; thus the velocities after the impact, ξ 1 and ξ 2 , are related to those before the impact, ξ 1 and ξ 2 , by ξ 1 = ξ 1 − n n · (ξ 1 − ξ 2 ) , ξ 2 = ξ 2 + n n · (ξ 1 − ξ 2 ) ,
(3.1)
where n is the unit vector along ξ 1 − ξ 1 . Note that the relative velocity V = ξ1 − ξ2
(3.2)
V = V − 2n(n · V),
(3.3)
satisfies
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C. Cercignani
i.e., undergoes a specular reflection at the impact. This means that if we split V at the point of impact into a normal component Vn , directed along n and a tangential component Vt (in the plane normal to n), then Vn changes sign and Vt remains unchanged in a collision. We can also say that n bisects the directions of V and −V = −(ξ 1 − ξ 2 ). Let us remark that, in the absence of collisions, P (1) would remain unchanged along the trajectory of a particle. Accordingly we must evaluate the effects of collisions on the time evolution of P (1) . Note that the probability of occurrence of a collision is related to the probability of finding another molecule with a center at exactly one diameter from the center of the first one, whose distribution function is P (1) . Thus, generally speaking, in order to write the evolution equation for P (1) we shall need another function, P (2) , which gives the probability density of finding, at time t, the first molecule at x1 with velocity ξ 1 and the second at x2 with velocity ξ 2 ; obviously P (2) = P (2) (x1 , x2 , ξ 1 , ξ 2 , t). Hence P (1) satisfies an equation of the following form: ∂P (1) ∂P (1) + ξ1 · = G − L. ∂t ∂x1
(3.4)
Here L dx1 dξ 1 dt gives the expected number of particles with position between x1 and x1 +dx1 and velocity between ξ 1 and ξ 1 +dξ 1 which disappear from these ranges of values because of a collision in the time interval between t and t + dt and G dx1 dξ 1 dt gives the analogous number of particles entering the same range in the same time interval. The count of these numbers is easy, provided we use the trick of imagining particle 1 as a sphere at rest and endowed with twice the actual diameter σ and the other particles to be point masses with velocity (ξ i − ξ 1 ) = Vi . In fact, each collision will send particle 1 out of the above range and the number of the collisions of particle 1 will be the number of expected collisions of any other particle with that sphere. Since there are exactly (N − 1) identical point masses and multiple collisions are disregarded, G = (N − 1)g and L = (N − 1)l, where the lower case letters indicate the contribution of a fixed particle, say particle 2. We shall then compute the effect of the collisions of particle 2 with particle 1. Let x2 be a point of the sphere such that the vector joining the center of the sphere with x2 is σ n, where n is a unit vector. A cylinder with height |V · n| dt (where we write just V for V2 ) and base area dS = σ 2 dn (where dn is the area of a surface element of the unit sphere about n) will contain the particles with velocity ξ 2 hitting the base dS in the time interval (t, t + dt); its volume is σ 2 dn|V · n| dt. Thus the number of collisions of particle 2 with particle 1 in the ranges (x1 , x1 + dx1), (ξ 1 , ξ 1 + dξ 1 ), (x2 , x2 + dx2), (ξ 2 , ξ 2 + dξ 2 ), (t, t + dt) occuring at points of dS is P (2) (x1 , x2 , ξ 1 , ξ 2 , t) dx1 dξ 1 dξ 2 σ 2 dn|V2 · n| dt. If we want the number of collisions of particle 1 with 2, when the range of the former is fixed but the latter may have any velocity ξ 2 and any position x2 on the sphere (i.e., any n), we integrate over the sphere and all the possible velocities of particle 2 to obtain: l dx1 dξ 1 dt = dx1 dξ 1 dt
R 3 B−
P (2) (x1 , x1 + σ n, ξ 1 , ξ 2 , t)|V · n|σ 2 dn dξ 2 ,
(3.5)
The Boltzmann equation and fluid dynamics
7
where B − is the hemisphere corresponding to V · n < 0 (the particles are moving one toward the other before the collision). Thus we have the following result: L = (N − 1)σ 2
R 3 B−
P (2) (x1 , x1 + σ n, ξ 1 , ξ 2 , t)(ξ 2 − ξ 1 ) · n dξ 2 dn. (3.6)
The calculation of the gain term G is exactly the same as the one for L, except for the fact that we have to integrate over the hemisphere B + , defined by V · n > 0 (the particles are moving away one from the other after the collision). Thus we have: G = (N − 1)σ 2
R 3 B+
P (2) (x1 , x1 + σ n, ξ 1 , ξ 2 , t)(ξ 2 − ξ 1 ) · n dξ 2 dn. (3.7)
We can now insert in Equation (3.4) the information that the probability density P (2) is continuous at a collision; in other words, although the velocities of the particles undergo the discontinuous change described by Equations (3.1), we can write: P (2) (x1 , ξ 1 , x2 , ξ 2 , t) = P (2) x1 , ξ 1 − n(n · V), x2 , ξ 2 + n(n · V), t if |x1 − x2 | = σ.
(3.8)
For brevity, we write (in agreement with Equations (3.1)): ξ 1 = ξ 1 − n(n · V),
ξ 2 = ξ 2 + n(n · V).
(3.9)
Inserting Equation (3.8) in Equation (3.5) we thus obtain: G = (N − 1)σ 2
R 3 B+
P (2) (x1 , x1 + σ n, ξ 1 , ξ 2 , t)(ξ 2 − ξ 1 ) · n dξ 2 dn (3.10)
which is a frequently used form. Sometimes n is changed into −n in order to have the same integration range as in L; the only change (in addition to the change in the range) is in the second argument of P (2) , which becomes x1 − σ n. At this point we are ready to understand Boltzmann’s argument. N is a very large number and σ (expressed in common units, such as, e.g., centimeters) is very small; to fix the ideas, let us consider a box whose volume is 1 cm3 at room temperature and atmospheric pressure. Then N ∼ = Nσ 2 ∼ = 104 cm2 = 1 m2 is a sizable = 1020 and σ ∼ = 10−8 cm. Then (N − 1)σ 2 ∼ quantity, while we can neglect the difference between x1 and x1 + σ n. This means that the equation to be written can be rigorously valid only in the so called Boltzmann–Grad limit, when N → ∞, σ → 0 with Nσ 2 finite.
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C. Cercignani
In addition, the collisions between two preselected particles are rather rare events. Thus two spheres that happen to collide can be thought to be two randomly chosen particles and it makes sense to assume that the probability density of finding the first molecule at x1 with velocity ξ 1 and the second at x2 with velocity ξ 2 is the product of the probability density of finding the first molecule at x1 with velocity ξ 1 times the probability density of finding the second molecule at x2 with velocity ξ 2 . If we accept this we can write (assumption of molecular chaos): P (2) (x1 , ξ 1 , x2 , ξ 2 , t) = P (1) (x1 , ξ 1 , t)P (1) (x2 , ξ 2 , t)
(3.11)
for two particles that are about to collide, or, letting σ = 0 P (2) (x1 , ξ 1 , x1 + σ n, ξ 2 , t) = P (1) (x1 , ξ 1 , t)P (1) (x1 , ξ 2 , t) for (ξ 2 − ξ 1 ) · n < 0.
(3.12)
Thus we can apply this recipe to the loss term (3.4) but not to the gain term in the form (3.5). It is possible, however, to apply Equation (3.12) (with ξ 1 , ξ 2 in place of ξ 1 , ξ 2 ) to the form (3.8) of the gain term, because the transformation (3.9) maps the hemisphere B + onto the hemisphere B − . If we accept all the simplifying assumptions made by Boltzmann, we obtain the following form for the gain and loss terms: P (1) (x1 , ξ 1 , t)P (1) (x1 , ξ 2 , t)(ξ 2 − ξ 1 ) · n dξ 2 dn, (3.13) G = Nσ 2 L = Nσ
R 3 B−
2 R 3 B−
P (1) (x1 , ξ 1 , t)P (1) (x1 , ξ 2 , t)(ξ 2 − ξ 1 ) · n dξ 2 dn.
(3.14)
By inserting these expressions in Equation (3.6) we can write the Boltzmann equation in the following form: ∂P (1) ∂P (1) + ξ1 · ∂t ∂x1 (1) = Nσ 2 P (x1 , ξ 1 , t)P (1) (x1 , ξ 2 , t) R 3 B−
−P
(1)
(x1 , ξ 1 , t)P (1) (x1 , ξ 2 , t) (ξ 2 − ξ 1 ) · n dξ 2 dn.
(3.15)
We remark that the expressions for ξ 1 and ξ 2 given in Equations (3.1) are by no means the only possible ones. In fact we might use a different unit vector ω, directed as V , instead of n. Then Equations (3.1) is replaced by: 1 ξ 1 = ξ¯ + |ξ 1 − ξ 2 |ω, 2 1 ξ 2 = ξ¯ − |ξ 1 − ξ 2 |ω, 2
(3.16)
The Boltzmann equation and fluid dynamics
9
where ξ¯ = 12 (ξ 1 + ξ 2 ) is the velocity of the center of mass. The relative velocity V satisfies V = ω|V|.
(3.17)
The recipes (3.13) and (3.14) can be justified at various levels of rigor [36,113,39,47]. We finally mention that we have for simplicity neglected any body force acting on the molecules, such as gravity. It is not hard to take them into account; if the force per unit mass acting on the molecules is denoted by X, then a term X · ∂P (1) /∂ξ 1 must be added to the left-hand side of Equation (3.8).
4. Molecules different from hard spheres In the previous section we have discussed the Boltzmann equation when the molecules are assumed to be identical hard spheres. There are several possible generalizations of this molecular model, the most obvious being the case of molecules which are identical point masses interacting with a central force, a good general model for monatomic gases. If the range of the force extends to infinity, there is a complication due to the fact that two molecules are always interacting and the analysis in terms of “collisions” is no longer possible. If, however, the gas is sufficiently dilute, we can take into account that the molecular interaction is negligible for distances larger than a certain σ (the “molecular diameter”) and assume that when two molecules are at a distance smaller than σ , then no other molecule is interacting with them and the binary collision analysis considered in the previous section can be applied. The only difference arises in the factor σ 2 |(ξ 2 − ξ 1 ) · n| which turns out to be replaced by a function of V = |ξ 2 − ξ 1 | and the angle θ between n and V ([39,35,42]). Thus the Boltzmann equation for monatomic molecules takes on the following form: ∂P (1) ∂P (1) + ξ1 · ∂t ∂x1 (1) P (x1 , ξ 1 , t)P (1) (x1 , ξ 2 , t) =N R3
−P
(1)
B−
(x1 , ξ 1 , t)P (1) (x1 , ξ 2 , t) B θ, |ξ 2 − ξ 1 | dξ 2 dθ dε,
(4.1)
where ε is the other angle which, together with θ , identifies the unit vector n. The function B(θ, V ) depends, of course, on the specific law of interaction between the molecules. In the case of hard spheres, of course B θ, |ξ 2 − ξ 1 | = cos θ sin θ |ξ 2 − ξ 1 |.
(4.2)
In spite of the fact that the force is cut at a finite range σ when writing the Boltzmann equation, infinite range forces are frequently used. This has the disadvantage of making the integral in Equation (4.1) rather hard to handle; in fact, one cannot split it into the difference of two terms (the loss and the gain), because each of them would be a divergent
10
C. Cercignani
integral. This disadvantage is compensated in the case of power law forces, because one can separate the dependence on θ from the dependence upon V . In fact, one can show [39, 35] that, if the intermolecular force varies as the n-th inverse power of the distance, then B θ, |ξ 2 − ξ 1 | = β(θ )|ξ 2 − ξ 1 |(n−5)/(n−1),
(4.3)
where β(θ ) is a non-elementary function of θ (in the simplest cases it can be expressed by means of elliptic functions). In particular, for n = 5 one has the so-called Maxwell molecules, for which the dependence on V disappears. Sometimes the artifice of cutting the grazing collisions corresponding to small values of |θ − π/2| is used (angle cutoff). In this case one has both the advantage of being able to split the collision term and of preserving a relation of the form (4.3) for power-law potentials. Since solving of the Boltzmann equation with actual cross sections is complicated, in many numerical simulations use is made of the so-called variable hard sphere model in which the diameter of the spheres is an inverse power law function of the relative speed V (see [43]). Another important case is when we deal with a mixture rather than with a single gas. In this case we have n unknowns, if n is the number of the species, and n Boltzmann equations; in each of them there are n collision terms to describe the collision of a molecule with other molecules of all the possible species [43,39]. If the gas is polyatomic, then the gas molecules have other degrees of freedom in addition to the translation ones. This in principle requires using quantum mechanics, but one can devise useful and accurate models in the classical scheme as well. Frequently the internal energy Ei is the only additional variable that is needed; in which case one can think of the gas as of a mixture of species [43,39], each differing from the other because of the value of Ei . If the latter variable is discrete we obtain a strict analogy with a mixture; otherwise we have a continuum of species. We remark that in both cases, kinetic energy is not preserved by collisions, because internal energy also enters into the balance; this means that a molecule changes its “species” when colliding. This is the simplest example of a “reacting collision”, which may be generalized to actual chemical species when chemical reactions occur. The subject of mixture and polyatomic gases will be taken up again in Section 16.
5. Collision invariants Before embarking in a discussion of the properties of the solutions of the Boltzmann equation we remark that the unknown of the latter is not always chosen to be a probability density as we have done so far; it may be multiplied by a suitable factor and transformed into an (expected) number density or an (expected) mass density (in phase space, of course). The only thing that changes is the factor in front of Equations (3.1) which is no longer N . In order to avoid any commitment to a special choice of that factor we replace NB(θ, V ) by B(θ, V ) and the unknown P by another letter, f (which is also the most commonly used letter to denote the one-particle distribution function, no matter what its
The Boltzmann equation and fluid dynamics
11
normalization is). In addition, we replace the current velocity variable ξ 1 simply by ξ and ξ 2 by ξ ∗ . Thus we rewrite Equation (4.1) in the following form: ∂f ∂f +ξ · = ∂t ∂x
B−
R3
(f f∗ − ff∗ )B(θ, V ) dξ ∗ dθ dε,
(5.1)
where V = |ξ − ξ ∗ |. The velocity arguments ξ and ξ ∗ in f and f∗ are of course given by Equations (3.1) (or (3.15)) with the suitable modification. The right-hand side of Equation (5.1) contains a quadratic expression Q(f, f ), given by: Q(f, f ) =
R3
S2
(f f∗ − ff∗ )B(θ, V ) dξ ∗ dθ dε.
(5.2)
This expression is called the collision integral or, simply, the collision term and the quadratic operator Q goes under the name of collision operator. In this section we study some elementary properties of Q. Actually it turns out that it is more convenient to study the slightly more general bilinear expression associated with Q(f, f ), i.e.: 1 Q(f, g) = 2
R3 S 2
(f g∗ + g f∗ − fg∗ − gf∗ )B(θ, V ) dξ ∗ dθ dε.
(5.3)
It is clear that when g = f , Equation (5.3) reduces to Equation (5.2) and Q(f, g) = Q(g, f ).
(5.4)
Our first aim is to indicate a basic property of the eightfold integral: Q(f, g)φ(ξ ) dξ R3
1 = 2
B−
R3 R3
(f g∗ + g f∗ − fg∗ − gf∗ )φ(ξ )B(θ, V ) dξ ∗ dξ dθ dε, (5.5)
where f, g and φ are functions such that the indicated integrals exist and the order of integration does not matter. Simple manipulations (see [43,39,35]) give the following result: Q(f, g)φ(ξ ) dξ R3
1 = 8
R3 R3
B−
(f g∗ + g f∗ − fg∗ − gf∗ )
× (φ + φ∗ − φ − φ∗ )B(θ, V ) dξ ∗ dξ dθ dε.
(5.6)
12
C. Cercignani
This relation expresses a basic property of the collision term, which is frequently used. In particular, when g = f , Equation (5.6) reads Q(f, f )φ(ξ ) dξ R3
=
1 4
R3
R3
B−
(f f∗ − ff∗ )(φ + φ∗ − φ − φ∗ )B(θ, V ) dξ ∗ dξ dθ dε. (5.7)
We now observe that the integral in Equation (5.6) is zero independent of the particular functions f and g, if φ + φ∗ = φ + φ∗
(5.8)
is valid almost everywhere in velocity space. Since the integral appearing in the left-hand side of Equation (5.7) is the rate of change of the average value of the function φ due to collisions, the functions satisfying Equation (5.8) are called “collision invariants”. It can be shown (see, e.g., [39]) that a continuous function φ has the property expressed by Equation (5.8) if and only if φ(ξ ) = a + b · ξ + c|ξ |2 ,
(5.9)
where a and c are constant scalars and b a constant vector. The assumption of continuity can be considerably relaxed [5,40,6]. The functions ψ0 = 1, (ψ1 , ψ2 , ψ3 ) = ξ , ψ4 = |ξ |2 are usually called the elementary collision invariants; they span the five-dimensional subspace of the collision invariants. 6. The Boltzmann inequality and the Maxwell distributions In this section we investigate the existence of positive functions f which give a vanishing collision integral: Q(f, f ) = (f f∗ − ff∗ )B(θ, V ) dξ ∗ dθ dε = 0. (6.1) R 3 B−
In order to solve this equation, we prove a preliminary result which plays an important role in the theory of the Boltzmann equation: if f is a nonnegative function such that log f Q(f, f ) is integrable and the manipulations of the previous section hold when φ = log f , then the Boltzmann inequality: log f Q(f, f ) dξ 0 (6.2) R3
holds; further, the equality sign applies if, and only if, log f is a collision invariant, or, equivalently: f = exp a + b · ξ + c|ξ |2 . (6.3)
The Boltzmann equation and fluid dynamics
13
To prove Equation (6.2) it is enough to use Equation (4.11) with φ = log f : log f Q(f, f ) dξ R3
=
1 4
R3
B−
log(ff∗ /f f∗ )(f f∗ − ff∗ )B(θ, V ) dξ dξ ∗ dε
and Equation (6.2) follows thanks to the elementary inequality (z − y) log(y/z) 0 y, z ∈ R + .
(6.4)
(6.5)
Equation (6.5) becomes an equality if and only if y = z; thus the equality sign holds in Equation (6.2) if and only if: f f∗ = ff∗
(6.6)
applies almost everywhere. But, taking the logarithms of both sides of Equation (6.6), we find that φ = log f satisfies Equation (5.8) and is thus given by Equation (5.9). f = exp(φ) is then given by Equation (6.3). We remark that in the latter equation c must be negative, since f must be integrable. If we let c = −β, b = 2βv (where v is another constant vector) Equation (6.3) can be rewritten as follows: f = A exp −β|ξ − v|2 , (6.7) where A is a positive constant related to a, c, |b|2 (β, v, A constitute a new set of constants). The function appearing in Equation (3.7) is the so called Maxwell distribution or Maxwellian. Frequently one considers Maxwellians with v = 0 (nondrifting Maxwellians), which can be obtained from drifting Maxwellians by a change of the origin in velocity space. Let us return now to the problem of solving Equation (6.1). Multiplying both sides by log f gives Equation (6.2) with the equality sign. This implies that f is a Maxwellian, by the result which has just been proved. Suppose now that f is a Maxwellian; then f = exp(φ) where φ is a collision invariant and Equation (6.6) holds; Equation (6.1) then also holds. Thus there are functions which satisfy Equation (6.1) and they are all Maxwellians, Equation (6.7). 7. The macroscopic balance equations In this section we compare the microscopic description supplied by kinetic theory with the macroscopic description supplied by continuum gas dynamics. For definiteness, in this section f will be assumed to be an expected mass density in phase space. In order to obtain a density, ρ = ρ(x, t), in ordinary space, we must integrate f with respect to ξ : ρ= f dξ . (7.1) R3
14
C. Cercignani
The bulk velocity v of the gas (e.g., the velocity of a wind), is the average of the molecular velocities ξ at a certain point x and time instant t; since f is proportional to the probability for a molecule to have a given velocity, v is given by 3 ξ f dξ v = R R 3 f dξ
(7.2)
(the denominator is required even if f is taken to be a probability density in phase space, because we are considering a conditional probability, referring to the position x). Equation (7.2) can also be written as follows: ρv =
R3
ξ f dξ
(7.3)
or, using components: ρvi =
R3
ξi f dξ
(i = 1, 2, 3).
(7.4)
The bulk velocity v is what we can directly perceive of the molecular motion by means of macroscopic observations; it is zero for a gas in equilibrium in a box at rest. Each molecule has its own velocity ξ which can be decomposed into the sum of v and another velocity c=ξ −v
(7.5)
called the random or peculiar velocity; c is clearly due to the deviations of ξ from v. It is also clear that the average of c is zero. The quantity ρvi which appears in Equation (7.4) is the i-th component of the mass flow or, alternatively, of the momentum density of the gas. Other quantities of similar nature are: the momentum flow mij =
R3
ξi ξj f dξ
(i, j = 1, 2, 3);
(7.6)
the energy density per unit volume: w=
1 2
R3
|ξ |2 f dξ ;
(7.7)
the energy flow: 1 ri = 2
R3
ξi |ξ |2 f dξ
(i, j = 1, 2, 3).
(7.8)
The Boltzmann equation and fluid dynamics
15
Equation (7.6) shows that the momentum flow is described by the components of a symmetric tensor of second order. The defining integral can be re-expressed in terms of c and v. We have [43,39,35]: mij = ρvi vj + pij ,
(7.9)
where: pij =
R3
ci cj f dξ
(i, j = 1, 2, 3)
(7.10)
plays the role of the stress tensor (because the microscopic momentum flow associated with it is equivalent to forces distributed on the boundary of any region of gas, according to the macroscopic description). Similarly one has [43,39,35]: 1 w = ρ|v|2 + ρe, 2
(7.11)
where e is the internal energy per unit mass (associated with random motions) defined by: 1 ρe = 2
R3
|c|2 f dξ ;
(7.12)
and: ri = ρvi
3 1 2 |v| + e + vj pij + qi 2
(i = 1, 2, 3),
(7.13)
j =1
where qi are the components of the heat-flow vector: qi =
1 2
ci |c|2 f dξ .
(7.14)
3
The decomposition in Equation (7.13) shows that the microscopic energy flow is a sum of a macroscopic flow of energy (both kinetic and internal), of the work (per unit area und unit time) done by stresses, and of the heat-flow. In order to complete the connection, as a simple mathematical consequence of the Boltzmann equation, one can derive five differential relations satisfied by the macroscopic quantities introduced above; these relations describe the balance of mass, momentum and energy and have the same form as in continuum mechanics. To this end let us consider the Boltzmann equation ∂f ∂f +ξ · = Q(f, f ). ∂t ∂x
(7.15)
16
C. Cercignani
If we multiply both sides by one of the elementary collision invariants ψα (α = 0, 1, 2, 3, 4), defined in Section 4, and integrate with respect to ξ , we have, thanks to Equation (1.15) with g = f and φ = ψα : ψα (ξ )Q(f, f ) dξ = 0, (7.16) R3
and hence, if it is permitted to change the order by which we differentiate with respect to t and integrate with respect to ξ : ∂ ∂t
ψα f dξ +
3
∂ ξi ψα f dξ = 0 ∂xi
(α = 1, 2, 3, 4).
(7.17)
i=1
If we take successively α = 0, 1, 2, 3, 4 and use the definitions introduced above, we obtain ∂ρ ∂ + (ρvi ) = 0, ∂t ∂xi 3
(7.18)
i=1
∂ ∂ (ρvj ) + (ρvi vj + pij ) = 0 ∂t ∂xi 3
(j = 1, 2, 3),
(7.19)
i=1
∂ 1 2 ρ|v| + ρe ∂t 2
3 3
1 2 ∂ + ρvi |v| + e + vj pij + qi = 0. ∂xi 2 i=1
(7.20)
j =1
The considerations of this section apply to all the solutions of the Boltzmann equation. The definitions, however, can be applied to any positive function for which they make sense. In particular if we take f to be a Maxwellian in the form (5.7), we find that the constant vector v appearing there is actually the bulk velocity as defined in Equation (7.2) while β and A are related to the internal energy e and the density ρ in the following way: β = 3/(4e),
A = ρ(4πe/3)−3/2.
(7.21)
Furthermore the stress tensor turns out to be diagonal (pij = ( 23 ρe)δij , where δij is the so-called “Kronecker delta” (= 1 if i = j ; = 0 if i = j )), while the heat-flow vector is zero. We end this section with the definition of pressure p in terms of f ; p is nothing else than 1/3 of the spur or trace (i.e., the sum of the three diagonal terms) of pij and is thus given by: 1 p= |c|2 f dξ . (7.22) 3 R3
The Boltzmann equation and fluid dynamics
17
If we compare this with the definition of the specific internal energy e, given in Equation (3.11), we obtain the relation: 2 p = ρe. 3
(7.23)
This relation also suggests the definition of temperature, according to kinetic theory, T = ( 23 e)/R, where R is the gas constant equal to the universal Boltzmann constant k divided by the molecular mass m. Thus: T=
1 3ρR
R3
|c|2 f dξ .
(7.24)
8. The H-theorem Let us consider a further application of the properties of the collision term Q(f, f ) of the Boltzmann equation: ∂f ∂f +ξ · = Q(f, f ). ∂t ∂x
(8.1)
If we multiply both sides of this equation by log f and integrate with respect to ξ , we obtain: ∂H ∂ + · J = S, ∂t ∂x
(8.2)
where H=
f log f dξ ,
(8.3)
ξ f log f dξ ,
(8.4)
log f Q(f, f ) dξ .
(8.5)
R3
J=
R3
S=
R3
Equation (8.2) differs from the balance equations considered in the previous section because the right-hand side, generally speaking, does not vanish. We know, however, that the Boltzmann inequality, Equation (5.2), implies: S 0 and S = 0 iff
f is a Maxwellian.
(8.6)
Because of this inequality, Equation (8.2) plays an important role in the theory of the Boltzmann equation. We illustrate the role of Equation (8.6) in the case of space-
18
C. Cercignani
homogeneous solutions. In this case the various quantities do not depend on x and Equation (8.2) reduces to ∂H = S 0. ∂t
(8.7)
This implies the so-called H -theorem (for the space homogeneous case): H is a decreasing quantity, unless f is a Maxwellian (in which case the time derivative of H is zero). Remember now that in this case the densities ρ, ρv and ρe are constant in time; we can thus build a Maxwellian M which has, at any time, the same ρ, v and e as any solution f corresponding to given initial data. Since H decreases unless f is a Maxwellian (i.e., f = M), it is tempting to conclude that f tends to M when t → ∞. This conclusion is, however, unwarranted from a purely mathematical viewpoint, without a more detailed consideration of the source term S in Equation (8.7), for which [47] should be consulted. If the state of the gas is not space-homogeneous, the situation becomes more complicated. In this case it is convenient to introduce the quantity H=
H dx,
(8.8)
Ω
where Ω is the space domain occupied by the gas (assumed here to be time-independent). Then Equation (8.2) implies dH dt
J · n dσ,
(8.9)
∂Ω
where n is the inward normal and dσ the surface element on ∂Ω. Clearly, several situations may arise (see [43] and [47] for a detailed discussion). It should be clear that H has the properties of entropy (except for the sign); this identification is strengthened when we evaluate H in an equilibrium state (see [43,39, 35]) because it turns out to coincide with the expression of a perfect gas according to equilibrium thermodynamics, apart from a factor −R. A further check of this identification is given by an inequality satisfied by the right-hand side of Equation (8.9) when the gas is able to exchange energy with a solid wall bounding Ω (see Section 11 and [43,39,47]).
9. Model equations When trying to solve the Boltzmann equation for practical problems, one of the major shortcomings is the complicated structure of the collision term, Equation (4.2). When one is not interested in fine details, it is possible to obtain reasonable results by replacing the collision integral by a so-called collision model, a simpler expression J (f ) which retains only the qualitative and average properties of the collision term Q(f, f ). The equation for the distribution function is then called a kinetic model or a model equation.
The Boltzmann equation and fluid dynamics
19
The most widely known collision model is usually called the Bhatnagar, Gross and Krook (BGK) model, although Welander proposed it independently at about the same time as the above mentioned authors [14,173]. It reads as follows: J (f ) = ν Φ(ξ ) − f (ξ ) ,
(9.1)
where the collision frequency ν is independent of ξ (but depends on the density ρ and the temperature T ) and Φ denotes the local Maxwellian, i.e., the (unique) Maxwellian having the same density, bulk velocity and temperature as f : Φ = ρ(2πRT )−3/2 exp −|ξ − v|2 /(2RT ) . Here ρ, v, T are chosen is such a way that for any collision invariant ψ we have ψ(ξ )Φ(ξ ) dξ = ψ(ξ )f (ξ ) dξ . R3
(9.2)
(9.3)
R3
It is easily checked that, thanks to Equation (9.3): (a) f and Φ have the same density, bulk velocity and temperature; (b) J (f ) satisfies conservation of mass, momentum and energy; i.e., for any collision invariant: ψ(ξ )J (f ) dξ = 0; (9.4) R3
(c) J (f ) satisfies the Boltzmann inequality log f J (f ) dξ 0
(9.5)
R3
the equality sign holding if and only if, f is a Maxwellian. It should be remarked that the nonlinearity of the BGK collision model, Equation (9.1), is much worse than the nonlinearity in Q(f, f ); in fact the latter is simply quadratic in f , while the former contains f in both the numerator and denominator of an exponential, because v and T are functionals of f , defined by Equations (6.2) and (6.27). The main advantage in the use of the BGK model is that for any given problem one can deduce integral equations for ρ, v, T , which can be solved with moderate effort on a computer. Another advantage of the BGK model is offered by its linearized form (see [43, 113,35]). The BGK model has the same basic properties as the Boltzmann collision integral, but has some shortcomings. Some of them can be avoided by suitable modifications, at the expense, however, of the simplicity of the model. A first modification can be introduced in order to allow the collision frequency ν to depend on the molecular velocity, more precisely on the magnitude of the random velocity c (defined by Equation (6.5)), while requiring that Equation (9.4) still holds. All the basic properties, including Equation (9.5), are retained, but the density, velocity and temperature appearing in Φ are not the local ones of the gas,
20
C. Cercignani
but some fictitious local parameters related to five functionals of f different from ρ, v, T ; this follows from the fact that Equation (9.3) must now be replaced by
R3
ν |c| ψ(ξ )Φ(ξ ) dξ =
R3
ν |c| ψ(ξ )f (ξ ) dξ .
(9.6)
A different kind of correction to the BGK model is obtained when a complete agreement with the compressible Navier–Stokes equations is required for large values of the collision frequency. In fact the BGK model has only one parameter (at a fixed space point and time instant), i.e., the collision frequency ν; the latter can be adjusted to give a correct value for either the viscosity µ or the heat conductivity κ, but not for both. This is shown by the fact that the Prandtl number Pr = µ/cp κ (where cp is the specific heat at constant pressure) turns out [39,35] to be unity for the BGK model, while it is about to 2/3 for a monatomic gas (according to both experimental data and the Boltzmann equation). In order to have a correct value for the Prandtl number, one is led [87,62] to replacing the local Maxwellian in Equation (9.1) by Φ(ξ ) = ρ(π)−3/2 (det A)1/2 exp −(ξ − v) · A(ξ − v) ,
(9.7)
where A is the inverse of the matrix A−1 = (2RT / Pr)I − 2(1 − Pr)p/(ρ Pr),
(9.8)
where I is the identity and p the stress matrix. If we let Pr = 1, we recover the BGK model. Only recently [3] this model (called ellipsoidal statistical (ES) model) has been shown to possess the property expressed by Equation (9.5). Hence the H -theorem holds for the ES model. Other models with different choices of Φ have been proposed [151,35] but they are not so interesting, except for linearized problems (see [43,39,35]). Another model is the integro-differential model proposed by Lebowitz, Frisch and Helfand [114], which is similar to the Fokker–Planck equation used in the theory of Brownian motion. This model reads as follows: J (f ) = D
1 ∂ + (ξk − vk )f , RT ∂ξk ∂ξk2
3 2
∂ f k=1
(9.9)
where D is a function of the local density ρ and the local temperature T . If we take D proportional to the pressure p = ρRT , Equation (9.9) has the same kind of nonlinearity (i.e., quadratic) as the true Boltzmann equation. The idea of kinetic models can be naturally extended to mixtures and polyatomic gases [151,127,81,43].
The Boltzmann equation and fluid dynamics
21
10. The linearized collision operator On several occasions we shall meet the so-called linearized collision operator, related to the bilinear operator defined in Equation (5.3) by Lh = 2M −1 Q(Mh, M),
(10.1)
where M is a Maxwellian distribution, usually with zero bulk velocity. When we want to emphasize the fact that we linearize with respect to a given Maxwellian, we write LM instead of just L. A more explicit expression of Lh reads as follows M∗ (h + h∗ − h∗ − h)B(θ, V ) dξ ∗ dn, (10.2) Lh = B+
3
where we have taken into account that M M∗ = MM∗ . Because of Equation (4.10) (with Mh in place of f , M in place of g and g in place of φ), we have the identity: MgLh dξ
3
1 =− 4
3
3 B+
(h + h∗ − h − h∗ )(g + g∗ − g − g∗ ) × B(θ, V ) dξ ∗ dξ dn.
(10.3)
This relation expresses a basic property of the linearized collision term. In order to make it clear, let us introduce a bilinear expression, the scalar product in the Hilbert space of square summable functions of ξ endowed with a scalar product weighted with M: ghM ¯ dξ , (10.4) (g, h) =
3
where the bar denotes complex conjugation. Then Equation (1.7) (with g¯ in place of g) gives (thanks to the symmetry of the expression in the right-hand side of Equation (10.3) with respect to the interchange g ⇔ h): (g, Lh) = (Lg, h).
(10.5)
(h, Lh) 0
(10.6)
Further:
and the equality sign holds if and only if h + h∗ − h − h∗ = 0, i.e., if and only if h is a collision invariant.
(10.7)
22
C. Cercignani
Equations (10.5) and (10.6) indicate that the operator L is symmetric and non-positive in the aforementioned Hilbert space.
11. Boundary conditions The Boltzmann equation must be accompanied by boundary conditions, which describe the interaction of the gas molecules with the solid walls. It is to this interaction that one can trace the origin of the drag and lift exerted by the gas on the body and the heat transfer between the gas and the solid boundary. The study of gas-surface interaction may be regarded as a bridge between the kinetic theory of gases and solid state physics and is an area of research by itself. The difficulties of a theoretical investigation are due, mainly, to our lack of knowledge of the structure of surface layers of solid bodies and hence of the effective interaction potential of the gas molecules with the wall. When a molecule impinges upon a surface, it is adsorbed and may form chemical bonds, dissociate, become ionized or displace surface molecules. Its interaction with the solid surface depends on the surface finish, the cleanliness of the surface, its temperature, etc. It may also vary with time because of outgassing from the surface. Preliminary heating of a surface also promotes purification of the surface through emission of adsorbed molecules. In general, adsorbed layers may be present; in this case, the interaction of a given molecule with the surface may also depend on the distribution of molecules impinging on a surface element. For a more detailed discussion the reader should consult [39,110] and [41]. In general, a molecule striking a surface with a velocity ξ reemerges from it with a velocity ξ which is strictly determined only if the path of the molecule within the wall can be computed exactly. This computation is very hard, because it depends upon a great number of details, such as the locations and velocities of all the molecules of the wall and an accurate knowledge of the interaction potential. Hence it is more convenient to think in terms of a probability density R(ξ → ξ ; x, t; τ ) that a molecule striking the surface with velocity between ξ and ξ + dξ at the point x and time t will re-emerge at practically the same point with velocity between ξ and ξ + dξ after a time interval τ (adsorption or sitting time). If R is known, then we can easily write down the boundary condition for the distribution function f (x, ξ , t). To simplify the discussion, the surface will be assumed to be at rest. A simple argument ([43,39,35]) then gives: f (x, ξ , t)|ξ · n| ∞ = dτ 0
ξ ·n<0
R(ξ → ξ ; x, t; τ )f (x, ξ , t − τ )|ξ · n| dξ
(x ∈ ∂Ω, ξ · n > 0).
(11.1)
The kernel R can be assumed to be independent of f under suitable conditions which we shall not detail here [39,110,41]. If, in addition, the effective adsorption time is small compared to any characteristic time of interest in the evolution of f , we can let τ = 0 in
The Boltzmann equation and fluid dynamics
23
the argument of f appearing in the right-hand side of Equation (3.4); in this case the latter becomes: f (x, ξ , t)|ξ · n| R(ξ → ξ ; x, t)f (x, ξ , t)|ξ · n| dξ = ξ ·n<0
(x ∈ Ω, ξ · n > 0),
(11.2)
where R(ξ → ξ ; x, t) =
∞
dτ R(ξ → ξ ; x, t; τ ).
(11.3)
0
Equation (11.2) is, in particular, valid for steady problems. Although the idea of a scattering kernel had appeared before, it is only at the end of 1960’s that a systematic study of the properties of this kernel appears in the scientific literature [35,110,41]. In particular, the following properties were pointed out [36,35,110, 41,34,108,48,109,37]: (1) Non-negativeness, i.e., R cannot take negative values: R(ξ → ξ ; x, t; τ ) 0
(11.4)
and, as a consequence: R(ξ → ξ ; x, t) 0.
(11.5)
(2) Normalization, if permanent adsorption is excluded; i.e., R, as a probability density for the totality of events, must integrate to unity:
∞
dτ 0
ξ ·n0
R(ξ → ξ ; x, t; τ ) dξ = 1
(11.6)
and, as a consequence:
ξ ·n0
R(ξ → ξ ; x, t) dξ = 1.
(11.7)
(3) Reciprocity; this is a subtler property that follows from the circumstance that the microscopic dynamics is time reversible and the wall is assumed to be in a local equilibrium state, not significantly disturbed by the impinging molecule. It reads as follows: |ξ · n|Mw (ξ )R(ξ → ξ ; x, t; τ ) = |ξ · n|Mw (ξ )R(−ξ → −ξ ; x, t; τ )
(11.8)
and, as a consequence: |ξ · n|Mw (ξ )R(ξ → ξ ; x, t) = |ξ · n|Mw (ξ )R(−ξ → −ξ ; x, t).
(11.9)
24
C. Cercignani
Here Mw is a (non-drifting) Maxwellian distribution having the temperature of the wall, which is uniquely identified apart from a factor. We remark that the reciprocity and the normalization relations imply another property: (3 ) Preservation of equilibrium, i.e., the Maxwellian Mw must satisfy the boundary condition (11.1):
∞
Mw (ξ )|ξ · n| =
dτ 0
ξ ·n<0
R(ξ → ξ ; x, t; τ )Mw (ξ )|ξ · n| dξ
(11.10)
equivalent to: Mw (ξ )|ξ · n| =
ξ ·n<0
R(ξ → ξ ; x, t)Mw (ξ )|ξ · n| dξ .
(11.11)
In order to obtain Equation (11.10) it is sufficient to integrate Equation (11.8) with respect to ξ and τ , taking into account Equation (11.6) (with −ξ and −ξ in place of ξ and ξ , respectively). We remark that one frequently assumes Equation (11.10) (or (11.11)), without mentioning Equation (11.8) (or (11.9)); although this is enough for many purposes, reciprocity is very important when constructing mathematical models, because it places a strong restriction on the possible choices. A detailed discussion of the physical conditions under which reciprocity holds has been given by Bärwinkel and Schippers [10]. The basic information on gas-surface interaction, which should be in principle obtained from a detailed calculation based on a physical model, is summarized in a scattering kernel. The further reduction to a small set of accommodation coefficients can be advocated for practical purposes, provided this concept is firmly related to the scattering kernel (see [43, 39,47] for further details). In view of the difficulty of computing the kernel R(ξ → ξ ) from a physical model of the wall, frequently one constructs a mathematical model in the form of a kernel R(ξ → ξ ) which satisfies the basic physical requirements expressed by Equations (11.5), (11.7), (11.9) and is not otherwise restricted except by the condition of not being too complicated. One of the simplest kernels is R(ξ → ξ ) = αMw (ξ )|ξ · n| + (1 − α)δ ξ − ξ + 2n(ξ · ξ ) .
(11.12)
This is the kernel corresponding to Maxwell’s model [121], according to which a fraction (1 − α) of molecules undergoes a specular reflection, while the remaining fraction α is diffused with the Maxwellian distribution of the wall Mw . This is the only model for the scattering kernel that appeared in the literature before the late 1960’s. We refer to original papers and standard treatises for details on more recent models [110,41,34,108,48,109,37, 10,121,135,91,100,112,99,68,174,38,54,53,22,43,39], among which the most popular in recent years has been the so-called Cercignani–Lampis (CL) model. It is remarkable that, for any scattering kernel satisfying the three properties of normalization, positivity and preservation of equilibrium, a simple inequality holds. The
The Boltzmann equation and fluid dynamics
25
latter was stated by Darrozès and Guiraud [70] who also sketched a proof. More details were given later [37,39]. It reads as follows Jn =
R3
ξ · nf log f dξ −(2RTw )−1
R3
ξ · n|ξ |2 f dξ
(x ∈ ∂Ω).
(11.13)
Equality holds if and only if f coincides with Mw (the wall Maxwellian) on ∂Ω (unless the kernel in Equation (11.2) is a delta function). We remark that if the gas does not slip upon the wall, the right-hand side of Equation (11.13) equals −qn /(RTw ) where qn is the heat-flow along the normal, according to its definition given in Section 5. If the gas slips on the wall, then one must add the power of the stresses pn · v to qn . In any case, however, the (w) (w) right-hand side equals qn , where qn is the heat-flow in the solid at the interface, because the normal energy flow must be continuous through the wall and stresses have vanishing power in the solid, because the latter is at rest. If we identify the function H introduced in Section 8 with −η/R (where η is the entropy of the gas), the inequality in Equation (11.13) is exactly what one would expect from the Second Principle of thermodynamics.
12. The continuum limit In this section we investigate the connection between the Navier–Stokes equations and the Boltzmann equation by using a method, which originated with Hilbert [85] and Enskog [74]. The discussion is made complicated by the various possible scalings. For example, if we denote by (¯x, t) the microscopic space and time variables (those entering in the Boltzmann equation) and by (x, t) the macroscopic variables (those entering in the fluid dynamical description), we can study scalings of the following kind x¯ = ε−1 x,
(12.1)
t = ε−α t,
(12.2)
where α is an exponent between 1 and 2. For α = 1, this is called the compressible scaling. If α > 1, we are looking at larger “microscopic” times. We now investigate the limiting behavior of solutions of the Boltzmann equation in this limit. Notice first that the compressible Euler equations, ∂t ρ + div(ρv) = 0, 2 ∂t (ρvi ) + div ρvvi + ρeei = 0, 3
1 2 1 2 5 + div ρv |v| + e = 0, ∂t ρ e + |v| 2 2 3
(12.3)
26
C. Cercignani
are invariant with respect to the scaling t → ε−1 t, x → ε−1 x. Here, ei denotes the unit vector in the i-th direction and p is related to ρ and e = 3RT /2 by the state equation for a perfect gas. To investigate how these equations change under the scalings (12.1)–(12.2), let vε (x, t) = εγ v ε−1 x, ε−α t , ρ ε (x, t) = ρ ε−1 x, ε−α t , T ε (x, t) = T ε−1 x, ε−α t ,
γ = α − 1, (12.4)
where (ρ, v, T ) solve the compressible Euler equations (12.3). We easily obtain ∂t ρ ε + div ρ ε vε = 0,
(12.5)
1 ∂t vε + vε · ∂x vε = − ε ∂x pε · ε2(1−α) , ρ
(12.6)
2 ∂t T ε + vε · ∂x T ε + T ε ∂x · vε = 0. 3
(12.7)
The scaling of the bulk velocity field vε in (12.4) is done in a dimensionally consistent way. We expect that the continuum limit of the Boltzmann equation under the scaling (12.1)– (12.2) will be given by the asymptotic behavior of (ρ ε , vε , T ε ), satisfying (12.3), in the limit ε → 0. We will now investigate this limit. To this end, let η = ε2(α−1) and expand vη ≡ vε = v0 + ηv1 + η2 v2 + · · · , ρ η ≡ ρ ε = ρ0 + ηρ1 + η2 ρ2 + · · · , T η ≡ T ε = T0 + ηT1 + η2 T1 + · · · . If we collect the terms of order η−1 in (12.4), we have ∂x p0 = 0
(12.8)
and the terms of order η0 give ∂t ρ0 + div(ρ0 v0 ) = 0, ∂t v0 + (v0 · ∂x )v0 = −
∂x p1 , ρ0
2 ∂t T0 + (v0 · ∂x )T0 + T0 div v0 = 0. 3
(12.9)
The Boltzmann equation and fluid dynamics
27
From (12.8) and the perfect gas law p0 = ρ0 T0 , which we assume to hold at zeroth order, it follows that ρ0 T0 is constant as a function of the space variables. The first and third equations in (12.9) now imply 2 ∂t (ρ0 T0 ) + div(ρ0 T0 v0 ) = − ρ0 T0 div v0 . 3
(12.10)
As ρ0 T0 is only a function of t, say A(t), Equation (12.10) implies that 3 div v0 = − A (t)/A(t). 5
(12.11)
Under suitable assumptions, Equation (12.11) implies that A (t) = 0 and then div v0 = 0. This is the case, for example, if we are in a box with nonporous walls, because then the normal component of v0 is 0 and we can use the divergence theorem. A similar argument applies to the case of a box with periodicity boundary conditions; or if we are in the entire space 3 and the difference between v0 and a constant vector decays fast enough at infinity. Assuming that we have conditions which imply that div v0 = 0, we easily get from the continuity equation that ρ0 will be independent of x if the initial value is, and the same for T0 . But with this knowledge Equations (12.9) then actually entail that ρ0 and T0 are constant. Therefore, if the initial conditions are “well prepared” in the sense that vε (x, 0) is a divergence-free vector field and ρ ε (x, 0), T ε (x, 0) are constant, we expect as a first-order approximation for ρ ε , vε , T ε the solution of the equations div v0 = 0, ∂t v0 + (v0 · ∂x )v0 = −
∂x p1 , ρ0
(12.12)
which are the incompressible Euler equations. This limit, known as the “low velocity limit”, is well known at the macroscopic level. We refer to Majda’s book [120] for references and a detailed discussion. The variable η−1 enters into the theory as the square of the speed of sound. If this parameter is large compared to typical speeds of the fluid, then the incompressible model is well suited to describe the time evolution, provided that the initial velocity field is divergence-free and the initial density and temperature are constant. The incompressible fluid limit was met in the last section in connection with the Stokes paradox. In fact, it seems that this limit and the derivation of the steady incompressible Navier–Stokes equations from the Boltzmann equation were first considered in connection with the flow past a body at small values of the Mach number [32]. We remark that in the steady case div v0 = 0 and ρ0 and T0 turn out to be constant without using the boundary conditions. Let us examine the kinetic picture as described by the Boltzmann equation. The above discussion suggests that if α ∈ (1, 2), then in the scaling (12.1)–(12.2) the solutions of the Boltzmann equation will converge to a local Maxwellian distribution whose parameters satisfy the incompressible Euler equations. This assertion can actually be proved rigorously [71,9].
28
C. Cercignani
For α = 2 something special happens. Of course, the incompressible Euler equations are invariant under the scaling (12.1)–(12.3); however, for α = 2 the incompressible Navier– Stokes equations, ∂t v + (v · ∂x )v = −∂x r + ν∆v, ∂x · v = 0
(12.13)
(where r = p/ρ and ν = µ/ρ is the kinematic viscosity) are also invariant under the same scaling. It is therefore of great interest to understand whether the Boltzmann dynamics “chooses” in this limit the Euler or the Navier–Stokes evolution. We can expect that the answer is Navier–Stokes. In other words, considering larger times than those typical for Euler dynamics (ε−2 t instead of ε−α t, α < 2), dissipation becomes nonnegligible. We can see an illustration of this behavior in the following example: consider two parallel layers of fluid moving with velocities v and v + δv. Suppose that we want to decide whether there is any momentum transfer between these layers (which is expected for the Navier–Stokes equations, but not for the Euler equations). The momentum transfer can in principle be affected by the trend to thermalization typical of the Boltzmann collision term, but a scaling argument shows that it is proportional to εα−2 δv, with the consequence that it remains only relevant for α = 2. These considerations can be put on a rigorous basis, and, of course, the viscosity coefficient can be computed in terms of kinetic expressions (see [43,39,35]). The incompressible Navier–Stokes equations can be derived from the Boltzmann equation if the time interval is such that smooth solutions of the continuum equations exist. The tool which yields this result is a truncated Hilbert expansion and one gets local convergence for the general situation [71] or global and uniform convergence if the data are small in a suitable sense [9]. Let us consider now the cases in which compressibility is not negligible. If we scale space and time in the same way, the scaled Boltzmann equation becomes: 1 ∂t f ε + ξ · ∂x f ε = Q f ε , f ε . ε
(12.14)
We will use the abbreviation Dt f := ∂t f + ξ · ∂x f . Of course, we expect that εDt f ε → 0
as ε → 0,
(12.15)
and if f ε → f 0,
(12.16)
the limit f 0 must satisfy Q f 0 , f 0 = 0.
(12.17)
The Boltzmann equation and fluid dynamics
29
This implies, as we know, that f 0 is a local Maxwellian distribution: ρ(x, t) |ξ − v(x, t)|2 f (x, ξ , t) ≡ M(x, ξ , t) = exp − . (2πRT (x, t))3/2 2RT (x, t) 0
(12.18)
The fields (ρ, x, T ), which characterize the behavior of the local Maxwellian distribution M in space and time, are expected to evolve according to continuum equations which we are going to derive. First, let us emphasize again that these fields are varying slowly on the space-time scales which are typical for the gas described in terms of the Boltzmann equation. From the conservation laws (I.12.36) ψα Q(f, f ) dξ = 0 (α = 0, . . . , 4),
(12.19)
we readily obtain, as we know, ψα (∂t f + ξ · ∂x f ) dξ = 0.
(12.20)
This is a system of equations for the moments of f which is in general not closed. However, if we assume f = M and use the identities (for M they are identities; for a general f they are definitions given in (I.12.3–7), where e = 32 RT ) ρ=
M dξ ,
(12.21)1
ρv =
Mξ dξ ,
1 3 1 w ≡ ρRT + ρ|v|2 = 2 2 2
(12.21)2 ξ 2 M dξ ,
(12.21)3
we readily obtain from (12.20) that ∂t ρ + div(ρv) = 0,
(12.22)
Mξ ξi = 0, ∂t (ρvi ) + div
(12.23)
∂t
1 3 1 ρRT + ρ|v|2 + div Mξ |ξ |2 = 0. 2 2 2
(12.24)
These equations are nothing but Equations (12.19)–(12.21), specialized to the case of a Maxwellian distribution. This is of crucial importance if we want to write
30
C. Cercignani
Equations (12.23) and (12.24) in closed form. To do so we have to express Mξ ξi and Mξ |ξ |2 in terms of the fields (ρ, v, T ). To this end, we use the elementary identities:
M(ξj − vj )(ξi − vi ) dξ = δij ρRT , (12.25) M(ξ − v)|(ξ − v)|2 dξ = 0,
which transform Equation (12.23) into ∂t (ρvi ) + div(ρvvi ) = −∂xi p,
(12.26)
p = ρRT .
(12.27)
with
Equation (12.27) is the perfect gas law. Obviously, the quantity p defined by Equation (12.27) has the meaning of a pressure. Recalling that the internal energy e is related to temperature T by 3 e = RT , 2
(12.28)
using this and Equation (12.25) we transform Equation (12.24) into: 1 2 1 2 ∂t ρ(e + |v| + div ρv e + |v| = −div(pv). 2 2
(12.29)
The set of Equations (12.22), (12.26) and (12.29) express conservation equations for mass, momentum and energy respectively and can be rewritten as the Euler equations (12.3). For smooth functions, an equivalent way of writing the Euler equations in terms of the field (ρ, v, T ) is ∂t ρ + div(ρv) = 0, ∂t v + (v · ∂x )v +
1 ∂x p = 0, ρ
(12.30)
2 ∂t T + (v · ∂x )T + T ∂x · v = 0. 3 However, in this form we lose the conservation form as given in (12.3), in which the time derivative of a field equals the negative divergence of a current which is a nonlinear function of this field. Before going on, some comments on our limits are in order, because one might suspect an inconsistency in the passage from a rarefied to a dense gas. Recall that the Boltzmann equation holds in the Boltzmann–Grad limit (Nσ 2 = O(1)). In the continuum limit, we
The Boltzmann equation and fluid dynamics
31
have to take Nσ 2 = 1/ε → ∞. This, at first glance, seems contradictory, but there is really no problem. The Boltzmann equation holds for a perfect gas, i.e., for a gas such that the density parameter δ = Nσ 3 /V , where V is the volume containing N molecules, tends to zero. The parameter Nσ 2 1 = 2/3 = N 1/3 δ 2/3 Kn V
(12.31)
may tend to zero, to ∞ or remain finite in this limit. These are the three cases which occur if we scale N as δ −m (m 0), for m < 2, m > 2 and m = 2 respectively. In the first case the gas is in free-molecular flow and we can simply neglect the collision term (Knudsen gas), in the second we are in the continuum regime which we are treating here, and we cannot simply “omit” the “small” term, i.e., the left-hand side of the Boltzmann equation, because the limit is singular. In the third case the two sides of the Boltzmann equation are equally important (Boltzmann gas) and this is the case dealt with before for solutions close to an absolute Maxwellian distribution. In spite of the fact that we face a singular perturbation problem, Hilbert [85] proposed an expansion in powers of ε. In this way, however, we obtain a Maxwellian distribution at the lowest order, with parameters satisfying the Euler equations and corrections to this solution which are obtained by solving inhomogeneous linearized Euler equations [85,43, 39,35]. In order to avoid this and to investigate the relationship between the Boltzmann equation and the compressible Navier–Stokes equations, Enskog introduced an expansion, usually called the Chapman–Enskog expansion [74,39,35]. The idea behind this expansion is that the functional dependence of f upon the local density, bulk velocity and internal energy can be expanded into a power series. Although there are many formal similarities with the Hilbert expansion, the procedure is rather different. As remarked by the author [35,39], the Chapman–Enskog expansion seems to introduce spurious solutions, especially if one looks for steady states. This is essentially due to the fact that one really considers infinitely many time scales (of orders ε, ε2 , . . . , ε n , . . .). The author [35,39] introduced only two time scales (of orders ε and ε2 ) and was able to recover the compressible Navier–Stokes equations. In order to explain the idea, we remark that the Navier–Stokes equations describe two kinds of processes, convection and diffusion, which act on two different time scales. If we consider only the first scale we obtain the compressible Euler equations; if we insist on the second one we can obtain the Navier–Stokes equations only at the price of losing compressibility. If we want both compressibility and diffusion, we have to keep both scales at the same time and think of f as f (x, ξ , t) = f εx, ξ , εt, ε2 t .
(12.32)
This enables us to introduce two different time variables t1 = εt, t2 = ε2 t and a new space variable x1 = εx such that f = f (x1 , ξ , t1 , t2 ). The fluid dynamical variables are functions
32
C. Cercignani
of x1 , t1 , t2 , and for both f and the fluid dynamical variables the time derivative is given by ∂f ∂f ∂ =ε + ε2 . ∂t ∂t1 ∂t2
(12.33)
In particular, the Boltzmann equation can be rewritten as ε
∂f ∂f + ε2 + εξ · ∂x f = Q(f, f ). ∂t1 ∂t2
If we expand f formally in a power series in ε, we find that at the lowest order f is a Maxwellian distribution. The compatibility conditions at the first order give that the time derivatives of the fluid dynamic variables with respect to t1 is determined by the Euler equations, but the derivatives with respect to t2 are determined only at the next level and are given by the terms of the compressible Navier–Stokes equations describing the effects of viscosity and heat conductivity. The two contributions are, of course, to be added as specified by (12.33) in order to obtain the full time derivative and thus write the compressible Navier–Stokes equations. It is not among the aims of this article to describe the techniques applied to and the results obtained from the computations of the transport coefficients, such as the viscosity and heat conduction coefficients, for given molecular interaction. For this we refer to standard treatises [64,86,75]. The results discussed in this section show that there is a qualitative agreement between the Boltzmann equation and the Navier–Stokes equations for sufficiently low values of the Knudsen number. There are however flows where this agreement does not occur. They have been especially studied by Sone [154]. New effects arise because the no-slip and no temperature jump boundary condition do not hold. In addition to the thermal creep induced along a boundary with a nonuniform temperature, discovered by Maxwell, two new kinds of flow are induced over boundaries kept at uniform temperatures. They are related to the presences of thermal stresses in the gas. The first effect [154,153,138] is present even for small Mach numbers and small temperature differences and follows from the fact that there are stresses related to the second derivatives of the temperature (see Section 15). Although these stresses do not change the Navier–Stokes equations, they change the boundary conditions; the gas slips on the wall, and thus a movement occurs even if the wall is at rest. This effect is particularly important in small systems, such as micromachines, since the temperature differences are small but may have relatively large second derivatives; it is usually called the thermal stress slip flow [154,153,138]. The second effect is nonlinear [101,155] and occurs when two isothermal surfaces do not have constant distance (thus in any situation with large temperature gradients, in the absence of particular symmetries). In fact, if we assume that in the Hilbert expansion the velocity vanishes at the lowest order, i.e., the speed is of the order of the Knudsen number, the terms of second order in the temperature show up in the momentum equation.
The Boltzmann equation and fluid dynamics
33
These terms are associated with thermal stresses and are of the same importance as those containing the pressure and the viscous stresses. A solution in which the gas does not move can be obtained if and only if: grad T ∧ grad |grad T |2 = 0. (12.34) Since |grad T | measures the distance between two nearby isothermal lines, if this quantity has a gradient in the direction orthogonal to grad T , the distance between two neighboring isothermal lines varies and we must expect that the gas moves. These effects may occur even for sufficiently large values of the Knudsen number; they cannot be described, however, in terms of the local temperature field. They rather depend by the configuration of the system. They should not be confused with flows due to the presence of a temperature gradient along the wall, such as the transpiration flow [139] and the thermophoresis of aerosol particles [162]. Numerical examples of simulations of this kind of flow are discussed in [43]. 13. Free-molecule and nearly free-molecule flows After discussing the behavior of a gas in the continuum limit, in this section we consider the opposite case in which the small parameter is the Knudsen number (or the inverse of the mean free path). By analogy with what we did in the previous section, we might be tempted to use a series expansion of the form (12.2), albeit with a different meaning of the expansion parameter. This, however, does not work in general, for a reason to be presently explained. The factor ξ multiplying the gradient of f in Equation (5.1) takes all possible values and hence also values of order ε; thus we should expect troubles from the molecules travelling with low speeds, because then certain terms in the left-hand side can become smaller than the righthand side, in spite of the small factor ε. This is confirmed by actual calculations, especially for steady problems. Let us now consider the limiting case when the collisions can be completely neglected. This, by itself, does not pose many problems. The Boltzmann equation (in the absence of a body force) reduces to the simple form Dt f = ∂t f + ξ · ∂x f = 0.
(13.1)
Since the molecular collisions are negligible, the gas-surface interaction discussed in Section 11 plays a major role. This situation is typical for artificial satellites, since the mean free path is 50 meters at 200 kilometers of altitude. The general solution of Equation (13.1) is in terms of an arbitrary function of two vectors g(·, ·): f (x, ξ , t) = g(x − ξ t, ξ ).
(13.2)
In the steady case, Equation (13.1) reduces to ξ · ∂x f = 0,
(13.3)
34
C. Cercignani
and the general solution becomes: f (x, ξ , t) = g(x ∧ ξ , ξ ).
(13.4)
Frequently it is easier to work with the property that f is constant along the molecular trajectories than with the explicit solutions given by Equations (13.3)–(13.4). The easiest problem to deal with is the flow past a convex body. In this case, in fact, the molecules arriving at the surface of the wall have an assigned distribution function f∞ , usually a Maxwellian distribution with the density ρ∞ , bulk velocity v∞ , and temperature T∞ , prevailing far away from the body, and the distribution function of the molecules leaving the surface is given by the boundary conditions. The distribution function at any other point P , if needed, is simply obtained by the following rule: if the straight line through P having the direction of ξ intersects the body at a point Q and ξ points from Q towards P , then the distribution function at P is the same as that at Q; otherwise it equals f∞ . Interest is usually confined to the total momentum and energy exchanged between the molecules and the body, which, in turn, easily yield the drag and lift exerted by the gas on the body and the heat transfer between the body and the gas. In practice, the temperature of a body is determined by a balance of all forms of heat transfer at the body surface. For an artificial satellite, a considerable part of heat is lost by radiation and this process must be duly taken into account in the balance. The results take a particularly simple form in the case of a large Mach number since we can let the latter go to infinity in the various formulas. One must, however, be careful, because the speed is multiplied by sin θ in many terms and thus the aforementioned limit is not uniform in θ . Thus the limiting formulas can be used, if and only if, the area where S sin θ 1 is small. The standard treatment is based on the definition of accommodation coefficients, but calculations based on other models are available [39,50,49]. The case of nonconvex boundaries is, of course, more complicated and one must solve an integral equation to obtain the distribution function at the boundary. If one assumes diffuse reflection according to a Maxwellian, the integral equation simplifies in a considerable way, because just the mass flow at the boundary must be computed [39]. In particular the latter equation can be used to study free-molecular flows in pipes of arbitrary cross section with a typical diameter much smaller than the mean free path (capillaries). If the cross section is circular the equation becomes particularly simple and is known as Clausing’s equation [39]. The perturbation of free-molecular flows is not trivial for steady problems because of the abovementioned non-uniformity in the inverse Knudsen number. If one tries a naïve iteration, the singularity arising in the first iterate may cancel when integrating to obtain moments (cancellation is easier, the higher is the dimensionality of the problem, because a first-order pole is milder, if the dimension is higher). The singularity is always present and, although it may be mild, it can build up a worse singularity when computing subsequent steps. The difficulties are enhanced in unbounded domains where the subsequent terms diverge at space infinity. The reason for the latter fact is that the ratio between the mean free path λ and the distance d of any given point from the body is a local Knudsen
The Boltzmann equation and fluid dynamics
35
number which tends to zero when d tends to infinity; hence collisions certainly arise in an unbounded domain and tend to dominate at large distances. On this basis we are led to expect that a continuum behavior takes place at infinity, even when the typical lengths characterizing the size of the body are much smaller than the mean free path; this is confirmed by the discussion of the Stokes paradox for the steady linearized Boltzmann equation (see [43,39,35]). Both difficulties are removed by the so-called collision iteration: the loss term is partly considered to be unknown in the iteration, thus building an exponential term which controls the singularity. The presence of the latter is still felt through the presence of logarithmic terms in the (inverse) Knudsen number. In higher dimensions this is multiplied by a power of (Kn)−1 which typically equals the number of space dimensions relevant for the problem under consideration in a bounded domain. In particular the dependence upon coordinates will show the same singularity (we can think of local Knudsen numbers based on the distance from the nearest wall); as a consequence first derivatives will diverge at the boundary in one dimension and the same will occur for second, or third derivatives, in two, or three, space dimensions, respectively. In an external domain we have, in addition to the low speed effects, the effect of particles coming from infinity, which actually dominates. In particular in one dimension (half-space problems) the terms coming from iterations are of the same order as the lowest order terms; actually for a half-space problem there is hardly a Knudsen number (the local one is an exception). In two dimensions the corrections in the moments are of order Kn−1 log Kn. In three dimensions a correction of order Kn−2 log Kn is preceded by a correction of order Kn−1 . Care must be exercised when applying the aforementioned results to a concrete numerical evaluation, as mentioned above. In fact, for large but not extremely large Knudsen numbers (say 10 Kn 100) log Kn is a relatively small number, although log Kn → ∞ for Kn → ∞. Hence terms of order log Kn/Kn, though mathematically dominating over terms of order 1/Kn are of the same order as the latter for practical purposes. As consequence, the two kinds of terms must be computed together if numerical accuracy is desired for the aforementioned range of Knudsen numbers. Related to this remark is the fact that any factor appearing in front of Kn in the argument of the logarithm is meaningless unless the term of order Kn−1 is also computed. This is particularly important when the factor under consideration depends upon a parameter which can take very large (or very small) values (typically a speed ratio). Thus Hamel and Cooper [70,85] have shown that the first iterate of the integral iteration is incapable of describing the correct dependence upon the speed ratio and have applied the method of matched asymptotic expansions [81] to regions near a body and far from a body. In particular, for the hypersonic flow of a gas of hard spheres past a two-dimensional strip, they find for the drag coefficient
ε log ε , CD = CDf.m. 1 + 2π
(13.5)
where the inverse Knudsen number ε is based on the mean free path λ = π 3/2 σ 2 n∞ Sw (σ is the molecular diameter and Sw = S∞ (Tw /T∞ ), whereas n∞ and S∞ are the number density at infinity).
36
C. Cercignani
If we consider infinite-range intermolecular potentials, then we have fractional powers rather than logarithms. All the considerations of this section have the important consequence that approximate methods of solution which are not able to allow for a nonanalytic behavior for Kn → ∞ produce poor results for large Knudsen numbers.
14. Perturbations of equilibria The first steady solutions other than Maxwellian to be investigated were perturbations of the latter. The method of perturbation of equilibria is different from the Hilbert method because the small parameter is not contained in the Boltzmann equation but in auxiliary conditions, such as boundary or initial conditions. The advantage of the method is that we can investigate problems in the transition regime, provided differences in temperature and speed are moderate. Let us try to find a solution of our problem for the Boltzmann equation in the form f=
∞
εn fn ,
(14.1)
n=0
where at variance with previous expansions ε is a parameter which does not appear in the Boltzmann equation. In addition f0 is assumed from the start to be a Maxwellian distribution. By inserting this formal series into Equation (5.1) and matching the various orders in ε, we obtain equations which one can hope to solve recursively: ∂t f1 + ξ · ∂x f1 = 2Q(f1 , f0 ),
(14.2)1
..., ∂t fj + ξ · ∂x fj = 2Q(fj , f0 ) +
j −1
Q(fi , fj −i ),
(14.2)j
i=1
..., where, as in Section 4, Q(f, g) denotes the symmetrized collision operator and the sum is empty for j = 1. Although in principle one can solve the subsequent equations by recursion, in practice one solves only the first equation, which is called the linearized Boltzmann equation. This equation can be rewritten as follows: ∂t h + ξ · ∂x h = LM h,
(14.3)
where LM denotes the linearized collision operator about the Maxwellian M, i.e., LM h = 2Q(M, Mh)/M, h = f1 /M (see Section 10). We shall assume, as is usually done with
The Boltzmann equation and fluid dynamics
37
little loss of generality, that the bulk velocity in the Maxwellian is zero and we shall denote the unperturbed density and temperature by ρ0 and T0 . Although the equation is now linear, and hence all the weapons of linear analysis are available, it is far from easy to solve for a given boundary value problem, such as Couette flow. Yet it is possible to gain an insight on the behavior of the general solution of Equation (14.3) (see [43,39,35]). This insight gives the following picture for a slab problem, provided the plates are sufficiently far apart (several mean free paths). There are two Knudsen layers near the boundaries, where the behavior of the solution is strongly dependent on the boundary conditions, and a central core (a few mean free paths away form the plates), where the solution of the Navier–Stokes equations holds (with a slight reminiscence of the boundary conditions). If the plates are close in terms of the mean free path, then this picture does not apply because the core and the kinetic layers merge. One can give evidence for the above statements just in the case of the linearized Boltzmann equation, but there is a strong evidence that this qualitative picture applies to nonlinear flows as well, with a major exception. In general, compressible flows develop shock waves at large speeds and these do not appear in the linearized description. As already remarked, these shocks are not surfaces of discontinuity as for an ideal fluid, governed by the Euler equations, but layers of rapid change of the solution (on the scale of the mean free path). One can obtain solutions for flows containing shocks from the Navier–Stokes equations, but, since they change significantly on the scale of the mean free path, they are inaccurate. Other regions where this picture is inaccurate are the zones of high rarefaction, where nearly free-molecular conditions may prevail, even if the rest of the flow is reasonably described in terms of Navier–Stokes equations, Knudsen layers and shock layers. The theory of Knudsen layers can be essentially described by the linearized Boltzmann equation. The main result concerns the boundary conditions for the Navier–Stokes equations. They turn out to be different from those of no-slip and no temperature jump. In fact, the velocity slip turns out to be proportional to the normal gradient of tangential velocity and the temperature jump to the normal gradient of temperature. When one can use the Navier–Stokes equations but must use the slip and temperature-jump boundary conditions, one talks of the slip regime; this typically occurs for Knudsen numbers between 10−1 and 10−2 . Subtler phenomena may occur if the solutions depend on more than one space coordinate. The most important change with respect to traditional continuum mechanics is the presence of the term with the second derivatives of temperature in the expression of the stress deviator and of the term with the second derivatives of bulk velocity in the expression of the heat flow. These terms were already known to Maxwell [121]. In recent times, their importance has been stressed by Kogan et al. [101] and by Sone et al. [155] (as already mentioned in Section 12). Even in fully three-dimensional problems the solution of the linearized Boltzmann equation reduces to the sum of two terms, one of which, hB , is important just in the Knudsen layers and the other, hA , is important far from the boundaries. The latter has a stress deviator and a heat flow with constitutive equations different from those of Navier– Stokes and Fourier. In spite of this, the bulk velocity, pressure, and temperature satisfy the Navier–Stokes equations when steady problems are considered. In fact, when we take
38
C. Cercignani
the divergence of the heat flow vector a term proportional to the Laplacian of v vanishes, thanks to the continuity equation, and thus just a term proportional to the temperature gradient survives; then, taking the divergence of the stress, a term grad(∆T ) vanishes, because of the energy equation. Yet, the new terms in the constitutive relations may produce physical effects in the presence of boundary conditions different from those of no-slip and no temperature jump. In fact, we must expect the velocity slip to be proportional to the shear stress and the temperature jump to the heat flow.
15. Approximate methods for linearized problems Linearization combined with the use of models lends itself to the use of analytical methods, which turn out to be particularly useful for a preliminary analysis of certain problems. Closed form solutions are not so frequent and are practically restricted to the case of halfspace problems [43,39,35]. The latter, in turn, are useful to investigate Knudsen layers and compute the slip and temperature jump coefficients. The use of BGK or similar models permits reducing the solution of Boltzmann’s integrodifferential equation in phase-space to solving integral equation in ordinary space. This is obtained because in the BGK model the distribution function f occurs only in two ways: explicitly in a linear, simple way and implicitly through a few moments (appearing in the local Maxwellian and the collision frequency). Then one can express f in terms of these moments by integrating a linear, simple partial differential equation; then, using the definitions of these moments and the expression of f one can obtain integral equations for the same moments [43,39,35]. These equations can be solved numerically in a much easier way than the Boltzmann equation. This is particularly true in the linearized case. The integral equation approach lends itself to a variational solution. The main idea of this method (for linearized problems) is the following. Suppose that we must solve the equation: Lh = S,
(15.1)
where h is the unknown, L a linear operator and S a source term. Assume that we can form a bilinear expression ((g, h)) such that ((Lg, h)) = ((g, Lh)), for any pair {g, h} in the set where we look for a solution. Then the expression (functional): ˜ = (h, ˜ Lh) ˜ − 2 (S, h) ˜ J (h)
(15.2)
has the property that if set h˜ = h + η, then the terms of first degree in η disappear and ˜ reduces to J (h) + ((η, Lη)) if and only if h is a solution of Equation (15.1). In other J (h) words if η is regarded as small (an error), the functional in Equation (15.2) becomes small of second order in the neighborhood of h, if and only if h is a solution of Equation (15.1). Then we say that the solutions of the latter equation satisfy a variational principle, or make the functional in Equation (15.2) stationary. Thus a way to look for solutions of Equation (15.1) is to look for solutions which make the functional in Equation (15.2) stationary (variational method).
The Boltzmann equation and fluid dynamics
39
The method is particularly useful if we know that ((η, Lη)) is non-negative (or nonpositive) because we can then characterize the solutions of Equation (15.1) as maxima or minima of the functional (15.2). But, even if this is not the case, the property is useful. First of all, it gives a non-arbitrary recipe to select among approximations to the solution in a given class. Second, if we find that the functional J is related to some physical quantity, we can compute this quantity with high accuracy, even if we have a poor approximation to h. If the error η is of order 10%, then J will be in fact computed with an error of the order ˜ from J (h) is of order η2 , as we have seen. of 1%, because the deviation of J (h) The integral formulation of the BGK model lends itself to the application of the variational method [58]. Thus in the case of Couette the functional is related to the stress component p12 which is constant and gives the drag exerted by the gas on each plate. Thus this quantity can be computed with high accuracy [58,43]. This method can be generalized to other problems and to the more complicated models [39]. It can also be used to obtain accurate finite ordinate schemes, by approximating the unknowns by trial functions which are piecewise constant [44]. In the case of the steady linearized Boltzmann equation, Equation (14.3), a similar method can be used. Let us indicate by Dh the differential part appearing in the left-hand side (Dh = ξ · ∂x h for steady problems) and assume that there is a source term as well (an example of a source occurs in linearized Poiseuille flow, see [43,39,35]) and write our equation in the form: Dh − Lh = S.
(15.3)
If we try the simplest possible bilinear expression (g, h) =
L 0
3
g(x, ξ )h(x, ξ ) dx dξ
(15.4)
and we use it with Lh = Dh − Lh we cannot reproduce the symmetry property ((Lg, h)) = ((g, Lh)). It works for Lh but not for Dh. There is however a trick [33] which leads to the desired result. Let us introduce the parity operator in velocity space, P , such that P [h(ξ )] = h(−ξ ). Then we can think of replacing Equation (15.3) by P Dh − P Lh = P S
(15.5)
because this is completely equivalent to the original equation. In addition, because of the central symmetry of the molecular interaction P Lh = LP h and the fact that we had no problems with L is not destroyed by the fact that we use P . On the other hand we have by a partial integration: (g, P Dh) = (P Dg, h) + g + , P h− B − P g − , h+ B .
(15.6)
40
C. Cercignani
Here g ± denote the restrictions of a function defined on the boundary to positive, respectively negative, values of ξ · n, where n is the unit vector normal to the boundary. In addition, we have put
g ± , h±
B
= ∂Ω
±ξ ·n>0
|ξ · n|g(x, ξ )h(x, ξ ) dξ dσ.
(15.7)
In the one-dimensional case, the integration over the boundary ∂Ω reduces to the sum of the boundary terms at x = 0 and x = L. Clearly the last two terms in Equation (15.6) do not fit in our description. We have two ways out of the difficulty. We first recall a property of the boundary conditions, discussed in Section 11. The boundary conditions must be linearized about the Maxwellian distribution M and this gives them the following form (see below): h+ = h0 + Kh− .
(15.8)
Because of reciprocity (Equation (11.9)), we have (P g − , Kh− ) B = (Kg − , P h− ) B .
(15.9)
Hence, if we assume that both g and h satisfy the boundary conditions, we have
g + , P h− B − P g − , h+ B = (h0 , P h− ) B − (P g − , h0 ) B + (Kg − , P h− ) B − (P g − , Kh− ) B = (h0 , P h− ) B − (P g − , h0 ) B . (15.10)
We remark that we can modify the solution of the problem by adding a combination of the collision invariants with constant coefficients. This does not modify the Boltzmann equation but can be used to modify the boundary conditions. Usually it is possible to dispose of the constant coefficients to make ((h0 , P h− ))B = 0 (and, at the same time, of course, ((h0 , P g − ))B = 0). We assume that this is the case. Using this relation and Equation (15.9), Equation (15.10) reduces to
g + , P h−
B
−
− + Pg ,h B = 0
(15.11)
and the variational principle holds with the operator Lh = P Dh − P Lh and the source P S. This variational principle is correct but not so useful, because it can be used only with approximations which exactly satisfy the boundary conditions and it can be complicated to construct these approximations. Thus we follow another procedure by incorporating the boundary conditions in the functional. It is enough to consider ˜ = (h, ˜ P D h˜ − P Lh) ˜ − 2 (P S, h) ˜ J (h) + (P h˜ − , h˜ + − K h˜ − − 2h0 ) B .
(15.12)
The Boltzmann equation and fluid dynamics
41
In fact, if we let h˜ = h + η, we find that the terms linear in η disappear from J and the variational principle holds. In agreement with what we said before, it is interesting to look at the value attained by J when h˜ = h. Equation (15.12) becomes (15.13) J (h) = − (P S, h) − (P h− , h0 ) B . This result acquires its full meaning only when we examine the expressions for h0 and S. In general S = 0 (an important case in which this is not true, is linearized Poiseuille flow). If we let S = 0, then we must look at the expression of h0 . The boundary source has a special form because it arises from the linearization, about a Maxwellian distribution M, of a boundary condition of the form: f+ = Kw f− ,
(15.14)
where Kw is an operator which has several properties, including Mw+ = Kw Mw− .
(15.15)
Now, if let f = M(1 + h) in Equation (15.14), we have: h+ =
Kw M− h− Kw M− + − 1. M+ M+
(15.16)
This relation is exact. We can now proceed to neglecting terms of order higher than first in the perturbation parameters. We can replace in Kw the temperature and velocity of the wall by those of the Maxwellian M (i.e., T0 and 0) and obtain a slightly different operator K0 . Thus we obtain the operator K, which we used before, by letting K0 M− h− /M+ = Kh− . Concerning the source, we have, using Equation (15.15): h0 =
Kw M− Kw M− Kw Mw− −1= − . M+ M+ Mw+
Since Mw and M differ by terms of first order, we can replace Kw by K0 because their difference is also of first order and would produce a term of second order in the expression of h0 . h0 =
K0 M− K0 Mw− K0 Mw− − =1− . M+ Mw+ Mw+
(15.17)
Now, if we neglect terms of higher than first order in the speed of the wall and the temperature difference Tw − T0 , we have Mw = M(1 + ψ), where (recalling that Mw is determined up to the density that we can choose to be the same as in M) ψ can be explicitly computed to give: |ξ |2 3 Tw − T0 ξ · vw + − ψ= RT0 2RT0 2 T0
(15.18)
42
C. Cercignani
and, finally, neglecting again terms of order higher than first: h0 = ψ+ − Kψ− .
(15.19)
Then Equation (15.13) with S = 0 gives J (h) = − (P h− , ψ+ ) B + (P h− , Kψ− ) B = − (P h− , ψ+ ) B + (Kh− , P ψ− ) B = − (P h− , ψ+ ) B + h+ − ψ + + Kψ − , P ψ− B = − (P h− , ψ+ ) B + h+ −, P ψ− B + Kψ − − ψ + , P ψ− B .
(15.20)
The last term is a known quantity, whereas the first and the second can be combined to give unknown quantities of physical importance. In fact, if we take into account the expression of ψ (Equation (15.18)), we obtain: − (P h− , ψ+ ) B + h+ −, P ψ− B ξ · vw =− ξ · nh(x, ξ )M dξ dσ RT0 3 Tw − T0 |ξ | − ξ · nh(x, ξ )M dξ dσ + 2RT0 2 T0 1 Tw − T0 =− dσ . pn · vw dσ + q(n) RT0 T0
(15.21)
Here pn is the normal stress vector and qn the normal component of the heat flow. The fact that the mass flow vanishes at the wall has been taken into account. Because of the linearity of the problem, it is possible and convenient, without loss of generality, to consider separately the two cases vw = 0 and Tw = T0 . Then the two terms in the expression above occur in two different problems. For some typical problems vw vanishes on just one part of the boundary, whereas it is a constant on the remaining part of the latter; then the factor multiplying this constant is the drag on the corresponding part of the boundary. Similarly one can consider the case in which the factor in front of qn vanishes on just one part of the boundary, whereas it is a constant on the remaining part of the latter, and relate the value of the functional to the heat transfer. The two variational principles which have been discussed are related to each other [39]. The integral equation approach and the variational method have been applied with great success to many linearized problems [43,39,35]. Among the most interesting results we mention the calculation of the minimum in the flow rate for Poiseuille flow [30,61,31, 44], first experimentally discovered by Knudsen [98], and the calculation of the drag on a sphere (at low Mach numbers) where the results agree very well [59] with the semiempirical formula derived by Millikan from his experimental data [126].
The Boltzmann equation and fluid dynamics
43
16. Mixtures As is well known, air at room pressure and temperature is a mixture, its main components being two diatomic gases, nitrogen and oxygen. This immediately calls for an immediate change in our basic equation, Equation (5.1), which is only suitable for a single monatomic gas. A remarkable feature of aerodynamics at the molecular level is that the evolution equations change in a significant way, when dealing with polyatomic rather than monatomic gases. This is not the case when the gas is treated as a continuum, where, at least at room conditions, only a few changes in the equations occur, the most remarkable being the change of the ratio of specific heats γ . If we consider a mixture of monatomic gases, the differences between the various species occur in the values of the masses and in the law of interaction between molecules of different species; in the simplest case, when the molecules are pictured as hard spheres, the second difference is represented by unequal values of the molecular diameters. In the mathematical treatment, a first difference will be in the fact that we shall need n distribution functions fi (i = 1, 2, . . . , n) if there are n species. The notation becomes complicated, but there is no new idea, except, of course, for the fact that we must derive a system of n coupled Boltzmann equations for the n distribution functions. The arguments are exactly the same, with obvious changes, and the result is ∂fi ∂fi ∂fi +ξ · + Xi · ∂t ∂x ∂ξ n
= (fi fk∗ − fi fk∗ )Bik (n · V, |V|) dξ ∗ dn, 3 k=1 R B+
(16.1)
where Bik is computed from the interaction law between the i-th and k-th species, while in the k-th term in the left-hand side, V = ξ − ξ ∗ is the relative velocity of the molecule of the i-th species (whose evolution we are following) with respect to a molecule of the k-th species (against which the former is colliding). The arguments ξ and ξ ∗ are computed, as before, from the laws of conservation of mass and energy in a collision with the following result: 2µik n (ξ − ξ ∗ ) · n , mi 2µik ξ ∗ = ξ ∗ + n (ξ − ξ ∗ ) · n , mk
ξ = ξ −
(16.2)
where µik = mi mk /(mi + mk ) is the reduced mass [39]. To prepare some material for the description of polyatomic gases and chemical reactions, we remark that Equation (16.1) can be rewritten as follows [39]: ∂fi ∂fi ∂fi +ξ · + Xi · ∂t ∂x ∂ξ
44
C. Cercignani
=
n
3 3 3 k=1 R ×R ×R
(fi fk∗ − fi fk∗ )Wik (ξ , ξ ∗ |ξ , ξ ∗ ) dξ ∗ dξ dξ ∗ .
(16.3)
Now ξ , ξ ∗ , ξ , ξ ∗ are independent variables (i.e. they are not related by the conservation laws) and Wik (ξ , ξ ∗ |ξ ξ ∗ ) = Sik n · V, |V| δ(mi ξ ∗ + mk ξ ∗ − mi ξ − mk ξ ∗ ) × δ mi |ξ ∗ |2 + mk |ξ ∗ |2 − mi |ξ |2 − mk |ξ ∗ |2 ,
(16.4)
where n = (ξ − ξ )/|ξ − ξ | and Bik (n · V, |V|) Sik n · V, |V| = (mi + mk )3 mi mk . 2n · V
(16.5)
Conservation of momentum and energy is now taken care of by the delta functions appearing in Equation (16.4) [39]. With a slight modification, Equation (16.3) can be extended to the case of a mixture in which a collision can transform the two colliding molecules of species j , l into two molecules of different species k, i (a very particular kind of chemical reaction). In this case the relations between the velocities before and after the encounter are different from the ones used so far, but we may still write a set of equations for the n species: ∂fi ∂fi ∂fi +ξ · + Xi · ∂t ∂x ∂ξ n
lj = (fl fj ∗ − fi fk∗ )Wik (ξ , ξ ∗ |ξ , ξ ∗ ) dξ ∗ dξ dξ ∗ , 3 3 3 k,l,j =1 R ×R ×R
(16.6)
where Wik gives the probability density that a transition from velocities ξ , ξ ∗ to velocities ξ , ξ ∗ takes place in a collision which transforms two molecules of species l, j , respectively, into two molecules of species i, k, respectively. It is clear how the previous model is included into the new one when the species change does not occur. The idea of kinetic models analogous to the BGK model can be naturally extended to mixtures and polyatomic gases [151,127,81]. A typical collision term of the BGK type will read lj
Ji (fr ) =
n
j =1
Jij (fr ) =
n
νij Φij (ξ ) − fi (ξ ) ,
(16.7)
j =1
where νij are the collision frequencies and Φij is a Maxwellian distribution to be determined by suitable conditions that generalize Equation (9.4). There are some important changes concerning the collision invariants and the definition of macroscopic functions in the case of mixtures. First of all, the collision invariants in
The Boltzmann equation and fluid dynamics
45
the case of n species have n components and are defined as follows: ψi (i = 1, . . . , n) is a collision invariant if and only if
i
R3
ψi Qi dξ = 0,
(16.8)
where Qi denotes the right-hand side of Equation (16.1). There are n + 4 rather than 5 linearly independent collision invariants. There are 3 invariants related to momentum conservation, ψ(n+α)i = mi ξα (α = 1, 2, 3), and one related to energy conservation, ψ(n+4)i = mi |ξ |2 ; the remaining n invariants are related to the conservation of the number of particles of each species ψij = δij (i, j = 1, . . . , n). This, of course, applies when there are no chemical reactions. Concerning the macroscopic quantities and their relation to the moments of the distribution functions, we remark that in the case of mixtures it is more convenient to think that the distribution function is normalized as a number density (this has been already taken into account when giving the expression of the collision invariants). Then the number densities of the single species are given by: n(i) =
fi dξ
(i = 1, . . . , n)
(16.9)
and the mass density ρ (i) is given by mi n(i) . The number and mass densities for the mixture are given by: n=
n
n(i) ,
(16.10)
ρ (i) .
(16.11)
i=1
ρ=
n
i=1
It is convenient to define the bulk velocities of the single species and the bulk velocity of the mixture as follows: 3 ξ fi dξ v(i) = R (i = 1, . . . , n), (16.12) R 3 fi dξ ρv =
n
ρ (i) v(i) .
(16.13)
i=1
It is usual to define the peculiar velocity c = ξ − v.
(16.14)
46
C. Cercignani
The stress tensor for the i-th species is given by: (i) pj k
= mi
cj ck fi dξ
(i = 1, . . . , n; j, k = 1, 2, 3);
(16.15)
R3
and the stress tensor for the mixture is the sum of the various stresses: pj k =
n
pj(i)k
(j, k = 1, 2, 3).
(16.16)
i=1
It is to be remarked that, though these definitions are the most common and natural, they are not used by all the authors. One might, e.g., define a peculiar velocity for each species and use it to define the partial stresses. Then it is no longer true that the stress tensor for the mixture is the sum of the partial stresses. Similarly the thermal energy per unit mass (associated with random motions) is defined for each species by: (i) (i)
n e
1 = 2
R3
|c|2 fi dξ
(i = 1, . . . , n)
(16.17)
and for the mixture by: ρe =
n
ρ (i) e(i) .
(16.18)
i=1
A similar procedure can be applied to the heat flow. The pressure is, as usual, 1/3 of the trace of the stress matrix and is related to the temperature by p = nkB T . Please remark that there is not a constant R such that p = ρRT .
17. Polyatomic gases A possible picture of a molecule of a polyatomic gas, suggested by quantum mechanics, is as follows [172]. The molecule is a mechanical system, which differs from a point mass by having a sequence of internal states, which can be identified by a label, assuming integral values. In the simplest cases these states differ from each other because the molecule has, besides kinetic energy, an internal energy taking different values Ei in each of the different states. A collision between two molecules, besides changing the velocities, can also change the internal states of the molecules and, as a consequence, the internal energy enters in the energy balance. From the viewpoint of writing evolution equations for the statistical behavior of the system, it is convenient to think of a single polyatomic gas as a mixture of different monatomic gases. Each of these gases is formed by the molecules corresponding to a given internal energy, and a collision changing the internal state of at least one molecule is considered as a reactive collision of the kind considered above,
The Boltzmann equation and fluid dynamics
47
Wik (ξ , ξ ∗ |ξ , ξ ∗ ) giving the probability density of a collision transforming two molecules with internal states l, j , respectively, and velocities ξ , ξ ∗ , respectively, into molecules with internal states i, k, respectively, and velocities ξ , ξ ∗ , respectively. This model is amply sufficient to discuss aerodynamic applications. We want to mention, however, that it requires nondegenerate levels of internal energy, if there are, e.g., strong magnetic fields which can act on the internal variables such as (typically) the spin of the molecules. In that case, if the molecule has spin s, the distribution function f becomes a square matrix of order 2s + 1 and the kinetic equation reflects the fact that matrices in general do not commute and, as remarked by Waldmann [170,171] and Snider [152], the collision term contains not simply the cross-section but the scattering amplitude, which may not commute with f . It is appropriate now to enquire why we started talking about quantum rather than classical mechanics. The main reason is not related to practice, but rather to history. Classical models of polyatomic molecules are regarded with suspicion since 1887 when Lorentz found a mistake in the proof of the H theorem of Boltzmann [23] for general polyatomic molecules. The question arises from the fact that when one proves the H theorem for a monatomic gas one usually does not explicitly underline (because it is irrelevant in that case) that the velocities ξ and ξ ∗ are not the velocities into which a collision transforms the velocities ξ and ξ ∗ , but the velocities which are transformed by a collision into the latter ones; this is conceptually very important, but the lack of a detailed discussion does not lead to any inconvenience because the expressions for ξ ∗ and ξ are invariant with respect to a change of sign of the unit vector n, which permits an equivalence between velocity pairs that are carried into the pair ξ ∗ , ξ and those which originate from the latter pair, as a consequence of a collision. The remarkable circumstance which we have just recalled is related to the particular symmetry of a collision described by a central force, which allows to associate to a collision [ξ ∗ , ξ ] → [ξ ∗ , ξ ] another collision, the socalled “inverse collision” [ξ ∗ , ξ ] → [ξ ∗ , ξ ], which differs from the former just because of the transformation of the unit vector n into −n. When polyatomic molecules are dealt with, the states before and after a collision require more than just the velocities of the mass centers to be described (the angular velocity, e.g., if the molecule is pictured as a solid body). Let us symbolically denote by [A, B] the state of the pair of molecules. Then there is no guarantee that one can correlate an “inverse collision” [A , B ] → [A, B], differing from the previous one just because of the change of n into −n with the collision [A, B] → [A , B ]. Now in the original proof of the H theorem for polyatomic molecules proposed by Boltzmann [23], the assumption was implicitly made that there is always such collision. Lorentz remarked [119,24,26] that this is not true in general. Boltzmann recognized his blunder and proposed another proof based on the so-called “closed cycles of collisions” [119,24,26,166]; the initial state [A, B] is reached not through a single collision but through a sequence of collisions. This proof, although called unobjectionable by Lorentz and Boltzmann [26], never satisfied anybody [167,171]. For a while the matter was forgotten till a quantum mechanical proof showed that the required property followed from the unitarity of the S matrix [171]. A satisfactory proof of the inequality required to prove the H theorem for a purely classical, but completely general, model was given only in 1981 [51]. lj
48
C. Cercignani
For aerodynamic applications all these aspects are not so relevant and, in fact, the main problem is to find a sufficiently handy model for practical calculations. Lordi and Mates [118] studied the “two centers of repulsion” model and found that a rather complicated numerical solution was required for a given set of impact parameters. The lack of closed form expressions makes the model impractical for applications, where the numerical solution describing the collision should be repeated many millions of times. Curtiss and Muckenfuss [132] developed the collision mechanics of the so-called spherocylinder model, consisting of a smooth elastic cylinder with two hemispherical ends. Whether or not two molecules collide depends on more than one parameter; in addition, there are several “chattering” collisions in a single collision event. The loaded-sphere model had been already developed by Jeans [94] in 1904 and was subsequently developed by Dahler and Sather [69] and Sandler and Dahler [146]. Although it is spherical in geometry, the molecules rotate about the center of mass, which does not coincide with the center of the sphere, with the consequence that it has essentially the same disadvantages as the sphero-cylinder model. The only exact model which is amenable to explicit calculations is the perfectly rough sphere model, first suggested by Bryan [28] in 1894. The name is due to the fact that the relative velocity at the point of contact of the two molecules is reversed by the collision. This model has some obvious disadvantages. First, a glancing collision may result in a large deflection; second, all collisions can produce a large interchange of rotational and translational energy, with the consequence that the relaxation time for rotational energy is unrealistically short; third, the number of internal degrees of freedom is three, rather than two, which makes the model inappropriate for a description of the main components of air, which are diatomic gases. One can disregard the first disadvantage and put a remedy to the second by assuming that a fraction of collisions follow the smooth-sphere rather that rough-sphere dynamics; but there is obviously no escape from the third difficulty. In practical calculations, one has learned, since long time, that one must compromise between the faithful adherence to a microscopic model and the computational time required to solve a concrete problem. This was true in the early days of rarefied gas dynamics (and may still be true nowadays when one tries to find approximate closed form solutions or spare computer time) even for monatomic gases, as we discussed in Section 9 (in connection with the BGK model) and shall discuss later (in connection with Direct Simulation Monte Carlo). As a matter of fact, when trying to solve the Boltzmann equation, one of the major shortcomings is the complicated structure of the collision term; if to this problem, present even in the simplest case, one adds the complication of the presence of internal degrees of freedom, any practical problem becomes intractable, unless one is ready to accept the aforementioned compromise. Fortunately, when one is not interested in fine details, it is possible to obtain reasonable results by replacing the collision integral by a phenomenological collision model, e.g., a simpler expression which retains only the qualitative and average properties of the true collision term. As computers become more and more powerful, the amount of phenomenological simplification diminishes and the calculations may more closely mimic the microscopic models. For polyatomic gases, the basic new fact with respect to the monatomic ones is that the total energy is redistributed between translational and internal degrees of freedom at each collision. Those collisions for which this redistribution is negligible are called elastic,
The Boltzmann equation and fluid dynamics
49
while the others are called inelastic. The simplest approach would be to calculate the effect of collisions as a linear combination of totally elastic and completely inelastic collisions, the second contribution being described by a model analogous to the BGK model which was described in Section 9. There are, of course, problems related to the molecule spins and their alignment; they are particularly important when we put the molecules in a magnetic field and peculiar phenomena, which go under the general name of Senftleben–Beenakker effects, arise. There is an entire book devoted to this topic [122] and we shall not deal with these problems. Let us now consider in more detail the case of a continuous internal-energy variable. In this case, it is convenient to take the unit vector n in the center-of-mass system and use the internal energy Ei and Ei∗ of the colliding molecules. As usual, the values before a collision will be denoted by a prime. Equation (1.2) is replaced by Q(f, f ) =
R3
dξ ∗
0
E−Ei
× 0
∞
n−2 2
Ei∗ dEi∗
(Ei∗ )
n−2 2
S2
E 0
(Ei )
n−2 2
dEi
dEi∗ (f f∗ − ff∗ )
× B(E; n · n ; Ei , Ei∗ → Ei , Ei∗ ).
(17.1)
Here E = m|ξ |2 /4 + Ei + Ei∗ is the total energy in the center-of-mass system which is conserved in a collision. The kernel B satisfies the reciprocity relation → Ei , Ei∗ ) |V|B(E; n · n ; Ei , Ei∗ = |V |B(E; n · n ; Ei , Ei∗ → Ei , Ei∗ ).
(17.2)
Here we follow a paper by Kušˇcer [111] and look for a one-parameter family of models, assuming that the scattering is isotropic in the center-of-mass system. The second crucial assumption will be that the redistribution of energy among the various degrees of freedom only depends upon the ratios of the various energies to the total energy E, εi = Ei /E, etc. This assumption is valid for collisions of rigid elastic bodies and can be considered as a good approximation for steep repulsive potentials. It is then possible to write B in the following form: → Ei , Ei∗ ) |V|B(E; n · n ; Ei , Ei∗
=
|V |2 σtot (E) θ (εi , εi∗ → εi , εi∗ ; τ ). |V| 4πE n
(17.3)
The denominator on the right takes care of normalization. Then the function θ satisfies the following relations:
1
ε 0
n−2 2
1−ε
dε 0
n−2
ε∗ 2 dε∗ θ (εi , εi∗ → εi , εi∗ ; τ ) = 1,
(17.4)
50
C. Cercignani (1 − εi − εi∗ )θ (εi , εi∗ → εi , εi∗ ; τ ) = (1 − εi − εi∗ )θ (εi , εi∗ → εi , εi∗ ; τ ).
(17.5)
The dependence of σtot on E makes it possible to adjust the model to the correct dependence of the viscosity on temperature. The parameter τ is chosen in such a way as to represent the degree of inelasticity of the collisions. τ = 0 corresponds to elastic collisions: → εi , εi∗ ; 0) = ε− θ (εi , εi∗
n−2 2
− n−2 2
ε∗
= δ(εi − εi )δ(εi − εi∗ ).
(17.6)
τ = ∞ corresponds to maximally inelastic collisions: → εi , εi∗ ; ∞) = ε− θ (εi , εi∗
n−2 2
− n−2 2
ε∗
Γ (n + 2) (1 − εi − ε∗i ). (Γ (n))2
(17.7)
A mixture of the two extreme cases gives the model first proposed by Borgnakke and Larsen [27] in 1975: θ (εi , εi∗ → εi , εi∗ ; τ )
→ εi , εi∗ ; 0) + 1 − e−τ θ (εi , εi∗ → εi , εi∗ ; ∞). = e−τ θ (εi , εi∗
(17.8)
Kušˇcer [111] notices an analogy between this model and Maxwell’s model for gas-surface interaction, as discussed Section 11, and introduces another model, called the theta model, which would correspond to the Cercignani–Lampis model in this analogy. The Larsen–Borgnakke model has become a customary tool in numerical simulations of polyatomic gases. It can be also applied to the vibrational modes through either a classical procedure that assigns a continuously distributed vibrational energy to each molecule, or through a quantum approach that assigns discrete vibrational levels to each molecule. It would be out of place here to discuss this point in more detail, for which we refer to the book of Bird [20]. We also refrain from discussing the interesting recent developments [12, 13] of an old idea of Boltzmann [25,42] to interpret, in the frame of classical statistical mechanics, the circumstance that at low temperatures the internal degrees of freedom appear to be frozen, as due to the extremely long relaxation times for the energy transfer process. The Larsen–Borgnakke model suffers from the limitation that it considers all the collisions as a mixture of the elastic or completely inelastic collisions, disregarding the possibility of a partially inelastic one. In order to construct a more general model we use a procedure which was first used to deal with models for boundary conditions [43,39,35]. We → ε , ε ), which is chosen on the basis start from a sensible approximate kernel θ0 (εi , εi1 i i1 of intuition, but does not satisfy the basic properties (17.4) and (17.5). At this point we add to it some other terms, which ensure that the three fundamental properties are satisfied, as follows: → εi , εi1 ) θ (εi , εi1 = θ0 (εi , εi1 → εi , εi1 )
The Boltzmann equation and fluid dynamics
+
51
Γ (n + 2) (1 − εi − εi1 ) 1 − H (εi , εi1 ) 1 − H (εi , εi1 ) /I, 2 (Γ (n))
H (εi , εi1 )=
1 0 0
1−εi
n−2 2
θ0 (εi , εi1 → εi , εi1 ) εi
n−2
εi12 dεi dεi1 ,
Γ (n + 2) (Γ (n))2 1 1−εi n−2 n−2 × (1 − εi − εi1 ) H (εi , εi1 ) εi 2 εi12 dεi dεi1 .
(17.9)
I =1−
0
(17.10)
0
As in the aforementioned case, Equation (17.9) may be interpreted as the linear ) is a sort combination of two normalized kernels θ1 (. . .) and θ2 (. . .), while H (εi , εi1 of accommodation coefficient depending on the energies of the impinging molecules ) must lay in the interval [0, 1]); we write (H (εi , εi1 )θ1 (. . .) + 1 − H (εi , εi1 ) θ2 (. . .), θ (. . .) = H (εi , εi1 ); θ0 (. . .) = θ1 (. . .)H (εi , εi1
θ2 (. . .) =
(17.11)
Γ (n + 2) (1 − εi − εi1 ) 1 − H (εi , εi1 ) /I. 2 (Γ (n))
The expression of θ2 (. . .) is suggested by the requirement that the product (1 − ))θ (. . .) must satisfy reciprocity; the definition of I is chosen in order to get H (εi , εi1 2 the normalization of θ2 (. . .). → ε , ε ; a, b), Cercignani and Lampis [52] proposed the following kernel θ0 (εi , εi1 i i1 containing two parameters a and b. θ0 (εi , εi1 → εi , εi1 ; a, b) =
1/2 ba − n−2 − n−2 − n−2 − n−2 (1 − εi − εi1 ) 4 εi 4 εi1 4 εi 4 εi1 π (1 − εi − εi1 )1/2 2 2 × exp −a(εi − ε ) − a(εi1 − εi1 ) . (17.12)
In order to avoid a singularity in the expression of the kernel (17.12) and therefore also in ), it is assumed that E − E − E = mc 2 /4 > τ , where τ is a little positive that of H (εi , εi1 r i i1 ) are limited. This approximate kernel θ (. . .) for parameter: then θ0 (. . .) and H (εi , εi1 0 , ε , ε ), so that the full kernel given by Equation (10) tends to a → ∞ tends to bθel (εi , εi1 i i1 the Larsen–Borgnakke model. For finite values of a, θ0 (. . .) describes a collision in which ), but may be different, according to εi is not exactly equal to εi (and similarly for εi1 , εi1 a Gaussian distribution. By fitting the theoretical expression for viscosity with some experimental data (from standard handbooks), the aforementioned authors [52] have obtained the values of the parameters in the case of N2 and O2 .
52
C. Cercignani
The aforementioned authors, in a joint paper with J. Struckmeyer [56,57,55], found that a good fitting of some experimental data given in [99] can be obtained. However, they were unable to obtain a unique determination of the parameters a and b. Some examples of possible choices of a and b have been given in [52], in the case of N2 and O2 . In those examples, low values of a, for instance, a = 1, a = 0.1, and for each of them a value of b close to its maximum value, were chosen, but it is also possible to choose much higher values of a. In order to obtain more information about the values of the parameters a and b, an obvious way would be to try to fit a second transport coefficient. Unfortunately, the experimental data for ηv versus temperature are very scanty [141,73]. The bulk viscosity of gases can only be measured by the attenuation and dispersion of an ultrasonic, acoustic signal. Moreover, it is a difficult method subject to experimental error [73]. Because of this, it is not possible to draw conclusions on the range of applicability of the model. In the application of the kernel to the calculation of transport coefficients, everything does work also without introducing the cut-off. The situation may be different in other problems, for instance in the application of DSMC. Therefore a similar model that does not present singularities and does not require a cut-off was introduced [56], based on the following kernel: → εi , εi1 ; a, b) θ0 (εi , εi1
=
2ba − n−2 (1 − εi − εi1 ) − n−2 − n−2 − n−2 4 εi 4 εi1 4 εi 4 εi1 ) π (1 − εi − εi1 ) + (1 − εi − εi1 2 2 ) . × exp −a(εi − ε ) − a(εi1 − εi1
(17.13)
Using this kernel, the authors repeated the calculation of heat conductivity, which follows the same procedure as before. The results given in [52] about heat conductivity versus temperature are identical to those calculated with the new kernel. We end this review of models for polyatomic gases by remarking that a model generalizing the ES model (see Section 9) to polyatomic gases has been discussed by Andries et al. [3], who have also shown that the H -theorem holds for this model for polyatomic gases as well. Concerning the boundary conditions for the distribution function for polyatomic gases, we remark that there is not much material published on this subject, perhaps because the theory is not so different from that holding in the monatomic case. Concerning specific models, one should mention an extension of the CL model to polyatomic molecules proposed by Lord [117] as a generalization of the Cercignani–Lampis model [48].
18. Chemistry and radiation Chemical reactions are important in high altitude flight because of the high temperatures which develop near a vehicle flying at hypersonic speed (i.e., at Mach numbers larger than 5). Up to 2000 K, the composition of air can be considered to be the same as at standard conditions. Beyond this temperature, N2 and O2 begin to react and form NO. At
The Boltzmann equation and fluid dynamics
53
2500 K diatomic oxygen begins to dissociate and form atomic oxygen O, till O2 completely disappears at about 4000 K. Nitrogen begins to dissociate at a slightly higher temperature (about 4250 K). NO disappears at about 5000 K. Ionization phenomena start at about 8500 K. As we implicitly remarked when we wrote Equation (16.6), the kinetic theory of gases is an ideal tool to deal with chemical reactions of a particular kind, i.e. bimolecular reactions, which can be written schematically as A + B ↔ C + D,
(18.1)
where A, B, C and D represent different molecular species. We already used the term “molecule”, as usual in kinetic theory, to mean also atom (a monatomic molecule); in this section we shall further enlarge the meaning of this term to include ions, electrons and photons as well, when we have to deal with ionization reactions and interaction with radiation. As long as the reaction takes place in a single step with the presence of no other species than the reactants, it is a well-known circumstance that the change of concentration of a given species (A, say) in a space-homogeneous mixture can be written as follows: dnA = kb (T )nC nD − kf (T )nA nB . dt
(18.2)
We remark that, in chemistry, one uses the molar density in place of the number density used here, the two being obviously related through Avogadro’s number. The rate coefficients kf and kb for the forward and backward (or reverse) reactions, respectively, are functions of temperature and are usually written by a semiempirical argument, which generalizes the Arrhenius formula, in the form: Ea k(T ) = ΛT η exp − , kT
(18.3)
where Λ and η (= 0 in the Arrhenius equation) are constants, and Ea is the so-called activation energy of the reaction. It is clear that these equations, though having a flavor of kinetic theory, are essentially macroscopic and can be assumed to hold when the distribution function is essentially Maxwellian. In fact, the above reaction theory can be obtained by assuming that the distribution functions are Maxwellians, whereas the role of internal degrees of freedom may be ignored and the reaction cross-section vanishes if the translational energy Et in the center-of-mass system is less than Ea and equals a constant σR if the energy is larger than Ea . A more accurate theory is obtained [18–20] by assuming that the ratio of the reaction cross-section to the total cross-section is zero when the total collision energy Ec (equal to the sum of Et and the total internal energy of the two colliding molecules Ei ) is less than Ea and proportional to the product of a power of Ec −Ea and a power of Ec . The exponents and the proportionality factor are essentially dictated by the number of internal degrees of freedom, the exponent of the temperature in the diffusion coefficient of species A in species B and
54
C. Cercignani
the empirical exponent η appearing in Equation (18.3). This theory provides a microscopic reaction model that can reproduce the conventional rate Equations (18.2)–(18.3) in the continuum limit. The model is, however, as in the case of gas-surface interaction and models for polyatomic gases, largely based on phenomenological considerations and mathematical tractability. The ideal microscopic model would consist of complete tabulations of the differential cross-sections as functions of the energy states and n. Some microscopic data, coming from extensive quantum-mechanical computations, supported by experiments, are available, but, unfortunately, not very much is known for reactions of engineering interest. When comparisons can be made, the reaction cross-section provided by the phenomenological model is of the correct order of magnitude. This provides some reasons of optimism about the validity of the results obtained with these models for the highly non-equilibrium rarefied gas flows. Termolecular reactions provide some difficulty to kinetic theory, because the Boltzmann equation essentially describes the effect of binary collisions. They are, however, of essential importance in high-temperature air, where the reverse (or backward) reaction of a dissociation one is a recombination reaction, which is necessarily termolecular, as we shall presently explain. A typical dissociation-recombination reaction can be represented as AB + X ↔ A + B + X,
(18.4)
where AB, A, B and X represent the dissociating molecule, the two molecules produced by the dissociation and a third molecule (of any species), respectively. The latter molecule, in the forward reaction, collides with AB and causes its dissociation. This process is described by a binary collision and is an endothermic reaction, requiring a certain amount of energy, the dissociation energy Ed . The recombination process is an exothermic reaction and it might seem that one could dispense with the “third body” X and consider it as a bimolecular reaction AB ← A + B.
(18.5)
However, one can easily see that the energy balance for this event cannot be satisfied in the presence of energy release. In fact if two molecules form an isolated system and are assumed to interact with a potential energy which is attractive at large distances and repulsive at short distances, they can come close enough to orbit one about the other but the repulsive part will eventually separate them. In fact, they cannot form a stable molecule; this is seen by writing the energy equation in the center-of-mass system. The final kinetic energy in this reference system should be zero for the molecule whereas it was positive when the two molecules approached each other and the potential energy is negative. A third molecule X is required to describe the recombination process. In order to keep the binary collision analysis, appropriate for a rarefied gas, we must think of the recombination process as a sequence of two binary collisions. The first of these forms an (unstable) orbiting pair P , that is stabilized by a second collision of this pair with X, as long as this collision occurs within a sufficiently small elapsed time. One can then extend the previous theory based on a binary collision analysis. If the activation energy is assumed to be zero, then the main change is that the cross section acquires a factor proportional to the number density of the species X.
The Boltzmann equation and fluid dynamics
55
This simple theory is based on a molecular interaction that is attractive at large distance and repulsive at short distances. One can assume a highly simplified scheme by taking a hard sphere core with diameter σ and a scattering of the square-well type at some larger distance σ∗ . The potential energy is, say, −Q (Q > 0) between σ and σ∗ . Then if m is the mass of a molecule A, for a distance r > σ∗ , the trajectory of one molecule with respect to the other before the interaction begins, will be a straight line r cos ϑ = b (in polar coordinates), where b is the impact parameter. The condition b < σ∗ must be satisfied if the molecules actually interact. Then, assuming for simplicity that A = B (as in the case of recombination of oxigen) the conservation of energy and angular momentum give
d 1 dϑ r
2 +
1 4Q 1 − = r 2 mV 2 b2
(18.6)
(σ < r < σ∗ ),
where V is, as usual, the relative speed. We easily verify that the trajectory has a corner point at a distance r = σ∗ and the molecule we are following is deflected toward the other by an angle −1
θ0 = cos
b σ∗
−1
− cos
b 4Q −1/2 1+ . σ∗ mV 2
(18.7)
The orbiting pair P has in this case a very simple motion: in fact the molecules A approach each other and then have a hard sphere collision. After that they tend to separate again; and the “molecule” P will disappear and two molecules A emerge again, unless a third molecule X collides with P , which is stabilized into a A2 . The unstable pair P is endowed with an internal energy EB =
m 2 V + Q. 4
(18.8)
This energy is stored to be, possibly, converted into kinetic energy (and hence, from a macroscopic viewpoint, heat of reaction) through the process (18.4). Even for this simple scheme we must write three Boltzmann equations: one for the species A, one for the species A2 and one for the unstable species P ; even if A is monatomic A2 and P are diatomic and hence have an internal energy. The species X used in the above argument can be any of the three aforementioned species. The species A loses particles when colliding with A (formation of P ) and P gains in the same process, but loses in most collisions undergone by a P molecule. This model can be slightly complicated by assuming that there is a potential barrier between σ∗ and σ∗∗ (σ∗∗ > σ∗ ). If the potential energy is Ea > 0 for these distances between the A molecules, then the formation can occur if, and only if, the relative speed V is larger than 4Ea /m. Then Ea plays the role of the activation energy. The resulting system of Boltzmann equations reads as follows: ∂f ∂f +ξ · = −2 ∂t ∂x
R3
B+
r ff∗ BAA dξ ∗ dn +
R3
(f F∗ − fg∗ )BAA2 dξ ∗ dn
B+
56
C. Cercignani
+ +
∂F ∂F +ξ · =2 ∂t ∂x
R 3 B+
R 3 B+
R 3 B+
+ +
∂g ∂g +ξ · = ∂t ∂x
R 3 B+
R3
R 3 B+
F g∗ BBA2 dξ ∗ dn +
R 3 B+
−
(f F∗ − f F∗ )BAA2 dξ ∗ dn,
−
e (f f∗ − ff∗ )BAA dξ ∗ dn
B+
R 3 B+
f g∗ BBA dξ ∗ dn
F g∗ BBA2 dξ ∗ dn (F f∗ − Ff∗ )BAA2 dξ ∗ dn (F F∗ − F F∗ )BA2 A2 dξ ∗ dn,
r f f∗ BAA dξ ∗ dn −
R 3 B+
(18.9)1
gF∗ BBA2 dξ ∗ dn,
(18.9)2
R 3 B+
gf∗ BBA dξ ∗ dn (18.9)3
where f , F , and g denote the distribution functions for species A, A2 , and B, whereas the superscripts r and e are used to discriminate between reactive and elastic collisions when necessary. For simplicity, we have omitted indicating the internal energies of particles A2 and B. The factors 2 take into account the fact that 2 particles of a species disappear or appear at the same time. Models to deal with a chemically reactive gas, akin to the BGK one have been discussed by Yoshizawa [178]. Ionization reactions involve the electronic states and it is unlikely that a purely classical theory will be successful in describing them, because of the selection rules. Yet, one can use the phenomenological approach to provide at least an upper bound for the reaction rates. As mentioned above, one can, in principle, think of interaction with radiation as if it were a reaction involving photons as “molecules”. Here spontaneous emission should also be taken into account. It becomes harder to develop phenomenological models, because one should consider as many species as there are excited levels for each molecule. Photons can be described by means of the so-called radiative transfer (or radiation transport) equation, which looks like a Boltzmann equation. The analogy is, however, in a sense, artificial, because the number of photons is not conserved. ∂f ∂f + cω · ∂t ∂x
The Boltzmann equation and fluid dynamics
=
R3
K(x, k → k)f (x, k ) dk − ν(x, k)f (x, k) + s(x, k).
57
(18.10)
Here K is the scattering probability from a wavevector k to a wavevector k. The direction of k is given by ω and its magnitude is the radiation frequency multiplied by the speed of light c, not to be confused with ν(x, k), the total frequency of scattering and absorption events. The term s(x, k) describes the volume radiation source, due to photon emission. If inelastic scattering effects, like fluorescence and stimulated emission, are neglected, then there is no interaction between photons of different frequency and we can replace the arguments by k and k by the corresponding unit vectors ω and ω . The emission term can be expressed as a product of the Planck distribution by the volume emission coefficient ε|k|,T . Boundary conditions for radiation can be described in a way similar to that used for gas-surface interaction. Of course, emission and absorption of radiation occur, along with reflection [149,96]. 19. The DSMC method Kinetic models and perturbation methods are very useful in obtaining approximate solutions and forming qualitative ideas on the solutions of practical problems, but in general they are not sufficient to provide detailed and precise answers for practical problems. Various numerical procedures exist which either attempt to solve for f by conventional techniques of numerical analysis or efficiently by-pass the formalism of the integrodifferential equation and simulate the physical situation that the equation describes (Monte Carlo simulation). Only recently proofs have been given that these partly deterministic, partly stochastic games provide solutions that converge (in a suitable sense) to solutions of the Boltzmann equation. Numerical solutions of the Boltzmann equation based on finite difference methods meet with severe computational requirements due to the large number of independent variables. The only method that has been used for space-inhomogeneous problems in more than one space dimension is the technique of Hicks, Yen and Nordsiek [136,177], which is based on a Monte Carlo quadrature method to evaluate the collision integral. This method was further developed by Aristov and Tcheremissine [4,164] and has been applied with some success to a few two-dimensional flows [165,45]. An additional difficulty for traditional numerical methods is the fact that chemically reacting and thermally radiating flows (and even simpler flows of polyatomic gases) are hard to describe with theoretical models having the same degree of accurateness as the Boltzmann equation for monatomic nonreacting and nonradiating gases. These considerations paved the way to the development of simulation schemes, which started with the work of Bird on the so-called Direct Simulation Monte Carlo (DSMC) method [17] and have become a powerful tool for practical calculations. There appear to be very few limitations to the complexity of the flow fields that this approach can deal with. Chemically reacting and ionized flows can be and have been analysed by these methods. A problem which arises in the applications of the DSMC method is the choice of a model for the molecular collisions. The issue is to have simple computing rules by discarding what
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is physically insignificant. In the case of monatomic gases, the relation of the deflection angle to the impact parameter and the relative speed appears to be the most important piece of physics. It turns out, however, that the scattering law has little effect and that the observable effects are strongly correlated with the cross-section change with relative speed. This realization led to the idea of the variable hard-sphere (VHS) model [19] which combines the scattering simplicity of the hard-sphere model with a variable cross-section based on a molecular diameter proportional to some power ω − 1/2 of the relative speed (ω being the power of absolute temperature ruling the change of the viscosity coefficient). This does not produce problems in a single gas, because the heat conductivity varies with approximately the same law as the viscosity coefficient, but problems arise for mixtures. If one wants the correct diffusion coefficient, another modification is needed. Koura and Matsumoto [107] developed the variable soft sphere (VSS) model, which introduces an additional power law parameter and gives the necessary flexibility for mixtures. More complicated models can be devised when the attractive part of the intermolecular force is taken into account [84]. A simpler method has also been proposed [97]. Before discussing the DSMC method and some of its applications in some detail, we remark that, although the DSMC has no rivals for practical computations, some other methods may turn out to be of interest in the future if much more powerful computers will be available. Thus, e.g., discrete velocity models have been an intensely studied subject for many years, before becoming a systematic method of approximating the Boltzmann equation. The DSMC method, the molecular collisions are considered on a probabilistic rather than a deterministic basis. The main aim is to calculate practical flows through the use of the collision mechanics of model molecules. In fact, the real gas is modeled by some hundred thousands or millions of simulated molecules on a computer. For each of them the space coordinates and velocity components (as well as the variables describing the internal state, if we deal with polyatomic molecules) are stored in the memory and are modified with time as the molecules are simultaneously followed through representative collisions and boundary interactions in the simulated region of space. In most applications, the number of simulated molecules is extremely small in comparison with the number of molecules that would be present in a real gas flow. Thus, in the simulation, each model molecule is representing the appropriate number of real molecules, The calculation is unsteady and the steady solutions are obtained as asymptotic limits of unsteady solutions. The flow field is subdivided into cells, which are taken to be small enough for the solution to be approximately constant through the cell. The time variable is advanced in discrete steps of size ∆t, small with respect to the mean free time, i.e., the time between two subsequent collisions of a molecule. This permits a separation of the inertial motion of the molecules from the collision process: one first moves the molecules according to collisionfree dynamics and subsequently the velocities are modified according to the collisions occurring in each cell. The rate of occurrence of collisions is given by (hard spheres): r=
i,j
rij ,
(19.1)
The Boltzmann equation and fluid dynamics
rij =
ρ n · (ξ i − ξ j ) dn, Nm B+
59
(19.2)
where N is the number of molecules in the sample and ξ i is the velocity of the i-th molecule (i = 1, . . . , N). Each time a collision occurs, the velocities of a collision pair are modified (and, as a consequence, r also varies). Let Tk be the length of the time interval between the (k − 1)-th and the k-th collision in the time interval [0, ∆t] (t = 0 is the time of the 0-th collision by definition). The time intervals Tk are chosen in this way: after the (k − 1)-th collision, one samples a pair (i, j ) on the basis of the probability distribution pij = rij /r (with a fixed velocity for each pair); then one takes Tk = 2N/[(N − 1)rij ]. Then one samples a direction ω of the post-collisional velocity ξ i − ξ j with the probability + distribution p(ω) = rij /r (fixed i, j and |ξ i − ξ j |) and replaces ξ i and ξ j by ξ + i and ξ j , given by ξ+ i =
1 ξ + ξ j + ω|ξ i − ξ j | , 2 i
(19.3)
1 ξ i + ξ j − ω|ξ i − ξ j | . (19.4) 2 The operation is repeated until k Tk exceeds ∆T. Some variations of Bird’s method have appeared, due to Koura [104], Belotserkovskii and Yanitskii [11], Deshpande [72]. They differ from Bird’s method because of the procedure used to sample the time interval between two subsequent collisions. In particular Koura [104] introduced the so-called “null collision technique”, which uses, for models different from hard spheres, the maximum of the cross-section, when estimating the possible collisions. Then for values of the cross-section less than the maximum, some collisions produce a null effect. A different method was proposed by Nanbu [133]. One does not subdivide ∆t and works with the probability Pi that the i-th molecule collides in [0, ∆t]. One has ξ+ i =
Pi = ∆t
N
(19.5)
rij .
j =1
One starts with rij evaluated at t = 0 and samples a random number ω in [0, 1]. According to whether ω is smaller or larger than Pi , the i-th molecule collides or does not collide in [0, ∆t]. If a collision occurs, one samples a collision partner of the i-th molecule from the probability distribution rij (i) Pj = N
k=1 rik
.
(19.6)
+ Then one samples a direction ω of the post-collisional velocity ξ + i − ξ j with the + probability distribution p(ω) = rij /r (for fixed values of i, j and |ξ + i − ξ j |) and replaces ξ i by ξ + i given by Equation (19.3). Then the procedure is repeated for all the values of i.
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The method of Nanbu was criticized by Koura [103], who asserted that momentum and energy are not conserved by collisions, because one does not change the velocity of the j -th molecule, when one changes the velocity of the i-th molecule. The criticism is not wellfounded, however, because, as Nanbu pointed out [134], the Boltzmann equation requires the overall conservation of momentum and energy of the system at each space point, not the conservation of the same quantities at each single simulated event. What is new in Nanbu’s method is precisely the circumstance that it does not try to simulate the N -body dynamics, but rather the description of the system described by the Boltzmann equation. Nanbu’s method is now well understood, from both a physical and mathematical standpoint and has been rigorously proven to yield approximations to solutions of the Boltzmann equation, provided the number of test molecules is sufficiently large. The relevant theorem reads as follows [7,8]: T HEOREM . If the Boltzmann equation with initial data f0 has a smooth, nonnegative solution f (x, ξ ) ∈ L1 , then the solution fˆ of Nanbu’s method converges weakly in L1 to f , in the sense that, for any test function φ(x, ξ ) ∈ L∞
φ fˆ dx dξ →
φf dx dξ
(19.7)
as N → ∞, ∆x, ∆t → 0. Subsequently, a similar result was proved for Bird’s method [169,142]. The fluctuations inevitably occurring in a DSMC calculation can cause problems if the number of molecules is not large enough to have a representative sample in each area of the flow domain. They can, however, have a physical significance and contain information on those occurring in a real gas. The comparisons between the fluctuations in DSMC and the predictions of fluctuating hydrodynamic theory have been reviewed by Garcia [76] and appear to be consistent with the fluctuations in a real gas. The computing task of a simulation method varies with the molecular model. For models other than Maxwell’s it is proportional to N for Bird’s method, while it is proportional to N 2 for Nanbu’s method. Babovsky [7] found a procedure to reduce the computing task of Nanbu’s method and make it proportional to N . His modification is based on the idea of subdividing the interval [0, 1] into N equal subintervals. If the random number ω lies in the j -th segment, one calculates only Pij = rij ∆t and there is no collision if ω < (j/n) − Pij , there is a collision with the j -th molecule if ω (j/n) − Pij . There is also a condition that Pij must satisfy, i.e., Pij < 1/N , but this is usually automatically verified, given the size of ∆t. Application of Nanbu’s method in the form modified by Babovsky shows that the the computing task is not only theoretically but also practically comparable to that of Bird’s method [80]. This modification eventually evolved into what is called the “Finite-Pointset” method. We remark that one may take advantage of flow symmetries in physical space, but all collisions are calculated as three-dimensional events. As for the boundary conditions, Maxwell’s model of diffuse reflection (see Section 11) is adequate for many problems. There are many cases. The CL model [48] has been
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adapted and extended by Lord [117] for application in DSMC studies. The resulting CLL model [117,150] has been shown to provide a realistic boundary condition with incomplete accommodation [176]. More complicated models would be required to describe chemical reactions which can occur at the surface for high impact energies.
20. Some applications of the DSMC method The first significant application of DSMC method dealt with the structure of a normal shock wave [121], but only a few years later Bird was able to calculate shock profiles [15] that allowed meaningful comparisons with the experimental results then available [16] and with subsequent experiments [147,2]. This long time span is understandable: the method is very demanding of computer resources. In 1964, even with the fastest computers, the restriction on the number of molecules which could be used was such that large random fluctuations had to be expected in the results, and it was difficult to arrive at definite conclusions. Thus the number of simulated molecules and the sample sizes in the computations that could be performed in those years were extremely small in comparison with those that have been routinely employed by an increasing number of workers. The problem of the shock wave structure has continued to be an important test case. Later studies have included comparisons of measured and computed velocity distribution functions within strong shock waves in helium [140]. Early DSMC studies were also devoted to the problem of hypersonic leading edge. This arises in connection with the flow of a gas past a very sharp plate, parallel to the oncoming stream. When the Reynolds number Re = ρ∞ V∞ L/µ∞ , based on the plate length is very large, the picture, familiar from continuum mechanics, of a potential flow plus a viscous boundary layer is valid everywhere except near the leading and the trailing edge. Estimates obtained already in the late 1960s by Stewartson [161] and Messiter [124] showed that the Knudsen number at the trailing edge is of order Ma∞ Re−3/4 , where Ma∞ is the upstream Mach number. As a consequence, kinetic theory is not needed (for large values of Re) at the trailing edge. For the leading edge, the Knudsen number is of order Ma∞ ; hence in supersonic, or, even more, hypersonic flow (Ma∞ 5), the flow in the region about the leading edge must be considered as a typical problem in kinetic theory. In particular, the viscous boundary layer and the outer flow are no longer distinct from each other, although [123,82,95] a shock-like structure may still be identified. It is in this connection that the name of merged-layer regime, mentioned in Section 1, arose. There are several methods based on simplified continuum models, represented by the papers of Oguchi [137], Shorenstein and Probstein [148], Chow [66,67], Rudman and Rubin [145], Cheng et al. [65], and Kot and Turcotte [102], which usefully predict surface and other gross properties in this regime. The good agreement between these approaches and experiment gave new evidence for the the importance of the Navier–Stokes equations. Nevertheless, if we go sufficiently close to the leading edge, the Navier–Stokes equations must be given up in favor of the Boltzmann equation. Huang and coworkers [90,88,89] carried out extensive computations based on discrete ordinate methods for the BGK model and were able to show the process of building the flow picture assumed in the simplified continuum models mentioned above.
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The first DSMC is due to Vogenitz et al. [168] and exhibits a flow structure qualitatively different from the predictions of earlier studies. Their results are supported by the experiments of Metcalf et al. [125]. Validation studies of the DSMC method were also conducted at the Imperial College [83]. Hypersonic flows past blunt bodies were also the object of many simulations, most of the calculations being those made for the Shuttle Orbiter re-entry, for which useful comparisons with measured data were possible [128]. This comparison was concerned with the windward centerline heating and employed an axially symmetric equivalent body. Later comparisons [143] with Shuttle data were for the aerodynamic characteristics of the full three-dimensional shape. Another interesting problem which has been simulated by Ivanov and his coworkers is the reflection on a plane wall of an oblique shock wave generated by a wedge [92,93]. Three-dimensional DSMC calculations have also been made for the flow past a delta wing [29]. The results compare well with wind-tunnel measurements [116] of the flow field under the same conditions. Other important problems are related to separated flows, especially wake flows and flows involving viscous boundary layer separation and reattachment. The first calculations referred to the two-dimensional flow over a sharp flat plate followed by an angled ramp [129]. The results were in a reasonably good agreement with wind tunnel studies, which is not truly two-dimensional because of inevitable sidewall effects. Similar experiments were therefore performed [63] for the corresponding axially symmetric flow, less subject to the aforementioned non-uniformity. The DSMC calculations for these cases [130] show excellent agreement with experimental results. In particular, separation and reattachment of a viscous boundary layer in the laminar regime are correctly predicted. The most remarkable wake flow simulation was for a 70◦ spherically blunted cone model that had been tested in several wind tunnels [1,115]. The results of the calculations [131] of the lee side flow that contains the vortex are in good agreement with the experiments and with Computational Fluid Dynamics (CFD) studies of the flow based on the Navier–Stokes equations. In the case of polyatomic gases one has several cross-sections, such as elastic, rotational, vibrational, and also reactive, if chemical reactions occur. Koura [105] has extended his null collision technique [104] to these cases and improved it later [106]. He applied this method to simulate the hypersonic rarefied nitrogen flow past a circular cylinder [106], with particular attention to the simulation of the vibrational relaxation of the gas; he also investigated the effect of changing the number of molecules in each (adaptive) cell and the truncation in the molecular levels. The Direct Simulation Monte Carlo method is not only a practical tool for engineers, but also a good method for probing into uncovered areas of the theory of the Boltzmann equation, such as stability of the solutions of this equation and the possible transition to turbulence [156,60,77,78,157,159,158,21,144,160,20]. We finally remark that the Direct Simulation Monte Carlo method has been used even to uncover the analytical nature of a singularity in a limiting solution of the Boltzmann equation, the structure of an infinitely strong shock wave. The latter arises when the temperature upstream of the shock is taken to be zero; then the solution of the Boltzmann equation is the sum of a delta function term and a more regular distribution. The latter was
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approximated by a Maxwellian by H. Grad [79] but turns out to go to infinity [46] when ξ equals the velocity upstream. The DSMC solution gives strong evidence on the nature of the singularity, which is confirmed by a deterministic method [163].
21. Concluding remarks The use of the Boltzmann equation to study rarefied flows has reached a mature stage. The qualitative features are well understood, new phenomena have been uncovered, powerful numerical methods have been developed. Further progress, such as the possibility to indicate that turbulence for gases has features different from turbulence in liquids, depends on the computing power available. The same can be said for the development of deterministic numerical methods as opposed to Monte Carlo. We have not treated all the possible subjects: among the most important omissions, we mention wave propagation, expansion into a vacuum and the application of the Boltzmann equation to the important problems of evaporation and condensation. For these flows, as well as for details on other topics we refer to relevant monographs [43,39] and the literature quoted therein.
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CHAPTER 2
A Review of Mathematical Topics in Collisional Kinetic Theory Cédric Villani UMPA, ENS Lyon, 46 allée d’Italie, F-69364 Lyon Cedex 07, France E-mail:
[email protected]
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2A. General Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Models for collisions in kinetic theory . . . . . . . . . . . . . . . . . . 2. Mathematical problems in collisional kinetic theory . . . . . . . . . . 3. Taxonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Basic surgery tools for the Boltzmann operator . . . . . . . . . . . . . 5. Mathematical theories for the Cauchy problem . . . . . . . . . . . . . 2B. Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Use of velocity-averaging lemmas . . . . . . . . . . . . . . . . . . . . 2. Moment estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Grad’s cut-off toolbox . . . . . . . . . . . . . . . . . . . . . . . . 4. The singularity-hunter’s toolbox . . . . . . . . . . . . . . . . . . . . . 5. The Landau approximation . . . . . . . . . . . . . . . . . . . . . . . . 6. Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2C. H Theorem and Trend to Equilibrium . . . . . . . . . . . . . . . . . . . . 1. A gallery of entropy-dissipating kinetic models . . . . . . . . . . . . . 2. Nonconstructive methods . . . . . . . . . . . . . . . . . . . . . . . . . 3. Entropy dissipation methods . . . . . . . . . . . . . . . . . . . . . . . 4. Entropy dissipation functionals of Boltzmann and Landau . . . . . . . 5. Trend to equilibrium, spatially homogeneous Boltzmann and Landau . 6. Gradient flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Trend to equilibrium, spatially inhomogeneous systems . . . . . . . . 2D. Maxwell Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Wild sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Contracting probability metrics . . . . . . . . . . . . . . . . . . . . . 3. Information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME I Edited by S.J. Friedlander and D. Serre © 2002 Elsevier Science B.V. All rights reserved 71
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2E. Open Problems and New Trends . . . . . . . . . . . . . 1. Open problems in classical collisional kinetic theory 2. Granular media . . . . . . . . . . . . . . . . . . . . 3. Quantum kinetic theory . . . . . . . . . . . . . . . . Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A review of mathematical topics in collisional kinetic theory
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Introduction The goal of this review paper is to provide the reader with a concise introduction to the mathematical theory of collision processes in (dilute) gases and plasmas, viewed as a branch of kinetic theory. The study of collisional kinetic equations is only part of the huge field of nonequilibrium statistical physics. Among other things, it is famous for historical reasons, since it is in this setting that Boltzmann proved his celebrated theorem about entropy. As of this date, the mathematical theory of collisional kinetic equations cannot be considered to be in a mature state, but it has undergone spectacular progress in the last decades, and still more is to be expected. I have made the following choices for presentation: (1) The emphasis is definitely on the mathematics rather than on the physics, the modelling or the numerical simulation. About these topics the survey by Carlo Cercignani will say much more. On the other hand, I shall always be concerned with the physical relevance of mathematical results. (2) Most of the presentation is limited to a small number of widely known, mathematically famous models which can be considered as archetypes – mainly, variants of the Boltzmann equation. This is not only for the sake of mathematics: also in modelling do these equations play a major role. (3) Two important interface fields are hardly discussed: one is the transition from particle systems to kinetic equations, and the other one is the transition from kinetic equations to hydrodynamics. For both problematics I shall only give basic considerations and adequate references. (4) Not all mathematical theories of kinetic equations (there are many of them!) are “equally” represented: for instance, fully nonlinear theories occupy much more space than perturbative approaches, and the Boltzmann equation without cut-off is discussed in about the same detail than the Boltzmann equation with cut-off (although the literature devoted to the latter case is considerably more extended). This partly reflects the respective vivacity of the various branches, but also, unavoidably, personal tastes and areas of competence. I apologize for this! (5) I have sought to give more importance to mathematical methods and ideas, than to results. This is why I have chosen a “transversal” presentation: for each problem, corresponding tools and ideas are first explained, then the various results obtained by their use are carefully described in their respective framework. As a typical example, and unlike most textbooks, this review does not treat spatially homogeneous and spatially inhomogeneous theories separately, but insists on tools which apply to both frameworks. (6) At first I have tried to give extensive lists of references, but soon realized that it was too ambitious . . . . The plan of the survey is as follows. First, a presentation chapter discusses models for collisional kinetic theory and introduces the reader to the various mathematical problems which arise in their study. A central position is given to the Boltzmann equation and its variants. Chapter 2B bears on the Cauchy problem for the Boltzmann equation and variants. The main questions here are propagation of regularity and singularities, regularization effects,
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decay and strict positivity of solutions. The influence of the Boltzmann collision kernel (satisfying Grad’s angular cut-off or not) is discussed with care. Chapter 2C considers the trend to equilibrium, insisting on constructive approaches. Boltzmann’s H theorem and entropy dissipation methods have a central role here. The shorter, but important Chapter 2D studies in detail the case of so-called Maxwell collision kernels, and several links between the theory of the Boltzmann equation and information theory. The ideas in this chapter crucially lie behind some of the most notable results in Chapter 2C, even though, strictly speaking, these two chapters are to a large extent independent. Finally, Chapter 2E discusses selected open problems and promising new trends in the field. Apart from the numerous references quoted in the text, the reader may find useful the short bibliographical notes which are included before the bibliography, to help orientate through the huge literature on the subject. Let me add one final word about conventions: it is quite customary in kinetic theory (just as in the field of hyperbolic systems of conservation laws) to use the vocable “entropy” for Boltzmann’s H functional; however the latter should rather be considered as the negative of an entropy, or as a “quantity of information”. In the present review I have followed the custom of calling H an entropy, however I now regret this choice and recommend to call it an information (or just the H functional); accordingly the “entropy dissipation functional” should rather be called “entropy production functional” or “dissipation of information” (which is both closer to physical intuition and maybe more appealing).
CHAPTER 2A
General Presentation Contents 1. Models for collisions in kinetic theory . . . . . . . . . . . 1.1. Distribution function . . . . . . . . . . . . . . . . . . 1.2. Transport operator . . . . . . . . . . . . . . . . . . . 1.3. Boltzmann’s collision operator . . . . . . . . . . . . 1.4. Collision kernels . . . . . . . . . . . . . . . . . . . . 1.5. Boundary conditions . . . . . . . . . . . . . . . . . . 1.6. Variants of the Boltzmann equation . . . . . . . . . . 1.7. Collisions in plasma physics . . . . . . . . . . . . . . 1.8. Physical validity of the Boltzmann equation . . . . . 2. Mathematical problems in collisional kinetic theory . . . 2.1. Mathematical validity of the Boltzmann equation . . 2.2. The Cauchy problem . . . . . . . . . . . . . . . . . . 2.3. Maxwell’s weak formulation, and conservation laws 2.4. Boltzmann’s H theorem and irreversibility . . . . . . 2.5. Long-time behavior . . . . . . . . . . . . . . . . . . 2.6. Hydrodynamic limits . . . . . . . . . . . . . . . . . . 2.7. The Landau approximation . . . . . . . . . . . . . . 2.8. Numerical simulations . . . . . . . . . . . . . . . . . 2.9. Miscellaneous . . . . . . . . . . . . . . . . . . . . . 3. Taxonomy . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Kinetic and angular collision kernel . . . . . . . . . . 3.2. The kinetic collision kernel . . . . . . . . . . . . . . 3.3. The angular collision kernel . . . . . . . . . . . . . . 3.4. The cross-section for momentum transfer . . . . . . 3.5. The asymptotics of grazing collisions . . . . . . . . . 3.6. What do we care about collision kernels? . . . . . . . 4. Basic surgery tools for the Boltzmann operator . . . . . . 4.1. Symmetrization of the collision kernel . . . . . . . . 4.2. Symmetric and asymmetric point of view . . . . . . 4.3. Differentiation of the collision operator . . . . . . . . 4.4. Joint convexity of the entropy dissipation . . . . . . . 4.5. Pre-postcollisional change of variables . . . . . . . . 4.6. Alternative representations . . . . . . . . . . . . . . . 4.7. Monotonicity . . . . . . . . . . . . . . . . . . . . . . 4.8. Bobylev’s identities . . . . . . . . . . . . . . . . . . 4.9. Application of Fourier transform to spectral schemes 5. Mathematical theories for the Cauchy problem . . . . . . 5.1. What minimal functional space? . . . . . . . . . . . 5.2. The spatially homogeneous theory . . . . . . . . . . 5.3. Maxwellian molecules . . . . . . . . . . . . . . . . . 5.4. Perturbation theory . . . . . . . . . . . . . . . . . . . 75
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The goal of this chapter is to introduce, and make a preliminary discussion of, the mathematical models and problems which will be studied in more detail thereafter. The first section addresses only physical issues, starting from scratch. We begin with an introduction to kinetic theory, then to basic models for collisions. Then in Section 2, we start describing the mathematical problems which arise in collisional kinetic theory, restricting the discussion to the ones that seem to us most fundamental. Particular emphasis is laid on the Boltzmann equation. Each paragraph contains at least one major problem which has not been solved satisfactorily. Next, a specific section is devoted to the classification of collision kernels in the Boltzmann collision operator. The variety of collision kernels reflects the variety of possible interactions. Collision kernels have a lot of influence on qualitative properties of the Boltzmann equation, as we explain. In the last two sections, we first present some basic general tools and considerations about the Boltzmann operator, then give an overview of existing mathematical theories for collisional kinetic theory.
1. Models for collisions in kinetic theory 1.1. Distribution function The object of kinetic theory is the modelling of a gas (or plasma, or any system made up of a large number of particles) by a distribution function in the particle phase space. This phase space includes macroscopic variables, i.e., the position in physical space, but also microscopic variables, which describe the “state” of the particles. In the present survey, we shall restrict ourselves, most of the time, to systems made of a single species of particles (no mixtures), and which obey the laws of classical mechanics (non-relativistic, nonquantum). Thus the microscopic variables will be nothing but the velocity components. Extra microscopic variables should be added if one would want to take into account nontranslational degrees of freedom of the particles: internal energy, spin variables, etc. Assume that the gas is contained in a (bounded or unbounded) domain X ⊂ RN (N = 3 in applications) and observed on a time interval [0, T ], or [0, +∞). Then, under the above simplifying assumptions, the corresponding kinetic model is a nonnegative function f (t, x, v), defined on [0, T ] × X × RN . Here the space RN = RN v is the space of possible velocities, and should be thought of as the tangent space to X. For any fixed time t, the quantity f (t, x, v) dx dv stands for the density of particles in the volume element dx dv centered at (x, v). Therefore, the minimal assumption that one can make on f is that for all t 0, f (t, ·, ·) ∈ L1loc X; L1 RN v ; or at least that f (t, ·, ·) is a bounded measure on K × RN v , for any compact set K ⊂ X. This assumption means that a bounded domain in physical space contains only a finite amount of matter.
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Underlying kinetic theory is the modelling assumption that the gas is made of so many particles that it can be treated as a continuum. In fact there are two slightly different ways to consider f : it can be seen as an approximation of the true density of the gas in phase space (on a scale which is much larger than the typical distance between particles), or it can reflect our lack of knowledge of the true positions of particles. Which interpretation is made has no consequence in practice.1 The kinetic approach goes back as far as Bernoulli and Clausius; in fact it was introduced long before experimental proof of the existence of atoms. The first true bases for kinetic theory were laid down by Maxwell [335,337,336]. One of the main ideas in the model is that all measurable macroscopic quantities (“observables”) can be expressed in terms of microscopic averages, in our case integrals of the form f (t, x, v)ϕ(v) dv. In particular (in adimensional form), at a given point x and a given time t, one can define the local density ρ, the local macroscopic velocity u, and the local temperature T , by ρ=
RN
f (t, x, v) dv,
ρ|u| + NρT =
ρu =
RN
2
f (t, x, v)v dv, (1)
2
RN
f (t, x, v)|v| dv.
For much more on the subject, we refer to the standard treatises of Chapman and Cowling [154], Landau and Lipschitz [304], Grad [250], Kogan [289], Uhlenbeck and Ford [433], Truesdell and Muncaster [430], Cercignani and co-authors [141,148,149].
1.2. Transport operator Let us continue to stick to a classical description, and neglect for the moment the interaction between particles. Then, according to Newton’s principle, each particle travels at constant velocity, along a straight line, and the density is constant along characteristic lines dx/dt = v, dv/dt = 0. Thus it is easy to compute f at time t in terms of f at time 0: f (t, x, v) = f (0, x − vt, v). 1 For instance, assume that the microscopic description of the gas is given by a cloud of n points x , . . . , x n 1 N in RN . . . , xn , vn ) of x , with velocities v1 , . . . , vn in Rv . A microscopic configuration is an element (x1 , v1 , n N n (RN x × Rv ) . The “density” of the gas in this configuration is the empirical measure (1/n) i=1 δ(xi ,vi ) ; N N it is a probability measure on Rx × Rv . In the first interpretation, f (x, v) dx dv is an approximation of the N n empirical measure. In the second one, there is a symmetric probability density f n on the space (RN x × Rv ) of all microscopic configurations, and f is an approximation of the one-particle marginal P1 f n (x1 , v1 ) = f n (x1 , v1 , . . . , xn , vn ) dx2 dv2 · · · dxn dvn .
Thus the first interpretation is purely deterministic, while the second one is probabilistic. It is the second interpretation which was implicitly used by Boltzmann, and which is needed by Landford’s validation theorem, see Section 2.1.
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In other words, f is a weak solution to the equation of free transport, ∂f + v · ∇x f = 0. ∂t
(2)
The operator v · ∇x is the (classical) transport operator. Its mathematical properties are much subtler than it would seem at first sight; we shall discuss this later. Complemented with suitable boundary conditions, Equation (2) is the right equation for describing a gas of noninteracting particles. Many variants are possible; for instance, v should be replaced 2 by v/ 1 + |v| in the relativistic case. Of course, when there is a macroscopic force F (x) acting on particles, then the equation has to be corrected accordingly, since the trajectories of particles are not straight lines any longer. The relevant equation would read ∂f + v · ∇x f + F (x) · ∇v f = 0 ∂t
(3)
and is sometimes called the linear Vlasov equation.
1.3. Boltzmann’s collision operator We now want to take into account interactions between particles. We shall make several postulates. (1) We assume that particles interact via binary collisions: this is a vague term describing the process in which two particles happen to come very close to each other, so that their respective trajectories are strongly deviated in a very short time. Underlying this hypothesis is an implicit assumption that the gas is dilute enough that the effect of interactions involving more than two particles can be neglected. Typically, if we deal with a threedimensional gas of n hard spheres of radius r, this would mean nr 3 1,
nr 2 1.
(2) Moreover, we assume that these collisions are localized both in space and time, i.e., they are brief events which occur at a given position x and a given time t. This means that the typical duration of a collision is very small compared to the typical time scale of the description, and also quantities such as the impact parameter (see below) are negligible in front of the typical space scale (say, a space scale on which variations due to the transport operator are of order 1). (3) Next, we further assume these collisions to be elastic: momentum and kinetic energy are preserved in a collision process. Let v , v∗ stand for the velocities before collision, and v, v∗ stand for the velocities after collision: thus v + v∗ = v + v∗ , |v |2 + |v∗ |2 = |v|2 + |v∗ |2 .
(4)
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Fig. 1. A binary elastic collision.
Since this is a system of N + 1 scalar equations for 2N scalar unknowns, it is natural to expect that its solutions can be defined in terms of N − 1 parameters. Here is a convenient representation of all these solutions, which we shall sometimes call the σ -representation: ⎧ v + v∗ |v − v∗ | ⎪ + σ, ⎨v = 2 2 ⎪ ⎩v = v + v∗ − |v − v∗ | σ. ∗ 2 2
(5)
Here the parameter σ ∈ S N−1 varies in the N − 1 unit sphere. Figure 1 pictures a collision in the velocity phase space. The deviation angle θ is the angle between pre- and postcollisional velocities. Very often, particles will be assumed to interact via a given interaction potential φ(r) (r = distance between particles); then v and v∗ should be computed as the result of a classical scattering problem, knowing v, v∗ and the impact parameter between the two colliding particles. We recall that the impact parameter is what would be the distance of closest approach if the two particles did not interact. (4) We also assume collisions to be microreversible. This word can be understood in a purely deterministic way: microscopic dynamics are time-reversible; or in a probabilistic way: the probability that velocities (v , v∗ ) are changed into (v, v∗ ) in a collision process, is the same that the probability that (v, v∗ ) are changed into (v , v∗ ). (5) And finally, we make the Boltzmann chaos assumption: the velocities of two particles which are about to collide are uncorrelated. Roughly speaking, this means that if we randomly pick up two particles at position x, which have not collided yet, then the joint distribution of their velocities will be given by a tensor product (in velocity space) of f with itself. Note that this assumption implies an asymmetry between past and future: indeed, in general if the pre-collisional velocities are uncorrelated, then post-collisional velocities have to be correlated2! 2 See, for instance, the discussion in Section 2.4.
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Under these five assumptions, in 1872 Boltzmann (cf. [93]) was able to derive a quadratic collision operator which accurately models the effect of interactions on the distribution function f : ∂f (t, x, v) = Q(f, f ) (t, x, v) (6) ∂t coll = dv∗ dσ B(v − v∗ , σ )(f f∗ − ff∗ ). (7) RN
S N−1
Here we have used standard abbreviations: f = f (t, x, v ), f∗ = f (t, x, v∗ ), f∗ = f (t, x, v∗ ). Moreover, the nonnegative function B(z, σ ), called the Boltzmann collision kernel, depends only on |z| and on the scalar product z/|z|, σ . Heuristically, it can be seen as a probability measure on all the possible choices of the parameter σ ∈ S N−1 , as a function of the relative velocity z = v − v∗ . But truly speaking, this interpretation is in general false, because B is not integrable . . . . The Boltzmann collision kernel is related to the cross-section Σ by the identity B(z, σ ) = |z|Σ(Z, σ ). By abuse of language, B is often called the cross-section. Let us explain (7) a little bit. This operator can formally be split, in a self-evident way, into a gain and a loss term, Q(f, f ) = Q+ (f, f ) − Q− (f, f ). The loss term “counts” all collisions in which a given particle of velocity v will encounter another particle, of velocity v∗ . As a result of such a collision, this particle will in general change its velocity, and this will make less particles with velocity v. On the other hand, each time particles collide with respective velocities v and v∗ , then the v particle may acquire v as new velocity after the collision, and this will make more particles with velocity v: this is the meaning of the gain term. It is easy to trace back our modelling assumptions: (1) the quadratic nature of this operator is due to the fact that only binary collisions are taken into account; (2) the fact that the variables t, x appear only as parameters reflects the assumption that collisions are localized in space and time; (3) the assumption of elastic collisions results in the formulas giving v and v∗ ; (4) the microreversibility implies the particular structure of the collision kernel B; (5) finally, the appearance of the tensor products3 f f∗ and ff∗ is a consequence of the chaos assumption. On the whole, the Boltzmann equation reads ∂f + v · ∇x f = Q(f, f ), ∂t
t 0, x ∈ RN , v ∈ RN ;
(8)
or, when a macroscopic force F (x) is also present, ∂f + v · ∇x f + F (x) · ∇v f = Q(f, f ), ∂t
t 0, x ∈ RN , v ∈ RN .
3 In the sense that ff = (f ⊗ f )(v, v ), f f = (f ⊗ f )(v , v ). ∗ ∗ ∗ ∗
(9)
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The deepest physical and mathematical properties of the Boltzmann equation are linked to the subtle interaction between the linear transport operator and the nonlinear collision operator. We note that this equation was written in weak formulation by Maxwell as early as 1866: as recalled in Section 2.3 below, Maxwell [335,337] wrote down the equation satisfied by the observables f (t, x, v)ϕ(v) dv (see in particular [335, Equation (3)]). However Boltzmann did a considerable job on the interpretation and consequences of this equation, and also made them widely known in his famous treatise [93], which was to have a lot of influence on theoretical physics during several decades. Let us make a few comments about the modelling assumptions. They may look rather crude, however they can in large part be completely justified, at least in certain particular cases. By far the deepest assumption is Boltzmann’s chaos hypothesis (“Stosszahlansatz”) which is intimately linked to the questions of irreversibility of macroscopic dynamics and the arrow of time. For a discussion of these subtle topics, the reader may consult the review paper by Lebowitz [293], or the enlightening treatise by Kac [284], as well as the textbooks [149,410] for the most technical aspects. We shall say just a few words about the subject in Sections 2.1 and 2.4.
1.4. Collision kernels Since the collision kernel (or equivalently the cross-section) only depends on |v − v∗ | and v−v∗ , σ , i.e., the cosine of the deviation angle, throughout the whole text we shall on |v−v ∗| abuse notations by writing
B(v − v∗ , σ ) = B |v − v∗ |, cos θ ,
v − v∗ cos θ = ,σ . |v − v∗ |
(10)
Maxwell [335] has shown how the collision kernel should be computed in terms of the interaction potential φ. In short, here are his formulas (in three dimensions of space), as they can be found for instance in Cercignani [141], for a repulsive potential. For given impact parameter p 0 and relative velocity z ∈ R3 , let the deviation angle θ be θ (p, z) = π − 2p
+∞ s0
ds/s 2 1−
p2 s2
− 4 φ(s) |z|2
=π −2 0
p/s0
du 1 − u2 −
4 φ pu |z|2
,
where s0 is the positive root of 1−
p2 s02
−4
φ(s0 ) = 0. |z|2
Then the collision kernel B is implicitly defined by p dp |z|. B |z|, cos θ = sin θ dθ
(11)
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It can be made explicit in two cases: • hard spheres, i.e., particles bounce on each other like billiard balls: in this case B(|v − v∗ |, cos θ ) is just proportional to |v − v∗ | (the cross-section is constant); • Coulomb interaction, φ(r) = 1/r (in adimensional variables and in three dimensions of space): then B is given by the famous Rutherford formula, B |v − v∗ |, cos θ =
1 |v − v∗
|3 sin4 (θ/2)
(12)
.
A dimensional factor of (e2 /4πε0 m) should multiply this kernel (e = charge of the particle, ε0 = permittivity of the vacuum, m = mass of the particle). Unfortunately, Coulomb interactions cannot be modelled by a Boltzmann collision operator; we shall come back to this soon. In the important4 model case of inverse-power law potentials, φ(r) =
1 , r s−1
s > 2,
then the collision kernel cannot be computed explicitly, but one can show that B |v − v∗ |, cos θ = b(cos θ )|v − v∗ |γ ,
γ=
s − (2N − 1) . s−1
(13)
In particular, in three dimensions of space, γ = (s − 5)/(s − 1). As for the function b, it is only implicitly defined, locally smooth, and has a nonintegrable singularity for θ → 0: sinN−2 θ b(cos θ ) ∼ Kθ −1−ν ,
ν=
2 s−1
(N = 3).
(14)
Here we have put the factor sinN−2 θ because it is (up to a constant depending only on the dimension) the Jacobian determinant of spherical coordinates on the sphere S N−1 . The nonintegrable singularity in the “angular collision kernel” b is an effect of the huge amount of grazing collisions, i.e., collisions with a very large impact parameter, so that colliding particles are hardly deviated. This is not a consequence of the assumption of inverse-power forces; in fact a nonintegrable singularity appears as soon as the forces are of infinite range, no matter how fast they decay at infinity. To see this, note that, according to (11),
π 0
B |z|, cos θ sin θ dθ = |z|
π
p 0
dp dθ = |z| dθ
0
pmax
p dp =
2 |z| pmax . 2
(15)
By the way, it seems strange to allow infinite-range forces, while we assumed interactions to be localized. This problem has never been discussed very clearly, but in 4 Inverse power laws are moderately realistic, but very important in physics and in modelling, because they are simple, often lead to semi-explicit results, and constitute a one-parameter family which can model very different phenomena. Van der Waals interactions typically correspond to s = 7, ion–neutral interactions to s = 5, Manev interactions [88,279] to s = 3, Coulomb interactions to s = 2.
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principle there is no contradiction in assuming the range of the interaction to be infinite at a microscopic scale, and negligible at a macroscopic scale. The fact that the linear Boltzmann equation can be rigorously derived from some particle dynamics with infinite range [179] also supports this point of view. As one sees from formula (13), there is a particular case in which the collision kernel does not depend on the relative velocity, but only on the deviation angle: particles interacting via a inverse (2N − 1)-power force (1/r 5 in three dimensions). Such particles are called Maxwellian molecules. They should be considered as a theoretical model, even if the interaction between a charged ion and a neutral particle in a plasma may be modelled by such a law (see, for instance, [164, Theorem 1, p. 149]). However, Maxwell and Boltzmann used this model a lot,5 because they had noticed that it could lead to many explicit calculations which, so did they believe, were in agreement with physical observations. Also they believed that the choice of molecular interaction was not so important, and that Maxwellian molecules would behave pretty much the same as hard spheres.6 Since the time of Maxwell and Boltzmann, the need for results or computations has led generations of mathematicians and physicists to work with more or less artificial variants of the collision kernels given by physics. Such a procedure can also be justified by the fact that for many interesting interactions, the collision kernel is not explicit at all: for instance, in the case of the Debye potential, φ(r) = e−r /r. Here are two categories of artificial collision kernels: – when one tames the singularity for grazing collisions and replaces the collision kernel by a locally integrable one, one speaks of cut-off collision kernel; – collision kernels of the form |v − v∗ |γ (γ > 0) are called variable hard spheres collision kernels. It is a common belief among physicists that the properties of the Boltzmann equation are quite a bit sensitive to the dependence of B upon the relative velocity, but very little to its dependence upon the deviation angle. True as it may be for the behavior of macroscopic quantities, this creed is definitely wrong at the microscopic level, as we shall see. In all the sequel, we shall consider general collision kernels B(|v − v∗ |, cos θ ), in arbitrary dimension N , and make various assumptions on the form of B without always caring if it corresponds to a true interaction between particles (i.e., if there is a φ whose associated collision kernel is B). Our goal, in a lot of situations, will be to understand how the collision kernel affects the properties of the Boltzmann equation. However, we shall always keep in mind the collision kernels given by physics, in dimension three, to judge how satisfactory a mathematical result is.
1.5. Boundary conditions Of course the Boltzmann equation has to be supplemented with boundary conditions which model the interaction between the particles and the frontiers of our domain X ⊂ RN (wall, etc.) 5 See Boltzmann [93, Chapter 3]. 6 Further recall that at the time, the “atomic hypothesis” was considered by many to be a superfluous
complication.
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The most natural boundary condition is the specular reflection: Rx v = v − 2 v · n(x) n(x), x ∈ ∂X,
f (x, Rx v) = f (x, v),
(16)
where n(x) stands for the outward unit normal vector at x. In the context of optics, this condition would be called the Snell–Descartes law: particles bounce back on the wall with an postcollision angle equal to the precollision angle. However, as soon as one is interested in realistic modelling for practical problems, Equation (16) is too rough . . . . In fact, a good boundary condition would have to take into account the fine details of the gas-surface interaction, and this is in general a very delicate problem.7 There are a number of models, cooked up from modelling assumptions or phenomenological a priori constraints. As good source for these topics, the reader may consult the books by Cercignani [141,148] and the references therein. In particular, the author explains the relevant conditions that a scattering kernel K has to satisfy for the boundary condition f (x, vout) =
K(vin , vout )f (x, vin ) dvin
to be physically plausible. Here we only list a few common examples. One is the bounce-back condition, f (x, −v) = f (x, v),
x ∈ ∂X.
(17)
This condition simply means that particles arriving with a certain velocity on the wall will bounce back with an opposite velocity. Of course it is not very realistic, however in some situations (see, for instance, [148, p. 41]) it leads to more relevant conclusions than the specular reflection, because it allows for some transfer of tangential momentum during collisions. Another common boundary condition is the Maxwellian diffusion, f (x, v) = ρ− (x)Mw (v),
v · n(x) > 0,
(18)
where ρ− (x) = v·n<0 f (x, v) dv and Mw is a particular Gaussian distribution depending only on the wall, |v|2
Mw (v) =
e− 2Tw (2π)
N−1 2
N+1
.
Tw 2
7 Here we assume that the fine details of the surface of the wall are invisible at the scale of the spatial variable, so that the wall is modelled as a smooth surface, but we wish to take these details into account to predict velocities after collision with the wall. Another possibility is to assume that the roughness of the wall results in irregularities which can be seen at spatial resolution. Then it is natural in many occasions to assume ∂X to be very irregular. For some mathematical works about this alternative approach, see [33,272,273].
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In this model, particles are absorbed by the wall and then re-emitted according to the distribution Mw , corresponding to a thermodynamical equilibrium between particles and the wall. Finally, one can combine the above models. Already Maxwell had understood that a convex combination of (16) and (18) would certainly be more realistic than just one of these two equations. No need to say, since the work of Maxwell, much more complicated models have appeared, for instance the Cercignani–Lampis (CL) model, see [148]. In mathematical discussions, we shall not consider the problem of boundary conditions except for the most simple case, which is specular reflection. In fact, most of the time we shall simply avoid this problem by assuming the position space to be the whole of RN , or the torus TN . Of course the torus is a mathematical simplification, but it is also used by physicists and by numerical analysts who want to avoid taking boundary conditions into account . . . .
1.6. Variants of the Boltzmann equation There are many variants. Let us only mention • relativistic models, see [50,158,199,200,233–235,14]; • quantum models, see [196,328,209] and the references therein. They will be discussed in Section 3 of Chapter 2E. Also we should mention that models of quantum Boltzmann equation have recently gained a lot of interest in the study of semiconductors, see in particular [65,66] and the works by Poupaud and coworkers on related models [386–388,244,360], also [354,355] (in two dimensions) and [15] (in three dimensions); • linear models, in particular the linear Boltzmann equation, ∂f + v · ∇x f = ∂t
RN ×S N−1
− v∗ , σ ) F (v∗ )f (v ) − F (v∗ )f (v) , dv∗ dσ B(v
where F is a given probability distribution. As a general rule, such linear equations model the influence of the environment, or background, on a test-particle (think of a particle in an environment of random scatterers, like in a random pinball game). The distribution F is the distribution of the background, and is usually assumed to be stationary, which means that the environment is in statistical equilibrium. Linear Boltzmann-like models are used in all areas of physics, most notably in quantum scattering [206] and in the study of transport phenomena associated with neutrons or photons [148, pp. 165–172]. A general mathematical introduction to linear transport equations can be found in [157, Chapter 21]; • diffusive models, like the Fokker–Planck equation, which is often used in its linear version, ∂f + v · ∇x f = ∇v · (∇v f + f v), ∂t
(19)
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or in its nonlinear form, ∂f + v · ∇x f = ρ α ∇v · T ∇v f + f (v − u) , ∂t
(20)
where α ∈ [0, 1], and ρ, u, T are the local density, velocity and temperature defined by (1). When α = 1, this model has the same quadratic homogeneity as the Boltzmann equation. Of course it is also possible to couple the equation only via T and not u, etc. A classical discussion on the use of the Fokker–Planck equation in physics can be found in the important review paper by Chandrasekhar [153]. The Landau equation, which is described in detail in the next section, is another diffusive variant of the Boltzmann equation; • energy-dissipating models, describing inelastic collisions. These models are particularly important in the theory of granular materials: see Section 2 of Chapter 2E; • model equations, like the (simplistic) BGK model, see, for instance, [141,148]. In this model one replaces the complicated Boltzmann operator by M f − f , where M f is the Maxwellian distribution8 with the same local density, velocity and temperature than f . Also variants are possible, for instance multiplying this operator by the local density ρ . . . . Another very popular model equation for mathematicians is the Kac model [283]. It is a one-dimensional caricature of the Boltzmann equation which retains some of its interesting features. The unknown is a time-dependent probability measure f on R, and the equation reads 1 ∂f = ∂t 2π
R
dv∗
2π
dθ 0
f f∗
− f,
(21)
where (v , v∗ ) is obtained from (v, v∗ ) by a rotation of angle θ in the plane R2 . This model preserves mass and kinetic energy, but not momentum; • discrete-velocity models: these are approximations of the Boltzmann equation where particles are only allowed a finite number of velocities. They are used in numerical analysis, but their mathematical study is (or once was) a popular topic. About them we shall say nothing; references can be found, for instance, in [230,141,148] and also in the survey papers [384,62]. Among a large number of works, we only mention the original contributions by Bony [94,95], Tartar [416,417], and the consistency result by Bobylev et al. [367], for the interesting number-theoretical issues that this paper has to deal with. We note [148, p. 265] that discrete-velocity models were once believed by physicists to provide miraculously efficient numerical codes for simulation of hydrodynamics. But these hopes have not been materialized . . . . Also some completely unrealistic discrete-velocity equations have been studied as simplified mathematical models, without caring whether they would approximate or not the true Boltzmann equation. These models sometimes have no more than two or three velocities! Some well-known examples are the Carleman equation, with two 8 See Section 2.5 below.
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velocities, the Broadwell model, with four velocities in the plane, or the Cabannes equation with fourteen velocities. For instance, the Carleman equation reads ⎧ ∂f1 ∂f1 ⎪ 2 ⎪ − f12 , + = f−1 ⎨ ∂t ∂x ⎪ ∂f ∂f ⎪ 2 ⎩ −1 − −1 = f12 − f−1 . ∂t ∂x
(22)
For a study of these models see [230,384,141,148,60,61,108,25] and references included. Of course, these models are so oversimplified9 that they cannot be considered seriously from the physical point of view, even if one may expect that they keep some relevant features of the Boltzmann equation. By the way, as noticed by Uchiyama [432], the Broadwell model cannot be derived from a fictitious system of deterministic four-velocity particles (“diamonds”) in the plane, see [149, Appendix 4C]. Only at the price of some extra stochasticity assumption can the derivation be fixed [117]. All the abovementioned models should be considered with appropriate boundary conditions. These conditions can also be replaced by the effect of a confinement potential V (x): this means that there is a macroscopic force of the form F (x) = −∇V (x) acting on the system. Finally, it is important to note that Boltzmann’s model is obtained as a result of the assumption of localized interaction; in particular, it does not take into account a possible interaction of long (macroscopic) range which would result in a macroscopic mean-field force, typically F (x) = −∇Φ(x),
Φ(x) = ρ ∗x φ,
where φ is the interaction potential and ρ the local density. The modelling of the interaction by such a coupled force is called a Vlasov description. When should one prefer a Vlasov, or a Boltzmann description? A dimensional analysis by Bobylev and Illner [88] shows that for inverse-power forces like 1/r s , and under natural scaling assumptions, • for s > 3, the Boltzmann term should prevail on the mean-field term; • for s < 3, the Boltzmann term should be negligible in front of the mean-field term. The separating case, s = 3, is the so-called Manev interaction [88,279]. There are subtle questions here, which are not yet fully understood, even at a formal level. Also the uniqueness of the relevant scaling is not clear. From a physicist’s point of view, however, it is generally accepted that a good description is obtained by adding up the effects of a mean-field term and those of a Boltzmann collision operator, with suitable dimensional coefficients. Another way to take into account interactions on a macroscopically significant scale is to use a description à la Povzner. In this model (see, for instance, [389]), particles interact 9 As a word of caution, we should add that even if they are so simplified, their mathematical analysis is not trivial at all, and many problems in the field still remain open.
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through delocalized collisions, so that the corresponding Boltzmann operator is integrated with respect to the position y of the test particle, and reads
RN x
dy
RN v
− v∗ , x − y) f (y, v∗ )f (x, v ) − f (y, v∗ )f (x, v) . dv∗ B(v
(23)
now depends on x − y. On the other hand, there is no collision Note that the kernel B parameter σ any more, the outgoing velocities being uniquely determined by the positions x, y and the ingoing velocities v , v∗ . It was shown by Cercignani [140] that this type of equations could be retrieved as the limit of a large stochastic system of “soft spheres”. A related model is the Enskog equation for dense gases, which has never been clearly justified. It resembles Equation (23) but the multiplicity of the integral is 2N − 1 instead of 2N , because there is no integration over the distance |x − y|. Mathematical studies of the Enskog model have been performed by Arkeryd, Arkeryd and Cercignani [24,27,28] – in particular, reference [24] provides well-posedness and regularity under extremely general assumptions (large data, arbitrary dimension) by a contraction method. See Section 2.1 in Chapter 2E for an inelastic variant which is popular in the study of granular material. The study of these models is interesting not only in itself, but also because numerical schemes always have to perform some delocalization to simulate the effect of collisions. This explains why the results in [140] are very much related to some of the mathematical justifications for some numerical schemes, as performed in [453,396].
1.7. Collisions in plasma physics The importance and complexity of interaction processes in plasma physics justifies that we devote a special section to this topic. A plasma, generally speaking, is a gas of (partially or totally) ionized particles. However, this term encompasses a huge variety of physical situations: the density of a plasma can be extremely low or extremely high, the pressures can vary considerably, and the proportion of ionized particles can also vary over several orders of magnitude. Nonelastic collisions, recombination processes may be very important. We do not at all try to make a precise description here, and refer to classical textbooks such as Balescu [46], Delcroix and Bers [164], the very nice survey by Decoster in [160], or the numerous references that can be found therein. All of these sources put a lot of emphasis on the kinetic point of view, but [160] and [164] are also very much concerned with fluid descriptions. A point which should be made now, is that the classical collisional kinetic theory of gases a priori applies when the density is low (we shall make this a little bit more precise later on) and when nonelastic processes can be neglected. Even taking into account only elastic interactions, there are a number of processes going on in a plasma: Maxwell-type interactions between ions and neutral particles, Van der Waals forces between neutral particles, etc. However, the most important feature, both from the mathematical and the physical point of view, is the presence of Coulomb
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interactions between charged particles. The basic model for the evolution of the density of such particles is the Vlasov–Poisson equation ⎧ ∂f ⎪ ⎨ ∂t + v · ∇x f + F (x) · ∇v f = 0, ⎪ ⎩F = −∇V ,
V=
e2 4πε0 r
∗x ρ,
ρ(t, x) = f (t, x, v) dv.
(24)
For simplicity, here we have written this equation for only one species of particles, and also we have not included the effect of a magnetic field, which leads to the Vlasov–Maxwell system (see, for instance, [191]). We have kept the physical parameters e = charge of the particle and ε0 = permittivity of vacuum for the sake of a short discussion about scales. Even though this is not the topic of this review paper, let us say just a few words on the Vlasov–Poisson equation. Its importance in plasma physics (including astrophysics) cannot be overestimated, and thousands of papers have been devoted to its study. We only refer to the aforementioned textbooks, together with the famous treatise by Landau and Lipschitz [304]. From the mathematical point of view, the basic questions of existence, uniqueness and (partial) regularity of the solution to the Vlasov equation have been solved at the end of the eighties, see in particular Pfaffelmoser [382], DiPerna and Lions [191], Lions and Perthame [318], Schaeffer [402]. Reviews can be found in Glassey [233] and Bouchut [96]. Stability, and (what is more interesting!) instability of several classes of equilibrium distributions to the Vlasov–Poisson equation have recently been the object of a lot of studies by Guo and Strauss [264–268]. Several important questions, however, have not been settled, like the derivation of the Vlasov–Poisson equation from particle systems (see Spohn [410] and Neunzert [356] for related topics) and the explanation of the famous and rather mysterious Landau damping effect. There is no doubt that the Vlasov–Poisson equation is the correct equation to describe a classical plasma on a short time scale. However, when one wants to consider long periods of time, it is necessary to take into account collisions between particles. For this it is natural to introduce a Boltzmann collision operator in the right-hand side of (24). However, the Boltzmann equation for Coulomb interactions does not make sense! Indeed, the collision integral would be infinite even for very smooth (or analytic) distribution functions. 10 This is due to the very slow decay of the Coulomb potential, and the resulting very strong angular singularity of the collision kernel given by Rutherford’s formula (12). A standard remedy to this problem is to assume that there is a screening (due to the presence of two species of particles, for instance), so that the effective interaction potential between charged particles is not the Coulomb interaction, but the so-called Debye potential φ(r) =
e−r/λD . 4πε0 r
(25)
10 More precisely, the natural definition of the collision operator would lead to the following nonsense:
whatever f , Q(f, f ) is an element of {−∞, 0, +∞}, see [450, Annex I, Appendix A].
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Here λD is the Debye length, i.e., a typical screening distance. In the classical theory of plasmas, λD =
ε0 kT , ρe2
where k is Boltzmann’s constant, T is the temperature of the plasma (rigorously speaking, it should depend on x . . .) and ρ its mean density (same remark). The resulting collision kernel is no longer explicit, but at least makes sense, because the very strong angular singularity in Rutherford’s formula (12) is tamed. The replacement of Coulomb by Debye potential can be justified by half-rigorous, halfheuristic arguments (see the references already mentioned). However, in most of the cases of interest, the Debye length is very large with respect to the characteristic length r0 for collisions, called the Landau length: r0 =
e2 . 4πε0 kT
More precisely, in so-called classical plasmas (those for which the classical kinetic description applies), one has r0 ρ −1/3 λD . This means first that the Landau distance is very small with respect to other scales (so that collisions can be considered as localized), and secondly that the plasma is so dilute that the typical distance between particles is very small with respect to the screening distance, which is usually considered as the relevant space scale. By a formal procedure, Landau [291] showed that, as the ratio Λ ≡ 2λD /r0 → ∞, the Boltzmann collision operator for Debye potential behaves as log Λ QL (f, f ), 2πΛ where QL is the so-called Landau collision operator: QL (f, f ) = ∇v ·
R3
dv∗ a(v − v∗ ) f∗ (∇f ) − f (∇f )∗ .
(26)
Here a(z) is a symmetric (degenerate) nonnegative matrix, proportional to the orthogonal projection onto z⊥ : aij (z) =
zi zj L δij − 2 , |z| |z|
and L is a dimensional constant.
(27)
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The resulting equation is the Landau equation. In adimensional units, it could be written as ∂f + v · ∇x f + F (x) · ∇v f = QL (f, f ). ∂t
(28)
Many mathematical and physical studies also consider the simplified case when only collisions are present, and the effect of the mean-field term is not present. However, the collision term, normally, should be considered only as a long-time correction to the meanfield term. We now present a variant of the Landau equation, which appears when one replaces the function L/|z| in (27) by |z|2 , so that aij (z) = |z|2 δij − zi zj .
(29)
This approximation is called “Maxwellian”. It is not realistic from the physical point of view, but has become popular because it leads to simpler mathematical properties and useful tests for numerical simulations. Under assumption (29), a number of algebraic simplifications arise in the Landau operator (26); since they are independent on the dimension, we present them in arbitrary dimension N 2. Without loss of generality, choose an orthonormal basis of RN in such a way that
RN
f dv = 1,
RN
f v dv = 0,
RN
f |v|2 dv = N
(unit mass, zero mean velocity, unit temperature), and assume moreover that RN
f vi vj dv = Ti δij
(the Ti ’s are the directional temperatures; of course with matrix (29) can be rewritten as
i
Ti = N ). Then the Landau operator
(N − Ti )∂ii f + (N − 1)∇ · (f v) + S f.
(30)
i
Here S stands for the Laplace–Beltrami operator, S f =
|v|2 δij − vi vj ∂ij f − (N − 1)v · ∇v f, ij
i.e., a diffusion on centered spheres in velocity space. Thus the Landau operator looks like a nonlinear Fokker–Planck-type operator, with some additional isotropisation effect due to the presence of the Laplace–Beltrami operator. The diffusion is enhanced in directions
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where the temperature is low, and slowed down in directions where the temperature is high: this is normal, because in the end the temperature along all directions should be the same. Formulas like (30) show that in the isotropic case, and under assumption (29), the nonlinear Landau equation reduces (in a well-chosen orthonormal basis) to the linear Fokker–Planck equation! By this remark [447] one can construct many explicit solutions (which generalize the ones in [296]). These considerations explain why the Maxwellian variant of the Landau equation has become a popular test case in numerical analysis [106]. Let us now review other variants of the Landau equation. First of all, there are relativistic and quantum versions of it [304]. For a mathematically-oriented presentation, see, for instance, [298]. There are also other, more sophisticated models for collisions in plasmas: see, for instance, [164, Section 13.6] for a synthetic presentation. The most famous of these models is the so-called Balescu–Lenard collision operator, whose complexity is just frightening for a mathematician. Its expression was established by Bogoljubov [90] via a so-called BBGKY-type hierarchy, and later put by Lenard under the form that we give below. On the other hand, Balescu [46,47] derived it as part of his general perturbative theory of approximation of the Liouville equation for many particles. Just as the Landau operator, the Balescu–Lenard operator is in the form
∂ ∂f ∂f∗ dv∗ aij (v, v∗ ) f∗ −f , (31) ∂vi R3 ∂vj ∂v∗j ij
but now the matrix aij depends on f in a strongly nonlinear way: aij (v, v∗ ) =
ki kj 1 δ k · (v − v∗ ) dk, |k|4 |ε|2 k∈R3 ,|k|Kmax
(32)
where ε is the “longitudinal permittivity” of the plasma, 8π 1 k · (∇f )∗ dv∗ . ε=1− 2 k · (v − v∗ ) − i0 |k| Here δ is the Dirac measure at the origin, and 1 1 1 = lim =P + iπδ x − i0 ε→0+ x − iε x is a complex-valued distribution on the real line (P stands for the Cauchy principal part). Moreover, Kmax is a troncature parameter (whose value is not very clearly determined) which corresponds to values of the deviation angles beyond which collisions cannot be considered grazing. This is not a Debye cut! Contrary to Landau, Balescu and Lenard derived the operator QL (26) as an approximation of (31). To see the link between these two operators, let us set k = µω, µ 0, ω ∈ S 2 ; then one can rewrite (32) as
S2
dω δ ω · (v − v∗ ) ωi ωj
Kmax 0
dµ . µ|ε|2
(33)
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But S2
dω δ(ω · z)ωi ωj =
zi zj π δij − 2 . |z| |z|
So one can replace the operator (31) by the Landau collision operator if one admits that the integral in dµ in (33) depends very little on v, v∗ , ω. Under this assumption, it is natural to replace f by a Maxwellian,11 and one finds for this integral an expression of the form 1 + (µ2D )/µ2 , where µD has the same homogeneity as the inverse of a Debye length. Then one can perform the integration; see Decoster [160] for much more details. In spite of its supposed accuracy, the interest of the Balescu–Lenard model is not so clear. Due to its high complexity, its numerical simulation is quite tricky. And except in very particular situations, apparently one gains almost nothing, in terms of physical accuracy of the results, by using it as a replacement for the Landau operator. In fact, it seems that the most important feature of the Balescu–Lenard operator, to this date, is to give a theoretical basis to the use of the Landau operator! There exist in plasma physics some even more complicated models, such as those which take into account magnetic fields (Rostoker operator, see, for instance, [465] and the references therein). We refrain from writing up the equations here, since this would require several pages, and they seem definitely out of reach of a mathematical treatment for the moment . . . . We also mention the simpler linear Fokker–Planck operator for Coulomb interaction, derived by Chandrasekhar. This is a linear operator of Landau-type, which describes the evolution of a test-particle interacting with a “bath” of Coulomb particles in thermal equilibrium. A formula for it is given in Balescu [46, §37], in terms of special functions.
1.8. Physical validity of the Boltzmann equation Experience has shown that the Boltzmann equation and its variants realistically describe phenomena which occur in dilute atmosphere, in particular aeronautics at high altitude, or interactions in dilute plasmas. In many situations, predictions based on the Navier–Stokes equation are not accurate for low densities; a famous historical example is provided by the so-called Knudsen minimum effect in a Poiseuille flow [148, p. 99]: if the difference of pressure between the entrance and the exit of a long, narrow channel is kept fixed, then the flow rate through a cross-section of this channel is not a monotonic function of the average pressure, but exhibits a minimum for a certain value of this parameter. This phenomenon, established experimentally by Knudsen, remained controversial till the 1960’s. Also the Boltzmann equation cannot be replaced by fluid equations when it comes to the study of boundary layers (Knudsen layer, Sone sublayer due to curvature . . .) and the gas-surface interaction. Nowadays, with the impressive development of computer power, it is possible to perform very precise numerical simulations which seem to fully corroborate the predictions based on the Boltzmann equation – within the right range of physical parameters, of course. 11 This is the natural statistical equilibrium, see Section 2.5.
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All these questions are discussed, together with many numerical simulations and experiments, in the recent broad-audience survey book by Cercignani [148] (see also the review paper [136] by the same author in the present volume). 2. Mathematical problems in collisional kinetic theory In this section, we try to define the most interesting mathematical problems which arise in the study of Boltzmann-like equations. At this point we should make it clear that the Boltzmann equation can be studied for the sake of its applications to dilute gases, but also as one of the most basic and famous models for nonequilibrium statistical mechanics. 2.1. Mathematical validity of the Boltzmann equation In the last section, we have mentioned that, in the right range of physical parameters, the physical validity of the Boltzmann equation now seems to be beyond any doubt. On the other hand, the mathematical validity of the Boltzmann equation poses a more challenging problem. For the time being, it has been investigated only for the hard-sphere model. Let us give a short description of the problem in the case of hard spheres. But before that, we add a word of caution about the meaning of “mathematical validity”: it is not a proof that the model is the right one in a certain range of physical parameters (whatever this may mean). It is only a rigorous derivation of the model, in a suitable asymptotic procedure, from another model, which is conceptually simpler but contains more information (typically: the positions of all the particles, as opposed to the density of particles). Of course, the Boltzmann equation, just like many models, can be derived either by mathematical validation, or by direct modelling assumptions, and the second approach is more arbitrary, less interesting from the mathematical point of view, but not less “respectable” if properly implemented! This is why, for instance, Truesdell [430] refuses to consider the problem of mathematical validity of the Boltzmann equation. The validation approach to be discussed now is due to Grad [249], and it is particularly striking because the starting point is nothing but the model given by Newton’s laws of classical mechanics. It was not before 1972 that Cercignani [139] showed Grad’s approach to be mathematically consistent, in the sense that it can be rigorously implemented if one is able to prove some “reasonable” estimates on the solutions.12 Grad’s approach. The starting point is the equation of motion, according to Newton’s laws, for a system of n spherical particles of radius r in R3 , bouncing elastically on each other with billiard reflection laws. The state of the system is described by the positions and velocities of the centers, x1 , v1 , . . . , xn , vn , and the phase space is the subset of (R3x × R3v )n (or (X × R3 )n ) such that |xi − xj | r (i = j ). On this phase space there is a flow (St )t 0 , well-defined up to a zero-probability set of initial configurations, which is neglected. We now consider symmetric probability densities f n (x1 , v1 , . . . , xn , vn ) on the phase space 12 This remark is important because many people doubted the possibility of a rigorous derivation, see the discussion in Section 2.4.
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(symmetry reflects the physical assumption of undiscernability of particles). Of course, the flow (St ) on the phase space induces a flow on such probability densities, the solution of which is denoted by (ftn )t 0 . Moreover, by integrating ftn over all variables but the k first position variables and the k first velocity variables, one defines the k-particle distribution function Pk f n (x1 , v1 , . . . , xk , vk ) (think of Pk as a projection operator). In probabilistic terminology, P1 f n is the first marginal of f n . Assume now that (i) n → ∞, r → 0 (continuum limit) in such a way that nr 2 → 1 [the gas is sufficiently dilute, but not too much, so that only binary interactions play a significant role, and a typical particle collides about once in a unit of time. This limit is called the Boltzmann–Grad limit]; (ii) P1 f0n → f0 , where f0 is a given distribution function [this assumption means that the one-particle function at time 0 can be treated as “continuous” as the number of particles becomes large]; (iii) P2 f0n → f0 ⊗ f0 , and more generally, for fixed k, Pk f0n → f0 ⊗ · · · ⊗ f0
(34)
[this is the chaos assumption at time 0]; the problem is then to prove that P1 ftn → ft , where ft = ft (x, v) is the solution of the Boltzmann equation with hard-sphere kernel, and with initial datum f0 . This problem seems exceptionally difficult. The main result in the field is the 1973 Lanford’s theorem [292]. He proved the result for small time, and under some strong assumptions on the initial probability distributions Pk f0n : they should be continuous, satisfy appropriate Gaussian-type bounds, and converge uniformly13 towards their respective limits. Later his proof was rewritten by Illner and Pulvirenti, and extended to arbitrarily large time intervals, under a smallness assumption on the initial datum, which enabled to treat the Boltzmann equation as a kind of perturbation of the free transport equation: see [275, 276] and the nice reviews in [149,394,410]. For sure, one of the outstanding problems in the theory of the Boltzmann equation is to extend Lanford’s result to a more general framework, without smallness assumption. Another considerable progress would be its extension to long-range interactions, which is not clear even from the formal point of view (see, for instance, Cercignani [137]). The Boltzmann–Grad limit is also often called the low-density limit, and presented in the following manner [149, p. 60]: starting from the equations of Newtonian dynamics, blow-up the scales of space and time by a factor ε−1 (thus ε is the ratio of the microscopic scale by the macroscopic scale), and require the number of particles to be of the order of ε−2 , then let ε go to 0. In particular, the density will scale as ε2/3 since the volume will scale as ε−3 , and this explains the terminology of “low density limit”. Remarks about the chaos assumption. 1. Heuristically, the relevance of the chaos assumption in Boltzmann’s derivation can be justified as follows: among all probability distributions f n which have a given marginal 13 On compact subsets of (R3 × R3 )k , which is the set obtained from (R3 × R3 )k by deleting all configurations = with xi = xj for some distinct indices i, j .
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f = P1 f n , the most likely, in some sense which we do not make precise here,14 is the tensor product f ⊗· · ·⊗f . And as n → ∞, this distribution becomes by far the most likely. Thus, Boltzmann’s chaos assumption may be justified by the fact that we choose the most likely microscopic probability distribution f n which is compatible to our macroscopic knowledge (the one-particle distribution, or first marginal, f ). 2. In fact, the chaos property is automatically satisfied, in weak sense, by all sequences of probability measures (f n ) on (R3 × R3 )n which are compatible with the density f . More precisely: let us say that a microscopic configuration z = (x1 , v1 , . . . , xn , vn ) ∈ (R3x × R3v )n is admissible if its empirical density, 1
δ(xi ,vi ) , n n
ωz =
i=1
is a good approximation to the density f (x, v) dx dv, and let (f n )n∈N be a sequence of symmetric probability densities on (R3x × R3v )n respectively, such that the associated measures µn give very high probability to configurations which are admissible. In a more precise writing, we require that for every bounded continuous function ϕ(x, v) on R3 × R3 , and for all ε > 0,
n µn z ∈ R3x × R3v ;
ϕ(x, v) ωz (dx dv) − f (t, x, v) dx dv > ε −−−→ 0. n→∞ 3
R3x ×Rv
Then, (f n ) satisfies the chaos property, in weak sense [149, p. 91]: for any k 1, Pk f n −−−→ f ⊗k n→∞
in weak-∗ measure sense. This statement expresses the fact that f n is automatically close to the tensor product f ⊗n in the sense of weak convergence of the marginals. However, weak convergence is not sufficient to derive the Boltzmann equation, because of the problem of localization of collisions. Therefore, in Lanford’s theorem one imposes a stronger (uniform) convergence of the marginals, or strong chaos property. 3. This is the place where probability enters the Boltzmann equation: via the initial datum, i.e., the probability density (f n )! According to [149, p. 93], the conclusion of Lanford’s theorem can be reformulated as follows: for all time t > 0, if ftn is obtained 14 This is related to the fact that the tensor product f ⊗n has minimal entropy among all n-particles probability distribution functions f n with given first marginal f , and to the fact that the negative of the entropy yields a measure of “likelihood”, see Section 2.4 below. [In this review, the entropy of a distribution function f is defined by the formula H (f ) = f log f ; note the sign convention.] For hard spheres, a subtlety arises from the fact that configurations in which two spheres interpenetrate are forbidden, so f n cannot be a tensor product. Since however the total volume of the spheres, nr 3 , goes to 0 in the limit, it is natural to assume that this does not matter. On the other hand, contact points in the n-particle phase space play a crucial role in the way the chaos property is propagated in time.
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from f0n by transportation along the characteristics of the microscopic dynamics, and if µnt is the probability measure on (R3x × R3v )n whose density is given by ftn , then for all ε > 0, and for all bounded, continuous function ϕ(x, v) on R3x × R3v , µnt
n z ∈ R3x × R3v ;
R3x ×R3v
ϕ(x, v) ωz (dx dv) − ft (x, v) dx dv > ε −−−→ 0, n→∞
where ft is the solution to the Boltzmann equation with initial datum f0 . In terms of µn : µn
n z ∈ R3x × R3v ;
R3x ×R3v
ϕ(x, v) ωSt z (dx dv) − ft (x, v) dx dv > ε −−−→ 0. n→∞
In words: for most initial configurations, the evolution of the density under the microscopic dynamics is well approximated by the solution to the Boltzmann equation. Of course, this does not rule out the existence of “unlikely” initial configurations for which the solution of the Boltzmann equation is a very bad approximation of the empirical measure. 4. If the chaos property is the crucial point behind the Boltzmann derivation, then one should expect that it propagates with time, and that ∀t > 0,
Pk ftn → ft ⊗ · · · ⊗ ft .
(35)
However, this propagation property only holds in a weak sense. Even if the convergence is strong (say, uniform convergence of all marginals) in (34), it has to be weaker in (35), say almost everywhere, see the discussion in Cercignani et al. [149]. The reason for this weakening is the appearance of microscopic correlations (under evolution by the microscopic, reversible dynamics). In particular, if the initial microscopic datum is “very likely”, this does not imply at all that the microscopic datum at later times should be very likely! On the contrary, it should present a lot of correlations . . . . 5. In fact, one has to be extremely cautious when handling (35). To illustrate this, let us formally show that for t > 0 the approximation P2 ftn (x, v; y, w) ft (x, v)ft (y, w)
(36)
cannot be true in strong sense, uniformly in all variables,15 as n → ∞ (the symbol here means “approaches, in L∞ norm, uniformly in all variables x, y, v, w, as n → ∞”). Indeed, assume that (36) holds true uniformly in x, y, v, w, and choose y = x + rσ , 15 Constrained by |x − y| r.
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v − w, σ > 0, i.e., an ingoing collisional configuration in the two-particle phase space. Then, presumably P2 ftn (x, v; x + rσ, w) ft (x, v)ft (x + rσ, w) ft (x, v)ft (x, w)
(37)
as n → ∞. But from the specular reflection condition, for any t > 0, P2 ftn (x, v; x + rσ, w) = P2 ftn (x, v ; x + rσ, w ), where v and w are post-collisional velocities, v = v − v − w, σ σ,
w = w + v − w, σ σ.
Applying (36) again, this would result in P2 ftn (x, v; x + rσ, w) = P2 ftn (x, v ; x + rσ, w ) ft (x, v )ft (x, w ), which is not compatible with (37) (unless ft solves Equation (53) below). This contradiction illustrates the fact that (36) cannot be propagated by the dynamics of hard spheres. It is actually property (37), sometimes called one-sided chaos, which is used in the derivation of the Boltzmann equation, and which should be propagated for positive times: it means that the velocities of particles which are just about to collide are not correlated. But it is a very difficult problem to handle Equation (37) properly, because it involves the restriction of f n to a manifold of codimension 1, and may be violated even for initial data which satisfy the conditions of Lanford’s theorem! So an appropriate generalized sense should be given to (37). Lanford’s argument cleverly avoids any discussion of (37), and only assumes (36) at time 0, the approximation being uniform outside collisional configurations. So he plainly avoids discussing one-sided chaos, and does not care what is propagated for positive times, apart from weak chaos.16 To sum up: the physical derivation of the Boltzmann equation is based on the propagation of one-sided chaos, but no one knows how this property should be expressed mathematically – if meaningful at all. An easier variant of the validation problem is the derivation of linear transport equations describing the behavior of a Lorentz gas: a test-particle in a random pinball game, with scatterers randomly distributed according to (say) a Poisson law. Under a suitable scaling, the law of this test-particle converges towards the solution of a linear Boltzmann equation, as was first formalized by Gallavotti [226], before several improvements appeared [409, 91]. See Pulvirenti [394] for a review and introduction of the subject. The convergence actually holds true for almost all (in the sense of Poisson measure) fixed configuration of scatterers, but fails for certain specific configurations, for instance a periodic array, as shown in Bourgain, Golse and Wennberg [102]. We also note that Desvillettes and Pulvirenti [179] are able to rigorously justify the linear Boltzmann equation for some interactions with infinite range. 16 This is possible because he uses a perturbative proof, based on an iterative Duhamel formula, in which everything is expressed in terms of the initial datum . . . .
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Kac’s approach. To conclude this section, we mention another line of approach towards the mathematical justification of the Boltzmann equation. It goes via the construction of some many-particle stochastic system, such that the first marginal of its law at a given time t should be an approximation of the solution to the Boltzmann equation if the initial datum is chaotic. This subject was initiated by Kac17 [283], and developed by Sznitman [412] in connection with the problem of propagation of chaos. Recent progress on this have been achieved by Graham and Méléard [256,344]. The main conceptual difference between both approaches lies in the moment where probability is introduced, and irreversibility18 as well. In Lanford’s approach, the starting point is a deterministic particle system; it is only the particular “chaotic” choice of the initial datum which leads to the macroscopic, irreversible Boltzmann equation in the limit. On the other hand, for Kac the microscopic particle system is already stochastic and irreversible from the beginning. Then the main effect of the limit is to turn a linear equation on a large n-particle phase space, into a nonlinear equation on a reduced, one-particle phase space. Of course Kac’s approach is less striking than Grad’s, because the starting point contains more elaborate modelling assumptions, since stochasticity is already built in. Kac formulated his approach in a spatially homogeneous19 setting, while this would be meaningless for Grad’s approach. In fact, it is as if Kac wanted to treat the positions of the particles (which, together with ingoing velocities, determine the outgoing velocities) as hidden probabilistic variables. Then, all the subtleties linked to one-sided chaos can be forgotten, and it is sufficient to study just propagation of (weak) chaos. Moreover, Kac’s approach becomes important when it comes to make an interpretation of the Monte Carlo numerical schemes which are often used to compute approximate solutions of the Boltzmann equation. These schemes are indeed based on large stochastic particle systems. See Pulvirenti [394,453,396] for references about the study of these systems, in connection with the validation problem. We do not develop here on the problem of the rigorous justification of numerical schemes, but this topic is addressed in the companion review [136] by Cercignani.
2.2. The Cauchy problem From the mathematical point of view, the very first problem arising in the study of the Boltzmann equation is the Cauchy problem: given a distribution function f0 (x, v) on N N RN x × Rv (or X × Rv ), satisfying appropriate and physically realistic assumptions, show that there exists a (unique) solution of ∂f + v · ∇x f = Q(f, f ), ∂t f (0, ·, ·) = f0 . 17 See Section 1.5 in Chapter 2E. 18 See Section 2.4. 19 See Section 5.2.
(38)
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Needless to say, the Boltzmann equation seems impossible to solve explicitly,20 except in some very particular situations: semi-explicit solutions by Bobylev [79], Bobylev and Cercignani [81]; self-similar solutions of infinite mass by Nikolskii, see [289, p. 286]; particular solutions in a problem of shear flow by Truesdell, see [430, Chapters 14–15], some simple problems of modelling with a lot of symmetries [148] . . . . Explicit solutions are discussed in the review paper [207]. These exact solutions are important in certain modelling problems, but they are exceptional. This justifies the study of a general Cauchy problem. Of course, the question of the Cauchy problem should be considered as a preliminary for a more detailed study of qualitative properties of solutions of the Boltzmann equation. The main qualitative properties in which one is interested are: smoothness and singularities, conservation laws, strict positivity, existence of Lyapunov functionals, long-time behavior, limit regimes. We shall come back on all of this in the next chapters. As recalled in Section 3, the properties of the solutions may depend heavily on the form of the collision kernel. As of this date, the Cauchy problem has still not received satisfactory answers. As we shall describe in Section 5, there are several “competing” theories which either concern (more or less) simplified cases, or are unable to answer the basic questions one may ask about the solutions. Yet this problem has spectacularly advanced since the end of the eighties. Another fundamental problem in many areas of modelling by Boltzmann equation, as explained, for instance, in Cercignani [148], is the existence of stationary solutions: given a box X, prove that there exists a (unique?) stationary solution of the Boltzmann equation in the box: v · ∇x f = Q(f, f ),
x ∈ X, v ∈ RN ,
together with well-chosen boundary conditions (ideally, dictated by physical assumptions). The stationary problem has been the object of a lot of mathematical studies in the past few years; see, for instance, [31,25,34,36,26,37,38]. We shall not consider it here, except for a few remarks. This is first because the theory is less developed than the theory of the Cauchy problem, secondly because we wish to avoid the subtle discussion of boundary conditions for weak solutions.
2.3. Maxwell’s weak formulation, and conservation laws The change of variables (v, v∗ , σ ) → (v , v∗ , k), with k = (v − v∗ )/|v − v∗ |, has unit Jacobian and is involutive. Since σ = (v − v∗ )/|v − v∗ |, one can abuse terminology by referring to this change of variables as (v, v∗ ) → (v , v∗ ). It will be called the prepostcollisional change of variables. As a consequence of microreversibility, it leaves the collision kernel B invariant. 20 Although no theorem of non-solvability has been proven!
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The fact that this change of variable has unit Jacobian is not a general feature of Boltzmann-like equations, actually it is false for energy-dissipating models21 . . . . Also the change of variables (v, v∗ ) → (v∗ , v) is clearly involutive and has unit Jacobian. As a consequence, if ϕ is an arbitrary continuous function of the velocity v, RN v
Q(f, f ) ϕ dv
= = =
RN ×RN
RN ×RN
1 2
dv dv∗ dv dv∗
RN ×RN
S N−1
S N−1
dσ B(v − v∗ , σ )(f f∗ − ff∗ )ϕ
(39)
dσ B(v − v∗ , σ )ff∗ (ϕ − ϕ)
(40)
dv dv∗
S N−1
dσ B(v − v∗ , σ )ff∗ (ϕ + ϕ∗ − ϕ − ϕ∗ ).
(41)
This gives a weak formulation for Boltzmann’s collision operator. From the mathematical point of view, it is interesting because expressions like (40) or (41) may be well-defined in situations where Q(f, f ) is not. From the physical point of view, it expresses the change in the integral f (t, x, v)ϕ(v) dv which is due to the action of collisions. Actually, this formulation is so natural for a physicist, that Equation (40) was written by Maxwell22 [335, Equation (3)] before Boltzmann gave the explicit expression of Q(f, f )! Let f be a solution of the Boltzmann equation (8), set in the whole space RN x to simplify. By the conservative properties of the transport operator, v · ∇x , d dt
f (t, x, v)ϕ(v) dx dv =
Q(f, f )ϕ dx dv,
(42)
and the right-hand side is just the x-integral of any one of the expressions in formulas (39)– (41). As an immediate consequence, whenever ϕ satisfies the functional equation ∀(v, v∗ , σ ) ∈ RN × RN × S N−1 ,
ϕ(v ) + ϕ(v∗ ) = ϕ(v) + ϕ(v∗ )
(43)
then, at least formally, d dt
f (t, x, v)ϕ(v) dx dv = 0
along solutions of the Boltzmann equation. The words “at least formally” of course mean that the preceding equations must be rigorously justified with the help of some integrability estimates on the solutions to the Boltzmann equation. 21 See Section 2 in Chapter 2E. 22 Actually it is not so easy to recognize the Boltzmann equation in Maxwell’s notations!
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It can be shown under very weak conditions23 [142,29], [149, pp. 36–42] that solutions to (43), as expected, are only linear combinations of the collision invariants: ϕ(v) = 1, vi ,
|v|2 , 2
1 i N.
This leads to the (formal) conservation laws of the Boltzmann equation, d dt
⎞ 1 f (t, x, v) ⎝ vi ⎠ dx dv = 0, ⎛
|v|2 2
1 i N,
meaning that the total mass, the total momentum and the total energy of the gas are preserved. These conservation laws should hold true when there are no boundaries. In presence of boundaries, conservation laws may be violated: momentum is not preserved by specular reflection, neither is energy if the gas is in interaction with a wall kept at a fixed temperature. See Cercignani [141,148] for a discussion of general axioms of the classical modelling of gas-surface interaction, and resulting laws. If one disregards this possible influence of boundaries, then the preservation of mass, momentum and energy under the action of the Boltzmann collision operator is clearly the least that one can expect from a model which takes into account only elastic collisions. Yet, to this date, no mathematical theory has been able to justify these simple rules at a sufficient level of generality. The problem is of course that too little is known about how well behaved are the solutions to the Boltzmann equation. Conservation of mass and momentum are no problem, but no one knows how to obtain an a priori estimate which would imply a little bit more integrability than just finite energy. Another crucial topic for a fluid description is the validity of local conservation laws, i.e., continuity equations obtained by integrating the Boltzmann equation with respect to v only. With notations (1), these equations are ⎧ ∂ρ ⎪ ⎪ + ∇x · (ρu) = 0, ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨∂ (ρu) + ∇x · f v ⊗ v dv = 0, ∂t ⎪ RN ⎪ ⎪ ⎪ ⎪ ∂ ⎪ 2 2 ⎪ ⎩ f |v| v dv = 0. ρ|u| + NρT + ∇x · ∂t RN
(44)
At this moment, only the first of these equations has been proven in full generality [308]. 23 This problem was first treated by Gronwall [259,260] and Carleman [119] under stronger conditions. Then
people started to study it under weaker and weaker assumptions. Its interest lies not only in checking that there are no hidden conservation laws in the Boltzmann equation, but also in solving the important Equation (53) below, for which simpler methods are however available.
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2.4. Boltzmann’s H theorem and irreversibility In this section, we discuss some of the most famous aspects of the Boltzmann equation. This will justify a few digressions to make the topic as clear as possible. Let us symmetrize the integral (39) once more, fully using all the symmetries of the collision operator. We obtain Q(f, f )ϕ dv =−
1 4
dv dv∗ dσ B(v − v∗ , σ )(f f∗ − ff∗ )(ϕ + ϕ∗ − ϕ − ϕ∗ ).
(45)
We shall refer to this formula as Boltzmann’s weak formulation. Without caring about integrability issues, we plug ϕ = log f into this equation, and use the properties of the logarithm, to find Q(f, f ) log f dv = −D(f ), (46) RN
where D is the entropy dissipation functional, 1 f f∗ D(f ) = dv dv∗ dσ B(v − v∗ , σ )(f f∗ − ff∗ ) log 0. 4 R2N ×S N−1 ff∗
(47)
That D(f ) 0 just comes from the fact that the function (X, Y ) → (X −Y )(log X −log Y ) is nonnegative. Next, we introduce Boltzmann’s H functional, f log f. (48) H (f ) = N RN x ×Rv
Of course, the transport operator −v · ∇x does not contribute in any change of the H functional in time.24 As a consequence, if f = f (t, x, v) is a solution of the Boltzmann equation, then H (f ) will evolve in time because of the effects of the collision operator: d H f (t, ·, ·) = − D f (t, x, ·) dx 0. (49) N dt Rx This is the famous Boltzmann’s H theorem: the H -functional, or entropy, is nonincreasing with time. This theorem is “proven” and discussed at length in Boltzmann’s treatise [93]. Before commenting on its physical implications, let us give a few analytical remarks: 1. For certain simplified models of the Boltzmann equation, McKean [342] has proven that the H -functional is, up to multiplicative and additive constants, the only “local” (i.e., of the form A(f )) Lyapunov functional. 24 More generally, the transport operator does not contribute to any change of a functional of the form A(f ) dx dv.
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2. There are some versions with boundary conditions; actually, it was emphasized by Cercignani that the H theorem still holds true for a modified H -functional (including the temperature of the wall, for instance, if the wall is kept at fixed temperature) as soon as a certain number of general axioms are satisfied. See [141] for precise statements. 3. The argument above, leading to formula (47), does not work for certain variants of the Boltzmann equation, like mixtures. Actually Boltzmann had also given a (false) proof in this case, and once the error was discovered, produced a totally obscure argument to fix it (see the historical references in [148]). As pointed out by Cercignani and Lampis [150], the most robust way to prove the H theorem is to use again Maxwell’s weak formulation, and to note that 1 f f∗ 1 f f∗ f f∗ dv dv∗ dσ = − + 1 0, (50) Bff∗ log Bff∗ log 2 ff∗ 2 ff∗ ff∗ because Q(f, f ) dv = 0 and log X − X + 1 0. This line of proof can be generalized to mixtures [148, §6.4] and other models. 4. The Landau equation also satisfies a H theorem. The corresponding entropy dissipation is (formally) DL (f ) =
1 2
R3 ×R3
2 ff∗ a(v − v∗ ) ∇(log f ) − ∇(log f ) ∗ dv dv∗ .
(51)
What is the physical sense of Boltzmann’s H theorem? First of all, we note that the H functional should coincide with the usual entropy of physicists up to a change of sign. Also, it is a dynamical entropy, in the sense that is defined for nonequilibrium systems. Thus Boltzmann’s H theorem is a manifestation of the second law of thermodynamics (Clausius’ law) which states that the physical entropy of an isolated system should not decrease in time. In particular, it demonstrates that the Boltzmann model has some irreversibility built in. This achievement (produce an analytical proof of the second law for some specific model of statistical mechanics) was one of the early goals of Boltzmann, and was later considered as one of his most important contributions to statistical physics. But the H theorem immediately raised a number of objections, linked to the fact that the starting point of the derivation of the Boltzmann equation is just classical, reversible mechanics – so where does the irreversibility come from? Of course, since the existence of atoms was controversial at the time, Boltzmann’s construction seemed very suspect . . . . About the controversy between Boltzmann and opponents, and the way to resolve apparent contradictions in Boltzmann’s approach, one may consult [202,284,149,293]. Here we shall explain in an informal way the main arguments of the discussion, warning the reader that the following considerations have not been put on a satisfactory mathematical basis. Zermelo pointed out that Boltzmann’s theorem seemed to contradict the famous Poincaré recurrence theorem.25 Boltzmann replied that the scales of time on which Poincaré’s theorem applied in the present setting were much larger than the age of the 25 For almost all choice of the initial datum, a conservative system with a compact phase space will always come arbitrarily close to its initial configuration for large enough times. This theorem applies to a system of n particles obeying the laws of classical mechanics, interacting via elastic collisions, enclosed in a box.
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universe, and therefore irrelevant. This answer is justified by the fact that the Boltzmann equation should be a good approximation of the microscopic dynamics, only on a time scale which depends on the actual number of particles – see the discussion in the end of Section 2.5. Then Loschmidt came out with the following paradox. Let be given a gas of particles, evolving from time 0 to time t0 > 0. At time t0 , reverse all velocities and let the gas evolve freely otherwise. If Boltzmann were right, the same Boltzmann equation should describe the behavior of the gas on both time intervals [0, t0 ] and [t0 , 2t0 ]. Since the reversal of velocities does not change the entropy of a distribution function, the entropy at time 2t0 should be strictly less than the entropy at time 0. But, by time-reversibility of classical mechanics, at time 2t0 the system should be back to its initial configuration, which would be a contradiction . . . . To this argument Boltzmann is reported to have replied “Go on, reverse the velocities!” The answer to Loschmidt’s paradox is subtle and has to do with the probabilistic content of the Boltzmann equation. Starting with the classical monograph by P. and T. Ehrenfest [202], it was understood that reversible microdynamics and irreversible macrodynamics are not contradictory, provided that the right amount of probability is used in the interpretation of the macroscopic model. This view is very well explained in the excellent book by Kac [284]. In the case of the Boltzmann derivation, everything seems deterministic: neither the microscopic model, nor the macroscopic equation are stochastic. But the probabilistic content is hidden in the choice of the initial datum. As we mentioned earlier,26 with very high probability the Boltzmann equation gives a good approximation to the evolution of the density of the gas. And here randomness is in the choice of the microscopic initial configuration, among all configurations which are compatible with the density f (see Section 2.1 for more precise formulations). But for exceptional configurations which are not chaotic, the derivation of the Boltzmann equation fails. Now comes the tricky point in Loschmidt’s argument. As we discussed in Section 2.1, the chaos property, in strong sense, is not preserved by the microscopic, reversible dynamics. What should be preserved is the one-sided chaos: ingoing collisional configurations are uncorrelated. More precisely, if the strong chaos assumption holds true at initial time, then chaos should be true for ingoing configurations at positive times, and for outgoing configurations at negative times. So, reversing all velocities as suggested by Loschmidt is an innocent operation at the level of the limit one-particle distribution (which has forgotten about correlations), but by no means at the level of the microscopic dynamics (which keeps all correlations in mind). It will transform a configuration which is chaotic as far as ingoing velocities are concerned, into a configuration which is chaotic as far as outgoing velocities are concerned27. . . . Then, the microscopic dynamics will preserve this property that outgoing velocities are not correlated, and, by repeating the steps of the derivation of the Boltzmann equation, we find that the correct equation, to describe the gas from time t0 on, should be the negative Boltzmann equation (with a minus sign in front of the collision operator). Therefore in 26 Recall the remarks about propagation of chaos in Section 2.1. 27 In particular, this configuration is very unlikely as an initial distribution. Thus Loschmidt’s paradox illustrates
very well the fact that the Boltzmann derivation works for most initial data, but not for all!
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Loschmidt’s argument, of course the entropy is unchanged at time t0 , but then it should start to increase, and be back at its original value at time 2t0 . At the moment, a fully satisfactorily mathematical discussion of Loschmidt’s paradox is not possible, since we do not know what one-sided chaos should really mean, mathematically speaking. But one can check, as is done in [149, Section 4.7], that strong chaos28 is not propagated in time – so that it will be technically impossible to repeat Lanford’s argument when taking as initial datum the microscopic configuration at time t0 , be it before or after reversal of velocities. With this in mind, one also easily answers to the objection, why would the Boltzmann equation select a direction of time? Actually, it does not,29 and this can be seen by the fact that if strong30 chaos is assumed at initial time, then the correct equation should be the Boltzmann equation for positive times, and the negative Boltzmann equation for negative times. We have selected a direction of time by assuming the distribution function to be “very likely” at time 0 and studying the model for positive times. In fact, in Boltzmann’s description, the entropy is maximum at time 0, and decreases for positive times, increases for negative times. As an amusing probabilistic reformulation: knowing the one-particle distribution function at some time t0 , with very high probability the entropy is a maximum at this time t0 ! Related considerations can be found in Kac [284, p. 79] for simpler models, and may explain the cryptic statement by Boltzmann that “the H -functional is always, almost surely, a local maximum”. Most of the explanations above are already included in Boltzmann’s treatise [93], in physicist’s language; in particular Boltzmann was very well aware of the probabilistic content of his approach. But, since so many objections had been raised against Boltzmann’s theory, many physicists doubted for a long time that a rigorous derivation of the Boltzmann equation, starting from the laws of classical mechanics, could be possible. This is one of the reasons why Lanford’s theorem was so spectacular. After this digression about irreversibility, let us now briefly comment on Boltzmann’s H -functional itself. Up to the sign, it coincides with Shannon’s entropy (or information) quantity, which was introduced in communication theory at the end of the forties.31 In the theory of Shannon, the entropy measures the redundancy of a language, and the maximal compression rate which is applicable to a message without (almost) any loss of information: see [156] and the many references therein. In this survey, we shall make precise some links between information theory and the kinetic theory of gases, in particular via some variants of famous information-theoretical inequalities first proven by Stam [165]. 28 Roughly speaking, in the sense of uniform convergence of the marginals towards the tensor product distributions, recall Section 2.1. 29 By the way, Boltzmann himself believed that the direction of positive times should be defined as the direction in which the H -functional has a decreasing behavior . . . . 30 “Double-sided” should be the right condition here! 31 Here is a quotation by Shannon, extracted from [331], which we learnt in [16]. “My greatest concern was how to call it. I thought of calling it ‘information’. But the word was overly used, so I decided to call it ‘uncertainty’. When I discussed it with John von Neumann, he had a better idea. He told me: “You should call it entropy, for two reasons. In first place your uncertainty has been used in statistical mechanics under that name, so it already has a name. In second place, and more important, no one knows what entropy really is, so in a debate you will always have the advantage.” ”
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From the physical point of view, the entropy measures the volume of microstates associated to a given macroscopic configuration.32 This is suggested by the following computation, due to Boltzmann (see [93]). Let us consider n particles taking p possible N different states; think of a state as a small “box” in the phase space RN x × Rv . Assume that the only information to which we have access is the number ni of particles in each state i. In other words, we are unable to distinguish particles with different states; or, in a probabilistic description, we only have access to the one-particle marginal. This macroscopic description is in contrast with the microscopic description, in which we can distinguish all the particles, and know the state of each of them. Given a macroscopic configuration (n1 , . . . , np ), the number of compatible microscopic configurations is Ω=
n! . n1 ! · · · np !
Let us set fi = ni /n, and let all ni ’s go to infinity. By Stirling formula (or other methods, see [156, p. 282]), one shows that, up to an additive constant which is independent of the fi ’s,
1 log Ω → − fi log fi . n p
i=1
This result explains the link between the H -functional and the original definition of the entropy by Boltzmann, as the logarithm of the volume of microstates.33 So we see that it is the exponential of the negative of the entropy, which plays the role of a “volume” in infinite dimension. Up to a normalization, this quantity is known in information theory as the entropy power34: N (f ) = exp −2H (f )/N .
(52)
More remarks about the physical content of entropy, or rather entropies, are formulated in Grad [251]. In the discussion of Boltzmann’s derivation and irreversibility, we have seen two distinct entropies: the macroscopic entropy H (f ), which is fixed by the experimenter at initial time, and then wants to decrease as time goes by; and the microscopic entropy, H (f n ), which is more or less assumed to be minimal at initial time (among the class of microscopic distributions f n which are compatible with f ), and is then kept constant in time by the microscopic dynamics. There is no contradiction between the fact that H (ft ) is decreasing 32 Here, the one-particle probability distribution f is the macroscopic description of the system, while the manyparticle probability distribution f n is the microscopic state. 33 This is the famous formula S = k ln Ω, which was written on Boltzmann’s grave. 34 The analogy between power entropy and volume can be pushed so far that, for instance, the Shannon–Stam entropy power inequality, N (f ∗ g) N (f ) + N (g), can be seen as a consequence of the Brunn–Minkowski inequality on the volume of Minkowski sums of compact sets |X + Y |1/d |X|1/d + |Y |1/d . This (very) nontrivial remark was brought to our attention by F. Barthe.
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and H (ftn ) is constant, because there is no link between both objects35 if there are correlations at the level of ftn . As a last comment, the decrease of the entropy is a fundamental property of the Boltzmann equation, but the H -functional is far from containing all the information about the Boltzmann equation. This is in contrast with so-called gradient flows, which are partial differential equations of the form ∂f/∂t = −grad E(f ), for some “entropy” functional E and some gradient structure. For such an equation, in some sense the entropy functional encodes all the properties of the flow . . . . The main, deep reason for the fact that the Boltzmann equation cannot be seen as a gradient flow, is the fact that the collision operator depends only on the velocity space; but even if we restrict ourselves to solutions which do not depend on space, then the Boltzmann equation is not (to the best of our knowledge) a gradient flow. Typical gradient flows, in a sense which will be made more precise later, are the linear Fokker–Planck equation (19), or the McNamara–Young model for granular media (see [70] and Section 2 in Chapter 2E). The lack of gradient flow structure contributes to the mathematical difficulty of the Boltzmann equation.
2.5. Long-time behavior Assume that B(v − v∗ , σ ) > 0 for almost all (v, v∗ , σ ), which is always the case in applications of interest. Then equality in Boltzmann’s H theorem occurs if and only if for almost all x, v, v∗ , σ , f f∗ = ff∗ .
(53)
Under extremely weak assumptions on f [149,307,377], this functional equality forces f to be a local Maxwellian,36 i.e., a probability distribution function of the form e−|v−u(x)| /2T (x) . (2πT (x))N/2 2
f (x, v) = ρ(x)
Thus it is natural to guess that the effect of collisions is to bring f (t, ·) closer and closer to a local Maxwellian, as time goes by. This is compatible with Gibbs’ lemma: among all distributions on RN v with given mass, momentum and energy, the minimum of the entropy is achieved by the corresponding Maxwellian distribution. 35 Except the inequality H (f ) lim inf H (f n )/n . . . . By the way [149, pp. 99–100], the function φ(X) = t t X log X is, up to multiplication by or addition of an affine function, the only continuous function a constant φ which satisfies the inequality φ(P1 f n ) φ(f n )/n for all f n ’s, with equality for the tensor product distribution. 36 The result follows from the characterization of solutions to (43), but can also be shown directly by Fourier transform as in [377]. However, no proof is more enlightening than the one due to Boltzmann (see Section 4.3 in Chapter 2C). His proof requires C 1 smoothness, but Lions [307, p. 423] gave a beautiful proof that L1 solutions to (53) have to be smooth. And anyway, one can always easily reduce to the case of smooth densities by the remarks in Section 4.7 of Chapter 2C.
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Of course, Equation (53) implies Q(f, f ) = 0. As a corollary, we see that solutions of the functional equation Q(f, f ) = 0,
(54)
where the unknown f (v) is a distribution function on RN v with finite mass and energy, are precisely Maxwellian distributions. As we have seen, local Maxwellian states are precisely those distribution functions for which the dissipation of entropy vanishes. But if the position space X is a bounded domain on RN (with suitable boundary conditions, like specular reflection, or Maxwellian reemission), then one can show that there are very few time-dependent local Maxwellian distributions which satisfy the Boltzmann equation. Except in particular cases (domains with symmetries, see [254]), such a solution has to take the form e−|v−u| /2T f (t, x, v) = ρ , (2πT )N/2 2
(55)
for some parameters ρ, u, T which depend neither on t nor on x. A state like (55) is called a global Maxwellian, or global equilibrium state. It is uniquely determined by its total mass, momentum and energy. The problem of the trend to equilibrium consists in proving that the solution of the Cauchy problem (38) converges towards the corresponding global equilibrium as t → +∞, and to estimate, in terms of the initial datum, the speed of this convergence. On this subject, the (now outdated) paper by Desvillettes [168] accurately surveys existing methods up to the beginning of the nineties. Since that time, new trends have emerged, with the research for constructive estimates and the development of entropy dissipation techniques. To briefly summarize recent trends, we should say that the problem of trend to equilibrium has received rather satisfactory answers in situations where the Cauchy problem is known to have well-behaved solutions. We shall make a detailed review in Chapter 2C. On the other hand, if more complicated boundary conditions are considered, it may happen that the global equilibrium be no longer Maxwellian; and that the mere existence of a global equilibrium already be a very difficult problem.37 Also, when the gas gets dispersed in the whole space, then things become complicated: in certain situations, and contrary to what was previously believed by many authors, solutions to the Boltzmann equation never get close to a local Maxwellian state [383,419, 327]. The main physical idea behind this phenomenon is that the dispersive effects of the transport operator may prevent particles to undergo a sufficient number of collisions. In the whole space setting, the relevant problem is therefore not trend to equilibrium, but rather dispersion: find estimates on the speed at which the gas is dispersed at infinity. We shall not develop on this problem; to get information the reader may consult Perthame [376] (see also [277] for similar estimates in the context of the Vlasov–Poisson equation). Dispersion estimates play a fundamental role in the modern theory of the Schrödinger equations, and there is a strong analogy with the estimates appearing in this field [135]. 37 See the references in Section 2.2.
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We conclude this section with an important remark about the meaning of +∞ in the limit t → +∞. It is only in a suitable asymptotic regime that the Boltzmann equation is expected to give an accurate description of, say, a system of n particles. But for a given, large number n of particles, say 1023 , the quality of this description cannot be uniform in time. To get convinced of this fact, just think that Poincaré’s recurrence theorem will apply to the n-particle system after a very, very long time. In fact, since the Boltzmann equation is established on physical scales such that each particle typically encounters a finite number of collisions in a unit of time, we may expect the Boltzmann approximation to break down on a time at most O(n), i.e., a time on which a typical particle will have collided with a nonnegligible fraction of its fellow particles, so that finite-size effects should become important. This means that any theorem involving time scales larger than 1023 is very likely to be irrelevant38 . . . . Such a conclusion would only be internal to Boltzmann’s equation and would not yield any information about physical “reality” as predicted by the model. So what is interesting is not really to prove that the Boltzmann equation converges to equilibrium as t → +∞, but rather to show that it becomes very close to equilibrium when t is very large, yet not unrealistically large. Of course, from the mathematical point of view, this may be an extremely demanding goal, and the mere possibility of proving explicit rates should already be considered as a very important achievement, as well as identifying the physical factors (boundary conditions, interaction, etc.) which should slow down, or accelerate the convergence.
2.6. Hydrodynamic limits The H theorem was underlying the problem of the trend to global equilibrium in the limit t → +∞. It also underlies the assumption of local thermodynamical equilibrium in the hydrodynamical limits. Generally speaking, the problem of the hydrodynamical limit can be stated as follows: pass from a Boltzmann description of a dilute gas (on microscopic scales of space and time, i.e., of the order of the mean free path and of the mean time between collisions, respectively) to a hydrodynamic description, holding on macroscopic scales of space and time. And the scaling should be such that f “looks like” a local Maxwellian, even if local Maxwellians cannot be solutions of the Boltzmann equation . . . . To make this more concrete, assume that one contracts the measurements of lengths and time by a factor ε, the velocity scale being preserved (ε can be thought of as the Knudsen number, which, roughly speaking, would be proportional to the ratio between the mean free path and a typical macroscopic length). Then, the new distribution function39 will be fε (t, x, v) = f
t x , ,v . ε ε
38 Besides being of no practical value, since these time scales are much much larger than reasonable physical scales. 39 Note that f is not a probability density. ε
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If f solves the Boltzmann equation, then fε solves the rescaled Boltzmann equation 1 ∂fε + v · ∇x fε = Q(fε , fε ). ∂t ε Hence the role of the macroscopic parameter ε is to considerably enhance the role of collisions. In view of the H theorem, one expects fε to resemble more and more a local Maxwellian when ε → 0: this is the assumption of local thermodynamical equilibrium, whose mathematical justification is in general a delicate, still open problem. Here is an equivalent, nonrigorous way of seeing the limit: the time scale of trend to local equilibrium should be of the order of the mean time between collisions, which should be much smaller than the macroscopic time. For fixed ε, the macroscopic quantities (density, momentum, temperature) associated to fε via (1) satisfy the equations ⎧ ∂ρε ⎪ ⎪ + ∇x · (ρε uε ) = 0, ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂(ρ u ) ε ε + ∇x · fε v ⊗ v dv = 0, ⎪ ∂t RN ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ⎪ 2 2 ⎪ ρ + ∇ |u | + Nρ T · f |v| v dv = 0. ⎩ ε ε ε ε x ε ∂t RN The assumption of local thermodynamical equilibrium enables one to close this system in the limit ε → 0, and to formally obtain ⎧ ∂ρ ⎪ ⎪ + ∇x · (ρu) = 0, ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂(ρu) + ∇x · (ρu ⊗ u + ρT IN ) = 0, ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ ρ|u|2 + NρT + ∇x · ρ|u|2 u + (N + 2)ρT u = 0 ∂t
(56)
with IN standing for the identity N × N matrix. System (56) is nothing but the system of the compressible Euler equations, when the pressure is given by the law of perfect gases, p = ρT . See [431]. Other scalings are possible, and starting from the Boltzmann equation one can get many other equations in fluid mechanics [56]. In particular, by looking at perturbations of a global equilibrium, it is possible to recover Navier–Stokes-type equations. This is one of several possible ways of interpreting the Navier–Stokes equation, see [467] for remarks about other interpretations. From a physicist’s point of view, the interesting aspect of this limit is the appearance of the viscosity from molecular dynamics. From a mathematician’s perspective, another interesting thing is that there are some well-developed mathematical theories for the Navier–Stokes equation, for instance the famous theory of weak solutions by Leray [299–301], see Lions [313,314] for the most recent developments – so one can hope to prove theorems!
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An interesting remark, due to Sone and coworkers [408,407], shows that sometimes a hydrodynamic equation which looks natural is actually misleading because some kinetic effects should have an influence even at vanishing Knudsen number; this phenomenon was called “ghost effect”. There are formal procedures for “solving” the Boltzmann equation in terms of a series expansion in a small parameter ε (like our ε above), which are known as Hilbert and Chapman–Enskog expansions. These procedures have never received a satisfactory mathematical justification in general, but have become very popular tools for deriving hydrodynamical equations. This approach is described in reference textbooks such as [154, 430,141,148,48], and particularly [250, Section 22 and following]. But it also underlies dozens of papers on formal hydrodynamical limits, which we do not try to review. By the way, it should be pointed out that equations obtained by keeping “too many” (meaning 3 or 4) terms of the Hilbert or Chapman–Enskog series, like the so-called Burnett or superBurnett equations, seem to be irrelevant (a discussion of this matter, an ad hoc recipe to fix this problem, and further references, can be found in Jin and Slemrod [282]). Also, these expansions are not expected to be convergent, but only “asymptotic”. In fact, a solution of the Boltzmann equation which could be represented as the sum of such a series would be a very particular one40: it would be entirely determined by the fields of local density, mean velocity and pressure associated with it. All these problems illustrate the fact that the Hilbert and Chapman–Enskog methods rely on very sloppy grounds. Rather violent attacks on their principles are to be found in [430]. In spite of this, these methods are still widely used. Without any doubt, their popularity lies in their systematic character, which enables one to formally derive the correct equations in a number of situations, without having to make any guess. An alternative approach, which is conceptually simpler, and apparently more effective for theoretical purposes, is a moment-based procedure first proposed by Grad [249]. For Maxwell molecules, this procedure leads to hydrodynamic equations whose accuracy could be expected to be of high order; the method has been developed in particular by Truesdell and collaborators [274,430] under the name of “Maxwellian iteration”. The problem of rigorous hydrodynamical limits has been studied at length in the literature, but most of these results have been obtained in a perturbative setting. Recently, some more satisfactory results (in the large) have been obtained after the development of a spectacular machinery (see [441] for a presentation). We shall discuss a few references on both lines of approach in Section 5, as we shall go along the presentation of the mathematical theories for the Cauchy problem. On the other hand, the Boltzmann equation, as a model, does not capture the full range of hydrodynamical equations that one would expect from dynamical systems of interacting particles. At the level of the Euler equation above, this can be seen by the fact that the pressure law is of the form p = ρT . See [363,397,467] for more general equations and partial results on the problem of the direct derivation of hydrodynamical equations from particle systems. This however does not mean that it is not worth working at the level of the Boltzmann equation; first because the limit should be simpler to rigorously perform, than the limit for “raw” particle systems; secondly because the Boltzmann equation is one 40 “Normal” in the terminology of Grad [250].
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of the very few models of statistical physics which have been derived from mechanical first principles. This kind of preoccupations meets those expressed by Hilbert in the formulation of his sixth problem41 about the axiomatization of physics: can one put the equations of fluid mechanics on a completely rigorous basis, starting from Newton’s laws of microscopic motion?
2.7. The Landau approximation This problem occurs in the kinetic theory of dilute plasmas, as briefly described in Section 1.7. Starting from the Boltzmann equation for screened Coulomb potential (Debye potential), can one justify the replacement of this operator by the Landau operator (26) as the Debye length becomes large in comparison with the space length scale? This problem can easily be generalized to an arbitrary dimension of space. A few remarks are in order as regards the precise meaning of this problem. First of all, this is not a derivation of the Landau equation from particle systems (such a result would be an outstanding breakthrough in the field). Instead, one considers the Boltzmann equation for Debye potential as the starting point. Secondly, if one wants to stick to the classical theory of plasmas, then – either one neglects the effects of the mean-field interaction; in this case the problem has essentially been solved recently, as we shall discuss in Section 5 of Chapter 2B; – or one takes into account this effect, and then the Landau equation should only appear as a long-time correction to the Vlasov–Poisson equation. To the best of our knowledge, no such result has ever been obtained even in simplified regimes.
2.8. Numerical simulations The literature about numerical simulation for the Boltzmann equation is considerable. All methods used to this day consider separately the effects of transport and collision. This splitting has been theoretically justified by Desvillettes and Mischler [178], for instance, but a thorough discussion of the best way to implement it seems to be still lacking. Dominant methods are based on Monte Carlo simulation, and introduce “particles” interacting by collisions. Of course, transport is no problem for a particles-based method: just follow the characteristics, i.e., the trajectories of particles in phase space. Then one has to implement the effect of collisions, and many variants are possible. Sometimes the dynamics of particles obey the Newton laws of collision only on the average, in such a way that their probability density still comply with the Boltzmann dynamics. We do not try to review the literature on Monte Carlo simulation, and refer to the very neat survey in Cercignani [148, Chapter 7], or to the review paper [136] by the same author. Elements of the theoretical justification of Monte Carlo simulation, in connection with chaos issues, are reviewed in Cercignani et al. [149, Chapter 10], Pulvirenti [394] and Graham and Méléard [257]. 41 Directly inspired by Boltzmann’s treatise, among other things.
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Let us however say a few words about deterministic methods which have emerged recently, thanks to the increase of computational capacity.42 For a long time these methods were just unaffordable because of the high computational cost of the (2N − 1)-fold integral in the Boltzmann collision operator, but they are now becoming more and more competitive. Deterministic schemes based on conservation laws have been devised in the last years by Buet, Cordier, Degond, Lemou, Lucquin [106,297,105]. In these works, the simulated distribution function is constrained to satisfy conservation of mass, momentum and energy, as well as decreasing of entropy. This approach implies very clever procedures, in particular to handle the discretization of the spheres appearing in the Boltzmann representation (this problem is pretty much the same than the consistency problem for discrete velocity models). For variants such as the Landau equation, these difficulties are less important, but then one has to hunt for possible undesirable symmetries, which may introduce spurious conservation laws, etc. Another deterministic approach is based on Fourier transform, and has been developed by Bobylev and Rjasanow, Pareschi, Perthame, Russo, Toscani [82,89,370,371,373,372]. We shall say a little bit more on these schemes in Section 4.8, when introducing Fourier transform tools. At the moment, both methods have been competing, especially in the framework of the Landau equation [107,215]. It seems that spectral schemes are useful to give extremely accurate results, but cannot beat conservation-based schemes in terms of speed and efficiency. Certainly more is to be expected on the subject.
2.9. Miscellaneous We gather here a few other basic questions about the Boltzmann equation, which, even though less important than the ones we have already presented, do have mathematical interest. Phenomenological derivation of the Boltzmann equation. We have seen two ways to introduce the Boltzmann equation: either by direct modelling assumptions (dilute gas, chaos, etc.) or by rigorous theorems starting from particle systems. There is a third way towards it, which is by making some “natural” phenomenological assumptions on the form of the collision operator, and try to prove that these assumptions uniquely determine the form of the collision operator. A classical discussion by Bobylev [84] exemplifies this point of view. To roughly sum up the very recent work by Desvillettes and Salvarani [180], it is shown that (essentially) the only smooth quadratic forms Q acting on probability densities such that (1) evolution by ∂t f = Q(f, f ) preserves nonnegativity, (2) Q has Galilean invariance, (3) Q(M, M) = 0 for any Maxwellian M, 42 The same increase of computational capacity seems to make discrete–velocity models less and less attractive.
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have to be linear combinations of a Boltzmann and a Landau operator. Note the inversion of the point of view: in Section 2.4 we have checked that the form of Boltzmann’s operator imposed a Maxwellian form to an equilibrium distribution, while here we see that the assumption of Maxwellian equilibrium states43 contributes to determine the structure of the Boltzmann operator. Image of Q. This problem is very simply stated: for any function f (v) with enough integrability, we know that Q(f, f )ϕ(v) dv = 0 whenever ϕ(v) is a linear combination of 1, vi (1 i N), |v|2 . Conversely, let h be a function satisfying hϕ = 0 for the same functions ϕ, is it sufficient to imply the existence of a probability distribution f such that Q(f, f ) = h? This problem apparently has never received even a partial answer for a restricted subclass of functions h. Divergence form of Boltzmann’s collision operator. Unlike several other operators, particularly Landau or Fokker–Planck, the Boltzmann operator is not written in divergence form, even though it is conservative. Physicists consider this not surprising, in as much as Boltzmann’s operator models sudden (as opposed to continuous) changes of velocities. However, generally speaking, any function Q whose integral vanishes can be written as the divergence of something. This writing is in general purely mathematical and improper to physical interpretation. But a nice feature of Boltzmann’s operator, as we have shown in [448], is that an explicit and (relatively) simple expression exists. We obtained it by going back to the physical interpretation in terms of collisions, as in Landau [291]. Since there is no well-defined “flux”, one is led to introduce fictitious trajectories, linked to the parametrization of the pre-collisional variables. Here is one possible expression for the flux, which is a priori not unique: Q(f, f ) = −∇v · J (f, f ), where J (f, f ) = − dv∗ dω B(v − v∗ , ω) (v−v∗ ,ω)>0
(v−v∗ ,ω)
× 0
=
dr f (v + rω)f (v∗ + rω)ω
(v∗ −v,v◦ −v)<0
(57)
dv◦ dv∗ f (v◦ )f (v∗ )
v − v◦ v − v◦ . × B v◦ − v∗ , |v − v◦ | |v − v◦ |N
(58)
The Boltzmann operator can even be written as a double divergence, Q(f, f ) =
ij
∂2 Aij (f, f ), ∂vi ∂vj
Aii (f, f ) = −∇v · J E (f, f ),
i
for some explicit functionals Aij and J E , 1 i, j N . Each of the “fluxes” J , Aj , J E is related to a conservation law. 43 Which can be taken as physically reasonable from outer considerations, like central limit theorem, etc.
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Besides casting some more light on the structure of Boltzmann’s operator and its relations with divergence operators, these formulas may be used for devising conservative numerical schemes via the “velocity diffusion” method (implemented in works by MasGallic and others). Eternal solutions. Let us introduce this problem with a quotation from Truesdell and Muncaster [430, p. 191] about irreversibility: “In a much more concrete way than Boltzmann’s H theorem, [the time-behavior of the moments of order 2 and 3 for the Boltzmann equation with Maxwellian44 molecules] illustrate[s] the irreversibility of the behavior of the kinetic gas. This irreversibility is particularly striking if we attempt to trace the origin of a grossly homogeneous condition by considering past times instead of future ones. Indeed the magnitude of each component of [the pressure tensor] and [the tensor of the moments of order 3] that is not 0 at t = 0 tends to ∞ as t → −∞. Thus any present departure from kinetic equilibrium must be the outcome of still greater departure in the past.” Appealing as this image may be, it is our conviction that it is actually impossible45 to let t → −∞. More precisely, heuristic arguments in [450, Annex II, Appendix] make the following conjecture plausible: let f = f (t, v) be a solution of the Boltzmann equation for v ∈ RN , t ∈ R, with finite kinetic energy. Then f is a Maxwellian, stationary distribution. In the case of the spatially inhomogeneous Boltzmann equation, the conjecture should be reformulated by saying that f is a local Maxwellian (which would allow “travelling Maxwellians”, of the form M(x − vt, v), where M(x, v) is a global Maxwellian in the x and v variables). This problem, which is reminiscent of Liouville-type results for parabolic equations [468], may seem academic. But in [450, Annex II, Appendix] we were able to connect it to the important problem of the uniformity of trend to equilibrium, and to prove this conjecture for some simplified models; Cabannes [109,110] also was able to prove it for some simplified discrete-velocity approximations of the Boltzmann equation. At this stage we mention that the nonnegativity of the solution is fundamental, since nontrivial, partially negative solutions may exist. By the way, the nonnegativity of the solutions to the Boltzmann equation is also very important from the mathematical point of view: if it is not imposed, then irrelevant blow-up of the distribution function may occur. Very recently, Bobylev and Cercignani [81] made a considerable progress in this problem, proving the conjecture for so-called Maxwellian collision kernels, under the additional requirement that the solution has all of its moments finite for all time. Simultaneously, they discovered some very interesting self-similar eternal solutions of the Boltzmann equation with infinite kinetic energy, which may describe the asymptotic behavior of the Boltzmann equation in some physical regimes where the energy is very large. 44 See Section 3. 45 Which may be an even more striking manifestation of irreversibility?
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3. Taxonomy After having introduced the main mathematical problems in the area, we are prepared for a more precise discussion of Boltzmann-like models. This section may look daunting to the non-specialist, and we shall try to spare him. The point is that there are, truly speaking, as many different Boltzmann equations as there are collision kernels. Hence it is absolutely necessary, before discussing rigorous results, to sketch a “zoological” classification of these. For simplicity, and because this case is much better understood, we limit the presentation to the classical, elastic case. For background about relativistic, quantum, nonelastic collision kernels, see Section 1.6, Chapter 2E, or the survey papers [209] and [146]. We shall conclude this section with a brief description of the main mathematical and physical effects that the choice of the collision kernel should have on the properties of the Boltzmann equation. 3.1. Kinetic and angular collision kernel Recall that the collision kernel B = B(v − v∗ , σ ) = B(|v − v∗ |, cos θ ) [by abuse of notations] only depends on |v − v∗ | and cos θ , the cosine of the deviation angle. Also recall that for a gas of hard spheres, B |v − v∗ |, cos θ = K|v − v∗ |, K > 0, while for inverse s-power forces, B factors up like B |v − v∗ |, cos θ = Φ |v − v∗ | b(cos θ ),
(59)
where Φ(|z|) = |z|γ , γ = (s − 5)/(s − 1) in dimension N = 3, and b(cos θ ) sinN−2 θ ∼ Kθ −(1+ν) , ν = 2/(s − 1) in dimension N = 3 also. Exactly what range of values of s should be considered is by no means clear in the existing literature. Many authors [111,18,170] have restricted their discussion to s > 3. Klaus [288, p. 895] even explains this restriction by the impossibility of defining the Boltzmann linearized collision operator for s 3. However, as we shall explain, at least a weak theory of the Boltzmann equation can be constructed for any exponent s ∈ (2, +∞). The limit value s = 2 corresponds to the Coulomb interaction, which strictly speaking does not fit into the framework of the Boltzmann equation, as we have discussed in Section 1.7. R EMARK . What may possibly be true, and anyway requires clarification, is that the derivation of the Boltzmann equation from particle systems may fail for s 3, because of the importance of the mean-field interaction. But even in this case, the Boltzmann description of collisions should be rehabilitated in the investigation of the long-time behavior. Even though one is naturally led to deal with much more general collision kernels, products like (59) are the basic examples that one should keep in mind when discussing
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assumptions. By convention, we shall call Φ the kinetic collision kernel, and b the angular collision kernel. We shall discuss both quantities separately. From the mathematical point of view, the control of Boltzmann’s collision operator is all the more delicate that the collision kernel is “large” (in terms of singularities, or behavior as |v − v∗ | → ∞). On the contrary, when one is interested in such topics as trend to equilibrium, it is good to have a strictly positive kernel because this means more collisions; then the difficulties often come from the vanishings of the collision kernel. In short, one should keep in mind the heuristic rule that the mathematical difficulties encountered in the study of the Cauchy problem often come from large values of the collision kernel, those encountered in the study of the trend to equilibrium often come from small values of the collision kernel.
3.2. The kinetic collision kernel It is a well-established custom to consider the cases Φ(|v − v∗ |) = |v − v∗ |γ , and to distinguish them according to • γ > 0: hard potentials; • γ = 0: Maxwellian potentials; • γ < 0: soft potentials. For inverse-power forces in dimension 3, hard potentials correspond to s > 5, soft potentials to s < 5. We shall stick to this convention, but insist that it is quite misleading. First of all, “hard potentials” are not necessarily associated to an interaction potential! It would be better to speak of “hard kinetic collision kernel”. But even this would not be a neat classification, because it involves at the same time the behavior of the collision kernel for large and for small values of the relative velocity, which makes it often difficult to appreciate the assumptions really needed in a theorem. Sometimes a theorem which is stated for hard potentials, would in fact hold true for all kinetic collision kernels which are bounded below at infinity, etc. As typical examples, trouble for the study of the Cauchy problem may arise due to large relative velocities for hard potentials, or due to small relative velocities for soft potentials . . . . How positive may γ be? For hard spheres, γ = 1, hence a satisfactory theory should be able to encompass this case. In many cases one is able to treat γ < 2 or γ 2, or even less stringent assumptions. Conversely, how negative may γ be? Contrarily to what one could think, critical values of the exponent s do not, in general, correspond to critical values of γ . As a striking example, think of Coulomb potential (s = N − 1), which normally should correspond to a power law γ = N/(N − 2) in dimension N . Besides the fact that this is meaningless when N = 2, this exponent is less and less negative as the dimension increases; hence the associated Cauchy problem is more and more easy because of the weaker singularity. The following particular values appear to be most critical: γ = −2, γ = −N . The appearance of the limit exponent −2 in the study of several mathematical properties [437, 247,248,446] has led us in [446] to suggest the distinction between moderately soft potentials (−2 < γ < 0) and very soft potentials (γ < −2). It is however not clear whether
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the border corresponds to a change of mathematical properties, or just an increase in difficulty. Note that dimension 3 is the only one in which the Coulomb potential coincides with the limit exponent −N , which makes its study quite delicate46 !
3.3. The angular collision kernel We now turn to the angular collision kernel b(cos θ ) = b(k · σ ), k = (v − v∗ )/|v − v∗ |. First of all, without loss of generality one may restrict the deviation angle to the range [0, π/2], replacing if necessary b by its “symmetrized” version, [b(cos θ ) + b(cos(π − θ ))]10θπ/2 . From the mathematical point of view, this is because the product f f∗ appearing in the Boltzmann collision operator is invariant under the change of variables σ → −σ ; from the physical point of view this reflects the undiscernability of particles. As mentioned above, for inverse-power law forces, the angular collision kernel presents a nonintegrable singularity as θ → 0, and is smooth otherwise. The fact that the collision kernel presents a nonintegrable singularity with respect to the angular variable is not a consequence of the choice of inverse-power forces, recall (15). By analogy with the examples of inverse-power forces, one would like to treat the following situations: b(cos θ ) sinN−2 θ ∼ Kθ −(1+ν)
as θ → 0,
(60)
0 ν < 2. Grad’s angular cut-off [250,141] simply consists in postulating that the collision kernel is integrable with respect to the angular variable. In our model case, this means S N−1
b(k · σ ) dσ = S N−2
π
b(cos θ ) sinN−2 θ dθ < ∞.
(61)
0
The utmost majority of mathematical works about the Boltzmann equation crucially rely on Grad’s cut-off assumption, from the physical point of view this could be considered as a short-range assumption.
3.4. The cross-section for momentum transfer Let M(|v − v∗ |) be defined by S N−1
1 dσ B(v − v∗ , σ )[v − v ] = (v − v∗ )M |v − v∗ | . 2
(62)
46 All the more that γ = −3 also seems to have some special, bad properties independently of the dimension, see [429].
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Indeed, by symmetry, the left-hand side is parallel to v − v∗ . The quantity M is called the collision kernel for momentum transfer. In our model case, M(|v − v∗ |) = µΦ(|v − v∗ |), where µ = S N−2
π
b(cos θ )(1 − cos θ ) sinN−2 θ dθ.
(63)
0
From the physical point of view, the cross-section for momentum transfer is one of the basic quantities in the theory of binary collisions (see, for instance, [405,164]) and its computation via experimental measurements is a well-developed topic. On the other hand, the mathematical importance of the cross-section for momentum transfer has not been explicitly pointed out until very recently. From the mathematical point of view, finiteness of M (for almost all v − v∗ ) is a necessary condition for the Boltzmann equation to make sense [450, Annex I, Appendix A], in the sense that if it is not satisfied, then Q(f, f ) should only take values in {−∞, 0, +∞}. On the other hand, together with Alexandre we have recently shown [12] that suitable assumptions on M are essentially what one needs to develop a coherent theory for the Boltzmann equation without Grad’s angular cut-off assumption. Note that in our model case, finiteness of M requires that b(cos θ )(1 − cos θ ), or equivalently b(cos θ )θ 2 , be integrable on S N−1 . This precisely corresponds to the range of admissible singularities ν ∈ [0, 2). When ν = 2, the integral (63) diverges logarithmically for small values of θ : this is one reason for the failure of the Boltzmann model to describe Coulomb collisions. Very roughly, the Debye truncature yields a finite µ, which behaves like the logarithm of the Debye length; this is what physicists call the Coulomb logarithm. And due to the fact that the divergence is only logarithmic, they expect the cross-section for momentum transfer to depend very little on the precise value of the Debye length.
3.5. The asymptotics of grazing collisions The mathematical links between the Boltzmann and the Landau collision operator can be made precise in many ways. As indicated for instance by Degond and Lucquin [162], for a fixed, smooth f , one can consider QL (f, f ) as the limit when ε → 0 of a Boltzmann collision operator for Coulomb potential, with a truncated angular collision kernel, bε (cos θ ) =
1 b(cos θ )1θε . log ε−1
The factor log ε−1 compensates for the logarithmic divergence of the Coulomb crosssection for momentum transfer. As for the parameter ε, it is proportional to the “plasma parameter”, which is very small for classical plasmas, and actually goes to 0 as the Debye length goes to infinity. Also in the case of non-Coulomb potentials can one define an asymptotic regime in which the Boltzmann equation turns into a Landau equation. Such a formal study was
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performed by Desvillettes [169]: he considered the limit of the Boltzmann operator when the collision kernel is of the form 1 θ N−2 N−2 θ θ bε (cos θ ) = 3 sin sin b cos , ε → 0. ε ε ε These two asymptotic procedures seem very different both from the mathematical and the physical point of view. Also, the truncation in the Coulomb case does not correspond to the Debye cut: indeed, for the Debye interaction potential, the collision kernel does not factor up. However, all limits of the type Boltzmann → Landau can be put into a unified formalism, first sketched in our work [446], then extended and made more precise in Alexandre and Villani [12]. The idea is that all that matters is that (1) all collisions become grazing in the limit, (2) the cross-section for momentum transfer keep a finite value in the limit. For simplicity we state precise conditions only in our model case where the collision kernel factors up as Φ(|v − v∗ |)bε (cos θ ): bε (cos θ ) −−−→ 0 uniformly in θ θ0 , ε→0 N−2 π µ ε ≡ S bε (cos θ ) (1 − cos θ ) sinN−2 θ dθ −−−→ µ. ∀θ0 > 0,
(64)
ε→0
0
This limit will be referred to as the asymptotics of grazing collisions. It can be shown that in this limit, the Boltzmann operator turns into the general Landau operator dv∗ a(v − v∗ ) f∗ (∇f ) − f (∇f )∗ , QL (f, f ) = ∇v · (65) RN
zi zj aij (z) = Ψ |z| δij − 2 , |z|
(66)
taking into account the identity Ψ |z| =
µ |z|2 Φ |z| . 4(N − 1)
(67)
According to (67), we shall use the terminology of hard, Maxwellian, or soft potentials for the Landau operator depending on whether Ψ (|z|) in (66) is proportional to |z|γ +2 with γ > 0, γ = 0, γ < 0 respectively. Let us insist that the most relevant situation is the particular case introduced in (26)–(27) to describe Coulomb collisions. The general Landau operator (65) can be considered in several ways, – either like an approximation of the Landau equation with Coulomb potential: from the (theoretical or numerical) study of the corresponding equations one establishes results which might be extrapolated to the Coulomb case; – either as an approximation of the effect of grazing collisions in the Boltzmann equation without cut-off: one could postulate that such an operator may be well approximated by Q1 + Q2 , where Q1 is a Boltzmann operator satisfying Grad’s cutoff assumption and Q2 is a Landau operator (see [141] for similar considerations);
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– or like a mathematical auxiliary in the study of the Boltzmann equation. This point of view will prove useful in Chapter 2C. To conclude this section, we mention that limits quite similar to the asymptotics of grazing collisions appear in the study of kinetic equations modelling quantum effects such as the Bose–Einstein condensation (like the Kompaneets equation, see Chapter 2E), as can be seen in the recent work of Escobedo and Mischler [209].
3.6. What do we care about collision kernels? We shall now try to explain in a non-rigorous manner the influence of the collision kernel (or equivalently, of the cross-section) on solutions to the Boltzmann equation. By the way, it is a common belief among physicists that the precise structure of the collision kernel (especially the angular collision kernel) has hardly any influence on the behavior of solutions. Fortunately for us mathematicians, this belief has proven to be wrong in several respects. We make it clear that the effects to be discussed only reflect the influence of the collision kernel, but may possibly come into conflict with boundary condition effects,47 for instance. We also point out that although some illustrations of these effects are known in many regimes, none of them has been shown to hold at a satisfactory degree of generality. Distribution tails. First of all, let us be interested in the behavior of the distribution function for large velocities: how fast does it decay? can large distribution tails occur? The important feature here is the behavior of the kinetic collision kernel as the relative velocity |v − v∗ | goes to infinity. If the collision kernel becomes unbounded at infinity, then solutions should be well-localized, and automatically possess finite moments of very high order, even if the initial datum is badly localized.48 On the other hand, if the collision kernel decreases at infinity, then a slow decay at infinity should be preserved as time goes by. This effect is best illustrated by the dichotomy between hard and soft potentials. In certain situations, it has been proven that for hard potentials, no matter how slowly the initial datum decays, then at later times the solution has finite moments of all orders. For soft potentials on the other hand, a badly localized initial datum leads to a badly localized solution at later times. Precise statements and references will be given in Chapter 2B. As a consequence, it is also expected that the trend to equilibrium be faster for hard potentials than for soft potentials. We shall come back to this in Chapter 2C. Regularization effects. We now turn to the smoothness issue. Two basic questions are in order: if the initial datum is smooth, does it imply that the solution remains smooth? If the initial datum is nonsmooth, can however the solution become smooth? This time the answer seems strongly dependent on the angular collision kernel. If the angular collision kernel is integrable (Grad’s cut-off assumption), then one expects that smoothness and 47 In particular, it is not clear for us whether specular reflection in a non-convex domain would not entail
appearance of singularities. The troubles caused by non-convex domains have been well-studied in linear transport theory, but not, to our knowledge, in the context of the Boltzmann equation. 48 See the heuristic explanations in Section 2.2 of Chapter 2B.
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lack of smoothness are propagated in time. In other words, the solution at positive times should have precisely the same smoothness as the initial datum. The understanding of this property has made very significant progress in the past years. On the other hand, when the collision kernel presents a nonintegrable singularity, then the solution should become infinitely smooth for positive times, as it does for solutions to the heat equation. This idea emerged only in the last few years, and its very first mathematical implementation was done by Desvillettes [171] for a one-dimensional caricature of the Boltzmann equation. Since then this area has been very active, and nowadays the regularization effect begins to be very well understood as well. Yet much more is to be expected in this direction. Effects of kinetic singularity. If we summarize our classification of collision kernels, there are only three situations in which they can become very large: small deviation angles (as illustrated by the cut-off vs. non-cutoff assumption), large relative velocities (as illustrated by hard vs. soft potentials), and small relative velocities (as illustrated by hard vs. soft potential, but in the reverse way). If the influence of the former two is now fairly well understood, it is not so at all for the latter. It is known that a singularity of the collision kernel at small relative velocities is compatible with propagation of some smoothness, but no one knows if it preserves all the smoothness or if it entails blow-up effects in certain norms, or conversely regularization phenomena. In Chapter 2E, we shall say a little bit more on this issue, so far inexistent, and which we believe may lead to very interesting developments in the future.
4. Basic surgery tools for the Boltzmann operator Here we describe some of the most basic, but most important tools which one often needs for a fine study of the Boltzmann operator, dv∗ dσ B(v − v∗ , σ )(f f∗ − ff∗ ). Q(f, f ) = RN
S N−1
Later on, we shall describe more sophisticated ingredients which apply in specific situations.
4.1. Symmetrization of the collision kernel In view of formulas (5), the quantity f f∗ − ff∗ is invariant under the change of variables σ → −σ . Thus one can replace (from the very beginning, if necessary) B by its “symmetrized” version B(z, σ ) = B(z, σ ) + B(z, −σ ) 1z·σ >0 .
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In other words, one can always assume the deviation angle θ to be at most π/2 in absolute value. This is why all spherical integrals could be written with an angular variable going from 0 to π/2, instead of π . From the physical point of view, this constatation rests on the undiscernability of particles (and this principle does not hold for mixtures). From the mathematical point of view, this trick is very cheap, but quite convenient when one wants to get rid of frontal collisions (deviation angle close to π , which almost amounts to an exchange between the velocities).
4.2. Symmetric and asymmetric point of view There are (at least) two entirely different ways to look at the Boltzmann operator Q(f, f ). The first is the symmetric point of view: the important object is the “tensor product” ff∗ = f ⊗ f , and the Boltzmann operator is obtained by integrating (f ⊗ f )(v , v∗ ) − (f ⊗ f )(v, v∗ ) with respect to the variable v∗ and the parameter σ . This point of view is often the most efficient in problems which have to do with the trend to equilibrium, because the H theorem rests on this symmetry. On the other hand, one can consider Q(f, f ) as the action upon f of a linear operator which depends on f : Q(f, f ) = Lf (f ). This introduces an asymmetry between f∗ (defining the operator) and f (the object on which the operator acts). This point of view turns out to be almost always the most effective in a priori estimates on the Boltzmann equation. For many asymmetric estimates, it is important, be it for the clarity of proofs or for the methodology, to work with the bilinear (but not symmetric!) Boltzmann operator Q(g, f ) =
RN
dv∗
S N−1
dσ B(v − v∗ , σ )(g∗ f − g∗ f ).
(68)
Note that we have reversed the natural order of the arguments to make it clear that Q(g, f ) should be understood as Lg (f ) . . . .
4.3. Differentiation of the collision operator The following simple identities were proven in Villani [445] (but certainly someone had noticed them before): ∇Q± (g, f ) = Q± (∇g, f ) + Q± (g, ∇f ).
(69)
These formulas enable one to differentiate the collision operator at arbitrary order via a Leibniz-type formula.
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4.4. Joint convexity of the entropy dissipation Remarkably, Boltzmann’s entropy dissipation functional D(f ) =
1 4
dv dv∗ dσ B(v − v∗ , σ )(f f∗ − ff∗ ) log
f f∗ ff∗
is a convex functional of the tensor product ff∗ – but not a convex functional of f ! This property also holds for Landau’s entropy dissipation, which can be rewritten as DL (f ) =
1 2
|Π(v − v∗ )(∇ − ∇∗ )(ff∗ )|2 , dv dv∗ Ψ |v − v∗ | ff∗
so that convexity of DL results from convexity of the function (x, y) → |x|2 /y in RN × R+ . Such convexity properties may be very interesting in the study of some weak limit process, because weak convergence is preserved by tensor product. But beware! ff∗ is a tensor product only with respect to the velocity variable, not with respect to the x variable.
4.5. Pre-postcollisional change of variables A universal tool in the Boltzmann theory is the involutive change of variables with unit Jacobian49 (v, v∗ , σ ) → (v , v∗ , k),
(70)
where k is the unit vector along v − v∗ , k=
v − v∗ . |v − v∗ |
Since σ = (v − v∗ )/|v − v∗ |, the change of variables (70) formally amounts to the exchange of (v, v∗ ) and (v , v∗ ). As a consequence, under suitable integrability conditions on the measurable function F , F (v, v∗ , v , v∗ )B |v − v∗ |, k · σ dv dv∗ dσ = =
F (v, v∗ , v , v∗ )B |v − v∗ |, k · σ dv dv∗ dk F (v , v∗ , v, v∗ )B |v − v∗ |, k · σ dv dv∗ dσ.
49 A way to see that this change of variables has unit Jacobian is to use the ω-representation of Section 4.6. In this representation, very clearly the pre-postcollisional change of variables has unit Jacobian; and the Jacobian from the σ -representation to the ω-representation is the same for pre-collisional and for post-collisional velocities.
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Here we have used |v − v∗ | = |v − v∗ |, σ · k = k · σ to keep the arguments of B unchanged; also recall the abuse of notations B(v − v∗ , σ ) = B(|v − v∗ |, k · σ ). Note that the change of variables (v, v∗ ) → (v , v∗ ) for given σ is illegal!
4.6. Alternative representations There are other possible parametrizations of pre- and post-collisional velocities. A very popular one is the ω-representation, v = v − v − v∗ , ω ω,
v∗ = v∗ + v − v∗ , ω ω,
ω ∈ S N−1 .
(71)
In this representation, the bilinear collision operator50 reads Q(g, f ) =
1 2
− v∗ , ω)(g∗ f − g∗ f ), dv∗ dω B(v
(72)
where N−2 z 1 , ω B(z, σ ). B(z, ω) = 2 2 |z| to recall that each pair (v , v∗ ) corresponds to two We have kept a factor 1/2 in front of B distinct values of ω. One of the advantages of the ω-representation is that it is possible to change variables (v, v∗ ) ↔ (v , v∗ ) for fixed ω and this is again an involutive transformation with unit Jacobian. Another advantage is that it is a linear change of variables. Yet, as soon as one is interested in fine questions where the symmetries of the Boltzmann operator play an important role, the σ -representation is usually more convenient. A third representation is the one introduced by Carleman [119], particularly useful for the study of the gain operator Q+ when the collision kernel satisfies Grad’s angular cutoff. The principle of Carleman’s representation is to choose as new variables v and v∗ , the pre-collisional velocities. Of course, not all values of v and v∗ are admissible. If v and v are given, then the set of admissible velocities v∗ is the hyperplane Evv , orthogonal to v − v and going through v. Using the identity v − v∗ = 2v − v − v∗ , one gets Q(g, f ) dv = RN
Evv
1 v − v∗ B 2v − v − v , ∗ |v − v |N−1 |v − v∗ | × g(v∗ )f (v ) − g(v + v∗ − v)f (v) . dv∗
(73)
50 All the representations formulas below for Q also work just the same for its gain and loss terms (Q+ and Q− ) separately, with obvious changes.
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To conclude this section, we mention Tanaka’s representation [415], which is equivalent to Maxwell’s weak formulation: + − Πv,v∗ ), (74) Q (g, f ) = dv dv∗ g∗ f (Πv,v ∗ where Πv,v∗ (resp. Πv,v ) is a measure on the sphere S N−1 , Πv,v∗ = B(v − v∗ , σ ) dσ δv ∗ (resp. Πv,v∗ = B(v − v∗ , σ ) dσ δv ).
4.7. Monotonicity Each time one has to handle an expression involving a nonnegative integrand and the collision kernel, it may be useful to consider it as a monotone function of the collision kernel. This point of view is particularly interesting for the entropy dissipation (47), which obviously is an increasing function of the collision kernel. Therefore, to bound (47) from below for a given collision kernel B, it is sufficient to bound it below for an auxiliary, simplified collision kernel B0 such that B B0 . Most of the time, the “simplified” collision kernel will be a Maxwellian one. As we shall see in Chapter 2D, Maxwellian collision kernel have specific properties.
4.8. Bobylev’s identities We now turn to more intricate tools introduced by Bobylev. Even though the Boltzmann operator has a nice weak formulation (Maxwell’s formula in Section 2.3), it is a priori quite painful to find out a representation in Fourier space. It turns out that such a representation is not so intricate, at least when the collision kernel is Maxwellian! This fact was first brought to the attention of the mathematical community by Bobylev, who was to make Fourier transform an extremely powerful tool in the study of the Boltzmann operator with Maxwellian collision kernel (see the review in [79]). Here is Bobylev’s identity: let b(cos θ ) be a collision kernel depending only on the cosine of the deviation angle, and let dv∗ dσ b(cos θ )[f f∗ − ff∗ ] Q(f, f ) = RN ×S N−1
be the associated Boltzmann operator. Then its Fourier transform is F Q(g, f ) (ξ ) =
S N−1
g(ξ ˆ − )fˆ(ξ + ) − g(0) ˆ fˆ(ξ ) b
ξ · σ dσ, |ξ |
(75)
where fˆ stands for the Fourier transform of f , ξ is the Fourier variable, and ξ± =
ξ ± |ξ |σ . 2
(76)
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Note that |ξ + |2 + |ξ − |2 = 1. A remarkable feature about (75) is that the integral is now (N − 1)-fold, instead of (2N − 1)-fold. This formula is actually a particular case of a more general one which does not assume Maxwellian collision kernel [10, Appendix]: F Q(g, f ) (ξ ) =
1 (2π)N/2
− # |ξ∗ |, ξ · σ g(ξ ˆ B + ξ∗ )fˆ(ξ + − ξ∗ ) |ξ | RN ×S N−1 (77) − g(ξ ˆ ∗ )fˆ(ξ − ξ∗ ) dξ∗ dσ,
# of B ≡ B(|z|, cos θ ) is with respect to the variable z where the Fourier transform B # ∗ |, cos θ ) = only. Of course, in the particular case B(|z|, cos θ ) = b(cos θ ), we have B(|ξ N/2 (2π) δ[ξ∗ = 0] b(cos θ ), and this entails formula (75). Thus we see that the reduction of the multiplicity in the integral is directly linked to the assumption of Maxwellian collision kernel. As a consequence of (75), results Bobylev’s lemma51: if Q is a Boltzmann operator with Maxwellian collision kernel, then, whatever the Maxwellian probability distribution M, Q(g ∗ M, f ∗ M) = Q(g, f ) ∗ M. This is a very useful regularization lemma when dealing with Maxwellian collision kernels.
4.9. Application of Fourier transform to spectral schemes Here we digress a little bit to briefly discuss numerical schemes based on Fourier transform, which are related to Bobylev’s ideas. Here are the main ideas of these “spectral schemes”: (1) truncate the support of the distribution function f , then extend f into a periodic function on RN ; (2) expand f in Fourier series, and compute the expression of the collision operator Q(f, f ) in terms of the Fourier coefficients of f . Special attention must be given to the way the support is truncated! As explained in [372], for instance, if the support of f is reduced to a compact √ set with diameter R, then it should be extended by periodicity with period T (2 + 2)R, in order to avoid overlap problems in the √ computation of the collision integral. Assume, for instance, T = π , R = λπ , λ = 2/(3 + 2). After passing in Fourier representation, fˆk =
1 (2π)N
[−π,π]N
f (v)e−ik·v dv,
51 This lemma was actually proven, for a constant collision kernel, by Morgenstern [351, Section 10] in the fifties! However, Bobylev was the author who devised a general proof, made this lemma widely known and linked it to other properties of Maxwellian collision kernels.
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and truncating high Fourier modes, a very simple expression is obtained for the k-th mode # of Q(f, f ): Q(k)
# = Q(k)
ˆ m), fˆ fˆm β( ,
|k| K,
+m=k;| |,|m|K
where ˆ m) = β( ,
|z|2λπ
z± =
dz S N−1
+ − dσ B(z, σ ) ei ·z +im·z − eiz·m ,
z ± |z|σ . 2
(78)
Of course, this formula is very much reminiscent of (77). Spectral schemes have several advantages: once in Fourier space, the numerical simulation is immediate (just a simple ˆ m) can be computed first-order system of coupled ODE’s). Moreover, all coefficients β( , once for all with extreme precision; this may demand some memory space, but is quite a gain in the speed of the computation. They are able to give extremely precise results. However, these schemes are rather rigid and do not allow for all the numerical tricks which can be used by other methods to reduce computational time. Moreover, since conservation laws are not built in the schemes, numerical simulations have to be conducted with a certain minimal precision in order to get realistic results. From the mathematical point of view, this method presents very interesting features, think that it simulates the Boltzmann equation in weak formulation. In particular, the method works just the same with or without Grad’s angular cut-off assumption. This can be used in theoretical works like [374] (analysis of the behavior of the method in the asymptotics of grazing collisions).
5. Mathematical theories for the Cauchy problem In this section, we try to survey existing mathematical frameworks dealing with the Cauchy problem (and also more quantitative issues) for Boltzmann-like equations. These theories are all connected, but more or less tightly. Before this, in the first section we describe the most apparent problems in trying to construct a general, good theory.
5.1. What minimal functional space? In the full, general situation, known a priori estimates for the Boltzmann equation are only those which are associated to the basic physical laws: • formal conservation of mass and energy, • formal decrease of the entropy. Of course, the latter means two estimates! an estimate on the entropy, and another one on the dissipation of entropy.
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When the position space is unbounded, say the whole of RN , then the properties of operator also make it rather easy to get local (in time) estimates on the transport f (t, x, v)|x|2 dx dv. For this it suffices to use the identity d dt
f (t, x, v)|x − vt|2 dx dv = 0.
On the whole, one gets for free the a priori estimates RN ×RN
+
f (t, x, v) 1 + |v|2 + |x|2 + log f (t, x, v) dx dv
t 0
RN x
RN ×RN
D f (τ, x, ·) dτ dx
f (0, x, v) 1 + 2|x|2 + 2t 2 + 1 |v|2 + log f (0, x, v) dx dv. (79)
Apart from the term in D(f ), estimate (79) does not depend on the collision kernel B. Disregarding this entropy dissipation estimate which is difficult to translate in terms of size and smoothness of the distribution function, all that we know for the moment about f is N N N f ∈ L∞ [0, T ]; L12 RN x × Rv ∩ L log L Rx × Rv . Here f L1 = 2
RN ×RN
f (x, v) 1 + |x|2 + |v|2 dx dv.
It is easily seen that the estimate in L log L, combined with the moment estimate, entails an L1 control of f (log f )+ . R EMARK . One may be interested in situations where the total mass is infinite (gas in the whole space). The estimates can then be adapted to this situation: see in particular Lions [310]. By the Dunford–Pettis compactness criterion, sequences of solutions to the Boltzmann equation with a uniform entropy estimate will be weakly compact, say in N Lp ([0, T ]; L1 (RN x × Rv )), and this ensures that their cluster points cannot be singular measures. This looks like a mathematical convenience: after all, why not try to handle the Boltzmann equation in a genuine measure setting, which seems to best reflect physical intuition? It turns out that the space of measures is not stable52 – and thus irrelevant – for the study of the Boltzmann equation. This can be seen by the following remark. Consider 52 On the other hand, the space of functions with bounded entropy is stable, as shown by the works of DiPerna and Lions discussed below.
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as initial datum a linear combination of Dirac masses approximating a continuous profile (think of each of these Dirac masses as clusters of particles with a common velocity, each cluster being located at a different position). Even though the meaning of the Boltzmann equation for so singular data is not a priori clear, it is easy to figure out what weak solution we should expect: each cluster should keep moving with constant velocity, until two clusters happen to be in the same position and collide. Should this eventuality occur, particles in each cluster would be scattered in all directions, with respective probabilities given by the Boltzmann collision kernel. But the point is, by modifying very slightly the initial positions of the clusters, one can always make sure (in dimension N 2) that no collision ever occurs! Therefore, this example shows that one can construct a sequence of weak solutions of the Boltzmann equation, converging in weak measure sense towards a continuous solution of the free transport equation. Thus there is no stability in measure space . . . . It would be tedious, but quite interesting, to extend this counterexample to absolutely continuous initial data, say linear combinations of very sharply peaked Maxwellians. The aforegoing discussion is strongly linked to the fact that both x and v variables are present in the Boltzmann equation. If one is interested in solutions which do not depend on the space variable, then it is perfectly possible to construct a meaningful theory of measure solutions, as first noticed by Povzner [389]. To come back to our original discussion, we have just seen that it is in general impossible to deal with arbitrary measures, and therefore the entropy estimate is welcome to prevent concentration. But is it enough to handle the Boltzmann operator? Very very roughly, it seems that to control the Boltzmann operator, we would like to have a control of dv∗ ff∗ , which is just f L1v f . The L1x,v norm of this quantity is just the norm of f in L2x (L1v ). This is the kind of spaces in which we would like to have estimates on f . But such estimates are a major open problem! It seems that the entropy dissipation estimate is not sufficient for that purpose, one of the troubles being caused by local Maxwellian states, which make the entropy dissipation vanish but can have arbitrarily large macroscopic densities. Thus the only a priori estimates which seem to hold in full generality do not even allow us to give a meaningful sense to the equation we wish to study . . . this major obstruction is one of the reasons why the Cauchy problem for the Boltzmann equation is so tricky – another reason being the intricate nature of the Boltzmann operator. Due to this difficulty, most theories only deal with more or less simplified situations. On the other hand, for simpler models like the BGK equation, in which the collision operator is homogeneous of degree 1, the Cauchy problem is much easier [375,378,398]. By the way, the lack of a priori estimates for the Boltzmann equation is also very cumbersome when dealing with boundary conditions, especially in the case of Maxwellian diffusion. Indeed, the treatment of boundary conditions in a classical sense requires the trace of the solution to be well-defined on the boundary, which is not trivial [271,149]. Weak formulations are available (see in particular Mischler [347]), but rather delicate. We distinguish six main theories, inequally represented and inequally active – and by no means hermetically independent, especially in recent research. By order of historical appearance: spatially homogeneous theory, theory of Maxwellian molecules, perturbative theory, solutions in the small, renormalized solutions, one-dimensional problems.
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5.2. The spatially homogeneous theory In the spatially homogeneous theory one is interested in solutions f (t, x, v) which do not depend on the x variable. This approach is rather common in physics, when it comes to problems centered on the collision operator: this looks reasonable, since the collision operator only acts on the velocity dependence. Moreover, the spatially homogeneous theory naturally arises in numerical analysis, since all numerical schemes achieve a splitting of the transport operator and the collision operator. Finally, it is expected that spatial homogeneity is a stable property, in the sense that a weakly inhomogeneous initial datum leads to a weakly inhomogeneous solution of the Boltzmann equation. Under some ad hoc smallness assumptions, this guess has been mathematically justified by Arkeryd et al. [32], who developed a theory for weakly inhomogeneous solutions of the full Boltzmann equation. Thus, the spatially homogeneous Boltzmann equation reads ∂f = Q(f, f ), ∂t
(80)
and the unknown f = f (t, v) is defined on R+ × RN . Note that in this situation, the only stationary states are Maxwellian distributions, by the discussion about Equation (54) in Section 2.5. The spatially homogeneous theory was the very first to be developed, thanks to the pioneering works by Carleman [118] in the thirties. Carleman proved existence and uniqueness of a solution to the spatially homogeneous problem for a gas of hard spheres, and an initial datum f0 which was assumed to be radially symmetric, continuous and decaying in O(1/|v|6 ) as |v| → +∞. He was also able to prove Boltzmann’s H theorem, and convergence towards equilibrium in large time. Later he improved his results and introduced new techniques in his famous treatise [119]. Then in the sixties, Povzner [389] extended the mathematical framework of Carleman and relaxed the assumptions. In the past twenty years, the theory of the Cauchy problem for spatially homogeneous Boltzmann equation for hard potentials with angular cut-off was completely rewritten and extensively developed, first by Arkeryd, then by DiBlasio, Elmroth, Gustafsson, Desvillettes, Wennberg, A. Pulvirenti, Mischler [17,20,23,186,187,204,269,270,170,458, 456,457,392,393,349]. It is now in a very mature state, with statements of existence, uniqueness, propagation of smoothness, moments, positivity . . . . Optimal conditions for existence and uniqueness have been identified by Mischler and Wennberg [349,463]. Recent works by Toscani and the author [428] have led to almost satisfactory results about the H theorem and trend to equilibrium, even though some questions remain unsettled. Also, one can work in a genuine measure framework. The study of singular kernels (be it soft potentials or potentials with nonintegrable angular singularities) is much more recent [18,172,171,173,446,449,10]. This area is still under construction, but currently very active. In view of the last advances, it is quite likely that very soon, we shall have a fairly complete picture of the spatially homogeneous theory, with or without cut-off, at least when the kinetic collision kernel is not too singular for small relative velocities. A review on the state of the art for the spatially homogeneous theory is performed in Desvillettes [174].
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A task which should be undertaken is to systematically extend all of these achievements to the framework of weakly inhomogeneous solutions [32].
5.3. Maxwellian molecules After spatial homogeneity, a further simplification is the assumption that the collision kernel be Maxwellian, i.e., do not depend on the relative velocity but only on the (cosine of the) deviation angle. The corresponding theory is of course a particular case of the preceding one, but allows for a finer description and presents specific features. It was first developed by Wild and Morgenstern in the fifties [464,350,351]. Then Truesdell [274] showed that all spherical moments of the solutions could be “explicitly” computed. Simple explicit solutions, important for numerical simulations, were produced independently by Bobylev [78] and Krook and Wu. Later, Bobylev set up and completed an ambitious program based on the Fourier transform (see the survey in [79]). Included were the classification of several families of semi-explicit solutions, and a fine study of trend to equilibrium. Key tools in this program were the identities of Section 4.8. As of now, the theory can be considered as complete, with the exception of some non-standard problems like the identification of the image of Q or the classification of all eternal solutions53 . . . .
5.4. Perturbation theory Another regime which has been extensively studied is the case when the distribution function is assumed to be very close to a global Maxwellian equilibrium, say the centered, unit-temperature Maxwellian M. Under this assumption, it is natural to try to linearize the problem, in such a way that quadratic terms in Boltzmann’s operator become negligible. In order to have a self-adjoint linearized Boltzmann operator, the relevant change of unknown is f = M(1 + h). Then, one can expand the Boltzmann operator Q(f, f ) by using its bilinearity, and the identity Q(M, M) = 0. This leads to the definition of the linearized Boltzmann operator, Lh = M −1 Q(M, Mh) + Q(Mh, M) . L is a symmetric, nonpositive operator in L2 (M) (endowed with the scalar product (h1 , h2 )M = h1 h2 M), nonpositivity being nothing but the linearized version of the H theorem. Moreover, the remainder in the nonlinear Boltzmann operator, Q(Mh, Mh), can be considered small if h is very close to 0 in an appropriate sense. Note that smallness of h in L2 (M) really amounts to smallness of f − M in L2 (M −1 ), which, by the standards of all other existing theories, is an extremely strong assumption! The linearized Boltzmann equations also have their own interest, of course, and the spectral properties of the linearized Boltzmann operator have been addressed carefully; 53 These problems have been explained in Section 2.9.
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see [141] for discussion and references. The different analysis required by the linearized Landau equation is performed in [161,298]. A remarkable feature is that when the collision kernel is Maxwellian, then the spectrum can be computed explicitly (and eigenfunctions too: they are Hermite polynomials). This calculation was first performed in a classical paper by Wang Chang and Uhlenbeck [454], then simplified by Bobylev [79, p. 135] thanks to the use of Fourier transform. Grad has set up the foundations for a systematic study of the linearized Boltzmann equation, see [250,252,255]. At the same time he initiated in this perturbative framework a “rigorous” study of hydrodynamical equations based on Chapman–Enskog or moments expansion [249,253]. Later came the pioneering works of Ukai [434,435] on the nonlinear perturbation [434,435], followed by a huge literature, among which we quote [333,359, 111,437,403,436,286]. This theory is now in an advanced stage, with existence and uniqueness theorems, and results of trend to equilibrium. The proofs often rely on the theory of linear operators, abstract theory of semigroups, abstract Cauchy–Kowalewski theorems. As we just said, with the development of this branch of the Cauchy problem for the Boltzmann equation, came the first rigorous discussions on the transition to hydrodynamical equations, in a perturbative framework, after a precise spectral analysis of the linearized operator was performed. On this approach we quote, for instance, [112,113, 159,203,358,438,44,334,58,43,212,213]. Actually, in the above we have mixed references dealing with stationary and with evolutionary problems, for which the setting is rather similar . . . . An up-to-date account of the present theory can be found in [214]. However, all known results in this direction deal with smooth solutions of the hydrodynamic equations. Also related to this linearized setting is a large literature addressing more qualitative issues, like half-space problems, to be understood as a modelization for kinetic layers,54 or the description of simple shock waves. Among many works, here are a few such papers: [42,357,115,258,52,51,332,155,151,188,214] (the work [37] is an exception, in the sense that it deals with the Milne problem in the fully nonlinear case). It is certainly mathematically and physically justified to work in a perturbative setting, as long as one keeps in mind that this only covers situations where the distribution function is extremely close to equilibrium. Thus this theory can in no way be considered as a general answer to the Cauchy problem. Even taking this into account, some criticisms can be formulated, for instance, the use of abstract spectral theory which often leads to nonexplicit results. Also, we note that the great majority of these works is only concerned with hard potentials with cut-off, and most especially hard spheres. Early (confusing) remarks on the Boltzmann operator without cutoff can be found in Pao [369] and Klaus [288], but this is all. Soft potentials with cut-off have been studied by Caflisch [111], Ukai and Asano [437]. 54 A kinetic layer describes the transition between a domain of space where a hydrodynamic description is relevant, and another domain in which a more precise kinetic description is in order; for general background see [148].
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Theoretical research in the area of linearized or perturbative Boltzmann equation is not so intensive as it used to be. We shall not come back to this theory and for more details we address the reader to the aforementioned references.
5.5. Theories in the small Another line of approach deals with short-time results. This may seem awkward for statistical equations which are mostly interesting in the long-time limit! but may be interesting when it comes to validation issues. Conversely – but this is more delicate –, it is possible to trade the assumption of small time for an assumption of small initial datum expanding in the vacuum. This case could be described as perturbation of the vacuum, and exploits the good “dispersive” properties of the transport operator. The modern approach starts with the classical papers of Kaniel and Shinbrot for the small-time result [285], Illner and Shinbrot for the small-datum result [278]. Also it should be clear that Lanford’s theorem55 contains a proof of local in time existence under rather stringent assumptions on the initial datum. It was computed that Lanford’s bounds allowed about 15% of the particles to collide at least once! The extensions of Lanford’s results by Illner and Pulvirenti [275,276] were also adaptations of small-datum existence theorems. Theories in the small were further developed by Toscani and Bellomo [64,425,418– 420,368], some early mistakes being corrected by Polewczak [385] who also proved smoothness in the x variable on this occasion. See the book of Bellomo, Palczewski and Toscani [63] for a featured survey of known techniques at the end of the eighties. Then the results were improved by Goudon [246,247] (introducing some new monotonicity ideas), and also Mischler and Perthame [348] in the context of solutions with infinite total kinetic energy. One of the main ideas is that if the initial datum is bounded from above by a well-chosen Maxwellian (or squeezed between two Maxwellians), then this property remains true for all times, by some monotonicity argument. Therefore, solutions built by these methods usually satisfy Gaussian-type bounds. To get an idea of the method, a very pedagogical reference is the short proof in Lions [308] which covers smooth, fast-decaying collision kernels. Bellomo and Toscani have also studied cases where the decay of the solution is not Gaussian, but only polynomial. It is in this framework that Toscani [419] was able to construct solutions of the Boltzmann equation in the whole space, which do not approach local equilibrium as time becomes large.56 From the mathematical point of view, these theories cannot really be considered in a mature stage, due to a certain rigidity. For instance, it is apparently an open problem to treat boundary conditions: only the whole space seems to be allowed. Also, since the Note added in proof : After completion of this review, two important works by Guo, about the Landau equation and the Boltzmann equation for soft potentials, in a close-to-equilibrium, periodic setting, popped up just to contradict this statement. 55 See Section 2.1. 56 See Section 2.5.
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proofs strongly rely on Grad’s splitting between the gain and loss part, the treatment of non-cutoff potentials is open. Finally, the limitations of small time, or small initial datum, are quite restrictive, even though not so much as for the linearized theory (the bounds here have the noticeable advantage to be explicit). However, as mentioned above, theories in the small have led to two quite interesting, and physically controversial, results: the validation of the Boltzmann equation for hard-spheres via the Boltzmann–Grad limit, at least in certain cases; and a simple construction of solutions of the Boltzmann equation which do not approach local equilibrium as time becomes large. Recent works by Boudin and Desvillettes [101] in this framework also resulted in interesting proofs of propagation of regularity and singularities, which were implicitly conjectured by physicists [148]. We shall not develop further on this line of approach, and address the reader to the above references for more.
5.6. The theory of renormalized solutions Introduced by DiPerna and Lions at the end of the eighties, this theory is at the moment the only framework where existence results for the full Boltzmann equation, without simplifying assumptions, can be proven [190,192,194,167,307–309,311,310,306,316,12, 13]. Apart from a high technical level, this theory mainly relies on two ingredients: • the velocity-averaging lemmas: under appropriate conditions, these lemmas, initiated by Golse, Perthame and Sentis [243] and further developed in [242,232,195,379, 99,312,315], yield partial regularity (or rather regularization) results for velocityaverages g(t, x, v)ϕ(v) dv of solutions of transport equations ∂g/∂t + v · ∇x g = S. The regularity is of course with respect to the time-space variables, and thus the physical meaning of these lemmas is that macroscopic observables enjoy some smoothness properties even when the distribution function itself does not. • the renormalization: this trick allows one to give a distributional sense to the Boltzmann equation even though there does not seem to be enough a priori estimates for that. It consists in formally multiplying (8) by the nonlinear function of the density, β (f ), where β belongs to a well-chosen class of admissible nonlinearities. By chainrule, the resulting equation reads ∂β(f ) + v · ∇x β(f ) = β (f )Q(f, f ). ∂t
(81)
Assume now that |β (f )| C/(1 + f ), for some C > 0. Then, since the Boltzmann operator Q(f, f ) is quadratic, one may expect β (f )Q(f, f ) to be a sublinear operator of f . . . in which case the basic a priori estimates of mass, energy and entropy57 would be enough to make sense of (81). Distribution functions satisfying (81) in distributional sense are called renormalized solutions. Strictly speaking, these solutions are neither weaker, nor stronger than distributional solutions. Typical choices for β are β(f ) = δ −1 log(1 + δf ) (δ > 0) or β(f ) = f/(1 + δf ). 57 See Section 5.1.
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Apart from the study of the Boltzmann equation, renormalization and velocity-averaging lemmas have become popular tools for the study of various kinetic equations [191, 375,318,176,229], ordinary differential equations with nonsmooth (Sobolev-regular) coefficients [193,97], or the reformulation of some hyperbolic systems of conservation laws as kinetic systems [319,320,280,381,380]. The idea of renormalization has even been exported to such areas of partial differential equations as nonlinear parabolic equations (see [76,77] and the references therein). In fact, renormalization is a general tool which can be applied outside the field of renormalized solutions; in this respect see the remark at the level of formula (130). As regards the Boltzmann equation, many fundamental questions are still unsolved: in particular uniqueness, propagation of smoothness, energy conservation, moment estimates, positivity, trend to equilibrium . . . . Therefore, as of this date, this theory cannot be considered as a satisfactory answer to the Cauchy problem. However, it provides a remarkable answer to the stability problem. The techniques are robust enough to adapt to boundary-value problems [271,144, 30,347,346] (be careful that some of the proofs in [271] are wrong and have been corrected in [30]; the best results are those of Mischler [347]). As an important application of the theory of renormalized solutions, Levermore [302] proved the validity of the linearization approximation if the initial datum is very close to a global Maxwellian. Also the hydrodynamical transition towards some models of fluid mechanics can be justified without assumption of smoothness of the limit hydrodynamic equations: see, in particular, Bardos, Golse and Levermore [57,55,54,53], Golse [239], Golse et al. [241], Golse and Levermore [240], Lions and Masmoudi [317], Golse and Saint-Raymond [245], SaintRaymond [401]. The high point of this program is certainly the rigorous limit from the DiPerna–Lions renormalized solutions to Leray’s weak solutions of the incompressible Navier–Stokes equation, which was performed very recently in [245]; see [441] for a review. The original theory of DiPerna and Lions heavily relied on Grad’s cut-off assumption, but recent progress have extended it to cover the full range of physically realistic collision kernels [12]. This extended theory has set a framework for the study of very general effects of propagation of “regularity”, in the form of propagation of strong compactness [308], or “smoothing”, in the form of appearance of strong compactness [311,316,12,13]. Moreover, even if a uniqueness result is not available, it appears that renormalized solutions are strong enough to prove some results of weak-strong uniqueness [308,324]: under certain assumptions on the collision kernel, if we know that there exists a strong solution to the Boltzmann equation, then there exists a unique renormalized solution, and it coincides with the strong solution. On the occasion of this study, Lions [308] pointed out the possibility to construct very weak solutions, called “dissipative solutions”, which are of very limited physical value, but have been used in various areas as a powerful tool for treating some limit regimes, be it in fluid mechanics for such degenerate equations as the three-dimensional Euler equation [313], in hydrodynamical limits [239,241,401] or stochastic fully nonlinear partial differential equations [321,322]. Thus ramifications of the DiPerna–Lions theory have been a source of inspiration for problems outside the field. Note added in proof : Recently, Bouchut has shown how to use velocity-averaging lemmas to study classical hypoellipticity in certain kinetic equations.
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In view of these achievements and of the current vitality of the theory of renormalized solutions, we shall come back to it in more detail in the next chapters.
5.7. Monodimensional problems It is of course impossible to speak of a monodimensional Boltzmann equation, since elastic collisions are meaningless in dimension 1. But in many problems of modelling [148], symmetry assumptions enable one to consider solutions depending on the position in space, x, through only one variable. From the mathematical point of view, such problems seem to present specific features, one of the reasons being that the dispersive power of the transport operator is very strong in dimension 1, so that dispersion estimates can be used to (almost) control the collision operator. In the end of the eighties, Arkeryd [22] was able to apply a contraction method similar to the one in [24] in order to get existence results for the Boltzmann equation in one dimension of space, however he needed a physically unrealistic damping in the collision operator for small relative velocities in the space direction. Then, building on original works by Beale [61] and especially Bony [94,95] on discrete-velocity Boltzmann equations, Cercignani [145,147] was able to extract some new estimates in this one-dimensional situation, and prove existence of “strong” solutions to the Boltzmann equation, under rather stringent assumptions on the collision kernel. Here “strong” means that Q± (f, f ) ∈ L1loc (Rx × RN v ). For some time this line of research was quite promising, but it now seems to be stalled . . . .
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Cauchy Problem Contents 1. Use of velocity-averaging lemmas . . . . . . . . . . . . . . . . . . . . 1.1. Reminders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. How to use velocity-averaging lemmas in the Boltzmann context? 1.3. Stability/propagation/regularization . . . . . . . . . . . . . . . . . 2. Moment estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Maxwellian collision kernels . . . . . . . . . . . . . . . . . . . . 2.2. Hard potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Soft potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Grad’s cut-off toolbox . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Control of Q+ by Q− and entropy dissipation . . . . . . . . . . 3.3. Dual estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Lions’ theorem: the Q+ regularity . . . . . . . . . . . . . . . . . 3.5. Duhamel formulas and propagation of smoothness . . . . . . . . 3.6. The DiPerna–Lions renormalization . . . . . . . . . . . . . . . . 3.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The singularity-hunter’s toolbox . . . . . . . . . . . . . . . . . . . . . 4.1. Weak formulations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Cancellation lemma . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Entropy dissipation estimates . . . . . . . . . . . . . . . . . . . . 4.4. Boltzmann–Plancherel formula . . . . . . . . . . . . . . . . . . . 4.5. Regularization effects . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Renormalized formulation, or Γ formula . . . . . . . . . . . . . . 4.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The Landau approximation . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Structure of the Landau equation . . . . . . . . . . . . . . . . . . 5.2. Reformulation of the asymptotics of grazing collisions . . . . . . 5.3. Damping of oscillations in the Landau approximation . . . . . . . 5.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Mixing effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The meaning of “Cauchy problem” in this chapter is to be understood in an extended sense: we shall not only be concerned in existence and uniqueness of solutions, but also in a priori estimates. Three main issues will be addressed: decay of the solutions at large velocities (and also at large positions, but large velocities are the main concern), smoothness, and strict positivity. As we explained above, the decay of the solutions mainly depends on the behavior of the kinetic collision kernel, while their smoothness heavily relies on the angular collision kernel. As for the strict positivity, the matter is not very clear yet. We have adopted the following presentation: first, we recall a bit about velocityaveraging lemmas, which have become a universal tool in the study of transport equations, and we shall comment on their use in the particular context of the Boltzmann equation. In Section 2, we address moment estimates, and discuss the influence of the kinetic collision kernel. Then in Section 3, we first enter the core of the study of Boltzmann’s operator, and we discuss issues of propagation of smoothness and propagation of singularities when the angular collision kernel is integrable (Grad’s angular cut-off). Conversely, in Section 4, we explain the structure of Boltzmann’s operator when the angular collision kernel presents a nonintegrable singularity for grazing collisions, and associated theorems of regularization. Since the Landau equation is linked to the Boltzmann equation via the emphasis on grazing collisions, this will lead us to discuss the Landau approximation in Section 5. We conclude in Section 6 with lower bound estimates. In many places the picture is incomplete, especially in the full, spatially inhomogeneous situation. Our discussion is mainly based on a priori estimates. We have chosen not to discuss existence proofs, strictly speaking. Sometimes these proofs follow from the a priori estimates by rather standard PDE arguments (fixed point, monotonicity, compactness), sometimes they are very, very complicated. In any case they are unlikely to be of much interest to the non-specialist reader, and we shall skip them all. Complete proofs of the most famous results can be found in [149]. Also, in this review we insist that a priori estimates should be explicit, but we do not care whether solutions are built by a constructive or non-constructive method. This is because we are mainly concerned with qualitative statements to be made about the solutions, and their physical relevance. If we were more concerned about practical aspects like numerical simulation, then it would be important that existence results be obtained by constructive methods. As a last remark, we note that we have excluded from the discussion all references which include nonstandard analysis [19,21,25] – just because we are not familiar with these techniques.
1. Use of velocity-averaging lemmas 1.1. Reminders Velocity-averaging lemmas express the local smoothness in macroscopic variables (t, x) of averages of the distribution function with respect to the microscopic variable (the velocity).
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Here is a basic, important example: assume that f satisfies ∂f + v · ∇x f = S, ∂t
f ∈ L2t,x,v ,
S ∈ L2t,x Hv−s .
(82)
Then, for any ϕ ∈ C0∞ (RN v ),
1
RN
2(1+s) f (t, x, v)ϕ(v) dv ∈ Ht,x .
Here H α is the Sobolev space of order α, and when we write “∈”, this really means “lies in a bounded subset of”. From the physical point of view, averaging lemmas express the fact that observables (typically, the local density) are smoother than the distribution function f itself. From the mathematical point of view, they are consequences of a “geometric” fact which we shall describe briefly. Consider the Fourier transform of f with respect to the variables t and x, write (τ, ξ ) for the conjugate variables, then (82) becomes (τ + v · ξ )fˆ = # S, so that |fˆ|2 =
|# S|2 . |τ + v · ξ |2
Since the numerator vanishes for well-chosen values of v, this does not tell us much about the decay of fˆ as τ and ξ go to infinity. But when v varies in a compact set of RN , the set of values of v such that τ + v · ξ is small will itself be very small; this is why on the average |fˆ| will decay at infinity faster than |# S|. Many variants are possible, see in particular [242,195]. A pedagogical introduction about velocity-averaging lemmas is provided by Bouchut [96]. Let us make a few comments: 1. The L2 a priori bound for f may be replaced by a Lp bound, p > 1 (then the regularization holds in some W k,q Sobolev space), but not by an L1 bound. Some replacements with L1 estimates can be found, e.g., in Saint-Raymond [400]. 2. It is possible to cover cases in which the right-hand side also lies in a negative Sobolev space with respect to the x variable, provided that the exponent of differentiation be less than 1. Obviously, if the exponent is greater than 1, then the transport operator, which is first-order differential in x, cannot regularize . . . . The case where the exponent is exactly 1 is critical, see Perthame and Souganidis [379]. 3. The above theorem considers time and space variables (t, x) ∈ R × RN , but there are local variants, see in particular [99]. 4. The transport operator v · ∇x may be replaced by a(v) · ∇x under various conditions on a. Note added in proof : See also a recent note by Golse and Saint-Raymond.
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5. Some vector-valued variants show that convolution products of the form f ∗v ϕ are smooth in all t, x, v variables. A remarkable aspect of averaging lemmas is that they do not rely on the explicit solution to the linear transport equation (82) (at least nobody knows how to use the explicit solution for that purpose!). Instead, they are usually based on Fourier transform, or more generally harmonic analysis. There are variants of averaging lemmas which do not lead to smoothness but to a gain p q of integrability, with estimates in Lt (Lx (Lrv )), and sometimes apply in a larger range of exponents. Developed by Castella and Perthame [135] with a view of applications to the Vlasov–Poisson equation, these estimates are analogous to a famous family of inequalities due to Strichartz for the Schrödinger equation. Even though these estimates also give more information about the transport operator (which appears to be much more complex than it would seem!), it is still not very clear what to do with them. A discussion of the links between these Strichartz-like estimates and velocity-averaging lemmas can be found in Bouchut [96]. In the next two sections, we briefly describe the interest of averaging lemmas in the context of the theory of renormalized solutions for the Boltzmann equation.
1.2. How to use velocity-averaging lemmas in the Boltzmann context? No need to say, it would be very useful to get regularity results on averages of solutions of the Boltzmann equation. Since the Boltzmann collision operator looks a little bit like a convolution operator with respect to the v variable, we could hope to recover partial smoothness for it, etc. However, if we try to rewrite the Boltzmann equation like (82), with S = Q(f, f ), we run into unsurmountable difficulties. First of all, we do not have the slightest a priori estimate on S! Something like integrability would be sufficient, since measures can be looked as elements of negative Sobolev spaces, but even this is not known in general.1 Next, we only have f ∈ L log L, and this seems to be a limit case where averaging lemmas do not apply2 . . . . As pointed out to us by F. Bouchut, L log1+ε L would be feasible, although extremely technical, but for L log L this seems to be linked with deep unsolved questions of harmonic analysis in Hardy spaces. This is the place where the clever DiPerna–Lions renormalization trick will save the game. After rewriting the Boltzmann equation in renormalized formulation, ∂β(f ) + v · ∇x β(f ) = β (f )Q(f, f ), ∂t we see an opportunity to apply averaging lemmas to the function β(f ), which lies, for instance, in L1 ∩ L∞ as soon as β(f ) Cf/(1 + f ). If we take it for granted that we shall find a meaningful definition of β (f )Q(f, f ), i.e., a renormalized formulation of the 1 Except when the x-variable is one-dimensional, see Section 5.7 in Chapter 2A. 2 Note however the following result by Golse and Saint-Raymond [245]. If ∂ f n + v · ∇ f n = S with (f n ) t x n weakly compact in L1 and (Sn ) bounded in L1 , then ϕ(v)f n dv is strongly compact in (t, x).
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−α (RN )), then Boltzmann operator, with bounds like β (f )Q(f, f ) ∈ L1 ([0, T ] × RN x ;H v we shall get a smoothness bound for β(f )ϕ dv, or β(f ) ∗v ϕ . . . . Of course, smoothness for averages of β(f ) does not mean smoothness for averages of f . But since β may be chosen to vary over a large range of admissible nonlinearities, by using approximation arguments combined with the a priori bounds on mass, energy, entropy, it is not difficult to show that averages like f (v) ϕ(v) dv, f ∗ ϕ (ϕ ∈ L∞ ) will lie in a uniformly strongly compact set of L1 . The interest of such approximation arguments is that they are robust and easy, and retain the softest information, which is gain of compactness. However, their nonexplicit nature is one of the drawbacks of the theory, and one can expect that important efforts will be devoted in the future to turn them into quantitative statements (as in [101,176], for instance).
1.3. Stability/propagation/regularization Some of the main principles in the theory of renormalized solutions are (1) as a starting point, try to work with just the basic known a priori estimates of mass, energy, entropy, entropy dissipation, (2) treat the Cauchy problem as a stability problem, and (3) replace smoothness by strong compactness. Point (2) means the following: consider a sequence f n (t, x, v)n∈N of solutions, or approximate solutions, satisfying uniform a priori estimates of mass, energy, entropy, entropy dissipation. Without loss of generality, f n → f weakly in Lp ([0, T ]; L1 (RN x × )), 1 p < ∞. Then one would like to prove that f also solves the Boltzmann RN v equation. As a corollary, this will yield a result of existence and stability of solutions. Since a priori estimates are so poor, this is a very bad, unsatisfactory existence result. But for the same reason, this is an extremely good stability result. Now for point (3), it consists in replacing the statement “f is a smooth function”, which is meaningless in a framework where so little information is available, by the statement “f n lies in a strongly compact set of L1 ”. For instance, • “smoothness propagates in time” is replaced by “if f n (0, ·, ·) lies in a strongly N n compact set of L1 (RN x × Rv ), then for all t > 0, so does f (t, ·, ·)”; n • “singularities propagate in time” is replaced by “if f (0, ·, ·) does not lie in a strongly N n compact set of L1 (RN x × Rv ), then for all t > 0, neither does f (t, ·, ·)”; • “there is an immediate regularization of the solution” is replaced by “for (almost all) N t > 0, f n (t, ·, ·) lies in a strongly compact set of L1 (RN x × Rv )”. Note that the second item in the list can be rephrased as “smoothness propagates backwards in time”. One of the nice features of the theory of renormalized solutions is that, with the help of averaging lemmas, these goals can be achieved by a good understanding of the structure of the Boltzmann operator alone. This approach has been developed by Lions, especially in [307] and in [311]. As a typical example, if we suspect some regularization effect due to collisions and wish to prove appearance of strong compactness, then, it essentially suffices to derive some smoothness estimate in the velocity variable, coming from an a priori estimate where the effect of collisions would be properly used,3 together with 3 Most typically, the entropy dissipation estimate.
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a meaningful renormalized formulation. Indeed, the velocity-smoothness estimate would imply that whenever ϕ is an approximation of a Dirac mass, then f n ∗v ϕ should be close to f n , in strong sense. On the other hand, from the use of averaging lemmas one would expect something like: f n ∗ ϕ is “smooth” in t, x, v. Then the strong compactness would follow. This strategy was introduced by Lions [311]. Of course, the technical implementation of these fuzzy considerations turns out to be very intricate. All of the statements of the previous lines are only approximately satisfied: for instance we will not know that f n ∗v ϕ is close to f n , but rather that γ (f n ) ∗v ϕ is close to γ (f n ), and we will not know whether this holds for almost all t, x, but only for those t, x at which the local mass, energy, entropy are not too high, etc. In all the sequel we shall conscientiously wipe out all of these difficulties and address the reader to the references above for details. On the other hand, we shall carefully describe the structure of the Boltzmann operator, its renormalized formulation, and how these properties relate to statements of propagation of smoothness or regularization. 2. Moment estimates Moment estimates are the first and most basic estimate for the Boltzmann equation. Since one wants to control the energy (= second moment), it is natural to ask for bounds on moments higher than 2. In fact, if one wants to rigorously justify the identity Q(f, f )|v|2 dv = 0, RN
and if the kinetic collision kernel in the Boltzmann operator behaves like |v − v∗ |γ , then it is natural to ask for bounds on the moments of order 2 + γ . Of course, once the question of local (in time) estimates is settled, one would like to have information on the long-time behavior of moments. In the spatially homogeneous situation, moment estimates are very well understood, and constitute the first step in the theory. In the case of the full, spatially-inhomogeneous Boltzmann equation there is absolutely no clue of how to get such estimates. This would be a major breakthrough in the theory. As for perturbative theories, they are not really concerned with moment estimates: by construction, solutions have a very strong (typically, Gaussian) decay at infinity. As for the long-time behavior of moments, it is also well controlled in the spatially homogeneous case. In the full setting, even for much simpler, linear variants of the Boltzmann equation, the problem becomes much trickier, and satisfactory answers are only beginning to pop out now. In all the sequel, we shall only discuss the spatially homogeneous situation. The starting point of most estimates [389,204,170,460,349] is the weak formulation Q(f, f )ϕ(v) dv RN v
1 = 2
R2N
dv dv∗ ff∗
S N−1
B |v − v∗ |, cos θ (ϕ + ϕ∗ − ϕ − ϕ∗ ) dσ,
(83)
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applied to ϕ(v) = |v|s , s > 2, or more generally to ψ(|v|2 ), where ψ is an increasing convex function.
2.1. Maxwellian collision kernels The most simple situation is when the collision kernel is Maxwellian. As noticed by Truesdell [274], integral expressions like S N−1 b(cos θ )ϕ(v ) dσ can be explicitly computed when ϕ is a homogeneous polynomial of the velocity variable. As a consequence, the integral in (83) can be expressed in terms of moments of f and angular integrals depending on b. This makes it possible to establish a closed system of differential equations for all “homogeneous” moments. So in principle, the exact values of all moments can be explicitly computed for any time. Then, Truesdell showed that all moments which are bounded at initial time converge exponentially fast to their equilibrium values. Moreover, if some moment is infinite at initial time, then it can never become finite.
2.2. Hard potentials In the case of hard potentials, or more generally when the kinetic collision kernel grows unbounded at infinity, then the solution to the Boltzmann equation is expected to be welllocalized at infinity, even if the decay at initial time is relatively slow. Heuristically, this can be understood as follows: if the collision kernel diverges for very large relative velocity, then very fast particles have a very high probability to collide with rather slow particles, which always constitute the majority of the gas. Thus, these fast particles will certainly be slowed down very quickly. At the level of weak formulations like (83), this means that the “dominant” part will be negative (as soon as ϕ is a convex function of |v|2 ). More precisely, if, say, B(|v − v∗ |, cos θ ) = |v − v∗ |γ b(cos θ ), γ > 0, then, for some constants K > 0, C < +∞ depending only on s, N, γ and b,
RN
Q(f, f )|v|s dv −K
RN
+C
RN
f dv
RN
f |v|s+γ dv
f |v|γ dv
RN
f |v|s dv .
(84)
This inequality is just one example among several possible ones. It easily follows from the Povzner inequalities or their variants, introduced in [389] and made more precise by Elmroth [204], Wennberg [460], Bobylev [85], Lu [328]. Here is a typical Povzner inequality from [328]: for any s > 2, and γ min(s/2, 2), |θ | π/2, |v |s + |v∗ |s − |v|s − |v∗ |s −κs (θ )|v|s + Cs |v|s−γ |v∗ |γ + |v|γ |v∗ |s−γ , where κs (θ ) is an explicit function of θ , strictly positive for 0 < θ π/2.
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Let us now look at the applications to the solutions of the spatially homogeneous Boltzmann equation. As a consequence of (84) and the conservation of mass ( f = 1), Elmroth [204] proved uniform boundedness of all moments which are finite at initial time. This kind of estimates has been simplified and lies at the basis of spatially homogeneous theory. Let us explain the argument without entering into details, or looking for best possible constants. Multiplying the inequality above by |v − v∗ |γ (0 < γ < 2, say), and integrating against the angular collision kernel, one easily gets 1 4
S N−1
b(cos θ )|v − v∗ |γ |v |s + |v∗ |s − |v|s − |v∗ |s dσ
−K|v|s+γ + C |v|s |v∗ |γ + |v∗ |s |v|γ
(85)
(additional terms, like |v|s−γ |v∗ |2γ , are easily absorbed into the last term in the right-hand side by Young’s inequality). Then, let us integrate (85) against ff∗ : after application of Fubini’s identity, we find (84), which can be rewritten as d dt
RN
f (t, v)|v|s dv
−K
RN
(86)
f dv
RN
f |v|
s+γ
dv + C
f |v| dv s
RN
f |v| dv . γ
RN
The last integral is bounded because of the energy bound and γ < 2. Since finds that the s-order moments, Ms (t) = f (t, v)|v|s dv,
f = 1, one
satisfy some system of differential inequalities d Ms −Ks Ms+γ + Cs Ms . dt Now, by Hölder’s inequality and f = 1 again, 1+γ /s
Ms+γ Ms
.
(87)
(88)
Since solutions of dX/dt −CX1+α + CX, α > 0, are uniformly bounded, Elmroth’s theorem follows. Desvillettes [170] pointed out that the conclusion is much stronger: if at least one moment of order s > 2 is finite at initial time, then all moments immediately become finite for positive times – and then of course, remain uniformly bounded as time goes to infinity. The result was further extended by Wennberg [460], Wennberg and Mischler [349], in particular the assumption of finite moment of order s > 2 at initial time can be dispended with. Moreover, these results hold for cut-off or non-cutoff angular collision kernels.
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Bobylev [85] has given a particularly clear discussion of such moment estimates, with various explicit bounds of Ms (t) in terms of Ms (0) and s. A very interesting byproduct of this study was the proof of Gaussian tail estimates. By precise estimates of the the bounds on Ms (t), he was able to prove that if the initial datum satisfies growth of exp(α0 |v|2 ) f0 (v) dv < +∞ for some α0 > 0, then, at least when γ = 1 (hard spheres), there exists some α > 0 such that exp α|v|2 f (t, v) dv < +∞. sup t 0 RN
Anticipating a little bit on precise results for the Cauchy problem, we can say that moment estimates have been a key tool in the race for optimal uniqueness results in the context of hard potentials with cut-off. In fact, progress in this uniqueness problem can be measured by the number of finite moments required for the initial datum: Carleman needed 6, Arkeryd [17] only 4, Sznitman [412] was content with 3, Gustafsson [270] with 2 + γ , Wennberg [458] needed only 2 + ε (ε > 0). Finally, Mischler and Wennberg [349] proved uniqueness under the sole assumption of finite energy. On this occasion they introduced “reversed” forms of Povzner inequalities, which show that the kinetic energy of weak solutions to the Boltzmann equation can only increase or stay constant; hence the uniqueness result holds in the class of weak solutions whose kinetic energy is nonincreasing. More surprisingly, these moment estimates can also be used for proving nonuniqueness results! in the class of weak solutions whose kinetic energy is not necessarily constant, of course. The idea, due to Wennberg [463], is quite simple: consider a sequence (f0n )n∈N of initial data, made up of a Maxwellian (equilibrium) distribution, plus a small bump centered near larger and larger velocities as n → ∞. The bump is chosen in such a way that its contribution to the total mass is negligible as n → ∞, but not its contribution to the kinetic energy; so that the total kinetic energy of f n is, say, twice the energy E of the Maxwellian. For each n, one can solve the corresponding Boltzmann equation with hard potentials, and it has energy 2E. One can check that, as n → ∞, this sequence of solutions converges, up to extraction of a subsequence, to a weak solution of the Boltzmann equation, with Maxwellian initial datum. But, by means of some precise uniform moment bounds, one can prove that for positive times, the kinetic energy passes to the limit:
∀t > 0,
lim
n→∞ RN
f n (t, v)|v|2 dv =
RN
f (t, v)|v|2 dv.
Hence this weak solution f of the Boltzmann equation has energy E at time 0, and energy 2E for any time t > 0, in particular it is not the stationary solution . . . .
2.3. Soft potentials When the kinetic collision kernel decays as |v − v∗ | → ∞, or more generally when it is uniformly bounded, then local in time moment estimates are much easier to get. On
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the other hand, the result is much weaker, since only those moments which are initially bounded, can be bounded at later times. When the collision kernel presents a singularity for zero relative velocity, say |v − v∗ |γ with γ < 0, then additional technical difficulties may arise. When γ −1, and even more when γ < −2, it is not a priori clear that power laws |v|s (or their mollified versions, (1 + |v|2 )s/2 ) are admissible test-functions for the spatially homogeneous Boltzmann equation. This difficulty is overcome for instance by the method in Villani [446], or the remarks in [10]. In all cases, anyway, one proves local in time propagation of moments. But now, due to the decay of the collision kernel at infinity, it becomes considerably more difficult to find good long time estimates. Obtaining polynomial bounds is quite easy, but this is not always a satisfactory answer. The main result on this problem is due to Desvillettes [170]: he showed that when the kinetic collision kernel behaves at infinity like |v − v∗ |γ , −1 < γ < 0, then all those moments which are initially bounded, can be bounded by C(1 + t), as the time t goes to infinity. By interpolation, if the initial datum has a very good decay at infinity (many finite moments), then the growth of “low”-order moments will be very slow. Thus, even if uniform boundedness is not proven, the “escape of moments at infinity” has to be slow. The results of Desvillettes can be extended to the case where γ > −2, though there is no precise reference for that (at the time where Desvillettes proved his result, weak solutions were not known to exist for γ −1). On the other hand, when γ −2, then it is still possible to prove local in time propagation of moments of arbitrary order, but the bounds are in general polynomial and quite bad . . . . In the case of the Landau equation where things are less intricate, one can derive the following estimate [429] when the decay at infinity is like |v − v∗ |γ , −3 γ −2: then the moment of order s grows no faster than O((1 + t)λ ), with λ = (s − 2)/3.
2.4. Summary We now sum up all the preceding discussion in a single theorem. As we said above, for the time being it is only in the spatially homogeneous setting that relevant moment estimates have been obtained. The conditions of the following theorem are enough to guarantee existence of weak solutions, but not necessarily uniqueness. |, cos θ ) = |v −v∗ |γ b(cos θ ) be a collision kernel, −3 γ 1, TπHEOREM 1. Let B(|v −v∗N−2 θ dθ < +∞. Let f0 ∈ L12 (RN v ) be an initial datum with 0 b(cos θ )(1 − cos θ ) sin finite mass and energy, and let f (t,v) be a weak solution of the Boltzmann equation, f (0, ·) = f0 , whose kinetic energy f (t, v)|v|2 dv is nonincreasing. Then this kinetic energy is automatically constant in time. Moreover, if Ms (t) ≡ then,
f (t, v)|v|s dv,
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(i) If γ = 0, then for any s > 2, ∀t > 0,
Ms (t) < +∞ ⇐⇒ Ms (0) < +∞ ,
Ms (0) < +∞ "⇒ sup Ms (t) < +∞. t 0
Moreover, under the sole assumption M2 < +∞, there exists a convex increasing function φ, φ(|v|) → ∞ as |v| → ∞, such that sup t 0
f (t, v)φ |v| |v|2 dv < +∞.
(ii) If γ > 0, then for any s > 2, ∀t0 > 0,
sup Ms (t) < +∞.
t t0
Moreover, if γ = 1, and the initial datum f0 satisfies some α0 > 0, then there exists some α > 0 such that sup t 0
exp(α0 |v|2 ) f0 (v) dv < +∞ for
2
eα|v| f (t, v) dv < +∞.
(iii) If γ < 0, then for any s > 2, ∀t > 0,
Ms (t) < +∞ ⇐⇒ Ms (0) < +∞ .
Moreover, (a) if γ > −2, then Ms (0) < +∞ "⇒ ∃C > 0, Ms (t) C(1 + t); in particular, for any ε > 0, Ms/ε < +∞ "⇒ ∃C > 0, Ms (t) C(1 + t)ε ; (b) if γ < −2, then Ms (0) < +∞ "⇒ ∃C > 0, ∃λ > 0, Ms (t) C(1 + t)λ (λ = 1 for s 4).
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R EMARKS . (1) All the constants in this theorem are explicit. (2) The statement about finiteness of f (t, v)φ(v)|v|2 dv in point (i) is interesting because it implies lim sup
R→∞ t 0 |v|R
f (t, v)|v|2 dv = 0;
in other words, no energy leaks at infinity. Such an estimate is obvious in situation (ii); it is a seemingly difficult open problem4 in situation (iii). (3) The range γ ∈ [−3, 1] has been chosen for convenience; it would be possible to adapt most of the proofs to larger values of γ , maybe at the expense of slight changes in the assumptions. Values of γ which would be less than −3 pose a more challenging problem, but do not correspond to any physical example of interest. The first part of point (i) is due to Truesdell [274], while the statement about point (ii) is mainly due to Desvillettes [170] and improved by Wennberg [458], Wennberg and Mischler [349]; the estimate for exponential moments is due to Bobylev [85]. As for point (iii), it is proven in Desvillettes [170] for γ > −1, and elements of the proof of the rest can be found in [446,444]. For the Landau equation (with Ψ (|v − v∗ |) = K|v − v∗ |γ +2 ) the very same theorem holds, with the following modification: point (ii) is known to hold only if there exists s0 > 2 such that Ms0 (0) < +∞ (see [182]). As for point (iii)(c), the more precise estimate λ = (s −2)/3 holds [429], at least if the collision kernel is replaced by a mollification which decreases at infinity like |v − v∗ |γ , but does not present a singularity for |v − v∗ | 0.
3. The Grad’s cut-off toolbox We now present several tools which are useful to the study of Boltzmann’s equation when the collision kernel satisfies Grad’s angular cut-off assumption. This means at least that whenever |v − v∗ | = 0, A |v − v∗ | ≡
B(v − v∗ , σ ) dσ < +∞.
(89)
Typical examples are |v − v∗ |γ b(cos θ ), where
π
b(cos θ ) sinN−2 θ dθ < +∞.
0
We shall mainly insist on two ingredients: the important Q+ regularity theorem, and the DiPerna–Lions renormalization. 4 See Section 5.3 in Chapter 2C for more, and some results.
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3.1. Splitting When Grad’s assumption holds true, then one can split the Boltzmann collision operator into the so-called “gain” and “loss” terms, and the loss term is then particularly simple. We give this splitting in asymmetric form: Q(g, f ) = Q+ (g, f ) − Q− (g, f ) = Q+ (g, f ) − f (A ∗ g).
(90)
Clearly, the delicate part in the study is to understand well enough the structure of the complicated integral operator dv∗ dσ B(v − v∗ , σ )g(v∗ )f (v ). Q+ (g, f ) = RN
S N−1
As early as in the thirties, this problem led Carleman to the alternative representation
+
Q (g, f ) =
RN
dv
Evv
dv∗
1 v − v∗ B 2v − v − v∗ , g(v∗ )f (v ), |v − v∗ | |v − v |N−1 (91)
with Evv standing for the hyperplane going through v, orthogonal to v − v. In Sections 3.3 and 3.4, we shall expand a little bit on the structure of the Q+ operator. Before that, we give an easy lemma about the control of Q+ by means of the entropy dissipation. 3.2. Control of Q+ by Q− and entropy dissipation Using the elementary identity X KY +
1 X (X − Y ) log , log K Y
K > 1,
with X = f f∗ and Y = ff∗ , we find, after integration against B dv∗ dσ , Q+ (f, f ) KQ− (f, f ) +
4 d(f ), log K
(92)
where d(f ) =
1 4
RN ×S N−1
(f f∗ − ff∗ ) log
f f∗ B dv∗ dσ ff∗
is a nonnegative operator satisfying d(f ) dv = D(f ), the entropy dissipation functional. Inequality (92) was first used by Arkeryd [21], and has proven very useful in the DiPerna–Lions theory [192].
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3.3. Dual estimates Many estimates for Q+ are best performed in dual formulation, with the help of the pre-postcollisional change of variables. For instance, to bound Q+ (g, f ) Lp (RN ) , it is sufficient to bound + Q (g, f )ϕ dv = dv dv∗ g∗ f B |v − v∗ |, cos θ ϕ(v ) dσ RN
R2N
S N−1
uniformly for ϕ Lp 1. So the meaningful object is the linear operator ϕ →
S N−1
B |v − v∗ |, cos θ ϕ(v ) dσ.
(93)
Pushing the method a little bit, one easily arrives at the following abstract result: let X, Y be two Banach spaces of distributions, equipped with a translation-invariant norm. Assume that the linear operator T : ϕ →
v + |v|σ B |v|, cos θ ϕ dσ 2 S N−1
is bounded (as a linear map) from Y to X. Then, the following estimate on Q+ holds, $ $ + $Q (f, g)$ C g 1 f X . L Y Actually, Y (resp. X ) does not really need to be the dual of Y (resp. X), it suffices that Q+ Y = sup{ Q+ g; g Y = 1} (resp. f ψ f X ψ X ). As an example of application of this result, consider the simple situation B(v − v∗ , σ ) = Φ(|v − v∗ |)b(cos θ ), where Φ is bounded and b(cos θ ) sinN−2 θ is integrable with support in [0, π/2]. Obviously, T is bounded L∞ → L∞ , and by the change of variables v → v (which is valid for fixed σ because we have restricted ourselves to θ ∈ [0, π/2]), one can prove that T is bounded L1 → L1 . By interpolation, T is bounded Lp → Lp for 1 p ∞, and therefore we obtain the estimate
$ + $ $Q (g, f )$ p C g 1 f Lp , L L
1 p ∞.
(94)
A variant of the argument when B(v − v∗ , σ ) = |v − v∗ |γ b(cos θ ), γ > 0, leads to $ + $ $Q (g, f )$ p C g 1 f p , Lγ Lγ L
1 p ∞,
(95)
where we use the notation f
p Ls
=
p f 1 + |v|s dv p
RN
1/p .
(96)
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More sophisticated variants of estimates like (95) have been studied by more complicated means in Gustafsson [269,270]. They constitute a first step in the Lp theory for the spatially homogeneous Boltzmann equation with hard potentials and cut-off. These estimates show that, in first approximation, the Q+ operator resembles a convolution operator. But we shall see in the next paragraph that a stronger property holds. To conclude this paragraph, we mention that the case where there is a singularity in the kinetic collision kernel for |v − v∗ | 0 (soft potentials . . .) has never been studied very precisely from the point of view of Lp integrability. 3.4. Lions’ theorem: the Q+ regularity One of the main ideas of Grad [252], when developing his linear theory and making clear his assumptions, was that the Q+ term should be considered as a “perturbation”. This may sound strange, but think that the linear counterpart of Q+ is likely to be an integral operator with some nice kernel, while the linear counterpart of Q− will contain a multiplicative, noncompact operator. Generally speaking, since Q+ is more “mixing” than Q− , we could expect it to have a smoothing effect. In a nonlinear context, the idea that Q+ should be smoother than its arguments was made precise by Lions [307]. In this paper he proved the following estimate. P ROPOSITION 2. Let B be a C ∞ collision kernel, compactly supported as a function of |v − v∗ |, vanishing for |v − v∗ | small enough, compactly supported as a function of θ ∈ (0, π). Then, there is a finite constant C depending only on B and N such that for any g ∈ L1 (RN ), f ∈ L2 (RN ), $ $ + $Q (g, f )$ N−1 C g 1 f 2 , (97) L L H
and similarly $ + $ $Q (g, f )$
H
2
N−1 2
C g L2 f L1 .
(98)
The proof was based on a duality method quite similar to the one above, and very sophisticated tools about Fourier integral operators; this estimate is actually linked to the regularity theory of the Radon transform. In fact, as a general rule [406], operators of the form b(x, y)ϕ(y) dσx (y), T ϕ(x) = Sx
where b is a smooth kernel and Sx is a hypersurface varying smoothly with x ∈ RN , satisfy an estimate like T ϕ H (N−1)/2 C ϕ L2 , under some nondegeneracy condition5 about the way Sx varies with x. 5 In the case of operator (93), S is the sphere with diameter [0, x] and σ is just the uniform measure on S . x x x The nondegeneracy condition is not satisfied at y = 0 (which is fixed), and this is why Lions’ theorem does not
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Lions’ theorem has become one of the most powerful tools in the study of fine properties of the Boltzmann equation with cut-off (see [101,307,308,349,429,393]). Wennberg [459] gave a simplified proof of this result, with explicit(able) constants, by using Carleman’s representation (91). Then, Bouchut and Desvillettes [98], and, independently, Lu [325] devised an elementary proof, with simple constants, of a slightly weaker result: $ $ + $Q (g, f )$ N−1 C g 2 f 2 . (99) L L H
2
However, the qualitative difference between (97) and (99) is significant in some applications: see, for instance, our a priori estimates in Lp norms for collision kernels which decay at infinity [429]. Also a relativistic variant of this result has been established by Andréasson [14]. In a more recent paper, Wennberg [462] has put both the classical and the relativistic estimates in a unified context of known theorems for the regularity of the Radon transform. He noticed that these two cases are the only ones, for a whole range of parameters, where these theorems apply. Of course, by Sobolev embedding (and interpolation), Lions’ theorem yields refinements of (94) when the collision kernel is smooth. A task which should be undertaken is to use a precise version of Lions’ theorem, like the one by Wennberg, to derive nice, improved weighted Lp bounds for realistic collision kernels. This, combined with the methods in [429], should enable one to recover the main results of Gustafsson in a much more elegant and explicit way.6 As a last remark, the fact that the collision kernel vanish at v − v∗ = 0 is essential in Lions’ theorem (not in the weaker version (99)). If this is not the case, then smoothness results similar to the ones obtained by (97) require additional integrability conditions on g. In the case of hard potentials however, it is still possible to prove a weaker gain of smoothness or integrability with respect to f , without further assumptions on g. 3.5. Duhamel formulas and propagation of smoothness The idea to consider Q+ as a perturbation can be made precise by the use of Duhamel-type formulas. For instance, in the spatially homogeneous case, the Boltzmann equation can be rewritten as ∂f + (Lf )f = Q+ (f, f ), ∂t
f (0, ·) = f0 ,
where we use the notation A ∗ f = Lf . Then the solution can be represented as f (t, v) = f0 (v)e−
t 0
Lf (τ,v) dτ
t
+
e−
t s
Lf (τ,v) dτ
Q+ (f, f )(s, v) ds.
(100)
0
apply in presence of frontal collisions, and needs the collision kernel to be vanishing close to θ = π . Another degeneracy arises when v − v∗ goes to 0, so the Q+ smoothness also needs vanishing of the collision kernel at zero relative velocity. 6 At the time of writing, this task has just been performed by C. Mouhot with the help of the author’s advice.
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In the spatially inhomogeneous case, one can write similarly
t t ∂ + v · ∇x f e 0 Lf (τ,x−(t −τ )v,v) dτ = Q+ (f, f )e 0 Lf (τ,x−(t −τ )v,v) dτ , ∂t
or t
f (t, x, v) = f0 (x − tv, v)e− 0 Lf (τ,x−(t −τ )v,v) dτ t t + Q+ (f, f ) s, x − (t − s)v, v e− s Lf (τ,x−(t −τ )v,v) dτ .
(101)
0
By these formulas, one can understand, at least heuristically, the phenomenon of propagation of regularity and singularities. Let us illustrate this in the case of the spatially homogeneous Boltzmann equation, starting with formula (100). If f0 ∈ L2 , then, at least for collision kernels which become very large at infinity, f (t, ·) is uniformly bounded in L2 . Now, assume for simplicity that the collision kernel B is sufficiently smooth that estimate (97) applies, then it becomes clear that the second term in the right-hand side of (100) is H 1 -smooth (when N = 3). Indeed, thanks to the convolution structure, Lf is also smooth, say C ∞ if B is C ∞ . On the other hand, the first term on the right-hand side has exactly the same smoothness as f0 . Thus both regularity and singularities are propagated in time. More precise theorems of this kind, for some realistic collision kernels, e.g., hard spheres, are to be found in Wennberg [459]. In principle one could actually prove that f (t, v) = G(t, v)f0 (v) + H (t, v),
(102)
where, at least if B is very smooth, G is a positive C ∞ function of t, v and H is smoother than f0 . This would follow by an iteration of Duhamel’s formula, in the spirit of [349]. Alternatively, one expects that f (t, ·) can be decomposed into the sum of a part which is smooth (at arbitrarily high order) and a part which is just as singular as f0 , but decays exponentially fast. In the spatially inhomogeneous situation, the same kind of results is expected. In particular, in view of (101) it is believed that singularities of the initial datum are propagated by the characteristics of free transport, (x, v) → (x + tv, v). Such a result was recently proven by Boudin and Desvillettes for small initial data: they showed the following generalization of (102), f (t, x, v) = G(t, x, v)f0 (x − tv, v) + H (t, x, v), where G and H are not necessarily smooth, but at least possess a fractional Sobolev regularity. Their proof is based on a combination of the Q+ smoothness and averaging lemmas. We should make an important point here about our statement above about propagation of singularities. If we consider an initial datum which is very smooth apart from some singular set S, then it is not expected that the solution stay very smooth apart from the image St of this singular set by the characteristic trajectories. It is possible that the smoothness of
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the solution deteriorate, even very far from St . But one expects that the worst singularities always lie on St . And in any case, one always expects that the singular part of the solution be decaying very fast as time goes by.
3.6. The DiPerna–Lions renormalization The Q+ smoothing effect cannot be applied directly to the spatially inhomogeneous equation, ∂f + v · ∇x f = Q(f, f ), ∂t
(103)
because of the lack of nice a priori estimates. As explained above, the idea of renormalization consists in giving a meaningful definition of β (f )Q(f, f ) under very weak a priori estimates. Solutions are then defined as follows: N ∞ + 1 N N D EFINITION 1. Let f ∈ C([0, T ]; L1 (RN x × Rv )) ∩ L (R , L+ (Rx × Rv )). It is said to be a renormalized solution of the Boltzmann equation if for any nonlinearity β ∈ C 1 (R+ , R+ ), such that β(0) = 0, |β (f )| C/(1 + f ), one has
∂β(f ) + v · ∇x β(f ) = β (f )Q(f, f ), ∂t
(104)
in distributional sense. The DiPerna–Lions renormalization [190,192] achieves this goal by using the splitting (90). First of all, β (f )Q− (f, f ) = fβ (f )(A ∗ f ). Following Section 5.1, let us assume that sup
0t T
R2N
f (t, x, v) 1 + |v|2 + |x|2 dx dv < +∞,
and that A(z) = o(|z|2 ). Then N 1 . A ∗ f ∈ L∞ [0, T ]; L1 RN x ; Lloc Rv Further assume that the nonlinearity β satisfies 0 β (f )
C . 1+f
(105)
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Then, fβ (f ) ∈ L∞ , and as a consequence (105) is well-defined in L1loc . As for the renormalized gain term, it can be handled easily because it is nonnegative. As a general “rule”, when one manages to give sense to all terms but one in some equation, and when this last term has a sign, then the equation automatically yields an a priori estimate. In our situation this is accomplished by integrating Equation (104) in all variables on N [0, T ] × RN x × Rv ; using the bounds on mass and energy, one gets T R2N
0
β (f )Q+ (f, f ) dt dx dv
R2N
β(f0 ) dx dv +
T 0
R2N
β (f )f Lf dt dx dv < +∞,
whence N β (f )Q+ (f, f ) ∈ L1 [0, T ] × RN x × Rv . By the way, there is another, more widely known, version of this estimate, based on the entropy dissipation [192]; but this variant is more complicated and has the disadvantage to use a symmetric estimate, while the renormalization procedure can also be done from an asymmetric point of view: β (f )Q(g, f ) = β (f )Q+ (g, f ) − β (f )f Lg. On the other hand, the entropy dissipation has been very useful, both in the proof of stability of renormalized solutions, and in certain refinements of the theory by Lions [308]. Also, by using the entropy dissipation√one can define a renormalized formulation with a stronger nonlinearity, namely β(f ) = 1 + f − 1. This is implicit in [308], and has been shown by an elementary method in [444, Part IV, Chapter 3], as an outgrow of the idea of H -solutions explained in Section 4.1. Renormalization and averaging lemmas were the two basic tools in the DiPerna–Lions theorem, which was the very first existence/stability result in the large for the spatially inhomogeneous Boltzmann equation. More precisely, these authors have proven [192] that the renormalized Boltzmann equation (104) is stable under weak convergence (a priori estimates of mass, energy and entropy being used). This theorem has remained a singular point in the field, due to the complexity of the proof and the use of technicalities which have no real counterpart for the rest of the theory . . . . Since then, Lions [307] found a simpler proof of existence, using the Q+ regularity. In fact, he proved that strong compactness propagates in time for the Boltzmann equation with cut-off: this is the analogue of the spatially homogeneous results discussed in the section above. Once strong compactness is established, passing to the limit is almost straightforward. By the way, it is rather easy to prove the converse property, namely that the sequence of initial data has to be strongly compact if the sequence of solutions is. This theorem is a (very weak) illustration of the principle of propagation of singularities.
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3.7. Summary To conclude this section, we give explicit theorems illustrating the discussion above. As of this date, all of them are best in their category, but not optimal. We begin with the spatially homogeneous situation. T HEOREM 3. Let B(v − v∗ , σ ) = |v − v∗ |γ b(cos θ ) be a collision kernel for hard potentials, satisfying Grad’s cut-off assumption: π b(cos θ ) sinN−2 θ dθ < +∞, γ > 0, 0
and let f0 be a probability distribution function with finite second moment, f0 (v)|v|2 dv < +∞. RN
Then, (i) there exists a unique energy-preserving solution to the Cauchy problem associated to B and f0 . This solution is unique in the class of weak solutions whose energy does not increase; p (ii) if f0 ∈ L1s1 ∩ Ls for some p ∈ [1, +∞], s1 > 2, 1 < s s1 − γ /p, then supt 0 f (t, ·) Lps < +∞; (iii) if moreover γ > 1/2, s 2 and f0 ∈ H 1 (RN ), N = 3, then supt 0 f (t, ·) H 1 < +∞; / H 1 (RN ), N = 3, then, for all t 0, (iv) if, on the other hand, γ > 1/2, s 2 and f0 ∈ 1 N f (t, ·) ∈ / H (R ). But f (t, ·) = g(t, ·) + h(t, ·) where g(t, ·) Lps = O(e−µt ) for some µ > 0, and supt 0 h(t, ·) H 1 < +∞. Point (i) is due to Mischler and Wennberg [349]; point (ii) to Gustafsson [270] for p < ∞ and to Arkeryd [20] for p = ∞; as for point (iii), is is due to Wennberg [459], point (iv) being an immediate consequence of the proof. A recent work by Mouhot and the author recovers the conclusion of (ii) under slightly different assumptions on s1 , with the advantage of getting explicit constants; we are working on extending the allowed range of exponents s1 . Further work is in progress to extend also the range of validity of the conclusion of (iii) (arbitrary dimension, more general collision kernels) as well as to treat propagation of H k smoothness for arbitrary k. We now turn to the inhomogeneous theory in the small. T HEOREM 4. Let B = B(z, σ ) be a collision kernel, B ∈ L∞ (S N−1 , W 1,∞ (RN )), N = 3. Let f0 (x, v) be a nonnegative initial datum satisfying the Maxwellian bound |x|2 + |v|2 , f0 (x, v) C0 exp − 2 where C0 = 1/(81 B L∞ (RN ;L1 (S N−1 )) ). Then,
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(i) there exists a global solution to the spatially inhomogeneous Boltzmann equation with collision kernel B and initial datum f0 , and for all t ∈ [0, T ] (T > 0), it satisfies a Maxwellian bound of the form |x − vt|2 + |v|2 f (t, x, v) CT exp − , 2
(106)
with CT depending only on T and C0 ; α (R+ × RN × RN ) for any α < 1/25, (ii) there exist “smooth” functions R and S in Hloc x v such that f (t, x, v) = f0 (x − vt)R(t, x, v) + S(t, x, v); (iii) if moreover f0 ∈ W k,∞ for some k ∈ N (or k = ∞), then k,∞ + N f ∈ Wloc R × RN x × Rv . This theorem is extracted from Boudin and Desvillettes [101]. Part (i), inspired from Mischler and Perthame [348], is actually an easy variation of more general theorems by Illner and Shinbrot [278]. One may of course expect the smoothness of R and S to be better than what this theorem shows! But this result already displays the phenomenon of propagation of singularities along characteristic trajectories. Moreover, it conveys the feeling that it will be possible to treat propagation of smoothness and singularity for very general situations, as soon as we have solved the open problem of finding nice integrability a priori estimates for large data. The main idea behind the bound (106) is the following: the left-hand side satisfies the differential inequality ∂f + v · ∇x f = Q(f, f ) Q+ (f, f ), ∂t while one can devise a right-hand side of the form |x − vt|2 + |v|2 g(t, x, v) = C(t) exp − 2 which satisfies the differential inequality ∂g + v · ∇x g Q+ (g, g), ∂t so that it is natural to expect f g if f0 g0 . Note that since g is a Maxwellian, Q+ (g, g) = Q− (g, g). Finally, we consider the DiPerna–Lions theory of renormalized solutions.
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T HEOREM 5. Let B(|v − v∗ |, cos θ ) = Φ(|v − v∗ |)b(cos θ ) be a collision kernel satisfying Grad’s angular cut-off, together with a growth condition at infinity: π b(cos θ ) sinN−2 θ dθ < +∞, 0
Φ ∈ L1loc RN ;
Φ |z| = o |z|2
as |z| → ∞.
Let (f0n )n∈N be a sequence of initial data with uniformly bounded mass, energy, entropy, sup n∈N
RN ×RN
f0n (x, v) 1 + |x|2 + |v|2 + log f0n (x, v) dx dv < +∞.
Let f n (t, x, v) be a sequence of solutions7 of the Boltzmann equation ⎧ n ⎨ ∂f + v · ∇x f n = Qf n , f n , ∂t ⎩ n f (0, ·, ·) = f0n .
t 0, x ∈ RN , v ∈ RN ,
(107)
Assume that the f n ’s satisfy uniform bounds of mass, energy, entropy and entropy dissipation: sup sup
n∈N t ∈[0,T ] RN ×RN
f n (t, x, v) 1 + |x|2 + |v|2 + log f n (t, x, v) dx dv < +∞, (108)
T
sup
D f n (t, x, ·) dx dt < +∞.
(109)
n∈N 0 N Without loss of generality, assume that f n → f in Lp ([0, T ]; L1 (RN x × Rv )), 1 p < ∞, T < ∞. Then, (i) f is a renormalized solution of the Boltzmann equation. It satisfies global conservation of mass and momentum, and the continuity estimate f ∈ C([0, T ]; L1 (RN x × RN v )); (ii) moreover, for all t > 0, f n → f strongly in L1 if and only if f0n → f0 strongly; in N this case the convergence actually holds in C([0, T ], L1 (RN x × Rv )); (iii) if moreover there exists a strong, classical solution of the Boltzmann equation with initial datum f0 , then f coincides with f0 .
C OROLLARY 5.1. Let f0 be an initial datum with finite mass, energy and entropy: f0 (x, v) 1 + |v|2 + |x|2 + log f0 (x, v) dx dv < +∞. RN ×RN
7 Renormalized solutions, or strong solutions, or approximate strong solutions.
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Then there exists a renormalized solution f (t, x, v) of the Boltzmann equation, with f (0, ·, ·) = f0 . R EMARKS . (1) A typical way of constructing approximate solutions is to solve the equation ∂f n Q(f n , f n ) + v · ∇x f n = , ∂t 1 + n1 f n dv which is much easier than the “true” Boltzmann equation because the collision operator is sublinear. (2) If the f n ’s are strong, approximate solutions, then the bounds (108)–(109) automatically hold, provided that the initial data have sufficient regularity. This remark, combined with the preceding, explains why the corollary follows from the theorem. (3) Point (iii) as stated above is slightly incorrect: for this point it is actually necessary to assume that the f n ’s are strong, approximate solutions, or are constructed as limits of strong, approximate solutions. Point (i) is due to DiPerna and Lions [192], points (ii) to Lions [307,308]. For the sake of simplicity, we have stated unnecessarily restrictive assumptions on the collision kernel in points (i) and (ii). Point (iii) was first proven by Lions under an assumption which essentially implies Φ ∈ L∞ , then extended by Lu [324]. We have not made precise what “classical” means in point (iii): in Lions’ version, g should satisfy the Boltzmann equation N almost everywhere on [0, T ] × RN x × Rv , and also satisfy the dissipative inequalities introduced in Lions [308]. The discussion of dissipative inequalities is subtle and we preferred to skip it; let us only mention that this concept is based on the entropy dissipation inequality, and that it led Lions to a clean proof of local conservation of mass, as well as to the concept of dissipative solutions. In Lu’s theorem, much more general collision kernels are included, at the price of slightly more restrictive (but quite realistic) assumptions imposed on the strong solution g. Lu also uses results from the theory of solutions in the small [63] to show existence of such strong solutions when Φ(|z|) = O(|z|γ ), γ > −1, and the initial datum is bounded by a well-chosen, small function. So these results bridge together the theory of renormalized solutions and the theory of solutions in the small. At the moment, point (ii) is the most direct way towards the corollary. The scheme of the proof is as follows. In a first step, one uses the uniform bounds and the Dunford–Pettis criterion to get weak compactness of the sequence of solutions in L1 . This, combined with the renormalized formulation and averaging lemmas, implies the strong compactness of velocity-averages of f n . Since the operators L = A∗ and Q+ are velocity averaging operators in some sense (remember the Q+ regularity theorem), one can then prove the strong compactness of Lf n and Q+ (f n , f n ). This is combined with a very clever use of Duhamel formulas to prove the strong compactness of the sequence f n itself, provided that the sequence of initial data f0n is strongly compact. Finally, one easily passes to the limit in the renormalized Q− operator, and then in the renormalized Q+ operator by a variant of the dominated convergence theorem which involves the domination of Q+ by Q− and (a little bit of) the entropy dissipation, as in Section 3.2.
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4. The singularity-hunter’s toolbox In this section, we now examine the situation when the collision kernel presents a nonintegrable angular singularity. This branch of kinetic theory, very obscure for quite a time, has undergone spectacular progress in the past few years, which is why we shall make a slightly more detailed exposition than in the case of Grad’s angular cut-off. The starting point for recent progress was the work by Desvillettes [171] on a variant of the Kac model, which was devised to keep some of the structure of the Boltzmann equation without cut-off. The study of these regularizing effects was first developed in the spatially homogeneous theory, and later in the theory of renormalized solutions, in the form of strong compactification. As explained previously, the main qualitative difference with respect to the situation where Grad’s assumption holds, is that one expects immediate regularization of the solution. From the mathematical point of view, the first clear difference is that the splitting Q(f, f ) = Q+ (f, f ) − Q− (f, f ) is impossible: both terms should be infinite. From the physical point of view, one can argue that when particles collide, there is an overwhelming probability for the change in velocity to be extremely small, hence the density in probability space should spread out, like it does in a diffusion process. The main analytical idea behind the regularization effect is that Q(f, f ) should look like a singular integral operator. As we shall see, it resembles a fractional diffusion operator; this illustrates the physically nontrivial fact that collision processes for longrange interactions are neither purely collisional in the usual sense, nor purely diffusive, but somewhat in between. There is an important body of work due to Alexandre about the study of the non-cutoff Boltzmann operator, with the help of pseudo-differential formalism [2–9], and on which we shall say almost nothing, the main reason being that most of the results there (some of which have been very important advances at the time of their appearance) can be recovered and considerably generalized by means of the simpler techniques described below. Generally speaking, there are two faces to singular operators in partial differential equations. On one hand one would like to control them, which means (i) find weak formulations, or (ii) find if, in some situations, they induce compensations due to symmetries. On the other hand, we would like to have (iii) simple estimates expressing the fact that they really are unbounded operators, and that the associated evolution equation does have a regularizing effect. To illustrate these fuzzy considerations, think of the Laplace operator, and the formulas (i) f ϕ = f ϕ, (ii) ( f ) ∗ ϕ = f ∗ ( ϕ), (iii) f ( f ) = − f 2H˙ 1 . Keeping this discussion in mind may help understanding the interest of the weak formulations in Section 4.1, the cancellation lemma in Section 4.2, and the entropy dissipation estimate in Section 4.3, respectively. Finally, as we already mentioned several times, another singularity problem will come into play: when one is interested in soft potentials, then the kinetic collision kernel presents a singularity for |v − v∗ | 0. When the strength of this singularity is high, this will entail additional technical difficulties, but it is not clear at the moment that this feature is related to physically relevant considerations.
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4.1. Weak formulations In presence of a nonintegrable singularity, Boltzmann’s collision operator is not a bounded operator between weigthed L1 spaces; it is not even clear that it makes sense almost everywhere. Thus one should look for a distributional definition. The most natural way towards such a definition (both from the mathematical and the physical points of view) is via Maxwell’s weak formulations:
Q(f, f )ϕ dv = dv dv∗ ff∗ B(v − v∗ , σ )(ϕ − ϕ) dσ . (110) S N−1
As pointed out by Arkeryd [18], if ϕ is a smooth test-function, then ϕ − ϕ will vanish when θ 0 (because then v v), and this may compensate for a singularity in B. This circumstance actually explains why one is able to compute relevant physical quantities, such as the cross-section for momentum transfer, even for non-cutoff potentials . . . . For the sake of discussion, we still consider the model case B(v − v∗ , σ ) = |v − v∗ |γ b(cos θ ). Using moment estimates, and the formula ϕ − ϕ = O(|v − v∗ |θ ) (which holds true when ϕ is smooth), Arkeryd [18] was able to prove existence of weak solutions for the spatially homogeneous Boltzmann equation as soon as γ −1, b(cos θ )θ sinN−2 θ dθ < ∞. The use of the more symmetric form obtained by the exchange of variables v ↔ v∗ does not a priori seem to help a lot, since one has only |ϕ + ϕ∗ − ϕ − ϕ∗ | C(ϕ)|v − v∗ |2 θ, so there is no gain on the angular singularity. But, as noted independently by several authors (see, for instance, [248,446]), an extra order of θ can be gained by integrating in spherical coordinates. More precisely, use the standard parametrization of σ in terms of θ , φ (φ ∈ S N−2 ), then C(ϕ)|v − v∗ |2 θ 2 . (ϕ + ϕ − ϕ − ϕ ) dφ (111) ∗ ∗ S N−2
This simple remark enables one to extend Arkeryd’s results to γ −2, b(cos θ )(1 − cos θ ) sinN−2 θ dθ < ∞. In dimension 3, these assumptions are fulfilled by inverse-power forces 1/r s when s 7/3 (to compare with s > 3 for Arkeryd’s original result . . . the reader may feel that the gain is infinitesimal, but remember that s = 2 should be the truly interesting limit exponent!). However, this point of view, which relies on the v ↔ v∗ symmetry, is in part misleading. The same control on the angular singularity (but worse in the kinetic singularity) can be
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obtained without using the symmetry v ↔ v∗ , as shown in Alexandre and Villani [12] by the use of more precise computations. When one is only interested in weak solutions in a spatially homogeneous problem, this remark is of no interest, but it becomes a crucial point in spatially inhomogeneous situations, or in the study of fine regularization properties. Here is a precise bound from [12]. Introduce the cross-section for momentum transfer, formulas (62) or (63). Then RN v
Q(g, f )ϕ dv
1 ϕ W 2,∞ 4
R2N
dv dv∗ g∗ f |v − v∗ | 1 + |v − v∗ | M |v − v∗ | .
(112)
To treat values of γ below −2 with the help of formula (111) and others, it seems that one should require nontrivial a priori estimates like R2N
f (v)f (v∗ ) dv dv∗ < +∞ |v − v∗ |−(γ +2)
(the exponent of |v − v∗ | in the denominator is positive!). As we shall see in Section 4.3, such estimates are indeed available in most cases of interest. But they are by no means easy! At the time when these extra estimates were not yet available, the search for a treatment of values of γ below −2 led the author [446] to introduce a new weak formulation (H solutions), based on the a priori bound
1/2 2 Q(f, f )ϕ 1 D(f )1/2 (ϕ + ϕ − ϕ − ϕ ) . Bff ∗ ∗ ∗ 2 Here D is Boltzmann’s entropy dissipation functional (47). This new bound was based on Boltzmann’s weak formulation (45), and the elementary estimate D(f )
RN ×RN ×S N−1
B(v − v∗ , σ )
f f∗ −
ff∗
2
dv dv∗ dσ
(113)
√ √ (which follows from (X − Y )(log X − log Y ) 4( X − Y )2 ). It enabled the author to prove existence of weak solutions under the assumptions γ > −4,
b(cos θ )(1 − cos θ ) sinN−2 θ dθ < ∞,
which allow the three-dimensional Coulomb potential as a limit (excluded) case. A main application was the first proof of the Landau approximation8 for realistic potentials in a spatially homogeneous setting. 8 See Section 5.
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This use of the entropy dissipation for the study of grazing collisions had the merit to display some interesting feature: a partial regularity estimate associated to the entropy dissipation. More precisely, finiteness of the entropy dissipation, when the collision kernel is singular, implies a partial smoothness estimate for ff∗ in the tensor velocity space RN × RN . This effect is best seen at the level of the Landau equation: Landau’s entropy dissipation can be rewritten as Π(v − v∗ )(∇ − ∇∗ ) ff∗ Ψ (|v − v∗ |)2 dv dv∗ . (114) DL (f ) = 2 RN ×RN
Recall that Π(v − v∗ ) is the orthogonal projection on (v − v∗ )⊥ . Equation (114) is a regularity estimate on the function ff∗ , but only in the variable v − v∗ , and only in directions which are orthogonal to v − v∗ . On the whole, this means N − 1 directions out of 2N . At the level of Boltzmann’s entropy dissipation, for each point (v, v∗ ) ∈ R2N , these N − 1 directions are precisely the tangent plane to the (N − 1)-dimensional manifold % & Svv∗ = (v , v∗ ) satisfying (5) . One may conclude to the simple heuristic rule: entropy dissipation yields a smoothness estimate along collisions. These entropy dissipation bounds have a lot of robustness in a spatially homogeneous context, due to the tensorial structure of the entropy dissipation functional. For instance, one can prove that if Dε is the entropy dissipation functional associated to a Boltzmann operator Qε “converging” in a suitable sense to Landau’s operator, and DL is Landau’s entropy disipation, and f ε → f in weak L1 , then DL (f ) lim inf Dε f ε . ε→0
On the other hand, precisely because they rely so much on the symmetry v ↔ v∗ and the tensor product structure, these methods turn out to be inadapted to more general problems. More efficient approaches will be presented in the sequel.
4.2. Cancellation lemma In this and the next two sections, we shall introduce more sophisticated tools for fine surgery on Boltzmann’s operator. As discussed above, integrals such as dv∗ dσ B(v − v∗ , σ )(g − g) (115) RN ×S N−1
are well-defined for a smooth function g, at least if the collision kernel is not too much singular. When g is not smooth (say L1 ), it is not clear at all that such an integral should converge. This is however true with great generality, due to symmetry effects. A precise
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quantitative version was introduced by the author in [449] (related estimates are to be found in Desvillettes [173] and Alexandre [6]). The estimate in [449] shows that when the collision kernel B in (115) depends smoothly on v − v∗ and presents an nonintegrable angular singularity of order ν < 2, then the integral (115) converges. The need to cover more singular situations motivated further refinement of this estimate; here we present the sharp version which is proven in [12]. It only requires finiteness of the cross-section for momentum transfer, M, and a very weak regularity assumption with respect to the relative velocity variable. P ROPOSITION 6. Let B(|z|, cos θ ) be a collision kernel with support in θ ∈ [0, π/2], and let S be defined by Sg ≡
dv∗ dσ B(v − v∗ , σ )(g∗ − g∗ ).
RN ×S N−1
Then, for any g ∈ L1 (RN v ), Sg = g ∗v S, where the convolution kernel S is given by S |z| = S N−2
×
π/2
dθ sinN−2 θ 0
1 |z| B , cos θ − B |z|, cos θ . cosN (θ/2) cos(θ/2)
(116)
Recall from Section 4.1 that the assumption about the deviation angle is no loss of generality. The proof of this lemma is rather elementary and relies on the change of variables v∗ → v∗ , which for fixed σ ∈ S N−1 is allowed if the integration domain avoids frontal collisions (θ ±π ). Here is an easy corollary: C OROLLARY 6.1. With the notations z = v − v∗ , k = (v − v∗ )/|v − v∗ |, let B (z, σ ) ≡
1<λ
M |z| =
|B(λz, σ ) − B(z, σ )| , (λ − 1)|z| 2
sup√
S N−1
B (z, σ )(1 − k · σ ) dσ.
Then S |z| CN M |z| + |z|M |z| ,
(117)
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where CN is a constant depending only on N . In particular, if |z|M |z| ∈ L1 RN ,
(118)
then S ∈ L1loc (RN ). What typical collision kernels are allowed by this lemma? It turns out that the quantity M measures the regularity of B with respect to the relative velocity variable in a very weak sense. For instance, assumption (118) is satisfied for all potentials of the form B(v − v∗ , σ ) = |v − v∗ |γ b(cos θ ),
γ > −N.
Of course, this excludes the borderline case where γ = −N , which may be the most interesting, because in dimension 3 it corresponds to Coulomb interactions . . . . However, now it is homogeneity which will save the game. A quick glance at formula (116) may give the impression that if B is homogeneous of degree −N in the relative velocity variable, then S = 0!! Of course, this is a trap: S should be defined as a principal value operator, and after a little bit of algebra, one finds the following corollary to Proposition 6: C OROLLARY 6.2. If B(v − v∗ , σ ) = |v − v∗ |−N β0 (cos θ ), then S = λ δ0 , where δ0 is the Dirac measure at the origin, and λ = −S N−2 S N−1
π/2
β0 (cos θ ) log cos(θ/2) sinN−2 θ dθ. 0
Note that λ is finite as soon as the cross-section for momentum transfer (63) with b = β0 , is. The compensation lemma of Proposition 6 has been a crucial tool, (1) to obtain sharp entropy dissipation estimates, see next section; (2) to derive a sharp renormalized formulation for the Boltzmann operator without cut-off, see Section 4.6.
4.3. Entropy dissipation estimates As we have explained in Section 4.1, under certain assumptions the entropy dissipation estimate yields a partial regularity bound on ff∗ when the collision kernel is singular. But does it imply a true regularity estimate on f itself? The first result in this direction is due to Lions [316]. He proved that if B(v − v∗ , σ ) Φ(|v − v∗ |)b(cos θ ), where Φ is smooth and positive, and sinN−2 θ b(cos θ ) Kθ −1−ν , then for all R > 0 there is a constant CR such that
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$ $ $ f (t, ·)$2
H s (|v|
CR D(f )
1/2
+ f L1 ,
ν s < s0 = 2
1 1+
ν N−1
171
.
(119)
The exponent s0 here is not optimal. The idea of the proof was clever, and a very unexpected application of the Q+ regularity in L2 form (formula (99), with explicit bounds needed). Starting fromthe entropy √ dissipation and formula (113), one finds some smoothness-type estimate on f f∗ − ff∗ in RN × RN × S N−1 . Then one integrates this estimate with respect to the variables v∗ and σ after multiplication √ √ by an artificial, #ε (v − v∗ , σ ). An estimate on Q #ε ( f , f ) follows, where well-chosen collision kernel B #ε is the Boltzmann operator associated with B #ε . Then one writes Q #ε ∗ f = Q #+ #ε f , f , f A f, f −Q ε √ √ #ε dσ . The regularity of Q #+ # #ε = N−1 B where A ε and the estimates on Qε ( f , f ) yield S the conclusion in the limit ε → 0, after quite a bit of intricate computations. A completely different method [449], based on the compensation lemma of last paragraph and on Carleman’s representation (91), led the author to a better estimate under more stringent assumption on f : assume that B(v − v∗ , σ ) Φ0 |v − v∗ | b0 (cos θ ),
(120)
where Φ0 |z| is continuous,
Φ0 |z| > 0
if |z| > 0,
(121)
and (as usual) b0 (cos θ ) sinN−2 θ Kθ −(1+ν),
K > 0, ν 0.
(122)
Further assume that f is positive, then $ $ $ f (t, ·)$2
H ν/2 (|v|
CR D(f ) + f 2L1 ,
(123)
2
where CR depends on f only via inf{f (v); |v| R}. Exponent ν/2 is optimal, as shown by a variant of the proof. But the assumption of local bound below for f is too strong. In a joint work with Alexandre, Desvillettes and Wennberg [10], we were able to extend the scope of (123), and eventually obtain sharp entropy dissipation estimates. The main difference with the previous argument was the replacement of Carleman’s representation for the use of Fourier transform. In the next paragraph we shall make this a little bit more precise, for the moment we precisely state our main result in [10]. To this date this may be considered as the most general manifestation of the regularizing effects of grazing collisions.
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P ROPOSITION 7. Let f ∈ L12 ∩ L log L(RN ) be a nonnegative distribution function, and let B be a collision kernel satisfying assumptions (120)–(122). Then, for all R > 0 there is a constant C = CR such that $ $2 $ f$
H ν/2 (|v|
where f L1 =
CR D(f ) + f 2L1 ,
(124)
2
f (v)(1 + |v|2 ) dv and CR (which is explicit) depends only on N , Φ0 , K, ν, R, a lower bound for f dv, and an upper bound for f (1 + |v|2 + | log f |). 2
R EMARKS . (1) In the limit case ν = 0, one recovers local estimates in log H . As a very general fact, as soon as the function b0 (cos θ ) sinN−2 θ is not integrable, i.e., when π Z : θ0 → b0 (cos θ ) sinN−2 θ dθ θ0
√ diverges as θ0 → 0, then one can estimate f in Xloc , where X is the functional space defined in Fourier representation by 2 N 1 2 # F (ξ ) Z X= F ∈L R ; dξ < +∞ . |ξ | |ξ |1 (2) This entropy dissipation estimate is asymmetric, and this is quite in contrast with the estimates which we shall discuss in the study of trend to equilibrium. In fact, the estimate holds just the same for Q(g, f ) log f (which is not always a nonnegative expression!), if one imposes B = Φ0 b0 instead of B Φ0 b0 , and replaces f L1 in (124) by f L1 + g L1 ; then the constant CR would 2 2 2 not depend on f but only on g. By (124), one can guess a precise heuristic point of view for the regularity properties associated with the non-cutoff Boltzmann operator: if the angular collision kernel is singular of order ν (assumption (122)), and g is a fixed distribution function with finite mass, energy and entropy, then the linear operator f → Q(g, f ) “behaves” in the same way as the fractional diffusion operator −(− )ν/2 . For Maxwellian molecules, the intuition of this result goes back to Cercignani [138], who had noticed, thirty years ago, that the eigenvalues of the linearized Boltzmann operator behaved like those of the power 1/4 of the Fokker–Planck operator.
4.4. Boltzmann–Plancherel formula A key step in the proof of Proposition 7 is the use of Fourier transform. As we said earlier, in the context of Boltzmann equation, it is only for Maxwellian collision kernels that the
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Fourier transform leads to simple expressions. So one step of the proof (based on a lot of fine surgery) is the reduction to the purely Maxwellian case: Φ0 ≡ 1 in the previous notations. Then, it all reduces to a sharp estimate from below of expressions of the form
R2N
dv dv∗
S N−1
2 dσ g(v∗ ) F (v ) − F (v) ,
where g is an approximation √ of f (say, f multiplied by a smooth cut-off function), and F is an approximation of f . The following Plancherel-type formula, established in [10] after the ideas of Bobylev, is the appropriate ingredient. P ROPOSITION 8. With the notations ξ ± = (ξ ± |ξ |σ )/2,
R2N
S N−1
1 (2π)N
=
b(k · σ )g∗ (F − F )2 dv dv∗ dσ
RN
(125)
2 + 2 ξ F (ξ ) + g(0) ˆ # F ξ · σ g(0) ˆ # |ξ | S N−1 − + #ξ F #(ξ ) dξ dσ, − 2 g(ξ ˆ )F
b
with standing for real part. This general formula can be used in many other regularity problems, in particular to establish Sobolev-regularity estimates for the spatially homogeneous Boltzmann equation without cut-off [185].
4.5. Regularization effects As a consequence of Proposition 7, one can derive some (rather weak) regularization theorems. This is immediately seen in the spatially homogeneous situation. Combining the entropy dissipation estimate
T
D f (t, ·) dt + H f (T , ·) H (f0 )
0
with Proposition 7, the a priori bound
ν/2 f ∈ L2 [0, T ]; Hloc RN v
(126)
follows at once. Weak as it is, this regularization estimate is already useful to the existence theory for singular collision kernels. Indeed, assume that B(v − v∗ , σ ) = |v − v∗ |γ b(cos θ ), where
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b satisfies the same assumptions as b0 in (122). As we saw in Section 4.1, when γ is very negative one runs into trouble to define relevant weak solutions of the spatially homogeneous Boltzmann equation. But now, with this new entropy dissipation bound, one can get sufficient a priori estimates if the angular singularity is strong enough compared to −γ . More precisely, if γ + ν + 2 0,
(127)
then, by the Hardy–Littlewood–Sobolev and Sobolev inequalities,
ff∗ |v − v∗ |γ +2 dv dv∗ dt C f L∞ 1 f L1 (Lq ) v t (Lv ) t
$ $2 C f0 L1 $ f $L2 (H ν/2 ) v
t
for some well-chosen q > 1 (everything being understood in local sense). It is worth pointing out that inequality (127) always holds for collision kernels coming out from inverse-power forces in dimension 3. We also note that the case which appears the most delicate to treat now, is the one of a collision kernel which is singular in the relative velocity variable but not in the angular variable; soon we shall encounter a similar problem in the spatially inhomogeneous setting. Other variants of these entropy dissipation estimates lead to (strong) compactness results. For instance, let (fn (t, v))n∈N be a sequence of probability distributions, satisfying sup n
T 0
Dn (fn ) dt + sup fn L log L + fn L1 < +∞, t ∈[0,T ]
2
where Dn is the entropy dissipation associated to a collision kernel Bn , approximating (in almost everywhere sense for instance) a singular collision kernel B. Then, (fn ) is strongly compact in L1 . This holds true even if there is not necessarily a uniform smoothness estimate. The smoothing effects which we just discussed are rather weak, but using the same kind of techniques one can bootstrap on the regularity again and again, at least if the collision kernel is smooth with respect to the relative velocity variable. The key inequality can be formally written as d f 2H α −K f 2H α+ν/2 + C f 2H α . dt This easily leads, after integration on [0, T ], to the immediate appearance of the H α+ν/2 norm of f if the H α norm of f is initially finite – and, by induction, to immediate C ∞ regularization.9 9 In fact this method is but an adaptation of the “energy method” in the study of parabolic regularity, where similar estimates would hold with the constant ν replaced by 2.
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Such a study is currently worked out by Desvillettes and Wennberg, who have announced a proof of C ∞ instantaneous regularization for solutions to the spatially homogeneous Boltzmann equation, starting from an initial datum which has finite entropy. This result had already been proven in certain particular cases by Desvillettes [172,171, 173], and his student Proutière [390], with the use of Fourier-transform techniques. 4.6. Renormalized formulation, or Γ formula The treatment of regularizing effects for the full, spatially inhomogeneous Boltzmann equation without cut-off requires an additional tool because of the difficulty of defining the collision operator. As we explained in Section 1.3, a renormalized formulation of the collision operator, together with entropy dissipation estimates (in the sharp form of Proposition 7), is enough to prove appearance of strong compactness. For a long time this problem stood open, until Alexandre came up with a very clever idea [6]. The implementation of Alexandre’s ideas, based on pseudo-differential theory, suffered from intricate computations and the impossibility to cover physically realistic collision kernels. In a joint work [12] with Alexandre, we have given a very general definition, based on the use of the cancellation lemma of Section 4.2, and the idea of using the asymmetric Boltzmann operator. Here is the renormalized formulation of [12], given in asymmetric formulation: dv∗ dσ B(v − v∗ , σ ) (g∗ − g∗ ) β (f )Q(g, f ) = fβ (f ) − β(f ) + Q g, β(f ) −
RN ×S N−1
RN ×S N−1
dv∗ dσ Bg∗ Γ (f, f ),
(128)
where Γ (f, f ) = β(f ) − β(f ) − β (f )(f − f ).
(129)
If β is a strictly concave function or strictly convex function, then Γ has a fixed sign. In the context of the study of renormalized solutions, it will be convenient to choose β to be concave (sublinear), satisfying β (f ) C/(1 + f ). Let us explain why each of the three terms in (128) is then well-defined. For the first one, we may assume fβ (f ) − β(f ) ∈ L∞ , and then this term will satisfy an L1loc bound as a result of the cancellation lemma of Section 4.2. As for the second term in (128), it can be given a distributional sense, by means of formula (112). The estimate works in a spatially inhomogeneous context because the arguments of the collision operator are g (∈ L1 , say) and β(f ) (∈ L∞ , say). This is the point where it is very important to have an asymmetric weak formulation! In the end, the third term is nonnegative as soon as β is concave, and since all other terms are well-defined, it satisfies an a priori estimate in L1loc for free – just as in the argument for the gain term in the DiPerna–Lions renormalization.10 10 See Section 3.6.
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Notice that this renormalization procedure can be understood as a “commutator” problem: find a nice expression for β (f )Q(g, f ) − Q g, β(f ) .
(130)
Such commutators are widely used in the study of linear diffusion operators. When L is a diffusion operator, then β (f )Lf − Lβ(f ) = −β (f )Γ (f ), where Γ is the “Dirichlet form” associated with L. This justifies our terminology of “Γ formula”. As a matter of fact, the renormalization procedure above presents some similarities with the renormalization of parabolic equations by Blanchard and Murat [76,77], and is also very close to the renormalization of the Landau operator given in Lions [311]. As an illustration of the drawbacks of “soft” theories, we note that the construction of renormalized solutions with the preceding definition is still an open problem. Instead, one is led to introduce the following, slightly weaker, definition: N ∞ + 1 N N D EFINITION 2. Let f ∈ C(R+ , D (RN x × Rv )) ∩ L (R , L+ (Rx × Rv )). It is said to be a renormalized solution of the Boltzmann equation with a defect measure, if for any nonlinearity β ∈ C 2 (R+ , R+ ) such that β(0) = 0, 0 β (f ) C/(1 + f ), β (f ) < 0, one has
∂ β(f ) + v · ∇x β(f ) β (f )Q(f, f ), ∂t
(131)
in distributional sense, and moreover f satisfies the law of mass-conservation:
∀t 0,
R2N
f (t, x, v) dx dv =
R2N
f (0, x, v) dx dv.
(132)
We insist that this is really a notion of weak solution, not just sub-solution. Indeed, if f were smooth, then the combination of (131) and (132) shows that there is equality in (131). See [12], and also DiPerna and Lions [190] for similar situations. Finally, we note that formula (128) is a general tool which finds applications outside the theory of renormalized solutions, for instance in the study of regularity for the spatially homogeneous equation (or even for the spatially inhomogeneous one, if suitable integrability bounds are assumed). In this context, it is convenient to choose β to be convex when studying regularization for initial data which belong to L log L or Lp spaces, and concave when studying regularization for initial data which are only assumed to be probability measures. In fact, in some sense the Γ formula plays for the Boltzmann equation the same role as integration by parts plays in the energy method for diffusion operators; therefore one should not be surprised of its great utility.
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4.7. Summary The following two theorems summarize our current knowledge of regularizing effects, respectively in the spatially homogeneous setting and in the framework of renormalized solutions. We restrict to the model cases B(v − v∗ , σ ) = Φ |v − v∗ | b(cos θ ),
(133)
where Φ(|z|) > 0 for |z| = 0, and b satisfies the usual singularity condition, sinN−2 θ b(cos θ ) Kθ −(1+ν)
as θ → 0.
(134)
T HEOREM 9. Let B satisfy Equations (133)–(134), and let f0 be a probability density on RN v , with bounded energy; f0 may have a singular part, but should be distinct from a Dirac mass.11 Then, (i) if Φ is smooth and bounded from above and below, then there exists a solution f (t, v) to the Boltzmann equation with initial datum f0 , which lies in C ∞ ((0, +∞) × RN v ); (ii) if Φ(|v − v∗ |) = |v − v∗ |γ where γ + ν > −2, and f0 has finite entropy, then there exists a weak solution f (t, v) to the Boltzmann equation with initial datum f0 , such that
ν/2 f ∈ L2loc R+ ; Hloc RN v ;
(iii) if Φ(|v − v∗ |) = |v − v∗ |γ where γ + ν > 0, then, without further assumptions on f0 there exists a weak solution f (t, v) to the Boltzmann equation with initial datum f0 , such that ∀t > 0,
f (t, ·) ∈ L log L RN v .
Point (i) of this theorem is work in progress by Desvillettes and Wennberg if one assumes that f0 has finite entropy. Then, in the case where one only assumes that f0 has finite mass and energy, work in progress by the author [440] shows that the entropy becomes finite for any positive time (actually, one proves estimates in L1loc (dt; Lp (RN v )), for arbitray large p). Key tools in these works are the Plancherel-like formula of Section 4.4 and the cancellation lemma of Section 4.2. Related to point (i) are probabilistic works by Fournier [217], Fournier and Méléard [219, 220] who prove immediate appearance of an L1 density if the initial datum is not a Dirac mass, and C ∞ smoothness for Maxwellian collision kernel in two dimensions [217]. The results by Fournier and Méléard are considerably more restricted because of strong decay assumptions on the initial datum, stringent assumptions on the smoothness of the kinetic collision kernel and restrictions on the strength of the singularity. However, they have the merit to develop on Tanaka’s approach [415] and to build a stochastic theory of the Boltzmann equation, whose solution is constructed via a complicated nonlinear stochastic 11 Because a Dirac mass is a stationary solution of the spatially homogeneous Boltzmann equation! so, starting from a Dirac mass does not lead to any regularization.
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jump process. These works constitute a bridge between regularization tools stated here, and Malliavin calculus. They also have applications to the study of stochastic particle systems which are used in many numerical simulations [177,221,223,222,256,257]. In particular, they are able to study the numerical error introduced in Monte Carlo simulations when replacing a non-cutoff Boltzmann equation by a Boltzmann equation with small cut-off.12 As for point (ii), it follows from the entropy dissipation estimates in Alexandre, Desvillettes, Villani and Wennberg [10] and by now standard computations which can be found, for instance, in Villani [446]. One would expect that when γ > 0, C ∞ smoothness still holds; current techniques should suffice to prove this, but it remains to be done. Point (iii) is from [440]. Uniqueness is still an open problem in this setting, on which the author is currently working. This question is related to smoothing: if one wants to use a classical Gronwall strategy, like in the proof of uniqueness for the spatially homogeneous Landau equation [182], then one sees that the key property to prove is that (essentially) the non-cutoff bilinear Boltzmann operator is not only “at least” as singular as the fractional Laplace operator of order ν, but also “at most” as singular as this one, in the sense that it maps L2 into H −ν/2 (locally). We do hope for rapid progress in this direction! In the case of the spatially homogeneous Landau equation, then the same regularization results hold true, and are easier to get because the Landau equation already looks like a nonlinear parabolic equation. Hence the smoothing effect can be recovered by standard estimates (only complicated), bootstrap and interpolation lemmas between weighted Sobolev and Lebesgue spaces. It is possible to go all the way to C ∞ smoothness even in cases where Ψ is not so smooth: for instance, Ψ (|v − v∗ |) = K|v − v∗ |γ +2 , γ > 0. This study was performed in Desvillettes and Villani [182]. For this case the authors proved immediate regularization in Schwarz space, and uniqueness of the weak solution, in the class whose energy is nonincreasing, as soon as the initial datum satisfies 2 of solutions f (v)(1 + |v|2s ) dv < +∞, 2s > 5γ + 12 + s (N = 3). By the way, this uniqueness theorem of a weak solution, building on ideas by Arsen’ev and Buryak [41], required some precise Schauder-type estimates for a linear parabolic equation whose diffusion matrix is not uniformly elliptic in the usual sense, and our work has motivated further research in this area [11]. We emphasize that the picture is much less complete in the case γ < 0. In particular, for the Landau equation with Coulomb potential (γ = −3 in dimension N = 3), nothing is known beyond existence of weak solutions (see Villani [446] or the remarks in [10]). We now turn to the spatially inhomogeneous setting. It is a striking fact that no theorem of existence of classical small solutions of the Boltzmann equation without cut-off has ever been proven to this day, except maybe for the isolated results in [9] which still need further clarification. So we only discuss renormalized solutions. 12 Apparently, Monte Carlo methods cannot be directly applied to the study of the non-cutoff Boltzmann equation. The only method which seems able to directly deal with non-cutoff collision kernels, without making some a priori truncation, is the Fourier-based deterministic scheme described in Section 4.9.
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T HEOREM 10. Assume that the collision kernel B is given by (133)–(134), and Φ(|v − v∗ |) = |v − v∗ |γ with 0 ν < 2,
γ −N,
γ + ν < 2.
(135)
Let (f n ) be a sequence of solutions13 of the Boltzmann equation, satisfying uniform estimates of mass, energy, entropy and entropy dissipation: sup sup
n∈N t ∈[0,T ]
RN ×RN
f n (t, x, v) 1 + |x|2 + |v|2 + log f n (t, x, v) dx dv < +∞. (136)
T
sup
D f n (t, x, ·) dx dt < +∞.
(137)
n∈N 0 N Without loss of generality, assume that f n → f weakly in Lp ([0, T ]; L1 (RN x × Rv )). Then, (i) f is a renormalized solution of the Boltzmann equation with a defect measure; (ii) automatically, f n → f strongly in L1 .
C OROLLARY 10.1. Let f0 be an initial datum with finite mass, energy and entropy: f0 (x, v) 1 + |v|2 + |x|2 + log f0 (x, v) dx dv < +∞. RN ×RN
Then there exists a renormalized solution with a defect measure, f (t, x, v), of the Boltzmann equation, with f (0, ·, ·) = f0 . This theorem is proven in Alexandre and Villani [12], answering positively a conjecture by Lions [308]. The result holds in much more generality, for instance, it suffices that the angular collision kernel be nonintegrable (no need for a power-law singularity), and the kinetic collision kernel need not either take the particular form of a power-law, if it satisfies some very weak regularity assumption with respect to the relative velocity variable. And also, it is not necessary that the collision kernel split into the product of a kinetic and an angular collision kernel. We mention all these extensions because they are compulsory when one wants to include realistic approximations of the Debye collision kernel, which is not cut-off, but not in product form . . . . The strategy of proof is the following. First, by Dunford–Pettis criterion, the sequence (f n )n∈N is weakly (relatively) compact in L1 . Then, by the renormalized formulation, and the averaging lemmas, one shows that velocity-averages of the f n ’s are strongly compact. Then the entropy dissipation regularity estimates yield bounds of regularity in the v variable, outside of a small set and outside of a set where the f n ’s are very small. As 13 Either renormalized solutions, or renormalized solutions with a defect measure, or approximate solutions, as in Theorem 5.
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a consequence, the sequence (f n )n∈N can be very well approximated by velocity-averages, and therefore it lies in a strongly compact set (as in [311]). Let us comment on the range of parameters in (135). The assumption γ + ν < 2 is just a growth condition on the kinetic collision kernel, and is a natural generalization of the assumption γ < 2 in the DiPerna–Lions theorem; by the way, for inverse s-powers in three dimensions, the inequality γ + ν < 1 always holds true. But now, we see that there are two extensions: first, the possibility to choose ν ∈ [0, 2) (which is the optimal range), secondly, the possibility to have a nonintegrable kinetic collision kernel, provided that the singularity be homogeneous of degree −N . This feature allows to deal with Coulomb-like cross-sections in dimension 3. By the way, a problem which is left open is whether the theorem applies when the collision kernel presents a nonintegrable kinetic singularity of order −N but no angular singularity. Such collision kernels are unrealistic, but sometimes suggested as approximations of Debye collision kernels [162]. The renormalized formulation above is able to handle this case (contrary to the DiPerna–Lions renormalization), but without angular singularity the regularizing effect may be lost – or is it implied by the nonintegrable singularity, as some heuristic considerations [12] may suggest? A result quite similar to Theorem 10 (actually simpler) holds for the Landau equation, see Lions [311], and also Alexandre and Villani [13]. To this day, no clean implementation of a regularization effect has been done in the framework of spatially inhomogeneous small solutions. Desvillettes and Golse [176] have worked on an oversimplified model of the Boltzmann equation without cut-off, for which L∞ solutions can be constructed for free. For this model equation they prove immediate H α regularization for some α which is about 1/30. In fact, regularization for the spatially inhomogeneous Boltzmann equation without cutoff may be understood as a hypoellipticity problem – with the main problem that the diffusive operator is of nonlocal, nonlinear nature. F. Bouchut has recently communicated to us some very general methods to tackle hypoelliptic transport equations in a Sobolev space setting, via energy-type methods; certainly that kind of tools will be important in the future. 5. The Landau approximation In this section, we address the questions formulated in Section 2.7. In short, how to justify the replacement of Boltzmann’s operator by Landau’s operator in the case of Debye (= screened Coulomb) potential when the Debye length is very large compared to the Landau length? 5.1. Structure of the Landau equation We recall here the structure of the Landau operator, in asymmetric form: dv∗ a(v − v∗ ) g∗ (∇f ) − f (∇g)∗ , QL (g, f ) = ∇v · RN
(138)
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zi zj aij (z) = Ψ |z| δij − 2 . |z|
181
(139)
The Landau operator can also be rewritten as a nonlinear diffusion operator,
¯ = ¯ − bf a¯ ij ∂ij f − cf, ¯ QL (g, f ) = ∇ · a∇f
(140)
ij
where b = ∇ · a, c = ∇ · b, or more explicitly bj =
c=
∂i aij ,
i
∂j bj ,
j
and a¯ = a ∗ g,
b¯ = b ∗ g,
c¯ = c ∗ g.
There is a weak formulation, very similar to Boltzmann’s, for instance,
RN
QL (g, f )ϕ =
R2N
g∗ f TL ϕ dv dv∗ ,
where [TL ϕ](v, v∗ ) = −2b(v − v∗ ) · ∇ϕ(v) + a(v − v∗ ) : D 2 ϕ(v).
(141)
Compare this with the following rewriting of Maxwell’s weak formulation of the Boltzmann equation:
RN
QB (g, f )ϕ =
R2N
g∗ f T ϕ dv dv∗ ,
where [T ϕ](v, v∗ ) =
S N−1
B(v − v∗ , σ )(ϕ − ϕ) dσ.
(142)
5.2. Reformulation of the asymptotics of grazing collisions As we explained in Section 3.5, one expects that the Boltzmann operator reduce to the Landau operator when the angular collision kernel concentrates on grazing collisions, the total cross-section for momentum transfer being kept finite. The first rigorous proofs concerned the spatially homogeneous situation: Arsen’ev and Buryak [41] for a smooth kinetic collision kernel, Goudon [248] for a kinetic singularity
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of order less than 2, Villani [446] for a kinetic singularity of order less than 4. All proofs were based on variants of the weak formulations above, and used the symmetry v ↔ v∗ . In order to extend these results to the spatially inhomogeneous setting, there was need for a renormalized formulation which would encompass at the same time the Boltzmann and Landau collision operators. This was accomplished with the results about the Boltzmann equation without cut-off in [12]. Here is the renormalized formulation of the Landau equation: ¯ ¯ ) − 2bβ(f ) β (f )QL (g, f ) = −c¯ fβ (f ) − β(f ) + ∇ · ∇ · aβ(f −
β (f ) a∇β(f ¯ ) ∇β(f ). β (f )2
(143)
Again, β stands for a concave nonlinearity, typically β(f ) = f/(1 + δf ). If one notes that the second term in the right-hand side of (143) can be rewritten as QL (g, β(f )), there is an excellent analogy between this renormalization and the renormalization of the Boltzmann operator which was presented in Section 4.6. This is what makes it possible to pass to the limit. The convergence of the first and second terms in the renormalized formulation can be expressed in terms of the kernels S (appearing in the cancellation lemma) and T . This allows one to cover very general conditions for the asymptotics of grazing collisions, and this generality is welcome to treat such cases as the Debye approximation. Here we only consider a nonrealistic model case. Let (Bn )n∈N be a sequence of collision kernels (144) Bn (v − v∗ , σ ) = Φ |v − v∗ | bn (cos θ ), where the kinetic collision kernel Φ satisfies Φ |z| −−−→ 0, |z|→∞
Φ |z| ,
sup√
1<λ 2
Φ(λ|z|) − Φ(|z|) ∈ L1loc RN z , λ−1
(145)
(146)
and the family of angular collision kernels (bn )n∈N concentrates on grazing collisions, in the sense ⎧ ∀θ0 > 0, sup bn (cos θ ) −−−→ 0, ⎪ ⎪ n→∞ ⎨ θθ0 (147) ⎪ ⎪ ⎩ bn (k · σ )(1 − k · σ ) dσ −−−→ µ > 0, |k| = 1. n→∞
S N−1
Let Sn be the kernel associated to Bn as in Section 4.2, and Tn be the linear operator associated to Bn as in formula (142). Moreover, let Ψ |z| =
µ |z|2 Φ |z| , 4(N − 1)
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and let QL , TL be the associated quantities entering the Landau operator. Then, Sn |z| −−−→(N − 1)∇ · n→∞
z Ψ |z| |z|2
in weak-measure sense, and Tn −−−→ TL n→∞
in distributional sense. In this sense one can say that the sequence of Boltzmann kernels Qn approaches QL . These lemmas are not enough to pass to the limit. It still remains (1) to gain strong compactness in the sequence of solutions to the Boltzmann equation, (2) to pass to the limit in the last term of the renormalized solution. Task (2) is a very technical job, based on auxiliary entropy dissipation estimates and quite intricate computations, from which the reader is unlikely to learn anything interesting. On the other hand, we explain a little bit about the strong compactness.
5.3. Damping of oscillations in the Landau approximation As we have seen earlier, entropy dissipation bounds for singular Boltzmann kernels entail the appearance of strong compactness, or immediate damping of oscillations. In the case of the Landau equation, this is the same. It turns out that it is also the same if one considers a sequence of solutions of Boltzmann equations in which the collision kernel concentrates on grazing collisions, in the sense of (147). This is a consequence of the following variant of our joint results in [10]: P ROPOSITION 11. Assume that Bn (v − v∗ , σ ) Φ0 (|v − v∗ |)b0,n (cos θ ), where Φ0 is continuous, Φ(|z|) > 0 for |z| > 0, and b0,n concentrates on grazing collisions, in the sense of (147). Then there exists µ > 0 and a sequence α(n) → 0 such that ⎧ α(n) ⎪ ⎪ ⎪ sinN−2 θ bn (cos θ ) (1 − cos θ ) dθ −−−→ µ > 0, ⎨ n→∞ 0 π ⎪ ⎪ ⎪ sinN−2 θ bn (cos θ ) dθ ≡ ψ(n) −−−→ +∞, ⎩ α(n)
(148)
n→∞
and there exists K > 0 such that
π 0
sinN−2 θ bn (cos θ ) θ 2 |ξ |2 ∧ 1 dθ K min ψ(n), |ξ |2 .
(149)
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√ In √ particular, for any distribution function f , let F = χ f be obtained by multiplication of f with a smooth cut-off function χ , then
1 1 F #(ξ )2 dξ C max , 2 Dn (f ) + f L1 , 2 ψ(n) R |ξ |R
where Dn is the entropy dissipation functional associated with Bn , and C depends on f only via a lower bound for f dv and an upper bound for f (1 + |v|2 + | log f |) dv. As a consequence of this proposition, strong compactness is automatically gained in the asymptotics of grazing collisions. By the way, this simplifies already existing proofs [446] even in the spatially homogeneous setting.
5.4. Summary Here we give a precise statement from [13]. T HEOREM 12. Let Bn be a sequence of collision kernels concentrating on grazing collisions, in the sense of (144)–(147). Further assume that Φ(|z|) > 0 as |z| > 0. Let (f n )n∈N be a sequence of renormalized solutions of the Boltzmann equation (with a defect measure) ∂f n + v · ∇x f n = Qn f n , f n , ∂t satisfying uniform bounds of mass, energy, entropy, entropy dissipation. Without loss of generality, assume that f n → f in weak L1 . Then, the convergence is automatically strong, and f is a renormalized solution (with a defect measure) of the Landau equation with Ψ |z| =
λ |z|2 Φ |z| . 4(N − 1)
R EMARK . Theorem 12 allows for kinetic collision kernels with a strong singularity at the origin, but does not allow collision kernels which are unbounded at large relative velocities. This theorem includes all preceding results in the field, however in a spatially homogeneous situation one could reasonably hope that present-day techniques would yield an explicit rate of convergence (as n → ∞) when Φ is not too singular. On the other hand, when Φ(|z|) = 1/|z|3 , an improvement of this theorem even in the spatially homogeneous setting would require a much deeper understanding of the Cauchy problem for the Landau equation for Coulomb interaction.14 14 See the discussion in Section 1.3 of Chapter 2E.
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6. Lower bounds We conclude this chapter with estimates on the strict positivity of the solution to the Boltzmann equation. Such results are as old as the mathematical theory of the Boltzmann equation, since Carleman himself proved one of them. At the present time, these estimates are limited to the spatially homogeneous setting, and it is a major open problem to get similar bounds in the full, x-dependent framework in satisfactory generality. Therefore, we restrict the ongoing discussion to spatially homogeneous solutions. Even in this situation, more work remains to be done in the non-cutoff case.
6.1. Mixing effects First consider the case when Grad’s angular cut-off is satisfied, and Duhamel’s formula (100) applies. Then one is allowed to write
t
f (t, v)
e−
t s
Lf (τ,v) dτ
Q+ (f, f )(s, v) ds,
(150)
0
f (t, v) e−
t 0
Lf (τ,v) dτ
(151)
f0 (v),
where Lf = A ∗ f , A(z) = B(z, σ ) dσ . As a trivial consequence of (151), if f0 is strictly positive (resp. bounded below by a Maxwellian), then the same property will be true for f (t, ·). But a much stronger effect holds true: whatever the initial datum, the solution will be strictly positive at later times. Just to get an idea of this effect, assume that A is bounded from above and below, so that e−
∀s, t ∈ [0, T ],
t s
Lf (τ,v) dτ
KT > 0
for some constant KT depending on T . Then, as a consequence,
t
f (t, v) KT
Q+ (f, f )(s, v) ds,
0 t T.
(152)
0
Further assume that f0 α1B ,
α > 0,
(153)
where 1B is the characteristic function of some ball B in velocity space, without loss of generality B is centered on 0. From (151) it follows that f (t, ·) αKT 1B ,
0 t T.
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Now, plug this inside (152), to find that f (t, v) α
2
KT3
t
Q+ (1B , 1B )(s, v) ds,
0 t T.
0
But Q+ (1B , 1B ) is positive and bounded below in all the interior of the ball (1 + δ)B for δ small enough. In particular, there is a positive constant β such that f (t, v) α 2 KT3 β 1(1+δ)B . By an immediate induction, ∀t > 0, ∀v ∈ RN ,
f (t, v) > 0.
Precise estimates of this type have enabled A. Pulvirenti and Wennberg [392,393] to prove optimal (Gaussian-type) bounds from below on f , for the spatially homogeneous Boltzmann equation with Maxwellian or hard potentials. In this respect they have improved 2+ε on the old results by Carleman [119], who obtained a lower bound like e−|v| (ε > 0) in the case of hard spheres. Assumption (153) can also be dispended with, by use of the Q+ regularity. Also the proofs in [393] are sharp enough to prove existence of a uniform (in time) Maxwellian lower bound.
6.2. Maximum principle The author suggests another explanation for the immediate appearance of strict positivity, which is the maximum principle for the Boltzmann equation. The study of this principle is still under progress, so we cannot yet display explicit lower bounds obtained with this method; the most important feature is that it applies in the non-cutoff case. Let us just give an idea of it. Rewrite the spatially homogeneous Boltzmann equation as ∂f = ∂t
RN ×S N−1
dv∗ dσ Bf∗ (f − f )
+f
RN ×S N−1
dv∗ dσ
B(f∗
− f∗ ) .
(154)
We assume that we deal with a C ∞ solution, which is reasonable when the kinetic collision kernel is nice and when there is a nonintegrable angular singularity. The good point about the decomposition (154) is that it is well-defined15 even in the non-cutoff case. Assume now, by contradiction, that there is some point (t0 , v0 ) (t0 > 0) such that f (t0 , v0 ) = 0. Obviously, ∂f/∂t = 0 at (t0 , v0 ). Thus the left-hand side of (154), and also 15 By cancellation lemma, for instance, see Section 4.2.
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the second term on the right-hand side vanish at (t0 , v0 ). But, when v = v0 , f − f 0, for all v . Thus the integrand in the first term on the left-hand side of (154) is nonnegative, but the integral vanishes, so f = f = 0, for all v . This entails that f is identically 0, which is impossible. In other words, we have recovered the weak result that f (t, ·) is strictly positive on the whole of RN for t > 0.
6.3. Summary T HEOREM 13. Let B be a collision kernel of the form B(v − v∗ , σ ) = |v − v∗ |γ b(cos θ ), where γ 0. Let f0 be an initial datum with finite mass and energy, and f (t, ·) be a solution of the spatially homogeneous Boltzmann equation. Then, (i) if Grad’s angular cut-off condition holds, then for any t0 > 0, there exists a Maxwellian distribution M(v) such that for all t t0 , f (t, v) M(v); (ii) if Grad’s angular cut-off condition does not hold, and f (t, v) is a C ∞ function on N (0, +∞) × RN v , then for any t > 0, v ∈ R , f (t, v) > 0. Point (i) is due to A. Pulvirenti and Wennberg [392,393]. Point (ii) was first proven by Fournier, using delicate probabilistic methods, in the special case of the Kac equation without cut-off [218], then also for the two-dimensional Boltzmann equation under technical restrictions [218]. Then it was proven in a much simpler way by the author, with the analytical method sketched above. Current work is aiming at transforming this estimate into a quantitative one. We note that in the case of the Landau equation with Maxwellian or hard potential [182], one can prove a theorem similar to that of A. Pulvirenti and Wennberg by means of the standard maximum principle for parabolic equations.16
16 Actually, in [182] the stated result is not uniform in time, but, as suggested to us by E. Carlen, a uniform bound is easily obtained by tracing back all the constants: since they are uniform for t ∈ (ε, 2ε) and do not depend on the initial datum, it follows that they are uniform in t > ε.
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CHAPTER 2C
H Theorem and Trend to Equilibrium Contents 1. A gallery of entropy-dissipating kinetic models . . . . . . . . . . . . 1.1. Spatially homogeneous models . . . . . . . . . . . . . . . . . . 1.2. Spatially inhomogeneous models . . . . . . . . . . . . . . . . . 1.3. Related models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. General comments . . . . . . . . . . . . . . . . . . . . . . . . . 2. Nonconstructive methods . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Classical strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Why ask for more? . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Digression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Entropy dissipation methods . . . . . . . . . . . . . . . . . . . . . . 3.1. General principles . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Entropy–entropy dissipation inequalities . . . . . . . . . . . . . 3.3. Logarithmic Sobolev inequalities and entropy dissipation . . . . 4. Entropy dissipation functionals of Boltzmann and Landau . . . . . . 4.1. Landau’s entropy dissipation . . . . . . . . . . . . . . . . . . . 4.2. Boltzmann’s entropy dissipation: Cercignani’s conjecture . . . . 4.3. Desvillettes’ lower bound . . . . . . . . . . . . . . . . . . . . . 4.4. The Carlen–Carvalho theorem . . . . . . . . . . . . . . . . . . . 4.5. Cercignani’s conjecture is almost true . . . . . . . . . . . . . . 4.6. A sloppy sketch of proof . . . . . . . . . . . . . . . . . . . . . . 4.7. Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Trend to equilibrium, spatially homogeneous Boltzmann and Landau 5.1. The Landau equation . . . . . . . . . . . . . . . . . . . . . . . . 5.2. A remark on the multiple roles of the entropy dissipation . . . . 5.3. The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . 5.4. Infinite entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Gradient flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Metric tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Convergence to equilibrium . . . . . . . . . . . . . . . . . . . . 6.3. A survey of results . . . . . . . . . . . . . . . . . . . . . . . . . 7. Trend to equilibrium, spatially inhomogeneous systems . . . . . . . 7.1. Local versus global equilibrium . . . . . . . . . . . . . . . . . . 7.2. Local versus global entropy: discussion on a model case . . . . 7.3. Remarks on the nature of convergence . . . . . . . . . . . . . . 7.4. Summary and informal discussion of the Boltzmann case . . . .
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In Chapter 2A we have discussed Boltzmann’s H theorem, and the natural conjecture that the solution of Boltzmann’s equation converges towards statistical equilibrium, which is a global Maxwellian distribution. In this chapter we shall study this problem of trend to equilibrium, and also enlarge a little bit the discussion to models of collisional kinetic theory which are variants of the Boltzmann equation: for instance, Fokker–Planck-type equations, or simple models for granular media. The Cauchy problem for these equations is usually not so challenging as for the Boltzmann equation, but the study of trend to equilibrium for these models may be very interesting (both in itself, and to enlighten the Boltzmann case). As a general fact, one of the main features of many collisional kinetic systems is their tendency to converge to an equilibrium distribution as time becomes large, and very often a thermodynamical principle underlies this property: there is a distinguished Lyapunov functional, or entropy, and the equilibrium distribution achieves the minimum of this functional under constraints imposed by the conservation laws. In Section 1 we shall review some of these models. For each example, we shall be interested in the functional of entropy dissipation, defined by the equation D(f0 ) = −
d E f (t) , dt t =0
where E is the Lyapunov functional, and (f (t))t 0 the solution to the equation under study, f (0) = f0 . We shall use the denomination “entropy dissipation” even when E is not the usual Boltzmann entropy. Traditional approaches for the study of trend to equilibrium rely on soft methods, like compactness arguments, or linearization techniques, which ideally yield rates of convergence. In Section 2 we briefly review both methods and explain why they cannot yield definitive answers, and should be complemented with other, more constructive methods. This will lead us to discuss entropy dissipation methods, starting from Section 3. In Section 4, we expose quantitative versions of the H theorem for the Boltzmann and Landau operators, in the form of some functional inequalities. Then in Section 5 we show how these inequalities can be used for the study of the trend to equilibrium for the spatially homogeneous Boltzmann and Landau equations. Section 6 is devoted to a class of collision models which exhibit a particular gradient structure. Specific tools have been devised to establish variants of the H theorem in this case. Finally, Section 7 deals with the subtle role of the position variable for spatially inhomogeneous models. The construction of this area is only beginning.
1. A gallery of entropy-dissipating kinetic models Let us first review some of the basic models and the associated entropy functionals, equilibria, entropy dissipation functionals. We shall not hesitate to copy-cut some of the formulas already written in our introductory chapter.
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1.1. Spatially homogeneous models These models read ∂f = Q(f ), ∂t
t 0, v ∈ RN ,
where the collision operator Q, linear or not, may be (1) the Boltzmann operator, Q(f ) = QB (f, f ) =
RN
dv∗
S N−1
dσ B(v − v∗ , σ )(f f∗ − ff∗ );
(155)
then there are three conservation laws: mass, momentum and energy. Moreover, the natural Lyapunov functional is the H -functional, H (f ) =
RN
f log f,
and its dissipation is given by the by now familiar functional D(f ) =
1 4
dv dv∗ dσ B(v − v∗ , σ )(f f∗ − ff∗ ) log
f f∗ 0. ff∗
(156)
Define ρ, u, T by the usual formulas (1), then the equilibrium is the Maxwellian |v−u|2
e− 2T . M(v) = M (v) = (2πT )N/2 f
I MPORTANT REMARK . We shall only consider here the case of the Boltzmann equation with finite temperature. In the case of infinite temperature, almost nothing is known, except for the very interesting recent contribution by Bobylev and Cercignani [81]. It should be noted that, since M has the same moments as f up to order 2, H (f ) − H (M) =
RN
f log
f , M
which is nothing but the Kullback relative entropy of f with respect to M, and that we shall denote by H (f |M). Generally speaking, the Kullback relative entropy between two probability densities (or more generally two nonnegative distributions) f and g is given by the formula f H (f |g) = f log . (157) g
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It is well-known1 that H (f |g) 0 as soon as f and g have the same mass; (2) the Landau operator, ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ Q(f ) = QL (f, f ) = ∇v ·
RN
dv∗ a(v − v∗ ) f∗ (∇f ) − f (∇f )∗ ,
⎪ ⎪ zz ⎪ ⎪ ⎩ aij (z) = Ψ |z| δij − i j ; |z|2
(158) (159)
in this case there are also three conservation laws, and the natural Lyapunov functional is also the H -functional. Now the entropy dissipation is
DL (f ) =
1 2
RN ×RN
2 ff∗ Ψ |v − v∗ | Π(v − v∗ ) ∇(log f ) − ∇(log f ) ∗ , (160)
where Π(z) stands for the orthogonal projector onto z⊥ . As for the equilibrium state, it is still the same as for the Boltzmann equation; (3) the linear Fokker–Planck operator, Q(f ) = QF P (f ) = ∇v · ∇v f + f v .
(161)
In this case there is only one conservation law, the mass (ρ = f dv), and the natural Lyapunov functional is the free energy: this is the sum of the H -functional and the kinetic energy,
E(f ) =
RN
f log f +
RN
f
|v|2 dv. 2
(162)
Moreover, the entropy dissipation is 2 ∇f DF P (f ) = + v dv, f f RN
which can be rewritten as the so-called relative Fisher information of f with respect to M, thereafter denoted by I (f |M). More generally, I (f |g) =
f 2 f ∇ log . g RN
f [− log(g/f ) + g/f − 1] (or as g[(f/g) log(f/g) − (f/g) + 1]) and to use the inequality log X X − 1 (or X log X X − 1). Compare with the Cercignani–Lampis trick of Equation (50). 1 The classical proof is to rewrite (157) as
(163)
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Compare with the definition of the relative Kullback entropy (157); (4) a coupled Fokker–Planck operator, like ρ α ∇v · T ∇v f + f (v − u) , where 0 α 1 and ρ, u, T are coupled to f by the usual formulas (1). In this case there are three conservation laws, the natural Lyapunov functional is the H -functional, and the entropy dissipation is ρ
α R
f f ∇v log f N M
2 = ρ α I f |M f .
The equilibrium is the same as for the Boltzmann operator. Other couplings are possible: one may decide to couple only T , or only u . . . ; (5) some entropy-dissipating model for granular flow, like the one-dimensional model proposed in [70], Q(f ) = ∇v · f ∇v (f ∗ U ) ,
(164)
where U (z) = |z|3 /3. Then there are two conservation laws, mass and momentum; and the natural Lyapunov functional is E(f ) =
1 2
R2N
f (v)f (w)U (v − w) dv dw,
(165)
while its dissipation is D(f ) =
RN
f |∇U ∗ f |2 .
Moreover the equilibrium is ρδu , i.e., a multiple of the Dirac mass located at the mean velocity. A particular feature of this model is its gradient flow structure. Generally speaking, models of the form ∂f δE =∇· f ∇ , (166) ∂t δf where E is some energy functional and δE/δf stands for its gradient with respect to the usual L2 structure, can be considered as gradient flows [364,365], via geometric and analytical considerations which are strongly linked with the Wasserstein distance.2 An 2 The Wasserstein distance is defined by Equation (244). The gradient structure is explained in Section 6.1.
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integration by parts shows that solutions of (166) admit E as a Lyapunov functional, and the dissipation is given by D(f ) =
δE 2 . f ∇ δf RN
Falling into this category is in particular the model for granular flow discussed in [68], in which one adds up the collision operators (164) and (161).
1.2. Spatially inhomogeneous models These models can be written in the general form ∂f + v · ∇x f + F (x) · ∇v f = Q(f ), ∂t
t 0, x ∈ RN , v ∈ RN ,
(167)
where F is the sum of all macroscopic forces acting on the system, and Q is one of the collision operators described in the previous paragraph (acting only on the velocity variable!). In the sequel we shall only consider the situation when the total mass of the gas is finite; without loss of generality it will be normalized to 1. We mention however that the case of infinite mass deserves interest and may be studied in the spirit of [310]. If the total mass is finite, then among the forces must be a confinement which prevents the system from escaping at infinity, and ensures the existence of a relevant equilibrium state. There are several possibilities: Potential confinement. Assume that the particles interact with the background environment via some fixed potential, V (x). Then the force is just F (x) = −∇V (x). The minimum requirement for V to be confining is e−V ∈ L1 . Since V is defined up to an additive constant, one can assume without loss of generality that RN
e−V (x) dx = 1.
The presence of the confining potential does not harm the conservation of mass, of course; on the other hand, when Q is a Boltzmann-type collision operator (with three conservation laws), it usually destroys the conservations of momentum and energy. Instead, there is conservation of the total mechanical energy,
|v|2 dx dv. f (x, v) V (x) + 2 RN ×RN
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And as far as the entropy is concerned, it is not changed for Boltzmann-type models: this is still the usual H -functional, the only difference being that now the phase space is N RN x × Rv : H (f ) =
RN ×RN
f log f.
This similarity is a consequence of the physical assumption that collisions are localized in space. For the linear Fokker–Planck equation, the situation is different: to the free energy one has to add the potential energy, so the natural Lyapunov functional is E(f ) =
RN ×RN
f log f +
RN ×RN
|v|2 f (x, v) V (x) + dx dv. 2
Box confinement. Another possible confinement is when the system is enclosed in a box X ⊂ RN x , with suitable boundary conditions. The most standard case, namely specular reflection, is a limit case of the preceding one: choose V = +∞ outside of X, V = const. within X. When specular reflection is imposed, then the energy conservation is restored (not momentum conservation), and the Lyapunov functional is the same as in the spatially homogeneous case, only integrated with respect to the x variable. For other boundary conditions such as diffusive, the entropy functional should be modified [141,143]. Torus confinement. This is the most convenient case from the mathematical point of view: set the system in the torus TN x , so that there are no boundaries. Physicists also use such models for discussing theoretical questions, and numerical analysts sometimes find them convenient. Additional force terms. Many models include other force terms, in particular selfconsistent effects described by mean-field interactions: typically, F (x) = −∇Φ(x),
Φ = φ ∗ ρ,
ρ=
RN
f dv,
where φ is an interaction potential between particles. As we already mentioned, from the physical point of view it is not always clear whether interactions should be modelled via collisions, or mean-field forces, or both . . . . A very popular model is the Vlasov–Fokker– Planck equation, in which the collision operator is the Fokker–Planck operator and the forces include both confinement and self-consistent interaction. Let us rewrite the model explicitly: ⎧ ∂f ⎪ ⎪ ⎨ ∂t + v · ∇x f + F (x) · ∇v f = ∇v · (∇v f + f v), ⎪ ⎪ ⎩ F = −∇(V + φ ∗ ρ), ρ = f dv. RN
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If the interaction is Coulomb, then one speaks of Vlasov–Poisson–Fokker–Planck model; this case is very singular, but it has a lot of additional structure because (by definition) the potential φ is the fundamental solution of the Laplace operator. In the self-consistent case, one has to add a term of interaction energy to the free energy: 1 2
RN ×RN
ρ(x)ρ(y)φ(x − y) dx dy.
Then the entropy dissipation is unchanged. Let us now turn to equilibrium states. Their classification in a spatially inhomogeneous context is quite a tedious task. Many subcases have to be considered, the dimension of the space comes into play, and also the symmetries of the problem. We are not aware of any systematic treatment; we shall only consider the most typical situations. • The Boltzmann (or Landau) equation in a box. Then, in dimension N = 2, 3 there is a unique steady state which takes the form of a global Maxwellian: f (x, v) = M(v). The mass and temperature of the Maxwellian are determined by the conservation laws, while the mean velocity is 0. This result holds true on the condition that the box be not circular in dimension N = 2, or cylindric in dimension N = 3 (i.e., with an axis of symmetry). For this one can consult [254,167,143]. • The Boltzmann (or Landau) equation in a confining potential. Then, the unique steady state has the form f (x, v) = e−V (x)M(v). Again, the mass and temperature of M are determined by the conservation laws, and the mean velocity is 0. This result holds true if the potential V is not quadratic; if it is, then there exist periodic (in time) solutions, which can be considered as stationary even if they are time-dependent. This was already noticed by Boltzmann (see, for instance, [143]). • The Boltzmann (or Landau) equation in a torus. Then, the unique steady state has the form f (x, v) = M(v) where M is an absolute Maxwellian. The mass and temperature, but also the mean velocity of M are determined by the conservation laws. • The Fokker–Planck equation in a confining potential. Then, the unique steady state is f (x, v) = e−V (x)M(v), where M is the Maxwellian with unit temperature and zero mean. The mass is of course determined by the conservation law. • The Vlasov–Fokker–Planck equation in a confining potential. In general there is a unique steady state in this situation, and it takes the form f (x, v) = ρ∞ (x)M(v), where M is the Maxwellian with unit temperature and zero mean. The density ρ∞ is nonexplicit, but solves a nonlinear equation of the form e−(V +φ∗ρ∞ ) . −(V +φ∗ρ∞ ) dx RN e
ρ∞ ∗ φ =
There are also variational formulations of this problem. In the case of the Vlasov– Poisson–Fokker–Planck equation, a detailed survey of the situation is given by Dolbeault [197]. In all the preceding discussion, we have avoided the models for granular collisions, Equation (164). A naive guess would be that the natural Lyapunov functional, in the spatially inhomogeneous case, is obtained by integrating its spatially homogeneous
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counterpart, Equation (165), in the x variable. This is false! Because the transport operator −v · ∇x may have an influence on the evolution of this functional.
1.3. Related models The following models are not kinetic models, but have come to be studied by members of the kinetic community because of the unity of methods and problematics. – The spatial Fokker–Planck equation, or Smoluchowski3 equation [399] ∂ρ = ∇x · ∇x ρ + ρ∇V (x) , ∂t
t 0, x ∈ RN .
(168)
One always assume e−V ∈ L1 , and without loss of generality e−V should be a probability measure, just as ρ. For this equation the natural Lyapunov functional is the free energy, or relative entropy, H (ρ|e−V ), and the entropy dissipation coincides with the relative Fisher information, I (ρ|e−V ). For a summary of recent studies concerning the trend to equilibrium for (168), the reader may consult Arnold et al. [39], or Markowich and Villani [330]. – Equations modelling porous medium with confinement: ∂ρ = ∇x · ∇x P (ρ) + ρ∇V (x) , ∂t
t 0, x ∈ RN ,
(169)
where P is a nonlinearity, P (ρ) standing for a pressure term, for instance P (ρ) = ρ γ . In this last case (power law), equations like (169), with a quadratic confinement potential, naturally arise as rescaled versions of their counterparts without confinement. The natural Lyapunov functional for (169) is
A(ρ) dx +
ρV (x) dx,
where P (ρ) = ρA (ρ) − A(ρ). The trend to equilibrium for (169) has been studied independently by Carrillo and Toscani [131], Dolbeault and Del Pino [163], Otto [364] for the power law case, then more generally by Carrillo et al. [129]. One of the most remarkable features of Equations (168) and (169) is that they have the form of a gradient flow, δE ∂ρ =∇ · ρ∇ . ∂t δρ For a general discussion of the implications, see, for instance, Otto and Villani [365], Markowich and Villani [330]. 3 There are several types of equations which are called after Smoluchowski!
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1.4. General comments In this section we shall informally discuss the features which may help the trend to equilibrium, or on the contrary make it more difficult – both from the physical and from the mathematical point of view. First of all, the distribution tails are usually at the origin of the worst difficulties. By distribution tails, we mean how fast the distribution function decreases as |v| → ∞, or |x| → ∞. This is not only a technical point; Bobylev has shown that large tails could be a true obstacle to a good trend to equilibrium for the Boltzmann equation, even in the spatially homogeneous case. More precisely, he proved the following result [79]. Consider the spatially homogeneous Boltzmann equation with Maxwell collision kernel (with or without cut-off), and fix the mass, momentum, energy of the initial datum f0 . Let M(v) be the corresponding equilibrium state. Then, for any ε > 0 one can construct an initial datum f0 = f0ε such that the associated solution f ε (t, v) of the Cauchy problem satisfies ∀t 0,
$ ε $ $f (t, ·) − M $ Kε e−εt ,
Kε > 0.
At this point we should make a remark to be honest: an eye observation of a plot of these particular solutions will show hardly any departure from equilibrium, because most of the discrepancy between f ε and M is located at very high velocities – and because the constant Kε is rather small. This illustrates the general fact that precise “experimental” information about rates of convergence to equilibrium is very difficult to have, if one wants to take into account distribution tails. Moreover, recent studies have shown that the Boltzmann equation, due to its nonlocal nature, is more sensitive to this tail problem than diffusive models like Landau or Fokker– Planck equations. For the latter equations, it is not possible to construct “pathological” solutions as Bobylev; the trend to equilibrium is typically exponential, with a rate which is bounded below. We shall come back to this point, which by the way is also folklore in the study of Markov processes: it is known that jump processes have more difficulties in going to equilibrium than diffusion processes. Next, it is clear that the more collisions there are, the more likely convergence is bound to be fast. This is why the size of the collision kernel does matter, in particular difficulties arise in the study of hard potentials because of the vanishing of the collision kernel at zero relative velocities; and also in the study of soft potentials because of the vanishing of the collision kernel for large relative velocities. A common belief is that the problem is worse for soft potentials than for hard. Also note that hard potentials are associated with a good control of the distribution tails, while soft potentials are not. Studies of the linearized operator show that in principle, one could expect an exponential decay to equilibrium for the spatially homogeneous Boltzmann equation with hard or Maxwellian potentials (under strong control of the distribution tails), while for soft α potentials the best that one could hope is decay like O(e−t ) for some α ∈ (0, 1) (see Caflisch [111]). This is of course related to the fact that there is a spectral gap in the first case, not in the second one. In the case of Boltzmann or Landau models (or some versions of coupled Fokker– Planck), the collision frequency also depends on the density of particles. This of course can
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be seen via the fact that Boltzmann and Landau operators are quadratic, while the linear Fokker–Planck is not. As a consequence, the trend to equilibrium should be extremely slow at places where the density stays low: typically, very large positions. Therefore, the trend to equilibrium is expected to hold on extremely long scales of times when one considers the Boltzmann equation in a confinement potential, as opposed to the Boltzmann equation in a finite box. In the x-dependent case, a strong mathematical difficulty arises: the existence of local equilibria. These are states which make the entropy dissipation vanish, but are not stationary states. In fact they are in equilibrium with respect to the velocity variable, but not with respect to the position variable; for instance they are local Maxwellians Mx (v), with parameters ρ, u, T depending on x. Of course the trend to equilibrium is expected to be slowed down whenever the system comes close to such a state. We shall discuss this problem in more detail in Section 7. Finally, a gradient flow structure often brings more tools to study the trend to equilibrium. We shall see this in the study of such models as (161) or (164). As we mentioned in Section 2.4 of Chapter 2A, in the case of the spatially homogeneous Boltzmann equation no gradient flow structure has been identified. Moreover, for all the spatially inhomogeneous equations which are considered here, the existence of the local equilibria rules out the possibility of such a structure.
2. Nonconstructive methods In this section, we briefly review traditional methods for studying the convergence to equilibrium.
2.1. Classical strategy A preliminary step of (almost) all methods is to identify stationary states by searching T for solutions of the functional equation D(f ) = 0, or more generally 0 D(f (t)) dt = 0. Once uniqueness of the stationary solution has been shown, then weak convergence of the solution towards equilibrium is often an easy matter by the use of compactness tools. Uniqueness may hold within some subclass of functions which is left invariant by the flow. For instance, in the case of the spatially homogeneous Boltzmann equation, it is easy to prove weak convergence as n → ∞ of f (n + t, v)n∈N towards the right Maxwellian distribution in weak-Lp ([0, T ] × RN ), as soon as lim lim sup
R→∞ t →∞
|v|R
f (t, v)|v|2 dv = 0.
(170)
Condition (170), thereafter referred to as “tightness of the energy”, ensures that there is no leak of energy at large velocities, and that f (t, ·) does converge towards the right Maxwellian distribution – and not towards a Maxwellian with too low temperature. In all
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the sequel, we will assume that the moments of f are normalized, so that the equilibrium distribution is the standard Maxwellian M with zero mean and unit temperature. As a typical result, under general conditions Arkeryd [17] proved that the solution to the spatially homogeneous Boltzmann equation with hard potentials does converge to M, weakly in L1 , as t → ∞. This result is facilitated by the fact that Equation (170) is very easy to prove for hard potentials, while it is a (seemingly very difficult) open problem for soft potentials. In the framework of the spatially homogeneous Boltzmann equation with Maxwellian collision kernel, other approaches are possible, which do not rely on the entropy dissipation. Truesdell [274] was the first one to use such a method: he proved that all spherical moments satisfy closed differential equations, and converge towards corresponding moments of M. This implies weak convergence of f (t, ·) towards M. Also contracting metrics4 can be used for such a purpose along the ideas of Tanaka [414, 415]. A refinement is to prove strong convergence of f (t, ·) towards M as t → ∞, for instance, as a consequence of some uniform (in time) smoothness estimates. The first result of this kind is due to Carleman [118]: he proved uniform equicontinuity of the family (f (t, ·))t 0 when f is the isotropic solution of the spatially homogeneous Boltzmann equation with hard spheres, assuming that the initial datum decays in O(1/|v|6 ). As a consequence, he recovered uniform convergence to equilibrium. This method was improved by Gustafsson [270] who proved strong Lp convergence for the solution of the spatially homogeneous Boltzmann equation with hard potentials, under an ad hoc Lp assumption on the initial datum. In the much more general framework of the spatially inhomogeneous Boltzmann equation, by use of the Q+ regularity, Lions [308] proved strong L1 compactness as t → ∞, say when the system is confined in a torus. This however is not sufficient to prove convergence, because there is no clue of how to prove the spatially-inhomogeneous variant of (170), dx f (t, v)|v|2 dv = 0. lim lim sup R→∞ t →∞
TN
|v|R
At this point we have to recognize that there is, to this date, no result of trend to equilibrium in the spatially inhomogeneous context, except in the perturbative framework of closeto-equilibrium5 solutions: see, for instance, [286] (perturbation setting in whole space) or [404] (in a bounded convex domain) – with just one exception: the case of a box with uniform Maxwellian diffuse boundary conditions, which was solved by Arkeryd and Nouri [35] in a non-perturbative setting. On the contrary, it is rather easy to prove convergence to equilibrium for, e.g., the spatially inhomogeneous linear Fokker–Planck equation. Once strong convergence to equilibrium has been established (for instance, in the case of the spatially homogeneous Boltzmann equation with hard potentials), a natural 4 See Section 2 in Chapter 2D. 5 Of course it is not a very satisfactory situation if one is able to prove convergence to equilibrium only when
one starts extremely close to equilibrium . . . .
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refinement is to ask for a rate of convergence. In the “good” cases, a hard work leads to exponential rates of decay thanks to linearization techniques and the study of the spectral gap of the linearized operator. This strategy was successfully applied by Arkeryd [23], Wennberg [456] to the spatially homogeneous Boltzmann equation with hard, or Maxwellian potentials. In the spatially inhomogeneous context, it was developed by the Japanese school under the assumption that the initial datum is already extremely close to equilibrium. In the case of soft potentials, though there is no spectral gap for the linearized operator, β Caflisch [111] was able to prove convergence to equilibrium like e−t for some exponent β ∈ (0, 1) – also under the assumption that the initial datum belong to a very small neighborhood of the equilibrium.
2.2. Why ask for more? The preceding results, as important as they may be, cannot be considered as a definitive answer to the problem of convergence to equilibrium. There are at least two reasons for that: (1) Non-constructiveness. The spectral gap (when it exists, which is not always the case!) is usually nonexplicit: for the Boltzmann equation with hard or soft potentials there is only one exception, the spatially homogeneous operator with Maxwellian collision kernel. What is more problematic, nobody knows how to get estimates on its size: usual arguments for proving its existence rely on Weyl’s theorem, which asserts that the essential spectrum is invariant under compact perturbation. But this theorem, which is based on a compactness argument, is nonexplicit . . . . Another problem arises because the natural space for the linearized operator (the space in which it is self-adjoint) is typically L2 (M −1 ), endowed with the norm f 2L2 (M −1 ) = 2 f /M, which is of course much narrower than the natural spaces for the Cauchy problem (say, Lebesgue or Sobolev spaces with polynomial weights). A new compactness argument is needed [457] to prove the existence of a spectral gap in these much larger spaces. R EMARK . This problem of functional space arises even for linear equations! For instance, if one considers the Fokker–Planck equation, then the spectral gap exists in the functional space L2 (M −1 ), but one would like to prove exponential convergence under the sole assumption that the initial datum possess finite entropy and energy. (2) Nature of the linearization procedure. In fact, even if linearization may predict an asymptotic rate of convergence, it is by nature unable to yield explicit results. Indeed, it only shows exponential convergence in a very small neighborhood of the equilibrium: a neighborhood in which nonlinearities are negligible in front of the linear terms. It cannot say anything on the time the solution needs to enter such a neighborhood . . . . This of course does not mean that linearization is in essence a bad method, but that it is a valuable method only for perturbations of equilibria.
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Entropy dissipation methods have been developed to remedy these problems, and yield explicit estimates of trend to equilibrium in a fully nonlinear context. We note that these methods are not the only effective methods in kinetic theory: other techniques, which have been developed in the particular framework of Maxwellian collision kernels, will be reviewed in Chapter 2D. Thus, the ideal mathematical situation, combining the power of both entropy methods and linearization techniques, would be the following. From a starting point which is far from equilibrium, an entropy method applies to show that the solution approaches equilibrium, possibly with a non-optimal rate (maybe not exponential . . .). After some explicit time, the solution enters a small neighborhood of equilibrium in which linearization applies, and a more precise rate of convergence can be stated. For this plan to work out, it would seem necessary to (1) refine linearization techniques to have explicit bounds on the spectral gap, (2) establish very strong a priori estimates, so that convergence in entropy sense imply a much stronger convergence, in a norm welladapted to linearization – or (2 ) show that the solution can be decomposed into the sum of an exponentially small part, and a part which is bounded in the sense of this very strong norm. 2.3. Digression At this point the reader may ask why we insist so much on explicit estimates. This of course is a question of personal mathematical taste. We do believe that estimates on the qualitative behavior of solutions should always be explicit, or at least explicitable, and that a compactness-based argument showing trend to equilibrium cannot really be taken seriously. First because it does not ensure that the result is physically realistic, or at least that it is not unrealistic by many orders of magnitude. Secondly because of the risk that the constants involved be so huge as to get out of the mathematical range which is allowed by the model. For instance, what should we think of a theorem predicting trend to equilibrium −1000 t ? The corresponding time scale is certainly much larger than the time scale like e−10 on which the Boltzmann description may be relevant.6 Of course, asking for realistic estimates may be a formidable requirement, and often one may already be very lucky to get just constructive estimates. Only when no such estimates are known, should one take into account nonexplicit bounds, and they should be considered as rough results calling for improvements. This is why, for instance, we have discussed the results of propagation, or appearance, of strong compactness in the context of the Cauchy problem for renormalized solutions . . . . 3. Entropy dissipation methods 3.1. General principles The main idea behind entropy dissipation methods is to establish quantitative variants of the mechanism of decreasing of the entropy: in the case of the Boltzmann equation, this 6 See the discussion at the end of Section 2.4.
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is the H theorem. This approach has the merit to stand upon a clear physical basis, and experience has shown its robustness and flexibility. RULE 1. The “discrepancy” between a distribution function f and the equilibrium f∞ should not be measured by the L1 norm, but rather by E[f |f∞ ] ≡ E(f ) − E(f∞ ), thereafter called relative entropy by abuse of language. Thus, one should not try to prove that f (t) converges to f∞ in L1 , but rather show that E(f (t)) → E(f∞ ) as t → ∞, which will be called “convergence in relative entropy”. A separate issue is to understand whether convergence in relative entropy implies convergence in some more traditional sense. RULE 2. One considers as a main object of study the entropy dissipation functional D. Of course, the definition of the entropy dissipation relies on the evolution equation; but it is important to consider D as a functional that can be applied to any function, solution or not of the equation. RULE 3. One tries to quantify the following idea: if, at some given time t, f (t) is far from f∞ , then E(f ) will decrease notably at later times. Before turning to less abstract considerations, we comment on the idea to measure the distance in terms of the entropy, rather than, say, in terms of the well-known L1 distance. A first remark is that there is no physical meaning, in the context of kinetic equations, in L1 distance. Some rather violent words by Truesdell will illustrate this. After proving exponential convergence of all moments in the framework of the spatially homogeneous Boltzmann equation with Maxwell collision kernel, he adds [274, p. 116] “Very likely it can be shown that [the solution] itself approaches Maxwellian form, but there is little interest in this refinement.” A justification of this opinion is given on p. 112: “Since apart from the entropy it is only the moments of the distribution function that have physical significance, the result sought is unnecessarily strong”. Thus, at the same time that he attacks the relevance of L1 results, Truesdell implicitly supports entropy results . . . . A second remark is that, very often, convergence of the entropy implies convergence in L1 sense. In the case of the H -functional, or more generally when E(f ) − E(f∞ ) takes the form of a relative Kullback entropy, this is a well-known result. Indeed, the famous (and elementary) Csiszár–Kullback–Pinsker inequality states that whenever f and g are two probability distributions, 1 f − g 2L1 2
f log
f = H (f |g). g
In many other instances, especially when a gradient flow structure is present, the quantity E(f ) − E(f∞ ) can also be shown to control some power of the Wasserstein distance.7 For this see in particular Otto and Villani [365]. A basic example is the Talagrand inequality, 1 W (f, M)2 H (f |M), 2 7 Equation (244) below.
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where M is the zero-mean, unit-temperature Maxwellian distribution. Usually, one can then obtain control of the L1 norm via some ad hoc interpolation procedure [130]. As a final remark, we comment on the entropy dissipation equality itself – say in the case of the Boltzmann equation. As we saw, formally, solutions of the spatially homogeneous Boltzmann equation satisfy the identity d H f (t, ·) = −D(f ), dt but when is this rigorous? It was actually proven by Lu [326] that this equality always holds for hard potentials with cut-off, under the sole assumptions that the initial datum has finite mass, energy and entropy. In fact, we shall always work under much stronger conditions. Thus, in all the sequel, we shall always consider situations in which the estimates for the Cauchy problem are strong enough, that the entropy dissipation identity can be made rigorous. Such is not the case, for instance, in the framework of the DiPerna–Lions theory of renormalized solutions.8
3.2. Entropy–entropy dissipation inequalities When trying to implement the preceding general principles, one can be lucky enough to prove an entropy–entropy dissipation inequality: this is a functional inequality of the type D(f ) Θ E[f |f∞ ] ,
(171)
where H → Θ(H ) is some continuous function, strictly positive when H > 0. The main idea is that “entropy dissipation controls relative entropy”. Such an inequality implies an immediate solution to the problem of trend to equilibrium. Indeed, let f (t) be a solution of the evolution equation. Since D(f (t)) = −(d/dt)E[f (t)|f∞ ], it follows that the relative entropy H (t) ≡ E[f (t, ·)|f∞ ] satisfies the differential inequality −
d H (t) Θ H (t) . dt
(172)
This implies that H (t) → 0 as t → +∞, and if the function Θ is known with enough details, one can compute an explicit rate of convergence. For instance, a linear bound like D(f ) 2λE[f |f∞ ] will entail exponential convergence to equilibrium, relative entropy converging to 0 like e−2λt . On the other hand, an exponent bigger than 1, D(f ) KE[f |f∞ ]1+α
(K > 0, α > 0)
8 In any case, this theory should be hopelessly excluded from any study of trend to equilibrium until energy conservation, and even local energy conservation, has been proven.
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will entail “polynomial” rate of convergence to equilibrium, the entropy going down like O(t −1/α ). Situations in which the exponent is lower than 1 are very rare; in such cases the system converges to equilibrium in finite time. This occurs in certain simple model equations for granular media [426]. Very often, one cannot hope for such a strong inequality as (171), but one can prove such an inequality in a restricted class of functions: D(f ) Θf E[f |f∞ ] ,
(173)
where the explicit form of Θf may depend on some features of f such as its size in some (weighted) Lebesgue spaces, its strict positivity, its smoothness, etc.: all kinds of a priori estimates which should be established independently. In collisional kinetic theory, there are many situations in which entropy–entropy dissipation inequalities cannot hold true, in particular for spatially inhomogeneous models when the collisions only involve the velocity variable. As we shall see, in such cases it is sometimes possible to use entropy–entropy dissipation inequalities from spatially homogeneous models. As a final remark, the interest of entropy–entropy dissipation inequalities is not restricted to proving theorems of trend to equilibrium. Entropy–entropy dissipation inequalities may also in principle be applied in problems of hydrodynamic (as opposed to long-time) limits, yielding rather explicit estimates. For this one may consult the work by Carlen et al. [124] on a baby model, the recent paper by Saint-Raymond [400] on the hydrodynamic limit for the BGK model, or the study by Berthelin and Bouchut [74] on a complicated variant of the BGK model. However, to apply this strategy to more realistic hydrodynamic limits, say starting from the Boltzmann equation, we certainly have to wait for very, very important progress in the field.
3.3. Logarithmic Sobolev inequalities and entropy dissipation We illustrate the preceding discussion on the simple case of the spatially homogeneous Fokker–Planck equation, ∂f = ∇v · (∇v f + f v). ∂t Recall that the entropy functional is the Kullback relative entropy of f with respect to the standard Gaussian M, H (f |M) =
RN
f log
f M
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(equivalently, the additive constant in the free energy has been chosen in such a way that the equilibrium state has zero energy). And the entropy dissipation functional is the relative Fisher information, I (f |M) =
f 2 f ∇v log . M RN
(174)
The archetype of (171) is the Stam–Gross logarithmic Sobolev inequality [411,261]. In an information-theoretical language, this inequality can be written most simply as I (f |M) 2H (f |M).
(175)
Inequality (175) was first proven, in an equivalent formulation, in a classical paper by Stam9 [411]. The links between the theory of logarithmic Sobolev inequalities and information theory have been pointed out for some time [45,120,165,16]. Of course, inequality (175) immediately implies that the solution to the Fokker–Planck equation with initial datum f0 satisfies H f (t)|M e−2t H (f0 |M). This is a complete10 and satisfactory solution to the problem of trend to equilibrium for the Fokker–Planck equation. Actually, the interplay between functional inequalities and diffusion equations goes in both directions [330]. As was noticed in a famous work by Bakry and Emery [45], some properties of trend to equilibrium for the Fokker–Planck equation can be used to prove inequalities such as (175). We shall discuss their approach in Section 6, together with recent developments. By the way, as a general rule, logarithmic Sobolev inequalities are stronger than spectral gap inequalities [261]. As a typical illustration: if one lets f = M(1 + εh) in (175), where Mh = 0, and then lets ε go to 0, one finds the inequality
Mh = 0 "⇒
M|∇h|2
Mh2 ,
(176)
which is the spectral gap inequality for the Fokker–Planck operator. Inequality (176) implies the following estimate for solutions of the Fokker–Planck equation: $ $ f0 ∈ L2 M −1 "⇒ $f (t, ·) − M $L2 (M −1 ) e−t f0 − M L2 (M −1 ) . 9 Stam proved the inequality N (f )I (f ) N , which is equivalent to (175) by simple changes of variables, in dimension 1. Here N is the entropy power functional of Shannon, formula (52). The proof of Stam was not completely rigorous, but has been fixed. 10 The assumption that the initial datum possess finite entropy can even be relaxed by parabolic regularization. For instance, one can prove [366] that H (f (t)|M) = O(1/t) as soon as f0 (v)|v|2 dv is finite.
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4. Entropy dissipation functionals of Boltzmann and Landau In this section, we discuss entropy–entropy dissipation inequalities for functionals (156) and (160). A common feature of both functionals is monotonicity: the Boltzmann entropy dissipation is a nondecreasing function of the collision kernel B, while the Landau entropy dissipation is a nondecreasing function of Ψ . This property makes it possible to only treat algebraically simplified cases where B (resp. Ψ ) is “small”. As a typical application, if we find a lower bound for D when the collision kernel is Maxwellian, then we shall have a lower bound for all collision kernels whose kinetic part is bounded below. This reduction is interesting because Maxwellian collision kernels do have many additional properties. We shall see some of these properties in a moment, and shall dig more deeply into them in Chapter 2D. All known lower bounds for the entropy dissipation functionals of Boltzmann or Landau have been obtained from a preliminary study of the Maxwellian case. As a consequence, it will be natural to define an “over-Maxwellian” collision kernel as a collision kernel B which is bounded below by a Maxwellian collision kernel. In the sequel, we shall assume without loss of generality that the first moments of the distribution function f are normalized:
1 f (v) v RN |v|2
dv =
1 0 N
,
(177)
and denote by M the associated Maxwellian.
4.1. Landau’s entropy dissipation We start with the case of the Landau equation, because its diffusive nature entails better properties of the entropy dissipation functional. Let us first state the main results, then we shall comment on them. T HEOREM 14. Let f be a probability distribution satisfying (177). (i) “Over-Maxwellian case”: Let Ψ (|z|) |z|2 , and let DL be the associated entropy dissipation functional, formula (160). Then there exists a constant λ(f ) > 0, explicit and depending on f only via an upper bound for H (f ), such that DL (f ) λ(f )I (f |M) 2λ(f )H (f |M).
(178)
More precisely, one can choose λ(f ) = (N − 1) inf
e∈S N−1 RN
f (v)(v · e)2 dv ≡ Tf .
(179)
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(ii) “Soft potentials”: Let Ψ (|z|) |z|2 (1 + |z|)−β , β > 0. Then, for all s > 0, there exists a constant Cs (f ), explicit and depending on f only via an upper bound for H (f ), such that β
− βs
DL (f ) Cs (f )H (f |M)1+ s Fs
,
(180)
where Fs = Ms+2 (f ) + Js+2 (f ), and Ms+2 (f ) =
RN
Js+2 (f ) =
RN
s+2 f (v) 1 + |v|2 2 dv,
2 s+2 ∇ f 1 + |v|2 2 dv.
(iii) “Hard potentials”: Let Ψ (|z|) |z|γ +2 , γ > 0. Then, there exists constants K1 (f ), K2 (f ), explicit and depending on f only via an upper bound for H (f ), such that γ DL (f ) K1 (f ) min I (f |M), I (f |M)1+ 2 γ K2 (f ) min H (f |M), H (f |M)1+ 2 .
(181) (182)
R EMARKS . (1) Note that the constant λ given by (179) has the dimensions of a temperature, and can vanish only if f is concentrated on a line. This is the typical degeneracy of the Landau equation; in particular, the operator in (30) is always strictly elliptic unless f is concentrated on a line. But the finiteness of the entropy prevents such a concentration, and allows one to get a bound from below on λ(f ). Of course, other estimates are possible: for instance, by use of some Lp , or L∞ , or smoothness bound on f . Or, if f is radially symmetric, then automatically λ(f ) = 1. (2) Also, as we shall see in the next section, it may sometimes be wiser to estimate from below λ(f ) in terms of the entropy dissipation of f ! (3) Further note that the inequalities on the right in (178) and in (182) are nothing but the logarithmic Sobolev inequality (175). In the preceding theorem, point (i) is the starting point for the remaining cases. It was established in Desvillettes and Villani [183] by two different methods. The first one relies on some explicit computations performed in Villani [443], which are recalled in formula (30). The second strategy is a variant of Desvillettes’ techniques, inspired by a method due to Boltzmann himself [93]. It consists in “killing”, with a well-chosen operator, the symmetries of the functional DL which correspond to the equilibrium state.11 To be just a little bit more precise, one writes D(f ) =
2 dv dv∗ ff∗ R(v, v∗ ) ,
11 See Boltzmann’s argument in Section 4.3.
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where R : R2N → RN , and one finds a linear operator T = T (v, v∗ ) : RN → RN such that T R is identically 0 if and only if D = 0; then DL (f )
1 T 2
2 dv dv∗ ff∗ T R(v, v∗ ) .
A careful choice of the operator T enables a very simple computation of the right-hand side in this inequality. Point (ii) is proven in [429]. The idea is that the vanishing of Ψ (|v − v∗ |)/|v − v∗ |2 as |v − v∗ | → ∞ can be compensated by some good estimates of decay at infinity, in the form of the constant Fs+2 (which involves both moments and smoothness). As for point (iii), it is rather easy to get by “perturbation” from point (i), see Desvillettes and Villani [183]. The idea is that the contribution of small |v − v∗ | is negligible. One writes |v − v∗ |γ +2 εγ |v − v∗ |2 − εγ +2 , then one estimates from below the contribution of εγ |v − v∗ |2 to the entropy dissipation, and from above the contribution of the small constant function εγ +2 . A few algebraic tricks [183] lead to the estimate (181) without further bounds on the concentration of f : the constant K in this estimate is essentially λ(f )1+γ /2 . Theorem 14 gives explicit and satisfactory answers to the quest of entropy–entropy dissipation estimates for the Landau equation; in the next section we shall see that they can be used efficiently for the study of the trend to equilibrium, at least in the spatially homogeneous situation. However, we should avoid triumphalism: it is abnormal that the exponent in the case of hard potentials (which is 1 + γ /2) be worse than the exponent in the case of soft potentials (1 + ε, with ε as small as desired, if f has a very good decay and smoothness at infinity). One would expect that for hard potentials, the inequality DL (f ) K(f )H (f |M) hold true.
4.2. Boltzmann’s entropy dissipation: Cercignani’s conjecture Now we turn to the more complicated case of the functional (156). Some parts of the following discussion are copied from [442]. An old conjecture by Cercignani, formulated at the beginning of the eighties, was that the Boltzmann equation would satisfy a linear entropy–entropy dissipation inequality. We state this conjecture here in a slightly more precise form than the original. There are two forms of it, a weak and a strong. C ERCIGNANI ’ S CONJECTURE . Let B 1 be a collision kernel, and (156) be the associated entropy dissipation functional. Let f (v) be a probability distribution on RN ,
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with unit temperature, and let M be the associated Maxwellian equilibrium. Then, (strong version) there exists λ > 0, independent of f , such that D(f ) 2λH (f |M);
(183)
(weak version) there exists λ(f ) > 0, depending on f only via some estimates of moments, Sobolev regularity, lower bound, such that D(f ) 2λ(f )H (f |M).
(184)
It soon appeared that the strong version of this conjecture had to be false. Indeed, it would have implied a universal exponential rate of convergence for solutions of the spatially homogeneous Boltzmann equation with a collision kernel B 1. But, as we mentioned in Section 1.4, Bobylev [79, p. 224] was able to produce a family of initial data (f0ε )ε>0 with unit mass and temperature, such that the associated solutions of the Cauchy problem (with Maxwellian collision kernel, say B ≡ 1) converge to equilibrium slowly, in the sense ∀t 0,
$ ε $ $f (t, ·) − M $ Kε e−εt ,
Kε > 0.
These initial data are constructed more or less explicitly with the help of the Fourier transform apparatus, and hypergeometric functions. Later, Wennberg [461] produced direct counterexamples to (183), covering the case of hard potentials as well. Finally, Bobylev and Cercignani [87] disproved even the weak version of the conjecture. They exhibited a family of distribution functions for which (184) does not hold for a uniform λ, while these distribution functions do have uniformly bounded Lp or H k norms (whatever p, k), uniformly bounded moments of order k (whatever k), and are bounded below by a fixed Maxwellian. These counterexamples are obtained by adding a very tiny (but very spread) bump, at very high velocities, to the equilibrium distribution. They again illustrate the principle that distribution tails are the most serious obstacle to a good trend to equilibrium for the Boltzmann equation. Thus, Cercignani’s conjecture is false. It may however be that (184) hold true under more stringent assumptions: – under very strong decay conditions, for instance, f ∈ L2 (M −1 ), as in the linearized theory;12 – or under an assumption of nonintegrable angular singularity, which may help. This conjecture would be supported by the good behavior of the Landau entropy dissipation. Note added in proof : To my own surprise, after completion of this review, I discovered that Cercignani’s conjecture does hold true when B(v − v∗ , σ ) 1 + |v − v∗ |2 . This is not in contradiction with the Bobylev– Cercignani counterexamples, because they assume B dσ C(1 + |v − v∗ |γ ), γ < 2! 12 A very recent, deep result by Ball and Barthe about the central limit theorem suggests that there is some hope if f satisfies a Poincaré inequality. This may be the first step towards identifying some “reasonable” conditions for Cercignani’s conjecture to be true.
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4.3. Desvillettes’ lower bound The first interesting lower bound for Boltzmann’s entropy dissipation functional was obtained by Desvillettes [166]. His idea was to go back to Boltzmann’s original argument for the identification of cases where the entropy dissipation vanishes. As many proofs of similar results, Boltzmann’s proof relies on some well-chosen linear operators which “kill symmetries”. Let us sketch this argument (slightly modified) in a nutshell, since it may enlighten a little bit the discussion of the most recent results in the field. B OLTZMANN ’ S THEOREM . Let N 2, and let f (v) be a smooth positive solution of the functional equation ∀(v, v∗ , σ ) ∈ RN × RN × S N−1 ,
f f∗ = ff∗ .
(185)
Then f is a Maxwellian distribution; in other words there exist constants λ ∈ R, µ ∈ RN such that ∇ log f (v) = λv + µ.
∀v ∈ RN ,
(186)
B OLTZMANN ’ S ARGUMENT. Average (185) over the parameter σ ∈ S N−1 , to find ff∗ =
1 |S N−1 |
S N−1
(f f∗ ) dσ.
(187)
It is easy to convince oneself that the function S N−1
f f∗ dσ ≡ G(v, v∗ )
depends only on the sphere S(v, v∗ ) with diameter [v, v∗ ]. Actually, up to a Jacobian factor (|v − v∗ |/2)N−1 , G is just the mean value of the function f (w)f (w) ˜ on this sphere, where w˜ stands for the velocity on S which is diametrically symmetric to w. The spheres S(v, v∗ ) are in turn parametrized by only N + 1 parameters, say (v + v∗ )/2 and |v − v∗ |; or, equivalently, by the physical variables ⎧ ⎪ ⎨m = v + v∗
[total momentum of colliding particles];
|v|2 |v∗ |2 ⎪ ⎩e = + 2 2
(188) [total kinetic energy of colliding particles].
Thus we shall abuse notations by writing G(v, v∗ ) = G(m, e). Now, introduce the linear differential operator T = (v − v∗ ) ∧ (∇ − ∇∗ ) (or, which amounts to the same, Π(v − v∗ )(∇ − ∇∗ ), where Π(v − v∗ ) is the orthogonal projection
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on (v − v∗ )⊥ ). Its kernel consists precisely of those functions that depend only on m and e. If we apply this operator to the equation log ff∗ = log G(m, e), we find (v − v∗ ) ∧ ∇ log f − (∇ log f )∗ ≡ 0. In words, for all v, v∗ there exists a real number λv,v∗ such that ∇ log f (v) − ∇ log f (v∗ ) = λv,v∗ (v − v∗ ).
(189)
This functional equation, set in RN , N 2, implies the conclusion at once. R EMARK . The very last part of the proof, starting from (189), is exactly what one needs to identify cases of equality for Landau’s entropy dissipation functional. This can make us suspect a deep connection between the entropy dissipations of Boltzmann and Landau. We shall soon see that there is indeed a hidden connection. With the help of the open mapping theorem, Desvillettes was able to produce a “quantitative” version of Boltzmann’s argument, leading to the T HEOREM 15. Let B 1, and let D be the associated entropy dissipation functional (47). Let f be a nonnegative density on RN , with finite mass and energy. Without loss of generality, assume that the first moments of f are normalized by (177). Then, for all R > 0 there is a constant KR > 0, depending only on R, such that D(f ) KR inf
m∈M |v|R
| log f − log m| dv,
where M is the space of all Maxwellian distributions. Note that the quantity on the right is always positive for some R > 0 if f is not Maxwellian. Several variants were obtained, with better estimates and simpler proofs, and recently Desvillettes [175] found a way to avoid the use of the open mapping theorem, and get explicit constants. Also Wennberg [455] extended the result to hard potentials. Although Desvillettes’ result is rather weak, it was important as the very first of its kind. Subsequent developments were partly motivated by the search for stronger estimates.
4.4. The Carlen–Carvalho theorem At the beginning of the nineties, Carlen and Carvalho [121,122] made a crucial contribution to the subject by using the tools of information theory and logarithmic Sobolev inequalities.
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They proved that there always exists an entropy–entropy dissipation inequality for Boltzmann’s collision operator as soon as one has some (very weak) control on the decay at infinity and smoothness of the distribution function. In their general result, decay at infinity of a distribution function f is measured by the decay of f (v)|v|2 dv χf : R → |v|R
as R ↑ ∞, while the smoothness is measured by the decay of ψf : λ → H (f ) − H (Sλ f ) as λ ↓ 0. Here (St )t 0 is as usual the semigroup generated by the Fokker–Planck operator; sometimes it is called the adjoint Ornstein–Uhlenbeck semigroup. Carlen and Carvalho’s general theorem [121] can be stated as follows: T HEOREM 16. Let B(v − v∗ , σ ) 1 be a collision kernel. Let χ0 , ψ0 be two continuous functions, decreasing to 0 as R ↑ +∞ and λ ↓ 0 respectively. Let then f be a probability distribution function with unit mass and temperature, and let M be the associated Maxwellian distribution. Assume that χf χ0 ,
ψf ψ0 .
(190)
Then, there exists a continuous function Θ = Θχ,ψ , strictly increasing from 0, depending on f only via χ0 and ψ0 , such that D(f ) Θ H (f |M) . R EMARKS . (1) This result crucially uses the special properties of Maxwellian collision kernels, explained in Chapter 2D. (2) The result in [121] is stated for a collision kernel which is bounded below in ωrepresentation.13 Recent works have shown that this assumption can be relaxed (see the references in Chapter 2D). The main ideas behind the proof of Theorem 16 are (1) the reduction to Maxwellian collision kernel by monotonicity, (2) the inequality14 D(f ) H (f ) − H Q+ (f, f ) 0, which holds true for a Maxwellian collision kernel b(cos θ ) such that b(cos θ ) sinN−2 θ dθ = 1, and (3) show that when f satisfies (190) and H (f ) − H (M) ε, then f lies in a compact set of probability measures on which H − H (Q+ ) attains its minimum value. 13 See Section 4.6 in Chapter 2A. 14 See Section 3.2 in Chapter 2D.
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One 2of the key ingredients is a study of the Fisher information functional I (f ) = |∇f | /f , and the representation formula H Q+ (f, f ) − H (M) =
+∞
I Q+ (Sλ f, Sλ f ) − I (Sλ f ) dλ.
(191)
0
This formula and related estimates are explained in Chapter 2D. A crucial point is to bound below the integrand in (191), for λ positive enough, by the method of Carlen [120]. We note that there is no assumption of lower bound on f in the Carlen–Carvalho theorem, though they actually use lower bounds in their estimates. There is no contradiction, because Maxwellian lower bounds are automatically produced by the semigroup (Sλ ). However, these lower bounds are rather bad, and so are the resulting estimates. Better bounds can be obtained if the probability density f is bounded below by some Maxwellian distribution. In a companion paper [122], Carlen and Carvalho showed how to extend their method to physically realistic cases like the hard-spheres kernel, B(v − v∗ , σ ) = |v − v∗ |, and gave a recipe for computing the function Θ. These results were the first entropy dissipation estimates which would find interesting and explicit applications to the Boltzmann equation, see Section 5. More importantly, they set new standards of quality, and introduced new tools in the field. However, the Carlen– Carvalho entropy–entropy dissipation inequalities are not very satisfactory because the function Θ is quite intricate, and usually very flat near the origin. 4.5. Cercignani’s conjecture is almost true As we mentioned earlier, the “linear” entropy–entropy dissipation inequality conjectured by Cercignani (Θ(H ) = const.H ) is in general false. Nevertheless, it was proven a few years ago by Toscani and Villani [428] that one can choose Θ(H ) = const.H 1+ε , with ε as small as desired. Here is a precise statement from [428]. We use the notation s/2 f (v) 1 + |v|2 dv f L1s = RN
and its natural extension f L1s log L =
RN
s/2 f (v) log 1 + f (v) 1 + |v|2 dv.
T HEOREM 17. (i) “Over-Maxwellian case”: Let B 1 be a collision kernel, and D be the associated entropy dissipation functional, Equation (156). Let f be a probability density on RN with unit temperature, and let M be the associated Maxwellian equilibrium. Let ε > 0 be arbitrary, and assume that for some δ > 0, A, K > 0, f L1
4+2/ε+δ
, f L1
2+2/ε+δ
log L
< +∞,
Note added in proof : All the results in this theorem have been improved in recent work by the author.
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f (v) Ke−A|v| . 2
(192)
, Then, there exists a positive constant Cε (f ), depending only on N , ε, δ, f L1 4+2/ε+δ f L1 log L , A and K, such that 2+2/ε+δ
D(f ) Cε (f )H (f |M)1+ε .
(193)
As an example (choosing δ = 1), the following more explicit constant works: D(f ) KTf Fε−ε H (f |M)1+ε ,
(194)
where K is an absolute constant (not depending on f ), Tf is the “temperature” given by (179), and 1 Fε = log + A f 2L1 f L1 log L . 3+2/ε K 5+2/ε (ii) “Soft potentials”: Assume now that −β B(v − v∗ , σ ) 1 + |v − v∗ | ,
β > 0.
Then, for all ε > 0, Equation (194) still holds with 1 Fε = log + A f 2L1 f L1 . 3+(2+β)/ε log L K 5+(2+β)/ε (iii) “Hard potentials”: Assume now that B(v − v∗ , σ ) |v − v∗ |γ ,
γ > 0.
p
Assume, moreover, that f ∈ Lκ , for some p > 1, and κ large enough. Then, there exists α > 1, C > 0, depending on N , γ , p, κ, f Lpκ , and on A, K in (192), such that D(f ) CH (f |M)α . Thus Cercignani’s conjecture is “almost” true, in the sense that any power of the relative entropy, arbitrarily close to 1, works for point (i), provided that f decays fast enough and satisfies a Gaussian lower bound estimate. This theorem is remindful of some results in probability theory, about modified logarithmic Sobolev inequalities for jump processes, see Miclo [345]. Even if the situation considered in this reference is quite different, and if the methods of proof have nothing in common, the results present a good analogy. From the physical point of view, this is not surprising, because the Boltzmann equation really models a (nonlinear) jump process. Let us briefly comment on the assumptions and conclusions.
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(1) The main improvement lies in the form of the entropy–entropy dissipation inequality, which is both much simpler and much stronger. p (2) The lower bound assumption can be relaxed into f (v) Ke−A|v| for some p > 2, provided that more moments are included in the estimate. (3) Strictly speaking, this theorem is not stronger than the Carlen–Carvalho theorem, because the assumptions of decay at infinity are more stringent. On the other hand, it does not require any smoothness condition. As regards the proof, it is completely different from that of the Carlen–Carvalho theorem, and relies strongly on Theorem 14, point (i). Since this is quite unexpected, we shall give a brief explanation in the next paragraph. Once again, the result for hard potentials is not so good as it should be, because the power in point (iii) cannot be chosen arbitrarily close to 1. We have hope to fix this problem by improving the error estimates for small relative velocities which were sketched in [428].15
4.6. A sloppy sketch of proof In this survey, we have chosen to skip all proofs, or even sketches of proof. We make an exception for Theorem 17 because of its slightly unconventional character, and also because of its links with Boltzmann’s original argument16 about cases of equality in the entropy dissipation – with ideas of information theory coming into play. Of course, we shall only try to give a flavor of the proof, and not go into technical subtleties, which by the way are extremely cumbersome. Also we only consider point (i), and set B = 1, or rather B = |S N−1 |−1 , so that B dσ = − dσ = 1. Thus the functional to estimate from below is 1 D(f ) = 4
R2N
dv dv∗ −
S N−1
dσ (f f∗ − ff∗ ) log
f f∗ . ff∗
The three main ingredients in Theorem 17 are – a precise study of symmetries for the Boltzmann collision operator, and in particular the fact that the entropy dissipation can be written as a functional of the tensor product f ⊗f; – a regularization argument à la Stam; – our preliminary estimate for the Landau entropy dissipation, Theorem 14. Stam’s argument. At the end of the fifties, Stam [411] had the clever idea to prove the so-called Shannon–Stam inequality, conjectured by Shannon: H
√ √ αX + 1 − αY αH (X) + (1 − α)H (Y ),
(195)
15 As this review goes to print, we just managed to prove the desired result, at the expense of very strong smoothness estimates (in all Sobolev spaces). 16 See Section 4.3.
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actually equivalent to (52), as a consequence of the Blachman–Stam inequality, which he introduced on that occasion: I
√ √ αX + 1 − αY αI (X) + (1 − α)I (Y ).
(196)
N In inequalities (195) and (196), X and Y are arbitrary independent random variables on R , and one writes H (X) = H (f ) = f log f , I (X) = I (f ) = |∇f |2 /f whenever f is the law of X. Stam found out that (196) is essentially an infinitesimal version of (195) under heat regularization. Think that I is nothing but the entropy dissipation associated to the heat equation . . . . A modern presentation of Stam’s argument is found in Carlen and Soffer [125]: these authors replace H and I by their relative counterparts with respect to the standard Gaussian M, and obtain (195) by integrating (196) along the adjoint Ornstein– Uhlenbeck semigroup (St )t 0 . More explicitly, since I (f |M) is the derivative of H (f |M) along regularization by St , and since also St f → M as t → ∞, one can write
+∞
H (f |M) =
I (St f |M) dt.
0
The strategy in [428] is inspired from this point of view: we would like to start from +∞
D(f ) =
−
0
d D(St f ) dt. dt
This identity is formally justified because St f → M as t → ∞, and D(M) = 0. Then one can hope that for some reason, the derivative −dD/dt will be easier to handle that the entropy dissipation functional D. This is the case in the proof of the Shannon–Stam inequality, and also here in the framework of the Boltzmann entropy dissipation. It actually turns out, rather surprisingly, that D(f )
K R2
+∞
DL (St f ) dt,
(197)
0
with K > 0, and R a typical size for the velocity. In other words, the entropy dissipation for the Landau equation is a kind of differential version of the entropy dissipation for the Boltzmann equation! Admit for a while (197), and combine it with the result of Theorem 14, in the form DL (St f ) (N − 1)TSt f I (St f |M) (N − 1)Tf I (St f |M). It follows that D(f ) C(f ) 0
+∞
I (St f |M) dt = C(f )H (f |M);
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which is the statement in Cercignani’s conjecture. Of course, we know that Cercignani’s conjecture is false, which means that (197) cannot rigorously hold true. A precise variant is established in [428]. The technical problem which prevents (197) is the presence of large velocities, as one could expect. Controlling the contribution of large velocities to the entropy dissipation is the most technical point in the proof presented in [428]. It means for instance establishing quantitative bounds on the tails of the entropy dissipation, like
+∞
dt
|X|R
0
|X|2 (St F − St G) log
Cs St F dX s St G R
for arbitrary probability densities F (X) and G(X) in R2N , where Cs is a constant depending on s and on suitable estimates on F and G (moments, lower bound . . .). In the next two pages, we shall skip all these technicalities and present a sketch of proof of (197) under the absurd assumption that all velocities are bounded, just to give the reader an idea of the kernel of the proof. S LOPPY SKETCH OF PROOF FOR (197). First we introduce the adjoint Ornstein– Uhlenbeck semigroup (St ), and we try to compute (−d/dt)D(St f ). At first sight this seems an impossible task to perform in practice, due to the number of occurrences of f in the entropy dissipation functional, and the complicated arguments v , v∗ . But a first observation will help: D(f ) is actually a functional of the tensor product f ⊗ f = ff∗ . And it is easily checked that the following diagram is commutative, with T standing for tensorization, T
f St
F = ff∗ (198)
St
T
St f
St F.
(Here we use the same symbol for the semigroups St in L1 (RN ) and in L1 (R2N ).) This enables to replace in computations (St f )(St f )∗ by St (ff∗ ). One could hope that, similarly, F = ff∗ St
St F
f f∗ St
St (f f∗ )
(199)
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is commutative. This is false! The point is that the angular variable σ is not intrinsic to the problem. To remove this flaw, we integrate with respect to the parameter σ . Since (x, y) → (x − y) log
x y
is a jointly convex function of its arguments, by Jensen’s inequality 1 ff∗ dv dv∗ ff∗ − − dσ f f∗ log . D(f ) D(f ) ≡ 4 R2N − dσ f f∗
(200)
Now it is true, even if not immediate at all,17 that T
f St
St f
A
F = ff∗ St
T
G=
dσf f∗ St
A
St F
(201)
St G
with A standing for the averaging operation over the sphere, is an entirely commutative diagram. This actually is a consequence of the fact that (St ) is a Gaussian regularization semigroup. This suggests to work with D instead of D, and to write D(St f ) in the form D(St F, St G), with the abuse of notations 1 F D(F, G) = (F − G) log dX, X = (v, v∗ ) ∈ R2N . 4 R2N G After these preliminaries, it is not hard to compute d 1 − D(St F, St G) = dt 4
∇(St F ) ∇(St G) 2 dX. (St F + St G) − St F St G R2N
(202)
Here, of course, ∇ = [∇v , ∇v∗ ] N is the gradient in RN v × Rv ∗ . Under suitable assumptions one can also prove that t → D(St f ) is a continuous function as t → 0, and goes to 0 as t → +∞. Then
1 D(f ) = 4
+∞ 0
∇(St F ) ∇(St G) 2 dX − dt (St F + St G) St F St G R2N
17 A weaker property, sufficient for the argument, is that S preserves the class of functions which only depend t on v + v∗ and |v|2 + |v∗ |2 .
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+∞ 0
221
∇(St F ) ∇(St G) 2 dX. − dt St F St F St G R2N
(203)
Since St G is a very complicated object, we would like to get rid of it. Recall from Boltzmann’s original argument that St G, being an average on spheres with diameter [v, v∗ ], does not depend on all of the variables v, v∗ , but only upon the reduced variables m = v + v∗ , e = |v|2 /2 + |v∗ |2 /2. Accordingly, we shall abuse notations and write St G(v, v∗ ) = St G(m, e). Now comes the key point: there is a conflict of symmetries between St G, which only depends on a low-dimensional set of variables, and St F , which is a tensor product. In Boltzmann’s argument, the Maxwellian distribution pops out because it is the only probability distribution which is compatible with both symmetries. Here these different structures of St F and St G reflect at the level of their respective gradients:
∇(St f ) (∇St f )∗ ∇(St F ) = , ; St F St f (St f )∗
(204)
∇(St G) 1 ∂St G ∂St G ∇m St G + v . = , ∇m St G + v∗ St G St G ∂e ∂e
(205)
In particular, ∇(St G) always lies (pointwise) in the kernel of the linear operator P : [A, B] ∈ R2N → Π(v − v∗ )[A − B] ∈ RN , where Π(z) is the orthogonal projection upon z⊥ . Of course P = operator,18 and so
√ 2 as a linear
∇(St F ) ∇(St G) 2 1 ∇St F 2 S F − S G P 2 P S F t t t
1 ∇(St f ) (∇St f )∗ 2 − = Π(v − v∗ ) . 2 St f (St f )∗
(206)
By combining (200), (203) and (206), 1 D(f ) 8
0
+∞
(∇St f )∗ 2 ∇St f − dt (St f )(St f )∗ Π(v − v∗ ) dv dv∗ . St f (St f )∗ R2N
The reader may have recognized a familiar object in the integrand of the right-hand side. Actually, apart from a factor |v − v∗ |2 , it is precisely the integrand in the Landau entropy dissipation, computed for St f ! If we now use our absurd assumption of boundedness of 18 In contrast with the linear operator appearing in Boltzmann’s proof, which was unbounded.
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all relative velocities, in the form |v − v∗ | R, we get +∞ 1 D(f ) dt (St f )(St f )∗ |v − v∗ |2 2N 8R 2 0 R
∇St f (∇St f )∗ 2 × Π(v − v∗ ) dv dv∗ − St f (St f )∗ +∞ 1 = dt DL (St f ). 8R 2 0
4.7. Remarks We shall point out a few remarks about the preceding argument. First of all, in the course of the rigorous implementation, it is quite technical to take into account error terms due to large velocities. One has to study the time-evolution of expressions like ϕ(X)(St F − St G) log(St F /St G). But the calculations are considerably simplified by a striking “algebraic” property: a local (not integrated) version of (202) holds true. Let F h(F, G) = (F − G) log , G
∇F ∇G 2 j (F, G) = (F + G) − . F G
Then, one can check that d [St , h] = j, dt t =0 in the sense that for all (smooth) probability distributions F and G, d St h(F, G) − h(St F, St G) = j (F, G). dt t =0 This property is somewhat reminiscent of the Γ calculus used for instance in Bakry and Emery [45] and Ledoux [294]. It yields another bridge between entropy dissipation inequalities and the theory of logarithmic Sobolev inequalities. Our second remark concerns the use of the Fokker–Planck semigroup regularization. As we have seen, the main point above was to estimate from below the negative of the timederivative of D(f ) along the semigroup (St )t 0 . As was already understood by Carlen and Carvalho, and even a long time ago by McKean [341] in the framework of the Kac model, this estimate has to do with the behavior of the Fisher information I (f ) = |∇f |2 /f along the Boltzmann semigroup. Note that I (f ) is the dissipation of the H -functional along the semigroup (St )t 0 . As we shall explain in Chapter 2D, the semigroup (Bt ), generated by
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the spatially homogeneous Boltzmann equation with Maxwell collision kernel, commutes with (St ), and it follows that d d − D(St f ) = − I (Bt f ). dt t =0 dt t =0
(207)
We shall see in Chapter 2D that the right-hand side of (207) is always nonnegative; this could be considered as an a priori indication that the functional D behaves well under Fokker–Planck regularization. Actually, in the simpler case of the Kac model,19 McKean [341, Section 7, Lemma d)] used relation (207) the other way round! He proved directly, with a very simple argument based on Jensen’s inequality, that the left-hand side of (207) is nonnegative for the Kac model. His argument can be transposed to the Boltzmann equation with Maxwell collision kernel in dimension 2, and also to the case where the collision kernel is constant in ωrepresentation: see [428, Section 8]. As a third remark, we insist that the above argument, besides being rather intricate, is certainly not a final answer to the problem. The use of the average over σ seems crucial to its implementation, while for some applications it would be desirable to have a method which works directly for arbitrary Maxwellian collision kernels b(cos θ ). There is no clue of how to modify the argument in order to tackle the problem of Cercignani conjecture (with exponent 1) for very strongly decaying distribution functions. It also does not manage to recover spectral gap inequalities for Maxwellian collision kernels, which are known to be true. Applied to simpler models than Boltzmann’s equation, it yield results which are somewhat worse than what one can prove by other, elementary means! However, in terms of lower bounds for Boltzmann’s entropy dissipation, at the moment this is by far the best that we have. Our final remark concerns the problem of solving (53). As mentioned in Section 2.5, many authors have worked to prove, under increasing generality, that these solutions are Maxwellian distributions. The problem with Boltzmann’s proof was that it needed C 1 smoothness. However, as suggested by Desvillettes, the use of the Gaussian semigroup (St ) (or just the simple heat regularization) allows one to save Boltzmann’s argument: let f be a L1 solution of (53) with finite energy; without loss of generality f has unit mass, zero mean and unit temperature. Average (53) over σ to get ff∗ = G(m, e) as in formula (187). Then apply the semigroup (St ) to find (St f )(St f )∗ = St G(m, e). Since St f is C ∞ for t > 0, Boltzmann’s proof applies and St f is a Gaussian, which has to be M by identification of first moments. Since this holds true for any t > 0, by weak continuity f = M. 19 Equation (21).
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5. Trend to equilibrium, spatially homogeneous Boltzmann and Landau As we already explained, in principle the trend to equilibrium is an immediate consequence of an entropy–entropy dissipation inequality and of suitable a priori estimates. However, there are some interesting remarks to make about the implementation.
5.1. The Landau equation By Theorem 14, one obtains at once convergence to equilibrium for the spatially homogeneous Landau equation – with explicit exponential rate if Ψ (|z|) K|z|2 ; – with explicit polynomial rate if Ψ (|z|) K|z|γ +2 , γ > 0. These results hold in the sense of relative entropy, but also in any Sobolev space, thanks to the regularization results which we discussed in Chapter 2B and standard interpolation inequalities. An interesting feature is that the rate of convergence given by the entropy–entropy dissipation inequality is likely to improve as time becomes large, by a “feedback” effect. Indeed, when f approaches equilibrium, then the constant Tf in (179) will approach the equilibrium value TM = 1. In the case Ψ (|z|) = |z|2 , this enables one to recover an asymptotically optimal rate of convergence [183]. The case of soft potentials (γ < 0) is more problematic, because the moment estimates are not uniform in time – and neither are the smoothness estimates which enter the constant Fs+2 in Theorem 14. The fact that we do not have any uniform moment estimate for some moment of order s > 2 may seem very serious. It is not clear that condition (170) should be satisfied. Compactness-based methods spectacularly fail in such a situation. However, and this is one of the greatest strengths of the entropy method, it is not necessary that the constant Fs+2 be uniformly bounded. Instead, it is sufficient to have some estimate showing that it does not grow too fast, say in O(t α ) for α small enough. With this idea in mind, Toscani and Villani [429] prove the following theorem: T HEOREM 18. Let Ψ (|z|) = |z|2 Φ(|z|), where Φ(|z|) is smooth, positive and decays like |z|−β at infinity, 0 < β < 3. Let f0 be an initial datum with unit mass and temperature, and let M be the associated Maxwellian distribution. Assume that f0 is rapidly decreasing, in the sense that ∀s > 0,
f0 L2s < +∞.
Then, for all ε > 0 there exists s0 > 0 and a constant Cε (f0 ), depending only on ε, N , Φ and f0 L2s , such that the unique smooth solution of the spatially homogeneous Landau 0 equation with initial datum f0 satisfies H f (t, ·)|M Cε (f0 ) t −1/ε .
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We note that this theorem does not cover the interesting case β = 3 (Coulomb potential in dimension 3): the proof in [429] just fails for this limit exponent. Including this case would be a significant improvement. We also note that this theorem deals with a smooth Ψ , while realistic Ψ ’s would present a singularity at the origin. This singularity cannot harm the entropy–entropy dissipation inequality, but may entail serious additional difficulties in getting the right a priori estimates.20
5.2. A remark on the multiple roles of the entropy dissipation Numerical applications for the constant Tf appearing in (179) are very disappointing (say, 10−20 . . . .) This is because the entropy is quite bad at preventing concentration. Much better estimates are obtained via L∞ bounds for instance (which can be derived from regularization). Another possibility is to use the entropy dissipation as a control of concentration for f . The idea is the following: if the entropy dissipation is low (which is the bad situation for trend to equilibrium), then the distribution function cannot be concentrated too much close to a hyperplane, because the entropy dissipation measures some smoothness. As a consequence, Tf cannot be too small. More explicitly, say if Ψ (|z|) |z|2 , then [183, Section 5] Tf
(N − 1)2 N+
DL (f ) N
.
By re-injecting this inequality in the proof of Theorem 14, one finds the following improvement (still under the assumption Ψ (|z|) |z|2 ) DL (f )
2N(N − 1)2 H (f |M) +
N4 N2 − . 4 2
This in turns implies exponential convergence to equilibrium with realistic bounds, which we give explicitly as an illustration. T HEOREM 19. Let Ψ (|z|) |z|2 , and let f0 be a probability distribution on RN , with zero mean velocity and unit temperature. Let M be the associated Maxwellian distribution. Let f (t, ·) be a classical solution of the Landau equation with initial datum f0 . Then, for all time t 0, $ $ $f (t, ·) − M $ 1 L √ 2 2(N−1)2 t NC0 12 C0 − (N−1)2 t C0 C0 2 N N N e e e− N , + √ e N −1 N (N − 1) 20 See the discussion in Section 1.3 of Chapter 2E.
(208)
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where C0 =
2N(N − 1)2 H (f0 |M) +
N4 N2 − . 4 2
This estimate shows that satisfactory bounds can sometimes be obtained by cleverly combining all elements at our disposal!
5.3. The Boltzmann equation Once again, one has to separate between over-Maxwellian collision kernel, hard potentials or soft potentials. In order to apply Theorem 17, we need Moment estimates. They hold true for hard potentials without any assumption on the initial datum, and for Maxwellian collision kernels if a sufficient number of moments are finite at the initial time; we have discussed all this in Chapter 2B. In the case of soft potentials, these estimates are only established locally in time, but in some situations one can control the growth well enough. Lower bound estimates. Uniform such bounds were proven by A. Pulvirenti and Wennberg for Maxwellian collision kernels or hard potentials with cut-off. In the case of soft potentials, uniform bounds are an open problem. But, still under the cut-off assumption, local (in time) bounds are very easy to obtain as a consequence of Duhamel’s formula (100), if the initial datum satisfies a lower bound assumption. Such a crude bound as f (t, v) Ke−At |v| , 2
At = (1 + t),
is sufficient in many situations [428, Section 4]. The case of non-cutoff collision kernels is still open. Lp estimates. In the model case of hard potentials with Grad’s cut-off assumption, such estimates are a consequence of the studies of Arkeryd and Gustafsson, as discussed in Section 3 of Chapter 2B. For instance, if the initial datum lies in L∞ with suitable polynomial decay, then the solution will be bounded, uniformly in time. Also the case of Maxwell collision kernel can be treated in the same way. However, when the collision kernel decays at infinity, things become more intricate. The search for robust estimates led the authors in [428] to a new way of controlling Lp norms by the Q+ smoothness,21 moment estimates, and a lot of interpolation. 21 For simplicity, kinetic collision kernels Φ(|v − v |) considered in [429] were smooth and bounded from ∗ above and below. The authors had forgotten that in such a case the Q+ smoothness could not apply directly, because Φ(0) > 0. The proof is however easy to fix by treating separately relative velocities which are close to 0; this has been done recently by Mouhot.
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On the other hand, in the case of non-cutoff collision kernels, Lp estimates are obtained via Sobolev estimates and regularizing effect. Here we see that many of the estimates which we discussed in Chapter 2B can be combined to yield a qualitative theorem for solutions of the Boltzmann equation: trend to equilibrium with some explicit rate. Since the general panorama of a priori estimates for the spatially homogeneous Boltzmann equation is not completely settled yet, we do not have a general theorem. Let us give one which encompasses the few cases that can be treated completely. We put very strong conditions on the initial datum so that a unified result can be given for different kinds of collision kernels. T HEOREM 20. Let B(v − v∗ , σ ) = Φ(|v − v∗ |)b(cos θ ) be a collision kernel satisfying Grad’s angular cut-off, let f0 be an initial datum with unit mass and temperature, and let f (t, ·) be a strong solution of the Boltzmann equation with initial datum f0 . Assume that f0 lies in L∞ and decays at infinity like O(|v|−k ) for any k 0. Assume moreover that 2 f0 (v) Ke−A|v| for some A, K > 0. Then (i) if Φ ≡ 1, then H (f |M) = O(t −∞ ); (ii) if Φ(|v − v∗ |) = |v − v∗ |γ , γ > 0, then H (f |M) = O(t −κ ) for some κ > 0; (iii) if Φ(|v − v∗ |) is bounded, strictly positive and decays at infinity like |v − v∗ |−β , with 0 < β < 2, then H (f |M) = O(t −∞ ). Moreover, all the constants in these estimates are explicitly computable. Of course, O(t −∞ ) means O(t −κ ) for any κ > 0. For parts (i) and (ii), see [428]; for part (iii), see [429]. Also we insist that result (i) also holds when Φ is bounded from above and below. In Chapter 2D, we shall see that the structure of the particular case Φ ≡ 1 allows a better result, in the form of an explicit exponential rate of convergence.
5.4. Infinite entropy We conclude this section with an interesting remark due to Abrahamsson [1]. Of course, it seems intuitive that entropy dissipation methods require an assumption of finiteness of the entropy. This is not true! In some situations one can decompose the solution of the spatially homogeneous Boltzmann equation into a part with infinite entropy, but going to zero in L1 sense, and a part with finite entropy, on which the entropy dissipation methods can be applied. If the estimates are done with enough care, this results in a theorem of convergence with explicit rate, even when the initial datum has infinite entropy. It is important here to have a good control of the entropy–entropy dissipation inequality which is used, in terms of the initial datum. With this technique, Abrahamsson [1] was able to prove convergence to equilibrium in L1 for the spatially homogeneous Boltzmann equation with hard spheres, assuming only that the initial datum has finite mass and energy. On this occasion he used the Carlen– Carvalho theorem, and also some iterated Duhamel formulas in the a priori estimates. Note that this problem mainly arises for cut-off collision kernels, because for most kernels with an angular singularity, entropy becomes finite for any positive time [440] by regularization effects.
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This ends our review of applications of entropy methods in the spatially homogeneous Boltzmann and Landau equations. Before discussing spatially inhomogeneous models, we shall briefly consider another class of spatially homogeneous systems, characterized by their gradient flow structure.
6. Gradient flows This section is a little bit outside the main stream of our review, but reflects active trends of research in kinetic theory, and may enlighten some of the considerations appearing here and there in this chapter. The main application to Boltzmann-like equations is Theorem 21 below, for simple models of granular flows.
6.1. Metric tensors As we explained before, several equations in kinetic theory have a gradient flow structure: they can be written δE ∂f = ∇v · f ∇v ∂t δf
(209)
for some energy functional f → E(f ), which we shall always call the entropy for consistency. Typical examples are the Fokker–Planck equation, for which 1 E(f ) = f log f + f |v|2 dv; 2 RN RN or the model from [68] for granular flow, ∂f = ∇v · f (f ∗ ∇U ) + σ v f + θ ∇v · (f v), ∂t with U (v) = |v|3 /3, and σ, θ > 0; then 1 E(f ) = f (v)f (w)U (v − w) dv dw 2 R2 θ f log f + f |v|2 dv. +σ 2 R R
(210)
(211)
Among examples outside kinetic theory, we have also mentioned the heat equation, the spatial Fokker–Planck equation, the porous medium equation . . . . Generally speaking, a gradient flow is an equation of the form dX = − grad E X(t) . dt
(212)
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Underlying the definition of the gradient operator is that of a Riemannian metric tensor on some “manifold” in which the unknown X lives. Thus, to explain why (209) is a gradient flow, we first have to explain how to define a meaningful metric tensor on the “manifold” of all probability measures. Of course this is a formal point of view, because infinite-dimensional Riemannian geometry usually does not make much sense, even if it is sometimes enlightening, as the well-known works by Arnold [40] in fluid mechanics illustrate. In our context, the relevant metric tensor is defined as follows. Let f be a probability density (assume that f is smooth and positive, since this is a formal definition). Let ∂f/∂s be a “tangent vector”: formally, this just means some function with vanishing integral. Then define $ $2 $ ∂f $ ∂f 2 $ $ = inf + ∇v · (f u) = 0 . f |u| dv; $ ∂s $ ∂s
(213)
The infimum in (213) is taken over all vector fields u on RN such that the linear transport equation ∂f/∂s + ∇v · (f u) = 0 is satisfied. By polarization, formula (213) defines a metric tensor, and then one is allowed to all the apparatus of Riemannian geometry (gradients, Hessians, geodesics, etc.), at least from the formal point of view. Then, an easy computation shows that (209) is the gradient flow associated with the energy E, on the “manifold” of all (smooth, positive) probability measures endowed with this Riemannian structure. The metric tensor defined by (213) has been introduced and studied extensively by Otto [364]. One of its important features is that the associated geodesic distance is nothing but the Wasserstein distance on probability measures.22 This is part of the whole area of mass transportation, whose connections with partial differential equations are reviewed in Villani [452].
6.2. Convergence to equilibrium A general property of gradient flows is that they make the entropy decrease. From formula (212) one sees that $ $2 d E X(t) = −$grad E X(t) $ . dt And the equilibrium positions of (212) are the critical points for E: typically, minima. In all the cases which we consider, there is a unique minimizer for E, which is therefore the only equilibrium state. Now, it is a general, well-known fact that the rate of convergence to equilibrium for a gradient flow is very much connected to the (uniform, strict . . .) convexity of the energy functional. A typical result is the following: assume that the energy E is uniformly convex, 22 See formula (244) below.
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in the sense that its Hessian is bounded below by some positive multiple of the identity tensor, Hess E λ Id. Then, E admits a unique minimizer X∞ , and the gradient flow (212) satisfies the linear entropy–entropy dissipation inequality grad E 2λ E(X) − E(X∞ ) .
(214)
A possible strategy to prove (214) is to go to the second derivative of the entropy functional with respect to time. From (212) and the definition of the Hessian, −
$ ' ( d$ $grad E X(t) $2 = 2 Hess(E) · ∇E, ∇E . dt
(215)
The functional which just appeared in the right-hand side is the dissipation of entropy dissipation. Therefore, the assumption of uniform positivity of the Hessian implies $ $ $ d$ $grad E X(t) $2 −2λ$grad E X(t) $2 . dt Integrating this inequality in time, one easily arrives at (214) if everything is wellbehaved.23 This remark shows that the trend to equilibrium for Equation (209) can in principle be studied via the properties of convexity of the underlying energy E. But the right notion of convexity is no longer the usual one: it should be adapted to the definition (213). This concept is known as displacement convexity, and was first studied by McCann [338,340], later by Otto [364], Otto and Villani [365]. D EFINITION 3. Let f0 , f1 be two (smooth, positive) probability measures on RN . By a classical theorem of Brenier [103,339] and others, there exists a unique gradient of convex function, ∇ϕ, such that ∇ϕ#f0 = f1 , meaning that the image measure24 of f0 by the mapping ∇ϕ is the measure f1 . Let us define the interpolation (fs )0s1 between f0 and f1 by fs = (1 − s) Id + s∇ϕ #f0 . 23 There are also other, simpler derivations of (214) based on Taylor formula. The above procedure was chosen because this is precisely a way to understand the famous Bakry–Emery method for logarithmic Sobolev inequalities. 24 By definition of the image measure: for all bounded continuous function h, (h ◦ ∇ϕ)f = hf . If ϕ is C 2 , 0 1 2 then for all v, f0 (v) = f1 (∇ϕ(v)) det(D ϕ)(v).
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Then, the functional E is said to be displacement convex if whatever f0 , f1 , s → E(fs )
is convex on [0, 1].
It is furthermore said to be uniformly displacement convex with constant λ > 0 if whatever f0 , f1 , d2 E(fs ) λW (f0 , f1 )2 ds 2
(0 s 1),
where W stands for the Wasserstein distance25 between f0 and f1 . R EMARKS . (1) To get a feeling of this interpolation procedure, note that the interpolation between δa and δb is δ(1−s)a+sb, instead of (1 − s)δa + sδb . Further note that the preceding definition reduces to the usual definition of convexity if the interpolation (fs )0s1 is replaced by the linear interpolation. (2) Let us give some examples. The functionals f log f , or f p (p 1) are displacement convex as one of the main results of McCann [340]. The functional f V is displacement convex if and only if the potential V is convex. Moreover, if V is uniformly convex with constant λ, the functional f V is uniformly displacement convex, with the same constant. Another interesting example is the case of functionals like R2N f (v)f (w)U (v − w) dv dw. Such a functional is never convex in the usual sense, except for some very peculiar potentials (power laws . . .). On the other hand, it is displacement convex as soon as U is convex. Among the results in Otto and Villani [365], we mention the following statement. If E is uniformly displacement convex, with constant λ > 0, then the associated gradient flow system (209) satisfies a linear entropy–entropy dissipation inequality of the form R
δE 2 dv 2λ E(f ) − E(f∞ ) , f ∇ N δf
where f∞ is the unique minimizer of the energy. This is not a true theorem, because the proof is formal, but this is a general principle which can be checked on each example of interest. A standard strategy of proof goes via the second derivative of the entropy, as we sketched above. In the context of the linear Fokker–Planck equation (168), this strategy of taking the second derivative of the entropy is known as the Bakry–Emery strategy, and goes back to the mid-eighties. To check the assumption of uniform displacement convexity, it is in principle sufficient to compute the Hessian of the entropy. When this is done, one immediately obtains the dissipation of entropy dissipation via formula (215). This calculation is however very intricate, as one may imagine. This is where Bakry and Emery [45] need their so-called 25 See formula (244) below.
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Γ2 calculus, which is a set of formal computation rules involving linear diffusion operators and commutators. On the other hand, the formalism developed by Otto [364], Otto and Villani [365] enables simpler formal computations, and can be adapted to nonlinear cases such as granular flows [130]. Just to give an idea of the complexity of the computations, and why it is desirable to have efficient formal calculus here, let us reproduce below the dissipation of entropy dissipation which is associated to the gradient flow for Equation (210): 2 ∂ ξ(v) dv ∂v R 2 + 2θ f (v)ξ(v) dv
DD(f ) = 2σ
f (v)
+
R2
R
' f (v)f (w) D 2 U (v − w)
( × ξ(v) − ξ(w) , ξ(v) − ξ(w) dv dw,
(216)
where ξ(v) =
|v|2 ∂ σ log f (v) + θ +U ∗f , ∂v 2
U (v) = |v|3 /3.
6.3. A survey of results Let us now review some results of trend to equilibrium which were obtained via the considerations above, or which can be seen as related. A survey paper on this subject is Markowich and Villani [330]. The first partial differential equation to be treated in this way was the spatial Fokker– Planck equation, ∂ρ = ∇x · ∇x ρ + ρ∇V (x) . ∂t The classical paper by Bakry and Emery [45] shows that the solution to this equation converges exponentially fast, in relative entropy sense, to the equilibrium e−V (assuming −V e = 1), at least if V is uniformly convex with constant λ. The decrease of the entropy is like e−2λt . Underlying entropy–entropy dissipation inequalities are known under the name of logarithmic Sobolev inequalities, and have become very popular due to their relationship with many other fields of mathematics (concentration of measure, hypercontractivity, information theory, spin systems, particle systems . . . see the review in [16]). The Bakry–Emery strategy, and the corresponding proof of the Stam–Gross logarithmic Sobolev inequality, were recently re-discovered by Toscani [422,423] in the case of the kinetic Fokker–Planck equation. Instead of Γ2 calculus, Toscani generalized a lemma by McKean [341] to compute the second derivative of the entropy functional. Though this
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work was mainly a re-discovery of already known results, it had several merits. First, it suggested a more physical way of understanding the Bakry–Emery proof, in terms of entropy dissipation and dissipation of entropy dissipation. Also, Toscani directly studied the Fokker–Planck equation ∂t f = f + ∇ · (f v), while previous authors mainly worked on the adjoint form, ∂t h = h − v · ∇h. Last but not least, his paper made these methods and techniques popular among the kinetic community, which began to work on this subject: see in particular the recent synthesis by Arnold, Markowich, Toscani and Unterreiter [39]. Then, these results were generalized to the porous medium equation with drift, ∂ρ = x ρ m + ∇x · (ρx). ∂t It was found that when m 1 − 1/N , solutions to this equation converge exponentially fast (with relative entropy decreasing like e−2t ) towards a probability density known as Barenblatt–Pattle profile. These results were obtained independently by Otto [364], Carrillo and Toscani [131], and Del Pino and Dolbeault [163]. All three papers established nonlinear analogues of the logarithmic Sobolev inequalities. The paper by Otto made the link with mass transportation problem. Various generalizations of all these results can be found in [129]. Let us now come back to kinetic, Boltzmann-like systems and display recent results about the equations for granular media suggested in [70,68]. These results were proven by Carrillo, McCann and Villani [130] by using the ideas above, and in particular those of gradient flows, Wasserstein distance and Bakry–Emery strategy. The theorem which we state is slightly more general: the dimension is arbitrary, and the interaction potential is not necessarily cubic. T HEOREM 21. Let U be a convex, symmetric potential on RN , and σ, θ 0. Let E(f ) = σ +
f log f + θ
1 2
RN ×RN
f (v)
|v|2 dv 2
f (v)f (w)U (v − w) dv dw.
Let moreover |v|2 +U ∗f . ξ = ∇ σ log f + θ 2 We consider the equation ∂f = ∇ · (f ξ ), ∂t
(217)
for which E is a Lyapunov functional, whose time-derivative is given by the negative of D(f ) = f |ξ |2 dv.
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Let f0 be an initial datum, and f (t) be the solution26 to (217). Let moreover f∞ be the unique minimizer of E (if θ = 0, the unique minimizer of E which has the same mean velocity as f0 ), and let E(f |f∞ ) = E(f ) − E(f∞ ). Then, (i) if θ > 0, then D(f ) 2θ E(f |f∞ ) for all probability distribution f , and E(f (t)|f∞ ) e−2θt E(f0 |f∞ ); (ii) if θ = 0 and U is uniformly convex, D 2 U λ, then D(f ) 2λE(f |f∞ ) for all f with the same mean velocity as f∞ , and E(f (t)|f∞ ) e−2λt E(f0 |f∞ ); (iii) if θ = 0 and U is strictly convex, in the sense D 2 U (z) K |z|α ∧ 1 ,
K, α > 0,
(218)
then D(f ) CE(f |f∞ )β for some positive constants C, β, and for any f with the same mean velocity as f∞ , and E(f (t)|f∞ ) converges to 0 at least in O(t −κ ) for some κ > 0; (iv) if θ = 0, U is strictly convex in the sense of (218), and moreover σ > 0, then for all f with the same mean velocity as f∞ , one has D(f ) λ(f )E(f |f∞ ) for some λ(f ) > 0 which only depends on an upper bound for E(f ). Moreover, E(f (t)|f∞ ) e−λ0 t E(f0 |f∞ ), for some positive constant λ0 which only depends on an upper bound for E(f0 ). R EMARKS . (1) The assumptions on the mean velocity reflect an important physical feature: in the cases in which they are imposed, the entropy is translation-invariant. (2) The motivation to study strictly convex, but not uniformly convex potentials comes from the physical model where U (z) = |z|3 /3. Then, lack of uniformity may come from small values of z. This difficulty is of the same type than in the study of hard potentials for the Boltzmann equations; this is not surprising since the model in [70] can be seen as a limit regime for some Boltzmann-type equation with inelastic hard spheres. (3) In general, point (iii) cannot be improved into exponential decay: when σ = θ = 0 and U (z) = |z|3 /3, then the decay of the energy is O(1/t) and this is optimal. (4) We note that part (iv) of this theorem is the most surprising, because in this case the energy functional is not uniformly displacement convex; yet there is a “linear” entropy– entropy dissipation inequality (not uniform in f ). This result raises hope that the entropy– entropy dissipation inequalities described in Section 4 for the Landau equation with hard potentials may be improved into inequalities of linear type. To conclude this section, we mention that the trend to equilibrium for Equation (217) has been studied by Malrieu [329] with the Γ2 calculus of Bakry and Emery. Even though his results are much more restrictive (only σ, θ > 0) and the constants found by Malrieu are not so good, on this occasion he introduced several interesting ideas about particle 26 We assume that U is sufficiently well-behaved that existence of a unique “nice” solution to (217) is guaranteed. This is quite a weak assumption.
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systems and his work provided yet another connection between the kinetic and probabilistic communities.
7. Trend to equilibrium, spatially inhomogeneous systems We now turn to the study of spatially inhomogeneous kinetic systems like the ones presented in Section 1.2. We first make several remarks. (1) Once again, we are mainly interested in explicit results, and wish to cover situations which are not necessarily perturbations of the equilibrium. Thus we do not want to use linearization tools, and focus on entropy dissipation methods. (2) For many of the spatially inhomogeneous models which we have introduced, the entropy and the entropy dissipation functionals are just the same as in the spatially homogeneous case, up to integration in x. Also, the transport part does not contribute to the entropy dissipation. Thus, one may think, the same entropy–entropy dissipation inequalities which we already used for the spatially homogeneous case will apply to the spatially inhomogeneous case. This is completely false, as we shall explain! And the obstruction is not a technical subtlety, but stands for a good physical reason. (3) Nevertheless, it is plainly irrelevant to ask for an x-dependent version of the entropy– entropy dissipation inequalities presented in Section 4, since the entropy dissipation does not make the x variable play any role. (4) The boundary conditions, and the global geometry of the spatial domain, are extremely important in this study. In this respect the problem of trend to equilibrium departs notably from the problem of the hydrodynamic limit, which fundamentally is a local problem. (5) To work on the trend to equilibrium, one should deal with well-behaved solutions, satisfying at least global conservation laws. In the sequel, we shall even assume that we deal with very well-behaved solutions, for which all the natural estimates of decay, smoothness and positivity are satisfied. Of course, for such equations as the Boltzmann or Landau equation, nobody knows how to construct such solutions under general assumptions . . . . Therefore, the results dealing about these equations will be conditional, in the sense that they will depend on some strong, independent regularity results which are not yet proven. It is however likely that such regularity bounds can be obtained with presentday techniques in certain particular situations, like weakly inhomogeneous solutions [32]. We wish to insist that even if we assume extremely good a priori estimates, the problem of convergence to equilibrium remains interesting and delicate, both from the mathematical and from the physical point of view!
7.1. Local versus global equilibrium When studying the trend to equilibrium in a spatially dependent context, a major obstacle to overcome is the existence of local equilibrium states, i.e., distribution functions which are Note added in proof : In a more recent, quite clever work, Malrieu was able to remove the condition θ > 0.
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in equilibrium with respect to the velocity variable, but not necessarily with respect to the position variable. For instance, for the Boltzmann or Landau equation, a local equilibrium is a local Maxwellian, |v−u(x)|2
e− 2T (x) f (x, v) = ρ(x) . (2πT (x))N/2
(219)
For the linear Fokker–Planck equation, a local equilibrium is a distribution function of the form |v|2
e− 2 f (x, v) = ρ(x) ≡ ρ(x)M(v). (2π)N/2
(220)
Local equilibria are not equilibrium distributions in general, but they make the entropy dissipation vanish. This shows that there is no hope to find an entropy–entropy dissipation inequality for the full x-dependent system. If the system ever happens to be in local equilibrium state at some particular time t0 , then the entropy dissipation will vanish at t0 , and it is a priori not clear that the entropy functional could stay (almost) constant for some time, before decreasing again. This may result in a strong slowing-down of the process of trend to equilibrium. This difficulty has been known for a long time (even to Boltzmann! as pointed out to us by C. Cercignani), and is discussed with particular attention by Grad [254], Truesdell [430, pp. 166–172] and, in the different but related context of hydrodynamic limits for particle systems, Olla and Varadhan [362]. On the other hand, whenever the solution happens to coincide with some local equilibrium state Mloc at some time, then the combined effect of transport and confinement will make it go out of local equilibrium again, unless Mloc satisfies some symmetry properties which ensure that it is a stationary state. In fact, in most situations these symmetry properties select uniquely the stationary state among the class of all local equilibria. Note the fundamental difference with the problem of hydrodynamical limit: in the latter, one wishes to prove that the solution stays as much as possible close to local equilibrium states, while here we wish to prove that if the solution ever happens to be very close to local equilibrium, then this property will not be preserved at later times. Thus, one can see the trend to equilibrium for spatially inhomogeneous systems as the result of a negociation between collisions on one hand, transport and confinement on the other: by dissipating entropy, collisions want to push the system close to local equilibrium, but transport and confinement together do not like local equilibria – except one. This is why transport phenomena, even if they do not contribute in entropy dissipation, play a crucial role in selecting the stationary state. Our problem is to understand whether these effects can be quantified.
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An answer to this question was recently obtained by the author in a series of collaborations with Desvillettes. In the sequel, we shall explain it on a simple case: the linear Fokker–Planck equation, with potential confinement, ∂f + v · ∇x f − ∇V (x) · ∇v f = ∇v · (∇v f + f v). ∂t
(221)
The trend to equilibrium for this model was studied by Bouchut and Dolbeault [100] with the help of compactness tools, so no explicit rate of convergence was given. Also, Talay [413] proved exponential decay with a probabilistic method (based on general theorems about recurring Markov chains) which does not seem to be entirely constructive. Other probabilistic approaches have been proposed to study this model, but they also strongly depend on the possibility to interpret (221) as the evolution equation for the law of the solution of some stochastic differential equation with particular properties. In the sequel, we shall explain how the entropy method of Desvillettes and Villani [184] leads to polynomial (as opposed to exponential), but fully explicit estimates. This method is robust in the sense that it can be generalized to smooth solutions of nonlinear equations, in particular Boltzmann or Vlasov–Poisson–Fokker–Planck equations. With respect to the Boltzmann equation, the model (221) has several pedagogical advantages. First, one can prove all the a priori estimates which are needed in the implementation of the method. What is more important, the local equilibrium only depends on one parameter (the density), instead of three (density, mean velocity and temperature). This entails significant simplifications in the computations and intermediate steps, which however remain somewhat intricate.
7.2. Local versus global entropy: discussion on a model case To use entropy methods in a spatially dependent context, the main idea is to work at the same time at the level of local and global equilibria; i.e., estimate simultaneously how far f is from being in local equilibrium and how far it is from being in global equilibrium. (1) One first introduces the local equilibrium associated with f , i.e., the one with the same macroscopic parameters as f . For instance, in the case of the linear Fokker–Planck equation, the local equilibrium is just (220), with ρ(t, x) = f (t, x, v) dv. In the case of the Boltzmann equation, the local equilibrium is the local Maxwellian (219), with ρ, u, T given by (1). How close f is from local or global equilibrium will naturally be measured by relative entropies. Thus one defines Hglo to be the relative entropy of f with respect to the global equilibrium, and Hloc to be the relative entropy of f with respect to the associated local equilibrium. In the Fokker–Planck (resp. Boltzmann) case, Hloc is H (f |ρM) (resp. N H (f |M f )); note that this is an integral over RN x × Rv now. Then, one looks for a system of differential inequalities satisfied by Hglo and Hloc . Note added in proof : A much more complete, very satisfactory study was recently performed by Hérau and Nier; see also the references provided in their work.
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(2) The first equation is given by an entropy–entropy dissipation estimate of the same type that the ones we discussed in Section 4. We just have to apply this inequality pointwise in x. For instance, for the linear Fokker–Planck equation, d D(f ) = − Hglo = dt
f I (f |ρM) dx = ρI M dx, ρ RN RN
(222)
and by the logarithmic Sobolev inequality (175), D(f ) 2
ρH RN x
f M dx = 2H (f |ρM) = 2Hloc ρ
(223)
(check the last-but-one equality to be convinced!). Note that the symbol H is used above in two different meanings: relative entropy of two probability distributions of the v variable, relative entropy of two probability distributions of the x and v variables. Similarly, if we have nice uniform a priori bounds for the solution of the Boltzmann equation, it will follow from our discussion in Section 4 that −
d α Hglo KHloc , dt
(224)
for some constants K > 0, α > 1. In a spatially homogeneous context, this inequality would be essentially sufficient to conclude by Gronwall’s lemma. Here, we need to keep much more information from the dynamics in order to recover a control on how the positivity of Hglo forces Hloc to go up again if it ever vanishes. (3) To achieve this, we now look for a differential inequality involving the time-behavior of Hloc . We start with a heuristic discussion. At a time when the entropy dissipation would vanish, then both the local relative entropy and its time derivative would vanish, since the relative entropy is always nonnegative. Therefore, one can only hope to control from below the second time derivative of the local relative entropy! Taking into account the first differential inequality about Hglo and Hloc , this more or less resembles to considering the third derivative of the entropy at an inflexion point. It is easy to compute (d2 /dt 2 )Hloc at a time t0 when f happens to be in local equilibrium. For instance, in the case of the linear Fokker–Planck equation, we have the remarkably simple formula 2 ∇ρ d2 + ∇V dx ≡ Ix ρ|e−V . H (f |ρM) = ρ N ρ dt 2 t =t0 Rx
(225)
Here Ix is the Fisher information, applied to functions of the x variable. We do not describe here the corresponding results for the Boltzmann equation, which are of the same nature but much, much more complicated [181]. Here we shall continue the discussion only for the Fokker–Planck equation, and postpone the Boltzmann case to the end of the next section.
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If V is well-behaved, the logarithmic Sobolev inequality, applied in the x variable, yields d2 H (f |ρM) KH ρ|e−V 2 dt t =t0
(226)
for some positive constant K depending only on V . This is the piece of information that was lacking! Indeed, for the linear Fokker–Planck equation, Hglo = Hloc + H ρ|e−V ; thus Equation (226) turns into d2 H (f |ρM) KHglo − KHloc. dt 2 t =t0
(227)
Note that the use of the logarithmic Sobolev inequality in the x variable is the precise point where the geometry of the boundary conditions (here replaced by a confinement potential) comes into play. The fact that this effect can be quantified by a functional inequality is very important for the method; see the remarks in the end of the chapter for the analogous properties in the Boltzmann case. Of course, the preceding calculations only apply at a time t0 where f happens to be in local equilibrium – which is a very rare event. Therefore, one establishes a quantitative variant of (227), in the form d2 K H (f |ρM) (Hglo − Hloc ) − J (f |ρM), dt 2 2
(228)
where J (f |ρM) is a complicated functional which vanishes only if f = ρM: 1 J (f |ρM) = 4
RN
+ +
RN
|∇x · (ρu)|2 dx ρ RN |∇x · (ρu ⊗ u)|2 |∇x [ρ(T − 1)]|2 dx + dx ρ ρ RN
RN
|∇x · S|2 dx + Iv (f |ρM) ρ
1 (ρu)2 dx + ρ 4
1 + Iv (f |ρ M)1/2 Ix (f |ρ M)1/2. 2
(229)
Here ρ, u, T are the usual macroscopic fields, and S is the matrix defined by the equation ρ(x)u(x) ⊗ u(x) + ρ(x)T (x)IN + S(x) =
RN
f (x, v)v ⊗ v dv.
(230)
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(4) The next step of the program is to control J in terms of Hloc , in order to have a closed system of differential inequalities on Hloc and Hglo . This is done by some ad hoc nonlinear interpolation procedure, which yields d2 K H (f |ρM) H (f |f∞ ) − Cε (f ) H (f |ρM)1−ε . 2 dt 2 Here ε is an arbitrary positive number in (0, 1) and Cε (f ) is a constant depending on f via moment bounds, smoothness bounds, and positivity estimates on f . All these bounds have to be established explicitly and uniformly in time, which turns out to be quite technical but feasible [184] (see also Talay [413]); then the constant Cε = Cε (f ) can be taken to be independent of t. In the case of the Boltzmann equation, it is possible to perform a similar interpolation procedure; the only missing step at the moment is establishing the a priori bounds. (5) Summing up, for solutions of the Fokker–Planck equation we have obtained the system of differential inequalities ⎧ d ⎪ ⎪ ⎨− dt Hglo 2Hloc, 2 ⎪ ⎪ ⎩ d Hloc K Hglo − Cε H 1−ε . loc dt 2 2
(231)
The last, yet not the easiest step, consists in proving that the differential system (231) alone implies that Hglo converges to 0 like O(t −κ ). Since there is apparently no comparison principle hidden behind this system, one has to work by hand . . . . The bound established in Desvillettes and Villani is Hglo = O t 1−1/ε , which is presumably optimal. Thus, the global entropy converges to 0 with some explicit rate, which was our final goal.
7.3. Remarks on the nature of convergence Solutions to (231) do have a tendency to oscillate, at least for a certain range of parameters. In fact, were it not for the positivity of relative entropies, system (231) would not imply convergence to 0 at all! We expect “typical” solutions of (231) to decrease a lot for small times, and then converge to 0 more slowly as t → +∞, with some mild oscillations in the slope. This kind of behavior is completely different from what one can prove in the context of spatially homogeneous kinetic equations.27 We think that it may reflect the physical nature of approach to equilibrium for spatially inhomogeneous systems. As time becomes large and the system approaches global equilibrium, it is more and more likely to “waste 27 The rate of convergence typically improves as t → +∞, see Section 5.1.
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time” fighting against local equilibria . . . . And this may result in oscillations in the entropy dissipation. But examination of a particular, “integrable” case, suggests that (1) these oscillations may be present only when the confinement potential is strong enough, (2) the decay should be exponential. This case corresponds to the quadratic confinement potential, V (x) = ω2 |x|2/2 + C. For this particular shape of the potential, the Fokker–Planck equation can be solved in semi-explicit form [399, Chapter 10], and the rate of decay is governed by the quantities exp(−λt), where 1−
√
1 − 4ω2 1 , ω2 , 2 4 √ 2 1 ± i 4ω − 1 1 λ= , ω2 > 2 4 λ=
(232)
(in [399], these equations are established only in dimension 1). Thus the decay is always exponential, the rate being given by the real part of λ. When the confinement is very tiny, then the convergence is very slow (think that there is no trend to equilibrium when there is no confinement); when the confinement becomes stronger then the rate increases up to a limit value 1/2. For stronger confinements, the rate does not improve, but complex exponentials appear in the asymptotics of the solution. Note that in the same situation, the rate of convergence for the spatially homogeneous equation would be equal to 1. Another integrable case is when there is no confinement potential, but x ∈ TN , the N dimensional torus. Then the decay is always exponential and the rate depends on the size of the periodic box. In dimension 1 of space, it is maximal (equal to 1) when the side of the box has length 2π [152]. It is yet an open problem to generalize the above considerations to nonintegrable cases, and to translate them at the level of entropy dissipation methods. In our opinion, these examples show that a lot of work remains to be done to get an accurate picture of the convergence, even in very simplified situations.
7.4. Summary and informal discussion of the Boltzmann case We now sum up the state of the art concerning the application of entropy dissipation methods to spatially inhomogeneous systems. The following theorem is the main result of Desvillettes and Villani [184]. T HEOREM 22. Let M(v) denote the standard Maxwellian probability distribution on RN with zero mean velocity and unit temperature. Let V be a smooth confining potential on RN , behaving quadratically at infinity: V (x) = ω2
|x|2 + Φ(x), 2
Note added in proof : For progress on these questions, the recent work by Hérau and Nier is recommended.
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where ω > 0 and Φ ∈ 1, and let
)
k0 H
k (RN ). Assume without loss of generality that
e−V (x) dx =
f∞ (x, v) = e−V (x) M(v) denote the unique global equilibrium of the Fokker–Planck equation ∂f + v · ∇x f − ∇V (x) · ∇v f = ∇v · (∇v f + f v). ∂t Let f0 = f0 (x, v) be a probability density such that f0 /f∞ is bounded from above and below, and let f (t) = f (t, x, v) be the unique solution of the Fokker–Planck equation with initial datum f0 . Then, for all ε > 0 there exists a constant Cε (f0 ), explicitly computable and depending only on V , f0 and ε, such that H f (t)|f∞ Cε (f0 ) t −1/ε . We already pointed out several shorthands of this result: in particular, the convergence should be exponential. We consider it as a major open problem in the field to compute the optimal rate of decay, in relative entropy, as a function of the confinement potential V . Let us now turn to nonlinear situations. The following result was recently obtained by the author in collaboration with Desvillettes [181]. T HEOREM 23. Let B be a smooth collision kernel, bounded from above and below. Let f0 = f0 (x, v) be a smooth probability density on Ωx × R3v , where Ω is a smooth bounded, connected open subset of R3 with no axis of symmetry, and let f (t) = f (t, x, v) be a smooth solution of the Boltzmann equation ∂f + v · ∇x f = Q(f, f ), ∂t
t 0, x ∈ Ω, v ∈ R3 ,
with specular reflection boundary condition. Let moreover f∞ (x, v) be the unique global equilibrium compatible with the total mass and kinetic energy of f0 . Assume that all the moments of f are uniformly bounded in time, that f is bounded in all Sobolev spaces, uniformly in time, and that it satisfies a lower bound estimate, f (t, x, v) ρ0 e−A0 |v|
p
for some p 2, ρ0 > 0, A0 > 0, uniformly in time. Then, for all ε > 0 there is a constant Cε , depending only on (finitely many of ) the requested a priori bounds, such that H f (t)|f∞ Cε t −1/ε . We do not display here the system of differential inequalities – much, much more complicated than (231) – which underlies this result, and we refer to [181] for more information. An unexpected feature was revealed by this study: not only is the entropy
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dissipation process indeed slowed down when the distribution function becomes a local Maxwellian state, but also, it is much more slowed down for some particular local Maxwellian states, in particular those of the form ρ(x)M(v), i.e., with constant temperature and zero velocity. More precisely, the entropy dissipation vanishes in time up to order 4 (instead of 2) when going through such a Maxwellian state. From the mathematical point of view, this entails a spectacular complication of the arguments, and the need for at least three differential inequalities: apart from the behavior of the global entropy, one studies at the same time the departure of f with respect to M f and the departure of f with respect to ρM. From the physical point of view, this additional degeneracy could be interpreted as an indication that the relaxation of the density typically holds on a longer time-scale, than the relaxation of the temperature and the local velocity – although we should be cautious about this. The proof of Theorem 23 combines the general method of Desvillettes and Villani [184] with the entropy dissipation results of Toscani and Villani [428]. In the computations, the natural functionals H (f |M f ), H (f |ρM) were traded for the simpler substitutes f − M f 2L2 , f − ρM 2L2 which enable one to weaken significantly the assumptions of Theorem 23. Of course, these assumptions are still very strong, even though rather natural after our discussions in Chapter 2B. The influence of the shape of the box is quantified by the values of several “geometric” constants related to it. One of these constants is the Poincaré constant P (Ω), defined in a scalar setting by $ $2 $ $ $ F − F ∇x F 2L2 (Ω) P (Ω)$ $ $ 2 Ω
,
L (Ω)
whenever F is a real-valued function on Ω, and in a vector setting by ∇x u 2L2 (Ω) P (Ω) u 2L2 (Ω) , whenever u is a vector field in Ω, tangent to ∂Ω. Another constant which appears in the proof is what we call Grad’s number, defined in [254]: G(Ω) =
∂vj 2 ∂vi + ; ∂xj ∂xi ω0 ∈S N−1 Ω inf
ij
∇ · v = 0, ∇ ∧ v = ω0 , v · n = 0 on ∂Ω , n standing for the unit normal on ∂Ω. The number G(Ω) is strictly positive if and only if Ω has no axis of symmetry. This number contributes to a lower bound for the constant K(Ω) in a variant of the Korn inequality which reads $ $ $ ∇x u + T ∇x u $2 $ $ $ 2 $ 2
L (Ω)
K(Ω) ∇x u 2L2 (Ω) .
(233)
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The Korn inequality, of paramount importance in elasticity theory [201], is naturally needed to establish the system of differential inequalities which we use to quantify the trend to equilibrium. Our proof of (233) is partly inspired by Grad [254]. The whole thing adapts to the case of the torus, or to the bounce-back boundary condition, with significant simplifications. On the other hand, in the case of domains with an axis of symmetry, additional global conservation laws (angular momentum) have to be taken into account, and the case of a spherical domain also has to be separated from the rest. These extensions are discussed by Grad [254], but have not yet been transformed in a quantitative variant along the lines above. R EMARK . As we have seen in Chapter 2B, if the initial datum is not very smooth and if the Boltzmann collision kernel satisfies Grad’s cut-off assumption, then the solution of the Boltzmann equation is not expected to be very smooth. But in this case, as we discussed in Section 3.5 of Chapter 2B, one can hope for a theorem of propagation of singularities in which a vanishingly small (as t → ∞) singular part could be isolated from a very smooth remainder, and, as in [1], the entropy dissipation strategy would still apply. Theorem 23 certainly calls for lots of improvement and better understanding. Yet, it already shows that, in theory, entropy dissipation methods are able to reduce the problem of trend to equilibrium for the full Boltzmann equation, to a problem of uniform a priori estimates on the moments, smoothness and strict positivity of its solutions. Moreover, it shows that there is no need for stronger a priori estimates than the ones which are natural in a nonlinear setting: in particular, no estimates in L2 (M −1 ) are needed. We hope that these results will also provide a further motivation for the improvement of known a priori bounds.
CHAPTER 2D
Maxwell Collisions Contents 1. Wild sums . . . . . . . . . . . . . . . . . . . . . . . . . 2. Contracting probability metrics . . . . . . . . . . . . . . 2.1. The Wasserstein distance . . . . . . . . . . . . . . 2.2. Toscani’s distance . . . . . . . . . . . . . . . . . . 2.3. Other Fourier-based metrics . . . . . . . . . . . . . 2.4. The central limit theorem for Maxwell molecules . 3. Information theory . . . . . . . . . . . . . . . . . . . . . 3.1. The Fisher information . . . . . . . . . . . . . . . . 3.2. Stam inequalities for the Boltzmann operator . . . 3.3. Consequence: decreasing of the Carlen–Carvalho ψ 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Summary . . . . . . . . . . . . . . . . . . . . . . . 4.2. A remark on sub-additivity . . . . . . . . . . . . . 4.3. Remark: McKean’s conjectures . . . . . . . . . . .
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In this chapter we focus on the Boltzmann collision operator when the collision kernel only depends on the deviation angle: B(v − v∗ , σ ) = b(cos θ ).
(234)
As recalled in Chapter 2A, the modelling of Maxwell molecules, i.e., more or less fictitious particles interacting via repulsive forces in 1/r 5 , in three dimensions of space, leads to a collision kernel B which satisfies (234). By extension, we shall call Maxwellian collision kernel any collision kernel of the form (234). Assumption (234) entails a number of particular properties. The gain part of the Boltzmann collision operator Q+ (g, f ) =
RN
dv∗
S N−1
dσ b(cos θ )g(v∗ )f (v ),
v − v∗ cos θ = ,σ , |v − v∗ |
(235)
shares many features with the (more symmetric) rescaled convolution operator, g f ≡ g1 ∗ f1 , 2
2
(236)
where the rescaling operation is defined by
v fλ (v) = N/2 f √ . λ λ 1
(237)
Note that if X and Y are independent √ random variables with respective law f and g, then f g is the law of (X + Y )/ 2. Therefore, with the analogy between the Q+ and operations in mind, the theory of the spatially homogeneous Boltzmann equation with Maxwellian collision kernel resembles that of rescaled sums of independent random variables. In the sequel, we shall insist on some peculiar topics which illustrate the originality of Maxwellian collision kernels: in Section 1 the Wild sum representation, which is an appealing semi-explicit representation formula for solutions in terms of iterated Q+ operators; in Section 2, the existence and applications of several contracting probability metrics compatible with the Boltzmann equation. Finally, in Section 3, we describe some interesting connections with information theory. For more standard issues concerning the Cauchy problem or the qualitative behavior of solutions, the best reference is the long synthesis paper by Bobylev [79], entirely based on the use of Fourier transform, which also reviews many contributions by various authors. Before embarking on this study, we recall that besides its specific interest, the study of Maxwellian collision kernels is often an important step in the study of more general properties of the Boltzmann operator.1 1 See, for instance, Section 4.4 in Chapter 2B or Sections 4.3 to 4.6 in Chapter 2C.
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1. Wild sums If the collision kernel is Maxwellian and Grad’s cut-off assumption is satisfied, then one can assume without loss of generality that for some (and thus any) unit vector k, π b(k · σ ) dσ = S N−2 b(cos θ ) sinN−2 θ dθ = 1. (238) S N−1
0
Then the spatially homogeneous Boltzmann equation can be rewritten as ∂f = Q+ (f, f ) − f. ∂t
(239)
However, when the collision kernel is nonintegrable, then one can only write the general form ∂f = dv∗ dσ b(cos θ )[f f∗ − ff∗ ]. (240) ∂t RN S N−1 As was noticed by Wild [464], given any initial datum f0 , Equation (239) can be solved recursively in terms of iterated Q+ operators (this is nothing but a particularly simple iterated Duhamel formula, if one considers (239) as a perturbation of ∂t f = −f ). One finds f (t, ·) = e−t
∞
n−1 + Qn (f0 ), 1 − e−t
(241)
n=1
where the n-linear operator Q+ n is defined recursively by Q+ 1 (f0 ) = f0 , Q+ n (f0 ) =
1 + + Q Qk (f0 ), Q+ n−k (f0 ) . n−1 n−1 k=1
The sum (241) can also be rewritten f (t, v) =
∞
n=1
n−1 e−t 1 − e−t
α(γ )Q+ (f ) , 0 γ
(242)
γ ∈Γ (n)
where Γ (n) stands for the set of all binary graphs with n leaves, each node having zero or two “children”, and Q+ γ (f0 ) is naturally defined as follows: if γ has two subtrees γ1 + + + and γ2 (starting from the root), then Q+ γ (f0 ) = Q (Qγ1 (f0 ), Qγ2 (f0 )). Moreover, α(γ ) are combinatorial coefficients. Wild sums and their combinatorial contents are discussed with particular attention by McKean [341], and more recently by Carlen, Carvalho and Gabetta [126].
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It follows from the Wild representation that a solution of the Boltzmann equation (239) can be represented as a convex combination, with time-dependent weights, of terms of the form f0 , Q+ (f0 , f0 ), Q+ Q+ (f0 , f0 ), f0 , Q+ Q+ (f0 , f0 ), Q+ Q+ (f0 , f0 ), Q+ (f0 , f0 ) , etc. This representation is rather intuitive because it more or less amounts to count collisions: the f0 term takes into account particles which have undergone no collision since the initial time, the term in Q+ (f0 , f0 ) corresponds to particles which have undergone only one collision with a particle which had never collided before, Q+ (f0 , Q+ (f0 , f0 )) to particles which have twice undergone a collision with some particle having undergone no collision before . . . . This point of view is also interesting in numerical simulations: in a seemingly crude truncation procedure, one can replace (241) by f˜(t, v) = e−t
N0
n−1 + N 1 − e−t Qn (f0 ) + 1 − e−t 0 M, n=1
where M is the Maxwellian distribution with same first moments as f0 . Later in this chapter, we shall explain why such a truncation is rather natural, how it is related to the problem of trend to equilibrium and how it can be theoretically justified.2 2. Contracting probability metrics In this section, probability metrics are just metrics defined on a subset of the space of probability measures on RN . We call a probability metric d nonexpansive along solutions of Equation (240) if, whenever f (t, ·) and g(t, ·) are two solutions of this equation, then d f (t), g(t) d(f0 , g0 ). (243) We also say that d is contracting if equality in (243) only holds for stationary solutions, i.e. (in the case of finite kinetic energy), when f0 , g0 are Maxwellian distributions. 2.1. The Wasserstein distance In his study of the Boltzmann equation for Maxwellian molecules, Tanaka [414,353,415] had the idea to use the Wasserstein (or Monge–Kantorovich) distance of order 2, & % (244) W (f, g) = inf E|X − Y |2 ; law(X) = f, law(Y ) = g . 2 This is the purpose of Theorem 24(v).
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Here the infimum is taken over all random variables X, Y with respective law f and g. It is always assumed that f and g have finite moments of order 2, which ensures that W (f, g) < +∞. In analytical terms, W can be rewritten as W (f, g) = inf
RN ×RN
|v − w|2 dπ(v, w); π ∈ Π(f, g) ,
where Π(f, g) stands for the set of probability distributions π on RN × RN which admit f and g as marginals, More explicitly, π ∈ Π(f, g) if and only if ∀(ϕ, ψ) ∈ C0 RN × C0 RN , ϕ(v) + ψ(w) dπ(v, w) = RN ×RN
RN
fϕ +
RN
gψ.
The Wasserstein distance and its variants are also known under the names of Fréchet, Höffding, Gini, Hutchinson, Tanaka, Monge or Kantorovich distances. The infimum in (244) is finite as soon as f and g have finite second moments. Moreover, it is well-known that convergence in Wasserstein sense is equivalent to the conjunction of weak convergence in measure sense, and convergence of the second moment: ⎧ N n ⎪ ⎪∀ϕ ∈ C0 R , f ϕ dv −−−→ f ϕ dv, ⎨ n→∞ (245) W f n , f −−−→ 0 ⇐⇒ n→∞ ⎪ ⎪ ⎩ f n |v|2 dv −−−→ f |v|2 dv. n→∞
Tanaka’s theorem [415] states that whenever f , g are two probability measures with the same mean, and b is normalized by (238), then W Q+ (f, f ), Q+ (g, g) W (f, g).
(246)
Tanaka’s representation (74) entered the proof of this inequality, which is formally similar to a well-known inequality for rescaled convolution, W (f f, g g) W (f, g). Thanks to the Wild formula, Tanaka’s theorem implies that W is a nonexpansive (in fact contractive) metric, say when restricted to the set of zero-mean probability measures. On this subject, besides Tanaka’s papers one may consult [391]. As a main application, Tanaka proved theorems of convergence to equilibrium for Equation (240) without resorting to the H theorem. In fact, convergence to equilibrium follows almost for free from the contractivity property, since the distance to equilibrium, W (f (t), M), has to be decreasing (unless f is stationary). So one can prove that W f (t), M −−−→ 0 t →∞
as soon as f0 has just finite energy, not necessarily finite entropy.
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Since that time, Tanaka’s theorem has been largely superseded: more convenient metrics have been found, and entropy methods have become so elaborate as to be able to cover cases where the entropy is infinite.3 Yet, Tanaka’s theorem reminds us that Boltzmann’s H theorem is not the only possible explanation for convergence to equilibrium. The next sections will confirm this.
2.2. Toscani’s distance Since Bobylev’s work, it was known that the Fourier transform provides a powerful tool for the study of the spatially homogeneous Boltzmann equation with Maxwellian collision kernel. To measure discrepancies in Fourier space, Toscani introduced the distance d2 (f, g) = sup
ξ ∈RN
|fˆ(ξ ) − g(ξ ˆ )| . |ξ |2
(247)
The supremum in (247) is finite as soon as f and g have finite second moments, and the same mean velocity:
RN
f (v)v dv =
RN
g(v)v dv,
RN
(f + g)|v|2 dv < +∞.
Also, convergence in d2 sense is equivalent to convergence in Wasserstein sense (245). It turns out [427] that, under the normalization (238), d2 Q+ (f, f ), Q+ (g, g) d2 (f, g),
(248)
with equality only if f , g are Maxwellian distributions. As a consequence, d2 is a contracting probability metric along solutions of (239) (when one restricts to probability measures with some given mean). As shown by Toscani and the author [427], this contracting property remains true for Equation (240) with a singular collision kernel. As a main consequence, the Cauchy problem associated with (240) admits at most one solution. This uniqueness theorem holds under optimal assumptions: it only requires finiteness of the energy and of the cross-section for momentum transfer (63). As other applications of the d2 distance, we mention – a simple proof of weak convergence to equilibrium under an assumption of finite energy only; – some partial results for the non-existence of nontrivial eternal solutions [450, Annex II, Appendix]; – some explicit estimates of rate of convergence in the central limit theorem [427], by refinement of the inequality d2 (f f, M) d2 (f, M). 3 Recall Section 5.4 in Chapter 2C.
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2.3. Other Fourier-based metrics Other useful Fourier metrics are defined by ds (g, f ) = sup
ξ ∈RN
|fˆ(ξ ) − g(ξ ˆ )| , s |ξ |
s > 2.
(249)
They are well-defined only when f , g have the same moments up to high enough order. f For instance, one cannot directly compare ff to the associated Maxwellian distribution M in distance d4 unless f vi vj vk dv = M vi vj vk dv for all i, j, k. But this drawback is easily fixed by subtracting from fˆ a well-chosen Taylor polynomial. The interest of using exponents s greater than 2 comes from the fact that the distances ds become “more and more contracting” as s is increased, and this entails better properties of decay to equilibrium. As soon as s > 2, one can prove [225] exponential decay to equilibrium in distance ds , if the initial datum has a finite moment of order s. If one only assumes that the initial datum has a finite moment of order 2, then the method also yields exponential decay in some distance of the form dφ (f, g) = sup
ξ ∈RN
|fˆ(ξ ) − g(ξ ˆ )| , 2 |ξ | φ(ξ )
for some well-chosen function φ with φ(0) = 0. By taking larger values of s, one improves the rate of convergence in ds metric; in particular, the choice s = 4 yields the optimal rate of convergence [128], which is the spectral gap of the linearized Boltzmann operator.4 This exponent 4 is related to the fact that the linearized Boltzmann operator admits Hermite polynomials as eigenfunctions, and the lowest eigenvalues are obtained for 4th-degree spherical Hermite polynomials. Of course, this result of optimal convergence is obtained in quite a weak sense; but, by interpolation, it also yields strong convergence in, say, L1 sense if one has very strong (uniform in time) smoothness and decay bounds at one’s disposal. Such bounds were established in [128], thanks to an inequality which can be seen as reminiscent of the Povzner inequalities, but from the point of view of smoothness, i.e., with moments in Fourier space, instead of velocity space5 : $ + $ $Q (f, f )$2 m 1 f 2 m + Cm , H H 2
m ∈ N.
(250)
This inequality holds true at least when f is close enough to M f in relative entropy sense. After establishing the optimal decay to equilibrium in d4 distance on one hand, and the uniform smoothness bound in H m on the other hand, Carlen, Gabetta and Toscani [128] 4 Recall that when the collision kernel is Maxwellian, then the spectrum of the linearized operator can be computed explicitly [79, p. 135]. 5 It is not rare in the theory of the spatially homogeneous Boltzmann equation that smoothness estimates and decay estimates bear a formal resemblance, and this may be explained by the fact that the Fourier transform of the spatially homogeneous Boltzmann equation is a kind of Boltzmann equation, see Equation (77) . . . .
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had no difficulty in interpolating between both partial results to prove convergence to equilibrium in L1 at exponential rate. The interpolation can be made at the price of an arbitarily small deterioration in the rate of convergence if m is very large. A precise theorem will be given in Section 4.1. R EMARK . This theorem of exponential trend to equilibrium with explicit rate is at the moment restricted to Maxwellian collision kernels. This is because only in this case are nice contracting probability metrics known to exist. Also note that the spectral gap of the linearized collision operator is known only for Maxwellian collision kernel. Accordingly, a lower bound on the spectral gap is known only when the collision kernel is bounded below by a Maxwellian collision kernel. The preceding problem of trend to equilibrium takes its roots on the very influencial 1965 work by McKean [341]. In this paper, he studied Kac’s equation (21), and at the same time proved exponential convergence to equilibrium, with rate about 0.016, suggested the central limit theorem for Maxwellian molecules6 and established the decrease of the Fisher information.7 The value 0.016 should be compared to the optimal rate 0.25, which is obtained, up to an arbitrarily small error, in [128]. McKean’s results have inspired research in the area until very recently, as the rest of this chapter demonstrates. Another related early work was Grünbaum [262].
2.4. The central limit theorem for Maxwell molecules Once again, let us consider the Boltzmann equation with Maxwellian collision kernel, fix an initial datum f0 with unit mass, zero mean and unit temperature, and denote by M the corresponding Maxwellian. Recall from Section 1 that the solution to the Boltzmann equation with initial datum f0 can be written as the sum of a Wild series, which is a convex combination of iterated Q+ operators acting on f0 . Let us be interested in the behavior of the terms of the Wild series as t → +∞. It is obvious that the terms of low order have less and less importance as t becomes large, and in the limit the only terms which matter are those which take into account a large number of collisions. But the action of Q+ is to decrease the distance to equilibrium; for instance, one has the inequality d2 Q+ (f, g), M max d2 (f, M), d2 (g, M) , as a variant of the inequalities discussed in Section 2.2. Therefore we can expect that terms of high order in the Wild series will be very close to M, and this may be quantified into a statement that f approaches equilibrium as time becomes large. This however is not true for all terms of high order in the Wild series, but only for those terms Q+ γ (f0 ) such that the corresponding tree γ is deep enough (no leaves of small 6 See next section. 7 See Section 3.1.
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height). Intuitively, small depth means that all particles involved have collided sufficiently many times. For instance, one would expect8 Q+ Q+ Q+ (f0 , f0 ), Q+ (f0 , f0 ) , Q+ Q+ (f0 , f0 ), Q+ (f0 , f0 ) to be rather close to M, but not Q+ Q+ Q+ Q+ Q+ Q+ Q+ (f0 , f0 ), f0 , f0 , f0 , f0 , f0 , f0 (think that even Q+ (f0 , M) is not very close to M . . .). Thus, to make the argument work, McKean [341] had to perform an exercise in combinatorics of trees, and show that the combined weight of “deep enough” trees approaches 1 as time becomes large. These ideas were implemented in a very clean, and more or less optimal way, by Carlen, Carvalho and Gabetta [126], thanks to Fourier-defined probability metrics. A precise result will be given in Section 4.1.
3. Information theory 3.1. The Fisher information Among the most important objects in information theory are the Shannon entropy and the Fisher information. Up to a change of sign, Shannon’s entropy9 is nothing but the Boltzmann H -functional. As for the Fisher information, it is defined as I (f ) =
RN
|∇f |2 =4 f
RN
2 ∇ f
(251)
(compare with the relative Fisher information (174)). The Fisher information √ is always well-defined in [0, +∞], be it via the L2 square norm of the distribution ∇ f or by the convexity of the function (x, y) → |x|2 /y. It is a convex, isotropic functional, lower semicontinuous for weak and strong topologies in distribution sense. Fisher [216] introduced this object as part of his theory of “efficient statistics”. The Fisher information measures the localization of a probability distribution function, in the following sense. Let f (v) be a probability density on RN , and (Xn ) a family of independent, identically distributed random variables, with law f (· − θ ), where θ is unknown and should be determined by observation. A statistic is a random variable θˆ = θˆ (X1 , . . . , Xn ), which is intended to give a “best guess” of θ . In particular, θˆ should converge towards θ with probability 1 as n → ∞; and also one often imposes (especially when n is not so large) that θˆ be unbiased, which means E θˆ = θ , independently of n. Now, 8 Draw the corresponding trees! 9 See the references in Section 2.4 of Chapter 2A.
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the Fisher information measures the best possible rate of convergence of θˆ towards θ in the sense of mean quadratic error, as n → ∞. More explicitly, if θˆ is unbiased, then N2 . Var θˆ nI (f )
(252)
Inequality (252) is called the Cramér–Rao inequality, but for an analyst it is essentially a variant of the Heisenberg inequality, which is not surprising since a high Fisher information denotes a function which is very much “localized” . . . . In fact, the standard Heisenberg inequality in RN can be written I (f )
2
∀v0 ∈ RN ,
RN
f (v)|v − v0 |2 dv N 2
RN
f dv
.
For given mass and energy, the Fisher information takes its minimum value for Maxwellian distributions – just as the entropy. And for given covariance matrix, it takes its minimum value for Gaussian distributions. This makes it plausible that the Fisher information may be used in problems such as the long-time behavior of solutions to the Boltzmann equation, or the central limit theorem. The idea of an information-theoretical proof of the central limit theorem was first implemented by Linnik [305] in a very confuse, but inspiring paper. His ideas were later put in a clean perspective by Barron [59] and others, see the references in [156]. The same paper by Linnik also inspired McKean [341] and led to the introduction of the Fisher information in kinetic theory.10 3.2. Stam inequalities for the Boltzmann operator As one of the key remarks made by McKean [341], the Fisher information is a Lyapunov functional for the Kac model (21). We already mentioned in Section 4.7 of Chapter 2C that his argument can be adapted to the two-dimensional Boltzmann equation. Also Toscani [421] gave a direct, different proof of this two-dimensional result. A more general result goes via Stam-type inequalities. The famous Blachman–Stam and Shannon–Stam inequalities11 [411,75,125] admit as particular cases I (f
f ) I (f ),
H (f
f ) H (f ).
These inequalities are central in information theory [156,165]. Their counterparts for the Boltzmann equation are H Q+ (f, f ) H (f ), (253) I Q+ (f, f ) I (f ), where Q+ is defined by (235), and the collision kernel has been normalized by (238). 10 McKean used the denomination “Linnik functional” for the Fisher information, which is why this terminology was in use in the kinetic community for some time. 11 Inequalities (196) and (195), respectively.
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Inequalities (253) immediately entail (by Wild sum representation when (238) holds, by approximation in the general case – or by an ad hoc application of the definition of convexity) that the Fisher information and the H -functional are Lyapunov functionals along the semigroup generated by the Boltzmann equation with Maxwellian collision kernel. In particular, this gives a new proof of the H theorem in this very particular situation. This remark is not so stupid as it may seem, because this proof of the H theorem is robust under time-discretization, and also applies for an explicit Euler scheme12 – apparently this is the only situation in which the entropy can be shown to be nonincreasing for an explicit Euler scheme. Inequalities (253) were proven in dimension 2 by Bobylev and Toscani [83], and in arbitrary dimension, but for constant collision kernel (in ω-representation), by Carlen and Carvalho [121]. Finally, the general case was proven by the author in [445]. Apart from rather classical ingredients, the proof relied on a new representation formula for ∇Q+ in the Maxwellian case: ∇Q+ (f, f ) 1 = dv∗ dσ b(k · σ ) f∗ (I + Pσ k )(∇f ) + f (I − Pσ k )(∇f )∗ , 2
(254)
where k = (v − v∗ )/|v − v∗ |, IN : RN → RN stands for the identity map and Pσ k : RN → RN is the linear mapping defined by Pσ k (x) = (k · σ )x + (σ · x)k − (k · x)σ. Formula (254) was obtained by an integration by parts on S N−1 , which crucially used the assumption of Maxwellian collision kernel. In the non-Maxwellian case, we could only obtain an inequality weaker by a factor 2: I (Q± (f, f )) 2 A L∞ I (f ), where A(z) = S N−1 B(z, σ ) dσ . Just as in the well-known Stam proof, the first inequality in (253) implies the second one via adjoint Ornstein–Uhlenbeck regularization. To be explicit, if (St )t 0 stands for the semigroup associated to the Fokker–Planck equation, and if f has unit mass, zero mean, unit temperature, then H Q+ (f, f ) − H (M) =
+∞
I Q+ (St f, St f ) − I (M) dt.
0
Underlying this formula is of course the commutation between St and Q+ , St Q+ (f, f ) = Q+ (St f, St f ), which follows from Bobylev’s lemma,13 for instance. R EMARKS . (1) The decreasing property of the Fisher information also holds for solutions of the Landau equation with Maxwell molecules. This can be seen by asymptotics 12 Also one may dream a little bit and imagine that this remark could end up with new lower bounds for the entropy dissipation! 13 See Section 4.8 in Chapter 2A.
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of grazing collisions, but a direct proof is also possible, and gives much better quantitative results [451]. In particular, one can prove that the Fisher information converges exponentially fast to its equilibrium value. (2) Still for the Landau equation, this decreasing property was compared with results from numerical simulations by Buet and Cordier [105]. In many situations, they observed a decreasing behavior even for non-Maxwell situations, e.g., Coulomb potential. They also suggested that this decreasing phenomenon was associated with entropic properties of the code: enforcing the decrease of the entropy in the numerical scheme would have a stabilizing effect which prevents the Fisher information to fluctuate. 3.3. Consequence: decreasing of the Carlen–Carvalho ψ functional In [121], Carlen and Carvalho introduced the function ψ : λ → H (f ) − H (Sλ f ) to measure the smoothness of the distribution function f , in the context of entropy– entropy dissipation inequalities. Here (St )t 0 is as usual the adjoint Ornstein–Uhlenbeck semigroup, i.e., the semigroup generated by the Fokker–Planck operator (161). Note that if the moments of f are normalized by (177), then ψ is a nonnegative function. One of the main points in the Carlen–Carvalho theorems14 was to obtain a control of ψ for λ close to 0. We now claim that for Maxwell collision kernel, the function ψ is pointwise nonincreasing: for each value of λ, H (f )−H (Sλ f ) is nonincreasing as a function of time. To see this, recall that in the case of Maxwellian collision kernel, the Boltzmann semigroup (Bt ) commutes with (St ), as a consequence of Bobylev’s lemma.15 Therefore, the time-derivative of H (f ) − H (Sλ f ) is D(Sλ f ) − D(f ). To prove that D(Sλ f ) − D(f ) is nonpositive, we just have to prove d D(Sλ f ) 0. dλ But,16 by commutation property, the dissipation along (St ), of the dissipation of entropy along (Bt ), is also the dissipation along (Bt ), of the dissipation of entropy along (St ). Hence, d d D(S f ) = I (Bt f ) 0. λ dλ λ=0 dt t =0 This proves the claim. 14 Recall Section 4.4 in Chapter 2C. 15 See Section 4.8 in Chapter 2A. 16 We already made this remark in Section 4.4 of Chapter 2C.
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4. Conclusions 4.1. Summary We now summarize most of our discussion about specific properties of the Boltzmann equation with Maxwellian collision kernel in a single theorem. As usual, we set f L1s (RNv ) =
RN
s/2 f (v) 1 + |v|2 dv.
T HEOREM 24. Let b(cos θ ) be a nonnegative collision kernel, satisfying finiteness of the cross-section for momentum transfer,
π
b(cos θ )(1 − cos θ ) sinN−2 θ dθ < +∞,
0
and let f0 ∈ L12 (RN v ) be an initial datum with finite mass and energy. Without loss of generality, assume that f0 has unit mass, zero mean velocity and unit temperature. Then, (i) there exists a unique (weak) solution (f (t))t 0 to the spatially homogeneous Boltzmann equation with initial datum f0 ; (ii) the quantities H (f (t)), I (f (t)), d2 (f (t), M), W (f (t), M) are nonincreasing as functions of t; (iii) d2 (f (t), M) and W (f (t), M) converge to 0 as t → +∞, and also H (f (t)) − H (M) if H (f0 ) < +∞; (iv) assume that the collision kernel satisfies Grad’s angular cut-off assumption, i.e.,
π
b(cos θ ) sinN−2 θ dθ < +∞.
0
Let λ be the spectral gap of the linearized Boltzmann operator. Then, for all ε > 0, there exists s > 0 and k ∈ N such that, if f0 ∈ L12+s ∩ H k (RN ), then there exists a constant C < +∞, explicit and depending on f only via f0 L1 and f0 H k , such that 2+s
∀t 0,
$ $ $f (t) − M $
L1
Ce−(λ−ε)t ;
(v) assume that the collision kernel satisfies Grad’s angular cut-off assumption, and is normalized by (238). Consider the Wild representation of f (t), and for any N0 1 let fN0 (t) be the truncation of the series at order N0 (take formula (241) and throw away all terms starting from the one in e−t (1 − e−t )N0 ). Define N gN0 (t) = fN0 (t) + 1 − e−t 0 M.
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Assume that f0 ∈ L12+δ ∩ H 2+δ (RN0 ) for some δ > 0. Then, there exist constants C < +∞ and α > 0, depending on f only via f0 L1 and f H 2+δ , such that 2+δ
∀N0 1, ∀t 0,
−t N0 $ $ $f (t) − gN (t)$ 1 C (1 − e ) . 0 L N0 α
Also, f (t) − M L1 converges exponentially fast to 0. Part (i) of this theorem is from Toscani and Villani [427]. As for part (ii), the statement about W2 is due to Tanaka [415], the one about d2 is in Toscani and Villani [427], the one about I is from Villani [445]. Point (iii) is due to Tanaka for W , the same proof applies for d2 . Actually Tanaka’s proof was given only under additional moment assumptions; for the general result one needs tightness of the energy, which is proven in [225]. The statement about the entropy is more delicate: in addition to the tightness of the energy, it requires the monotonicity property of the function ψ, as described in Section 3.3. These two estimates make it possible to use the main result in Carlen and Carvalho [121] and conclude that the relative entropy satisfies a closed differential equation which implies its convergence to 0 at a computable rate. This argument is explicitly written in Carlen, Carvalho and Wennberg [127] in the particular case when the collision kernel is constant in ω-representation. Next, point (iv) is the main result of Carlen, Gabetta and Toscani [128], while point (v) is the main result of Carlen, Carvalho and Gabetta [126]. Point (v) can be seen as a bound (essentially optimal) on the error which is performed when replacing the solution of the Boltzmann equation by a truncation of the Wild sum. Note that the result of exponential convergence in (v) is much more general than the one in (iv), but the rate of convergence is a priori worse. We note that the proof of point (iv) uses the last part of (iii). Indeed, the convergence to equilibrium is shown to be exponential only in a certain neighborhood of the equilibrium;17 to make the constant C explicit it remains to estimate the time needed to enter such a neighborhood, which is what entropy methods are able to do. In conclusion, one can say that the theory of spatially homogeneous Maxwell molecules is by now essentially complete. The links between information theory and kinetic theory have been completely clarified in the last years, this being due in large part to the contributions by Carlen and coworkers. Among the few questions still open, we mention the classification of all nontrivial eternal solutions18 – which certainly can be attacked more efficiently in the Maxwellian case, thanks to the many additional tools available, as demonstrated by the advances made by Bobylev and Cercignani [81] – and the problems which are mentioned in the next two sections. Also, it would be extremely interesting to know how point (iv) above generalizes to a spatially inhomogeneous setting, even from the formal point of view, and even assuming on the solutions all the smoothness one can dream of. 17 In particular, because the bound (250) is only proven when f is close enough to M. 18 See Section 2.9 in Chapter 2A.
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4.2. A remark on sub-additivity An interesting problem is the classification of all Lyapunov functionals for the Boltzmann equation. Recall the following result by McKean19 [342]: for the Kac model, the entropy, or H -functional is, up to addition of an affine function or multiplication by a constant, the only Lyapunov functional of the form A(f ). McKean also conjectured that the Fisher information would be the only Lyapunov functional of the form A(f, ∇f ). A related problem is to classify all functionals J , say convex and isotropic, which satisfy J (Q+ ) J , in a way similar to (253), under the normalization (238). Such functionals are particular Lyapunov functionals for the Boltzmann equation (240). In dimension 2 of velocity space, Bobylev and Toscani [83] have obtained the following sufficient condition: for all probability distributions f and g on R2 , and for all λ ∈ [0, 1], J (fλ ∗ g1−λ ) λJ (f ) + (1 − λ)J (g).
(255)
Whatever the dimension, this criterion is satisfied by all functionals that we have encountered so far: H , I , d2 (·, M), W (·, M)2 . However, nobody knows if it is sufficient in dimension higher than 2. As a consequence of our remarks on the Landau equation with Maxwellian collision kernel [443] and the asymptotics of grazing collisions, any Lyapunov functional J has to satisfy (255) in the particular case when f is radially symmetric and g is the Maxwellian distribution with zero mean, and same energy as f . 4.3. Remark: McKean’s conjectures In his seminal 1965 work [341], McKean also formulated several conjectures. Even though they all seem to be false, they have triggered interesting developments. Let us mention two of these conjectures. The “super-H theorem” postulates that the entropy is a completely monotone function of time: 2 2 dH /dt 0, d H / dt 0, . . . (−1)n dn H / dt n 0. For some time this was a popular subject among a certain group of physicists. This conjecture is however false, as shown by Lieb [303] with a very simple argument. Strangely, for the particular Bobylev–Krook–Wu explicit solutions, this “theorem” holds true for n 101 and breaks downs afterwards [361]. The “McKean conjecture”, strictly speaking. Let Mδ stand for the Maxwellian distribution with zero mean and δ temperature, and consider the formal expansion H (f ∗ Mδ ) = −
∞
In (f ) δ n n=0
n!
2
,
(256)
19 This result is somewhat reminiscent of the axiomatic characterization of entropy by Shannon, see, for instance, [156, pp. 42–43] and references therein.
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so that I0 (f ) = −H (f ), I1 (f ) = I (f ), I2 (f ) = − ij f [∂ij (log f )]2 , etc. Knowing that dI0 /dt 0 and dI1 /dt 0, McKean conjectured the more general inequality (−1)n dIn /dt 0. This conjecture seems to be false in view of the formal study realized by Ledoux [295] for the Fokker–Planck equation. Keeping in mind that the entropy measures volume in infinite dimension, the successive terms in (256) could be seen as infinite-dimensional analogues of the mixed volumes arising in convex geometry. It is not even clear that they have alternate signs for n 1 . . . . However, this conjecture has inspired a few works in kinetic theory, see, for instance, Gabetta [224], or the discussion of the Kac model in Toscani and Villani [428, Section 7]. These ideas have also been used by Lions and Toscani [323] to establish certain strengthened variants of the central limit theorem.
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CHAPTER 2E
Open Problems and New Trends Contents 1. Open problems in classical collisional kinetic theory . . . . 1.1. Strong solutions in a spatially inhomogeneous setting . 1.2. Derivation issues . . . . . . . . . . . . . . . . . . . . . 1.3. Role of the kinetic singularity . . . . . . . . . . . . . . 1.4. Improved entropy–entropy dissipation estimates . . . . 1.5. Approach to equilibrium for Kac’s master equation . . 1.6. Influence of the space variable on the equilibration rate 2. Granular media . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Derivation issues: problems of separation of scales . . 2.2. Spatial inhomogeneities . . . . . . . . . . . . . . . . . 2.3. Trend to equilibrium . . . . . . . . . . . . . . . . . . . 2.4. Homogeneous Cooling States . . . . . . . . . . . . . . 3. Quantum kinetic theory . . . . . . . . . . . . . . . . . . . . 3.1. Derivation issues . . . . . . . . . . . . . . . . . . . . . 3.2. Trend to equilibrium . . . . . . . . . . . . . . . . . . . 3.3. Condensation in finite time . . . . . . . . . . . . . . . 3.4. Spatial inhomogeneities . . . . . . . . . . . . . . . . .
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The goal of this chapter is to present some of the main open problems in collisional kinetic theory, then to discuss some of the new questions arising in two developing branches of the field: the study of granular media on one hand, quantum kinetic theory on the other. Other choices could have included semiconductors (whose modelling is very important for industrial applications), modelling of biological interactions (in which problems have not been very clearly identified up to now), the study of aerosols and sprays (which naturally involve the coupling of kinetic equations with fluid mechanics), etc. Also, we only discuss problems associated with the qualitative behavior of solutions, and do not come back on less traditional issues like those which were presented in Section 2.9 of Chapter 2A. Selecting “important” problems is always dangerous because of subjectivity of the matter, and changes in mathematical trends and fashions. To illustrate this, let us quote Kac himself [283, p. 178, footnote 5]: “Since the master equation1 is truly descriptive of the physical situation, and since existence and uniqueness of the solutions of the master equation are almost trivial, the preoccupation with existence and uniqueness theorems for the Boltzmann equation appears to be unjustified on grounds of physical interest and importance.”
1. Open problems in classical collisional kinetic theory 1.1. Strong solutions in a spatially inhomogeneous setting The theory of the Cauchy problem for Boltzmann-like equations is by now fairly advanced under the assumption of spatial homogeneity. For instance, in the case of hard potentials, it seems reasonable to expect that this theory will soon be completed with the help of already existing tools. On the other hand, essentially nothing is known concerning the general, spatially inhomogeneous case in a non-perturbative context. Progress is badly needed on the following issues: – moment estimates, – regularity estimates (propagation of regularity/singularity, regularization), – strict positivity and lower bounds. This lack of a priori estimates is a limiting factor in many branches of the field. A priori estimates would enable one to • prove uniqueness of solutions, and energy conservation; • perform a simple treatment of boundary conditions (walls, etc.); • give estimates of speed of convergence to equilibrium along the lines presented in Chapter 2C (for this one needs uniform estimates as times goes to +∞); • justify the linearization procedure which is at the basis of so many practical applications of kinetic theory, see, for instance, [148]. Such a priori estimates also would be very useful, even if not conclusive, to • prove the validity of the Landau approximation in plasma physics, viewed as a largetime correction to the Vlasov–Poisson equation; 1 See Section 1.5 below.
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• prove the validity of the fluid approximation to the Boltzmann equation. For this, local conservation laws seem to be the minimum one can ask for in order to prove the hydrodynamic limit.2 Even in the more modest framework of solutions in the small, where smooth solutions can often be built, many gaps remain, like the treatment of singular collision kernels, or the derivation of uniform smoothness estimates as t → +∞.
1.2. Derivation issues As already mentioned, Lanford’s theorem is limited to a perturbative framework (small solutions), and concerns only the hard-sphere interaction. The treatment of more general interactions is almost completely open, and would be of considerable interest. As an oustanding problem in the field is of course the formidable task of rigorously deriving collisional kinetic equations for Coulomb interaction. As for the problem of extending Lanford’s theorem to a nonperturbative setting, one of the main difficulties is certainly related to the fact that there is no good theory for the Cauchy problem in the large – but we expect much, much more obstacles to overcome here!
1.3. Role of the kinetic singularity Let us consider a Boltzmann collision kernel, say of the form B(v − v∗ , σ ) = Φ |v − v∗ | b(cos θ ). In Chapter 2B we have seen how the properties of the Boltzmann equation depend on whether b is integrable or singular. On the other hand, what remains unclear even in the spatially homogeneous case, is the influence of the kinetic collision kernel Φ. When Φ is singular, does it induce blow-up effects, and in which sense? Does it help or harm regularizing effects induced by an angular singularity? The most important motivation for this problem comes from the modelling of Coulomb collisions in plasma physics: the collision kernel given by the Rutherford formula presents a singularity like |v − v∗ |−3 in dimension N = 3. Here are two questions which arise naturally. (1) Consider the Boltzmann equation with truncated Rutherford collision kernel, of the form |v − v∗ |−3 b(cos θ )1θε , which is physically unrealistic but used in certain modelling papers [162]. This collision kernel presents a nonintegrable (borderline) kinetic singularity. Does it entail that the equation induces smoothness, or even just compactifying effects? Some formal arguments given in Alexandre and Villani [12] may support a positive answer, but the situation seems very intricate. As explained in [12, Section 5], the geometry of the 2 This is what the author believed, until very recently some proofs of hydrodynamic limit appeared [54,53,240, 245], covering situations in which local conservation laws are not known to hold for fixed Knudsen number, but are asymptotically recovered in the limit when the Knudsen number goes to 0!!
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problem is “dual”, in some sense, to the one which appears in the study of nonintegrable, borderline angular singularities . . . . If the answer is negative, this suggests that such collision kernels should be used with a lot of care! (2) On the other hand, consider the Landau approximation for Coulomb collisions, or Landau–Coulomb equation, which is more realistic from the physical point of view. Does this equation have smooth solutions? In the spatially homogeneous situation, the Landau–Coulomb equation can be rewritten as ∂f
∂ 2f a¯ ij + 8πf 2 , = ∂t ∂vi ∂vj
t 0, v ∈ R3 ,
(257)
ij
where
a¯ ij =
vi vj 1 ∗ f. δij − |v| |v|2
If f is smooth, then the matrix (a¯ ij ) is locally positive definite, but bounded, and (257) is reminiscent of the nonlinear heat equation ∂f = f + f 2 ∂t
(258)
which has been the object of a lot of studies [468] and generically blows up in finite time, say in L∞ norm. The common view about (258) is that the diffusive effects of the Laplace operator are too weak to compensate for blow-up effects induced by the quadratic source term. And if the diffusion matrix (a¯ ij ) is bounded, this suggests that (257) is no more diffusive than (258). Weak solutions to (257) have been built in Villani [446]; they satisfy the a priori estimate √ f ∈ L2t (Hv1 ) locally. This estimate is however, to the best of the knowledge of the author, compatible with known a priori estimates for (258). These considerations may suggest that blow-up in finite time may occur for solutions of (257). If there is blow-up, then other questions will arise: how good is the Landau approximation at a blow-up time? What happens to blow-up if the physical scales are such that the Landau effects should only be felt as t → +∞? However, blow-up has never been reported by numerical analysts. And after seeing some numerical simulations by F. Filbet, the author has changed his mind on the subject, to become convinced that blow-up should indeed not occur. All this calls for a wide clarification.
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1.4. Improved entropy–entropy dissipation estimates In this section we only consider very nice distribution functions, say smooth and rapidly decaying, bounded below by a fixed Maxwellian. We saw in Chapter 2C that such probability distributions satisfy entropy–entropy dissipation inequalities of the form α D(f ) KH f |M f , where M f is the Maxwellian equilibrium associated with f , K and α are positive constants, H is the relative entropy functional, and D is the entropy dissipation for either Boltzmann or Landau’s equation. In several places do our results call for improvement: Landau equation with hard potentials. In the case of the Landau equation, α = 1 is admissible when Ψ (|z|) |z|2 ; α = 1 + ε (ε arbitrarily small) is admissible when Ψ (|z|) |z|2+γ , γ < 0. It is natural to conjecture that also α = 1 be admissible for hard potentials (γ > 0). More generally, this should be true when Ψ (|z|) = |z|2 ψ(|z|) with ψ continuous and uniformly positive for |z| δ > 0. This conjecture is backed by the spectral analysis of the linearized Landau operator [161], and also by the similar situation appearing in Carrillo, McCann and Villani [130] in the study of entropy–entropy dissipation inequalities for variants of granular media models. At the moment, the best available exponent for hard potentials is α = 1 + 2/γ , from Desvillettes and Villani [183]. Boltzmann equation with hard potentials. In the case of the Boltzmann equation, α = 1 + ε is admissible for Maxwellian or soft potentials. It is accordingly natural to think that α = 1 + ε is also admissible3 for hard potentials. Recall that Cercignani’s conjecture (α = 1) is false in most cases according to Bobylev and Cercignani [87]. Cercignani’s conjecture revisited? Counterexamples in [87] leave room for Cercignani’s conjecture to hold true in two situations of interest: • when the collision kernel is noncut-off and presents an angular singularity. This would be plausible since grazing collisions behave better with respect to large velocities, as the example of the Landau equation demonstrates4; • when f ∈ Lp ((M f )−1 ) for some p 1. Of special interest are the cases p = 1 (cf. Bobylev’s estimate for hard spheres, in Theorem 1(ii)); p = 2 (natural space for linearization) and p = ∞ (when f/M f is bounded from above). Maybe a Maxwellian bound from below is also needed for proving such theorems. As we mentioned when discussing Cercignani’s conjecture in Chapter 2C, about this topic one also has to make the connection with the recent Ball and Barthe result about the central limit theorem. 3 As this review goes to print, the author just managed to prove precisely this result, under the assumption that the density be bounded in all Sobolev spaces. 4 Similar results in the theory of linear Markov processes would also be interesting.
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1.5. Approach to equilibrium for Kac’s master equation A related topic is the Kac spectral gap problem and its entropy dissipation variant. This subject is a little bit in digression with respect to those which we discussed so far, but we wish to explain it briefly because of its intimate (and not well-known) connections with Cercignani’s conjecture. These connections were brought to our attention by E. Carlen. In his famous paper [283], Kac introduced a stochastic model which he believed to be a way of understanding the spatially homogeneous Boltzmann equation. His equation models the behavior of n particles interacting through binary elastic collisions occurring at random Poissonnian times, with collision parameter σ randomly chosen on the sphere. It reads 1
∂fn = n (259) − dσ B(vi − vj , σ ) Aij σ fn − fn , ∂t S N−1 2 i<j
where the summation runs over all pairs of distinct indices (i, j ) in {1, . . . , n}, and fn is a symmetric probability distribution on the manifold (actually a sphere) of codimension N + 1 in (RN )n defined by the relations n
|vi |2 = 2nE > 0,
i=1
n
vi = nV ∈ RN .
i=1
We use the notation − for the normalized integral on the sphere, |S N−1 |−1 . Moreover the ij linear operator Aσ represents the result of the collision of the spheres with indices i and j , Aij σ fn (v1 , . . . , vn ) = f (v1 , . . . , vi , . . . , vj , . . . , vn ),
vi =
|vi − vj | vi + vj + σ, 2 2
vj =
|vi − vj | vi + vj − σ. 2 2
As explained by Kac, the spatially homogeneous Boltzmann equation can be recovered, at least formally, as the equation governing the evolution of the one-particle marginal of fn in the limit n → +∞. In this limit, time has to be sped up by a factor n. See [283,412,256] for a study of this and related subjects. A simplified version, which is commonly called Kac’s master equation, is given by 1 2π ∂fn ij = Ln fn = n − dθ fn ◦ Rθ − fn , ∂t 0 2
(260)
i<j
where fn is a probability distribution on the sphere in Rn , defined by
vi2 = 2nE.
(261)
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Moreover, Rθ v = (v1 , . . . , vi , . . . , vj , . . . , vn ), ij
where (vi , vj ) = Rθ (vi , vj ) is obtained from (vi , vj ) by a rotation of angle θ in the (i, j )√plane. Without loss of generality, we set E = 1/2 in (261), so that the sphere has radius n. With this choice, the image measure of the uniform probability measure on the sphere, under projection onto some axis of coordinate, becomes the standard Gaussian measure as n → ∞ (Poincaré’s lemma, actually due to Maxwell). Moreover, we shall use the uniform probability measure on the sphere as reference measure, so that probability distributions are normalized by −√
nS n−1
fn dσ = 1.
Among the problems discussed by Kac is that of establishing an asymptotically sharp lower bound on the spectral gap λn of Ln as n → +∞. Recently, Diaconis and Saloff3 Coste [189] proved λ−1 n = O(n ), then Janvresse [281] proved Kac’s conjecture that −1 λn = O(n); she used Yau’s so-called martingale method. Finally, a complete solution was given very recently by Carlen, Carvalho and Loss [123], who managed to compute the spectral gap by a quite unexpected method (also by induction on the dimension). This work also extends to Equation (259) if the collision kernel B is Maxwellian. Since time should be sped up by a factor n in the limit n → ∞, the corresponding evolution equation will satisfy estimates like $ $ $ $ $fn (t, ·) − 1$ 2 √ n−1 e−λt $fn (0, ·) − 1$ 2 √ n−1 L ( nS ) L ( nS )
(262)
for some λ > 0, which can be chosen uniform as n → ∞ according to Janvresse’s theorem. Here 1 is √ the equilibrium state, i.e., the density of the uniform probability measure on the sphere nS n−1 . Inequality (262) conveys a feeling of uniform trend to equilibrium as n → ∞, which was Kac’s goal. However, it is not very clear in which sense (262) is a uniform estimate. Since all the functions fn ’s are defined on different spaces, one should be careful in comparing them. In particular, think that if fn satisfies the chaos property, then fn L2 is roughly of order C n for some constant C > 0 (which in general is not related to the L2 (or L2 (M −1 )) norm of the limit one-particle marginal f , see [283, Equation (6.44)]). And fn − 1 2L2 = fn 2L2 − 1 is also of order C n . Having this in mind, it would be natural to 1/n
compare distances in dimension n by the quantity · L2 . But if we do so, we find $ $ $1/n λ $ $fn (t, ·) − 1$1/n e− n t $fn (0, ·) − 1$L2 (S n−1 ) , L2 (S n−1 )
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which does not behave well in the limit5 ! A way to circumvent the difficulty would be to compare all first marginals, which all live in L1 (R), and prove that under some precise conditions on the sequence (fn ), ∃λ > 0, ∀n 1,
$ $ $P1 fn (t, ·) − M $
L2 (M −1 )
Ce−λt .
Now, a problem which looks more natural and more interesting in this context is the problem of the entropy–entropy dissipation estimate for Kac’s master equation. Again, we state this problem assuming without loss of generality that E = 1/2, so that vi2 = n in (261) and we use the uniform probability measure as reference measure for the definition of the entropy: fn log fn dσ. H (fn ) = − √ nS n−1
Note that H (1) = 0. P ROBLEM . Find Kn optimal such that for all symmetric probability distribution fn on √ nS n−1 , −
(Ln fn ) log fn Kn H (fn ).
(263)
If Kn−1 = O(n), then (263) entails the following entropy estimate for solutions of the Kac equation: H fn (t, ·) − H (1) e−µt H fn (0, ·) − H (1) , for some µ > 0. Since H (fn ) typically is O(n), this would lead to the satisfactory estimate
H (fn (t, ·)) − H (1) −µt H (fn (0, ·)) − H (1) e , n n
(264)
and also one-particle marginals of all fn ’s could be compared easily as a consequence an adequate chaos assumption for fn (0, ·). But from the counterexamples due to Bobylev and Cercignani [87], one expects that Kn−1 = O(n) is impossible. Indeed, by passing to the limit as n → ∞ in (263), under a chaos assumption, one would have a proof of Cercignani’s entropy dissipation conjecture for Kac’s model, which should be false (although this has never been checked explicitly) . . . . On the other hand, the author [450, Annex III, Appendix B] was able to prove Kn−1 = O(n2 ) by the same method as in Section 4.6 of Chapter 2C. This leads to two open questions: • What is the optimal estimate? 5 Or should the relevant scaling be f (1 + h/√n)⊗n , which would mean that we are interested in fluctuations n of the equilibrium state? This ansatz formally leads to fn − 1 2 2 = O(n) . . . . L
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• Does an estimate like O(n) hold for a well-chosen sub-class of probability distributions?
1.6. Influence of the space variable on the equilibration rate Let us now consider trend to equilibrium in a spatially inhomogeneous context. Diffusive models. The strategy of Desvillettes and Villani, exposed in Chapter 2C, shows trend to equilibrium like O(t −∞ ) for many entropy-dissipating systems when good smoothness a priori estimates are known. However, an exponential rate of convergence would be expected, at least for the linear Fokker–Planck, or Landau equation. The following issues would be of particular interest: • admitting that the solution of the linear Fokker–Planck equation with confinement potential V goes to equilibrium in relative entropy like O(e−αt ), what is the optimal value of α and how does it depend on V ? Can one obtain this result by an entropy method? • admitting that the solution of the Landau equation in a box (periodic, or with appropriate boundary condition) goes to equilibrium in relative entropy like O(e−αt ), what is the optimal value of α and how does it depend on the boundary condition or the size of the box? Can one devise an entropy method to obtain exponential decay? Boltzmann equation. When Boltzmann’s equation with a Maxwellian collision kernel is considered, then one can compute the spectral gap of the linearized operator. In the spatially homogeneous case, as we have seen in Chapter 2D, this spectral gap essentially governs the rate of decay to equilibrium, even in a non-linearized setting. At the moment, entropy methods seem unable to predict such a result,6 but a clever use of contracting probability metrics saves the game. Now, how does all this adapt to a spatially inhomogeneous context and how is the rate of convergence affected by the box, boundary conditions, etc.? Of course, in a preliminary investigation one could take for granted all the a priori bounds that one may imagine: smoothness, decay, positivity . . . . This concludes our survey of open problems for the classical theory of the Boltzmann equation. As the reader has seen, even if the field is about seventy years old, a lot remains to be done! Now we shall turn to many other problems, arising in less classical contexts which have become the object of extensive studies only recently.
2. Granular media Over the last years, due to industrial application and to the evolution of the trends in theoretical physics, a lot of attention was given to the modelling of granular material (sand, powders, heaps of cereals, grains, molecules, snow, or even asteroids . . .). The literature on Note added in proof : On this problem see the recent progress by Hérau and Nier. 6 Because of the obstruction to Cercignani’s conjecture . . . .
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the subject has grown so fast that some journals are now entirely devoted to it! And also the number of involved physicists has become extremely large. Among the main motivations are the understanding of how granular material behaves under shaking, how flows are evolving or how to prevent them, how to facilitate mixing, how to prevent violent blow-up of a silo, or avalanches, how matter aggregates in a newborn solar system, etc. One popular model for these studies is a kinetic description of a system of particles interacting like hard spheres, but with some energy loss due to friction. Friction is a universal feature of granular material because of the roughness of the surface of particles. Many studies have been based on variants of the Boltzmann or Enskog equations which allow energy loss. This subject leads to huge difficulties in the modelling; see, for instance, the nice review done by Cercignani [146] a few years ago. Also some mathematical contributions have started to develop, most notably the works by Pulvirenti and collaborators [70–72,68,69,395]. Here we point out some of the most fundamental mathematical issues in the field. Since the physical literature is considerable, we only give a very restricted choice of physicists’ contributions. Thanks are due to E. Caglioti for explaining us a lot about the subject and providing references.
2.1. Derivation issues: problems of separation of scales The separation between microscopic and macroscopic scales in the study of granular media is not at all so clear as in the classical situation, and this results in many problems when it comes to derivation of the relevant equations. To illustrate this, we mention the astonishing numerical experiments described in [237]: a gas of inelastic particles is enclosed in a one-dimensional box with specular reflection (elastic wall) on one end, Maxwellian re-emission (heating wall) on the other. It is found that, basically, just one particle keeps all the energy, while all the others remain slow and stay close to the elastic wall. In other words, the wall is unable to heat the gas, and moreover it is plainly impossible to define meaningful macroscopic quantities! Enskog equation. At the basis of the derivation of the Boltzmann equation, be it formal or more rigorous, is the localization of collisions: the length scale for interaction is much smaller than the length scale for spatial fluctuations of density. The Boltzmann–Grad limit n → ∞, r → 0, nr 2 → 1 (n = number of particles, r = radius of particles, dimension = 3) is a way to formalize this for a gas of elastic hard spheres. On the other hand, in the case of granular media, the size of particles is generally not negligible in front of the typical spatial length. This is why many researchers use an Enskog-like equation, with delocalized collisions: for instance, ∂f + v · ∇x f = r 2 ∂t
R3
dv∗
S2
dσ |v − v∗ |
× J G(x, x + rσ ; ρ)f (t, x, v)f ˜ (t, x − rσ, v˜∗ ) − G(x, x − rσ ; ρ)f (t, x, v)f (t, x − rσ, v∗ ) .
(265)
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(See Cercignani [146] and Bobylev et al. [86].) Here r > 0 is the radius of particles, v, v∗ are post-collisional velocities and v, ˜ v˜∗ are pre-collisional velocities, given by the formulas ⎧ v + v∗ 1 − e 1+e ⎪ ⎪ − (v − v∗ ) + |v − v∗ |σ, ⎨v˜ = 2 4e 4e ⎪ v + v∗ 1 − e 1+e ⎪ ⎩v˜∗ = + (v − v∗ ) − |v − v∗ |σ. 2 4e 4e
(266)
Moreover e is an inelasticity parameter: when e = 1 the preceding equations are the usual equations of elastic collisions, while the case e = 0 correspond to sticky particles. In general e may depend on |v − v∗ | but we shall take it constant to simplify. Finally, to complete the explanation of (265), J is the Jacobian associated to the transformation (266), J=
1 |v − v∗ | , e2 |v˜ − v˜∗ |
(267)
and G is the famous but rather mysterious correlation function which appears in the Enskog equation. Roughly speaking, G relates the 2-particle probability density with the 1-particle probability density, as follows: f2 (t, x, v, y, w) = G x, y; ρ(t, x), ρ(t, y) f (t, x, v)f (t, y, w), where ρ(t, x) = f dv. In the Boltzmann case this term did not appear because of the chaos assumption . . . . Exactly which function G should be used is not clear and a little bit controversial: see Cercignani [146] for a discussion. If the reader finds the complexity of (265) rather frightening, we should add that we did not take into account variables of internal rotation, which may possibly be important in some situations since the particles are not perfectly spherical; see [146] for the corresponding modifications. There are no clear justifications for Equation (265), even at a formal level. In the limit of rarefied gases with a chaos assumption, or in the spatially homogeneous case, one formally recovers an inelastic Boltzmann equation, which is more simple: the collision operator just reads, with obvious notations, Qe (f, f ) =
R3
dv∗
S2
dσ |v − v∗ |(f˜f˜∗ − ff∗ ).
(268)
This collision operator also has a nice weak formulation,
Qe (f, f )ϕ =
R3 ×R3 ×S 2
ff∗ [ϕ − ϕ] dv dv∗ dσ,
(269)
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where the post-collisional velocities v , v∗ (with respect to v, v∗ taken as pre-collisional velocities) are ⎧ v + v∗ 1 − e 1+e ⎪ ⎪ + (v − v∗ ) + |v − v∗ |σ, ⎨v = 2 4 4 ⎪ v + v∗ 1 − e 1+e ⎪ ⎩v∗ = − (v − v∗ ) − |v − v∗ |σ. 2 4 4
(270)
As in the elastic case, formula (269) can be symmetrized once more by exchange of v and ˜ v˜∗ do not coincide with v , v∗ because collisions are not reversible. Also v∗ . Note that v, note that Qe (f, f ) dv = 0, Qe (f, f ) v dv = 0, but Qe (f, f )|v|2 dv 0. Now let us enumerate some more models. Starting from (268), many variants can be obtained by • reducing the dimension of phase space by considering only 2-dimensional, or even 1-dimensional models; • adding a little bit of diffusion, which is presumably realistic in most situations (heat bath, shaking . . .); • adding some drift term θ ∇v · (f v), to model some linear friction acting on the system; • change the “hard-sphere-like” collision kernel |v − v∗ | for |v − v∗ |γ with, say −1 γ 1. This provides an equation with some inelastic features, and which may be easier to study . . . . The dimensional homogeneity of the equation can be preserved by multiplying the collision operator by a suitable power of the temperature. For instance, Bobylev, Carrillo and Gamba [86] have performed a very detailed study of the case γ = 0 (“pseudo-Maxwellian collision kernel”) along the general lines of the theory developed by Bobylev for elastic Maxwellian collisions; • only retain grazing collisions by an asymptotic procedure similar to the Landau approximation7 [424,426]. Actually, some physicists mention that grazing collisions do occur very frequently in granular material [236] . . . . The operator which pops out of this limit procedure is of the form QL (f, f ) + ∇v · f ∇v (f ∗ U ) , where U (z) is proportional to |z|3 (or more generally to |z|γ +2 ), and QL is an elastic Landau operator with Ψ (|z|) proportional to |z|3 (or |z|γ +2 ). In particular, 7 See Section 2.7 in Chapter 2A.
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in dimension 1 of phase space, this elastic Landau operator vanishes and the resulting collision operator is just a nonlinear friction operator ∇v · f ∇v (f ∗ W ) . The resulting evolution equation is ∂f + v · ∇x f = ∇v · f ∇v (f ∗ W ) , ∂t
(271)
with U (z) = |z|3 . The very same equation can also be obtained as the mean-field limit of a one-dimensional system of particles colliding inelastically, as first suggested (so it seems) by McNamara and Young [343], and rigorously proven by Benedetto, Caglioti and Pulvirenti [70]. Note that (271) is a Boltzmann-like, not an Enskog-like equation! Of course, we have only presented some of the most mathematically oriented models. In the physical literature one encounters dozens of other equations, derived from physical or phenomenological principles, which we do not try to review. Hydrodynamics. Another topic where the separation of scales is problematic is the hydrodynamic limit. This subject is important for practical applications, but nobody really knows, even at a formal level, what hydrodynamic equations should be used. Due to the possibility of long-range correlations, there generally seems to be no clear separation of scales between the kinetic and hydrodynamic regimes [236]. The Chapman– Enskog expansion works terribly bad: each term of the series should be of order 1! Resummation methods have been tried to get some meaningful fluid equations [236]. From the mathematical viewpoint this procedure is rather esoteric, which suggests to look for alternative methods . . . . To add to the confusion, due to the tendency of granular media to cluster, it is not clear what should be considered as local equilibrium! See the discussion about Homogeneous Cooling States in Section 2.4 below. Also, to be honest we should add that even in the most favorable situation, i.e., a simplified model like (271), with some additional diffusion term to prevent clustering as much as possible, and under the assumption of separation of scales, then limit hydrodynamic equations can be written formally [69], but rigorous justification is also an open problem, mainly due to the absence of Lyapunov functional . . . .
2.2. Spatial inhomogeneities In the classical, elastic theory of the Boltzmann equation, spatial homogeneity is a mathematical ad hoc assumption. However, it is not so unrealistic from the physical viewpoint in the sense that it is supposed to be a stable property: weakly inhomogeneous initial data should lead to weakly inhomogeneous solutions (recall [32]). On the other hand, in the case of granular media, some physicists think that severe inhomogeneities may develop from weakly inhomogeneous states, particularly because
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of the possibility of collapse by loss of energy [198]. Also, numerical experiments seem to indicate the unstability of a homogeneous description (see the references and comments in [70, Section 4]). Besides shedding more doubts on mathematical studies based on the assumption of spatial homogeneity, these remarks raise a very interesting mathematical challenge, namely prove that weak inhomogeneity may break down in finite time for some realistic initial configurations. It is not very clear whether this study should be performed with a Boltzmann-like, or an Enskog-like equation . . . . Some related considerations about blow-up: first of all, there are initial configurations of n inelastic particles which lead to collapse in finite time, and they are not exceptional (some nice examples are due to Benedetto and Caglioti [67]). Next, for the inelastic Boltzmann equation, there is no entropy functional to prevent dramatic collapse, as in the elastic case. Even the DiPerna–Lions theory cannot be adapted to inelastic Boltzmann equations, so the Cauchy problem in the large is completely open in this case! We further note that Benedetto, Caglioti and Pulvirenti prove that there is no blow-up for Equation (271) when W (z) = λ|z|3 with λ very small; in this case Equation (271) can be treated as a perturbation of vacuum. Of course it is precisely when λ is of order 1 that one could expect blow-up effects. Also, Benedetto and Pulvirenti [73] study the onedimensional Boltzmann equation for a gas of inelastic particles with a velocity-dependent inelasticity parameter e = e(|v − v∗ |) behaving like (1 + a|v − v∗ |γ )−1 for some a, γ > 0, and they show, by an adaptation of the one-dimensional techniques of Bony [95], that blow-up does not occur in that situation.
2.3. Trend to equilibrium For the moment, and in spite of our remarks in the previous paragraph, we restrict to the spatially homogeneous setting (or say that we are only interested in trend to local equilibrium, in some loose sense). We consider two cases: (1) the inelastic Boltzmann collision operator alone; then, because of energy loss, the equilibrium is a Dirac mass at some mean velocity; (2) the inelastic Boltzmann collision operator together with some diffusion. Then there is a nontrivial, smooth stationary state (see Gamba, Panferov and Villani [228]). It is not explicit but some qualitative features can be studied about smoothness, tails, etc. The same topic has also been recently studied by Carrillo and Illner in the case of the pseudoMaxwellian collision kernel. In both cases entropy methods do not seem to apply because no relevant Lyapunov functional has been identified (apart from the energy). What is worse, in the second case, even uniqueness of the stationary state is an open problem. The study of trend to equilibrium is therefore open. However, such a study was successfully performed for two simplified models: – in the pseudo-Maxwellian variant of the inelastic Boltzmann operator, with or without diffusion [86,80]. Then the behavior of all moments can be computed, and trend to equilibrium can be studied from the relaxation of moments;
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– in the simplified model considered by Benedetto, Caglioti and Pulvirenti [70], or its diffusive variant. In this case, the appearance of a new Lyapunov functional, with an interaction energy 1 2
f (v)f (w) R×R
|v − w|3 dv dw 3
enables a study of rate of convergence by entropy dissipation methods.8 In the spatially inhomogeneous case, the situation is still worse. In the non-diffusive case, all states of the form ρ∞ (x)δ0 (v), for instance, are global equilibria. But are there some preferred profiles ρ∞ ? Even for the simplified Equation (271) or its diffusive variants, the Lyapunov functional which worked fine in the spatially homogeneous context, now fails to have any particular behavior. At a more technical level, the method of Desvillettes and Villani [184] cannot be applied because of the non-smoothness of the equilibrium distribution. On the whole, trend to equilibrium for granular media is a really challenging problem.
2.4. Homogeneous Cooling States For most physicists, a Dirac mass is not a relevant steady state, and the role played in the classical theory by Maxwellian distributions should rather be held by particular solutions which “attract” all other solutions.9 They often agree to look for these particular solutions in the self-similar form v − v0 1 fS (t, v) = N F (272) , t 0, v ∈ RN . α (t) α(t) In the sequel, we set v0 = 0, i.e., restrict the discussion to centered probability distributions. A solution of the spatially homogeneous inelastic Boltzmann equation which takes the form (272) is called a Homogeneous Cooling State (HCS). Though the existence of HCS is often taken for granted, it is in general a considerable act of faith. Sometimes HCS are considered under some scaling where also the elasticity coefficient e goes to 1 as t → +∞ . . . . For physical studies of these questions one can consult [104,238]. For the pseudo-Maxwellian variant of the inelastic Boltzmann operator, Bobylev, Carrillo and Gamba [86] have shown that HCS do not exist: one can construct a self-similar distribution function which captures the behavior of all moments of all solutions, but this distribution function is not nonnegative! HCS exist only in the following weakened sense: for any integer n0 , there exists a self-similar solution of the inelastic Boltzmann equation which gives the right behavior for all moments of order n0 of all solutions. And yet, this 8 See the discussion in Section 6.3 of Chapter 2C. 9 By the way, in the classical setting it was once conjectured that the Bobylev–Krook–Wu explicit solutions
would attract all solutions of the spatially homogeneous Boltzmann equation with Maxwellian molecules. But this has been shown to be false . . . except, in some sense, in the unphysical regime of negative times [81]!
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weaker statement is still false for some particular values of the inelasticity parameter e. See [86] for more details, in particular a discussion of the stability of the HCS description. For model (271), in a spatially homogeneous setting, i.e., ∂f = ∇v · f ∇v (f ∗ U ) , ∂t
(273)
with U (z) = |z|3 /3, then HCS do exist and are just made of a combination of two Dirac masses: up to some change of scales, fS (t, v) =
1 δ− 1 + δ 1 . 2t 2t 2
This solution is obtained via the search for steady states to the rescaled equation ∂f = ∇v · f ∇v (f ∗ W ) − ∇v · (f v). ∂t
(274)
One finds that a distinguished steady state is FS = (δ− 1 + δ 1 )/2. 2 2 It is also true [70] that this HCS is a better approximation to solutions of (273), than just the Dirac mass (at least if the initial datum has no singular part). This means that solutions of (274) do converge towards the steady state FS . But this approximation is in general quite bad! It was shown by Caglioti and Villani [116] that the improvement in the rate of convergence is essentially no better than logarithmic in time. For instance, if W stands for the Wasserstein distance (244), then solutions of (273) satisfy
+∞
W f (t), fS (t) dt = +∞.
1
Since also W (f (t), δ0 ) = O(1/t), this shows that the improvement in the rate of convergence cannot be O(log1+ε t), for any ε > 0 – which means negligible by usual standards.10 The preceding considerations cast further doubts about the mathematical relevance of HCS, which however are at the basis of several hydrodynamical equations for granular media. Further clarification is still badly needed; an attempt is done in [86] for the simplified pseudo-Maxwellian model. This concludes our brief discussion of the kinetic theory of granular media. In the sequel, we shall enter a completely different physical world.
3. Quantum kinetic theory Very recently, Lu [328], Escobedo, and Mischler [210,211] have begun to apply the techniques of the modern theory of the spatially homogeneous Boltzmann equation, to 10 Or maybe should one use an even weaker notion of distance to measure the rate of convergence??
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quantum kinetic models, thus opening up the path to a promising new direction of research. This is part of a general trend which has become increasingly active over last years: the mathematical derivation and study of quantum statistical models. This also coincides with a time when the interest of physicists in Bose condensation is enhanced by the possibility of experiments with very cold atoms. Most of the explanations which follow come from discussions with Escobedo and Mischler, and also from Lu’s paper [328]. First of all, we should clarify the meaning of quantum kinetic theory: it does not rest on the traditional quantum formalism (wave function, Wigner transforms, etc.). Instead, it is rather a classical description of interacting particles with quantum features. This approach was initiated by the physicists Nordheim, and Uehling and Uhlenbeck, in the thirties. Thus, the basic equation still looks just like a Boltzmann equation: ∂f + v(p) · ∇x f = Q(f, f ), ∂t
t 0, x ∈ R3 , p ∈ R3 .
(275)
Here p stands for the impulsion of the particle and v is the corresponding velocity: v(p) = ∇p E(p), where E(p) is the energy corresponding to the impulsion p. The unknown is a time-dependent probability density on the phase space of positions and impulsions. When dealing with massive particles, we shall consider a non-relativistic setting (to simplify) and identify v with p. On the other hand, when dealing with photons, which have no mass, we assume that the energy is proportional to |p|, and the velocity to p/|p|. Now, all the quantum features are encoded at the level of the collision operator Q in (275). One traditionally considers three types of particles: fermions, which satisfy Pauli’s exclusion principle. In this case the transition from state p to state p is easier if f (p) is low. Accordingly, the “Boltzmann–Fermi” collision operator reads QF (f, f ) =
R3
dp∗
S2
dσ B(v − v∗ , σ ) f f∗ (1 + εf )(1 + εf∗ ) − ff∗ (1 + εf )(1 + εf∗ ) ,
(276)
where ε is a negative constant.11 Up to change of units, we shall assume that ε = −1. Moreover, as in the classical case, f = f (p ) and so on, and p , p∗ are given by the formulas ⎧ p + p∗ |p − p∗ | ⎪ ⎪ + σ, ⎨p = 2 2 ⎪ p + p∗ |p − p∗ | ⎪ ⎩p∗ = − σ. 2 2
(277)
11 In physical units, ε should be −(h/m)3 /g, where h is Planck’s constant, m the mass of a particle and g the socalled “statistical weight” of this species of particles. For the derivation of (276) see Chapman and Cowling [154, Chapter 17].
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Equation (275) with Q = QF will be called the Boltzmann–Fermi equation. It is supplemented with the a priori bound 0 f 1 [= −1/ε]; bosons, which, on the contrary to fermions, do like to cluster. The collision operator, QB , is just the same as (276), but now ε = +1. The corresponding equation will be called the Boltzmann–Bose equation; photons, which are mass-free particles exchanging energy. Usually they are considered only in interaction with bosons or fermions. For instance, here is the Boltzmann–Compton model:
∞
QC (f, f ) =
2 b(k, k ) f k 2 + f e−k − f k + g e−k dk ,
0
t 0, k 0.
(278)
Here the phase space is just R+ , the space of energies, because the distribution of photons is assumed to be spatially homogeneous and isotropic. Thus the corresponding evolution equation is just ∂f = QC (f, f ), ∂t
t 0, k 0.
(279)
We quote from Escobedo and Mischler [211]: Equation (279) describes the behavior of a low-energy, spatially homogeneous, isotropic photon gas interacting with a lowtemperature electron gas with Maxwellian distribution of velocities, via Compton scattering. This model will be called Boltzmann–Compton. We now survey some of the main problems in the field.
3.1. Derivation issues The derivation of equations like Boltzmann–Fermi or Boltzmann–Bose is not a tidy business (see Chapman and Cowling [154] . . .). Therefore, the expected range of applicability and the precise form of the equations are not so clear. A better understood situation is the linear setting: description of the effect of a lattice of quantum scatterers on a density of quantum particles. Not only is the exact equation well understood, but also a theoretical basis, in the spirit of Lanford’s theorem, can be given. Starting from the many-body Schrödinger equation as microscopic equation, Erd˝os and Yau [205,206] have been able to retrieve the expected linear Boltzmann equation as a macroscopic description. Related works are performed in [132–134] . . . . In the sequel we do not consider these issues and restrict the discussion to the nonlinear equations written above. Here are a few problematic issues about them.
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Cross-sections. It seems, nobody really knows what precise form of the cross-section, or equivalently of the collision kernel B(v − v∗ , σ ) in (275) (or b(k, k ) in (278)) should be used – except in some particular cases with photon interaction . . . . Some formulas can be found in [154] but they are not very explicit. This makes it difficult to give an interpretation of some of the mathematical results, as we shall see. It would be desirable to identify some model collision kernels playing the same role as the ones associated with inverse-power interactions in the classical theory. According to certain physicists, it would be not so bad to understand the case of a simple hard-sphere collision kernel. Grazing collisions. Some variants of these equations are obtained by a grazing collision asymptotics. To this class belong the quantum Landau equation (see Lemou [298] and references therein), or the well-known Kompaneets equation12 [290],
2 ∂ ∂f ∂F 2 ∂f 2 = k + k − 2k f + f ≡ , ∂t ∂k ∂k ∂k
t 0, k 0.
(280)
A flux condition must be added at the boundary: lim F (k) = 0.
k→0
(281)
Equation (280) describes the same kind of phenomena as the Boltzmann–Compton equation, and can in fact be obtained from it by an asymptotic procedure similar to the one leading from the Boltzmann to the Landau equation (see Escobedo and Mischler [211]) under some assumptions on the initial datum. However, the validity of this approximation cannot be universally true, because the Kompaneets equation has some strange “blow-up” properties: Escobedo, Herrero and Velazquez [208] have shown that the flux condition (281) may break down in finite time for arbitrarily small initial data. Also the long-time behavior of the Kompaneets equation can be non-conventional; this is consistent with the remark by Caflisch and Levermore [114] that for large enough mass there are no stationary states . . . . Besides physical interest, all these considerations illustrate the fact that the asymptotics of grazing collisions may destroy (or create?) some important features of the models. Consistency with classical mechanics. All these quantum models involve the Planck constant as a parameter; of course when one lets the Planck constant go to 0 (which would in fact be the formal consequence of a change in physical scales, from microscopic to macroscopic), one expects to recover the Boltzmann-like equations of classical mechanics. This can be justified in some cases, see, for instance, [196]. Hydrodynamics. Physicists expect that some hydrodynamic limit of the Boltzmann–Bose equation leads to the Gross–Pitaevski, or Ginzburg–Landau, model (based on a cubic nonlinear Schrödinger equation) for the evolution of the Bose condensate13 part. Again, 12 This equation is often written with k 2 f as unknown. 13 See the next section.
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this would need clarification . . . . We note however that the justification of this limit would look more interesting if the derivation of the Boltzmann–Bose equation was first put on a more rigorous basis.
3.2. Trend to equilibrium Equations such as Boltzmann–Fermi, Boltzmann–Bose or Boltzmann–Compton all satisfy entropy principles, and equilibrium states are entropy minimizers14: (1) For the Boltzmann–Fermi equation, the entropy is HBF (f ) =
f log f − (1 − f ) log(1 − f ) .
Equilibrium states are of the form F (p) =
1 2 eα|p−p0 | +β
+1
(α > 0, β ∈ R)
(282)
or F (p) = 1|p−p0 |R
(R > 0).
A state like (282) is called a Fermi–Dirac distribution. Here p0 is the mean impulsion. (2) For the Boltzmann–Bose equation, the entropy is
HBB (f ) =
f log f − (1 + f ) log(1 + f )
(here ε = +1) and the shape of equilibrium states depends on the temperature. There is a critical condensation temperature Tc such that the equilibrium state B takes the form B(p) = B(p) =
1 2 eα|p−p0 | +β
−1
1 2 eα|p−p0 |
−1
(α > 0, β 0) when T Tc ,
+ µδp0
(α > 0, µ > 0) when T < Tc .
(283)
(284)
These distributions are called Bose–Einstein distributions. The singular part of (284) is called a Bose condensate. (3) Finally, for the Boltzmann–Compton equation, the entropy is given by
∞
k 2 + f log k 2 + f − f log f − kf − k 2 log k 2 dk,
HBC (f ) = 0
14 Under the constraint 0 f 1 for the Boltzmann–Fermi model.
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and, according to Caflisch and Levermore [114], the minimizers are of the form B(k) =
1 ek+λ
−1
+ αδ0 ,
where λ and α are nonnegative numbers, at least one of them being 0. For λ > 0 this is a Bose distribution, for λ = 0 it is called a Planck distribution. As in the Boltzmann case, these distributions, obtained by a minimization principle, also coincide with the probability distributions which make the collision operator vanish. There are some technicalities associated with the fact that singular measures should be allowed: they have recently been clarified independently by Escobedo and Mischler, and by Lu. Now, let us consider the problem of convergence to equilibrium, in a spatially homogeneous setting for simplicity. As far as soft methods (compactness and so on) are concerned, Pauli’s exclusion principle facilitates things a great deal because of the additional L∞ bound. Therefore, convergence to equilibrium in a (very) weak sense is not very difficult [211]. However, no constructive result in this direction has ever been obtained, neither has any entropy–entropy dissipation inequality been established. In the Bose case, this is an even more challenging problem since also soft methods fail, due to the lack of a priori bounds. The entropy is now sublinear and fails to prevent concentration, which is consistent with the fact that condensation may occur in the longtime limit. Actually, as soon as T < Tc , a given solution cannot stay within a weakly compact set of L1 as t → +∞ . . . . Lu [328] has attacked this problem with the welldeveloped tools of modern spatially homogeneous theory, and proven that – when the temperature is very large (T ' Tc ), solutions of the spatially homogeneous Boltzmann–Bose equation are weakly compact in L1 as t → +∞, and converge weakly towards a Bose distribution of the form (283); – when the temperature is very low (T < Tc ), solutions are not weakly compact in L1 , but converge to equilibrium in the following extremely weak sense [328]: if (tn ) is a sequence of times going to infinity, then from f (tn , ·) one can extract a subsequence converging in biting-weak L1 sense towards a Bose distribution of the form (284). In this theorem, not only is biting-weak L1 convergence a very weak notion (weaker than distributional convergence), but also the limit may depend on the sequence (tn ). Furthermore, it is not known whether weak L1 compactness as t → ∞ holds true when T is greater than Tc , but not so large. Lu’s theorem is proven for isotropic homogeneous solutions. Isotropy should not be a serious restriction, but seems compulsory to the present proof. What is more, Lu’s work relies on a strong cut-off assumption for the kernel B: essentially, B |v − v∗ |, cos θ Cθ |v − v∗ |3 ∧ |v − v∗ | ,
C > 0,
(285)
where θ is as usual the deviation angle. This assumption enables a very good control of the Q+ part, but may do some harm for other, yet to be found, a priori estimates.
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3.3. Condensation in finite time Physical experiments with very cold atoms have recently become possible, and have aroused a lot of interest. For instance, a few years ago it became possible to experimentally create and study Bose condensates. Among other phenomena, physicists report the formation of a condensate in finite time. However, Lu has proven that there is no finite time clustering for the Boltzmann–Bose equation studied in [328]. This seems to leave room for two possibilities, both of which may lead to exciting new research directions: • either the Boltzmann–Bose equation should be discarded for a more precise model when trying to model Bose condensation; • or the Bose condensation is excluded by the strong cut-off assumption (285), which penalizes interactions with v v∗ (supposedly very important in condensation effects). On this occasion we strongly feel the need to have a better idea of what collision kernels would be physically realistic. Proving the possibility (or genericity) of finite-time condensation for “bigger” collision kernels (say B ≡ 1?) would be a mathematical and physical breakthrough for the theory of the Boltzmann–Bose model.
3.4. Spatial inhomogeneities So far we have only considered spatially homogeneous quantum Boltzmann equations, now what happens for spatially inhomogeneous data? Due to the additional L∞ bound, the Boltzmann–Fermi model seems easier to study than the classical Boltzmann equation; in particular existence results can be obtained without too much difficulty [196,309]. The situation is completely different for the Boltzmann–Bose model, since one would like to consider singular measures as possible data. A completely new mathematical theory would have to be built! A particularly exciting problem would be the understanding of the spacetime evolution for the Bose condensate. It was communicated to us by Lu that for small initial data in the whole of R3 , one can prove that Bose–Einstein condensation never occurs . . . . This should be taken as a clue that the underlying mathematical phenomena are very subtle.
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Bibliographical notes General references. Standard references about the kinetic theory of rarefied gases and the Boltzmann equation are the books by Boltzmann [93], Carleman [119], Chapman and Cowling [154], Uhlenbeck and Ford [433], Truesdell and Muncaster [430], Cercignani [141,148], Cercignani, Illner and Pulvirenti [149], as well as the survey paper by Grad [250]. The book by Cercignani et al., with a very much mathematically oriented spirit, may be the best mathematical reference for nonspecialists. The book by Uhlenbeck and Ford is a bit outdated, but a pleasure to read. There is no up-to-date treatise which would cover the huge progress accomplished in the theory of the Boltzmann equation over the last ten years. For people interested in more applied topics, and practical aspects of modelling by the Boltzmann equation, Cercignani [148] is highly recommended. We may also suggest the very recent book by Sone [407], which is closer to numerical simulations. The book by Glassey [233] is a good reference for the general subject of the Cauchy problem in kinetic theory (in particular for the Vlasov–Poisson and Vlasov– Maxwell equations, and for the Boltzmann equation near equilibrium). Also the notes by Bouchut [96] provide a compact introduction to the basic tools of modern kinetic theory, like characteristics and velocity-averaging lemmas, with applications. To the best of our knowledge, there is no mathematically-oriented exposition of the kinetic theory of plasma physics. Among physicists’ textbooks, Balescu [46] certainly has the most rigorous presentation. The very clear survey by Decoster [160] gives an accurate view of theoretical problems arising nowadays in applied plasma physics. There are many, many general references about equilibrium and non-equilibrium statistical physics; for instance, [49,227]. People who would like to know more about information theory are advised to read the marvelous book by Cover and Thomas [156]. A well-written and rather complete survey about logarithmic Sobolev inequalities and their links with information theory is [16] (in french). Historical references. The founding papers of modern kinetic theory were those of James Clerk Maxwell [335,336] and Ludwig Boltzmann [92]. It is very impressive to read Maxwell’s paper [335] and see how he made up all computations from scratch! The book [93] by Boltzmann has been a milestone in kinetic theory. References about the controversy between Boltzmann and his peers can be found in [149, p. 61], or Lebowitz [293]. Some very nice historical anecdotes can also be found in Balian [49]. Certainly the two mathematicians who have most contributed to transform the study of the Boltzmann equation into a mathematical field are Torsten Carleman in the thirties, and Harold Grad after the Second World War. Derivation of the Boltzmann equation. For this subject the best reference is certainly Cercignani, Illner and Pulvirenti [149, Chapters 2 and 4]. A pedagogical discussion of slightly simplified problems is performed in Pulvirenti [394]. Another excellent source is the classical treatise by Spohn [410] about large particle systems. These authors explain in detail why reversible microdynamics and irreversible macrodynamics are not
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contradictory – a topic which was first developed in the famous work by Ehrenfest and Ehrenfest [202], and later in the delightful book by Kac [284]. Further information on the derivation of macroscopic dynamics from microscopic equations can be found in Kipnis and Landim [287]. Hydrodynamic limits. A very nice review of rigorous results about the transition from kinetic to hydrodynamic models is Golse [239]. No prerequisite in either kinetic theory, or fluid mechanics is assumed from the reader. Note the discussion about ghost effects, which is also performed in Sone’s book [407]. The important advances which were accomplished very recently by several teams, in particular, Golse and co-workers, were reviewed by the author in [441]. There is a huge probabilistic literature devoted to the subject of hydrodynamic limits for particle systems, starting from a vast program suggested by Morrey [352]. Entropy methods were introduced into this field at the end of the eighties, see in particular the founding works by Guo, Papanicolaou and Varadhan (the GPV method, [263]), and Yau [466]. For a review on the results and methods, see the notes by Varadhan [439], the recent survey by Yau [467] or again, the book by Kipnis and Landim [287]. Mathematical landmarks. Here are some of the most influential works in the mathematical theory of the Boltzmann equation. The very first mathematical steps are due to Carleman [118,119] in the thirties. Not only was Carleman the very first one to state and solve mathematical problems about the Boltzmann equation (Cauchy problem, H theorem, trend to equilibrium), but he was also very daring in his use of tools from pure mathematics of the time. In the seventies, the remarkable work by Lanford [292] showed that the Boltzmann equation could be rigorously derived from the laws of reversible mechanics, along the lines first suggested by Grad [249]. This ended up a very old controversy and opened new areas in the study of large particle systems. Yet much remains to be understood in the Boltzmann–Grad limit. At the end of the eighties, the classical paper by DiPerna and Lions [192] set up new standards of mathematical level and dared to attack the problem of solutions in the large for the full Boltzmann equation, which to this date has still received no satisfactory answer. A synthetical review of this work can be found in Gérard [231]. Finally, we also mention the papers by McKean [341] in the mid-sixties, and Carlen and Carvalho [121] in the early nineties, for their introduction of information theory in the field, and the enormous influence that they had on research about the trend to equilibrium for the Boltzmann equation.
Acknowledgements It is a pleasure to thank D. Serre for his suggestion to write up this review, and E. Ghys for his thorough reading of the first version of the manuscript. I also warmly thank L. Arkeryd, E. Caglioti, C. Cercignani, L. Desvillettes, X. Lu, F. Malrieu, N. Masmoudi, M. Pulvirenti, G. Toscani for providing remarks and corrections, and J.-F. Coulombel for his job of
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tracking misprints. The section about quantum kinetic theory would not have existed without the constructive discussions which I had with M. Escobedo and S. Mischler. Research by the author on the subjects which are described here was supported by the European TMR “Asymptotic methods in kinetic theory”, ERB FMBX-CT97-0157. Finally, the bibliography was edited with a lot of help from the MathSciNet database.
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CHAPTER 3
Viscous and/or Heat Conducting Compressible Fluids Eduard Feireisl∗ ˇ Žitná 25, 115 67 Praha 1, Czech Republic Institute of Mathematics AV CR,
Contents 1. Basic equations of mathematical fluid dynamics . . . . . . . . . . . . . . 1.1. Balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Barotropic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Bibliographical comments . . . . . . . . . . . . . . . . . . . . . . 2. Mathematical aspects of the problem . . . . . . . . . . . . . . . . . . . . 2.1. Global existence for small and smooth data . . . . . . . . . . . . . 2.2. Global existence of discontinuous solutions . . . . . . . . . . . . . 2.3. Global existence in critical spaces . . . . . . . . . . . . . . . . . . 2.4. Regularity vs. blow-up . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Large data existence results . . . . . . . . . . . . . . . . . . . . . . 2.6. Bibliographical comments . . . . . . . . . . . . . . . . . . . . . . 3. The continuity equation and renormalized solutions . . . . . . . . . . . . 3.1. On continuity of the renormalized solutions . . . . . . . . . . . . . 3.2. Renormalized and weak solutions . . . . . . . . . . . . . . . . . . 3.3. Renormalized solutions on domains with boundary . . . . . . . . 4. Weak convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Weak compactness of bounded solutions to the continuity equation 4.2. On compactness of solutions to the equations of motion . . . . . . 4.3. On the effective viscous flux and its properties . . . . . . . . . . . 4.4. Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . 5. Mathematical theory of barotropic flows . . . . . . . . . . . . . . . . . . 5.1. Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Pressure estimates for isentropic flows . . . . . . . . . . . . . . . . 5.3. Density oscillations for barotropic flows . . . . . . . . . . . . . . . 5.4. Propagation of oscillations . . . . . . . . . . . . . . . . . . . . . . 5.5. Approximate solutions . . . . . . . . . . . . . . . . . . . . . . . . 6. Barotropic flows: large data existence results . . . . . . . . . . . . . . . ∗ Work supported by Grant 201/98/1450 of GA CR. ˇ
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6.1. Global existence of classical solutions . . . . . . . 6.2. Global existence of weak solutions . . . . . . . . 6.3. Time-periodic solutions . . . . . . . . . . . . . . . 6.4. Counter-examples to global existence . . . . . . . 6.5. Possible generalization . . . . . . . . . . . . . . . 7. Barotropic flows: asymptotic properties . . . . . . . . . 7.1. Bounded absorbing balls and stationary solutions 7.2. Complete bounded trajectories . . . . . . . . . . . 7.3. Potential flows . . . . . . . . . . . . . . . . . . . . 7.4. Highly oscillating external forces . . . . . . . . . 7.5. Attractors . . . . . . . . . . . . . . . . . . . . . . 7.6. Bibliographical remarks . . . . . . . . . . . . . . 8. Compressible–incompressible limits . . . . . . . . . . . 8.1. The spatially periodic case . . . . . . . . . . . . . 8.2. Dirichlet boundary conditions . . . . . . . . . . . 8.3. The case γn → ∞ . . . . . . . . . . . . . . . . . . 9. Other topics, directions, alternative models . . . . . . . 9.1. Models in one space dimension . . . . . . . . . . 9.2. Multi-dimensional diffusion waves . . . . . . . . 9.3. Energy decay of solutions on unbounded domains 9.4. Alternative models . . . . . . . . . . . . . . . . . 10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Local existence and uniqueness, small data results 10.2. Density estimates . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Basic equations of mathematical fluid dynamics 1.1. Balance laws Let Ω ⊂ R N be a domain in two- or three-dimensional space (N = 2, 3) filled with a fluid. We shall assume the fluid is a continuous medium the state of which at a time t ∈ I ⊂ R and a spatial point x ∈ Ω is characterized by the three fundamental macroscopic quantities; the density # = #(t, x), the velocity u = u(t, x), and the temperature θ = θ (t, x). The fluid motion is governed by a system of partial differential equations expressing the basic principles of classical continuum mechanics. Conservation of mass: ∂# + div(#u) = 0 ∂t
(1.1)
Balance of momentum (Newton’s second law of motion): ∂(#u) + div(#u ⊗ u) + ∇p = div Σ + #f ∂t
(1.2)
Conservation of energy (the first law of thermodynamics): ∂E + div (E + p)u = div(Σu) − div q + #f · u ∂t
(1.3)
Here p is the pressure, Σ denotes the viscous stress tensor, E stands for the specific energy, q is the heat flux, and f denotes a given external force density. We have chosen the spatial description where attention is focused on the present configuration of the fluid and the region of physical space currently occupied. This description was introduced by d’Alembert and is usually called Eulerian in hydrodynamics. There is an alternative way – the referential description – introduced in the eighteenth century by Euler that is called Lagrangean. In this description the Cartesian coordinate X of the position of the particle at the time t = t0 is used as label for the particle X (see, e.g., Truesdell and Rajagopal [105]).
1.2. Constitutive relations The general system (1.1)–(1.3) of N + 2 equations must be complemented by constitutive relations reflecting the diversity of materials in nature. An important class of fluids that occupies a central place in fluid mechanics is the linearly viscous or Newtonian fluid, whose viscous stress tensor Σ takes the form Σ = Σ(∇u) = µ ∇u + (∇u)t + λ div u Id,
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where µ and λ are the viscosity coefficients assumed to be constant unless otherwise specified. The full stress characterized by the Cauchy stress tensor T is related to Σ by the Stokes law T = Σ − p Id, where the pressure p = p(#, θ ) is a general function of the independent state variables # and θ . The specific energy E can be written in the form E = Ekinetic + Einternal ,
1 Ekinetic = #|u|2 , 2
Einternal = #e,
where e is the specific internal energy related to the density and the temperature by a general constitutive law e = e(#, θ ). In accordance with the basic principles of thermodynamics, we postulate the existence of a new state variable – the specific entropy S = S(#, θ ) – satisfying ∂S 1 ∂e p = − . ∂# θ ∂# #2
1 ∂e ∂S = , ∂θ θ ∂θ
(1.4)
Consequently, one can replace (1.3) by the entropy equation Σ(∇u) : ∇u q · ∇θ q = − , ∂t (#S) + div #Su + θ θ θ2
(1.5)
where, by virtue of the second law of thermodynamics, the right-hand side should be nonnegative which yields the restriction µ 0,
λ+
2 µ 0 together with q · ∇θ 0. N
We focus on viscous fluids assuming always µ > 0,
λ+
2 µ 0. N
(1.6)
Finally, the heat flux q is related to the temperature by the Fourier law q = −κ∇θ,
κ 0,
where the heat conduction coefficient κ may depend on θ , # and even on ∇θ though it is assumed constant in most of the cases we shall deal with.
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1.3. Barotropic models The flow is said to be barotropic if the pressure p depends solely on the density #. There are several situations when such a hypothesis seems appropriate. For instance, the ideal gas consitutive relation for the pressure reads p = (γ − 1)#e,
e = cv θ, cv > 0,
(1.7)
where γ > 1 is the adiabatic constant. Accordingly, the entropy S takes the form S = log(e) + (1 − γ ) log(#). Substituting µ = λ = κ = 0 in (1.5) and assuming a spatially homogeneous distribution S0 of the entropy at a time t0 ∈ I we easily deduce S(t) = S0 for any t ∈ I and, consequently, p(#) = a#γ ,
a = (γ − 1) exp(S0 ) > 0.
Under such circumstances, Equations (1.1), (1.2) represent a closed system describing the motion of an isentropic compressible viscous fluid. A similar situation occurs in the isothermal case when we suppose θ (t) = θ0 and (1.7) reduces to p(#) = rθ0 #,
r > 0.
For a general barotropic flow, the specific energy E can be taken in the form 1 E[#, u] = #|u|2 + P (#) 2
(1.8)
with P (z)z − P (z) = p(z). The energy of a barotropic flow satisfies the equality ∂t E + div (E + p)u = div(Σu) − Σ : ∇u + #f · u
(1.9)
which is now a direct consequence of (1.1), (1.2). From the mathematical point of view, the barotropic flows represent an interesting class of problems for which an existence theory with basically no restriction on the size of data is available (see Section 6 below). 1.4. Boundary conditions To obtain mathematically well-posed problems, the equations introduced above must be supplemented by initial and/or boundary conditions. The boundary ∂Ω is assumed to
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be an impermeable rigid wall, i.e., the fluid does not cross the boundary but may move tangentially to the boundary. Accordingly, we require u · n = 0 on I × ∂Ω,
(1.10)
where n stands for the outer normal vector. For both experimental and mathematical reasons, (1.10) should be accompanied by a condition for the tangential component of the velocity. As observed in experiments with viscous fluids, the tangential component approaches zero at the boundary to a high degree of precision. This can be expressed by the no-slip boundary conditions: u=0
on I × ∂Ω.
(1.11)
On the other hand, in vessels with frictionless boundary (cf. Ebin [25]), condition (1.10) is usually complemented by the requirement that the tangetial component of the normal stress is zero, which can be written in the form of the no-stick boundary conditions: u · n = 0,
(Σn) × n = 0
on I × ∂Ω.
(1.12)
Similarly, one prescribes either the heat flux or the temperature. For a thermally insulated boundary, the condition reads q · n = 0 on I × ∂Ω while θ = θb
on I × ∂Ω
when the boundary distribution of the temperature is known. If Ω is unbounded, it is customary to prescribe also the limit values of the state variables for large x ∈ Ω, e.g., # → #∞ ,
u → u∞ ,
θ → θ∞
as |x| → ∞.
In the in-flow and/or out-flow problems, the homogeneous Dirichlet boundary conditions (1.11) are replaced by a more general stipulation u = ub
on I × ∂Ω.
Moreover, the density distribution must be given on the in-flow part of the boundary, i.e., #(t, x) = #b (x) for (t, x) ∈ I × ∂Ω,
ub (x) · n(x) < 0.
Other types of boundary conditions including unilateral constraints and free boundary problems are treated in the monograph by Antontsev et al. [4, Chapters 1, 3].
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1.5. Bibliographical comments An elementary introduction to the mathematical theory of fluid mechanics can be found in the book by Chorin and Marsden [12]. More extensive material is available in the monographs by Batchelor [7], Meyer [77], Serrin [93], or Shapiro [94]. A more recent treatment including the so-called alternative models is presented by Truesdell and Rajagopal [105]. A rigorous mathematical justification of various models of viscous heat conducting fluids is given by Šilhavý [96]. Mainly mathematical aspects of the problem are discussed by Antontsev et al. [4], Málek et al. [68], and more recently by Lions [61,62]. 2. Mathematical aspects of the problem The first and most important criterion of applicability of any mathematical model is its well-posedness. According to Hadamard, this issue comprises a thorough discussion of the following topics. • Existence of solutions for given data. The data for the problem in question are usually the values of the state variables #, u, and θ specified at a given time t = t0 and/or the driving force f together with the boundary values of certain quantities as the case may be. The problem is whether or not there exist solutions for any choice of the data on a given time interval I . • Uniqueness. The model is to be deterministic, specifically, the time evolution of the system for t > t0 must be uniquely determined by its state at the time t0 . • Stability. Small perturbations of the data should result in small variation of the corresponding solution at least on a given compact time interval. On the other hand, experience with much simpler systems of ordinary differential equations suggests that chaotic behaviour may develop with growing time. Roughly speaking, the solutions may behave in a drastically different way in the long run no matter how close they might have been initially. To begin, let us say honestly that a rigorous answer to most of the issues mentioned above is very far from being complete. Global existence and uniqueness of solutions to the system (1.1)–(1.3) is still a major open problem and only partial results shed some light on the amazing complexity of the problem. In this introductory section, we review the presently available results on the existence of classical as well as weak or distributional solutions to the full system of equations of a compressible Newtonian and heat-conducting fluid. Equations (1.1)–(1.3) written in Cartesian coordinates take the form ∂# ∂(#uj ) + = 0; ∂t ∂xj ∂(#ui ) ∂(#uj ui ) ∂p + + ∂t ∂xj ∂xi j ∂ui ∂ ∂u ∂ µ + (λ + µ) + #f i , = ∂xj ∂xj ∂xi ∂xj
(2.1)
i = 1, . . . , N;
(2.2)
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j ∂(#θ ) ∂(#θ uj ) ∂u cv +p + ∂t ∂xj ∂xj j 2 ∂ ∂uk 2 ∂u ∂θ µ ∂uj = + +λ . κ + ∂xj ∂xj 2 ∂xk ∂xj ∂xj
(2.3)
Here and always in what follows, the summation convention is used. The term classical solution means that the state variables have as many derivatives as necessary to give meaning to (2.1)–(2.3) on Ω × I and are continuous up to the boundary ∂Ω to satisfy the boundary conditions as the case may be. Usually, we make no distinction between classical and strong solutions whose generalized derivatives are locally integrable functions and satisfy the equations almost everywhere in the sense of the Lebesgue measure. Typically, any strong solution is a classical one provided some additional smoothness of the data is assumed. Multiplying (2.1)–(2.3) by a compactly supported and smooth test function ϕ and integrating the resulting expressions by parts, we get the integral identities: # I
Ω
∂ϕ ∂ϕ + #uj dx dt = 0; ∂t ∂xj
#ui I
Ω
(2.4)
∂ϕ ∂ϕ ∂ϕ ∂ui ∂ϕ + #ui uj +p −µ ∂t ∂xj ∂xi ∂xj ∂xj
∂uj ∂ϕ + #f i ϕ dx dt = 0, i = 1, . . . , N; ∂xj ∂xi j ∂ϕ ∂ϕ ∂u ∂θ ∂ϕ + #θ uj cv #θ −p ϕ−κ ∂t ∂x ∂x ∂x j j j ∂xj I Ω j µ ∂u ∂uk 2 ∂uj 2 + + ϕ+λ ϕ dx dt = 0. 2 ∂xk ∂xj ∂xj − (λ + µ)
(2.5)
(2.6)
We shall say that Equations (2.1)–(2.3) hold in D (I × Ω) (in the sense of distributions) or, equivalently, that #, u, and θ is a weak solution of the problem if the integral identities (2.4)–(2.6) hold for any test function ϕ ∈ D(I × Ω). The symbol D(Q) denotes the space of infinitely differentiable functions with compact support in an open set Q. It is not difficult to observe that the local formulation (2.1)–(2.3) and the integral formulation (2.4)–(2.6) are in fact equivalent provided the solution is smooth enough. On the other hand, (2.4)–(2.6) make sense under much weaker assumptions, namely, when the quantities #, #ui , #ui uj , p, ∂ui /∂xj , #f i , #θ , #θ uj , p∂uj /∂xj , ∂θ/∂xi , |∂ui /∂xj |2 , i, j = 1, . . . , N are locally integrable on I × Ω. We shall always tacitly suppose that this is the case whenever speaking about weak solutions.
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2.1. Global existence for small and smooth data Following the pioneering work of Matsumura and Nishida [72] we consider the system (2.1)–(2.3) where µ = µ(#, θ ),
λ = λ(#, θ )
are smooth functions satisfying µ > 0, λ + 2/3µ 0;
(2.7)
the pressure p is given by a general constitutive relation p = p(#, θ )
and p,
κ = κ(#, θ ),
κ > 0.
∂p ∂p , > 0; ∂# ∂θ
(2.8) (2.9)
In addition and in accordance with (1.4), we suppose p(#, θ ) =
∂p(#, θ ) θ. ∂θ
(2.10)
The problem (2.1)–(2.3) is complemented by the Dirichlet boundary conditions ui |∂Ω = 0,
i = 1, 2, 3,
θ |∂Ω = θb ,
(2.11)
where θb > 0; and the initial conditions #(0, x) = #0 (x) > 0, θ (0, x) = θ0 (x),
ui (0, x) = ui0 (x),
x ∈ Ω.
i = 1, 2, 3, (2.12)
The following global existence theorem holds. T HEOREM 2.1. Let Ω ⊂ R 3 be a domain with compact and smooth boundary. Let the quantities µ, λ, p, and κ comply with the hypotheses (2.7)–(2.10). Moreover, let the initial data #0 , ui0 , θ0 belong to the Sobolev space W 3,2 (Ω) and satisfy the compatibility conditions ui0 = 0,
θ0 = θb ,
∂ p(#0 , θ0 ) ∂xi ∂uj0 ∂ui ∂ ∂ µ(#0 , θ0 ) 0 + λ(#0 , θ0 ) + µ(#0 , θ0 ) − #0 f i , = ∂xj ∂xj ∂xi ∂xj
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E. Feireisl j
p(#0 , θ0 ) =
∂u0 ∂xj
j j 2 ∂u0 µ(#0 , θ0 ) ∂u0 ∂uk0 2 ∂ ∂θ0 κ(#0 , θ0 ) + + + λ(#0 , θ0 ) , ∂xj ∂xj 2 ∂xk ∂xj ∂xj
i = 1, 2, 3, on the boundary ∂Ω. Finally, let f i = ∂F /∂xi , i = 1, 2, 3 where F belongs to the Sobolev space W 5,2 (Ω). Then there exists ε > 0 such that the initial-boundary value problem (2.1)–(2.3), (2.11), (2.12) posseses a unique solution #, u, θ on the time interval t ∈ (0, ∞) provided the initial data satisfy ¯ W 3,2 (Ω) + u0 W 3,2 (Ω) + θ0 − θb W 3,2 (Ω) + F W 5,2 (Ω) < ε, #0 − # where #¯ =
1 |Ω|
#0 dx. Ω
Theorem 2.1 in its present form is taken over from Matsumura and Nishida [73] (cf. also [72]). The proof, which is rather lengthy and technical, is based on a priori estimates resulting from energy relations. Their method has been subsequently adapted by many authors to attack various problems with non-homogeneous boundary conditions (cf., e.g., Valli and Zajaczkowski [109]) as well as the barotropic models (see Valli [108]). The common feature of all these results is that they apply only to problems where the data are small and regular. The solutions the existence of which is claimed in Theorem 2.1 belong to the space #, u, θ ∈ C [0, T ]; W 3,2 (Ω) , where W k,p (Ω) denotes the Sobolev space of functions whose derivatives up to order k lie in the Lebesgue space Lp (Ω) (for basic properties of Sobolev spaces see, e.g., the monograph of Adams [1]). The scale W k,2 forms a suitable function spaces framework because of the variational structure of the problem. The a priori estimates are obtained in the Hilbertian scale W k,2 in a very natural way via the “energy method” used in the pioneering paper by Matsumura [71]. Assuming more regularity of the initial data and F one could prove the same result with W 3,2 replaced by W k,2 with k sufficiently large. It follows then from the standard embedding theorems that the solution would be classical. Alternatively, one can use the smoothing effect of the diffusion semigroup to deduce that the solutions constructed in Theorem 2.1 are, in fact, classical for t > 0 (cf. Matsumura and Nishida [72]).
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2.2. Global existence of discontinuous solutions Discontinuous solutions are fundamental both in the physical theory of nonequilibrium thermodynamics and in the mathematical theory of models of inviscid fluids. It seems natural, therefore, to have a rigorious mathematical theory for the system (2.1)–(2.3) which would accommodate discontinuities in solutions. Of course, one has to abandon the classical concept of solution as a differentiable function and turn to the weak solutions which satisfy the integral identities (2.4)–(2.6). It follows from (2.4)–(2.6) that the density #, the momenta #ui , i = 1, 2, 3, and the specific internal energy #θ considered as vector functions of time are weakly continuous, i.e., the quantities
i
#φ dx, Ω
#u φ dx,
i = 1, . . . , N,
Ω
#θ φ dx Ω
belong to C(I ) for any fixed φ ∈ D(Ω). Consequently, it makes sense to prescribe the initial conditions even in the class of weak solutions. Pursuing this path Hoff [47] examined the system (2.1)–(2.3) on the whole space R 3 where the pressure p and the internal energy e obey the ideal gas constitutive relations (1.7) and the initial data #0 , u0 , θ0 satisfy #0 − #¯ ∈ L∞ ∩ L2 R 3 , θ0 − θ¯ ∈ L2 R 3 ,
3 u0 ∈ W s,2 R 3 ,
s ∈ (1/3, 1/2), (2.13)
for certain positive constants #, ¯ θ¯ . The Sobolev spaces W s,2 (R 3 ) for a general real parameter s may be defined in terms of the Fourier transform (see Adams [1]). We report the following result (see [47, Theorem 1.1]). T HEOREM 2.2. Let Ω = R 3 . Assume that λ, µ and κ are constants satisfying µ > 0,
√ µ/3 < λ + µ < 1 + 1/3 15 µ,
κ > 0.
Let the pressure p obey the ideal gas constitutive relation p = (γ − 1)cv #θ,
γ > 1, cv > 0,
(2.14)
¯ θ¯ . and the initial data #0 , u0 , θ0 satisfy (2.13) for certain positive constants #, ¯ Let positive constants 0 < #1 < #¯ < #2 , 0 < θ2 < θ1 < θ be given. Finally, set f ≡ 0 in (2.2). Then there exists ε > 0 depending on #i , θi , i = 1, 2, and s such that the initial-value problem (2.1)–(2.3), (2.12) possesses a weak solution #, u, θ on the set (0, ∞) × R 3
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E. Feireisl
provided ¯ L2 ∩L∞ (R 3 ) + θ0 − θ¯ L2 (R 3 ) + u0 W s,2 ∩L4 (R 3 ) < ε, #0 − # ess inf θ0 θ1 . Moreover, the solution satisfies #1 #(t, x) #2 ,
θ (t, x) θ2
for a.a. (t, x) ∈ (0, ∞) × R 3 ,
and #(t) → #, ¯
θ (t) → θ¯
u(t) → 0,
in Lp R 3 as t → ∞
for any 2 < p ∞. A similar result under slightly more restrictive hypotheses on the data can be proved for Ω = R 2 (see Hoff [47, Theorem 1.1]). The proof of Theorem 2.2 leans, among other things, on the regularity properties of the quantity p −(λ+2µ) div u termed the effective viscous flux. More specifically, this quantity is shown to be free of jump discontinuities. This is the first indication of the important role played by the effective viscous flux in the mathematical theory of compressible fluids. We will address this issue in detail in Section 4.3.
2.3. Global existence in critical spaces The solutions obtained in Theorem 2.2 solve the problem for a very general class of initial data but are not known to be unique. On the other hand, Theorem 2.1 yields a unique solution at the expense of higher regularity imposed on the data. A natural question to ask is how far one can get from the hypotheses of Theorem 2.1 to those of Theorem 2.2 to save uniqueness or, more precisely, what is a critical space of data for which the weak solutions are unique. Such a question was already studied for the incompressible Navier–Stokes equations by Fujita and Kato [41]. The same problem for the full system (2.1)–(2.3) under rather general constitutive relations has been addressed only recently by Danchin [16]. He obtains existence and uniqueness of global solutions in a functional space setting invariant by the natural scaling of the associated equations: #ν (t, x) = # ν 2 t, νx ,
uν = νu ν 2 t, νx ,
θν (t, x) = ν 2 θ ν 2 t, νx ,
where the pressure law p is changed to ν 2 p. A functional space for the triple [#, u, θ ] is termed critical if the associated norm is invariant under the transformation [#, u, θ ] → [#ν , uν , θν ]
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up to a constant independent of ν. Accordingly, the well-posedness for the problem (2.1)– s (R 3 ) whose exact definition goes (2.3) can be stated in terms of the Besov spaces B2,1 beyond the framework of the present paper (see [16]). Let us only remark that the final result is of the same character as Theorem 2.1, namely, global existence and uniqueness of (weak) solutions of the problem (2.1)–(2.3) for data which are a small perturbation of a given equilibrium state. 2.4. Regularity vs. blow-up Since the celebrated work of Leray, it has been a major open problem of mathematical fluid mechanics to prove or disprove that regular solutions of the incompressible Navier–Stokes equations in three space dimensions exist for all time. Clearly, the same problem for the general system (2.1)–(2.3) seems even more delicate. As a matter of fact, there is a negative result of XIN [110, Theorem 1.3]. He considers the system (2.1)–(2.3) posed on the whole space R 3 with zero thermal conductivity κ = 0 and the initial density #0 compactly supported: T HEOREM 2.3. Let Ω = R 3 and m > 3 be a given number. Consider the system (2.1)–(2.3) complemented by the initial conditions (2.12) where the viscosity coefficients λ, µ are constant and satisfy (1.6), and p obeys the constitutive law (2.14). Moreover, let κ = 0, f = 0, and #0 , u0 , θ0 ∈ W m,2 R 3 , supp #0 compact in R 3 ,
θ0 θ > 0.
Then there is no solution of the initial value problem (2.1)–(2.3), (2.12) such that #, u, θ ∈ C 1 [0, ∞); W m,2 R 3 . It seems interesting to compare the conclusion of Theorem 2.3 with the existence result of Theorem 2.1. Obviously, the above theorem does not seem to solve (in a negative way) the question of regularity for the compressible Navier–Stokes equations because of the hypothesis of compactness of the support of #0 . We remark in this regard that the Navier– Stokes system is a model of nondilute fluids in which the density is bounded below away from zero. It is natural, therefore, to expect the problem to be ill-posed when vacuum regions are present at the initial time. 2.5. Large data existence results To begin with, one should say there are practically no global existence results for the full system (2.1)–(2.3) when the data are allowed to be large. The question of local existence of classical solutions for regular initial data was addressed by Nash [79]. There is no indication, however, whether or not these solutions exist for all times.
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E. Feireisl
Note that the problem here is of different nature than for systems of nonlinear conservation laws without diffusion terms. Indeed the equations (2.2), (2.3) are parabolic in u, θ respectively provided the density # is kept away from zero. Accordingly, one can anticipate these state variables to be regular provided uniform estimates of # were available. On the other hand, the density solves the hyperbolic equation (2.1) which is, however, only linear with respect to #. Consequently, no shock waves should develop in # provided they were not present initially and the velocity field u was sufficiently regular. Formally, one can use the standard method of characteristics to deduce: d # t, X(t) + # t, X(t) div u t, X(t) = 0, dt where X (t) = u t, X(t) ,
X(0) = X0 ∈ Ω.
We end up in a “vicious circle” as we need uniform bounds on div u to estimates the amplitude (and positivness) of # but those are not available from the standard energy estimates. As indicated by Choe and Jin [11, Theorems 1.3, 1.4], the following three questions are intimately interrelated: • uniform (on compact time intervals) upper bounds on the density #; • uniform boundedness below away from zero of #; • uniform bounds on u. Answering one of these questions would certainly lead to a rigorous large data existence theory in the framework of distributional (weak) solutions for the problem (2.1)–(2.3) (cf. also Lions [62]). The above mentioned difficulties made several authors to search for a completely different approach to the problem. Motivated by the pioneering work of DiPerna [22], the theory of measure-valued solutions was developed by Málek et al. [68]. Roughly speaking, the “value” of each state variable at a fixed point (t, x) is no longer a number (or a finite component vector) but a probability measure (the Young measure) characterizing possible oscillations in a sequence of approximate solutions used to construct this particular variable. The numerical values of #, u, θ are centers of gravity of the corresponding Young measures and the nonlinear constitutive relations are expressed in a very simple way. These solutions are of course more general quantities than the distributional solutions and coincide with them provided one can show that the Young measures are concentrated at one point, i.e., they are Dirac masses for each value of the independent variables (t, x). One can expect positive existence results in the class of measure valued solutions whenever suitable a priori estimates are available so that the nonlinear compositions are equi-integrable and consequently weakly compact in the space of Lebesgue integrable functions. This is of course considerably less than compactness of the state variables in the strong L1 -topology – an indispensable ingredient of any existence proof of distributional solutions. The major shortcoming of measure-valued solutions is certainly the almost insurmountable problem of uniqueness solved only in the case of a scalar conservation law in [22].
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This is, of course, the price to be paid for the relatively simple existence theory and one might feel tempted to say it is the same situation as when the weak solutions were introduced. However, this gap between existence and uniqueness, accepted for the weak solutions, seems to be simply too large in the class of measure-valued solutions and the approach is slowly being abandoned. 2.6. Bibliographical comments Besides the results mentioned above, the small data existence problems were treated by Solonnikov [97], Tani [101], Valli [108] and others. The existence theory in critical spaces for barotropic flows was developed by Danchin [15]. As pointed out several times, the main obstacle to obtain large data existence results is the lack of suitable a priori estimates. Formal compactness results for the full system (1.1)–(1.3) were obtained by Lions [62] on condition of uniform boundedness of all state variables. 3. The continuity equation and renormalized solutions Motivated by the work of Kruzkhov on scalar conservation laws, DiPerna and Lions [23] introduced the concept of renormalized solutions as a new class of solutions to general linear transport equations. They play a similar role as the entropy solutions in the theory of nonlinear conservation laws – they represent a class of physically relevant solutions in which the corresponding initial value problems admit a unique solution. Multiplying (1.1) by b (#), where b is a continuously differentiable function, we obtain the identity ∂b(#) + div b(#)u + b (#)# − b(#) div u = 0. ∂t
(3.1)
Obviously, any strong (classical) solution of (1.1) satisfies automatically (3.1). For the weak solutions, however, (3.1) represents an additional constraint which may not be always satisfied. Following [23] we shall say that # is a renormalized solution of (1.1) on the set I × Ω if #, u, ∇u are locally integrable and (3.1) is satified in the sense of distributions (in D (I × Ω)), i.e., the integral identity ∂ϕ + b(#)u · ∇ϕ + b(#) − b (#)# div uϕ dx dt = 0 b(#) (3.2) ∂t I Ω holds for any test function ϕ ∈ D(I × Ω) and any b ∈ C 1 (R) such that b (z) = 0
for all z large enough, say, |z| M.
(3.3)
Let us emphasize here that unlike the entropy solutions that can be characterized as satisfying a certain type of admissible jump conditions on discontinuity curves, the renormalized solutions characterize the so-called concentration phenomena (cf. Section 3.2 below).
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E. Feireisl
3.1. On continuity of the renormalized solutions The renormalized solutions enjoy many remarkable properties most of which can be proved by means of the regularization technique developed by DiPerna and Lions [23]. The following auxilliary assertion is classical (cf. Lions [61, Lemma 3.2]). L EMMA 3.1. Let Ω ⊂ R N be a domain and p
q
u, ∇u ∈ Lloc (Ω),
σ ∈ Lloc (Ω),
where 1 p, q ∞,
1/r = 1/p + 1/q 1.
Let ϑε be a regularizing sequence, i.e., ϑε ∈ D(R N ), ϑε radially symmetric and radially decreasing, RN
ϑε dx = 1,
ϑε (x) → 0 as ε → 0 for any fixed x ∈ R N \ {0}.
Then $ $ $ϑε ∗ div(σ u) − div [ϑε ∗ σ ]u $ r c(K) u W 1,p (K) σ Lq (K) L (K) for any compact K ⊂ Ω and rε = ϑε ∗ div(σ u) − div [ϑε ∗ σ ]u → 0
in Lrloc (Ω) as ε → 0,
where ∗ stands for convolution on R N . Now one can regularize (3.1), more precisely, take φ(x) = ϑε (x − y) in (3.2) to deduce ∂ϑε ∗ b(#) + div [ϑε ∗ b(#)]u + ϑε ∗ b (#)# − b(#) div u = rε ∂t
(3.4)
for t ∈ I and x ∈ Ω such that dist[x, ∂Ω] > ε. Here b is an arbitrary function satisfying (3.3) and rε (t) as in Lemma 3.1, i.e., rε → 0 in L1loc (I × Ω) as ε → 0 provided u is locally integrable. The first consequence of (3.4) is continuity in time of the renormalized solutions. P ROPOSITION 3.1. Let u, ∇u be locally integrable on I × Ω where I ⊂ R is an open time interval and Ω ⊂ R N a domain. Let # – a locally integrable function – be a renormalized solution of the continuity equation (2.1) on I × Ω.
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Then for any compact B ⊂ Ω and any function b as in (3.3), the composition b(#) : t ∈ I → b(#)(t) is a continuous function of t with values in the Lebesgue space L1 (K), i.e., b(#) ∈ C J ; L1 (B) for any compact J ⊂ I. Moreover, we have the following corollary. C OROLLARY 3.1. In addition to the hypotheses of Proposition 3.1, assume that I ⊂ R, Ω ⊂ R N are bounded; and # ∈ L∞ 0, T ; Lp (Ω) for a certain p > 1, u, ∇u ∈ L1 (I × Ω). Then # as a function of t ∈ I is continuous with values in L1 (Ω): # ∈ C I¯; L1 (Ω) . The proof of both Proposition 3.1 and Corollary 3.1 can be done via the regularization technique as in [23]. 3.2. Renormalized and weak solutions Another conclusion which can be deduced from Lemma 3.1 and (3.4) is that the class of weak and renormalized solutions coincide provided # or ∇u or both are sufficiently integrable. P ROPOSITION 3.2. Assume p
q
# ∈ Lloc (I × Ω),
u, ∇u ∈ Lloc (I × Ω),
1 p, q ∞,
1/p + 1/q 1.
where
Then # is a renormalized solution of (2.1) if and only if # satisfies (2.1) in D (I × Ω), i.e., the integral identity (2.4) holds for any test function ϕ ∈ D(I × Ω). Integrating (1.9) we can see that the typical regularity class for the velocity gradient is ∇u ∈ L2loc (I × Ω). Accordingly, to apply Proposition 3.2, one needs # ∈ L2loc (I × Ω). There is another reason why the density “should be” square integrable. The continuity equation can be (formally) written in the form Dt # + # div u = 0,
where Dt =
∂ + u · ∇# ∂t
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is the so-called material derivative. The quantity # div u plays the role of a forcing term in the above equation. Thus if we want to keep, at least in a certain weak sense, the structure given by characteristics (cf. Section 2.5), we should have # div u locally integrable. Taking the square integrability of ∇u for granted we are led to require # ∈ L2loc (cf. also DiPerna and Lions [23]). However, as we will see in Proposition 4.1 below, the square integrabilty of the density is not necessary for a weak solution of (1.1) to be a renormalized one.
3.3. Renormalized solutions on domains with boundary Assume that Ω ⊂ R N is a domain with Lipschitz boundary. As for the velocity field u, q q we suppose u ∈ Lloc (I × Ω ), ∇u ∈ Lloc (I × Ω ) for a certain q > 1. Although u need not be continuous, one can still consider the no-slip boundary conditions (1.11) in the sense of traces. Accordingly, assuming (1.11) and extending u to be zero outside Ω, one 1,q has u ∈ Wloc (R N ). Equivalently, by virtue of the Hardy inequality (see, e.g., Opic and Kufner [85]), one can replace (1.11) by the following stipulation: |u| q ∈ Lloc I × Ω . dist[x, ∂Ω]
(3.5)
Using (3.5) we can show a continuation theorem for renormalized solutions. q
P ROPOSITION 3.3. Let Ω ⊂ R N be a Lipschitz domain and let u, ∇u belong to Lloc (I × Ω ) for a certain q > 1, and let (1.11) be satisfied in the sense of traces. Let # be a renormalized solution of (2.1) on I × Ω. Then # is a renormalized solution of (2.1) on I × R N provided #, u are extended to be zero outside Ω. Proposition 3.3 together with Propositions 3.1, 3.2 yield an interesting corollary, namely, the principle of total mass conservation for the weak solutions of (2.1). Consider a bounded domain Ω ⊂ R N with Lipschitz boundary on which u satisfies the no-slip boundary condition (1.11). Formally, one can integrate (2.1) over Ω to deduce d dt
# dx = 0, Ω
i.e., the total mass m=
# dx Ω
is a constant of motion. By virtue of Propositions 3.1–3.3, we have the same result for distributional solutions:
Viscous and/or heat conducting compressible fluids
325
P ROPOSITION 3.4. Let Ω ⊂ R N be a bounded Lipschitz domain. Let # ∈ L∞ I ; Lp (Ω) ,
u, ∇u ∈ Lq (I × Ω),
1 < p, q ∞, 1/p + 1/q 1, solve (2.1) in D (I × Ω) and u|∂Ω = 0. Then the total mass #(t) dx m= Ω
is constant for t ∈ I . 4. Weak convergence results Many of the most important techniques set forth in recent years for studying the problem (2.1)–(2.3) are based on weak convergence methods. To establish the existence of a solution, an obvious idea is first to invent an appropriate collection of approximating problems, which can be solved; and then to pass to the limit in the sequence of approximate solutions to obtain a solution of the original problem. The overall impediment of this approach is of course the nonlinearity. Whereas it is very often true that one can find certain uniform estimates on the family of approximate solutions, the bounds on oscillations of these quantites are usually in short supply. This is, for instance, the case of the density # solving the hyperbolic equation (2.1). In this section, we shall investigate the compactness properties of weakly convergent sequences of solutions of the continuity equation (2.1) and the momentum equations (2.2). More precisely, we consider a family of weak solutions {#n }, {un } of the system (2.1), (2.2), i.e., the integral identities #n I
Ω
∂ϕ j ∂ϕ + #n un dx dt = 0, ∂t ∂xj
I
Ω
#n uin
(4.1)
∂ϕ ∂ϕ ∂ui ∂ϕ j ∂ϕ + #n uin un + pn −µ n ∂t ∂xj ∂xi ∂xj ∂xj j
− (λ + µ)
∂un ∂ϕ + #fni ϕ dx dt = 0, ∂xj ∂xi
i = 1, . . . , N, n = 1, 2, . . . ,
(4.2)
hold for any test function ϕ ∈ D(I × Ω). Since all results we shall discuss are of local nature, we assume that both the time interval I and the spatial domain Ω are bounded.
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Moreover, we suppose that #n , un , ∇un , pn , and fn are locally integrable and weakly convergent, specifically, ⎫ #n → # ⎪ ⎪ ⎪ ⎪ un → u ⎪ ⎪ ⎬ ∇un → ∇u ⎪ ⎪ ⎪ pn → p ⎪ ⎪ ⎪ ⎭ fn → f
weakly in D (I × Ω) as n → ∞.
Here vn → v weakly means that v is locally inegrable on I × Ω and
vn ϕ dx dt →
I
Ω
vϕ dx dt I
for any ϕ ∈ D(I × Ω).
Ω
Our goal in this section is to identify the limit problem solved by the quantities #, u, p, and f. The best possible result is, of course, they satisfy the same system of equations. If this is the case, the problem enjoys the property of compactness with respect to the weak topology. Given relatively feeble a priori estimates (cf. Section 5), the weak compactness of the problem plays a decisive role in the larga data existence theory for barotropic flows presented in Section 6.
4.1. Weak compactness of bounded solutions to the continuity equation Although hyperbolic, the continuity equation exhibits the best properties as far as the weak compactness of solutions is concerned. Consider a sequence #n , un of renormalized solutions of (1.1) on I × Ω, i.e., in addition to (4.1), we assume (3.2) holds for any b as in (3.3). Moreover, we shall assume that un L1 (I ×Ω) , ∇un Lq (I ×Ω) c
for a certain q > 1;
(4.3)
and that the family #n is equi-bounded and equi-integrable, i.e., #n dx dt c, I
Ω
lim
|Q|→0 Q
#n dx dt = 0
uniformly with respect to n = 1, 2, . . . .
(4.4)
The failure of weak convergence to imply strong convergence is usually recorded by certain measures called defect measures. To this end, we introduce the cut-off operators Tk = Tk (z), z Tk (z) = kT , k
k 1,
(4.5)
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327
where T ∈ C 1 (R) is an odd function such that T (z) = z
for 0 z 1,
T (z) = 2
for z 3,
T concave on [0, ∞).
The amplitude of possible oscillations in the density sequence will be measured by the quantity . $ $ oscp [#n − #](Q) = sup lim sup$Tk (#n ) − Tk (#)$Lp (Q) . n→∞
k1
Unlike the defect measures introduced by DiPerna and Majda [24], osc vanishes on any set on which #n tends to # strongly in the L1 -topology regardless possible concentration effects. Now we shall address the following question: Under which conditions do the limit functions #, u solve (2.1)? Taking a function b as in (3.3) one deduces easily from (3.2), (4.3), and (4.4) that p (4.6) b(#n ) → b(#) in C I ; Lweak (Ω) , 1 p < ∞. Here and in what follows, we shall use the standard notation g(v) for a weak (Lp ) limit of a sequence g(vn ) where vn tends weakly to v. The possibility to find a subsequence of vn such that the composition b(vn ) is weakly convergent for any continuous b satisfying certain growth conditions is the basic statement of the theory of Young measures (cf. Tartar [102,103], Pedregal [88]). Such a limit, however, need not be unique unless the convergence of vn is strong. p A sequencevn converges to v in C(I ; Lweak (Ω)) if it is bounded in L∞ (I ; Lp (Ω)), the function t → Ω vn (t, x)φ(x) dx can be identified with a continuous function on I , and vn (t, x)φ(x) dx → v(t, x)φ(x) dx uniformly with respect to t ∈ I Ω
Ω
for any test function φ ∈ D(Ω). By virtue of the Aubin–Lions lemma (see, e.g., Lions [59, Theorem 5.1], the relations (4.3), (4.6) imply q
b(#n )un → b(#)u weakly in Lloc (I × Ω).
(4.7)
Indeed taking p large enough in (4.6) we get Lp (Ω) compactly imbedded in W −1,q (Ω); whence b(#n ) → b(#) in C I ; W −1,q (Ω) which together with (4.3) yields the desired conclusion. The distributions lying in the “negative” Sobolev space W −1,q can be identified with generalized derivatives of vector functions in Lq (see, e.g., Adams [1]). In particular, if we knew that #n is bounded in p Lloc (I × Ω), we could conclude that #, u solve (2.1) in the sense of distributions. In a general case, we report the following result (cf. [29, Proposition 7.1]):
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E. Feireisl
P ROPOSITION 4.1. In addition to (4.1), assume #n , un satisfy (2.1) in the sense of renormalized solutions on I × Ω. Moreover let (4.3), (4.4) hold, and oscp [#n − #](Q) c(Q) for any compact Q ⊂ I × Ω, where 1 1 + < 1. p q Then #, u is a renormalized solution of (2.1) on I × Ω. The main advantage of Proposition 4.1 is that the sequence #n itself need not be bounded in Lp . As we will see later, this is particularly convenient when barotropic fluids are studied (cf. Proposition 5.3 below). 4.2. On compactness of solutions to the equations of motion In this part, we shall assume that un L2 (I ×Ω) , ∇un L2 (I ×Ω) c
for all n = 1, 2, . . . .
In particular, the products ui uj , i, j = 1, . . . , N , are bounded in L1 (I, L2 ∗ is the Sobolev exponent for the embedding W 1,2 ⊂ L2 to hold, i.e., 2∗ is arbitrary finite for N = 2 and 2∗ =
2N N −2
(4.8) ∗ /2
(Ω)) where 2∗
if N = 3, 4, . . . .
j
Consequently, for the cubic quantity #n uin un to be at least integrable, it is neccessary that $ $ ess sup $#n (t)$Lp (Ω) c for a certain p N/2 (4.9) t ∈I
provided N > 2. Here again, we face one of the major obstacles to build up a rigorous mathematical theory for the full system (2.1)–(2.3), namely, the lack of suitable a priori estimates. The only available bounds on the density are those deduced from boundedness of the total energy. In general, these “energy” estimates are not sufficient to get (4.9). Of course, the barotropic case offers a considerable improvement as the energy is given by formula (1.8) and, consequently, the desired estimates follow provided γ N/2. The situation is more delicate in the physically relevant two-dimensional case. Here, the Sobolev space W 1,2 is embedded in the Orlicz space LΦ generated be the function Φ(z) = exp z2 − 1
Viscous and/or heat conducting compressible fluids
(see Adams [1]). Consequently, condition (4.9) should be replaced by Ψ (#n ) dx c with Ψ (z) z log(z). ess sup t ∈I
329
(4.10)
Ω
Since #n , un satisfy also the continuity equation (4.1) we deduce from (4.9), (4.10) respectively that #n → #
p in C I ; Lweak (Ω) , p N/2 if N = 3, . . . ,
(4.11)
#n → #
in C I ; LΨ weak (Ω) for N = 2.
(4.12)
or
In both cases this implies compactness of #n in L2 (I, W −1,2 (Ω)), and we get #n un → #u weakly in, say, L1 (I × Ω). j
The weak compactness of the cubic term #n uin un represents a more difficult problem. In addition to the above hypotheses, we assume the kinetic energy to be bounded uniformly in n, i.e., #n |un |2 is bounded in L1 (I × Ω) uniformly with respect to n. Supposing (4.9) holds for p > N/2 we have, similarly as above, #n un → #u
2N , N 2; in C I ; Lrweak (Ω) for a certain r > N +2
(4.13)
whence j
#n uin un → #ui uj
weakly in, say, L1 (I × Ω), i, j = 1, . . . , N, N 2.
As a matter of fact, the result is not optimal for N = 2; in that case one could use directly (4.12) provided Ψ was a function dominating z log(z). Summing up the previous considerations we get the following conclusion: P ROPOSITION 4.2. Let the quantities #n , un satisfy the estimates (4.8), (4.9) with p > N/2. Moreover, let the kinetic energy be bounded, specifically, 2 ess sup #n (t)un (t) dx c for all n = 1, 2, . . . . (4.14) t ∈I
Ω
Finally, assume fn are bounded and fn → f
uniformly on I × Ω.
(4.15)
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E. Feireisl
Then the limit functions #, u, p, and f satisfy (2.1), (2.2) in D (I × Ω), i.e., the integral identities (2.4), (2.5) hold for any test function ϕ ∈ D(I × Ω). Let us repeat once more that it is an open problem whether or not the bounds required for the density component are really available. As we have seen in Section 2.5, uniform bounds on the density are equivalent to uniform boundedness of u – a situation reminiscent of the classical regularity problem for the incompressible Navier–Stokes equations. 4.3. On the effective viscous flux and its properties Consider the quantity p − (λ + 2µ) div u called usually the effective viscous flux. Formally, assuming all the functions in (2.2) smooth and vanishing for |x| → ∞ we can compute pn − (λ + 2µ) div un
j = −1 div(#n fn ) − −1 div(#n un )t − Ri,j #n uin un
(4.16)
(summation convention). The symbol Ri,j denotes the pseudodifferential operator Ri,j = ∂xi −1 ∂xj or, in terms of symbols, Ri,j [v] = F −1
ξi ξj F [v](ξ ) , |ξ |2
where F denotes the Fourier transform in the x-variable. The effective viscous flux enjoys certain weak compactness properties discovered by Lions [62] which represent the key point in the global existence proof for barotropic flows. Following [62] we can use (formally) (4.16) to obtain pn − (λ + 2µ) div un b(#n ) = b(#n ) −1 div(#n fn ) − ∂t b(#n ) −1 div(#n un ) j (4.17) + b(#n )t −1 div(#n un ) − b(#n )Ri,j #n uin un , where b is as in (3.3). One should keep in mind that #n , un here are defined on a bounded domain Ω and, consequently, a localization procedure is needed to justify this argument. Using Proposition 4.2 we can now pass to the limit for n → ∞ in (4.16) and multiply the resulting expression by b(#) to deduce p − (λ + 2µ) div u b(#) = b(#) −1 div(#f) − ∂t b(#) −1 div(#u) + b(#)t −1 div(#u) − b(#)Ri,j #ui uj . (4.18)
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331
Assuming, in addition to the hypotheses already made, that #n , un are also renormalized solutions of (2.1), we can use (4.7), (4.18) together with the smoothing properties of −1 , to pass to the limit in (4.17) for n → ∞ to obtain:
lim
n→∞ I
pn − (λ + 2µ) div un b(#n )ϕ dx dt
Ω
− I
p − (λ + 2µ) div u b(#)ϕ dx dt
Ω
= lim
n→∞ I
− I
Ω
j j b(#n ) uin Ri,j #n un − Ri,j #n uin un ϕ dx dt
b(#) ui Ri,j #uj − Ri,j #ui uj ϕ dx dt
(4.19)
Ω
for any test function ϕ ∈ D(I × Ω). It is a remarkable result of Lions [62] that the right-hand side of (4.19) is in fact zero. To see this we offer two rather different techniques in hope to illuminate a bit the connection of such a result with the theory of compensated compactness. Following the proof in [62] one can make use of the regularity properties of the commutator ui Ri,j #uj − Ri,j #ui uj discovered by Coifman and Meyer [13]. Specifically, this quantity belongs to the Sobolev space W 1,q provided ui ∈ W 1,2 and #uj ∈ Lr with r > 2 in which case 1/q = 1/p + 1/2. Of course, this hypothesis requires # ∈ Lp with p > 3 for N 3 which is too strong for our purposes, but a simple interpolation argument shows one can treat the general case # ∈ Lp , p > N/2, by the same method. Pursuing a different path we can write I
Ω
j j b(#n ) un Ri,j #n uin − Ri,j #n uin un ϕ dx dt
j un Xj (#n ) · Y(#n un ) − U(#n un ) · Vj (#n ) dx dt,
= I
Ω
where the vector fields Xj , Y, U, Vj are given by formulas j Xk (#n ) = ϕb(#n )δj,k − Rk,l ϕb(#n )δl,j , Uk (#n un ) = #n ukn − Rk,i #n uin , j Vk (#n ) = Rk,l ϕb(#n )δl,j ,
j = 1, . . . , N,
and δi,j stands for the Kronecker symbol.
Yk (#n un ) = Rk,i #n uin ,
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E. Feireisl
Now, it is easy to check that div Xj = div U = 0 and Y = ∇ −1 div(#n un ) , Vj = ∇ −1 div(ϕb(#n )δl,j , i.e., curl Y = curl Vj = 0. Applying the Lp –Lq version of Div-Curl Lemma of the compensated compactness theory (cf. Murat [78] or Yi [111]) together with (4.6), (4.13), we conclude Xj (#n ) · Y(#n un ) → ϕb(#)δj,k − Rk,l ϕb(#)δl,j · Rk,i #ui in L2 I ; W −1,2 (Ω) , and, similarly, in L2 I ; W −1,2 (Ω) U(#n un ) · Vj (#n ) → #uk − Rk,i #ui Rk,l ϕb(#)δl,j provided p > N/2. This yields, similarly as above, the desired conclusion, namely, the right-hand side of (4.19) equals zero. Thus we have obtained the following important result (see [62]): P ROPOSITION 4.3. Let the quantities #n , un , and fn satisfy the hypotheses (4.8), (4.9) for p > N/2, together with (4.14), (4.15). Let, moreover, pn Lr (I ×Ω) c
for a certain r > 1.
Then we have lim pn − (λ + 2µ) div un b(#n )ϕ dx dt n→∞ I
Ω
p − (λ + 2µ) div u b(#)ϕ dx dt
= I
Ω
for any b satisfying (3.3) and any test function ϕ ∈ D(I × Ω). 4.4. Bibliographical remarks The theory of compensated compactness has played a crucial role in the development of the first large data existence results for systems of nonlinear conservation laws (see Dafermos [14], DiPerna [21], Tartar [102]). A good survey on weak convergence methods can be found in the monograph by Evans [26]. One of the well-known results is the socalled Div-Curl Lemma refered to above (cf. Murat [78]):
Viscous and/or heat conducting compressible fluids
333
L EMMA 4.1. Let Un , Vn be two sequences of vector functions defined on some open set Q ⊂ R N such that Un → U
weakly in Lp (Q),
Vn → V
weakly in Lq (Q);
and div Un precompact in W −1,p (Q),
curl Vn precompact in W −1,q (Q),
where 1 1 + 1. p q
1 < p, q < ∞, Then Un · Vn → U · V
in D (Q).
Note that the situation in Proposition 4.3 is particularly simple as div Un = curl Vn = 0 and the proof of Lemma 4.1 is elementary. There is yet another way to show Proposition 4.3 presented in [27, Lemma 5]. The defect measures similar to osc were introduced by DiPerna and Majda [24] in their study of the Euler equations.
5. Mathematical theory of barotropic flows We review the recent development of the mathematical theory of barotropic flows, specifically, we shall discuss some large data existence and related results originated by the pioneering work of Lions [62]. Accordingly, the crucial hypothesis we cannot dispense with is that the pressure p and the density # are functionally dependent and the relation between them is given by formula p = p(#) with p : [0, ∞) → [0, ∞) – a nondecreasing and continuous function.
(5.1)
As a matter of fact, most of the results will be stated for the simpler isentropic pressuredensity relation p(#) = a#γ ,
a > 0, γ 1,
and possible generalizations discussed afterwards.
(5.2)
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E. Feireisl
The temperature θ being eliminated from the pressure constitutive law, the system (2.1)– (2.3) reduces to ∂# + div(#u) = 0, ∂t ∂#u + div(#u ⊗ u) + ∇p(#) = µ u + (λ + µ)∇(div u) + #f. ∂t
(5.3) (5.4)
The spatial variable x will belong to a regular bounded domain Ω ⊂ R N , N = 2, 3, and the velocity u will satisfy the no-slip boundary conditions u|∂Ω = 0.
(5.5)
Taking (formally) the scalar product of (5.4) with u and integrating by parts we obtain the energy inequality: d dt
µ|∇u|2 + (λ + µ)|div u|2 dx
E(t) dx + Ω
Ω
#f · u dx,
(5.6)
Ω
where the specific energy E satisfies (1.8). If p is given by (5.2), we have 1 E = #|u|2 + # log(#) 2
1 a #γ E = #|u|2 + 2 γ −1
for γ = 1,
if γ > 1.
As already agreed on in Section 1, the fluids under consideration are viscous, i.e., and λ + µ 0.
µ>0
Note that the restrictions imposed on λ allow for all physically relevant situations. In what follows, we consider the finite energy weak solutions of the problem (5.3)–(5.5) on the set I × Ω, more specifically, #, u will meet the following set of conditions: • the density # and the velocity u satisfy # ∈ L∞ I ; Lγ (Ω) ,
# 0,
N u ∈ L2 I ; W01,2 (Ω) ;
• the specific energy E belongs to L1loc (I ; L1 (Ω)) and the energy inequality (5.6) holds in D (I ), i.e.,
E dx dt −
∂t ψ I
Ω
#f · u dx dt
ψ I
µ|∇u| + (λ + µ)|div u| dx dt
ψ I
2
Ω
holds for any function ψ ∈ D(I ), ψ 0;
Ω
2
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335
• the functions #, u extended to be zero outside Ω solve the continuity equation (5.3) in D (I × R N ) (cf. (2.4)); moreover, (5.3) is satisfied in the sense of renormalized solutions, i.e., (3.2) holds for any b as in (3.3); • the equations of motion (5.4) are satisfied in D (I × Ω) (cf. (2.5)). As the reader will have noticed in Section 4, the value of the adiabatic constant γ will play an important role in the analysis. In most cases, we shall assume γ > N/2, where N = 2, 3 are the physically relevant situations. The external force density f is assumed to be a bounded and measurable function such that ess sup f(t, x) F. t ∈I, x∈Ω
In what follows, we shall give an outline of the large data existence results in the class of finite energy weak solutions. We shall also discuss the long-time behaviour and related asymptotic problems. To this end, we pursue the classical scheme for solving nonlinear problems: • First of all, we find a priori estimates, i.e., the bounds imposed formally on any classical solution and depending only on the data (cf. Sections 5.1, 5.2). • Given a family of solutions satisfying the bounds induced by a priori estimates, we examine the question of compactness, i.e., whether or not any accummulation point of this family in suitable topologies is again a solution of the original problem (see Sections 5.3, 5.4). • Finally, one has to find a suitable approximation scheme solvable, say, by a classical fixed-point technique, and compatible with both the estimates and compactness properties mentioned above (Section 5.5). To conclude this introduction, let us note that any finite energy weak solution satisfies γ # ∈ C I ; Lweak (Ω) ∩ C I ; Lα (Ω) , 1 α < γ , 2γ γ +1 #u ∈ C I ; Lweak (Ω)
(5.7)
provided γ > N/2 (cf. Proposition 3.1). In particular, the density and the momenta are well defined at any specific time t ∈ I . Moreover, the total mass m= # dx is independent of t ∈ I ; (5.8) Ω
and the total energy E defined for any t ∈ I by formula
E(t) = E #, (#u) (t) =
a 1 |(#u)|2 (t) + #γ (t) dx 2 # γ − 1 #(t )>0
is a lower semi-continuous function of t ∈ I (see [27, Corollary 2]).
(5.9)
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E. Feireisl
5.1. Energy estimates Besides the total mass m, the total energy E is another quantity which can be shown bounded in terms of the data at least on compact time intervals. P ROPOSITION 5.1. Let Ω ⊂ R N be a bounded Lipschitz domain. Let #, u be a finite energy weak solution of (5.3)–(5.5) where the pressure satisfies the isentropic constitutive law (5.2) with γ > N/2. Then $ $ $#(t)$γ γ + L (Ω)
2 #(t)u(t) dx + Ω
c E0 , m, F, t − inf{I } ,
t
inf{I } Ω
|∇u|2 dx ds (5.10)
where the quantity c is bounded for bounded values of arguments and E0 = lim sup E #, (#u) (t). t →inf{I }+
The bound (5.10), which can be easily obtained combining the energy inequality (5.6) and the Gronwall lemma, can be viewed as an a priori estimate though it holds for any finite energy weak solution of the problem. It is not difficult to see that similar results can be derived provided p is given by a general constitutive relation (5.1) and satisfies suitable growth conditions for large values of the density. On the other hand, as already mentioned in Section 2.5, uniform a priori estimates of # seems to be out of reach of the standard techniques and represent a major open problem of the present theory.
5.2. Pressure estimates for isentropic flows By virtue of (5.10), the isentropic pressure p(#) belongs automatically to the set L1 (I ×Ω) at least for bounded time intervals I . On the other hand, the weak compactness results like Proposition 4.3 require p in a weakly complete (reflexive) space Lr (I × Ω) with r > 1. Such a bound is indeed available as the following result shows: P ROPOSITION 5.2. Assume Ω ⊂ R N , N 2, is a bounded Lipschitz domain. Let #, u be a finite energy weak solution to the problem (5.3)–(5.5) on I × Ω where the isentropic pressure p is given by (5.2) with γ > N/2. Let 1 1 2γ 0 < η < min , −1 4 γ N be given. Denote by m =
Ω
# dx the (conserved) total mass and let F = ess supI ×Ω |f|.
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337
Then for any bounded time interval J ⊂ I , we have J
. γ +1 γ #γ +η dx dt c m, F, η, |J | 1 + sup E(t) . t ∈J
Ω
(5.11)
A local version of the above estimates was obtained by Lions [62]. In fact, the bounds on η in Proposition 5.2 are not optimal. Similarly as in the local case (see [62]), one could verify the best values for η: 0<η
2 γ − 1. N
However, for further purposes, it is convenient to have the integrals containing #η bounded in terms of the total mass m equivalent to the L1 -norm rather than the total energy E proportional to the Lγ -norm of #. The validity of (5.11) up to the boundary of Ω was proved in [39] by means of a multiplier technique. More specifically, the main idea is to take the quantities φi (t, x) = ψ(t)Bi ϑε ∗ Tk (#η ) ,
i = 1, . . . , N,
as test functions for (2.5). Here ψ ∈ D(I ), 0 ψ 1,
|∂t ψ| dt 2, I
ϑε (x) is a regularizing sequence as in Lemma 3.1, Tk are the cut-off operators introduced in (4.5), and, most importantly, the symbol B stands for an inverse of the div operator, i.e., v = B[g] solves the equation div v = g −
1 |Ω|
g dx,
v|∂Ω = 0.
(5.12)
Ω
Equation (5.12) has been studied by many authors. Here, we have adopted the approach which is essentially due to Bogovskii [8]. It can be shown that the problem (5.12) admits a solutions operator B : g → v enjoying the following properties: 1,p • B = [B1, . . . , BN ] is a bounded linear operator from Lp (Ω) into [W0 (Ω)]N , specifically, $ $ $B[g]$ 1,p c(p) g Lp (Ω) W (Ω) 0
for any 1 < p < ∞.
• The function v = B[g] solves the problem (5.12). • If, moreover, g ∈ Lp (Ω) can be written in the form g = div h where h ∈ [Lr (Ω)]N , h · n = 0 on ∂Ω, then $ $ $B[g]$ r c(p, r) h Lr (Ω) . L (Ω)
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The proof of the existence of the operator B as well as the above properties can be found in Galdi [42] or Borchers and Sohr [9]. An alternative approach to show (5.11) based on properties of the Stokes operator was proposed by Lions [63]. The estimates (5.8), (5.10), (5.11) are the only available for the problem (5.3)– (5.5) unless some smallness assumptions are imposed on the data, i.e., on E0 , m, and f. Thus in accordance with Proposition 4.2, the main stumbling block to show compactness of solutions is the pressure term. Indeed the above estimates guarantee only weak compactness of the density # while for p = p(#) to be compact we need strong compactness of # in Lp . As we shall see later, the way out this vicious circle is provided by Proposition 4.3, namely, by the compactness properties of the effective viscous flux.
5.3. Density oscillations for barotropic flows We show how Proposition 4.3 can be used to describe the amplitude of density oscillations for barotropic flows discussed in Section 4.1. In addition to (5.1), we suppose p(0) = 0,
p(#) 0 for # 0, p convex on [0, ∞).
It is easy to verify that p(y) − p(z) p(y − z) for all 0 z y which yields immediately p(y) − p(z) Tk (y) − Tk (z) p Tk (y) − Tk (z) Tk (y) − Tk (z)
for all y, z 0.
(5.13)
Under the hypotheses (and notation) of Proposition 4.3, we have p = p(#) and we can write lim p(#n )Tk (#n ) − p(#) Tk (#) dx dt n→∞ Q
= lim
n→∞ Q
+
p(#n ) − p(#) Tk (#n ) − Tk (#) dx dt
p(#) − p(#) Tk (#) − Tk (#) dx dt
Q
for any bounded Q ⊂ I × Ω.
(5.14)
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As p is convex and Tk concave on [0, ∞), the second integral on the right-hand side of (5.14) is non-negative and, making use of (5.13) we infer p(#n )Tk (#n ) − p(#) Tk (#) dx dt lim n→∞ Q
p Tk (#n ) − Tk (#) Tk (#n ) − Tk (#) dx dt.
lim sup n→∞
(5.15)
Q
On the other hand, we have lim div un Tk (#n ) − div uTk (#) dx dt n→∞ Q
= lim
n→∞ Q
div un Tk (#n ) − Tk (#) dx dt
$ $ sup div un L2 (Q) lim sup $Tk (#n ) − Tk (#)$L2 (Q) .
(5.16)
n→∞
n1
Thus if the pressure is superlinear at infinity, the relations (5.15), (5.16) together with Proposition 4.3 enable to estimate the amplitude of oscillations oscp [#n − #](Q) introduced in Section 4.1. In particular, the following result holds (see [29, Proposition 6.1]). P ROPOSITION 5.3. Let Ω ⊂ R N , N 2 be a bounded Lipschitz domain and I ⊂ R a bounded interval. Let the pressure p be given by the formula (5.1), p(0) = 0, p convex, and p(#) a#γ ,
a > 0, γ > N/2.
Let #n , un be a sequence of finite energy weak solutions of the problem (5.3)–(5.5) with f = fn and such that mn = #n m, Ω
lim sup E #n , (#n un ) (t) E0 ,
t →inf{I }+
and ess sup |fn | F I ×Ω
independently of n. Then .1/γ oscγ +1 [#n − #](Q) c(Q) sup div un L2 (Q) n1
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for any weak limit # of the sequence #n and any bounded Q ⊂ I × Ω. As a straightforward consequence of Propositions 4.1, 5.3 we get the following: C OROLLARY 5.1. Under the hypotheses of Proposition 5.3, let #n → # un → u
weakly star in L∞ I ; Lγ (Ω) , weakly in L2 I ; W01,2 (Ω) .
Then #, u solve (5.3) in the sense of renormalized solutions, i.e., Equation (3.2) holds for any b satisfying (3.3).
5.4. Propagation of oscillations For simplicity, we suppose the pressure p is given by the isentropic constitutive relation (5.2) with γ > N/2. Similarly as in Proposition 5.3, let #n , un be a sequence of finite energy weak solutions of (5.3)–(5.5) on some bounded time interval I such that lim sup E #n , (#n un ) E0 ,
t →inf{I }+
fn L∞ (I ×Ω) F
uniformly in n. The issue we want to address now is the time propagation of oscillations in the density component. To begin with, it seems worth-observing that any reasonable solution operator we could associate with the finite energy weak solutions cannot be compact with respect to #. This is due to the hyperbolic character of the continuity equation (5.3). In accordance with the observations made by Lions [60], the oscillations should propagate in time. Serre [92] studied this phenomenon and showed the amplitude of the Young measures associated to the sequence #n (t) is a non-increasing function of time. His proof is complete in the dimension N = 1 and formal for N 2 taking the conclusion of Proposition 4.3 for granted. Having proved Proposition 4.3 Lions [62] completed the proof for N 2. The fact that oscillations cannot be created in #n unless they were present initially plays the crucial role in the existence theory developed in [62]. Here we go a step further by showing that the amplitude of possible oscillations decays with time at uniform rate depending solely on the value of the initial energy E0 (see [37]). In particular, the time images of bounded energy initial data are asymptotically compact with respect to the density component. This is precisely what is needed to develop a meaningful dynamical systems theory associated to the problem. In accordance with our hypotheses, we can show γ in C I ; Lweak (Ω) , Tk (#n ) → Tk (#) in C I ; Lαweak (Ω) for any α 1, k 1, un → u weakly in L2 I ; W01,2 (Ω) . #n → #
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To measure the amplitude of oscillations of the sequence #n , we introduce a defect measure dft, dft[#n − #](t) =
where ν = # log(#) − # log(#).
ν(t, x) dx,
(5.17)
Ω
By virtue of Corollary 5.1, both #n and # are renormalized solutions of (5.3) and we have #n log(#n ) → # log(#) in C J ; Lαweak(Ω) , 1 α < γ , # log(#) ∈ C J ; Lαweak(Ω) , 1 α < γ . Consequently, dft[#n − #] is a continuous function of t ∈ I . Mainly for technical reasons, we are not able to deal directly with the function dft. We consider instead a family of approximate functions: Lk (z) =
z log(z) z log(k) + z
k z
for 0 z k, Tk (s)/s 2 ds
for z k.
It is easy to observe that Lk (z) = βk z + bk (z) where bk satisfy (3.3) and Lk (z)z − Lk (z) = Tk (z). Since both #n , # are renormalized solutions of (5.3) on I × R 3 , we deduce
Lk (#n ) − Lk (#) (t2 ) dx −
Ω
=
Ω
t2 t1
+
Lk (#n ) − Lk (#) (t1 ) dx
Tk (#) div u − Tk (#) div un dx dt Ω
t2 t1
Tk (#) − Tk (#n ) div un dx dt
Ω
for any t1 , t2 ∈ I . Letting n → ∞ and using Proposition 4.3 together with (5.15), we obtain
Lk (#) − Lk (#) (t2 ) dx −
Ω
t2
Lk (#) − Lk (#) (t1 ) dx
Ω
Tk (#n ) − Tk (#)γ +1 dx dt
a lim sup λ + 2µ n→∞ t1 Ω t2 Tk (#) − Tk (#) div u dx dt +
t1
Ω
for any t1 t2 , t1 , t2 ∈ I.
(5.18)
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Our aim now is to pass to the limit for k → ∞ in (5.18). Clearly, Lk (#) − Lk (#) (t) dx dt → dft[#n − #](t) for k → ∞ Ω
while $ $ $Tk (#) − Tk (#) $ 2 L (I ×Ω) $ $β 1−β $Tk (#) − Tk (#) $L1 (I ×Ω) oscγ +1 [#n − #](I × Ω) ,
β=
γ −1 . 2γ
By virtue of Proposition 5.3, the right-hand side of the above inequality tends to zero for k → ∞ and so does the right-hand side of (5.18). Finally, we have t2 Tk (#n ) − Tk (#)γ +1 dx dt lim sup n→∞
t1
|Ω|
Ω
α−γ +1 α
lim sup n→∞
$ $Tk (#n ) − Tk (#)$γ +1 dx dt, Lα (Ω)
t2 $
t1
and (5.18) yields: dft[#n − #](t2 ) − dft[#n − #](t1 ) α−γ +1 t2 a|Ω| α γ +1 + #n − # Lα (Ω) dx dt 0 lim sup λ + 2µ n→∞ t1 for any t1 t2 , t1 , t2 ∈ I and 1 α < γ . To conclude, we shall need the following auxiliary result (cf. [28, Lemma 2.1]). L EMMA 5.1. Given α ∈ (1, γ ) there exists c = c(α) such that z log(z) − y log(y) 1 + log+ (y) (z − y) + c(α) |z − y|1/2 + |z − y|α for any y, z 0. In accordance with Lemma 5.1, we can write 1 + log+ (#) (#n − #) dx #n log(#n ) dx − Ω
c(α) |Ω|
Ω 2α−1 2α
α #n − # 2α Lα (Ω) + #n − # Lα (Ω)
which together with (5.19) yields dft[#n − #](t2 ) − dft[#n − #](t1 ) +
t2 t1
Φ dft[#n − #](t) dt 0,
(5.19)
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where the nonlinear function Φ depends only on the structural properties of the logarithm and can be chosen independently of the data to satisfy Φ : R → R
is continuous and strictly increasing, Φ(0) = 0.
(5.20)
Summing up the above considerations we have arrived at the following conclusion: P ROPOSITION 5.4. Let Ω ⊂ R N , N 2 be a bounded Lipschitz domain and I ⊂ R a bounded interval. Let #n , un be a sequence of finite energy weak solutions of the problem (5.3)–(5.5) on I × Ω, where pressure p is given by the isentropic constitutive relation p = a#γ ,
a > 0, γ >
N , 2
and f = fn . Let lim sup E #n , (#n un ) (t) E0 ,
t →inf{I }+
fn L∞ (I ×Ω) F
independently of n. Let # be a weak limit of the sequence #n . Then dft[#n − #](t2 ) χ(t2 − t1 )
for any t1 , t2 ∈ I, t1 t2 ,
where χ is the unique solution of the initial-value problem χ (t) + Φ χ(t) = 0,
χ(0) = dft[#n − #](t1 )
and Φ is a fixed function satisfying (5.20). It can be shown that Φ has a polynomial growth for values close to zero and, consequently, the quantity dft[#n − #](t) behaves like t −β for a certain β > 0 when t → ∞.
5.5. Approximate solutions The a priori estimates derived in Sections 5.1, 5.2 together with the compactness results in Propositions 4.2, 5.4 form a suitable platform for a larga data existence theory for the problem (5.3)–(5.5). The final task, as usual, is to find a suitable approximation scheme compatible with both a priori estimates and the compactness results claimed above. Needless to say there are many ways to do it. Here we pursue the approach of [32] and consider the approximate problem: ∂# + div(#u) = ε #, ∂t
(5.21)
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∂#u + div(#u ⊗ u) + ∇p(#) + δ∇#β + ε∇u · ∇# ∂t = µ u + (λ + µ)∇ div u + #f
(5.22)
complemented by the boundary conditions u|∂Ω = ∇# · n|∂Ω = 0.
(5.23)
The parameters ε > 0, δ > 0 are “small” and β > 0 “large”. The system (5.21)–(5.23) can be solved by means of the standard Faedo–Galerkin method to obtain approximate solutions #ε,δ , uε,δ (cf. [32, Proposition 2.1]). Then one can pass to the limit, first for ε → 0 and then for δ → 0, to obtain a finite energy weak solution of the problem (5.3)– (5.5) (see [32]). The reason for introducing two parameters ε and δ is that the energy estimates presented in Section 5.1 and the pressure estimates in Section 5.2 are compatible only if β > N . An alternative approach is the approximation scheme introduced by Lions [62] or the method of time-discretization based on solving a family of stationary problems (see also Lions [62]). 6. Barotropic flows: large data existence results The mathematical theory presented in Section 5 can be used to obtain rigorous existence results for barotropic flows with essentially no restriction on the size of the data. We start with a very particular case posed in two space dimension where one can show even existence of strong (classical) solutions. 6.1. Global existence of classical solutions The result we are going to present is due to Vaigant and Kazhikhov [107]. Consider the system ∂# + div(#u) = 0, ∂t
(6.1)
∂(#u) + div(#u ⊗ u) + a∇#γ = µ u + ∇ (λ(#) + µ) div u , ∂t
(6.2)
where (t, x) ∈ (0, T ) × R 2 . The functions #, u are for simplicity considered spatially periodic, i.e., #(t, x + ω) = #(t, x),
u(t, x + ω) = u(t, x).
(6.3)
The problem is complemented by the initial conditions #(0, x) = #0 (x) # > 0,
u(0, x) = u0 (x).
(6.4)
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Under the hypotheses µ > 0,
a > 0,
γ 0,
and λ(#) = b#β ,
b > 0, β 3,
(6.5)
Vaigant and Kazhikhov [107] proved the following result. T HEOREM 6.1. In addition to the hypotheses (6.5), let 2 #0 ∈ L∞ per R ,
2 1,2 u0 ∈ Wper R .
Then the initial-value problem (6.1)–(6.4) possesses a global (T = ∞) weak solution. The continuity equation (6.1) holds in D ((0, T ) × R 2 ) and the equations of motion (6.2) are satisfied a.a. on (0, T ) × R 2 . If, moreover, 1,q #0 ∈ Wper R 2 ,
2,q u0 ∈ Wper R 2
for some q > 2,
then there is a unique strong solution satisfying the equations a.e. on (0, T ) × R 2 . Finally, if 2 1+α #0 ∈ Cper R ,
2 2+α u0 ∈ Cper R
for some α > 0,
then the strong solution is classical (smooth). Theorem 6.1 is a remarkable result since it solves both the problem of existence and uniqueness as well as regularity of solutions. The obvious restrictions of applicability are due to the rather unnatural hypotheses (6.5), i.e., the viscosity coefficient µ must be constant while λ depends on # in a very specific way. The proof of Theorem 6.1 is based on very strong a priori estimates – much better than presented in Sections 5.1, 5.2. These estimates are available thanks to the particular form of the constitutive relations and the fact the problem is posed in two space dimensions.
6.2. Global existence of weak solutions We consider the problem (5.3)–(5.5) posed on a bounded regular domain Ω ⊂ R N , N = 2, 3. We prescribe the initial conditions #(0) = #0 0,
(#u)(0) = q0
(6.6)
satisfying a compatibility condition q0 (x) = 0 whenever #0 (x) = 0
(6.7)
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E. Feireisl
and such that #0 ∈ Lγ (Ω),
|q0 |2 ∈ L1 (Ω). #0
(6.8)
The assumption (6.8) is nothing else but the requirement the initial data to be of finite energy. The following theorem asserts the existence of the finite energy weak solutions to the problem (5.3)–(5.5), (6.6) introduced in Section 5. T HEOREM 6.2. Let Ω ⊂ R N , N = 2, 3, be a bounded regular domain and T > 0 given. Consider the system (5.3), (5.4) complemented by (5.5), (6.6), where p is given by the isentropic constitutive law (5.2) with γ>
N , 2
f is a bounded measurable function on (0, T ) × Ω, and the initial data #0 , q0 satisfy (6.7), (6.8). Then the problem (5.3)–(5.5) posseses a finite energy weak solution #, u on (0, T ) × Ω satisfying the initial conditions (6.6). As already remarked in (5.7), both the density # and the momenta (#u) are continuous functions of t with respect to the Lp -weak topology, and, consequently, the initial conditions (6.6) make sense. The existence result stated in Theorem 6.2 was first proved by Lions [62] for γ 3/2 if N = 2 and γ 9/5 for N = 3. The proof needs some modifications presented in [39] and [63] to accommodate the Dirichlet boundary conditions. The present version including the full range of γ > N/2 was shown in [32, Theorem 1.1]. Given the weak compactness results, namely Propositions 4.1, 4.2, for solutions of (5.3), (5.4) respectively, the main ingredient of the proof of Theorem 6.2 is the strong compactness of the density stated in Proposition 5.4, the proof of which requires, among other things, convexity of the pressure. It is easy to see, however, that the same result can be obtained for a general barotropic pressure p as in (5.1) that can be written in the form p(#) = a#γ + p0 (#),
a > 0, γ > N/2,
where p0 is a globally Lipschitz function. Note that the proof for the range γ 9/5, N = 3, γ 3/2, N = 2 can be modified to include a general constitutive law (5.1) where p(#) a#γ for all # large enough (cf. Lions [62]). It seems interesting to note that the physically relevant isothermal case where γ = 1 seems to be completely open even if N = 2. The only large data existence result is that of Hoff [46] where the initial data (as well as the solutions) are radially symmetric. The general case γ 1, N = 3 for radially symmetric data was solved only recently by Jiang and Zhang [55].
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6.3. Time-periodic solutions Similarly as above, we consider the system (5.3)–(5.5) driven by a volume force f which is periodic in time, i.e., f is a bounded measurable vector function on R × Ω satisfying f(t + ω, x) = f(t, x) for a.a. t ∈ R, x ∈ Ω for a certain period ω > 0. We are interested in the existence of a finite energy weak solution #, u enjoying the same property, i.e., #(t + ω) = #(t),
(#u)(t + ω) = (#u)(t)
for all t ∈ R
and such that # dx = m, Ω
where m is a given positive total mass. There are three main obstacles making this problem rather delicate. Given the existence results for the initial-boundary value problem presented above, only weak solutions are available, for which the question of uniqueness is highly nontrivial and far from being solved. This excludes all the so-called indirect methods based on fixed-point arguments for the corresponding period map. While the former difficulty might seem only technical, there is another feature of the problem, mentioned already in Section 5.4 namely, there is no “solution operator” or “period map” which would be compact due to possible time propagation of oscillations in the density. Last but not the least, fixing the total mass m, we have to look for solutions lying on a sphere in the space L1 which excludes the possibility of using any fixed-point technique in a direct fashion. In the light of the above arguments, the only possibility to get positive results is to work directly in the space of periodic solutions that means to consider a genuine boundary-value problem for the evolutionary system (5.3), (5.4). This approach has been used in [31] to prove the existence of the time periodic solutions to (5.3), (5.4) on a cube in R 3 complemented by the no-stick boundary conditions (1.12). Combining the method of [31] with the existence theory [32] one can prove the following result. T HEOREM 6.3. Let Ω ⊂ R N , N = 2, 3, be a bounded regular domain. Consider the problem (5.3)–(5.5) where p is given by the isentropic constitutive law (5.2) with γ > 5/3
if N = 3,
γ >1
for N = 2,
and f is a bounded measurable function on R × Ω such that f(t + ω, x) = f(t, x) for a.a. t ∈ R, x ∈ Ω and a certain ω > 0.
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E. Feireisl
Then, given m > 0, there exists a finite energy weak solution #, u of (5.3)–(5.5) on R × Ω such that #(t + ω) = #(t),
(#u)(t + ω) = (#u)(t)
for all t ∈ R
and # dx = m. Ω
The condition γ > 5/3 in the three-dimensional case seems rather strange compared with γ > 3/2 required for solving the initial-value problem. This is related to the problem of ultimate boundedness or resonance phenomena for global in time solutions. We will discuss this interesting topic in the next section.
6.4. Counter-examples to global existence It is not clear to which extent the hypothesis γ > N/2 is really necessary for global existence results. Several attempts have been made to show that the barotropic model does not admit globally defined strong or even weak solutions but the results are still not very convincing in either positive or negative sense. Following the method of Vaigant [106], Desjardins [19, Proposition 1] studied the integrability properties of the density # in the system (5.3)–(5.5). P ROPOSITION 6.1. Let Ω = B(1) ⊂ R 3 be a unit ball and let p satisfy (5.2) with 1 < γ < 3. Let q>
11γ − 2 . 6 2γ
Then there exist f ∈ L1 (0, T ; L γ −1 (Ω)) and a globally defined weak solution #, u of (5.3)–(5.5) such that T 0
#(t, x)q dx dt = ∞. B(1/2)
The weakness of this result stems from the necessity to use the forcing term f which is singular at t = T . It is still an open problem whether or not the uniform upper bounds on the density can be obtained independently of the choice of γ .
6.5. Possible generalization We shall comment shortly on possible improvements of Theorem 6.1 lying in the scope of the present theory.
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To begin with, Theorem 6.1 still holds when Ω is a general (not necessarily bounded) domain with compact boundary on which the no-slip boundary conditions for the velocity are prescribed. As far as the other boundary conditions discussed in Section 1.4 are concerned, the possibility to show positive existence results seems to be closely related to the question of the boundary estimates of the pressure discussed in Section 5.2. Similarly, the hypothesis that f is bounded can be replaced by a more general condition 2γ
f ∈ L1 (I ; L γ −1 (Ω)). Other possibilities and suggestions are discussed by Lions [62].
7. Barotropic flows: asymptotic properties Similarly as in the preceding section, we focus on the system (5.2)–(5.5) considered on a bounded regular domain Ω ⊂ R N , N = 2, 3. We shall assume that the driving force f is a bounded measurable function defined, for simplicity, for all t ∈ R, x ∈ Ω such that f(t, x) F
for a.a. t ∈ R, x ∈ Ω.
(7.1)
In accordance with Section 5, the total energy defined as E[#, #u](t) =
a 1 |#u|2 (t) + #γ (t) dx γ −1 #(t )>0 2 #
is a lower-semicontinuous function of t.
7.1. Bounded absorbing balls and stationary solutions We shall address the problem of ultimate boundedness of global in time finite energy weak solutions, the existence of which is guaranteed by Theorem 6.2. We shall see that the total energy E is the right quantity to play the role of a “norm” in these considerations. If the driving force f is uniformly bounded as in (7.1), the “dynamical system” generated by the finite energy weak solutions of the problem (5.3)–(5.5) is ultimately bounded or dissipative in the sense of Levinson with respect to the energy “norm” provided that the adiabatic constant satisfies γ > 1 for N = 2 and γ > 5/3 if N = 3. Specifically, we report the following result (see [38, Theorem 1.1]), the proof of which is based on the pressure estimates obtained in Proposition 5.2: T HEOREM 7.1. Let Ω ⊂ R N , N = 2, 3, be a bounded Lipschitz domain and I ⊂ R an interval such that inf{I } > −∞. Consider the system (5.3)–(5.5) with the isentropic pressure p given by (5.2) with γ >1
if N = 2,
and f satisfying (7.1).
γ > 5/3 for N = 3,
(7.2)
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E. Feireisl
Then there exists a constant E∞ , depending solely on the amplitude of the driving force F and the total mass m, with the following property: Given E0 , there exists a time T = T (E0 ) such that E #, (#u) (t) E∞
for all t ∈ I, t > T + inf{I }
for any #, u – a finite energy weak solution of the problem (5.3)–(5.5) – satisfying lim sup E[#, u](t) E0 ,
t →inf{I }+
# dx = m. Ω
It seems interesting to compare Theorem 7.1 with the result of Lions [62, Theorem 6.7] on the existence of stationary solutions of (5.3)–(5.5) to shed some light on the role of the hypothesis (7.2). T HEOREM 7.2. Let Ω ⊂ R N , N = 2, 3, be a bounded regular domain, f = f(x) a function belonging to L∞ (Ω), and m > 0. Assume p = p(#) is given by (5.2) with γ satisfying (7.2). Then there exists a pair of functions # = #(x) ∈ Lp (Ω), p > γ , u = u(x) ∈ W01,2 (Ω) solving the stationary problem div(#u) = 0, div(#u ⊗ u) + a∇#γ = µ u + (λ + µ)∇ div u + #f, # dx = m Ω
in D (Ω). As we will see later, Theorem 7.2 can be deduced from Proposition 5.4, Theorem 6.3, and Theorem 7.1. The property stated in Theorem 7.1 is evidence of the dissipative nature of the system (5.3), (5.4). In finite-dimensional setting, J.E. Billoti and J.P. LaSalle proposed it as a definition of dissipativity. Unfortunately, however, some difficulties inherent to infinitedimensional dynamical systems make it, in that case, less appropriate.
7.2. Complete bounded trajectories ∞ We suppose that the driving force f belongs to F – a bounded subset of L∞ loc (R; L (Ω)). To bypass the possible problem of non-uniqueness of finite energy weak solutions, we introduce a quantity U (t0 , t) playing the role of the evolution operator related to the
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351
problem (5.3)–(5.5). U [E0 , F ](t0 , t) =
/
#(t), (#u)(t) | #, u is a finite energy weak solution of the problem (5.3)–(5.5) defined on an open interval I, (t0 , t] ⊂ I, with f ∈ F
0 and such that lim sup E[#, u](t) E0 . t →t0 +
We start with the concept of the so-called short trajectory in the spirit of Málek and Neˇcas [67]: / U s [E0 , F ](t0 , t) = #(t + τ ), (#u)(t + τ ) , τ ∈ [0, 1] | #, u is a finite energy weak solution of the problem (5.3)–(5.5) on an open interval I, (t0 , t + 1] ⊂ I, with f ∈ F ,
0 and such that lim sup E(t) E0 . t →t0
The following result can be viewed as a corollary of Proposition 5.4 and Theorem 7.1 (cf. [37, Theorem 1.1] or [27, Proposition 10]). P ROPOSITION 7.1. Let Ω ⊂ R N , N = 2, 3, be a bounded domain with Lipschitz boundary. Let the pressure p be given by (5.2) with γ >1
for N = 2,
γ > 5/3 if N = 3.
Let F be bounded in L∞ (R × Ω). Consider a sequence [#n , (#n un )] ∈ U s [E0 , F ](a, tn ) for a certain tn → ∞. Then there is a subsequence (not relabeled) such that #n → # in Lγ (0, 1) × Ω and in C [0, 1]; Lα (Ω) for 1 α < γ , 2γ
#n un → (#u)
γ +1 in Lp ((0, 1) × Ω) and in C([0, 1]; Lweak (Ω)) for any 1 p <
E #n , (#n un ) → E #, (#u)
2γ γ +1 ,
and
in L1 (0, 1),
where #, u is a finite energy weak solution of the problem (5.3)–(5.5) defined on the whole real line I = R such that E ∈ L∞ (R) and with f ∈ F + where / F + = f | f = lim hn (· + τn ) weak star in L∞ (R × Ω) τn →∞
0 for a certain hn ∈ F and τn → ∞ .
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E. Feireisl
Proposition 7.1 shows the importance of the complete bounded trajectories, i.e., the finite energy weak solutions defined on I = R whose total energy E is uniformly bounded on R. Let us define As [F ] =
%
#(τ ), (#u)(τ ) , τ ∈ [0, 1] | #, u is a finite energy weak solution of the problem (5.3)–(5.5) on the interval I = R, & with f ∈ F + and E #, (#u) ∈ L∞ (R) .
The next statement is a straighforward consequence of Proposition 7.1 (see also [28, Theorem 3.1]). T HEOREM 7.3. Let Ω ⊂ R N be a bounded Lipschitz domain. Let p be given by (5.2) with γ >1
for N = 2,
γ > 5/3 if N = 3.
Let F be a bounded subset of L∞ (R × Ω). Then the set As [F ] is compact in Lγ ((0, 1) × Ω) × [Lp ((0, 1) × Ω)]3 and 1 sup
[#,#u]∈U [E0 ,F ](t0 ,t )
inf
# − # ¯ Lγ ((0,1)×Ω)
[#, ¯ #¯ u¯ ]∈As [F ]
$ $ 2 + $(#u) − (#¯ u¯ )$Lp ((0,1)×Ω) → 0 as t → ∞
for any 1 p < 2γ /(γ + 1). Theorem 7.3 says that the set As [F ] is a global attractor on the space of “short” trajectories. This is a result in the spirit of Málek and Neˇcas [67] or Sell [89]. In particular, the set As [F ] is compact non-empty provided F is non-empty. Consider the special case when f = f(x) is a driving force independent of time. Accordingly, we can take F = F + = {f}. By virtue of Theorem 6.3, the problem (5.3)–(5.5) possesses a time-periodic solution #n , un for any period ωn = 2−n such that #n dx = m. Ω
Moreover, Theorem 7.1 implies that the restriction #n , #n un to the time interval [0, 1] belongs to As . As As is compact, the sequence #n , un has an accummulation point which is a complete global solution of (5.3)–(5.5). Moreover, this solution is clearly independent of t, i.e., it is a stationary solution of a given total mass m. In other words, we have proved Theorem 7.2.
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7.3. Potential flows We shall examine the flows driven by a potential force, i.e., we assume f = f(x) = ∇F (x), where F is a Lipschitz continuous function. In this case, the term on the right-hand side of the energy inequality (5.6) can be rewritten as dH , where H(t) = (#u) · ∇F dx = #F dx, dt Ω Ω and, consequently, (5.6) reads as follows: d E(t) − H(t) + µ|∇u|2 + (λ + 2µ)|div u|2 dx dt 0. dt Ω
(7.3)
We denote EH∞ = ess lim E(t) − H(t) . t →∞
By virtue of (7.3) and the Poincaré inequality, the integral ∞ u 2 1,2 dt is convergent, W0 (Ω)
1
in particular,
lim
T →∞ T
and
T +1
Ekin(t) dt = 0,
T +1
lim
T →∞ T
Ω
Ekin =
1 2
(7.4)
#|u|2 dx, Ω
a #γ − #F dx dt = EH∞ . γ −1
(7.5)
Similarly as in Proposition 7.1, one can show that any sequence tn → ∞ contains a subsequence such that
tn +1
#(t) − #s Lγ (Ω) dt → 0,
tn
where, in view of (7.4), (7.5), #s is a solution of the stationary problem γ a∇#s = #s ∇F in Ω, #s dx = m, Ω
Ω
a γ #s − #s F dx = EH∞ . γ −1
(7.6)
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E. Feireisl
Consequently, it is of interest to study the structure of the set of the static solutions, i.e., the solutions of the problem (7.6); in particular, whether or not they form a discrete set. If this is the case, any finite energy weak solution of (5.3)–(5.5) is convergent to a static state. A partial answer was obtained in the case of potentials with at most two “peaks” ([36, Theorem 1.1] and [33, Theorem 1.2]). T HEOREM 7.4. Let Ω ⊂ R N be an arbitrary domain. (i) Assume F is locally Lipschitz continuous on Ω and such that all the upper level sets % & [F > k] = x ∈ Ω | F (x) > k are connected in Ω for any k. Then given m > 0, the problem (7.6) possesses at most one nonnegative solution #s . (ii) If F is locally Lipschitz continuous and Ω can be decomposed as Ω = Ω 1 ∪ Ω 2,
Ω1 ∩ Ω2 = ∅,
where Ωi are two subdomains (one of them possibly empty) so that [F > k] ∩ Ωi
is connected in Ωi for i = 1, 2 for any k ∈ R,
(7.7)
then, given m, EH∞ , the problem (7.6) admits at most two distinct non-negative solutions. Making use of Theorem 7.4, one can show the following result on stabilization of global solutions for potential flows (cf. [34, Theorem 1.1], [27, Theorem 15]). T HEOREM 7.5. Let Ω ⊂ R N , N = 2, 3, be a bounded Lipschitz domain. Let the pressure p satisfy the constitutive relation (5.2) with γ > N/2. Let f = f(x) = ∇F (x) where F is globally Lipschitz potential on Ω. Moreover, assume that Ω can be decomposed as in Theorem 7.4 so that (7.7) holds. Then for any finite energy weak solution #, u of the problem (5.3)–(5.5) defined on a time interval I = (t0 , ∞), there exists a solution #s of the stationary problem (7.6) such that #(t) → #s
strongly in Lγ (Ω) as t → ∞,
Ekin(t) dx = Ω
1 2
|#u|2 (t) dx → 0 #(t )>0 #
as t → ∞.
The conclusion of Theorem 7.5 still holds if Ω is a general (not necessarily bounded) domain with compact boundary provided F satisfies the stronger hypothesis of Theorem 7.4, namely, all upper level sets [F > k] must be connected. Similar problems on the exterior of an open ball and for radially symmetric solutions were investigated by Matuš˚u-Neˇcasová et al. [76]. Related results can be found in Novotný and Straškraba [83,84].
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It is an interesting open problem if the conclusion of Theorem 7.5 still holds when the hypothesis on the upper level sets of F is relaxed. If Ω ⊂ R N is a bounded domain and the potential F nontrivial (nonconstant), there always exists an m – the total mass – small in comparison with F such that the solutions of the static problem (7.6) contain vacuum zones (cf. [34, Section 5]). Thus for any nonconstant F the global solutions approach rest states with vacuum regions as time goes to infinity. One should note in this context there are many formal results on convergence of isentropic flows to a stationary state under various hypotheses including uniform (in time) boundedness of the density away from zero (see, e.g., Padula [86]). As we have just observed, this can be rigorously verified only for solutions representing small perturbations of strictly positive rest states.
7.4. Highly oscillating external forces There seems to be a common belief that highly oscillating driving forces of zero integral mean do not influence the long-time dynamics of dissipative systems. Averaging a function over a short time interval should be considered analogous to making a macroscopic measurement in a physical experiment. The result of such an experiment being close to zero, the effect on the solutions to a sufficiently robust dynamical systems, if any, should be negligible at least in the long run. From the mathematical point of view, these ideas have been made precise by Chepyzhov and Vishik [10] dealing with trajectory attractors of evolution equations. They showed that the trajectory attractors of certain dissipative dynamical systems perturbed by a highly oscillating forcing term are the same as for the unperturbed system. Their results apply to a vast set of equations including the damped wave equations and the Navier–Stokes equations for incompressible fluids. Our goal now is to present comparable results for the problem of isentropic compressible flows dynamics. Highly oscillating sequences converge in the weak topology, i.e., the topology of convergence of integral means. Consider a ball BG of radius G centered at zero in the space L∞ ((0, 1) × Ω). The weak-star topology on BG is metrizable and we denote the corresponding metric dG . We report the following result (see [30, Theorem 1.2]). T HEOREM 7.6. Assume Ω ⊂ R N , N = 2, 3 is a bounded Lipschitz domain. Consider the system (5.3)–(5.5) where the pressure p is given by (5.2) with γ >1
for N = 2,
γ > 5/3 if N = 3,
and f(t, x) = ∇F (x) + g(t, x), where F is globally Lipschitz continuous and such that the upper level sets [F > k] are connected for any k. Then given G > 0, ε > 0 there exists δ = δ(G, ε) > 0 such that $ $ lim sup #(t) − #s Lγ (Ω) + $#u(t)$L1 (Ω) < ε t →∞
356
E. Feireisl
for any finite energy weak solution #, u of the problem (5.3)–(5.5) provided $ $ lim sup $g(t)$L∞ ((t,∞)×Ω) < G, t →∞ lim sup dG g(t + s)|s∈[0,1], 0 < δ. t →∞
Here #s is the unique solution of the stationary problem (7.6). 7.5. Attractors For a general dynamical system a set A is called a global attractor if it is compact, attracting all trajectories, and minimal in the sense of inclusion in the class of sets having the first two properties. The theory of attractors for incompressible flows is well developed. We refer the reader to the monographs of Babin and Vishik [5], Hale [44], and Temam [104] for this interesting subject. A global or universal attractor describes all possible dynamics of a given system, and, as an aspect of dissipativity, the attractor usually has a finite fractal dimension. There seems to be at least one essential problem to develop a sensible dynamical systems theory for compressible fluids, namely, the finite energy weak solutions we deal with are not known to be uniquely determined by the initial data. On the other hand, the notion of global attractor itself does not require uniqueness or even the existence of a “solution semigroup” and plausible results in this respect can be obtained. Let % A[F ] = #(0), (#u)(0) | #, u is a finite energy weak solution & of the problem (5.3)–(5.5) on I = R with f ∈ F + and E ∈ L∞ (R) . Roughly speaking, the set A contains all global and globally bounded trajectories where global means defined on the whole time axis R. The next statement shows that A[F ] is a global attractor in the sense of Foias and Temam [40] (cf. [28, Theorem 4.1]). T HEOREM 7.7. Let Ω ⊂ R N , N = 2, 3, be a bounded Lipschitz domain, and let p be given by (5.2) with γ >1
if N = 2,
γ > 5/3
for N = 3.
Let F be a bounded subset of L∞ (R × Ω). 2γ
γ +1 (Ω) and Then A[F ] is compact in Lα (Ω) × Lweak
inf # − # ¯ Lα (Ω) + (#u − #¯ u¯ ) · φ dx → 0 sup
¯ #¯ u]∈ ¯ A[F ] [#,#u]∈U [E0 ,F ](t0 ,t ) [#,
as t → ∞ 2γ
for any 1 α < γ and any φ ∈ [L γ −1 (Ω)]3 .
Ω
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The apparent shortcomming of this result is that A is only a “weak” attractor with respect to the momentum component. Pursuing the idea of Ball [6], one can show a stronger result on condition that some additional smoothness of A is known (see [27, Theorem 17]). T HEOREM 7.8. In addition to the hypotheses of Theorem 7.7, assume the total energy E defined by (5.9) and considered as a function the density # and the momenta #u is (sequentially) continuous on A[F ], specifically, for any sequence [#n , qn ] ∈ A[F ] such that #n → # in L1 (Ω), qn → q weakly in L1 (Ω) one requires E[#n , qn ] → E[#, q]. Then 1 sup
inf
2 ¯ L1 (Ω) → 0 # − # ¯ Lγ (Ω) + #u − #¯ u
¯ #¯ u¯ ]∈A[F ] [#,#u]∈U [E0 ,F ](t0 ,t ) [#,
as t → ∞. 7.6. Bibliographical remarks The existence of global attractors for the problem (5.3)–(5.5) with γ = 1 and N = 1 was studied by Hoff and Ziane [50,51]. In this case, any forcing term f is of potential type so the only situation which is not covered by Section 7.3 is the case when f is time dependent. Similar results for the full system (1.1)–(1.3), still in one space dimension, were obtained recently by Zheng and Qin [112].
8. Compressible–incompressible limits It is well-accepted in fluid mechanics that one can derive formally incompressible models as Navier–Stokes equations from compressible ones. Such a situation can be expected when letting the Mach number go to zero in the isentropic compressible Navier–Stokes equations. Following Lions and Masmoudi [64] we consider a system ∂#ε + div(#ε uε ) = 0, ∂t
(8.1)
∂#ε uε a + div(#ε uε ⊗ uε ) + 2 ∇#εγ = µε uε + (λε + µε )∇ div uε ∂t ε
(8.2)
complemented by the initial conditions #ε (0) = #ε0 0,
(#ε uε )(0) = qε
(8.3)
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E. Feireisl
satisfying (6.7). We shall always assume µε → µ > 0,
λε → λ > −µ as ε → 0.
8.1. The spatially periodic case In addition to the above hypotheses, assume the initial data are spatially periodic as in (6.3). Moreover, let qε → U0 #ε0
weakly in L2per (R N ) as ε → 0,
(8.4)
|qε |2 1 + 2 (#ε0 )γ − γ #ε0 (m0ε )γ −1 + (γ − 1)(m0ε )γ dx c 0 #ε ε
(8.5)
and
ω1
ωN
···
0
0
where m0ε
=
N 3
−1
ω1
ωi
0
i=1
ωN
··· 0
#ε0 dx → 1
as ε → 0
independently of ε. Let us denote, as usual, the total energy E #ε , (#ε uε ) =
{#ε >0}
1 |#ε uε |2 a (#ε )γ dx. + 2 2 #ε ε (γ − 1)
The following result is due to Lions and Masmoudi [64]: T HEOREM 8.1. Assume γ > N/2. Let #ε , uε be a (spatially periodic) finite energy weak solution of the problem (8.1)–(8.3) on the time interval (0, ∞) where the data satisfy (8.4), (8.5). Moreover, let
ω1
ess sup E #ε , (#ε uε ) t →0+
ωN
···
0
0
0 γ 1 |qε |2 a # + 2 dx. 0 2 #ε ε (γ − 1) ε
Then γ #ε → 1 in C [0, T ]; Lper R 2 1,2 and uε is bounded in L2 (0, T ; Wper (Ω)) for arbitrary T > 0.
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Moreover, passing to a subsequence as the case may be we have uε → U
2 1,2 weakly in L2 0, T ; Wper R ,
where U solves the incompressible Navier–Stokes equations ∂t U + div(U ⊗ U) = µ U + ∇P ,
div U = 0
(8.6)
with the initial condition U(0) = PU0 where P is the projection on the space of divergencefree functions.
8.2. Dirichlet boundary conditions Now we focus on the system (8.1)–(8.3) posed on a bounded domain Ω ⊂ R N and complemented by the no-slip boundary conditions for the velocity: uε |∂Ω = 0.
(8.7)
Consider the following (overdetermined) problem: − Φ = νΦ
in Ω,
∇Φ · n|∂Ω = 0,
Φ constant on ∂Ω.
(8.8)
A solution of (8.8) is trivial if ν = 0 and Φ is a constant. The domain Ω will be said to satisfy condition (H) if all solutions of (8.8) are trivial. The following result was proved by Desjardins et al. [20]: T HEOREM 8.2. Let Ω ⊂ R N , N = 2, 3, be a bounded regular domain. In addition to the hypotheses of Theorem 8.1, assume that uε satisfies the no-slip condition (8.7). Then #ε tends to 1 strongly in C([0, T ]; Lγ (Ω)) and, passing to a subsequence if necessary, uε → U
weakly in L2 (0, T ) × Ω
for all T > 0 and the convergence is strong if Ω satisfies condition (H). In addition, U satisfies the incompressible Navier–Stokes system (8.6) complemented by the no-slip boundary conditions on ∂Ω and with U(0) = PU0 . 8.3. The case γn → ∞ Let us consider the isentropic system in the case when γ = γn → ∞. We follow the presentation of Lions and Masmoudi [65]. Let Ω ⊂ R 3 be a bounded regular domain. Consider the system ∂t #n + div(#n un ) = 0,
(8.9)
360
E. Feireisl γ
∂t (#n un ) + div(#n un ⊗ un ) + a∇#nn = µ un + (λ + µ)∇ div un
(8.10)
with the no-slip boundary conditions for the velocity un |∂Ω = 0
(8.11)
and complemented by the initial conditions #n (0) = #n0 0,
(#n un )(0) = qn ,
(8.12)
where 0 γ n # γ
n L n (Ω)
cγn ,
#n bounded in L1 (Ω),
|qn |2 bounded in L1 (Ω), #0
(8.13)
independently of n. We are interested in the limit of the sequence #n , un of finite energy weak solutions of the problem (8.9)–(8.12) when γn → ∞. To this end, let us first formulate the limit problem: ∂t # + div(#u) = 0,
0 # 1,
(8.14)
∂t (#u) + div(#u ⊗ u) + ∇P = µ u + (λ + µ)∇ div u,
(8.15)
div u = 0 a.a. on the set {# = 1},
(8.16)
P = 0 a.a. on {# < 1},
P 0
a.a. on {# = 1}.
(8.17)
The following result is due to Lions and Masmoudi [65]. T HEOREM 8.3. Let Ω ⊂ R 3 be a bounded regular domain. Let #n , un be a sequence of finite energy weak solutions of the problem (8.9)–(8.12) on (0, T ) × Ω where the data satisfy (8.13). Let #n0 converge weakly to some #0 and qn converge weakly to q. Then #n , un contain subsequences such that [#n − 1]+ → 0 in L∞ 0, T ; Lα (Ω) for any 1 α < ∞, #n → #
weakly star in L∞ 0, T ; Lα (Ω) , 1 α < ∞,
where 0 # 1. γ
Moreover, #nn is bounded in L1 ((0, T ) × Ω) and γ
#nn → P
weakly star in M (0, T ) × Ω .
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If, in addition, #n0 → #0 strongly in L1 (Ω), then #, u, P solve the problem (8.14)–(8.17) in D ((0, T ) × Ω) where u is a weak limit of un in L2 (0, T , W01,2 (Ω)). Here M denotes the space of Radon measures.
9. Other topics, directions, alternative models 9.1. Models in one space dimension In the above analysis, we have systematicaly and deliberately avoided the case of one space dimension. Note that for compressible fluids such a situation can be physically relevant as well as interesting. From the mathematical point of view, these problems exhibit a rather different character due to the particularly simple topological structure of the underlying spatial domain. The question of global existence is largely settled in the case of one space dimension. The basic result in this direction is that of Kazhikhov [56], a more extensive material can be found in the monograph of Antontsev et al. [4]. The discontinuous (weak) solutions were studied by Hoff [45], Serre [90,91] and Shelukhin [95]. The results are quite satisfactory with respect to the criteria of well-posedness discussed in Section 2. Jiang [54] proved global existence for the full system in one space dimension when the viscosity coefficients depend on the density. Probably the most general result as well as an extensive list of relevant literature is contained in the recent paper by Amosov [2]. There is a vast amount of literature concerning the qualitative properties of solutions. Straškraba [98,99], Straškraba and Valli [100] and Zlotnik [113] studied the long timebehaviour of global solutions in the barotropic case driven by a nonzero external force. Similar results for the full system were obtained in [35]. More information can be found in Amosov and Zlotnik [3], Hsiao and Luo [53], Matsumura and Yanagi [74] and many others. A complete list of references goes beyond the scope of the present paper.
9.2. Multi-dimensional diffusion waves A more detailed description of the long-time behaviour for the barotropic case in several space dimensions was obtained by Hoff and Zumbrun [52]. Following their presentation we consider the system (5.3), (5.4) with f = 0 on the whole space Ω = R 3 . The initial data #(0) = #0 ,
(#u)(0) = q0
(9.1)
are smooth and close to the constant state #∗ = 1, q0 = 0. Under these circumstances, the problem (5.3), (5.4) admits a global solution and the following theorem holds.
362
E. Feireisl
T HEOREM 9.1. Assume that the initial data satisfy #0 − 1 L1 ∩W 1+d,2 (R 3 ) + q0 L1 ∩W 1+d,2 (R 3 ) < ε, where ε > 0 is sufficiently small and d 3 is an integer. Then the initial-value problem (5.3), (5.4), (9.1) possesses a global solution #, u satisfying $ $ $ $ α $∂ (#(t) − 1)$ p 3 + $∂ α (#u)(t)$ p 3 x x L (R ) L (R ) for 2 p ∞, (1 + t)−rα,p c(d)ε (1 + t)−rα,p +1/p−1/2 if 1 p < 2 for any multi-index |α| (d − 3)/2 where rα,p = |α|/2 + 3/2(1 − 1/p). Theorem 9.1 shows that perturbations of the constant state decay at the rate of a heat kernel for p 2 but less rapidly if p < 2; in fact, the bound may even grow with time in the latter case. A more detailed picture of the long-time behaviour in the Lp -norm for p 2 is provided by the following result. T HEOREM 9.2. Under the assumptions of Theorem 9.1, we have $ $ $ α $ $∂ (# − 1)(t)$ p 3 + $∂ α (#u)(t) − Kµ ∗ [P q0 ] $ p 3 x x L (R ) L (R ) c(d)ε(1 + t)−rα,p +1/p−1/2 , p 2, where P is the projection on the space of divergence free functions and Kµ is the standard heat kernel, i.e., the fundamental solution of the problem ∂t v − µ v = 0. Thus the dynamics in Lp , p > 2 is dominated by a term with constant density and a nonconstant divergence-free momentum field decaying at the rate of the heat kernel. In other words, for p > 2, all smooth, small amplitude solutions are asymptotically incompressible. Finally, consider an auxiliary problem: 1 ∂t # + div(#u) = (λ + 2µ) #, 2 1 ∂t (#u) + p (1)∇# = µ (#u) + λ∇ div(#u). 2 Let us denote U (t) = #(t), (#u)(t)
(9.2)
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the solution of the linear problem (9.2) with the initial data #(0) = #0 − 1,
(#u)(0) = q0 .
The long-time dynamics in Lp , p < 2, is described as follows. T HEOREM 9.3. Under the hypotheses of Theorem 9.1, we have $ α $ $∂ #(t) − 1, (#u)(t) − U (t) $ p 3 x L (R ) c(l, σ )ε(1 + t)−rα,p +3/4(2/p−1)−1/2+σ ,
1 p < 2,
for any positive σ . All results in this part are taken over from [52].
9.3. Energy decay of solutions on unbounded domains Various authors have considered the long-time behaviour of solutions on unbounded domains. Following Kobayashi and Shibata [58] we consider the full system (2.1)–(2.3) on an exterior domain Ω ⊂ R 3 where the pressure p = p(#, θ ) is given by a general constitutive law conform with the basic thermodynamical principles expressed in (1.4). As for the boundary conditions, we take u|∂Ω = 0,
θ |∂Ω = θb ,
lim u(t, x) = 0,
|x|→∞
lim θ (t, x) = θb .
|x|→∞
Assuming the initial data #(0) = #0 ,
u(0) = u0 ,
θ (0) = θ0
are closed to a constant state [#, ¯ 0, θb ], Kobayashi and Shibata [58, Theorem 2] show the following decay rates: $ $ $ $ $ $ $#(t) − #¯ $ 2 3 + $u(t)$ 2 3 + $θ (t) − θb $ 2 3 ct −3/4 , L (R ) L (R ) L (R ) $ $ $ $ $ $ $#(t) − #¯ $ ∞ 3 + $u(t)$ ∞ 3 + $θ (t) − θb $ ∞ 3 ct −5/4 . L (R )
L (R )
L (R )
Related results were obtained by Deckelnick [17], Kobayashi [57], Padula [87] and many others.
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E. Feireisl
9.4. Alternative models Up to now, we have considered only Newtonian fluids where the viscous stress tensor Σ was a linear function of the velocity gradient ∇u. However, some experimental results show that in nature there exist stronger dissipative mechanisms not captured by the classical Stokes law. Let us shortly discuss this interesting and rapidly developing area of modern mathematical physics which gives an alterantive and, given the enormous amount of open problems in the classical theory, mathematically attractive way to describe the fluid motion. In the linear theory of multipolar fluids, the constitutive laws, in particular, the viscous stress tensor Σ depend not only on the first spatial gradients of the velocity field u but also on the higher order gradients up to order 2k − 1 for the so-called k-polar fluids. In the work of Neˇcas and Šilhavý [82], an axiomatic theory of viscous multipolar fluids was developed in the framework of the theory of elastic non-viscous multipolar materials due to Green and Rivlin [43]. Accordingly, the viscous stress tensor Σ takes a general form: Σ=
k−1
(−1)j µj j ∇u + (∇u)t + λj j div u Id
j =0
+ ω ∇u + (∇u)t + β div u Id,
(9.3)
where, in the nonlinear component, β = β |∇u|, div u, det(∇u) ,
ω = ω |∇u|, div u, det(∇u) .
The existence of the so-called measure-valued solutions of the initial value problem for isothermal flows, i.e., for the system (2.1), (2.2) with Σ given as in (9.3) and the pressure satisfying p = rθ0 #, was proved by Matuš˚u-Neˇcasová and Novotný [75]. The weak solutions for linear multipolar fluids were obtained in a series of papers by Neˇcas et al. [81,80]. Recently, new results concerning the so-called power-law fluids, i.e., when k = 1 in (9.3), were shown by Mamontov [69,70].
10. Conclusion Despite the enormous progress during the last two decades, we still seem to be very far from a satisfactory rigorous mathematical theory of viscous compressible and/or heat conducting fluids. There are good an bad news according to the degree of complexity of the problems considered but we still wait, for instance, for a large data existence result for, say, the isothermal flow in two and three space dimensions. On the point of conclusion, let us discuss shortly the major mathematcal difficulties presently encountered.
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10.1. Local existence and uniqueness, small data results As we have seen in Section 2, the initial value problem for the full system (1.1)–(1.3) complemented by physically relevant constitutive relations admits a unique global in time classical solution. From the mathematical point of view, this is nothing less or more than to say that the linearized system is well-posed. This fact seems to be the primary criterion of applicability of any mathematical model. Indeed there is only a little to say should the linearized problem be ill-posed. However, there still can remain an essential gap between “linear” and “nonlinear” provided there is no dissipative meachanism present as it is the case for nonlinear hyperbolic systems. The possibility to construct classical though only “small” solutions reveals the dissipative character of the problem, namely, the effect of the diffusion terms present in the parabolic equations (1.2), (1.3). Another aspect of dissipativity is the existence of bounded absorbing sets discussed in Section 7.1 and the existence of global attractors mentioned in Section 7.5. Although we still do not know if the attractor has a finite fractal dimension, there are strong indications (cf. Hoff and Ziane [50]) it might be the case.
10.2. Density estimates Unlike (1.2), (1.3), the continuity equation (1.1) governing the time evolution of the density is hyperbolic and linear with respect to #. As a consequence, one cannot expect any smoothing effect as for parabolic problems or compactification phenomena as it is the case for genuinely nonlinear hyperbolic equations. We have made it clear several times in this paper that the major obstacle to develop a rigorous large data theory for our problem is the lack of a priori estimates on the density #. The density being a non-negative function there are two aspects of the problem – boundedness from below away from zero and uniform upper bounds. Let us remark that the system (1.1)–(1.3) and, in particular, the constitutive relations for Newtonian fluids hold for nondilute fluids with no vacuum zones. Let us review the results of Desjardins [18] illuminating the role of upper bounds on # in the well-posedness problem. Consider the isentropic model represented by the system (5.3), (5.4) where the pressure p satisfies (5.2) with γ > 1. For simplicity, we consider the case of spatially periodic boundary conditions in two space dimensions. The following result is proved by Desjardins [18, Theorem 2]. T HEOREM 10.1. Consider the system (5.3), (5.4) where p(#) = a#γ ,
a > 0, γ > 1,
in two space dimensions and with spatially periodic data 2 #(0) 0 ∈ L∞ per R ,
2 1,2 u(0) ∈ Wper R ,
f ≡ 0.
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Then there exists T0 > 0 and a weak solution #, u of the problem such that for all T < T0 2 # ∈ L∞ 0, T ; L∞ per R and √ #∂t u ∈ L2 0, T ; L2per R 2 , 2 1,2 p − (λ + 2µ) div u ∈ L2 0, T ; Wper R , ∇u ∈ L∞ 0, T ; L2per R 2 , 2 2,2 Pu ∈ L2 0, T ; Wper R , where P denotes the projection on the space of divergence free functions. Moreover, the regularity properties stated above hold as long as $ $ sup $#(t)$L∞ (R 2 ) < ∞.
t ∈[0,T ]
per
Desjardins [18] proved also that the weak solutions constructed in Theorem 10.1 enjoy the weak-strong uniqueness property well-known from the theory of incompressible flows. Specifically, the above weak solution coincides with a strong one as long as the latter exists (cf. [18, Theorem 3]). The lower bounds on the density represent an equally delicate issue. As we have seen in Section 7.3, one cannot avoid vacuum states provided we accept the isentropic model as an adequate description for the long time behaviour of solutions. On the other hand, the density should remain strictly positive for any finite time t provided its initial distribution enjoys this property. Unfortunately, however, this is not known in the class of weak solutions provided N 2. To reveal the pathological character of the problem when vacua are present, we follow Liu et al. [66] and consider the isentropic model in one space dimension where the initial distribution of the density is given as #(0) = #0 (x − 2r) + #0 (x + 2r), where #0 is a compactly supported smooth function with support contained in the ball {|x| < r}. Should the model correspond to physical intuition, one would expect, at least on a short time interval, the solution to be given as #(t, x) = #(t, ˜ x − 2r) + #(t, ˜ x + 2r), u(t, x) = u(t, ˜ x − 2r) + u(t, ˜ x + 2r), where #, ˜ u˜ solve the problem for the initial data #(0) ˜ = #0 . However, as shown in [66], this is not the case. Of course, this apparent difficulty is due to the discrepancy between
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the finite speed of propagation property which holds for the hyperbolic equation (5.3) and the instantaneous propagation due to the diffusion character of (5.4) (for other unusual features of the problem we refer also to Hoff and Serre [48]). In fact, one should consider the viscosity coefficients µ and λ depending on the density # in this case (see Jiang [54]). The formation or rather non-formation of vacua has been studied in a recent paper by Hoff and Smoller [49]. They prove that the weak solutions of the Navier–Stokes equations for compressible fluid flows in one space dimension do not exhibit vacuum states in a finite time provided that no vacuum is present initially under fairly general conditions on the data. Unfortunately, however, such a result is not known in higher space dimensions even when the data exhibit some sort of symmetry, say, they are radially symmetric with respect to origin.
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CHAPTER 4
Dynamic Flows with Liquid/Vapor Phase Transitions Haitao Fan Department of Mathematics, Georgetown University, Washington DC 20057, USA
and Marshall Slemrod Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Initial value problems of the inviscid system (1.3) and admissibility criteria . . . . . . . . . . . Existence of solutions of the Riemann problem (3.1) . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Existence of solutions of the Riemann problem (1.3) . . . . . . . . . . . . . . . . . . . . . 4.2. A-priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Solutions constructed by vanishing similarity viscosity are also admissible by traveling criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The purpose of this paper is to review some recent results on Navier–Stokes equations with van der Waals type constitutive relation for the pressure: ∂w ∂u = (conservation of mass), ∂t ∂x ∂u ∂ ∂ 2w ∂u = − ε2 A 2 −p(w, θ ) + ε ∂t ∂x ∂x ∂x (conservation of linear momentum), ∂E ∂ ∂u ∂ 2w u −p + ε = − ε2 A 2 ∂t ∂x ∂x ∂x ∂θ ∂u ∂w + ε2 A + κε (conservation of energy), ∂x ∂x ∂x
(1.1)
where w is the specific volume, u the fluid velocity, θ the fluid temperature, ε the viscosity, κ the coefficient of thermal conductivity, A an assumed constant capillarity coefficient. The constitutive relations for the the pressure p and the specific entropy η are derived from the thermodynamic relationship p=−
∂f , ∂w
η=−
∂f , ∂θ
where f (w, θ ) is the specific free energy given by the relationship f = Rθ ln(w − b) − a/w + F (θ ), where F is in general an arbitrary function of θ . Since the specific internal energy e must satisfy e = f + θ η, it follows that e = −a/w + F (θ ) − θ F (θ ). As a simplifying assumption, we set F (θ ) = −cv θ ln θ + some constant,
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where cv is the assumed constant specific heat at constant volume. Then the specific internal energy e is given by e = −a/w = cv θ + constant and the specific total energy E in (1.1) is E = u2 /2 + e + Awx2 /2. That is, the total energy is the sum of contributions from kinetic energy, internal energy, and interfacial energy. The isothermal version of (1.1) is clearly ∂w ∂u = , ∂t ∂x ∂ ∂ 2w ∂u ∂u = − ε2 A 2 , −p(w) + ε ∂t ∂x ∂x ∂x
(1.2)
while its inviscid case is covered by ∂w ∂u − = 0, ∂t ∂x ∂ ∂u + p(w) = 0. ∂t ∂x
(1.3)
The constitutive equation for a van der Waals fluid at fixed temperature below the critical temperature θc =
8a 27Rb
has the shape depicted in Figure 1. In the isothermal case, we are interested in subcritical temperatures and hence assume p ∈ C 1 (R) and p (w) < 0
p (w) > 0
if w ∈ / [α, β], if w ∈ (α, β).
(1.4)
The regions w < α and (w > β) correspond, for van der Waals fluids, liquid and vapor phase region respectively. The line joining (m, p(m, T )) and (M, p(M, T )) is called Maxwell line where two equilibrium phases can coexist. The region α < w < β is called the spinodal region. If the fluid ever enters the spinodal region, the fluid will quickly decompose to liquid or vapor or their mixture. In other words, the spinodal region is a highly unstable region. To see this intuitively, we consider a ball of such fluid with w in spinodal region, see Figure 2. Pressures in the ball and its surrounding medium are set
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377
Fig. 1.
w0 , p = p(w0 )
Fig. 2. If the fluid inside the ball is in the spinodal region, w ∗ ∈ (α, β) then the ball is unstable.
equal, so that the system is in equilibrium mechanically. We perturb the fluid in the ball by decreasing the pressure of the surrounding medium a little bit. Then the fluid inside the ball will expand. If the fluid inside the ball is regular, in the sense that an increase in the volume results in the decrease in pressure, the ball will expand a little bit and the pressure inside the ball will drop to the level of that of the surrounding medium and the system will settle down to a new equilibrium close to the one before perturbation. However, when the liquid inside the ball is in spinodal region, such a little increase in w will result in an increase in pressure in the ball and hence the ball will further expand. Beside the instability in the spinodal region, there is another phenomena associated to phase transitions in a typical van der Waals type: metastability. For example, suppose vapor is initially set at rest, and we start to compress it with some w > M. When we reach w = M, the vapor should start to condensate in an ideal equilibrium world. But in the real world, the condensation will not start until we continue to compress so that the vapor enters into the region β < w < M. The vapor in this region can stay as vapor for long time until enough many nuclei of liquid are created and then rapid condensation takes place.
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Systems (1.1), (1.2), (1.3) coupled with (1.4) not only serve as prototype models for studying the dynamics of phase transitions, but also are interesting mathematical objects in its own right. For example, system (1.2) are of hyperbolic–elliptic mixed type with α < w < β as its elliptic region. It is well known that the initial value problems for elliptic systems are ill-posed. Systems of hyperbolic type have been extensively studied. The presence of both hyperbolic and elliptic region in (1.2) certainly leads to new phenomena and new issues. The isothermal system (1.2), having physical background and being one of the simplest systems of conservation laws of hyperbolic–elliptic mixed type, certainly qualifies to be a prototype model for studying such systems. In this paper, we shall review some recent results related to (1.1), (1.2), (1.3) with (1.4). Although we tried our best to cover as much related results as possible, it is possible that we missed some. The rest of this paper is arranged as follows: in Section 2, we derive Equation (1.1). In Section 3, we review some results on the initial value problem of (1.3) and related admissibility criteria. In Section 4, we recall the proof of the existence of solutions of (1.3) satisfying the traveling wave criterion via the vanishing similarity-viscosity approach. Although these results and proofs appeared in our earlier works, we present here a revised version which is more readable.
2. The equations of motion We consider the one-dimensional motion of fluid processing a free energy f (w, θ ) = f0 (w, θ ) +
ε2 A ∂w 2 . 2 ∂x
(2.1)
Here w is the specific volume, θ the absolute temperature, A > 0 a constant, and x the Lagrangian coordinate. The term ε2 A ∂w 2 , 2 ∂x where ε > 0 is a small parameter, is the specific interfacial energy introduced by Korteweg [52]. The graph of f0 as a function of w for fixed θ will vary smoothly from a single well potential for θ > θcrit to double well potential for θ < θcrit . The θcrit is called the critical temperature. Discussions of such free energy formulations may be found in [3, 10,13,14,12,25,35,66,67,78]. The stress corresponding to the free energy (2.1) is given by T=
∂f ∂f0 ∂ 2w = (w, θ ) − ε2 A 2 . ∂w ∂w ∂x
(2.2)
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Note that there is no viscous force in (2.2). Addition of a viscous stress term gives us the stress of the form T = −p(w, θ ) + ε
∂u ∂ 2w − ε2 A 2 ∂x ∂x
(2.3)
suggested by Korteweg’s theory of capillarity [52]. In (2.3), u(x, t) denotes the velocity of the fluid, ε > 0 is the viscosity and p = ∂f0 /∂w is the pressure. The one-dimensional balance laws of mass and linear momentum are easily written down: ∂w ∂u = ∂t ∂x
(mass balance),
(2.4a)
∂u ∂T = ∂t ∂x
(linear momentum balance).
(2.4b)
The equation for balance of energy is more subtle. While a thorough examination of the energy equation appears in Dunn and Serrin [25] it is the conceptually simple approach of [32] we recall here. Let e(w, θ ) denote the internal energy. Felderhof’s postulate is that the internal energy is influenced only by the component of internal stress τ = ∂u −p(w, θ ) + ε ∂x , i.e., the balance of energy is given by ∂u ∂h ∂e =τ + , ∂t ∂x ∂x
(2.4c)
where h is the heat flux. Unlike Equations (2.4a, b), Equation (2.4c) is not in divergence form. To alleviate this difficulty we consider the specific total energy E=
u2 ε2 A ∂w 2 + e(w, θ ) + 2 2 ∂x
made up the specific kinetic, internal, and interfacial energy. Now compute the time rate of change of E: ∂u ∂E ∂w ∂ 2 w =u + et + ε 2 A ∂t ∂t ∂x ∂x∂t =u
∂u ∂ 2 w ∂u ∂h ∂ ∂w ∂u ∂T +T + ε2 A 2 + + ε2 A , ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 2
where we have used the relation T = τ − ε2 A ∂∂xw2 . We easily see that the balance of energy can be written as ∂E ∂ ∂h ∂ ∂u ∂w 2 = (uT ) + ε A + . (2.5) ∂t ∂x ∂x ∂x ∂x ∂x
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∂w The term ε2 A ∂u ∂x ∂x represents the “interstitial working” [25]. For simplicity we constitute ∂θ where κε > 0 is the (assumed constant) thermal conductivity. h by Fourier’s law: h = κε ∂x Then we may collect the balance laws and write them as
∂w ∂u = (mass), ∂t ∂x 2 ∂ ∂u ∂u 2 ∂ w = −p(w, θ ) + ε −ε A 2 (linear momentum), ∂t ∂x ∂x ∂x 2 ∂E ∂ ∂u 2 ∂ w = −ε A 2 u −p + ε ∂t ∂x ∂x ∂x ∂θ ∂u ∂w 2 +ε A + κε (energy). ∂x ∂x ∂x
(2.6a)
(2.6b)
(2.6c)
The isothermal case of (2.6) is ∂w ∂u = , ∂t ∂x 2 ∂ ∂u ∂u 2 ∂ w = −ε A 2 . −p(w) + ε ∂t ∂x ∂x ∂x
(2.7a)
(2.7b)
3. Initial value problems of the inviscid system (1.3) and admissibility criteria In this section, we recall recent results on the initial value problems of inviscid system (1.3). Most results on the initial value problem of (1.3) are on Riemann problems. The Riemann problem of (1.3) is the initial value problem wt − ux = 0, ut + p(w)x = 0,
(u− , v− ), u(x, 0), v(x, 0) = (u+ , v+ ),
(3.1) if x < 0, if x > 0.
Through the study of the Riemann problem, we gain understanding on the behavior of solutions of (1.3). Based on knowledge about solutions to Riemann problems, Glimm’s scheme can be used to construct solutions of (1.3) for general initial data. Compare to the viscous system (1.2), the inviscid system (1.3), as an approximation of (1.2), offer the following advantages: the structure of solutions are clearer. Solutions of Riemann problems may be constructed by solving a few algebraic equations. However, these advantages come with a price to pay: solutions of initial value problems of the
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inviscid system (1.3) are usually weak solutions with jump discontinuities. Such solutions are nonunique unless further restrictions on weak solutions are applied. These restrictions are called admissibility criteria. The admissibility criterion should pick “good” solutions suitable for the problem under consideration: here we are considering phase transitions modeled by (1.2). The inviscid system (1.3) is used as an approximation of (1.2). Thus, to make solutions of (1.3) to mimic those of (1.2), it is natural to require that admissible solutions of (1.3) to be ε → 0+ limits of solutions of (1.2) with the same initial value. This is called the vanishing viscosity criterion. However, enforcing the vanishing viscosity criterion is usually very difficult and expensive. For example, to implement the vanishing viscosity criterion, one have to be able to (a) prove that solutions of (1.3) satisfying the criterion exist and (b) verify whether a given solution of (1.3) satisfy the criterion or not. These tasks are usually very difficult and expensive. So far, the part (a) is carried out for strictly hyperbolic systems of conservation laws [11,24,23]. For the (b), some results and techniques are given in [34] for piece-wise smooth solutions, with small shocks, of stricly hyperbolic systems. Thus, simpler admissibility criteria are called for. An internal layer asymptotic analysis on solutions of the viscous system (1.2) indicates that jump discontinuities of solutions of (1.3) must have traveling wave profiles in order for the solution of (1.3) to approximate that of (1.2). Traveling waves of (1.2) are solutions which are functions of the form g(x − st), where the constant s is the speed of the traveling wave. The traveling wave equations corresponding to (1.1) with ξ = (x − st)/ε, w = w(ξ ), u = u(ξ ), θ = θ (ξ ) are dw = v, dξ dv A = −s 2 (w − w− ) − p(w, θ ) + p(w− , θ− ) − sv, dξ dθ = −s e(w, θ ) − e(w− , θ− ) κ dξ −
(3.2)
s2 Asv 2 (w − w− )2 − − p(w− , θ− )(w − w− ) , 2 2
(w, u, θ )(−∞) = (w− , u− , θ− ), (w, u, θ )(+∞) = (w+ , u+ , θ+ ), where s is the speed of the traveling wave. For the isothermal case (1.2), above becomes dw = v, dξ dv A = −s 2 (w − w− ) + p(w− ) − p(w) − sv, dξ (w, u)(−∞) = (w− , u− ),
(w, u)(+∞) = (w+ , u+ ).
(3.3)
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Fig. 3.
A shock solution of (1.3) (u, w)(x, t) =
(u+ , w+ ), (u− , w− ),
if x − st > 0, if x − st < 0,
(3.4)
where s is the speed of the shock, is said to have a traveling wave profile if the traveling wave equation (3.3) has a solution. This leads to the traveling wave admissibility criterion: Traveling wave criterion states that a shock (3.4) is admissible if the system of traveling wave equations (3.3) has a solution. When (3.3) has a solution, we also say that there is a connection between (w− , u− ) and (w+ , u+ ). If all singular points of a solution of (1.3) are jump discontinuities and these jump discontinuities are admissible by the traveling wave criterion, we say that the solution is admissible by the traveling wave criterion. The solvability of the traveling wave equation (3.3). We are particularly interested in the case w− < α, w+ > β, since this data involves phase changes. Indeed, solutions of Riemann problems of (1.3) cannot take values inside the spinodal region (α, β) and hence must have a shock jumping over the spinodal region, at least for the case A 1/4 [27, Lemma 2.3(i)]. The solvability of the connecting orbit problems (3.3) were studied by Slemrod [74–76] and Hagan and Slemrod [40], Hagan and Serrin [39] and Shearer [70– 72]. Let w− α and s 0. For simplicity, we assume the ray starting from (w− , p(w− )), with slope −s 2 , to the right can intersect the graph of p at most at three points (cf. Figure 3). We denote the w-coordinates of these points by w2 (w− , s),
w3 (w− , s)
and w4 (w− , s),
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respectively. Points w− and wk (w− , s), k = 2, 3, 4, are equilibrium points of (3.2). w− and w3 (w− , s) are saddle points of (3.2) while w4 (w− , s) is a node of (3.2). By [27], Riemann solvers of (1.3) cannot have values in the spinodal region (α, β) at least for the case A 1/4, thus, traveling waves connecting w− and w2 (w− , s) is of no use in this case. Now we consider the existence of a solution of (3.2) connecting w− and w3 (w− , s), i.e., w(−∞) = w− , w(+∞) = w3 (w− , s). For w− ∈ [γ , m], if there is a s¯ 0 such that the signed area between the graph of p and the chord connecting (w− , p(w− )) and (w3 (w− , s¯), p(w3 (w− , s¯))) is 0 (cf. Figure 3), then there is a speed s ∗ 0 such that 0 s ∗ s¯
(3.5)
and the problem (3.2) with s = s ∗ , w2 = w3 (w− , s ∗ ) has a solution, which satisfies w # (ξ ) > 0 and is a saddle–saddle connection, i.e., 0 s∗ <
−p (w− ),
s∗ <
−p (w3 (w− , s ∗ )).
(3.6)
We note that this saddle–saddle connection accounts for the usual liquid-vapor phase transitions, including the coexistence of two phase equilibria in the case s = 0. In (3.4), equality holds if and only if s¯ = 0. Furthermore, for any 0 < s < s ∗ the trajectory of (3.2) emanating from (w− , 0) will overshoot w3 (w− , s) and flow to (w4 (w− , s), 0) as ξ → ∞. In other words, for all w2 > w4 (w− , s ∗ ), there is a traveling wave solution of (3.2). Furthermore, this traveling wave solution is a saddle–node connection, i.e.,
−p (w− ) > s >
−p (w4 (w− , s)).
(3.7)
These statements were proved in Hagan and Slemrod’s paper [40]. Grinfeld [36] proved that if s¯ exists, then for any N ∈ Z+ , there is a number AN > 0 such that for all A AN , the system (3.2) has N saddle–saddle connection solutions, wj (ξ ), j = 0, 1, 2, . . . , N − 1, such that wj (ξ ) intersects w = 0-axis traversely j times. If s¯ does not exist but s0 := min s: w3 (s, w− ) > 0, then there is at least one saddle–saddle connection for all A > 0. If p(w) further satisfies p (w)(w − w0 ) > 0 for w = w0 ,
(3.8)
for some w0 ∈ (α, β) then, for γ w− m, there is a unique speed s ∗ 0 such that w− can be connected to w3 (w− , s ∗ ) by a traveling wave solution of (3.2) with w2 = w3 (w− , s ∗ ), which is a saddle–saddle connection [69–71]. We notice that when (3.8) holds there is no w4 (w− , s) for w− ∈ (γ , m]. In fact, uniqueness of s ∗ holds for all w− < α. In the above paragraph, w− is fixed. However, if we fix w+ = w3 (w− , s), there can be two w− < α such that there are connections with s > 0 between w− and w3 [8].
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We note when p(w) is a cubic polynomial, we can have an explicit solution for (3.2). Let m+M p(w) = p0 − p1 (w − m)(w − M) w − , (3.9) 2 where m and M are Maxwell constants. Then a solution of (3.2) is (cf. [79,80]) w(ζ ) =
4 p1 w+ − w− w− + w+ w+ − w− + tanh (ζ − ζ0 ) . 2 2 2A 2
(3.10)
For each w− fixed, w+ in (2.11) is determined by equations: 3(1 − 6A)(2y − z + 1)2 + z2 = 1, DSy = (M − w+ )/(M − m),
(3.11)
z = (w+ − w− )/(M − m). The number of solutions of (3.11) ranges from zero to two. When (3.11) has two solutions, we get two solutions of (3.2) of the form (3.10); one of them has positive speed and the other negative. This is, of course, consistent with Theorem 3.1. In fact, the nonuniqueness of traveling waves connecting a fixed w3 to some w− is true in general [8]. In addition, Grinfeld [37] and Mischaikow [59] conducted studies on the full system (2.6) using Conley’s index theory. Stability of traveling waves is an important topic for (1.2). In fact, having a stable or metastable shock profile, which is a traveling wave solution of (1.2), is a necessary condition for the shock of (1.3) to be admissible. Hoff and Khodja [47] proved the dynamic stability of certain steady-state solutions of the Navier–Stokes equations for compressible van der Waals fluids vt − ux = 0,
ut + p(v, e)x = ε(x)ux /v x , 2 u /2 + e t + up(v, e) x = ε(x)uux /v + λ(x)T (v, e)x /v x .
(3.12)
The steady-state solutions consist of two constant states, corresponding to different phases, separated by a convecting phase boundary. They showed that such solutions are nonlinearly stable in the sense that, for nearby, perturbed initial data, the Navier–Stokes system has a global solution that tends to the steady-state solution uniformly as time goes to infinity. Benzoni-Gavage [7] studied the linear stability of planar phase jumps satisfying the traveling wave criterion (3.3) in Eulerian coordinates with viscosity neglected, called capilarity admissibility criterion. She showed that the such phase boundaries are linearly stable. Although neglecting the viscosity is unphysical, such a result served as the base from which she studied the case when the viscosity is small [9] to yield similar results.
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Zumbrun [85] proved the linear stability of slow heteroclinic traveling waves of (2.7) under localized perturbation. He also showed that homoclinic traveling waves near Maxwell line involving multiple phase transitions are exponentially unstable. This implies that the slow heteroclinic traveling waves of (2.7) are stable if they are monotone. The method used are spectrum analysis framework [33,86] and some energy estimates. Motion of phase boundary under perturbation and the effect of boundary conditions of (1.2) was studied by Chen and Wang [15]. The initial data is a perturbation of the stationary phase boundary, the Maxwell line. They found the ordinary differential equations describing the motion of the phase boundary under perturbation by an asymptotic expansion and a matching analysis. They conclude that the phase boundary will approach a well defined location as time goes to infinity. Existence of solutions of the Riemann problem for (1.3) satisfying the traveling wave criterion. One method for solving (3.1) is construction of wave and shock curves that are admissible according to some criteria and then construct a wave fan of centered waves and shocks that matches the initial data. Being constructive, this approach yields very detailed structure of the solutions if successfully carried out. The difficulty is that it is hard to know all the admissible shocks to enable such a construction. When A = 0 in (3.3), James [49] considered the Riemann problem. Shearer [69], Hsiao [48] proved the existence of solutions of the Riemann problem. In this case, a phase boundary is admissible if and only if the speed of the phase boundary is 0. The uniqueness of such Riemann solutions is proved by Hsiao [48]. See also [50,51]. In the case A > 0, the only stationary phase boundary is the one connecting (m, 0) and (M, 0) [74]. This is in perfect agreement with the Maxwell equal area rule. When the Riemann data are in different phase region, e.g., w− < α and w+ > β, Shearer [72] proved that solutions of Riemann problem exist if |w− − m| + |w+ − M| + |u+ − u− | is small, where m, M are the Maxwell line constants. He first studied the behavior of traveling waves near Maxwell line, then constructed the Riemann solvers accordingly. To extend his approach to more general Riemann data, one will have to know explicitly, for any given w1 , what w2 can be connected to w1 by a traveling wave. This is almost impossible in general. Another approach is to construct the solution of (1.3) as the ε → 0+ limit of the viscous system (1.2), or simply that of the solutions of wt − ux = εwxx , ut + p(w)x = εuxx ,
(3.13)
with the same initial data. Although this approach has been carried out successfully for strictly hyperbolic 2 × 2 systems [24,23] for hyperbolic–elliptic mixed type system (1.3), this approach seems quite difficult at present. Thus, Slemrod [77] and Fan [26,28,29] used the vanishing similarity viscosity approach pioneered by Dafermos [20] and Tupciev [81].
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(a)
(b) Fig. 4.
The idea of this approach is to construct the weak solution of (3.1) as the ε → 0+ limit of the solution of wt − ux = εtwxx , ut + p(w)x = εtuxx , (u− , v− ), u(x, 0), v(x, 0) = (u+ , v+ ),
(3.14) if x < 0, if x > 0.
This approach enables us to establish the existence of weak solutions of (3.1) for general Riemann data not in the spinodal region. The condition required is p(w) → ±∞ as w → ∓∞. Although the form of the viscosity used in (3.12) is often criticized as unphysical, it turns out that solutions constructed through ε → 0+ limiting process of (3.12) are also admissible by the traveling wave criterion derived from (3.11) [26], see Section 4.3. Furthermore, when p (w)(w − w0 ) < 0 for w = w0 ∈ (α, β) and with w− < α < β < w+ (or w+ < α < β < w− ), the solution of the Riemann problem admissible by the traveling wave criterion is unique. Thus, under above condition, if one obtains a solution via the vanishing viscosity method (3.12), it would be the same as what we obtained by the vanishing similarity viscosity approach (3.12). When the Riemann data are on the same side of the spinodal region, w± < m (or w± > M), Shearer [71] showed that when the Riemann data |w± − m| (or |w± − M|) are small, then for some u± , there are at least two solutions for the Riemann problem. An example of the nonuniqueness is illustrated by Figure 3. Shearer’s result raised the question that which of the two solutions is physically “good”. We think that both solutions are good, but at different times: consider the shock tube experiment corresponding to the Riemann data (w± , u± ) depicted in Figure 3. We note that −u+ = u− > u∗ = 0 and β < w− = w+ < M. This data describe the shock tube experiment where vapor moves from both sides towards the center x = 0 where the fluid is at rest. The pressure at the center part increases due to the compression from both sides. The data in Figure 3 are such that the pressure at the center part is above the equilibrium pressure and hence the vapor in the middle part is metastable vapor. The metastable vapor will stay as vapor until enough liquid drops are initiated due to random fluctuation. Thus, in the early stage, the first solution in Figure 4(a) is the good solution. As time increases,
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enough many liquid drops are initiated in metastable vapor, more likely in the center part where pressure is higher. Then phase changes occur rapidly via the growth of these liquid drops. In this late stage, we expect to see the second solution, shown in Figure 4(b). We note that the center of the wave in the early time of the second stage may be different from that of the first solution due to the randomness in location of liquid drop initiated. From the above consideration, we see that both solutions in Figure 3 are “good”, but at different times. In fact, the two-phase-boundary solution depicted in Figure 4(b) is visually quite stable in numerical simulations once it is initiated [73]. As to when the solution in Figure 4(a) changes to that in Figure 4(b), the viscosity method does not provide answer. This is because system (3.1) with higher-order derivative terms, such as (1.2) is based on interfacial energy. The Maxwell equal area rule is derived from the consideration of interfacial energy which is the reason why viscosity–capillarity type of high-order derivative terms in (1.2) agrees with the Maxwell equal area rule. However, such high-order derivative terms do not cover the mechanism for creating of liquid drops due to random fluctuations. Thus, (1.2) is designed to describe the motion of phase boundaries, not the initiation of new phases. This is why some selection criteria, called initiation criteria, are used to help to decide when new phases are initiated in many classical theory of quasistatic problems, cf. [38]. We imagine that the transition from the solution without phase boundary to the one with two phase boundary is a dynamic process that takes some time for enough many nuclei of new phase to form and grow to complete the phase change. Typically, nucleation process are slow in metastable states unless near the Wilson line, or spinodal limits, which is why we have metastable states. Proper choice of initiation criteria used in (1.3) to correctly describe the physical process is an open problem. Initiation criteria should reflect (a) that if the fluid is in metastable state, then after a sufficiently long time, nucleation process will initiate enough many fluid drops of the stable phase and eventually, the fluid change to the stable phase, and (b) that the further away the metastable fluid is from the equilibrium, the faster the new, stable phase will be initiated. From above consideration, we see that initiation criteria must involve the time spend in metastable state and the distance of the pressure from the equilibrium pressure. So far, initiation criteria used in most works on (3.1) do not include above factors. Rather, these criteria typically specify that if the distance of the pressure from the equilibrium pressure is less than a fixed barrier, then the one-phase solution is picked, otherwise the two-phase solution is picked. This ignored the fact that no matter how close the pressure is to the equilibrium pressure, transition to the two-phase solution will happen later, even though slower. Kinetic relation admissibility criteria. Above considerations demonstrate that the system (3.1) is not complete by itself. It must be augmented by some selection criteria that helps to find the solution relevant to the problem under consideration. For phase transition problems, such criteria may depend on materials in a complicated way. This leads to another point of view on admissibility criteria for phase boundaries: rather than tracing the admissibility criteria for phase boundaries to something more elementary such as viscosity and capillarity, one can consider admissibility criteria as constitutive relations controlling the speed of the phase boundary, determined by the materials and to be measured in laboratory. Such restriction are considered by Truskinovskii [79] and Abeyaratne
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and Knowles [1] and are called kinetic relations, [1]. Consider an interval [x1 , x2 ] of fluids in the Lagrangian coordinate. The total mechanical energy associated with the interval is E(t) =
x1
where P =
w 0
1 2 −P (w)(x, t) + u (x, t) dx, 2
x2
(3.15)
p(η) dη. A calculation based on (1.3) shows that
p(w1 )u1 − p(w2 )u2 −
d E(t) = f (w1 , w2 )s(t) dt
(3.16)
if (u, w)(x, t) is a shock solution of (3.1) and x2 > s(t) > x1 . Here in (3.16), f (w1 , w2 ) = −
w2
w1
p(η) dη +
1 p(w2 ) + p(w1 ) (w2 − w1 ). 2
(3.17)
We can see that the left-hand side of (3.16) is the excess of rate of work of the external forces over the rate of increase of mechanical energy. Since the motion (3.1) described is isothermal, the well known Clausius–Duhem inequality requires that the instantaneous rate of mechanical dissipation to satisfy f (w1 , w2 )s 0,
(3.18)
which is the classical entropy criterion. A subsonic phase boundary {(u1 , w2 ), (u2 , w2 ); s} is called admissible by kinetic relation criterion if, besides the entropy criterion (3.18) and Rankine–Hugoniot condition, the function f defined by (3.17), also satisfies f (w1 , w2 ) = ϕ(s)
(3.19)
for a function ϕ predetermined by the material. The kinetic relation criterion (3.19) actually is a stricter version of the entropy criterion: ∂t E(u, w) + ∂x F (u, w) = µ 0,
(3.20)
where µ is a given nonpositive measure [55]. Abeyaratne and Knowles [1] implemented the kinetic relation criterion onto trilinear materials, i.e., the function p(w) is three-piecewise linear and of the shape depicted in Figure 1. They proved that the Riemann solver satisfying their kinetic relation, the initiation criterion, and the entropy inequality is unique. Later in [2], they extended above result to nonisothermal case with heat conduction taken into consideration. The speed of the phase boundary is not constant. They found an integro-differential equation for the speed. LeFloch [55] proved the L1 continuous dependence of the Riemann solver, admissible by kinetic relation criterion, on the Riemann data. Abeyaratne and Knowles [1] also showed that, at least for trilinear materials, the traveling wave criterion and the entropy rate criterion, etc., when applied to subsonic phase boundaries, is a kind of kinetic relation
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criterion. Later, Fan [30] showed that traveling wave criteria are kinetic relation criteria if p(w) is symmetric around the point (w∗ = (α + β)/2, p(w∗ )). Natalini and Tang [60] considered some discrete kinetic models with the objective of providing a practical tool encompassing various kinetic relations for the phase boundaries. Similar to Shearer’s results on nonuniqueness of the Riemann solver, the Riemann problem (3.1) has two solutions admissible by kinetic relation criterion for some Riemann data. One solution entirely lies in one of the phase region {(u, w); w < α} (or {(u, w); w > β}) with no phase boundary while the other solution has two phase boundaries and hence takes values in both phase regions. For convenience, we shall call the first solution the one-phase solution and the latter the two-phase solution. In fact, this nonuniqueness phenomenon is common for a large class of local admissibility criteria which are local restrictions on points of discontinuities of solutions of (3.1) [30]. To handle this nonuniqueness, Abeyaratne and Knowles [1] used an initiation criterion which specifies a critical value for a function h and assert that phase transitions will occur and hence the two-phase solution is “good” if the value of h exceeds the critical value. Otherwise, the one-phase solution should be picked. LeFloch [55] considered the (1.3) with initial data being a BV perturbation of an admissible phase boundary. He constructed solutions of (3.1)1 and (3.1)2 with p being trilinear. The selection criteria used are the kinetic relation criterion, initiation criterion and entropy inequality. He constructed solutions by Glimm’s scheme and proved these solutions are admissible by the above selection criteria. In recent papers Bedjaoui and LeFloch has investigated the relation between the kinetic relation and viscosity– capillarity [5,6]. The instability from Glimm scheme when applied to (1.3) is discussed by Pego and Serre [63]. Asakura [4] studied the Cauchy problem for (1.3) with initial data being Maxwell stationary phase boundary plus a small perturbation. He showed that there exists a global in time propagating phase boundary which is admissible in the sense that it satisfies the kinetic relation criterion; the states outside the phase boundary tend to the Maxwell states as time goes to infinity. Colombo and Corli [16] constructed Riemann semigroup of (1.3) admissible by Φ-relation, a generalized kinetic relation criterion. In particular, this result allows the authors to build a complete theory of existence (via front tracking) and continuous dependence on the initial data of the solutions. Related papers have been done by Corli [17–19]. Entropy rate criteria. Another interesting admissibility criterion is the entropy rate criterion proposed by Dafermos [21]. This criterion asserts that the weak solution of (3.1) which dissipates the total entropy the fastest is the admissible solution. The total entropy is typically the total mechanical energy: E(t) :=
∞
−∞
1 2 1 2 −P (w) + u + P (w0 ) − u0 dx, 2 2
(3.21)
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where (w0 , u0 ) is the initial value. The dissipation of the total energy when the solutions are piecewise smooth is measured by
dE(t) =− χ (t)f w(χ(t)−, t), w(χ(t)+, t) , dt
(3.22)
shocks
where f is given in (3.17). Dafermos [22] further justified this admissibility criterion by proving that in strictly hyperbolic systems, wave fans satisfying Liu’s shock admissibility criterion [56] consisting of rarefaction waves and shocks of moderate strength do maximize the rate of entropy production. In elastodynamics, this statement holds for arbitrary shock strength. Hattori [41,42,44] and Pence [64] applied this criterion in their study of Riemann problems of systems of conservation laws of mixed type. Hattori [43] further studied initial value problems of (1.3) using the entropy rate admissibility criterion and Glimm’s scheme. He proved the existence of weak solutions when initial data is a BV perturbation of Riemann data and the perturbation is compactly supported. Among admissibility criteria mentioned in the above, the vanishing viscosity criteria (1.3) and (3.13) and the entropy rate criterion are of global nature. Others, such as traveling wave criteria and the kinetic relation criterion are local restrictions at points of jump discontinuity. It is a hope that these local restrictions can characterize completely admissible solutions of (3.1). If this hope fails, more conditions, probably conditions of global nature, should be imposed. Thus it is important to experiment with various criteria with global authority, especially those motivated by physics. For example, when the Riemann solvers admissible by a local admissibility criteria are not unique, which one does a global admissibility criteria pick? A comparison of the effect of vanishing viscosity criterion (3.13), entropy rate criterion and traveling wave criterion derived from (3.13) in the context of (3.1) is made in [30]. In [30], initial data is chosen such that there are two solutions of (3.1), admissible by the traveling wave criterion, one being the one-phase solution and the other the two-phase solution. It is found that only the one-phase solution is admissible by the vanishing viscosity criterion (3.13), at least for the special pressure function p given in [30]. However, the entropy rate criterion picks the two-phase solution. We note that by the vanishing viscosity criterion, the two-phase solution cannot be initiated from the initial data in (3.1). Once the two-phase solution is initiated, it is quite stable, at least visually in numerical simulations [73]. More results on the viscous system (1.2). Hattori and Mischaikow [46] studied the initial boundary value problem ut t = σ (ux )x + νuxxt − ηuxxxx , u(0, t) = 0, σ ux (1, t) + νuxt (1, t) − ηuxxx (1, t) = P ,
(3.23)
uxx (1, t) = uxx (0, t) = 0. Note that Equation (3.23)1 can be tranformed to (1.2) under the variable change v = ut , w = ux . They proved the existence and uniqueness of the global solution of (4.1) under certain growth condition on σ (q). They also proved the existence of global compact
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attractor. They obtained the complete bifurcation diagram. The existence and large time behavior of initial boundary value problem of (4.1)1 with η = 0 is studied by Pego [62]. He showed that discontinuities of the solution are stationary, that the energy of the system cannot be minimized as t → ∞, among other things. Hattori and Li [45] studied the initial value problems of a fluid dynamic model for materials of Korteweg type in two-dimension: ρt + (ρu)x + (ρv)y = 0, (ρu)t + ρu2 x + (ρuv)y + p(ρ)x = (T11 )x + (T12 )y , (ρv)t + (ρuv)x + ρv 2 y + p(ρ)y = (T21 )x + (T22 )y .
(3.24)
They proved the existence of the unique local solution. Their proof does not depend on the monotonicity of the pressure function and hence can be used for van der Waals pressure. Nicolaenko [61] showed the existence of inertial manifolds for (1.2). The proof used the slightly dissipative Hamiltonian structure of the system. Milani, Eden and Nicolaenko [58] established the existence of local attractors and of exponential attractors of finite fractal dimension. This showed that even in regions of mixed type, the initial value problem exhibits finite-dimensional dynamical behavior. Serre [68] used a very interesting approach computing the formal oscillatory limit, in the spirit of Whitham [84], of the thermo-visco-capillarity system as the small parameters tends to zero. He then obtained a system of modulation equations for the limiting motion.
4. Existence of solutions of the Riemann problem (3.1) In this section, we recall our proofs for the following results on the existence of solutions of (3.1) in [77,28]. Their approach is to use the vanishing similarity viscosity (3.14) to establish the existence of weak solutions of (3.1) that are also admissible by traveling wave criterion derived from (3.13). Reasonable shock admissibility criteria should be compatible with translations and dilations of coordinates, under which the system is invariant. Dafermos [22] argued that admissibility should be tested in the framework of Riemann problem, i.e., in the context of solutions of the form U (x, t) = V (x/t) which represent wave fans emanating from the origin at time t = 0. Thus we utilize the system (3.14), which is invariant under translations and dilations of coordinates, to handle the Riemann problem (3.1). This approach has been pursued by many authors in their stydies of Riemann problems [20,54,27,53,81,77,82]. For convenience, we shall call solutions of (3.1) constructed in this way admissible according to the similarity viscosity criterion. The main results are as follows: T HEOREM 4.1. Assume in (3.1) that w± ∈ / [α, β] and p(w) → ±∞ as w → ∓∞.
(4.1)
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Then there is a sequence {εn }, εn → 0+ as n → ∞ such that the solution of (3.14) with initial data (3.1)3 converges almost everywhere to a weak solution u(x, t) of (3.1). Furthermore this solution also satisfies the traveling wave criterion derived from (3.13). The structure of solutions of (3.1) when w− < α < β < w+ constructed in Slemrod [77] and [26,28,29] by the similarity viscosity approach is as follows: each of these solutions can be embedded on a continuous curve in (u, w) the phase plane. Solutions must have a phase boundary, i.e., w(ξ ) ∈ / (α, β) for any ξ ∈ R. Solutions consist of two wave fans: ξ < 0 the first kind wave fan and ξ > 0 the second kind wave fan. A first (second) kind wave fan consists of 1-shocks and (2-shocks) and 1-simple waves (2-simple waves) and possibly the phase boundary and constant states. ξ = 0 is either a constant state or the phase boundary (cf. [26]). Most of above results are generalized by Lee [54] to the system ut − f (v)x = 0,
(4.2)
vt − g(u)x = 0,
where f is strictly increasing and convex, and g is increasing (and either concave or convex) except in a finite interval where it is decreasing and so the system is hyperbolic–elliptic mixed type. Under stricter restrictions on p(w), we have the following uniqueness results: T HEOREM 4.2. Assume conditions in Theorem 4.1. If w− < α < β < w+ (or w+ < α < β < w− ) and that p (w) > 0
for w < α,
and p (w) < 0
for w > β,
(4.3)
then (i) the solution of (1.3) satisfying the traveling wave criterion based on (1.2) is unique, and (ii) the solution (uε , wε )(x, t) of (3.14) converges almost everywhere to the unique solution of (1.3) as ε → 0+. The statement (i) of Theorem 4.2 is proved in [27]. The statement (ii) follows immediately from (i) and Theorem 4.1. The rest of Section 4 is devoted to the proof of Theorem 4.1.
4.1. Existence of solutions of the Riemann problem (1.3) To take the advantage of the invariance of (3.14) under dilatation of coordinates, we make variable change ξ = x/t in (3.14). A simple computation shows that (3.14) reduces to the
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following system εu = −ξ u + p(w) , εw = −ξ w − u , (u, w)(−∞) = (u− , w− ),
(4.1.0) (u, w)(+∞) = (u+ , w+ ).
Our program for proving Theorem 4.1 is to show that there is a solution of (4.1.1) with total variation bounded uniformly in ε > 0. Then the first statement in Theorem 4.1 follows. The proof for the second statement of Theorem 4.1 will be given in Section 4.3. To this end, we consider, instead of (4.1.0), the following altered system εu = −ξ u + µp(w) , εw = −ξ w − µu , u(±L), w(±L) = (u± , w± ),
(4.1.1)
where L > 1, 0 µ 1. L EMMA 4.1.1 [77]. Let (uε (ξ ), wε (ξ )) be the solution of (4.1.0). Then one of the following holds on any subinterval (a, b) for which p (wε (ξ )) < 0. (1) Both uε (ξ ) and wε (ξ ) are monotone on (a, b). (2) One of the uε (ξ ) and wε (ξ ) is a strictly increasing (decreasing) function with no critical point on (a, b) while the other has at most one critical point that is necessarily a local maximum (minimum) point. (3) If the critical point in (2) is of w(ξ ), then the condition p (w(ξ )) < 0 can be relaxed to p (w(ξ )) 0 for ξ ∈ (a, b). Now, we rewrite Lemma 2.2 of [77] which describes the shape of a solution of (4.1.0) in the elliptic region {(u, w) ∈ R2 : w ∈ (α, β)}. L EMMA 4.1.2. Let (u(ξ ), w(ξ )) be a solution of (4.1.1) with µ > 0. Then on any interval (l1 , l2 ) ⊂ (−L, L) for which p (w(ξ )) > 0 the graph of u(ξ ) versus w(ξ ) is convex at points where w (ξ ) > 0 and concave at points where w (ξ ) < 0. By considering (4.1.1), the existence of the connecting orbit problem (4.1.0) can be proved, as shown in the following theorem. T HEOREM 4.1.3. Suppose u− < u+ and w± < α. Then there is a solution of (4.1.0) satisfying that
u(ξ1 ), w(ξ1 ) = u(ξ2 ), w(ξ2 ) for any ξ1 , ξ2 ∈ (−∞, +∞), ξ1 = ξ2
and w(ξ1 ) w¯ := max(w− , w+ )
(4.1.2)
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and that there are at most two disjoint open intervals (a, b) such that w(ξ ) ∈ (w, ¯ α)
for ξ ∈ (a, b)
(4.1.3) (4.1.3a)
and either w(a) = w, ¯
w(b) = α
or w(a) = α,
w(b) = w, ¯
(4.1.3b)
provided that the possible solution of (4.1.1) satisfying (4.1.2) and (4.1.3) is bounded in C 1 ([−L, +L]), $ $ $(u, w)$ 1 < M, (4.1.3c) C ([−L,L];R2 ) for some M > 0 independent of µ ∈ [0, 1] and L > 1. P ROOF. We rewrite (4.1.1) as εy (ξ ) = µf (y) − ξy (ξ ),
(4.1.4)
where y(ξ ) =
u(ξ ) , w(ξ )
f y(ξ ) =
p(w) . −u(ξ )
Multiplying (4.1.4) by the factor exp(−ξ 2 /(2ε)) and integrate twice, we can rewrite (4.1.4) as the integral equation: 2 −τ µ ξ exp f y(τ ) dτ y(ξ ) = y(−L) + z(y) dτ + 2ε ε −L −L 2 ξ ζ 2 µ τ −ζ − τf y(τ ) exp dτ dζ, ε −L −L 2ε
ξ
(4.1.5a)
where
1 µ L f Y (τ ) dτ y(+L) − y(−L) − z(Y ) = L −ξ 2 ε −L −L exp 2ε dξ 2 L ζ τ − ζ2 µ τf Y (τ ) exp + dτ dζ ε −L −L 2ε = z1 (Y ) + µz2 (Y ).
(4.1.5b)
Choose η ∈ (w, ¯ α).
(4.1.6)
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We are interested in those functions (u(ξ ), w(ξ )) ∈ C 1 ([−L, +L]; R2 ) satisfying
u(ξ1 ), w(ξ1 ) = u(ξ2 ), w(ξ2 ) for any ξ1 , ξ2 ∈ [−L, +L], ξ1 = ξ2 and w(ξ1 ) η
(4.1.7)
and that there are at most two disjoint open intervals (a, b) such that w(ξ ) ∈ (η, α) for ξ ∈ (a, b), and either w(a) = η, w(b) = α or w(a) = α, w(b) = η.
(4.1.8)
We note that (4.1.7) and (4.1.8) is invariant under small C 1 perturbations. The subset in C 1 ([−L, +L]; R2 ) $ % $ Ω := (u, w) ∈ C 1 [−L, L]; R2 : $(u, w)$C 1 ([−L,L];R2) < M + 1, & and (4.1.7) and (4.1.8) are satisfied
(4.1.9)
is open. We define an integral operator T : Ω × [0, 1] → C 1 [−L, L]; R2 by 2 −ζ µ ξ T (Y, µ)(ξ ) = y(−L) + z(Y ) exp f Y (ζ ) dζ dζ + 2ε ε −L −L 2 ξ ζ 2 µ τ −ζ dτ dζ, (4.1.10) − τf Y (τ ) exp ε −L −L 2ε
ξ
where z(Y ) is given by (4.1.5b). It is clear that a fixed point of T (Y, µ) is a solution of (4.1.1). It is a matter of routine analysis to show that T maps Ω × [0, 1] continuously into C 1 ([−L, L]; R2 ). Furthermore, we can verify, by taking d/dξ twice on (4.1.10), that T maps Ω × [0, 1] into a bounded, with bound independent of µ, subset of C 2 ([−L, L]; R2). Thus T is a compact operator from C 1 ([−L, L]; R2 ) × [0, 1] into C 1 ([−L, L]; R2). We recall the following fixed point theorem ([57], Theorem IV.1). P ROPOSITION 4.1.4. Let X be a real normed vector space and Ω a bounded open subset of X. Let T : Ω × [0, 1] → X be a compact operator. If (i) T (x, µ) = x for x ∈ ∂Ω, µ ∈ [0, 1], and (ii) the Leray–Shauder degree DI (T (·, 0) − I, Ω) = 0, where I is the identity operator, then T (x, 1) = x has at least one solution in Ω.
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To solve our problem, we take X = C 1 ([−L, +L]; R2). We can see that (ii) is satisfied. Indeed, T (Y, 0) − Y = Y0 − Y,
(4.1.11)
where y(L) − y(−L) Y0 := L −ζ 2 −L exp 2ε dζ
2 ζ dζ + y(−L). exp − 2ε −L ξ
We note that T (Y, 0) = Y0 ∈ Ω is the solution of (4.1.4) when µ = 0. It is a fixed function, independent of Y and µ. Then we have DI (T (·, 0) − I, Ω) = DI (Y0 − I, Ω) = 1, as desired. Now, we preceed to verify (i) of Proposition 4.1.4. We assume, for contradiction, that there is a fixed point of T (Y, µ), Y = (u, w)(ξ ) ∈ ∂Ω.
(4.1.12)
Then one of the following cases must hold: Case A. (u(ξ ), w(ξ )) C 1 ([−L,L];R2) = M + 1. This case is impossible to occur under the condition (4.1.3c). Case B. The condition (4.1.7) is violated. In this case, there are ξ1 , ξ2 ∈ [−L, +L], ξ1 < ξ2 , such that (u(ξ1 ), w(ξ1 )) = (u(ξ2 ), w(ξ2 )) and w(ξ1 ) η. The curve (u(ξ ), w(ξ )) in (u, w)-plane near ξ = ξ1 and ξ = ξ2 cannot go across each other.1 This is because if otherwise, the curve (u(ξ ), w(ξ )) in (u, w)-plane plus a C 1 ([−L, +L]; R2 ) perturbation still intersects itself and hence is not in Ω. Thus, (u(ξ ), w(ξ )) is not in ∂Ω which yields a contradiction. From Lemma 4.1.1, we know that if (u(ξ ), w(ξ )) stays inside the region w α, the curve (u(ξ ), w(ξ )) cannot intersect itself. Thus, w(ξ3 ) > α for some ξ3 ∈ [−L, +L]. 1 Here, we clarify the meaning of “two curves go across each other”: For two curves, (u , w )(ξ ) and 1 1 (u2 , w2 )(ζ ), to cross each other in (u, w)-plane, they have to intersect each other first:
(u1 , w1 )(ξ1 ) = (u2 , w2 )(ζ2 ) at some points ξ1 , ζ2 . For convenience, we parameterize the two curves by the length of curve s with s = 0 denoting above point of intersection. If the two curves coincide with each other near s = 0, then the orientation of the parameterization should be such that (u1 , w1 )(s) = (u2 , w2 )(s)
(1)
over [s− , s+ ] with 0 ∈ [s− , s+ ]. In particular, if no such coincidence is present, then s− = s+ = 0. We further let the interval [s− , s+ ] be the largest on which (1) holds. We use the following notations: Tj (s) is the tangential direction of the j -th curve, j = 1, 2, k the normal direction of the (u, w)-plane, which is a constant vector in R3 . We say that curves (u1 , w1 )(s) and (u2 , w2 )(s) go across each other if (1) holds and (T1 (s) × T2 (s)) · k does not change sign on an open interval containing [s− , s+ ].
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We can further describe the curve (u(ξ ), w(ξ )) in the (u, w)-plane as follows. There is an interval [−L, θ1 ] such that w(ξ ) α and w(θ1 ) = α, and by Lemma 4.1.1, w (θ1 ) > 0. As ξ increases from θ1 , (u(ξ ), w(ξ )) moves into the region α < w < β. As long as w (ξ ) > 0 and w(ξ ) ∈ (α, β), the curve (u(ξ ), w(ξ )) in the (u, w)-plane is convex with respect to w. Let (θ1 , θ2 ) be the largest interval such that w (ξ ) > 0 and w(ξ ) ∈ (α, β).
Fig. 5.
Then either w(θ2 ) = β or w(θ2 ) ∈ (α, β) and w (θ2 ) = 0 holds. For definiteness, we assume that w(θ2 ) = β and w (θ2 ) > 0, since the other case is simpler. In view of Lemma 4.1.1, this interval is followed by another interval [θ2 , θ3 ] in which w(ξ ) β and u (ξ ) > 0 while w(ξ ) has one and only one critical point which is a local maximum / [−L, θ3 ). Then there is the maximum interval point and w(θ3 ) = β. This shows that ξ2 ∈ [θ3 , θ4 ) in which w(ξ ) ∈ [α, β], w (ξ ) < 0 and the curve (u(ξ ), w(ξ )) in the (u, w)-plane is concave with respect to w. We see that at the right end of the interval, either w(θ4 ) = α,
w (θ4 ) < 0
w(θ4 ) α,
w (θ4 ) = 0
or
holds. We claim that w (θ4 ) = 0 is impossible because if otherwise the concavity would make w (ξ ) > 0 for ξ > θ4 and near θ4 . This would force the curve (u, w)(ξ ) to go across itself in the region w α in order to reach w(L) = w+ . This is contradictory to (u, w)(·) ∈ ∂Ω. Thus, w(θ4 ) = α and w (θ4 ) < 0 hold. This also shows that if ξ2 ∈ [θ3 , θ4 ), then ξ1 ∈ [θ1 , θ2 ] and w (ξ1 ) 0,
w (ξ2 ) 0,
w(ξ ) > w(ξ1 )
for ξ ∈ (ξ1 , ξ2 ).
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Fig. 6.
This, however, will lead to a contradiction by integrating (4.1.1b): 0<
w(ξ ) − w(ξ2 ) dξ = ε w (ξ2 ) − w (ξ1 ) 0.
ξ2 ξ1
Above description shows that the point of self intersection ξ2 ∈ / [θ1 , θ4 ]. Following [θ3 , θ4 ] is the interval [θ4 , θ5 ) in which η < w(ξ ) α. Let [θ4 , θ5 ) be the largest of such interval. Then w(θ5 ) = η,
w (θ5 ) < 0
(4.1.13)
holds because if otherwise, w(ξ ) would have a local minimum point in [θ4 , θ5 ) and w(θ5 ) = α, see Figure 6. By Lemma 4.1.1, u(ξ ) would decrease over the interval [θ4 , θ5 ). After ξ = θ5 , w(ξ ) would enter the w < α region. Then the curve (u, w)(·) in (u, w)-plane would have to go across itself in order to connect to (u+ , w+ ), which is prohibited. We further claim that over the interval (θ5 , L], w(ξ ) < η. Indeed, if otherwise, either w(ξ ) > α for some ξ > θ5 or w(ξ ) is less than α and has multiple extreme points in [θ5 , L]. The case of multiple extreme points are impossible in view of Lemma 4.1.1. The other case that w(ξ4 ) > α for some ξ4 ∈ (θ5 , L] is also impossible since it and (4.1.13) imply that there are at least three disjoint open intervals (a, b), bounded away from each other, such that w(a) = α (or η) and w(b) = η (or α). But this is impossible for a function (u, w)(·) ∈ ∂Ω. This claim implies that the points of self-intersection satisfies ξ2 ∈ [θ4 , θ5 ], ξ1 ∈ [−L, θ1 ] and w(ξ2 ) ∈ [η, α]. Thus, we have w (ξ2 ) < 0 and w (ξ1 ) > 0, and w(ξ ) w(ξ2 ) = w(ξ1 ) for ξ ∈ [ξ1 , ξ2 ]. Integrating (4.1.1b) over [ξ1 , ξ2 ] and using (u(ξ1 ), w(ξ1 )) = (u(ξ2 ), w(ξ2 )), we obtain 0<
ξ2
w(ξ ) − w(ξ2 ) dξ = ε w (ξ2 ) − w (ξ1 ) < 0,
ξ1
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which is a contradiction. Thus, Case B cannot happen. Case C. The condition (4.1.8) is violated. That is, there are more than two disjoint open intervals (a, b) such that η < w(ξ ) < α for ξ ∈ (a, b) and w(a) = η (or α), w(b) = α (or η). Since (u, w)(ξ ) ∈ ∂Ω, the number of disjoint open connected component intervals (a, b) with w(a) = η (or α) and w(b) = α is four or more. Two of such intervals are (a1 , b1 ), (a2 , b2 ) with a2 = b1 and w (b1 ) = 0. If w(b1 ) = η and hence w(a1 ) = w(b2 ) = α, then ξ = b1 is a local minimum point of w(ξ ). According to our discussion of Case B, it is necessary that b1 < θ4 and it is impossible that w(b2 ) = α. This contradiction shows that w(b1 ) = η. We claim that the other possibility w(b1 ) = α cannot happen either. Indeed, if w(b1 ) = α and hence w(a1 ) = w(b2 ) = η < α, the point ξ = b1 is a local maximum point for w(ξ ). We see that w(ξ ) α for all ξ ∈ [−L, L] since if otherwise, w(ξ ) would have multiple extreme points in one of the connected component of {ξ ∈ [−L, L]: w(ξ ) α} which is impossible according to Lemma 4.1.1. Then, the number of disjoint open connected component intervals (a, b) with w(a) = η (or α) and w(b) = α is just two, not four or more. This contradiction shows that Case C cannot occur. Summarizing our analysis for above three cases, we find that if (u(ξ ), w(ξ )) ∈ ∂Ω, then Y = (u(ξ ), w(ξ )) cannot be a fixed point of T (Y, µ) for µ ∈ [0, 1]. Applying Proposition 4.1.4, we see that T (Y, 1) has a fixed point. To prove the existence of solutions of (4.1.0), we need to pass to the limit L → ∞. We follow Dafermos [20] and extend (u(ξ ), w(ξ )) as follows (u+ , w+ ), if ξ > L, u(ξ ; L), w(ξ ; L) = (u− , w− ), if ξ < −L. By the hypothesis (4.1.4), we see that {(u(·; L), w(·; L))} is precompact in C((−∞, ∞); R2 ). So, there is a sequence Ln → ∞ as n → ∞ such that (u(ξ ; Ln ), w(ξ ; Ln )) → (u(ξ, ∞), w(ξ, ∞)) uniformly as n → ∞. By integrating (4.1.1a, b) with µ = 1 twice from ξ0 , we can prove the limit (u(ξ, ∞), w(ξ, ∞)) satisfies (4.1.1a, b). It remains to prove that (u(±∞, ∞), w(±∞, ∞)) = (u± , w± ). To this end, we manipulate (4.1.1a, b) to obtain 2
1 ξ d 2 exp ξ /2ε y (ξ ) = f y(ξ ) exp dξ ε 2ε or 1 exp ξ 2 /2ε y (ξ ) = y (0) + ε
ξ 0
2 ζ dζ. ∇f (y)y (ζ ) exp 2ε
(4.1.14)
Applying Gronwall’s inequality on (4.1.15), we obtain 2 y (ξ ) y (0) exp 2R|ξ | − ξ 2ε 2R|ξ | − ξ 2 , M exp 2ε
(4.1.15)
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where R > 0 depend at most on M, ν and ε > 0. Inequality (4.1.21) holds for y(ξ ; L) also. Then u(±∞, ∞), w(±∞, ∞) = (u± , w± ) follows from (4.1.21) easily. It remains to prove that the solution (u(ξ, ∞), w(ξ, ∞)) constructed above satisfies (4.1.2) and (4.1.3). Indeed, the same reasoning for Case B and C implies that (u(ξ, ∞), w(ξ, ∞)) satisfies (4.1.8) and (4.1.9) also. Since η ∈ (w, ¯ α) is chosen arbitrarily, (4.1.2) and (4.1.3) hold for (u(ξ, ∞), w(ξ, ∞)). C OROLLARY 4.1.5. Let (u(ξ ), w(ξ )) be a solution of (4.1.1) or (4.1.0) satisfying (4.1.2), (4.1.3). Then, (i) The subset of [−L, +L] & % ξ ∈ [−L, +L]: w(ξ ) α has at most two connected components. Furthermore, each components must have −L or +L as one of its endpoints. (ii) The set % & ξ ∈ [−L, +L]: w(ξ ) ∈ (α, β) consists of at most two connected components. (iii) The set % & ξ ∈ [−L, +L]: w(ξ ) β , if nonempty, is an interval. P ROOF. This is proved in our discussion in the proof of Theorem 4.1.3, Case B.
The assumption (4.1.4) in above theorem can be replaced by a weaker one, as stated in the following theorem. T HEOREM 4.1.6. The conclusion of Theorem 4.1.3 remains valid if (4.1.4a) is replaced by sup u(ξ ) + w(ξ ) M1 , −Lξ L
where M1 is independent of µ ∈ [0, 1] and L > 1. P ROOF. The proof is the same as that of Theorem 1.3 in [77].
Theorems 4.1.3 and 4.1.6 give the conditions under which (4.1.0) has a connecting orbit for w± < α and u− < u+ . Slemrod [77] proved the following theorem for the case w± < α and u− < u+ :
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T HEOREM 4.1.7. Assume that w± < α and u− < u+ . Then, there is a solution of (4.1.0) satisfying w(ξ ) α,
(4.1.16)
if every possible solution of (4.1.1) satisfies $ $ $ u(ξ ), w(ξ ) $
C([−L,+L];R2)
C
(4.1.17)
for some constant C independent of µ ∈ [0, 1] and L > 1.
4.2. A-priori estimates In this section, we shall prove the a-priori estimates needed in Theorems 4.1.3 and 4.1.7 as well as some ε-independent estimates. Let denotes a solution of (4.1.1) with the properties (4.1.2) and (4.1.3). For clarity, we shall use (u(ξ ), w(ξ )) instead of (uε (ξ ), wε (ξ )) in this section if no confusion is expected. T HEOREM 4.2.1. Suppose w± < α and u− < u+ . Let (uε (ξ ), wε (ξ )) be a solution of (4.1.1) with the properties (4.1.2) and (4.1.3). Then, $ $ $uε (ξ )$ C, C([−L,+L];R2 )
(4.2.1)
where C is, throughout this section, a constant independent of ε > 0, µ ∈ [0, 1] and 1 < L +∞. P ROOF. When µ = 0, our assertion can be easily verified. Thus, we assume µ > 0 in the rest of the proof. We first prove uε (ξ ) C. Let ξε be a local minimum point of uε (ξ ). Then either wε (ξε ) ∈ / (α, β),
wε (ξε ) < 0
(4.2.2)
wε (ξε ) ∈ (α, β),
wε (ξε ) > 0
(4.2.3)
or
hold. Case A. (4.2.2) holds. In this case, by Lemma 4.1.1, w(ξε ) < β since if otherwise both uε (ξ ) and wε (ξ ) would have critical points in the set {ξ ∈ [−L, +L]: wε (ξ ) β}, which is an interval by Corollary 4.1.5. Thus, w(ξε ) α and hence % & ξε ∈ ξ ∈ [−L, +L]: wε (ξ ) α = [−L, θ1 ] ∪ [θ4 , +L],
(4.2.4)
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where θ1 θ4 . If θ1 < θ4 and ξε ∈ [−L, θ1 ], then w (ξε ) < 0 implies that wε (ξ ) also has a critical point in [−L, θ1 ] which is prohibited by Lemma 4.1.1. Thus, ξε ∈ [θ4 , L]. We can regard the curve (u, w)(ξ ) in the (u, w)-plane as a function u(w). Then we have duε (ξ ) u (ξ ) = . dwε (ξ ) w (ξ ) Performing a calculation on (4.1.1), we obtain ε
d dξ
duε (ξ ) dwε (ξ )
duε (ξ ) − −p (wε (ξ )) dwε (ξ ) duε (ξ ) × + −p (wε (ξ )) . dwε (ξ )
=µ
(4.2.5a)
if |duε (ξ )/dwε (ξ )| This implies that, as ξ increases, duε (ξ )/dwε (ξ ) is decreasing −p (wε (ξ )) and is increasing if |duε (ξ )/dwε (ξ )| −p (wε (ξ )). Thus the “initial” condition duε (ξ ) =0 (4.2.5b) dwε (ξ ) ξ =ξε leads to that for ξ ∈ [θ4 , +L] duε (ξ ) −p (w) dw (ξ ) w+max wβ ε
(4.2.6)
and hence uε (ξ ) u+ + (α − w− ) max
w∈[w+ ,α]
−p (w) .
(4.2.7)
Case B. (4.2.3) holds. By Corollary 4.1.5, [−L, L] can be divided as [−L, L] = [−L, θ1 ] ∪ (θ1 , θ2 ) ∪ [θ2 , θ3 ] ∪ (θ3 , θ4 ) ∪ [θ4 , +L],
(4.2.8)
where, of course θ1 θ2 θ3 θ4 , and & % ξ ∈ [−L, +L]: wε (ξ ) α = [−L, θ1 ] ∪ [θ4 , L],
(4.2.9a)
% & ξ ∈ [−L, +L]: wε (ξ ) ∈ (α, β) = (θ1 , θ2 ) ∪ (θ3 , θ4 ),
(4.2.9b)
& % ξ ∈ [−L, +L]: wε (ξ ) β = [θ2 , θ3 ].
(4.2.9c)
It is clear that when (4.2.3) holds, ξε ∈ (θ1 , θ2 ) = ∅ (cf. Figure 5).
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According to the sign of ξε , we have two cases: Case B(1). ξε 0.
(4.2.10)
Since ξε is a local minimum point of u(ξ ), uε dx > 0 for ξ ∈ (ξε , ξε + δ) for some δ > 0. Then we can define % & η1 := sup ζ > ξε : uε dx > 0 for ξ ∈ (ξε , ζ ) .
(4.2.11)
Since wε (ξε ) α by (4.2.3), and wε (ξε ) > 0, there is a local maximum point η2 of wε (ξ ) with η2 > ξε . We can further require that η2 is the least of such points, i.e., % & η2 := sup ζ > ξε : wε (ζ ) > 0 .
(4.2.12)
Then, by Lemmas 4.1.1 and 4.1.2, η1 ∈ / (ξε , η2 ) and hence (cf. Figure 5) (4.2.13)
η1 > η2 > ξε . By integrating (4.1.1a) on (ξε , ξ ) where ξ ∈ (ξε , η2 ), we obtain 0 < εuε dx =
ξ ξε
−ζ uε (ζ ) dζ + µ p wε (ξ ) − p wε (ξε ) .
It follows from (4.2.10) and (4.2.11) that −ξ uεn dx < 0 for ξ ∈ (ξε , η1 ). Thus, in view of (4.2.3), we have 0 < εuε (ξ ) µ p wε (ξ ) − p wε (ξε ) µ p wε (ξ ) − p(α) for ξ ∈ (ξε , η2 ).
(4.2.14)
Therefore, α < wε (η2 ) w1
(4.2.15)
holds, where w0 := γ ,
w1 := ν
in Figure 1. Equation (4.2.13) also yields a useful inequality 0 < εuε (ξ ) µ p(β) − p(α) for ξ ∈ [ξε , η1 ].
(4.2.16)
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Using (4.1.1), we can obtain
dwε (ξ ) 2 d2 wε −µ (ξ ) = . 1 + p wε (ξ ) du2ε εuε dx duε (ξ )
(4.2.17)
Hence, if dwε (ξ ) 1 du (ξ ) 2 max ε w∈[w0 ,w1 ] ( |p (w)|) and ξ ∈ [ξε , η1 ], then 1 −µ d2 wεn − < 0. (ξ ) 2 2εuε dx 2(p(β) − p(α)) duεn
(4.2.18)
Thus, as ξ decreases from η2 to ξε , dw/du will increases from 0 and eventually 1 dwε (ξ ) = duε (ξ ) ξ =η3 2 maxw∈[w0 ,w1 ] ( |p (w)|)
(4.2.19)
for some η3 ∈ (η2 , ξε ). Let % & η4 := sup η3 ∈ (ξε , η2 ): (4.2.19) is satisfied .
(4.2.20)
Then, dwε (ξ ) dwε (ξ ) = − duε (ξ ) ξ =η4 duε (ξ ) ξ =η2 2 maxw∈[w0 ,w1 ] ( |p (w)|) uε (η4 ) 2 d wεn = (ξ ) d uε (ξ ) 2 du uε (η2 εn 1
uε (η2 ) − uε (η4 ) 2(p(β) − p(α))
or 0 uε (η2 ) − uε (η4 )
p(β) − p(α) . maxw∈[w0 ,w1 ] ( |p (w)|)
From (4.2.18), we also see that 1 dwε (ξ ) duε (ξ ) 2 maxw∈[w0 ,w1 ] ( |p (w)|)
(4.2.21)
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or duε (ξ ) max |p (w)| dw (ξ ) 2 w∈[w 0 ,w1 ] ε for ξ ∈ (ξε , η4 ). Thus, 0 uε (η2 ) − uε (ξε ) = uε (η2 ) − uε (η4 ) +
wε (η4 ) wε (ξε )
duε (ξ ) dwε dwε (ξ )
max |p (w)| (w1 − w0 ),
p(β) − p(α) +2 w∈[w0 ,w1 ] maxw∈[w0 ,w1 ] ( |p (w)|)
(4.2.22)
where we used (4.2.15) and η4 ∈ (ξε , η1 ). Similarly, we can prove that 0 uε (η1 ) − uε (η2 )
p(β) − p(α) |p (w)| (w1 − w0 ). + 2 max w∈[w0 ,w1 ] maxw∈[w0 ,w1 ] ( |p (w)|)
(4.2.23)
Then we obtain 2(p(β) − p(α)) maxw∈[w0 ,w1 ] ( |p (w)|) max |p (w)| (w1 − w0 ).
uε (ξε ) uε (η1 ) − −4
w∈[w0 ,w1 ]
(4.2.24)
If uε (η1 ) u+ , then, (4.2.24) shows that uε (ξ ) is bounded from below uniformly in ε > 0, µ ∈ [0, 1] and L > 1. Now, we devote our attention to the case when uε (η1 ) < u+ . Then, η1 < L because uε (L) = u+ . By the definition (4.2.11), of η1 , uε (η1 ) = 0. Then, by Lemma 4.1.1 and 4.1.2, η1 has to be an extreme point for uε (ξ ). Since uε dx > 0 for ξ ∈ (ξε , η1 ), η1 is a local maximum point. Lemmas 4.1.2 and 4.1.1 implies that either wε (η1 ) > 0 and wε (η1 ) ∈ / (α, β)
(4.2.25)
wε (η1 ) < 0 and wε (η1 ) ∈ (α, β).
(4.2.26)
or
The case (4.2.25) cannot happen because it implies that η1 ∈ [−L, θ1 ] which violates the known fact that η1 > ξε ∈ [θ1 , θ2 ). Then (4.2.26) infers that there is a local minimum point η5 > η4 of uε (ξ ) which satisfies uε (η5 ) = 0 and wε (η5 ) α.
(4.2.27)
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Then our argument for the Case A applies and gives us u+ − uε (η5 ) (α − w+ ) max −p (w) . w∈[w+ ,α]
(4.2.28)
Using (4.2.28) in (4.2.24), we obtain the desired result 2(p(β) − p(α)) maxw∈[w0 ,w1 ] ( |p (w)|) max |p (w)| (w1 − w0 )
uε (ξε ) uε (η1 ) − −4
w∈[w0 ,w1 ]
2(p(β) − p(α)) maxw∈[w0 ,w1 ] ( |p (w)|) max |p (w)| (w1 − w0 )
uε (η5 ) − −4
w∈[w0 ,w1 ]
u+ − (α − w+ ) max
w∈[w+ ,α]
−p (w)
2(p(β) − p(α)) maxw∈[w0 ,w1 ] ( |p (w)|) − 4 max |p (w)| (w1 − w0 ), −
w∈[w0 ,w1 ]
(4.2.29)
which proves that uε (ξ ) is bounded from below uniformly in ε > 0, µ ∈ [0, 1] and L > 1. Case B(2) ξε < 0. The proof is similar to Case B(1). The only difference is that instead of (4.2.11), we define % & η1 = inf ζ < ξε : u (ξ ) > 0 for ξ ∈ (ζ, ξε ) and change the rest of the proof accordingly. Similarly, we can also prove that uε (ξ ) is bounded from above uniformly in ε > 0, µ ∈ [0, 1] and L > 1. In the remainder of this section, we adopt the following notation: % & u∗ := sup uε (ξ ) | ξ ∈ R, ε ∈ (0, 1) ,
(4.2.30a)
% & u∗ := inf uε (ξ ) | ξ ∈ R, ε ∈ (0, 1) .
(4.2.30b)
Once we established the a-priori estimates for uε (ξ ), we can proceed to prove the following results for wε (ξ ) by using the similar argument used in [20]. T HEOREM 4.2.2. Assume w± < α and u− < u+ . Let (uε (ξ ), wε (ξ )) be a solution of (4.1.1) satisfying (4.1.2) and (4.1.3). Then
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(i) wε (ξ ) C([−l,+l];R2) C(ε) where C(ε) is independent of µ ∈ [0, 1], L > 1. (ii) If |p(w)| → ∞, as |w| → ∞, and if µ = 1, then wε (ξ ) C([−L,+L];R2) C where C is independent of L > 1 and ε > 0. P ROOF. We only prove that w(ξ ) is bounded from above uniformly. The other part of the proof is similar and is omitted. (i) Without loss of generality, we assume wε (ξ ) has a local maximum point τε . We further assume that τε 0.
(4.2.31)
The proof for the other case is similar. By Lemmas 4.1.1 and 4.1.2, uε (τε ) > 0.
(4.2.32)
& % η := inf ξ < τε : wε dx > 0 .
(4.2.33)
We define
It is clear wε (η) 0. Integrating (4.1.1b), we obtain 0 −εwε (η) = −
τε
ξ wε dx dξ + µ uε (η) − uε (τε ) .
(4.2.34)
η
By the definition (4.2.33), we see that ξ wε dx 0 on (η, τε ) and hence τε ξ wε dx dξ µ uε (η) − uε (τε ) u∗ − u∗ . 0
(4.2.35)
η
If ξ min(−1, τε ), then
ξ η
ζ wε (ζ ) dζ −
η
ξ
wε (ζ ) dζ = wε (η) − wε (ξ ).
From the definition (4.2.33), we know that either η = −L or η is a local minimum point of uε (ξ ), In view of Lemmas 4.1.1 and 4.1.2, wε (η) ∈ [min(w− , w+ ), β]. Then above inequality yields wε (ξ ) − η
ξ
ζ wε (ζ ) dζ + wε (η) u∗ − u∗ + β
(4.2.36)
for all ξ min(−1, τε ). In other words, wε (ξ ) is bounded from above uniformly in ε > 0, µ ∈ [0, 1] and L > 1 if ξ min(−1, τε ). For ξ ∈ (−1, τε ], we have, from (4.1.1b), that τε 0 −εwε dx = −ζ wε (ζ ) dζ + µ uε (ξ ) − uε (τε ) u∗ − u∗ . ξ
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This implies that wε (τε ) wε (−1) + C1 (ε) u∗ − u∗ + β + C1 (ε). Thus, the statement (i) is proved. (ii) It remains to consider the case when τε ∈ (−1, 0] and µ = 1. For each ε, we can choose θ ∈ (−2, −1) such that uε (θ ) u∗ − u∗ . By integrating (4.1.1a), with µ = 1, on [θ, τε ], we obtain τε p wε (τε ) = εuε (τε ) − εuε (θ ) + p wε (θ ) − ξ uε dx dξ
−εuε (θ ) + p wε (θ ) −
θ
τε
ξ uε dx dξ.
(4.2.37)
θ
Every term on the right-hand side of (4.2.33) is bounded uniformly in ε > 0 and L > 1. Thus, by virtue of the assumption on p in the theorem, wε (τε ) are bounded from below uniformly in ε > 0 and L > 1. T HEOREM 4.2.3. Assume w± < α and u− > u+ . Let (uε (ξ ), wε (ξ )) be a possible solution of (4.1.1) satisfying wε (ξ ) α. Then (i) uε (ξ ) C([−L,+L];R2 C where C is a constant independent of L > 1, µ ∈ [0, 1] and ε > 0. (ii) wε (ξ ) C([−L,+L];R2 C(ε) where C(ε) is independent of µ ∈ [0, 1], L > 1. (iii) If |p(w)| → ∞, as |w| → ∞, and if µ = 1, then wε (ξ ) C([−L,+L];R2 C where C is independent of L > 1 and ε > 0.
P ROOF. The proof is almost the same as that of Theorem 4.2.2. Combining Theorems 4.1.3, 4.1.4, 4.2.1 and 4.2.2, we obtain the following result.
T HEOREM 4.2.4. (i) Assume w± < α and u− < u+ . There is a solution (uε (ξ ), wε (ξ )) of (4.1.0) satisfying uε (ξ1 ), wε (ξ1 ) = uε (ξ2 ), wε (ξ2 ) for any ξ1 , ξ2 ∈ (−∞, +∞), ξ1 = ξ2
(4.2.38)
and wε (ξ1 ) w¯ := min(w− , w+ ) and that there are at most two disjoint open intervals (a, b) such that ¯ α), wε (ξ ) ∈ (w,
and
¯ wε (b) = α either wε (a) = w, wε (b) = w. ¯ or wε (a) = α,
(4.2.39) (4.2.39a) (4.2.39b)
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(ii) For the case w± < α and u− > u+ , there is a solution of (4.1.0) satisfying wε (ξ ) α. (iii) There is a subsequence {εn }, εn → 0+ as n → ∞, such that (uεn (ξ ), wεn (ξ )) given in (i) and (ii) converges a.e. to a weak solution (u(ξ ), w(ξ )) of the Riemann problem (3.1). Furthermore, the solutions we constructed have at most two phase boundaries. P ROOF. (i) and (ii) Theorems 4.1.6, 4.2.1, 4.2.2 and 4.2.3 provide the a-priori estimates needed by Theorems 4.1.3 and 4.1.7 Thus, parts (i), (ii) are established. (iii) From Corollary 4.1.5, we know that the solutions of (4.1.0) provided in (i), uε (ξ ) and wε (ξ ) are piecewise monotone. Thus, ({uε (ξ ), wε (ξ )) given in (i) has total variation bounded uniformly in ε > 0. Then the classical Helly’s theorem states that there is a sequence {εn }, εn → 0+ as n → ∞, such that (uε (ξ ), wε (ξ )) converges almost everywhere. Apply this limit to the weak form of (3.14), we see that the limit is a weak solution of (3.1). 4.3. Solutions constructed by vanishing similarity viscosity are also admissible by traveling wave criterion In this section, we shall prove that solutions constructed by vanishing similarity viscosity is also admissible by traveling wave criterion. Let (uε (ξ ), wε (ξ )) denote the solution of (4.1.0). From last section, we see that (uε (ξ ), wε (ξ )) are bounded uniformly in ε > 0. Let us denote the upper and lower bounds of uε (ξ ) by u∗ and u∗ respectively. Similarly, the upper and lower bounds of wε (ξ ) is denoted by w∗ and w∗ respectively. For simplicity of presentation, we restrict ourselves to the case w− < α < β < w+ . In this case, solutions (uε (ξ ), wε (ξ )) of (4.1.0) have the following shapes: there are two points ξ = θ1 < θ2 , depending on ε, such that wε (ξ ) α,
for ξ ∈ (−∞, θ1 ],
α < wε (ξ ) < β
for ξ ∈ (θ1 , θ2 ),
(4.3.1a) wε (θ1 ) = α,
wε (θ2 ) = β
(4.3.1b)
and wε (ξ ) β
for ξ θ2 .
(4.3.1c)
According to Lemmas 4.1.1 and 4.1.2, over each of the intervals (−∞, θ1) and (θ2 , ∞), there are three possibilities: (i) The function wε (ξ ) has one local extreme point and uε (ξ ) is monotone. (ii) The function uε (ξ ) has one local extreme point and wε (ξ ) is monotone. (iii) Both uε (ξ ) and wε (ξ ) are monotone. Over the interval (θ1 , θ2 ), inequalities wε (ξ ) > 0 and d2 uε /dwε2 > 0 hold. According to shapes (i)–(iii), there are nine different combinations of shapes for (uε (ξ ), wε (ξ )). We also see that over each of the regions w α, α < w < β and w β, we can consider the curve (uε (ξ ), wε (ξ )) in (u, w)-plane as the curve of the function Uε (w) or Wε (u), depending on which of uε (ξ ) and wε (ξ ) is monotone.
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L EMMA 4.3.1. Let δ > 0 be some fixed small number so that w− < α − δ < β + δ < w+ .
(4.3.2)
(a) In the region w α −δ (or w β +δ) either dUε (w)/dw or dWε (u)/du is uniformly bounded in ε. (b) In the region α − δ w β + δ, dUε (w)/dw is uniformly bounded. P ROOF. (a) According to the possibilities for shapes of (uε (ξ ), wε (ξ )) in the region w α and w β, there are three cases: Case A. There is a critical point for uε (ξ ) in the region w α (or w β). We can calculate from (4.1.0) to get
ε
d dξ
dUε (w) dw
=
dUε (w) dw
2
+ p wε (ξ ) .
(4.3.3)
At the critical point ξ = τε of uε (ξ ), dUε (w)/dw|ξ =τε = 0. Then Equation (4.3.3) says that dUε (w) dw ξ =ξ is decreasing as ξ increases from τε . But dUε (w)/dw cannot decrease to below − maxw∗ ww∗ |p (w)|since d/dξ(dUε (w)/dw) will become positive if dUε (w)/dw reaches − maxw∗ ww∗ |p (w)|. Similarly, as ξ decreases from τε , dUε (w)/dw will ∗ increase but never reaches maxw∗ ww |p (w)| because if it does, d/dξ(dUε (w)/dw) will become negative. This shows that dUε (w) max ∗ |p (w)| dw w∗ ww
(4.3.4)
if uε (ξ ) has a critical point τε with wε (τε ) α or wε (τε ) β. Case B. The function wε (ξ ) has a critical point in the region w α (or w β). Then, uε (ξ ) is necessarily monotone in the region. If the critical point is in the region w α, then it is the absolute minimum point of wε (ξ ), according to Lemma 4.1.1. If the critical point τε is in the region w β, then it is the absolute maximum point of wε (ξ ). Thus, wε (τε ) is in the region w α − δ (or w β + δ). From (4.1.0), we derive that d ε dξ
dWε (u) du
dWε (u) 2 = − 1 + p wε (ξ ) . du
(4.3.5)
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At the critical point ξ = τε , we have ing (4.3.3), we can prove that
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411
= 0. Similar to our analysis follow-
dWε (u) 1 max du w∈[w∗ ,α−δ]∪[β+δ,w ∗] |p (w)|
(4.3.6)
for w ∈ [w∗ , α − δ] ∪ [β + δ, w∗ ]. Case C. There is no critical point for uε (ξ ) or wε (ξ ) in the region w α (or w β). For this case, we claim that one of |dUε (w)/dw| and |dWε (u)/du| are bounded uniformly. To prove this claim, it suffices to prove if one of them is not bounded uniformly, then the other is. There are the following three possibilities: Subcase C1. The function dUε (w)/dw is not bounded from below uniformly in ε in the region w α. Let the absolute minimum point of dUε (w)/dw in the closed region w α be ξ = τε . Then there is a sequence εn , such that duεn → −∞ as n → ∞. dw ξ =τεn
(4.3.7)
For simplicity, we denote this sequence by ε. In the region w α, it is necessary that dwεn / dξ > 0 in order to connect to w(∞) = w+ > β > α. Therefore, we have duεn / dξ < 0 in the region w α due to (4.3.7). Evaluating Equation (4.3.5) around the point ξ = τε , we find that dUε (w)/dw is increasing at ξ = τε , the absolute minimum point of dUε (w)/dw over w− w α. This implies wε (τε ) = w− and hence τε = −∞. Now, applying our reasoning from (4.3.5)–(4.3.6) and using (4.3.15): dWε (u) → 0 as n → ∞, du ξ =τεn we can obtain (4.3.6). Subcase C2. The function dUε (w)/dw is not bounded from above uniformly in ε in the region w α. Let the absolute maximum point of dUε (w)/dw in the closed region {ξ : wε (ξ ) α} = [−∞, θ1] be ξ = τε . Then there is a sequence εn , such that dUε (w) → ∞ as n → ∞. dw ξ =τεn
(4.3.8)
For simplicity, we denote this sequence by ε. Similar to Subcase C1, when w α, it is necessary that dwεn / dξ > 0. Therefore, we have duεn / dξ > 0 in the region w α. By Equation (4.3.5), we find that dUε (w)/dw is increasing at ξ as long as dUε (w) max ∗ |p (w)|. dw ξ =ξ w∗ ww
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This implies wε (τε ) = α. Recalling that wε (θ1 ) = α and wε (θ2 ) = β, we can see that τε = θ1 and dUε (w) dUε (w) dw dw ξ =θ1
for θ1 ξ θ2
(4.3.9)
due to Lemma 4.1.2. Then we have uεn (θ2 ) − uεn (θ1 ) =
w(θ2 ) w(θ1 )
duεn dUε (w) dw (β − α) → ∞ dw dw ξ =θ1
as n → ∞. This violates the uniform boundedness of uε (ξ ), Theorem 4.2.1. This contradiction shows that Subcase C2 cannot occur. Subcase C3. The function dUε (w)/dw is not bounded uniformly in ε in the region w β. The proof for this case is similar to Subcases C1 and C2. Combining the Cases A–C, we complete the proof of (a). (b) Since Wε (u) is convex in the region α w β, the absolute extreme values of dWε (u)/du over the region α − δ w β + δ must occur in the region [α − δ, α] ∪ [β, β + δ]. Our proof for Subcase C2 for (a) shows that dUε (w)/dw is bounded uniformly from above when w ∈ [α − δ, α]. Now we shall prove that dUε (w)/dw is also bounded uniformly from below when w ∈ [α − δ, α]. To this end, we assume its contrary, i.e., there is a sequence {εn } such that dUε (w) → −∞ dw ξ =τε
(4.3.10)
as n → ∞ for some τε ∈ R with wεn (τε ) ∈ [α − δ, α]. Equation (4.3.3) implies that dUε (w)/dw is decreasing as ξ decreases when dUε (w) − max ∗ |p (w)|. dw w∗ ww Let ξ = ξ1 −∞ be the point such that w(ξ1 ) = w− and w− wε (ξ ) wε (τε ) for ξ ∈ (ξ1 , τε ]. Then, we have uεn (τε ) − uεn (ξ1 ) =
w(τε ) w(ξ1 )
duεn dUε (w) dw (α − δ − w− ) → −∞ dw dw ξ =τε
as n → ∞. This violates the uniform boundedness of wε (ξ ), Theorem 4.2.2. This contradiction proves that dUε (w)/dw is uniformly bounded from above in the region [α − δ, α]. The same proof can be used to prove that dUε (w)/dw is uniformly bounded when w ∈ [β, β + δ].
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Now, we consider the ε → 0+ limit of a convergent subsequence of (uε (ξ ), wε (ξ )). We denote the convergent subsequence of (uε (ξ ), wε (ξ )) by (uεn (ξ ), wεn (ξ )) and the limit by (u(ξ ), w(ξ )). L EMMA 4.3.2. Let (uεn (ξ ), wεn (ξ )) be a convergent sequences of (uε (ξ ), wε (ξ )). Then, there is a subsequence of {εn }, denoted by {εn } again, such that {Uεn (w)} converges to a locally Lipschitz continuous (in u or w) curve. Furthermore, the limit (u(ξ ), w(ξ )) lies on this curve for every ξ ∈ R. P ROOF. By further extracting subsequences, we can make all duεn /dw or all dwεn /du to be bounded uniformly in εn over the region w α − δ. The same can be achieved for the region w β + δ. For definiteness and simplicity of presentation, we consider the case where all dwεn /du are bounded uniformly in the region w α − δ, and all duεn /dw are bounded uniformly in the region w β + δ. Then, the curve (uεn (ξ ), wεn (ξ )) in (u, w)plane, can be regarded as a function of w in the region w∗ w β + δ and a function of u in the region w α − δ. These two pieces of curves are connected by the part of the curve (uεn (ξ ), wεn (ξ )) over the interval α − δ1 w β + δ1 ,
(4.3.11)
δ1 > δ
(4.3.12)
where
and satisfies (4.3.2). This middle piece can be considered, by Lemma 4.3.1(b), as a function of w. Each of these three pieces of curves are uniformly bounded in C 1 over their domains of definition which are intervals bounded uniformly in ε. Thus, there is a subsequence of {εn }, denoted by {εn } again, such that all these three pieces converges as n → ∞ to Lipschitz continuous (in variable u or w) curves in (u, w)-plane. Due to the overlaps of the middle curve with the other two pieces, (4.3.11)–(4.3.12), the three pieces of the limit curves form a continuous curve in (u, w)-plane. This curve is locally Lipschitz in u or w with Lipschitz constant uniformly bounded in ε. We call this curve the base curve. Now, we prove that the limit (u(ξ ), w(ξ )) is on the base curve. Fix a ξ ∈ R. For definiteness, we shall assume that w(ξ ) α − δ. All other cases can be handled similarly. Either w(ξ ) < α − δ or w(ξ ) = α − δ > α − δ1 . In either cases, duεn /dw is bounded uniformly in εn and the base curve is parameterized as U (w) in the region w∗ w β. Then we have U w(ξ ) − u(ξ ) = lim Uε w(ξ ) − uε (ξ ) n n n→∞ = lim Uεn w(ξ ) − Uεn wεn (ξ ) n→∞ lim C w(ξ ) − wεn (ξ ) = 0. n→∞
Thus, the point (u(ξ ), w(ξ )) is on the base curve U (w).
(4.3.13)
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For convenience, we parameterize the base curve (U (s), W (s)) where s is the length of the arc joining points (u− , w− ) and (U (s), W (s)). In this kind of parameterization, the s defined by (u(ξ ), w(ξ )) = (U (s), W (s)) increases when ξ increases. Now, we study the discontinuities of (u(ξ ), w(ξ )). Let ξ0 be a point of discontinuity of (u(ξ ), w(ξ )). We use Cξ0 to denote the portion of the base curve in the (u, w)-plane that connects points (u(ξ0 −), w(ξ0 −)) and (u(ξ0 +), w(ξ0 +)). We fix (u, ¯ w) ¯ ∈ Cξ0 . Similar to [20], we define, for n large, ξεn (w; u, ¯ w) ¯ to be the branch of the inverse function of w = wεn (ξ ) for which ¯ u, ¯ w) ¯ → u¯ uεn ξεn (w;
(4.3.14)
as n → ∞. We further define, for n large, ξεn , uˆ εn , w #εn by the relations ¯ + εζ, ξεn := ξεn (w)
(4.3.15)
uˆ εn (ζ ) := uεn (ξεn ),
(4.3.16)
w #εn (ζ ) := wεn (ξεn ).
(4.3.17)
#εn (ζ )) L EMMA 4.3.3. Let ξ0 be a point of discontinuity of (u(ξ ), w(ξ )). For (uˆ εn (ζ ), w defined above, there is a subsequence of {εn }, also denoted by {εn }, such that #εn (ζ ) → u(ζ ˆ ), w #(ζ ) ∈ C 1 R; R2 uˆ εn (ζ ), w
as n → ∞
(4.3.18)
uniformly for ζ in a compact subset of R. (u(ζ ˆ ), w #(ζ )) satisfies the following initial value problem: du(ζ ˆ ) = −ξ0 u(ζ ˆ ) − u(ξ0 −) + p w #(ζ ) − p w(ξ0 −) , dζ
(4.3.19a)
d# w(ζ ) = −ξ0 w #(ζ ) − w(ξ0 −) − u(ζ ˆ ) − u(ξ0 −) , dζ
(4.3.19b)
u(0) ˆ = u¯
(4.3.19c)
w #(0) = w. ¯
Furthermore, (u(ζ ˆ ), w #(ζ )) lies on Cξ0 . #εn (ζ )) have uniformly bounded total variation since (uε (ξ ), P ROOF. Clearly, (uˆ εn (ζ ), w wε (ξ )) do. Thus, there is a subsequence of {εn }, again denoted by {εn }, such that uˆ εn (ζ ), w #εn (ζ ) → u(ζ ˆ ), w #(ζ ) as n → ∞ for any ζ ∈ R.
(4.3.20)
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By Lemma 4.3.1, we can choose a small neighborhood Vξ0 of (u(ξ0 −), w (ξ0 −)) in the (u, w)-plane such that duεn ξ ∈ R such that uεn (ξ ), wεn (ξ ) ∈ Vξ0 (4.3.21a) dw or
dwεn du
ξ ∈ R such that uεn (ξ ), wεn (ξ ) ∈ Vξ 0
(4.3.21b)
is bounded uniformly in n. Since in each of the region w α, α w β, w β, at least one of uε (ξ ) and wε (ξ ) is monotone, in each of above three regions, one of U (s) and W (s) is monotone. We can further choose Vξ0 small and (uδ , wδ ) ∈ Cξ0 ∩ Vξ0 such that U (s) or W (s) is monotone in Vξ0 . For definiteness, we can assume, without loss of generality, that (4.3.21a) holds and both wεn (ξ ) and W (s) is monotone in Vξ0 . The proof for the other case is similar. There is, for n large, √ θεn ∈ ξεn (wδ ) − εn , ξεn (wδ ) ,
(4.3.22)
such that u (θεn ) √3 T V (uε ) 3M/√εn , εn εn
(4.3.23a)
w (θε ) √3 T V (wε ) 3M/√εn . n εn εn
(4.3.23b)
From (4.3.7) and
uεn (ξ ), wεn (ξ ) → u(ξ ), w(ξ ) ,
it is easily seen that ξεn (wδ ) → ξ0
(4.3.24)
and hence θεn → ξ0 . Since W (s) and wεn (ξ ) are monotone, the limit lim infn→∞ wεn (θεn ) lies between w(ξ0 −) and wδ . Thus, extracting, if necessary, another subsequence, we deduce wεn (θεn ) → w2
as n → ∞
(4.3.25)
for some w2 between w(ξ0 −) and wδ . Then, by (2.8a), we have that limn→∞ uεn wεn (θεn ) = U (w2 ) = u2 ,
(4.3.26)
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where (u2 , w2 ) ∈ Vξ0 . For simplicity, we shall write ε instead of εn in the rest of this section. Integrating Equations (1.3) from θε to τε := ξε (w) ¯ + εζ , we get duˆ ε (ζ ) = −ξ0 uˆ ε (ζ ) − uε (θε ) + p w #ε (ζ ) − p wε (θε ) dζ τε + εuε (θε ) − (ξ − ξ0 )uε (ξ ) dξ,
(4.3.27a)
d# wε (ζ ) = − uˆ ε (ζ ) − uε (θε ) − ξ0 w #ε (ζ ) − wε (θε ) dζ τε + εwε (θε ) − (ξ − ξ0 )wε (ξ ) dξ.
(4.3.27b)
θε
θε
By (4.3.23) εwε (θε ) and εuε (θε ) approach 0 as ε → 0 uniformly in ζ . Recalling that θε → ξ0 , τε → ξ0 as n → ∞, uniformly in ζ for ζ in compact subsets of R, we see that the last term in (4.3.27a, b) vanish, as n → ∞, uniformly in ζ in a compact set. A classical theorem of the theory of ordinary differential equations implies that (uˆ ε (ζ ), w #ε (ζ )) → (u(ζ ˆ ), w #(ζ )), as n → ∞, uniformly on compact subsets of R, and that du(ζ ˆ ) = −ξ0 u(ζ ˆ ) − u2 + p w #(ζ ) − p(w2 ), dζ
(4.3.28a)
d# w(ζ ) = − u(ζ ˆ ) − u2 − ξ0 w #(ζ ) − w2 , dζ
(4.3.28b)
u(0) ˆ = u, ¯
(4.3.28c)
w #(0) = w. ¯
By letting Vξ0 shrink to (u(ξ0 −), w(ξ0 −)) so as to force (u2 , w2 ) → (u(ξ0 −), w(ξ0 −)), we obtain (4.3.19). Similar to our proof of Lemma 4.3.2, we can prove that (u(ζ ˆ ), w #(ζ )) is on the curve (U (s), W (s)) for all ζ ∈ R. We note that (u(0) ˆ =w #(0)) = (u, ¯ w) ¯ ∈ Cξ0 , and that as ζ increases (or decreases) from ζ = 0, the point (u(ζ ˆ ), w #(ζ )) moves toward the end point (u(ξ0 +), w(ξ0 +)) ((u(ξ0 −), w(ξ0 −))) along Cξ0 . The point (u(ζ ˆ ), w #(ζ )) cannot cross (u(ξ0 +), w(ξ0 +)) and (u(ξ0 −), w(ξ0 −)) to go outside of Cξ0 . This is because if it did go out of Cξ0 , there would be a point ζ1 ∈ R such that u(ζ ˆ 1 ), w #(ζ1 ) = u(ξ0 −), w(ξ0 −) or u(ξ0 +), w(ξ0 +) . Then the Rankine–Hugoniot condition, satisfied by any jump solution of (3.1) with speed ξ0 , −ξ0 u(ξ0 +) − u(ξ0 −) + p w(ξ0 +) − p w(ξ0 −) = 0,
(4.3.29a)
Dynamic flows with liquid/vapor phase transitions
−ξ0 w(ξ0 +) − w(ξ0 −) − u(ξ0 +) − u(ξ0 −) = 0
417
(4.3.29b)
yields that at (u(ζ ˆ ), w #(ζ )) ≡ (u(ξ0 −), w(ξ0 −)) or (u(ζ ˆ ), w #(ζ )) ≡ (u(ξ0 +), w(ξ0 +)) by the uniqueness of solutions of initial value problems of systems of ordinary differential equations. This, however, violates the (u(0), ˆ w #(0)) = (u, ¯ w) ¯ ensured by (4.3.14)–(4.3.17). This contradiction proves that (u(ζ ˆ ), w #(ζ )) ∈ Cξ0 for all ζ ∈ R. C OROLLARY 4.3.4. Let (u(ξ ), w(ξ )) be a weak solution of (3.1) constructed as the limit of a convergent sequence {uεn (ξ ), wεn (ξ )} of solutions of (3.14) with the same initial data (3.1)3 . Then, (u(ξ ), w(ξ )) is also admissible by the traveling wave criterion based on (3.13), which is the same as that based on (1.2) when A = 1/4. P ROOF. The limit (u(ξ ), w(ξ )) has bounded total variation. Then points of discontinuity of (u(ξ ), w(ξ )) are points of jump discontinuity. Let ξ0 be a point of jump discontinuity of (u(ξ ), w(ξ )). Lemma 4.3.3 states that (4.3.19) has a solution. We note that the system (4.3.19)1 and 2 is equivalent to the traveling wave equation (3.13) and the speed s = ξ0 . Indeed, the speed of the jump discontinuity of (u(ξ ), w(ξ )) at x/t = ξ = ξ0 is ξ0 . We note that Cξ0 in last lemma is the portion of the base curve connecting the points (u(ξ0 −), w(ξ0 −)) and (u(ξ0 +), w(ξ0 +)). As ζ increases from 0 to ∞, the point (u(ζ ˆ ), w #(ζ )) moves monotonically toward (u(ξ0 +), w(ξ0 +)) along the curve Cξ0 . In the ζ → ∞ limit, (u(ζ ˆ ), w #(ζ )) must approach to an equilibrium point of (4.3.19) on Cξ0 . Equilibrium points of (4.19) are points (u1 , w1 ) that satisfies the Rankin–Hugoniot condition (4.3.27) with (u(ξ0 +), w(ξ0 +)) replaced by (u1 , w1 ). Similarly, as ζ decreases from 0 to −∞, the point (u(ζ ˆ ), w #(ζ )) will move toward (u(ξ0 −), w(ξ0 −)) and approaches an equilibrium point in the ζ → ∞ limit. Thus, when there are only finitely many equilibrium points for each fixed speed ξ0 , the jump discontinuity (u(ξ0 −), w(ξ0 −)), (u(ξ0 +), w(ξ0 +)) can be connected together by finitely many traveling waves of the same speed ξ0 . References [1] R. Abeyaratne and J. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal. 114 (2) (1991), 119–154. [2] R. Abeyaratne and J. Knowles, Dynamics of propagating phase boundaries: thermoelastic solids with heat conduction, Arch. Rational Mech. Anal. 126 (3) (1994), 203–230. [3] E.C. Aifantis and J. Serrin, The mechanical theory of fluid interfaces and Maxwell’s rule, J. Colloidal Interface Sci. 96 (1983), 517–529 [4] F. Asakura, Large time stability of propagating phase boundaries, Hyperbolic Problems: Theory, Numerics, Applications, Vol. I (Zürich, 1998), Internat. Ser. Numer. Math., Vol. 129, Birkhäuser, Basel (1999), 21–29. [5] N. Bedjaoui and P.G. LeFloch, Difussive-dispersive traveling waves and kinetic relations: an hyperbolic– elleptic model of phase transitions, Preprint (2000). [6] N. Bedjaoui and P.G. LeFloch, Difussive-dispersive traveling waves and kinetic relations, Part I: nonconvex hyperbolic conservation laws, Preprint (2000). [7] S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid, Nonlinear Anal. 31 (1–2) (1998), 243–263. [8] S. Benzoni-Gavage, Nonuniqueness of phase transitions near the Maxwell line, Proc. Amer. Math. Soc. 127 (4) (1999), 1183–1190.
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[64] T.J. Pence, On the mechanical dissipation of solutions to the Riemann problem for impact involving a twophase elastic material, Arch. Rational Mech. Anal. 117 (1) (1992), 1–52. [65] V. Roytburd and M. Slemrod, An application of the method of compensated compactness to a problem in phase transitions, Material Instabilities in Continuum Mechanics (Edinburgh, 1985–1986), 427–463. [66] J. Serrin, Phase transitions and interfacial layers for van der Waals fluids, Proc. SAFA IV Conference, Recent Methods in Nonlinear Analysis and Applications, Naples, A. Canfora, S. Rionero, C. Sbordone and C. Trombetti, eds, Liguori, Naples (1980), 169–176. [67] J. Serrin, The form of interfacial surfaces in Korteweg’s theory of phase equilibria, Quart. Appl. Math. 41 (1983), 351–364. [68] D. Serre, Entrpie du mlange liqquide-vapour d’un fluide thermo-capillaire, Arch. Rational Mech. Anal. 128 (1994), 33–73. [69] M. Shearer, Riemann problem for a class of conservation laws of mixed type, J. Differential Equations 46 (1982), 426–443. [70] M. Shearer, Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type, Proc. Roy. Soc. Edinburgh 93 (1983), 233–244. [71] M. Shearer, Nonuniqueness of admissible solutions of Riemann initial value problem for a system of conservation laws of mixed type, Arch. Rational Mech. Anal. 93 (1986), 45–59. [72] M. Shearer, Dynamic phase transitions in a van der Waals gas, Quart. Appl. Math. 46 (1988), 631–636. [73] C.-W. Shu, Private communication. [74] M. Slemrod, Admissibility criterion for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), 301–315. [75] M. Slemrod, Dynamic phase transitions in a van der Waals fluid, J. Differential Equations 52 (1984), 1–23. [76] M. Slemrod, Dynamics of first order phase transitions, Phase Transitions and Material Instabilities in Solids (1984), 163–203. [77] M. Slemrod, A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase, Arch. Rational Mech. Anal. 105 (1989), 327–365. [78] L.M. Truskinovskii, Equilibrium phase interfaces, Dokl. Akad. Nauk SSSR 265 (1982), 306–310. [79] L.M. Truskinovskii, Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium, Prikl. Mat. Mekh. 51 (1987), 777–784; English translation: J. Appl. Math. Mech. 51 (1987), 1009–1019. [80] L.M. Truskinovskii, Structure of an isothermal phase jump, Dokl. Akad. Nauk SSSR 285 (1985), 2. [81] V.A. Tupciev, On the method of introducing viscosity in the study of problems involving the decay of discontinuity, Dokl. Akad. Nauk. SSSR 211 (1973), 55–58. [82] A.E. Tzavaras, Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws, Arch. Rational Mech. Anal. 135 (1) (1996), 1–60. [83] V.A. Weigant, Global solutions to the Navier–Stokes equations of a compressible fluid with functions of state of van der Waals type, Siberian Adv. Math. 6 (2) (1996), 103–150. [84] G.B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York (1974). [85] K. Zumbrun, Dynamical stability of phase transitions in the p-system with viscosity–capillarity, SIAM J. Appl. Math. 60 (2000), 1913–1924. [86] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of shock waves, Indiana Univ. Math. J. 47 (1998), 741–871.
CHAPTER 5
The Cauchy Problem for the Euler Equations for Compressible Fluids Gui-Qiang Chen Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA E-mail:
[email protected]
and Dehua Wang Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA E-mail:
[email protected]
Contents 1. 2. 3. 4.
5. 6.
7.
8.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Local well-posedness for smooth solutions . . . . . . . . . . Global well-posedness for smooth solutions . . . . . . . . . Formation of singularities in smooth solutions . . . . . . . . 4.1. One-dimensional Euler equations . . . . . . . . . . . 4.2. Three-dimensional Euler equations . . . . . . . . . . 4.3. Other results . . . . . . . . . . . . . . . . . . . . . . . Local well-posedness for discontinuous solutions . . . . . . Global discontinuous solutions I: Riemann solutions . . . . 6.1. The Riemann problem and Lax’s theorems . . . . . . 6.2. Isothermal Euler equations . . . . . . . . . . . . . . . 6.3. Isentropic Euler equations . . . . . . . . . . . . . . . 6.4. Non-isentropic Euler equations . . . . . . . . . . . . . Global discontinuous solutions II: Glimm solutions . . . . . 7.1. The Glimm scheme and existence . . . . . . . . . . . 7.2. Decay of solutions . . . . . . . . . . . . . . . . . . . 7.3. L1 -stability of Glimm solutions . . . . . . . . . . . . 7.4. Wave-front tracking algorithm and L1 -stability . . . . Global discontinuous solutions III: entropy solutions in BV 8.1. Generalized characteristics and decay . . . . . . . . . 8.2. Uniqueness of Riemann solutions . . . . . . . . . . .
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8.3. Large-time stability of entropy solutions . . . . . . . . . . . . . . . . 9. Global discontinuous solutions IV: entropy solutions in L∞ . . . . . . . . . 9.1. Isentropic Euler equations . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Entropy–entropy flux pairs . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Compactness framework . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Convergence of the Lax–Friedrichs scheme and the Godunov scheme 9.5. Existence and compactness of entropy solutions . . . . . . . . . . . . 9.6. Decay of periodic entropy solutions . . . . . . . . . . . . . . . . . . . 9.7. Stability of rarefaction waves and vacuum states . . . . . . . . . . . . 9.8. Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Global discontinuous solutions V: the multidimensional case . . . . . . . . . 10.1. Multidimensional Euler equations with geometric structure . . . . . . 10.2. The multidimensional Riemann problem . . . . . . . . . . . . . . . . 11. Euler equations for compressible fluids with source terms . . . . . . . . . . 11.1. Euler equations with relaxation . . . . . . . . . . . . . . . . . . . . . 11.2. Euler equations for exothermically reacting fluids . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract Some recent developments in the study of the Cauchy problem for the Euler equations for compressible fluids are reviewed. The local and global well-posedness for smooth solutions is presented, and the formation of singularity is exhibited; then the local and global wellposedness for discontinuous solutions, including the BV theory and the L∞ theory, is extensively discussed. Some recent developments in the study of the Euler equations with source terms are also reviewed.
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1. Introduction The Cauchy problem for the Euler equations for compressible fluids in d space dimensions is the initial value problem for the system of d + 2 conservation laws ∂t ρ + ∇ · m = 0, m⊗m ∂t m + ∇ · + ∇p = 0, ρ m (E + p) = 0, ∂t E + ∇ · ρ
(1.1)
d+1 d for (x, t) ∈ Rd+1 + , R+ := R × (0, ∞), with initial data
(ρ, m, E)|t =0 = (ρ0 , m0 , E0 )(x),
x ∈ Rd ,
(1.2)
where (ρ0 , m0 , E0 )(x) is a given vector function of x ∈ Rd . System (1.1) is closed by the constitutive relations p = p(ρ, e),
E=
1 |m|2 + ρe. 2 ρ
(1.3)
In (1.1) and (1.3), τ = 1/ρ is the deformation gradient (specific volume for fluids, strain for solids), v = (v1 , . . . , vd ), is the fluid velocity, with ρv = m the momentum vector, p is the scalar pressure, and E is the total energy, with e the internal energy which is a given function of (τ, p) or (ρ, p) defined through thermodynamical relations. The notation a ⊗ b denotes the tensor product of the vectors a and b. The other two thermodynamic variables are the temperature θ and the entropy S. If (ρ, S) are chosen as the independent variables, then the constitutive relations can be written as (e, p, θ ) = e(ρ, S), p(ρ, S), θ (ρ, S) ,
(1.4)
governed by θ dS = de + p dτ = de −
p dρ. ρ2
(1.5)
For a polytropic gas, p = Rρθ,
e = cv θ,
γ =1+
R , cv
(1.6)
and p = p(ρ, S) = κρ γ eS/cv ,
e=
κ ρ γ −1 eS/cv , γ −1
(1.7)
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where R > 0 may be taken to be the universal gas constant divided by the effective molecular weight of the particular gas, cv > 0 is the specific heat at constant volume, γ > 1 the adiabatic exponent, and κ > 0 can be any constant under scaling. As will be shown in Section 4, no matter how smooth the Cauchy data (1.2) are, solutions of (1.1) generally develop singularities in a finite time. Hence, system (1.1) is complemented by the Clausius inequality (1.8) ∂t ρa(S) + ∇ · ma(S) 0 in the sense of distributions for any a(S) ∈ C 1 , a (S) 0, in order to single out physically relevant discontinuous solutions, called entropy solutions. The Euler equations for a compressible fluid that flows isentropically take the following simpler form: ∂t ρ + ∇ · m = 0, m⊗m ∂t m + ∇ · + ∇p = 0, ρ
(1.9)
where the pressure is regarded as a function of density, p = p(ρ, S0 ), with constant S0 . For a polytropic gas, p(ρ) = κ0 ρ γ ,
γ > 1,
(1.10)
where κ0 > 0 is any constant under scaling. This system can be derived as follows. It is well known that, for smooth solutions of (1.1), the entropy S(ρ, E) is conserved along fluid particle trajectories, i.e., ∂t (ρS) + ∇ · (mS) = 0.
(1.11)
If the entropy is initially a uniform constant and the solution remains smooth, then (1.11) implies that the energy equation can be eliminated, and the entropy S keeps the same constant in later time, in comparison with non-smooth solutions (entropy solutions) for which only S(x, t) min S(x, 0) is generally available (see [297]). Thus, under constant initial entropy, a smooth solution of (1.1) satisfies the equations in (1.9). Furthermore, it should be observed that solutions of system (1.9) are also a good approximation to solutions of system (1.1) even after shocks form, since the entropy increases across a shock to third-order in wave strength for solutions of (1.1) (cf. [120]), while in (1.9) the entropy is constant. Moreover, system (1.9) is an excellent model for isothermal fluid flow with γ = 1, and for shallow water flow with γ = 2. In the one-dimensional case, system (1.1) in Eulerian coordinates is ∂t ρ + ∂x m = 0, 2 m + p = 0, ∂t m + ∂x ρ m ∂t E + ∂x (E + p) = 0, ρ
(1.12)
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2
with E = 12 mρ + ρe. The system above can be rewritten in Lagrangian coordinates in oneto-one correspondence so long as the fluid flow stays away from vacuum ρ = 0: ∂t τ − ∂x v = 0, ∂t v + ∂x p = 0, v2 + ∂x (pv) = 0, ∂t e + 2
(1.13)
with v = m/ρ, where the coordinates (x, t) are the Lagrangian coordinates, which are different from the Eulerian coordinates for (1.12); for simplicity of notations, we do not distinguish them. For the isentropic case, systems (1.12) and (1.13) reduce to: ∂t ρ + ∂x m = 0, 2 m + p = 0, ∂t m + ∂x ρ
(1.14)
∂t τ − ∂x v = 0, ∂t v + ∂x p = 0,
(1.15)
and
respectively, where the pressure p is determined by (1.10) for the polytropic case, p = p(ρ) = p(τ ˜ ), τ = 1/ρ. The Cauchy problem for all the systems above fits into the following general conservation form: ∂t u + ∇ · f(u) = 0,
u ∈ Rn , x ∈ R d ,
(1.16)
with initial data: u|t =0 = u0 (x),
(1.17)
where f = (f1 , . . . , fd ) : Rn → (Rn )d is a nonlinear mapping with fi : Rn → Rn , i = 1, . . . , d. Besides (1.1)–(1.15), many partial differential equations arising in the physical or engineering sciences can also be formulated into the form (1.16) or its variants. The hyperbolicity of system (1.16) requires that, for any ω ∈ S d−1 , the matrix (∇f(u) · ω)n×n have n real eigenvalues λi (u, ω), i = 1, 2, . . . , n, and be diagonalizable. One of the main difficulties in dealing with (1.16) and (1.17) is that solutions of the Cauchy problem (even those starting out from smooth initial data) generally develop singularities in a finite time, because of the physical phenomena of focusing and breaking of waves and the development of shock waves and vortices, among others. For this reason, attention focuses on solutions in the space of discontinuous functions. Therefore, one can not directly use the classical analytic techniques that predominate in the theory of partial differential equations of other types.
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Another main difficulty is nonstrict hyperbolicity or resonance of (1.16), that is, there exist some ω0 ∈ S d−1 and u0 ∈ Rn such that λi (u0 , ω0 ) = λj (u0 , ω0 ) for some i = j . In particular, for the Euler equations, such a degeneracy occurs at the vacuum states or from the multiplicity of eigenvalues of the system. The correspondence of (1.8) in the context of hyperbolic conservation laws is the Lax entropy inequality: ∂t η(u) + ∇ · q(u) 0
(1.18)
in the sense of distributions for any C 2 entropy–entropy flux pair (η, q) : Rn → R × Rd , q = (q1 , . . . , qd ), satisfying ∇ 2 η(u) 0,
∇qi (u) = ∇η(u)∇fi (u),
i = 1, . . . , d.
Most sections in this paper focus on the Cauchy problem for one-dimensional hyperbolic systems of n conservation laws ∂t u + ∂x f(u) = 0,
u ∈ Rn , x ∈ R, t > 0,
(1.19)
with Cauchy data: u|t =0 = u0 (x).
(1.20)
The Euler equations can describe more complicated physical fluid flows by coupling with other physical equations. One of the most important examples is the Euler equations for nonequilibrium thermodynamic fluids. In local thermodynamic equilibrium as we discussed above, system (1.1) is closed by the constitutive relation (1.3). When the temperature varies over a wide range, the gas may not be in local thermodynamic equilibrium, and the pressure p may then be regarded as a function of only a part e of the specific internal energy, while another part q is governed by a rate equation: ∂t (ρq) + ∇x · (mq) =
Q(ρ, e) − q , εs(ρ, e)
(1.21)
p = p(ρ, e),
|m|2 + ρ(e + q), 2ρ
(1.22)
and E=
where ε > 0 is a parameter measuring the relaxation time, which is small in general, and Q(ρ, e) and s(ρ, e) are given functions of (ρ, e). The equations in (1.1) and (1.21) with (1.22) define the Euler equations for nonequilibrium fluids, which model the nonequilibrium thermodynamical processes.
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Another important example is the inviscid combustion equations that consist of the Euler equations in (1.1) adjoined with the continuum chemistry equation: φ(θ ) = Ke−θ0 /θ ,
∂t (ρZ) + ∇ · (mZ) = −φ(θ )ρZ,
(1.23)
where θ0 and K are some positive constants, Z denotes the mass fraction of unburnt gas so that 1 − Z is the mass fraction of burnt gas. Here we assume that there are only two species present, the unburnt gas and the burnt gas, and the unburnt gas is converted to the burnt gas through a one-step irreversible exothermic chemical reaction with an Arrhenius kinetic mechanism. As regards the equations in (1.1), a modification of the internal energy e is the only change in these equations. The internal energy of the mixture, e(ρ, S, Z), is defined within a constant by e(ρ, S, Z) = Zeu (ρ, S) + (1 − Z)eb (ρ, S), with eu and eb the internal energies of the unburnt and burnt gas, respectively. For simplicity, we assume that both of the burnt and unburnt gas are ideal with the same γ -law so that eu (ρ, S) = cv θ + q0 ,
eb = cv θ,
with q0 > 0 the normalized energy of formation at some reference temperature for the unburnt gas for an exothermic reaction. Then e(ρ, S, Z) = cv θ (ρ, S) + q0 Z,
θ (ρ, S) =
p(ρ, S) . Rρ
(1.24)
Then the equations in (1.1) and (1.23) with (1.24) define the inviscid combustion equations, which model detonation waves in combustion. This paper is organized as follows. In Section 2, we present a local well-posedness theory for smooth solutions and then in Section 3 a global well-posedness theory for smooth solutions. In Section 4, we exhibit the formation of singularity in smooth solutions, the main feature of the Cauchy problem for the Euler equations. In Section 5, we present a local well-posedness theory for discontinuous entropy solutions. From Section 6 to Section 10, we discuss global well-posedness theories for discontinuous entropy solutions. In Section 6, we present a global theory for discontinuous entropy solutions of the Riemann problem, the simplest Cauchy problem with discontinuous initial data. First we recall two Lax’s theorems for the local behavior of wave curves in the phase space and the existence of global solutions of the Riemann problem, respectively, for general onedimensional conservation laws with small Riemann data. Then we discuss the construction of global Riemann solutions and their behavior for the isothermal, isentropic, and nonisentropic Euler equations in (1.12)–(1.15) with large Riemann data, respectively. In Section 7, we focus on the global discontinuous solutions obtained from the Glimm scheme [130], called Glimm solutions. We first describe the Glimm scheme for hyperbolic
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conservation laws and a global well-posedness theory for the Glimm solutions, including the existence, decay, and L1 -stability of the Glimm solutions. The Glimm scheme is also applied to the construction of global entropy solutions of the isothermal Euler equations with large initial data. We also present an alternative method, the wave-front tracking method, to construct global discontinuous solutions, which can be identified with a trajectory of the standard Riemann semigroup, and to yield the L1 -stability of the solutions. In Section 8, our focus is on general global discontinuous solutions in L∞ ∩ BVloc satisfying the Lax entropy inequality and without specific reference on the method for construction of the solutions. We first describe a theory of generalized characteristics and its direct applications to the decay problem of the discontinuous solutions under the assumption that the traces of the solutions along any space-like curves are functions of locally bounded variation. Then we study the uniqueness of Riemann solutions and the asymptotic stability of entropy solutions in BV for gas dynamics, without additional a priori information on the solutions besides the natural Lax entropy inequality. In Section 9, our focus is on the one-dimensional system of the isentropic Euler equations and its global discontinuous solutions in L∞ satisfying only the weak Lax entropy inequality. We first carefully analyze the system and its entropy–entropy flux pairs. Then we describe a general compactness framework, with a proof for the case γ = 5/3, for establishing the existence, compactness, and decay of entropy solutions in L∞ , and the convergence of finite-difference schemes including the Lax–Friedrichs scheme and the Godunov scheme. We discuss the stability of rarefaction waves and vacuum states even in a broader class of discontinuous entropy solutions in L∞ . We also record some related results for the system of elasticity and the non-isentropic Euler equations. In Section 10, we discuss global discontinuous solutions for the multidimensional case. We describe a shock capturing difference scheme and its applications to the multidimensional Euler equations for compressible fluids with geometric structure. Then we present some classifications and phenomena of solution structures of the twodimensional Riemann problem, especially wave interactions and elementary waves, for the Euler equations and some further results in this direction. In Section 11, we consider the Euler equations for compressible fluids with source terms. Our focus is on two of the most important examples: relaxation effect and combustion effect. Some new phenomena are reviewed. We remark that, in this paper, we focus only on some recent developments in the theoretical study of the Cauchy problem for the Euler equations for compressible fluids. We refer the reader to other papers in these volumes, as well as Glimm and Majda [134], Godlewski and Raviart [138], LeVeque [189], Lions [201], Perthame [255], Tadmor [296], Toro [306], and the references cited therein for related topics including various kinetic formulations and approximate methods for the Cauchy problem for the Euler equations.
2. Local well-posedness for smooth solutions Consider the three-dimensional Euler equations in (1.1) and (1.7) for polytropic compressible fluids staying away from the vacuum, which are rewritten in terms of the density ρ ∈ R,
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the velocity v ∈ R3 , and the entropy S ∈ R (taking κ = cv = 1 without loss of generality) in the form: ∂t ρ + ∇ · (ρv) = 0, ∂t (ρv) + ∇ · (ρv ⊗ v) + ∇p = 0, ∂t S + v · ∇S = 0,
(2.1)
with the equation of state: p = p(ρ, S) = ρ γ eS , γ > 1. System (2.1) is a 5 × 5 system of conservation laws. It can be written in terms of the variables (p, v, S) in the equivalent form in the region where the solution is smooth: ∂t p + v · ∇p + γp ∇ · v = 0, ρ(∂t v + v · ∇v) + ∇p = 0, ∂t S + v · ∇S = 0,
(2.2)
with ρ = ρ(p, S) = p1/γ e−S/γ . The norm of the Sobolev space H s (Rd ) is denoted by
D α g 2 dx. g 2s = d |α|s R
For g ∈ L∞ ([0, T ]; H s ), define $ $ g = sup $g(·, t)$ . s,T s 0t T
For the Cauchy problem of (2.2) with smooth initial data: (p, v, S)|t =0 = (p0 , v0 , S0 )(x),
(2.3)
the following local existence theorem of smooth solutions holds. T HEOREM 2.1. Assume (p0 , v0 , S0 ) ∈ H s ∩ L∞ (R3 ) with s > 5/2 and p0 (x) > 0. Then there is a finite time T ∈ (0, ∞), depending on the H s and L∞ norms of the initial data, such that the Cauchy problem (2.2) and (2.3) has a unique bounded smooth solution (p, v, S) ∈ C 1 (R3 × [0, T ]), with p(x, t) > 0 for all (x, t) ∈ R3 × [0, T ], and (p, v, S) ∈ C([0, T ]; H s ) ∩ C 1 ([0, T ]; H s−1). Consider the Cauchy problem (1.16) and (1.17) for a general hyperbolic system of conservation laws with the values of u lying in the state space G, an open set in Rn . The state space G arises because physical quantities such as the density should be positive. Assume that (1.16) has the following structure of symmetric hyperbolic systems: For all u ∈ G, there is a positive definite symmetric matrix A0 (u) that is smooth in u and satisfies c0−1 In A0 (u) c0 In
(2.4)
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with a constant c0 uniform for u ∈ G1 , for any G1 ⊂ G1 G, such that Ai (u) = A0 (u)∇fi (u) is symmetric, where ∇fi (u), i = 1, . . . , d, are the n × n Jacobian matrices and In is the n × n identity matrix. A consequence of this structure for (1.16) is that the linearized problem of (1.16) and (1.17) is well-posed (see Majda [223]). The matrix A0 (u) is called the symmetrizing matrix of system (1.16). Multiplying (1.16) by the matrix A0 (u) and denoting A(u) = (A1 (u), . . . , Ad (u)) yield the system: A0 (u)∂t u + A(u)∇u = 0.
(2.5)
An important observation is that almost all equations of classical physics of the form (1.16) admit this structure. For example, the equations in (2.2) for polytropic gases are symmetrized by the 5 × 5 matrix A0 (p, S) =
(γp)−1 0 0
0 ρ(p, S)I3 0
0 0 . 1
Therefore, Theorem 2.1 is a consequence of the following theorem on the local existence of smooth solutions, with the specific state space G = {(p, v, S), : p > 0} ⊂ R5 , for the general symmetric hyperbolic system (1.16). T HEOREM 2.2. Assume that u0 : Rd → G is in H s ∩ L∞ with s > d/2 + 1. Then, for the Cauchy problem (1.16) and (1.17), there exists a finite time T = T ( u0 s , u0 L∞ ) ∈ (0, ∞) such that there is a unique bounded classical solution u ∈ C 1 (Rd × [0, T ]) with u(x, t) ∈ G for (x, t) ∈ Rd × [0, T ] and u ∈ C([0, T ]; H s ) ∩ C 1 ([0, T ]; H s−1). The proof of this theorem proceeds via a classical iteration scheme. An outline of the proof of Theorem 2.2 (thus Theorem 2.1) is given as follows. To prove the existence of the smooth solution of (1.16) and (1.17), it is equivalent to construct the smooth solution of (2.5) and (1.17) by applying the symmetrizing matrix A0 (u). Choose the standard mollifier j (x) ∈ C0∞ (Rd ), supp j (x) ⊆ {x: |x| 1}, j (x) 0, Rd j (x) dx = 1, and set jε (x) = ε−d j (x/ε). For k = 0, 1, 2, . . . , take εk = 2−k ε0 , where ε0 > 0 is a constant, and define uk0 ∈ C ∞ (Rd ) by uk0 (x) = Jεk u0 (x) =
Rd
jεk (x − y)u0 (y) dy.
We construct the solution of (2.5) and (1.17) through the following iteration scheme: Set u0 (x, t) = u00 (x) and define uk+1 (x, t), for k = 0, 1, 2, . . . , inductively as the solution of the linear equations: A0 uk ∂t uk+1 + A uk ∇uk+1 = 0,
uk+1 |t =0 = uk+1 0 (x).
(2.6)
From the well-known properties of the mollification: uk0 − u0 s → 0, as k → ∞, and uk0 − u0 0 C0 εk u0 1 , for some constant C0 , it is evident that uk+1 ∈ C ∞ (Rd × [0, Tk ])
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is well-defined on the time interval [0, Tk ]. Here Tk > 0 denotes the largest time where the estimate | uk − u00 |s,Tk C1 holds for some constant C1 > 0. Then there is a constant T∗ > 0 such that Tk T∗ (T0 = ∞) for k = 0, 1, 2, . . . , which follows from the following estimates: $ k+1 $ $ k+1 $ $u $ $u − u00 $s,T C1 , C2 , (2.7) t s−1,T ∗
∗
for all k = 0, 1, 2, . . . , with some constant C2 > 0. From (2.6), we obtain A0 uk ∂t uk+1 − uk + A uk ∇ uk+1 − uk = Ek ,
(2.8)
where Ek = − A0 uk − A0 uk−1 ∂t uk − A uk − A uk−1 ∇uk . Use the standard energy estimate method for the linearized problem (2.8) to obtain $ k+1 $ $ $ $ $ $u − uk $0,T CeCT $uk+1 − uk0 $0 + T $Ek $0,T . 0 The property of mollification, (2.7), and Taylor’s theorem yield $ k+1 $ $ $ Ek C $uk − uk−1 $ . $u − uk0 $0 C2−k , 0 0,T 0,T For small T such that C 2 T exp(CT ) < 1, one obtains ∞
$ k+1 $ $u − uk $0,T < ∞, k=1
which implies that there exists u ∈ C([0, T ]; L2 (Rd )) such that $ $ lim $uk − u$0,T = 0. k→∞
(2.9)
From (2.7), we have | uk |s,T + | ukt |s−1,T C, and uk (x, t) belongs to a bounded set of G for (x, t) ∈ Rd × [0, T ]. Then the interpolation inequalities imply that, for any r with 0 r < s, $ $ k $u − ul $
r,T
$ $ $1−r/s $ $r/s $1−r/s Cs $uk − ul $0,T $uk − ul $s,T C $uk − ul $0,T . (2.10)
From (2.9) and (2.10), limk→∞ | uk − u |r,T = 0 for any 0 r < s. Thus, choosing r > d/2 + 1, Sobolev’s lemma implies uk → u in C [0, t]; C 1 Rd .
(2.11)
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From (2.8) and (2.11), one can conclude that uk → u in C([0, T ]; C(Rd )), u ∈ C 1 (Rd × [0, T ]), and u(x, t) is the smooth solution of (1.16) and (1.17). To prove u ∈ C([0, T ]; H s ) ∩ C 1 ([0, T ]; H s−1), it is sufficient to prove u ∈ C([0, T ]; s H ), since it follows from the equations in (2.5) that u ∈ C 1 ([0, T ]; H s−1). The proof can be further reduced to verifying that u(x, t) is strongly right-continuous at t = 0, since the same argument works for the strong right-continuity at any other t ∈ [0, T ), and the strong right-continuity on [0, T ) implies the strong left-continuity on (0, T ] because the equations in (2.5) are reversible in time. R EMARK 2.1. Theorem 2.2 was established by Majda [223] which relies solely on the elementary linear existence theory for symmetric hyperbolic systems with smooth coefficients (Courant and Hilbert [77]), as we illustrated above. Moreover, a sharp continuation principle was also proved there: For u0 ∈ H s , with s > d/2 + 1, the interval [0, T ) with T < ∞ is the maximal interval of the classical H s existence for (1.16) if and only if either (ut , Du) L∞ → ∞ as t → T , or, as t → T , u(x, t) escapes every compact subset K G. The first catastrophe in this principle is associated with the formation of shock waves in the smooth solutions, and the second is associated with a blow-up phenomenon. Kato also gave a proof of Theorem 2.2, in [164], which uses the abstract semigroup theory of evolution equations to treat appropriate linearized problems. In [165], Kato also formulated and applied this basic idea in an abstract framework which yields the local existence of smooth solutions for many interesting equations of mathematical physics. See Crandall and Souganidis [78] for related discussions. In [226], Makino, Ukai and Kawashima established the local existence of classical solutions of the Cauchy problem with compactly supported initial data for the multidimensional Euler equations, with the aid of the theory of quasilinear symmetric hyperbolic systems; in particular, they introduced a symmetrization which works for initial data having compact support or vanishing at infinity. There are also discussions on the local existence of smooth solutions of the three-dimensional Euler equations (2.1) in Chemin [35]. R EMARK 2.2. For the one-dimensional Cauchy problem (1.19) and (1.20), it is known from Friedrichs [122], Lax [175], and Li and Yu [195] that, if u0 (x) is in C 1 for all x ∈ R with finite C 1 norm, then there is a unique C 1 solution u(x, t), for (x, t) ∈ R × [0, T ], with sufficiently small T . As a consequence, the one-dimensional Euler equations in (1.12)– (1.15) admit a unique local C 1 solution provided that the initial data are in C 1 with finite C 1 norm and stay away from the vacuum.
3. Global well-posedness for smooth solutions Consider the Cauchy problem for the one-dimensional isentropic Euler equations of gas dynamics in (1.14), for x ∈ R and t > 0, with initial data: (ρ, m)|t =0 = (ρ0 , m0 )(x),
(3.1)
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and γ -law for pressure: p(ρ) = ρ γ /γ ,
γ > 1.
(3.2)
For the case 1 < γ 3, which is of physical significance, system (1.14) is genuinely nonlinear in the sense of Lax [181] in the domain {(x, t): ρ(x, t) 0}. For ρ > 0, consider the velocity v = m/ρ and v0 (x) = m0 (x)/ρ0 (x). The eigenvalues of (1.14) are λ1 = v − c, where c = ρ θ , with θ = are
λ2 = v + c, γ −1 2
∈ (0, 1], is the sound speed. The Riemann invariants of (1.14)
w1 = w1 (ρ, v) := v +
ρθ , θ
w2 = w2 (ρ, v) := v −
ρθ . θ
Set w10 (x) := w1 ρ0 (x), v0 (x) ,
w20 (x) := w2 ρ0 (x), v0 (x)
as the initial values of the Riemann invariants. With the aid of the method of characteristics (see Lax [178]), the following global existence theorem of smooth solutions of (1.14) and (3.1) can be proved. T HEOREM 3.1. Suppose that the initial data (ρ0 , v0 )(x), with ρ0 (x) > 0, are in C 1 (R), with finite C 1 norm and (x) 0, w10
w20 (x) 0,
(3.3)
for all x ∈ R. Then the Cauchy problem (1.14) and (3.1) has a unique global C 1 solution (ρ, v)(x, t), with ρ(x, t) > 0 for all x ∈ R and t > 0. P ROOF. First we show that, if ρ0 (x) > 0, no vacuum will develop at any time t > 0 for the smooth solution. From the first equation of (1.14), d ρ = −ρ∂x v, dt
(3.4)
where d = ∂t + v(x, t)∂x dt denotes the directional derivative along the direction dx = v(x, t). dt
(3.5)
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For any point (x, ¯ t¯) ∈ R2+ := {(x, t): x ∈ R, t ∈ R+ }, R+ = (0, ∞), the integral curve of (3.5) through (x, ¯ t¯) is denoted by x = x(t; x, ¯ t¯). At t = 0, it passes through the point ¯ t¯), 0) := (x(0; x, ¯ t¯), 0). Along the curve x = x(t; x, ¯ t¯), the solution of the ordinary (x0 (x, differential equation (3.4) with initial data: ¯ t¯) ρ|t =0 = ρ0 x0 (x, is t¯ ¯ t¯) exp − ∂x v x(t; x, ¯ t¯), t dt > 0. ρ(x, ¯ t¯) = ρ0 x0 (x, 0
To prove the global existence of the C 1 solution (ρ, v)(x, t), given the local existence from Remark 2.2, it is sufficient to prove the following uniform a priori estimate: For any fixed T > 0, if the Cauchy problem (1.14) and (3.1) has a unique C 1 solution (ρ, v)(x, t) for x ∈ R and t ∈ [0, T ), then the C 1 norm of (ρ, v)(x, t) is bounded on R × [0, T ]. For a smooth solution (ρ, v) of system (1.14), one can verify by straightforward calculations that the derivatives of the Riemann invariants w1 and w2 along the characteristics are zero: w1 = 0,
w2 = 0,
(3.6)
where = ∂t + λ2 ∂x and = ∂t + λ1 ∂x are the differentiation operators along the characteristics. Differentiate the equation w1 = 0 in (3.6) with respect to the spatial variable x to obtain 2 w1 + ∂w1 λ2 (∂x w1 )2 + ∂w2 λ2 ∂x w1 ∂x w2 = 0. ∂t2x w1 + λ2 ∂xx
Since 0 = w2 = w2 − 2c∂x w2 , by setting r = ∂x w1 and noticing λ2 = λ2 (w1 , w2 ) =
1+θ 1−θ w1 + w2 , 2 2
∂x w2 =
one has r +
1+θ 2 1−θ r + w r = 0. 2 4c 2
s=
θ −1 θ −1 ln ρ = ln(w1 − w2 ). 2 2θ
Set
Then ∂w2 s =
1−θ 4c
and s = w2 ∂w2 s =
1−θ w . 4c 2
w2 , 2c
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Thus r +
1+θ 2 r + s r = 0. 2
Set g = es r = ρ (θ−1)/2∂x w1 . Then g = −
1+θ 2
θ |w1 − w2 | 2
1−θ 2θ
g2 .
(3.7)
h2 .
(3.8)
Similarly, for h = ρ (θ−1)/2 ∂x w2 , one has 1+θ h =− 2
θ |w1 − w2 | 2
1−θ 2θ
Let x = x(β, t) be the forward characteristic passing through any fixed point (β, 0) at t = 0, defined by dx(β, t) = λ2 w1 x(β, t), t , w2 x(β, t), t , dt
x(β, 0) = β.
According to (3.6), w1 is constant along characteristics, and thus w1 (x(β, t), t) = w1 (β, 0) = w10 (β) and sup |w1 (x, t)| = sup |w10 (x)|. Similarly, w2 is constant along the backward characteristics corresponding to the eigenvalue λ1 , and sup |w2 (x, t)| = sup |w20 (x)|. For any given point (x(β, t), t) on the forward characteristic x = x(β, t), there exists a unique α = α(β, t) β such that w2 (x(β, t), t) = w20 (α). Therefore, along the characteristic x = x(β, t), one has from (3.7) that 1−θ 2θ 2 1 + θ θ dg(x(β, t), t) w10 (β) − w20 α(β, t) =− g x(β, t), t , dt 2 2 θ −1 (β). g|t =0 = ρ0 (β) 2 w10
(3.9)
Then θ −1
(β) ρ0 (β) 2 w10 , g x(β, t), t = t 1 + 0 K(β, τ ) dτ
(3.10)
where 1+θ K(β, t) = 2
θ w10 (β) − w20 α(β, t) 2
1−θ 2θ
ρ0 (β)
θ −1 2
w10 (β).
(3.11)
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From (3.3), K(β, t) 0. Thus, g(x(β, t), t) is bounded, and ∂x w1 x(β, t), t =
θ w10 (β) − w20 α(β, t) 2
1−θ 2θ
g x(β, t), t
is also bounded. Similarly, ∂x w2 is also bounded from (3.8). As a consequence, the C 1 norms of ρ = (θ (w1 − w2 )/2)1/θ and v = (w1 + w2 )/2 are bounded on R × [0, T ]. The proof is complete. R EMARK 3.1. In the proof of Theorem 3.1, the second-order derivatives of the Riemann invariants are formally used. However, the final equality (3.10) does not involve these second-order derivatives. Some appropriate arguments of approximation or weak formulation can be used to show that the conclusion is still valid for C 1 solutions. R EMARK 3.2. For the global existence of smooth solutions of general one-dimensional hyperbolic systems of conservation laws, we refer the reader to Li [194] which contains some results and discussions on this subject. Also see Lin [197,198] and the references cited therein for the global existence of Lipschitz continuous solutions for the case that discontinuous initial data may not stay away from the vacuum. For the three-dimensional Euler equations for polytropic gases in (2.1), Serre and Grassin in [141,142,273] studied the existence of global smooth solutions under appropriate assumptions on the initial data for both isentropic and non-isentropic cases. It was proved in [141] that the threedimensional Euler equations for a polytropic gas in (2.1) have global smooth solutions, provided that the initial entropy S0 and the initial density ρ0 are small enough and the initial velocity v0 forces particles to spread out, which are of similar nature to the condition (3.3).
4. Formation of singularities in smooth solutions The formation of shock waves is a fundamental physical phenomenon manifested in solutions of the Euler equations for compressible fluids, which are a prototypical example of hyperbolic systems of conservation laws. This phenomenon can be explained by mathematical analysis by showing the finite-time formation of singularities in the solutions. For nonlinear scalar conservation laws, the development of shock waves can be explained through the intersection of characteristics; see the discussions in Lax [180, 181] and Majda [223]. For systems in one space dimension, this problem has been extensively studied by using the method of characteristics developed in Lax [178], John [161], Liu [206], Klainerman and Majda [170], Dafermos [83], etc. For systems with multidimensional space variables, the method of characteristics has not been proved tractable. An efficient method, involving the use of averaged quantities, was developed in Sideris [282] for hyperbolic systems of conservation laws and was further refined in Sideris [283] for the three-dimensional Euler equations. Also see Majda [223].
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4.1. One-dimensional Euler equations Consider the Cauchy problem (1.14) and (3.1) for the one-dimensional Euler equations of isentropic gas dynamics. With the notations in Section 3, the following result on the formation of singularity in smooth solutions of (1.14) and (3.1) follows. T HEOREM 4.1. The lifespan of any smooth solution of (1.14) and (3.1), staying away from the vacuum, is finite, for C 1 initial data (ρ0 , v0 )(x), with ρ0 (x) > 0 and finite C 1 norm satisfying w10 (β) < 0,
or w20 (β) < 0,
(4.1)
for some point β ∈ R. Furthermore, if there exist two positive constants δ and ε such that min w10 (x) − max w20 (x) := δ > 0, x
(4.2)
x
and, for some point β ∈ R, (β) −ε, w10
or w20 (β) −ε,
(4.3)
then the lifespan of any smooth solution of (1.14) and (3.1) does not exceed θ −1 θ −1 2θ θ 2 2 δ T∗ = ρ0 C(R) . (1 + θ )ε 2
(4.4)
P ROOF. For a smooth solution (ρ, v)(x, t) of system (1.14), one can verify, as in the proof of Theorem 3.1, that ρ(x, t) > 0, and g =
1+θ 2
θ (w1 − w2 ) 2
1−θ 2θ
g2 ,
with g = −ρ (θ−1)/2∂x w1 . By defining the characteristic x = x(β, t) passing through the point (β, 0), β ∈ R, as in the proof of Theorem 3.1, we have, as in (3.10) and (3.11), θ −1
(β) ρ0 (β) 2 w10 g x(β, t), t = , t 1 + 0 K(β, τ ) dτ
with 1+θ K(β, t) = 2 =
θ w10 (β) − w20 α(β, t) 2
1−θ 2θ
1−θ θ −1 1+θ ρ x(β, t), t 2 ρ0 (β) 2 w10 (β). 2
ρ0 (β)
θ −1 2
w10 (β)
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If the smooth solution stays away from the vacuum, i.e., the density ρ has a positive lower bound, then one concludes that g(x(β, t), t) will blow up at a certain finite time (β) < 0. Under the condition (4.2) and if w (β) −ε in (4.3), g(x(β, t), t) will if w10 10 (β) 0 blow up at some finite time which is less than or equal to T∗ defined in (4.4). If w20 or further w20 (β) −ε, similar consequence can be obtained from (3.8). This completes the proof of Theorem 4.1. R EMARK 4.1. The argument was developed in Lax [178] for 2 × 2 hyperbolic systems of conservation laws with genuine nonlinearity. The implication of the result is that the first derivatives of solutions blow up in a finite time, while the solutions stay themselves bounded and away from the vacuum. This is in agreement with the phenomenon of shock waves. See also Majda [223] and Lin [197,198] for further discussions. The formation of singularities for n × n genuinely nonlinear hyperbolic systems of one-dimensional conservation laws (1.19) was discussed in John [161]. It was shown in [161] that, if the initial data are sufficiently small (but not identically zero), then the first derivatives of the solution will become infinite in some finite time. T HEOREM 4.2. Consider the Cauchy problem (1.19) and (1.20) of n × n genuinely nonlinear hyperbolic systems. Assume the initial data u0 (x) are a C 2 function with compact support. Then there exists a positive constant δ such that, if 0 < supx |u0 (x)| δ, the solution u(x, t) cannot exist in the class C 2 for all positive t. This result was generalized in Liu [206] to include systems with linearly degenerate characteristic fields such as the Euler equations.
4.2. Three-dimensional Euler equations Consider the Cauchy problem of the three-dimensional Euler equations for polytropic gases in (2.1) with smooth initial data: (ρ, v, S)|t =0 = (ρ0 , v0 , S0 )(x),
ρ0 (x) > 0, x ∈ R3 ,
(4.5)
satisfying (ρ0 , v0 , S0 )(x) = ρ, ¯ 0, S ,
for |x| R,
where ρ¯ > 0, S, and R are given constants. The equations in (2.1) possess a unique local C 1 solution (ρ, v, S)(x, t) with ρ(x, t) > 0 provided the initial data (4.5) are sufficiently ¯ v0 (x), regular (Theorem 2.1). The support of the smooth disturbance (ρ0 (x) − ρ, S0 (x) − S ) propagates with speed at most σ = (ρ, v, S)(x, t) = ρ, ¯ 0, S ,
pρ (ρ, ¯ S ) (the sound speed), that is,
if |x| R + σ t.
(4.6)
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The proof of this essential fact of finite speed of propagation for the three-dimensional case can be found in John [162], as well as in Sideris [282], established through local energy estimates. Take p¯ = p(ρ, ¯ S). Define P (t) = = F (t) =
R3
R3
p(x, t)1/γ − p¯ 1/γ dx ρ(x, t) exp S(x, t)/γ − ρ¯ exp(S/γ ) dx,
R3
x · ρv(x, t) dx,
which, roughly speaking, measure the entropy and the radial component of momentum. The following theorem on the formation of singularities in solutions of (2.1) and (4.5) is due to Sideris [283]. T HEOREM 4.3. Suppose that (ρ, v, S)(x, t) is a C 1 solution of (2.1) and (4.5) for 0 < t < T , and P (0) 0,
(4.7)
F (0) > ασ R 4 max ρ0 (x),
(4.8)
x
where α = 16π/3. Then the lifespan T of the C 1 solution is finite. P ROOF. Set M(t) =
R3
ρ(x, t) − ρ¯ dx.
From the equations in (2.1), combined with (4.6), and integration by parts, one has M (t) = −
R3
P (t) = −
∇ · (ρv) dx = 0,
R3
∇ · ρv exp(S/γ ) dx = 0,
which implies M(t) = M(0),
P (t) = P (0);
(4.9)
and F (t) =
R3
x · (ρv)t dx =
=
B(t )
R3
ρ|v|2 + 3(p − p) ¯ dx
ρ|v|2 + 3(p − p) ¯ dx,
(4.10)
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where B(t) = {x ∈ R3 : |x| R + σ t}. From Hölder’s inequality, (4.7), and (4.9), one has
1 p dx |B(t)|γ −1 B(t )
γ p
1/γ
dx
B(t )
γ 1 1/γ = p¯ dx p¯ dx, P (0) + |B(t)|γ −1 B(t ) B(t )
where |B(t)| denotes the volume of the set B(t). Therefore, by (4.10),
F (t)
R3
ρ|v|2 dx.
(4.11)
By the Cauchy–Schwarz inequality and (4.9),
2
F (t) =
x · ρv dx
2
B(t )
(R + σ t)
2
B(t )
ρ|v| dx M(t) +
2
B(t )
2
ρ¯ dx
ρ0 (x) − ρ¯ dx +
ρ|v| dx
4π (R + σ t)5 max ρ0 (x) x 3
B(t )
2
B(t )
ρ|x|2 dx
ρ|v| dx B(t )
2
(R + σ t)
B(t )
ρ¯ dx B(t )
ρ|v|2 dx. B(t )
Then (4.11) implies that
F (t)
4π (R + σ t)5 max ρ0 (x) x 3
−1
F (t)2 .
(4.12)
Since F (0) > 0 by (4.8), F (t) remains positive for 0 < t < T , as a consequence of (4.12). Dividing by F (t)2 and integrating from 0 to T in (4.12) yields F (0)−1 > F (0)−1 − F (T )−1 (ασ max ρ0 )−1 R −4 − (R + σ T )−4 . Thus, (R + σ T )4 < R 4 F (0)/ F (0) − ασ R 4 max ρ0 . This completes the proof of Theorem 4.3.
R EMARK 4.2. The method of the proof above, which is a refinement of Sideris [282], applies equally well in one and two space dimensions. In the isentropic case (S is a constant), the condition P (0) 0 reduces to M(0) 0.
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R EMARK 4.3. To illustrate a way in which the conditions (4.7) and (4.8) may be satisfied, consider the initial data: ρ0 = ρ, ¯ S0 = S. Then P (0) = 0, and (4.8) holds if |x|
x · v0 (x) dx > ασ R 4 .
Comparing both sides, one finds that the initial velocity must be supersonic in some region relative to the sound speed at infinity. The formation of a singularity (presumably a shock wave) is detected as the disturbance overtakes the wave front forcing the front to propagate with supersonic speed. R EMARK 4.4. Another result was established in Sideris [283] on the formation of singularities, without condition of largeness such as (4.8). The result says that, if S0 (x) S and, for some 0 < R0 < R,
|x|>r
|x|>r
2 |x|−1 |x| − r ρ0 (x) − ρ¯ dx > 0, |x|
−3
2 |x| − r 2 x · ρ0 (x)v0 (x) dx 0,
(4.13)
for R0 < r < R, then the lifespan T of the C 1 solution of (2.1) and (4.5) is finite. The assumption (4.13) means that, in an average sense, the gas must be slightly compressed and outgoing directly behind the wave front. For the proof, some important technical points were adopted from Sideris [281] on the nonlinear wave equations in three dimensions. R EMARK 4.5. The result in Theorem 4.3 indicates that the C 1 regularity of solutions breaks down in a finite time. It is believed that in fact only ∇ρ and ∇v blow up in most cases; see a proof in Alinhac [2] for the case of axisymmetric initial data for the Euler equations for compressible fluids in two space dimensions.
4.3. Other results The method of characteristics has been used to establish the finite-time formation of singularities for one-dimensional hyperbolic systems of conservation laws and related equations; see Lax [178], John [161], Liu [206], Klainerman and Majda [170], Dafermos [83], Keller and Ting [169], Slemrod [287], Lin [197,198], etc. A technique was introduced in Dafermos [83] to monitor the time evolution of the spatial supremum norms of first derivatives and was further applied in Dafermos and Hsiao [90], Hrusa and Messaoudi [153], and Chen and Wang [320] for the problems with thermal diffusion. Contrary to the formation of singularities, global smooth solutions may exist for conservation laws with certain dissipation mechanisms including friction damping, heat diffusion, and memory effects, provided the initial data are smooth and small. That is,
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the smoothing effect from the dissipation may prevent the development of shock waves in solutions with small smooth initial data. See the survey paper by Dafermos [82]. In the case of damping, this property has been justified for certain one-dimensional equations; see Nishida [242], Hsiao [154], and the references cited therein for the existence of global smooth solutions with small smooth initial data to the one-dimensional Euler equations with damping. For the multidimensional Euler equations, it has been proved by Sideris and Wang [286] that the damping can also prevent the formation of singularities in smooth solutions with small initial data. For related discussions, see Wang [319] for a spherically symmetric smooth Euler–Poisson flow and Guo [149] for a smooth irrotational Euler–Poisson flow in three space dimensions. In the case of heat diffusion, the global existence of smooth solutions was established in Slemrod [288] for nonlinear thermoelasticity with smooth and small initial data. Although the smoothing effect from damping or heat diffusion alone can prevent the breaking of smooth waves of small amplitude, the combined effect of damping and heat diffusion may still not be strong enough to prevent the formation of singularities in large smooth solutions, as shown in Chen and Wang [320]. A preliminary study of the so called critical threshold phenomena associated with the Euler–Poisson equations was made in Engelberg, Liu and Tadmor [110], where the answer to questions of global smoothness vs. finite-time breakdown depends on whether the initial configuration crosses an intrinsic critical threshold. The damping induced by memory effects can also preserve the smoothness of small initial data; see Dafermos and Nohel [91] and MacCamy [220]. For multidimensional scalar conservation laws, the formation of shock waves was proved in Majda [223] by using characteristics for solutions with smooth initial data. Some general discussions on the formation of shock waves in plane wave solutions of multidimensional systems of conservation laws can also be found in Majda [223]. The method of Sideris [282,283] has been effective for multidimensional systems of Euler equations. A similar technique was employed by Glassey [128] in the case of nonlinear Schrödinger equations (see also Strauss [294]). It has been adopted to prove the formation of singularities in solutions of many other multidimensional problems; see Makino, Ukai and Kawashima [226] and Rendall [263] for a compressible fluid body surrounded by the vacuum, Rammaha [261,262] for two-dimensional Euler equations and magnetohydrodynamics, Perthame [254] for the Euler–Poisson equations for spherically symmetric flows, and Guo and Tahvildar-Zadeh [150] for the Euler–Maxwell equations for spherically symmetric plasma flows, etc. For the multidimensional Euler equations for compressible fluids with smooth initial data that are a small perturbation of amplitude ε from a constant state, the lifespan of smooth solutions is at least O(ε−1 ) from the theory of symmetric hyperbolic systems (Friedrichs [123], Kato [163]). Results on the formation of singularities show that the lifespan of a smooth solution is no better than O(ε−2 ) in the two-dimensional −2 case (Rammaha [261]) and O(eε ) (Sideris [283]) in the three-dimensional case. See Alinhac [2] and Sideris [284,285] for additional discussions in this direction. There have been many studies on the blow-up of smooth solutions for nonlinear wave equations; see the results collected in Alinhac [3], John [162], and the references cited therein. Other related discussions about the formation of singularities for conservation laws
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can be found in Brauer [17], Chemin [35,36], Kosinski [171], Wang [316], as well as the references cited therein.
5. Local well-posedness for discontinuous solutions The formation of singularities, especially shock waves, discussed in Section 4 indicates that one should seek discontinuous entropy solutions of the Euler equations for general initial data. Usually, it is difficult to construct the discontinuous solutions especially in the multidimensional case. We focus on the local existence of discontinuous entropy solutions in this section. We first consider the local existence of the simplest type of discontinuous solutions, i.e., the shock front solutions of the multidimensional Euler equations. Shock front solutions are the most important discontinuous nonlinear progressing wave solutions in compressible Euler flows and other systems of conservation laws. For a general multidimensional hyperbolic system of conservation laws (1.16), shock front solutions are discontinuous piecewise smooth entropy solutions with the following structure: (a) There exists a C 2 space-time hypersurface S(t) defined in (x, t) for 0 t T with space-time normal (νx , νt ) = (ν1 , . . . , νd , νt ) as well as two C 1 vector-valued functions: u+ (x, t) and u− (x, t), defined on respective domains S + and S − on either side of the hypersurface S(t), and satisfying ∂t u± + ∇ · f u± = 0,
in S ± ;
(5.1)
(b) The jump across the hypersurface S(t) satisfies the Rankine–Hugoniot condition: % + & νt u − u− + νx · f u+ − f u− S = 0.
(5.2)
For the quasilinear system (1.16), the surface S is not known in advance and must be determined as part of the solution of the problem; thus the equations in (5.1) and (5.2) describe a multidimensional, highly nonlinear, free-boundary value problem for the quasilinear system of conservation laws. The initial data yielding shock front solutions are defined as follows. Let S0 be a smooth hypersurface parametrized by α, and let ν(α) = (ν1 (α), . . . , νn (α)) be a unit normal to S0 . Define the piecewise smooth initial values for respective domains S0+ and S0− on either side of the hypersurface S0 as u0 (x) =
+ u+ 0 (x), x ∈ S0 ,
− u− 0 (x), x ∈ S0 .
(5.3)
It is assumed that the initial jump in (5.3) satisfies the Rankine–Hugoniot condition, i.e., there is a smooth scalar function σ (α) so that + − − −σ (α) u+ 0 (α) − u0 (α) + ν(α) · f u0 (α) − f u0 (α) = 0,
(5.4)
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and that σ (α) does not define a characteristic direction, i.e., σ (α) = λi u± 0 ,
α ∈ S0 , 1 i n,
(5.5)
where λi , i = 1, . . . , n, are the eigenvalues of (1.16). It is natural to require that S(0) = S0 . Consider the two-dimensional isentropic Euler equations in (1.9), away from the vacuum, which can be rewritten in the form: ∂t ρ + ∇ · (ρv) = 0, ρ 0, v ∈ R2 , x ∈ R2 , t > 0, ∂t (ρv) + ∇ · (ρv ⊗ v) + ∇p = 0, p = p(ρ) = ρ γ /γ , γ > 1,
(5.6)
with piecewise smooth initial data: + + ρ , v (x), x ∈ S0+ , (ρ, v)|t =0 = 0− 0− ρ0 , v0 (x), x ∈ S0− .
(5.7)
The following local existence of discontinuous entropy solutions is taken from Majda [222]. T HEOREM 5.1. Assume that S0 is a smooth closed curve and that (ρ0+ , v+ 0 )(x) belongs to the uniform local Sobolev space Huls (S0+ ), while (ρ0− , v− )(x) belongs to the Sobolev space 0 − s H (S0 ), for some fixed s 10. Assume also that there is a function σ (α) ∈ H s (S0 ) so that (5.4) and (5.5) hold, and the compatibility conditions up to order s − 1 are satisfied on S0 by the initial data, together with the entropy condition + θ − θ < σ (α) < v− , v+ 0 · ν(α) + ρ0 0 · ν(α) + ρ0
θ = (γ − 1)/2,
(5.8)
and the stability condition p(ρ0+ ) − p(ρ0− ) ρ0+
− ρ0−
γ −1 − 2 < ρ0− + v0 · ν(α) − σ (α) .
(5.9)
Then there is a C 2 hypersurface S(t) together with C 1 functions (ρ ± , v± )(x, t) defined for t ∈ [0, T ], with T sufficiently small, so that ρ + , v+ (x, t), (ρ, v)(x, t) = − − ρ , v (x, t),
(x, t) ∈ S + , (x, t) ∈ S − ,
(5.10)
is the discontinuous shock front solution of the Cauchy problem (5.6) and (5.7) satisfying (5.1) and (5.2).
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In Theorem 5.1, the uniform local Sobolev space Huls (S0+ ) is defined as follows: Let w ∈ C0∞ (Rd ) be a function so that w(x) 0; and w(x) = 1 when |x| 1/2, and w(x) = 0 when |x| > 1. Define x−y . wr,y (x) = w r A vector function u is in Huls , provided that there exists some r > 0 so that max wr,y u H s < ∞.
y∈Rd
R EMARK 5.1. There are extensive studies in Majda [221–223] on the local existence and stability of shock front solutions. The compatibility conditions in Theorem 5.1 are defined in [222] and needed in order to avoid the formation of discontinuities in higher derivatives along other characteristic surfaces emanating from S0 . Once the main condition in (5.4) is satisfied, the compatibility conditions are automatically guaranteed for a wide class of initial data. Theorem 5.1 can be extended to the full Euler equations in three space dimensions (d = 3) in (1.1) (see Majda [222]). See Métivier [229] for the uniform existence time of shock front solutions in the shock strength. Also see Blokhin and Trakhinin [14] in this volume for further discussions. The proof of Theorem 5.1 can be found in [222]. The idea of the proof is similar to that of the proof of Theorem 2.2, but the technical details are quite different due to the unusual features of the problem considered in Theorem 5.1. The shock front solutions are defined as the limit of a convergent classical iteration scheme based on a linearization by using the theory of linearized stability for shock fronts developed in [221]. The technical condition s 10, instead of s > 1 + d/2 = 2 (d = 2), is required because pseudo-differential operators are needed in the proof of the main estimates. Some improved technical estimates regarding the dependence of operator norms of pseudo-differential operators on their coefficients would lower the value of s. For the one-dimensional Euler equations in (1.12), away from the vacuum, m = ρv and ∂t ρ + ∂x (ρv) = 0, x ∈ R, ∂t (ρv) + ∂x ρv 2 + p = 0, 1 ∂t E + ∂x v(E + p) = 0, E = ρv 2 + ρe, 2
(5.11)
some stronger existence results of local discontinuous solutions can be found in [148,195] for the Cauchy problem with piecewise smooth initial data + + + ρ , v , e (x), x > 0, (5.12) (ρ, v, e)|t =0 = 0− 0− 0− ρ0 , v0 , e0 (x), x < 0, where (ρ0± , v0± , e0± )(x) are bounded smooth functions for x 0 and x 0, respectively, and (ρ0+ , v0+ , e0+ )(0) = (ρ0− , v0− , e0− )(0). Then the following theorem holds.
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T HEOREM 5.2. Suppose that the amplitude |(ρ0+ − ρ0− , v0+ − v0− , e0+ − e0− )(0)| is sufficiently small, then the Cauchy problem (5.11) and (5.12) has a unique piecewise smooth solution (ρ, v, e)(x, t) for x ∈ R and t ∈ [0, T ], for sufficiently small T . R EMARK 5.2. For the one-dimensional Euler equations for (isentropic or non-isentropic) polytropic gases (2.1) or (1.14) (d = 1), the assumption of small amplitude is not needed. See [148,195] for the proofs of Theorem 5.2 and related results. R EMARK 5.3. The piecewise smooth solution (ρ, v, e)(x, t) of the Cauchy problem (5.11) and (5.12) possesses a structure in a neighborhood of the origin similar to the solution of the corresponding Riemann problem of (5.11) with initial data (ρ, v, e)|t =0 =
ρ0+ , v0+ , e0+ (0), x > 0, − − − ρ0 , v0 , e0 (0), x < 0.
(5.13)
See Section 6, as well as Chang and Hsiao [33], Courant and Friedrichs [76], Dafermos [88], Serre [277], and Smoller [291], for the discussion of the solution structure of the Riemann problem. R EMARK 5.4. There are some discussions in [76,195] on the local existence of spherically symmetric discontinuous solutions with spherically symmetric initial data. See Section 10.1 for some recent results on the global existence of spherically symmetric discontinuous entropy solutions.
6. Global discontinuous solutions I: Riemann solutions In this section, we present a global theory of discontinuous entropy solutions of the Riemann problem, the simplest Cauchy problem with discontinuous initial data.
6.1. The Riemann problem and Lax’s theorems We first introduce two Lax’s theorems for the local behavior of wave curves in the phase space and the existence of global entropy solutions of the Riemann problem, respectively, for one-dimensional strictly hyperbolic systems of conservation laws (1.19) with Riemann data: uL , x < 0, (6.1) u|t =0 = u0 (x) := uR , x > 0, where uL and uR are two constant states. This theorem applies to the Euler equations with small Riemann data. Since both system (1.19) and the Riemann initial data (6.1) are invariant under uniform stretching of coordinates: (x, t) → (αx, αt), the Cauchy problem (1.19) and (6.1) admits
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self-similar solutions, defined on the space-time plane, and constant along straight-line rays emanating from the origin: u(x, t) = R(x/t),
x ∈ R, t ∈ R+ ,
(6.2)
where R(ξ ) is a bounded measurable function in ξ ∈ R, which satisfies the ordinary differential equation: d(f(R(ξ )) − ξ R(ξ )) + R(ξ ) = 0 dξ
(6.3)
in the sense of distributions. To solve the Riemann problem, it is more instructive to present first the rarefaction curves and the shock curves in the phase space. Rarefaction curves. Given a state u− , we consider possible states u that can be connected to the state u− , on the right, by a centered rarefaction wave of the i-th characteristic field, which is genuinely nonlinear, that is, ∇λi · ri = 1, where ∇f · ri = λi ri , 1 i n. Consider the self-similar Lipschitz solutions V(ξ ), ξ = x/t, of the Riemann problem (1.19) and (6.1) as above. Then we have ξ = λi (V)(ξ ), ∇f V(ξ ) − ξ I V (ξ ) = 0,
(6.4)
with boundary condition: V|ξ =λi (u− ) = u− ,
(6.5)
and, on the i-centered rarefaction waves, ∂V 1 dV 1 = = ri V(x/t) . ∂x t dξ t
(6.6)
Then we conclude: P ROPOSITION 6.1. Let the i-th characteristic field of system (1.19) be genuinely nonlinear in N ⊂ Rn . Let u− be any point in N . Then there exists a one-parameter family of states u = u(ε), ε 0, u(0) = u− , which can be connected to u− on the right by an ˙ i-centered rarefaction wave. The parametrization can be chosen so that u(0) = ri (u− ) and ¨ u(0) = r˙ i (u− ). Shock curves. Given a state u− , we consider possible states u that can be connected to the state u− , on the right, by a shock or contact discontinuity. The Rankine–Hugoniot condition for discontinuities with speed σ is σ [u] = f(u) .
(6.7)
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Here and in what follows we use the notation [H ] = H+ − H− , where H− and H+ are the values of any function H on the left-hand side and the right-hand side of the discontinuity, respectively. A discontinuity satisfying (6.7) is called an i-shock if it satisfies the Lax entropy conditions: λi−1 (u− ) < σ < λi (u− ),
λi (u) < σ < λi+1 (u).
(6.8)
First we consider the case which the i-field is genuinely nonlinear. Given u− ∈ N , we can view (6.7) as n-equations for the (n + 1)-unknowns u and σ . P ROPOSITION 6.2. Let the i-th characteristic field of system (1.19) be genuinely nonlinear in N . Let u− be any point in N . Then there exists a one-parameter family of states u = u(ε), ε 0, u(0) = u− , which can be connected to u− on the right by an i¨ ˙ = r˙ i (u− ), and shock. The parametrization can be chosen so that u(0) = ri (u− ) and u(0) σ (0) = λi (u− ), σ˙ (0) = 1/2. Contact discontinuities. If the i-th characteristic field is linearly degenerate, then λi is an i-Riemann invariant. P ROPOSITION 6.3. Let the i-th characteristic field of system (1.19) be linearly degenerate. If u− and u+ have the same i-Riemann invariants with respect to the linearly degenerate field, then they are connected to each other by a contact discontinuity of speed σ = λi (u− ) = λi (u+ ). Propositions 6.1, 6.2, and 6.3 can be combined into the following Lax’s theorem [177] (also see [181]). T HEOREM 6.1. Given a state u− , it can be connected to a one-parameter family of states u+ = u(ε), −ε0 < ε < ε0 , on the right of u− through a centered i-wave, i.e., an i-shock, or an i-rarefaction wave, or an i-contact discontinuity; u(ε) is twice continuously differentiable with respect to ε. Then, using Theorem 6.1 and the implicit function theorem leads to the Lax’s existence theorem [177] (also see [181]) for the Riemann problem (1.19) and (6.1). T HEOREM 6.2. Assume that system (1.19) is strictly hyperbolic and each characteristic field is either genuinely nonlinear or linearly degenerate. For sufficiently small |uL − uR |, there exists a unique self-similar solution (6.2) of the Riemann problem (1.19) and (6.1), with small total variation. This solution comprises n + 1 constant states uL = u0 , u1 , . . . , un−1 , un = uR . When the i-th characteristic field is linearly degenerate, ui is joined to ui−1 by an i-contact discontinuity; when the i-th characteristic field is genuinely nonlinear, ui is joined to ui−1 by either an i-centered rarefaction wave or an i-compressive shock.
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6.2. Isothermal Euler equations Consider the isothermal Euler equations in (1.15), that is, γ = 1 and p = p(τ ˜ ) = 1/τ,
(6.9)
with Riemann data: (τ, v)|t =0 =
(τL , vL ),
x < 0,
(τR , vR ),
x > 0.
(6.10)
System (1.15) and (6.9) has the eigenvalues ±1/τ and the Riemann invariants v ± ln τ . The shock curves Si , i = 1, 2, and rarefaction curves Ri , i = 1, 2, with left state (i = 1) or right state (i = 2) (τ− , v− ) have the following forms, respectively: q − q− , q > q− , Si (τ− , v− ): v − v− = (−1)i 2 sinh 2 Ri (τ− , v− ):
v − v− = (−1)i (q − q− ),
q < q− ,
where q = − ln τ and q− = − ln τ− . Define a function W (s) =
s,
2 sinh 2s ,
s 0,
(6.11)
s 0.
Then the equations for the i-wave curves, i = 1, 2, can be rewritten into the form: i-wave curve: v − v− = (−1)i W (q − q− ).
(6.12)
The function W (s) in (6.11) satisfies W (s) > 0, i.e., W (s) is increasing. It is easy to verify that W (s1 + s2 ) W (s1 ) + W (s2 ),
for s1 , s2 0,
W (s1 + s2 ) = W (s1 ) + W (s2 ),
for s1 , s2 0.
For any s, let s ± = (|s| ± s)/2. Then W s + + W (s − ) = W |s| W (s).
(6.13)
(6.14)
If (vm , qm ) is the intermediate state in the Riemann problem of (1.15) and (6.9) connecting the two states (vL , qL ) = (v1 , q1 ) and (vR , qR ) = (v2 , q2 ), then W (qm − q1 ) + W (qm − q2 ) = v1 − v2 .
(6.15)
Without ambiguity, we denote D(q1 , q2 ) := |q1 − qm | + |q2 − qm |,
(6.16)
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although D also depends on v1 and v2 . Then we have P ROPOSITION 6.4. For any qi , i = 1, 2, 3, D(q1 , q3 ) D(q1 , q2 ) + D(q2 , q3 ).
(6.17)
P ROOF. Let qij be the intermediate states between qi and qj , i, j = 1, 2, 3, i = j . Then, from (6.15), one has W (q13 − q1 ) + W (q13 − q3 ) = W (q12 − q1 ) + W (q12 − q2 ) + W (q23 − q2 ) + W (q23 − q3 ). Set x = q13 − q1 , y = q13 − q3 , a = q12 − q1 , b = q12 − q2 , c = q23 − q2 , and d = q23 − q3 . Then x − y = a − b + c − d, and W (x) + W (y) = W (a) + W (b) + W (c) + W (d).
(6.18)
If xy 0, then D(q1 , q3 ) = |x| + |y| = |x − y| = |a − b + c − d| |a| + |b| + |c| + |d| D(q1 , q2 ) + D(q2 , q3 ). If x > 0 and y > 0, by (6.14) and (6.18), W (x) + W (y) W a + + W (b− ) + W c+ + W (d − ) + W (a − ) + W b + + W (c− ) + W d + , and then either W (x) W a + + W (b− ) + W c+ + W (d − ),
(6.19)
W (y) W (a − ) + W b+ + W (c− ) + W d + .
(6.20)
or
If (6.19) is true, then, by (6.13), W (x) W (a + + b− + c+ + d − ). The monotonicity property of W yields x a + + b− + c+ + d − , and thus D(q1 , q3 ) = x + y = 2x − (x − y) 2 a + + b− + c+ + d − − (a − b + c − d) |a| + |b| + |c| + |d| = D(q1 , q2 ) + D(q2 , q3 ). Similarly, if (6.20) is true, then (6.17) also holds.
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The Riemann solutions of (1.15), (6.9), and (6.10) have three constant states (τj , vj ), j = 1, 2, 3, connected by two of the elementary waves: 1-wave (S1 wave or R1 wave) and 2-wave (S2 wave or R2 wave). R EMARK 6.1. The proof of (6.17) given above is due to Poupaud, Rascle and Vila [258]. 6.3. Isentropic Euler equations Consider the Riemann problem for the isentropic Euler equations in (1.14) with Riemann data: (ρL , mL ), x < 0, (ρ, m)|t =0 = (6.21) (ρR , mR ), x > 0, which may contain the vacuum states, where ρJ 0 and mJ are the constants, and |mJ /ρJ | C0 < ∞, J = L, R. As usual, assume that the pressure function p(ρ) satisfies that, when ρ > 0, p(ρ) > 0,
p (ρ) > 0
ρp (ρ) + 2p (ρ) > 0
(hyperbolicity),
(genuine nonlinearity),
(6.22)
and, when ρ tends to zero, p(ρ), p (ρ) → 0,
(6.23)
which is different from the isothermal case. The eigenvalues of system (1.14) are λi = m/ρ + (−1)i p (ρ), i = 1, 2,
(6.24)
and the corresponding right-eigenvectors are ,
ri = αi (ρ)(1, λi ) ,
2ρ p (ρ) , αi (ρ) = (−1) ρp (ρ) + 2p (ρ) i
so that ∇λi · ri = 1, i = 1, 2. The Riemann invariants are ρ p (s) m i−1 wi = + (−1) ds, i = 1, 2. ρ s 0 From (6.23) and (6.24), λ2 − λ1 = 2 p (ρ) → 0,
(6.25)
(6.26)
ρ → 0.
Therefore, system (1.14) is strictly hyperbolic in the nonvacuum states {(ρ, v): ρ > 0, |v| C0 }. However, strict hyperbolicity fails near the vacuum states {(ρ, m/ρ): ρ = 0, |m/ρ| C0 }.
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Shock wave curves. From the Rankine–Hugoniot condition (6.7) and the Lax entropy condition (6.8), we obtain that the i-shock wave curves Si (ρ− , m− ), i = 1, 2, are m− Si (ρ− , m− ): m − m− = (ρ − ρ− ) + (−1)i ρ−
ρ p(ρ) − p(ρ− ) (ρ − ρ− ), ρ− ρ − ρ−
(−1)i (ρ − ρ− ) < 0, ρ− > 0. It is easy to check that the curves Si (ρ− , m− ), i = 1, 2, are concave and convex, respectively, with respect to (ρ− , m− ) in the ρ–m plane. Rarefaction wave curves. Given a state (ρ− , m− ), the i-centered rarefaction wave curves Ri (ρ− , m− ), i = 1, 2, are Ri (ρ− , m− ):
m− m − m− = (ρ − ρ− ) + (−1)i ρ ρ−
ρ ρ−
p (s) ds, s
(−1) (ρ − ρ− ) > 0. i
Then the curves Ri , i = 1, 2, are concave and convex, respectively, in the ρ–m plane. For the Riemann problem (1.14) and (6.21) satisfying (6.22) and (6.23), there exists a unique, globally defined, piecewise smooth entropy solution R(x/t), which may contain the vacuum states on the upper half-plane t > 0, satisfying w2 R(x/t) w2 (uL ), w1 R(x/t) w1 (uR ), w1 R(x/t) − w2 R(x/t) 0. These Riemann solutions can be constructed for the case: wi (uR ) wi (uL ), i = 1, 2, as follows. If ρL > 0 and ρR = 0, then there exists a unique vc such that ⎧ x/t < λ1 (uL ), ⎨ uL , R(x/t) = V1 (x/t), λ1 (uL ) x/t vc , ⎩ vacuum, x/t > vc , where V1 (ξ ) is the solution of the boundary value problem V1 (ξ ) = r1 V1 (ξ ) ,
ξ > λ1 (uL );
V1 |ξ =λ1 (uL ) = uL .
If ρL = 0 and ρR > 0, then there exists a unique v˜c such that ⎧ ⎨ vacuum, x/t < v˜c , R(x/t) = V2 (x/t), v˜c x/t λ2 (uR ), ⎩ x/t > λ2 (uR ), uR ,
(6.27)
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where V2 (ξ ) is the solution of the boundary value problem V2 (ξ ) = r2 V2 (ξ ) ,
ξ < λ2 (uR );
V2 |ξ =λ2 (uR ) = uR .
(6.28)
If ρL , ρR > 0, there are two subcases: (a) There exist unique vc1 , vc2 , vc1 < vc2 , such that the Riemann solution has the form: ⎧ uL , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V1 (x/t), R(x/t) := vacuum, ⎪ ⎪ ⎪ V2 (x/t), ⎪ ⎪ ⎩ uR ,
x/t < λ1 (uL ), λ1 (uL ) x/t vc1 , vc1 < x/t < vc2 , vc2 x/t λ2 (uR ),
(6.29)
x/t > λ2 (uR ),
where V1 (ξ ) and V2 (ξ ) are the solutions of the boundary value problems (6.27) and (6.28), respectively. (b) There exists a unique uc = (ρc , mc ), ρc > 0, such that the Riemann solution has the form: ⎧ uL , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V1 (x/t), R(x/t) := uc , ⎪ ⎪ ⎪ V2 (x/t), ⎪ ⎪ ⎩ uR ,
x/t < λ1 (uL ), λ1 (uL ) x/t λ1 (uc ), λ1 (uc ) < x/t < λ2 (uc ), λ2 (uc ) x/t λ2 (uR ), x/t > λ2 (uR ),
where V1 (ξ ) and V2 (ξ ) are also the solutions of the boundary value problems (6.27) and (6.28), respectively. For the subcase (a), although the Riemann data are nonvacuum states at t = 0, the vacuum states occur in the Riemann solutions instantaneously as t becomes positive. Therefore, the vacuum states are generic in inviscid compressible fluid flow (except the isothermal case). Riemann solutions for the other cases can be constructed similarly. See Chang and Hsiao [33], Dafermos [88], Serre [277], and Smoller [291] for the details. P ROPOSITION 6.5. The regions % & Σ w10 , w20 = (ρ, m): w1 w10 , w2 w20 , w1 − w2 0 are invariant regions of the Riemann problem (1.14) and (6.21). That is, if the Riemann data lie in Σ(w10 , w20 ), the solution of the Riemann problem also lies in Σ(w10 , w20 ). This can be checked directly from the explicit formulas known for the Riemann solutions.
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6.4. Non-isentropic Euler equations For convenience, in this section we focus on the non-isentropic Euler equations in (1.13), (1.6), and (1.7) in Lagrangian coordinates. We first analyze the global behavior of shock curves in the phase space and the singularity of centered rarefaction waves in the physical plane, and then construct global solutions of the Riemann problem (6.1) for (1.13), (1.6), and (1.7). These are essential for determining the uniqueness of Riemann solutions with arbitrarily large oscillation in Section 8.2. Shock curves. for (1.13) is
The Rankine–Hugoniot condition (6.7) for a discontinuity with speed σ
σ [v] = [p],
σ [τ ] = −[v],
1 σ e + v 2 = [pv]. 2
(6.30)
If σ = 0, then the discontinuity is a contact discontinuity which corresponds to the second characteristic field. If σ = 0, then the discontinuity is a shock, which corresponds to either the first or third characteristic field. The Lax entropy inequality (6.8) and the Rankine–Hugoniot condition (6.30) imply that, on a 1-shock, [p] > 0,
[τ ] < 0,
[v] < 0,
[τ ] > 0,
[v] < 0.
and, on a 3-shock, [p] < 0,
From (6.30), we have 1 e − e− + (p + p− )(τ − τ− ) = 0. 2
(6.31)
Set s = p/p− . Then (6.31) becomes τ γ −1 (s + 1) −1 , pτ = p− τ− 1 − 2 τ− which implies τ s+β , = τ− βs + 1
with β =
γ +1 . γ −1
Note that [v] = −σ [τ ] = − −[p][τ ].
(6.32)
The Cauchy problem for the Euler equations for compressible fluids
Then, denoting the sound speed by c, i.e., c = v − v− = (−1)
i−1 2
c−
455
√ γpτ , one has
1−s 2 . √ γ (γ − 1) βs + 1
(6.33)
Let s = e−x . From (6.32) and (6.33), we obtain that the i-shock is determined by p = e−x , p−
(−1)
1 + βex τ = , τ− β + ex
i−1 2
x 0,
(6.34) (6.35)
i−1 v − v− = (−1) 2 c−
1 − e−x 2 , γ (γ − 1) 1 + βe−x
(6.36)
with speed σ = (−1)
i+1 2
c− τ−
1 + βe−x . β +1
(6.37)
Now we choose the speed σ as a parameter for the shock curve, that is, x is a function of σ : x = x(σ ), and compute the derivatives of x(σ ) in σ < 0 (1-shock) and in σ > 0 (3-shock). We use the notations = d/dx and ˙= d/dσ . Since σ2 =
2 c− 1 + βe−x(σ ) , 2 β +1 τ−
(6.38)
we take the derivative on both sides of (6.38) in σ and use (6.38) to deduce x(σ ˙ ) = (−1)
i−1 2
β + 1 τ− x(σ ) 1 + βe−x(σ ) . 2 e β c− β +1
(6.39)
We take the second-order derivative on both sides of (6.38) in σ to have x(σ ˙ )2 − x(σ ¨ )=
β x(σ ˙ )2 > 0. 2(β + ex(σ ) )
(6.40)
Then x(σ ¨ )=
ex(σ ) + β/2 x(σ ˙ )2 > 0. ex(σ ) + β
(6.41)
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We take the third-order derivative on both sides of (6.38) in σ to have ˙¨x(σ ) − 3x(σ ˙ )x(σ ¨ ) + x(σ ˙ )3 = 0.
(6.42)
On the other hand, we have from (6.34) that p = −p,
p = p,
p = −p,
and then 2 p˙ = −px, ˙ p¨ = p (x) ˙ − x¨ > 0, ˙¨ p = p −(x) ˙ 3 + 3x˙ x¨ −˙¨x = 0.
(6.43)
From (6.35), we similarly have τ˙ = τ x, ˙
2 ˙ − x¨ , τ¨ = 3τ (x)
˙¨τ =
6βτ x˙ 2 ( x) ˙ − x ¨ . β + ex
(6.44)
Furthermore, we note that S/cv = ln( κ1 pτ γ ). Then τ˙ β(ex − 1)2 x˙ S˙ p˙ , = +γ =− cv p τ (β + ex )(1 + βex ) x S¨ τ˙ 2 p¨ p˙ 2 τ¨ (x) ˙ 2 = − 2 +γ −γ 2 = P e , cv p p τ τ (β + ex )2 (1 + βex )2 where 3 2 1 2 β 3 2 β + β + 2 y − β + 5β y + P (y) = β(y − 1) −βy − , 2 2 2
y > 0.
The following proposition is taken from Chen, Frid and Li [52]. P ROPOSITION 6.6. Along any shock curve, S = S(σ ) satisfies ˙ ) + σ S(σ ¨ ) 0. 2S(σ P ROOF. This can be seen from a direct calculation, which yields ˙ ) + σ S(σ ¨ )= 2S(σ
cv x(σ ˙ )(1 − ex(σ )) Q ex(σ ) , x(σ ) x(σ ) 2 (β + e )(1 + βe )
while Q(y) = −2βy 3 − β 2 + 2β + 4 y 2 − 3β(β + 1)y − β < 0. Since x(σ ˙ )(1 − ex(σ )) is always nonnegative, the result follows.
(6.45)
The Cauchy problem for the Euler equations for compressible fluids
Rarefaction waves.
457
Consider the self-similar solutions V(ξ ) = (τ, v, e + v2 )(ξ ), ξ = x/t,
of (1.13) with left state u− = (τ− , v− , e− +
2 v− 2 ).
Then we have
ξ = λi (V)(ξ ), i = 1, 3, dτ dv +ξ = 0, dξ dξ de dτ +p = 0, dξ dξ with boundary condition V|ξ =λi (u− ) = u− and, on the i-centered rarefaction waves, x ∂V 1 dV 1 , i = 1, 3. (6.46) = = ri V ∂x t dξ t t In particular, we have x ∂W 1 = r˜ i W , ∂x t t
i = 1, 3,
(6.47)
√ √ i−1 v (v,S) ( −pv (v, S), (−1) 2 , 0), . where W = (τ, v, S) and r˜ i = 2 p−p vv (v,S) Similar to the argument for shocks, we can also obtain centered rarefaction wave curves in the phase space for the first and third characteristic fields. For a rarefaction wave V(x/t) with right state u+ , denoting ∞ i−1 −pτ (s, S± ) ds, i = 1, 3, wi = v + (−1) 2 τ
with w1 (u− ) − w3 (u+ ) > 0, one has w1 (u− ) w1 V(x/t) w1 (u+ ), w1 V(x/t) − w3 V(x/t) > 0,
w3 (u− ) w3 V(x/t) w3 (u+ ), (6.48) S(x/t) = S+ = S− .
These rarefaction waves are identical to those of the isentropic case with the 2-field in the isentropic case corresponding to the 3-field in the non-isentropic case. Solvability. For the Riemann problem (1.13) and (6.1), we have P ROPOSITION 6.7. Given states WL = (vL , τL , SL ) and WR = (vR , τR , SR ), there exists a unique global Riemann solution R(x/t) in the class of self-similar piecewise smooth solutions consisting of shocks, rarefaction waves, and contact discontinuities, provided that the Riemann data satisfy vR − vL <
2 c(τL , SL ) + c(τR , SR ) , γ −1
√ where c(τ, S) = τ −pτ (τ, S).
(6.49)
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The proof of Proposition 6.7 can be found in [290,291,33]. The condition (6.49) is necessary and sufficient for Riemann solutions staying away from the vacuum; without this condition, Riemann solutions may contain δ-masses at the vacuum states and become measure solutions (see Wagner [314] and Chen and Frid [51]).
7. Global discontinuous solutions II: Glimm solutions We now discuss the Glimm solutions that are the entropy solutions, obtained via the Glimm random choice method, of the Cauchy problem for hyperbolic systems of conservation laws, which apply to the Euler equations for compressible fluids. A related method, the wave-front tracking algorithm, is also discussed.
7.1. The Glimm scheme and existence We first discuss the Glimm scheme in [130] which uses the solutions of the Riemann problem to construct a global entropy solution in BV of the Cauchy problem (1.19) and (1.20) for hyperbolic systems of n conservation laws, provided that u0 (x) has small total variation on R. For the isothermal Euler equations, the Glimm scheme yields a global entropy solution with initial data of arbitrarily large total variation. Glimm scheme. Assume that system (1.19) is strictly hyperbolic and each characteristic field is either genuinely nonlinear or linearly degenerate in a neighborhood of a constant ¯ Denote by λ1 (u) < · · · < λn (u) the eigenvalues of the Jacobian matrix ∇f(u). The state u. solution u(x, t) of the Cauchy problem is obtained as the limit of the approximate solutions uh (x, t), when h → 0, constructed by the Glimm scheme, as described below. Fix h > 0, a space-step size, and determine the corresponding time-step size ∆t = h/Λ satisfying the Courant–Friedrichs–Lewy condition, where Λ is an upper bound of the characteristic speeds |λi |, i = 1, 2, . . . , n. Then we partition the upper half-plane R2+ := {(x, t): x ∈ R, t 0} into the strips S k = {(x, t): x ∈ R, k∆t t < (k + 1)∆t}, k ∈ Z+ , and identify the mesh points (j h, k∆t) with k ∈ Z+ , j ∈ Z, and j + k even. Choose any random sequence of numbers a = {a0 , a1 , a2 , . . .} ⊂ (−1, 1) which is equidistributed in (−1, 1) in the following sense: for any subinterval I ⊂ (−1, 1) of length |I |, lim
l→∞
2 Nl = |I | l
uniformly with respect to I , where Nl is the number of indices k l with ak ∈ I . Set the sampling points as Pjk = ((j + ak )h, k∆t) with j + k odd. Denote the approximate solution by uh (x, t). It is defined by induction on k = 0, 1, 2, . . . in each strip S k . Define u0j = u0 ((j + a0 )h) and ukj = uh (j + ak )h − 0, k∆t − 0
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for j + k odd and k 1. Set uh (x, k∆t) = ukj for x ∈ ((j − 1)h, (j + 1)h) with j + k odd. Define the solution uh (x, t) for x ∈ [(j − 1)h, (j + 1)h], t ∈ [k∆t, (k + 1)∆t), j + k even, as the solution of the Riemann problem of the system with initial data u|t =k∆t =
ukj −1 , x < j h, ukj +1 , x > j h.
Then uh (x, t) is well defined: it is the exact entropy solution in each strip S k , it is continuous at the interfaces x = j h, k∆t t < (k +1)∆t with j +k odd, and it experiences jump discontinuities across the lines t = k∆t, k = 0, 1, 2, . . . . The waves emanating from the neighboring discontinuing mesh points (j h, k∆t) and ((j + 2)h, k∆t), j + k even, do not intersect. If it is proved that uh (x, t) is uniformly bounded in h in R2+ , Λ can be chosen, and the Glimm approximate solutions are constructed for all t 0. Then the limit of the approximate solutions is the entropy solution of the Cauchy problem (1.19) and (1.20) as in the following theorem. T HEOREM 7.1. Assume that system (1.19) is strictly hyperbolic and each characteristic field is either genuinely nonlinear or linearly degenerate in a neighborhood of a constant ¯ Then there exist two positive constants δ1 and δ2 such that, for initial data u0 state u. satisfying ¯ L∞ (R) δ1 , u0 − u
TVR (u0 ) δ2 ,
(7.1)
the Cauchy problem (1.19) and (1.20) has a global entropy solution u(x, t) for (x, t) ∈ R2+ , satisfying the entropy inequality (1.18) (d = 1) in the sense of distributions for any convex entropy–entropy flux pair and $ $ $u(·, t) − u¯ $ ∞ C0 u0 − u ¯ L∞ (R) , L (R) TVR u(·, t) C0 TVR (u0 ),
for any t ∈ [0, ∞),
for any t ∈ [0, ∞),
$ $ $u(·, t1 ) − u(·, t2 )$ 1 C0 |t1 − t2 | TVR (u0 ), L (R)
(7.2) (7.3)
for any t1 , t2 ∈ [0, ∞), (7.4)
for some constant C0 > 0. In order to show that the approximate solutions uh (x, t) converge to a solution of the Cauchy problem (1.19) and (1.20), it is required to establish (i) The compactness of the approximate solutions in order to ensure that a convergent subsequence (still denoted by) uh (x, t) may be selected such that uh (x, t) → u(x, t), a.e. for (x, t) ∈ R2+ ;
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(ii) The consistency of the scheme in order to guarantee that the limit u(x, t) is indeed a solution of the Cauchy problem (1.19) and (1.20). For the compactness of the Glimm approximate solutions under the assumption (7.1), the following estimates can be established: $ h $ $u (·, t) − u¯ $ ∞ C0 u0 − u ¯ L∞ (R) , for any t ∈ [0, ∞), L (R) TVR uh (·, t) C0 TVR (u0 ), for any t ∈ [0, ∞), $ $ h $u (·, t1 ) − uh (·, t2 )$ 1 C0 |t1 − t2 | + h TVR (u0 ), L (R) for any t1 , t2 ∈ [0, ∞),
(7.5) (7.6)
(7.7)
for some constant C0 > 0. Estimate (7.5) guarantees that the approximate solutions uh (x, t) can be constructed globally for all t over [0, ∞) if δ1 in (7.1) is sufficiently small. These compactness estimates imply that the family of approximate solutions uh (x, t) has uniformly bounded variation and thus converges almost everywhere, by the Helly theorem, to a function u(x, t) in BV. It can be shown that, for any equidistributed random sequence of numbers a = {a0 , a1 , a2 , . . .} ⊂ (−1, 1), the limit function u(x, t) is an entropy solution of the Cauchy problem (1.19) and (1.20), which also satisfies the entropy condition. For the compactness estimates, (7.7) is an immediate consequence of (7.6) since the waves emanating from each mesh point propagate with speed not exceeding Λ. To establish (7.6), one first notes that, for any t ∈ (k∆t, (k + 1)∆t), TVR (u(·, t)) is constant and can be measured by the sum of the strengths of waves that emanate from the mesh points (j h, k∆t) with j + k even. To estimate how the sum of wave strengths changes from the strip S k to the strip S k+1 , consider the family of diamond shaped regions ♦j k , k+2 j + k odd, with vertices Pjk , Pjk+1 , and Pjk+1 +1 , Pj −1 . A wave fan of n waves (ε1 , . . . , εn ) emanates from the mesh point Pjk+1 inside ♦j k . Through the side of ♦j k connecting the
two vertices Pjk and Pjk+1 −1 , there crosses a fan of waves (α1 , . . . , αn ) which is part (possibly none or all, as some of the components αi could be zero) of the wave fan emanating from the mesh point Pjk−1 , and through the side of ♦j k connecting the two vertices Pjk and
Pjk+1 +1 there crosses a fan of waves (β1 , . . . , βn ) which is part (possibly none or all, as some of the components βi could be zero) of the wave fan emanating from the mesh point Pjk+1 . Indeed, the wave fan (ε1 , . . . , εn ) approximates the wave pattern that would have resulted if the wave fans (α1 , . . . , αn ) and (β1 , . . . , βn ) had been allowed to propagate beyond t = (k + 1)∆t and thus interact. It can be shown that the strengths of incoming and outgoing waves are related by n
i=1
n
|αi | + |βi | + O(Qj k ) |εi | =
(7.8)
i=1
with Qj k = i,j {|αi ||βj |: αi and βj interacting}. If the quadratic term Qj k were not present, the total variation of uh (·, t), as measured by the strengths of waves, would not increase from S k to S k+1 .
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The effect of the quadratic term can be controlled as follows. Consider the polygonal k+1 curve Jk whose arcs connect nodes Pjk , Pjk+1 −1 , and Pj +1 , j + k odd. Define the Glimm functional associated with the curve Jk as F (Jk ) = L(Jk ) + MQ(Jk ),
(7.9)
where L(Jk ) :=
% & |α|: any wave α crossing Jk
is the linear part measuring the total variation, Q(Jk ) :=
% & |α||β|: α, β interacting waves crossing Jk
is the quadratic part measuring the potential wave interaction, and M is a large positive constant. The functional F (Jk ) is well defined and essentially equivalent to TVR (uh (·, t)) for k∆t t < (k + 1)∆t. It can be shown from (7.8) that F (Jk ) is nonincreasing in k as long as the total variation remains small, which implies the estimate (7.6). For the details of the proof of Theorem 7.1, see Glimm [130] and Liu [211,214]; also see Dafermos [88], Serre [277], and Smoller [291]. For extensions of the Glimm scheme to nonhomogeneous balance laws, see Dafermos and Hsiao [89], Liu [207], and Chen and Wagner [64]. The proof of Theorem 7.1 is based on the estimate showing that the effect of interactions is of second-order for the general system of n conservation laws, that is, the change in magnitude of waves due to interaction is of second-order in the magnitude of waves before interaction. For a system of two conservation laws, there exists a coordinate system of Riemann invariants, and the effect of interaction is of third-order, that is, the system is uncoupled modulo the third-order of the total variation of the solution. Therefore, Theorem 7.1 holds for the initial data of small oscillation but of larger total variation in the case of two conservation laws. This, in particular, applies to the isentropic Euler equations in (1.14) and (1.15). For the isothermal Euler equations, γ = 1, the condition of small oscillation can also be removed. Isothermal gas dynamics. For the Euler equations for isothermal gas dynamics, global entropy solutions can be constructed by the Glimm scheme with any large initial data of bounded variation due to the special structure of the wave curves (6.12). For (1.15) in Lagrangian coordinates, the one-dimensional isothermal motion of gases has the equation of state (6.9). Consider the Cauchy problem of (1.15) and (6.9) for x ∈ R and t 0 with initial data: (τ, v)|t =0 = (τ0 , v0 )(x),
x ∈ R.
Then we have the following theorem due to Nishida [242].
(7.10)
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T HEOREM 7.2. Suppose that τ0 (x) and v0 (x) are bounded functions with bounded variation over R and infx∈R τ0 (x) > 0. Then the Cauchy problem (1.15), (6.9), and (7.10) has a global entropy solution (τ, v)(x, t) with bounded total variation in x ∈ R for any t 0. P ROOF. The solution (τ, v)(x, t) in Theorem 7.2 is obtained as the limit of the approximate solutions (τ h , v h )(x, t) constructed by the Glimm scheme as in Theorem 7.1. In order to prove Theorem 7.2, it suffices to show that there exists a constant C0 > 0 such that TVR τ h , v h (·, t) C0 TVR (τ0 , v0 ).
(7.11)
To establish the compactness estimate (7.11), one needs to use the special structure of the wave curves of (1.15) and (6.9), established in Section 6.2, and show that the linear part L of the Glimm functional is decreasing. To see this, we first notice that the Riemann solution of (1.15), (6.9), and (6.10) has three constant states (τi , vi ), i = 1, 2, 3, connected by two of the elementary waves: 1-wave (S1 wave or R1 wave) and 2-wave (S2 wave or R2 wave). Denote these two waves by a vector α = (α1 , α2 ), and denote the strength of i-wave, i = 1, 2, by |α1 | = |q2 − q1 |, |α2 | = |q3 − q2 |, and |α| = |α1 | + |α2 |. The approximate solutions (τ h , v h ) will be estimated along the piecewise linear curves J defined as follows. Let the curve J0 be composed of the all segments joining Pj0 to Pj1+1 and Pj1+1 to Pj0+2 for all odd j . An immediate successor curve J2 of curve J1 is composed k−1 k of the same line segments except two segments joining Pjk to Pjk−1 +1 and Pj +1 to Pj +2 ,
k+1 k which are replaced by those joining Pjk to Pjk+1 +1 and Pj +1 to Pj +2 . Then all curves J are obtained by taking successively immediate successors, starting out from the curve J0 . Define the functional L(J ) on the approximate solutions restricted to each curve J by
L(J ) =
|α|,
where the summation is taken over all vectors of two elementary waves α = (α1 , α2 ) in the approximate solutions crossing the curve J . If J2 is an immediate successor of the curve J1 , Proposition 6.4 (i.e., (6.17)) implies L(J2 ) L(J1 ). By induction, L(J ) L(J0 ) for any curve J , which implies TVR q h (·, t) TVR (q0 ) for any t 0, with q0 (x) = ln τ0 (x), and thus |q h (x, t)| K for some positive constant K, since τ0 ∈ L∞ (R). Then TVR τ h (·, t) C1 TVR q h (·, t) C1 TVR (q0 ), and one has TVR v h (·, t) C2 TVR q h (·, t) C2 TVR (q0 ),
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463
from the equations of elementary wave curves (6.12) and |W (s)| C2 for |s| K. Therefore, TVR τ h , v h (·, t) C3 TVR (q0 ) C0 TVR (τ0 , v0 ). This completes the proof of Theorem 7.2.
Theorem 7.2 was originally established by Nishida [242]. For extensions to other isothermal flows, see [258] for the Euler–Poisson flow and [224] for the spherically symmetric Euler flow. For non-isentropic gas dynamics (1.13), consider the following Cauchy problem: (τ, v, S)|t =0 = (τ0 , v0 , S0 )(x).
(7.12)
The following existence theorem is due to Liu [212] (also see Temple [302]). T HEOREM 7.3. Let K ⊂ {(τ, v, S): τ > 0} be a compact set in R+ × R2 , and let N 1 be any positive constant. Then there exists a constant C0 = C0 (K, N), independent of γ ∈ (1, 5/3], such that, for every initial data (τ0 , v0 , S0 )(x) ∈ K with TVR (τ0 , v0 , S0 ) N , when (γ − 1)TVR (τ0 , v0 , S0 ) C0 ,
(7.13)
for any γ ∈ (1, 5/3], the Cauchy problem (1.13) and (7.12) has a global entropy solution (τ, v, S)(x, t) which is bounded and satisfies TVR (τ, v, S)(·, t) CTVR (τ0 , v0 , S0 ), for some constant C > 0 independent of γ . For the isentropic case: S = constant, the existence result of Theorem 7.3 was proved in Nishida and Smoller [245] (also see DiPerna [99]). For extensions to the initial-boundary value problems, see [246,213]. A similar theorem to Theorem 7.3, for general pressure law, was established in Temple [302]. In the direction of relaxing the requirement of small total variation, see Peng [253], Temple and Young [303,304], and Schochet [269]. For additional further discussions and references to the Glimm scheme, see Dafermos [88] and Serre [277].
7.2. Decay of solutions In this section we discuss the decay properties of Glimm solutions in BV of hyperbolic systems of conservation laws (1.19). Any system of two conservation laws (1.19) (n = 2) is endowed with a coordinate system (w1 , w2 ) of Riemann invariants corresponding to the two eigenvalues λ1 and λ2 .
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The system is genuinely nonlinear if ∂wi λi = 0, i = 1, 2. The isentropic Euler equations staying away from the vacuum are an important example of a 2 × 2 genuinely nonlinear and strictly hyperbolic system. We focus our attention on the decay properties of Glimm solutions with large total variation for genuinely nonlinear and strictly hyperbolic systems of two conservation laws, which are valid, in particular, for the isentropic Euler equations away from the vacuum. First, for the Glimm solution u(x, t) of (1.19), one has the following decay law: TVR u(·, t) Ct −1/2 ,
(7.14)
for some constant C > 0, which holds for any genuinely nonlinear system of two conservation laws with initial data of small oscillation (see Glimm and Lax [133]) and of n conservation laws with initial data of small total variation (see Liu [204]). DiPerna also proved in [103] that the total variation decays to zero, with no rate of convergence, for a more general system of n conservation laws which admits linearly degenerate characteristic fields such as the non-isentropic Euler equations staying away from the vacuum (n = 3). For any genuinely nonlinear and strictly hyperbolic system of two conservation laws, the Glimm solution with periodic initial data decays to the mean-value of the initial data over the period, and with initial data of compact support decays to an N-wave. Periodic solutions. First we consider Glimm solutions of the system of two conservation laws (1.19) with periodic initial data. The following fundamental decay behavior is due to Glimm and Lax [133]. T HEOREM 7.4. For the genuinely nonlinear and strictly hyperbolic system of two conservation laws (1.19) with n = 2, if the initial data u0 ∈ L∞ (R) have small oscillation and are periodic with period L, then there exists a solution u(x, t) which is periodic with respect to x with period L for all t > 0 and satisfies CL TV[x,x+L] u(·, t) , t u(x, t) − u¯ CL , t
for any x ∈ R,
(7.15)
(7.16)
where u¯ is the mean-value of u0 (x) over the space period and C > 0 is some constant. To illustrate the ideas involved in the proof of this theorem, we first consider the scalar conservation law ([133,180]): ∂t u + ∂x f (u) = 0,
(7.17)
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where f (u) is strictly convex, f (u) c0 > 0, and thus f (u) is strictly monotone increasing. Any differentiable solution u(x, t) is constant along the characteristic x = x(t) defined by dx = f u(x, t) . dt
(7.18)
The characteristics are straight lines and generally intersect. At a point of intersection, the solution becomes discontinuous. Along the curve of discontinuity with propagation speed σ , the Lax entropy condition f (u− ) > σ > f (u+ )
(7.19)
is satisfied, which implies that u− > u+ ,
(7.20)
since f (u) is increasing, where u− and u+ are the values of the solution u(x, t) on the left side and right side of curve of discontinuity, respectively. Let x1 (t) and x2 (t) be a pair of characteristics for 0 t T . Then there is a whole one-parameter family of characteristics connecting the points of the interval [x1 (0), x2 (0)] at t = 0 with points of the interval [x1 (T ), x2 (T )] at t = T . Since u(x, t) is constant along these characteristics, u(x, 0) on the interval [x1(0), x2 (0)] and u(x, T ) on the interval [x1 (T ), x2 (T )] are equivariant, i.e., they take on the same values in the same order, and thus the total increasing and decreasing variations of u(x, t) on these two intervals are the same. Denote by D(t) = x2 (t) − x1 (t) > 0 the width of the strip bounded by x1 and x2 . Then, from (7.18), D (t) = f (u2 ) − f (u1 ), where u1 and u2 are constant along the characteristics x1 (t) and x2 (t), respectively, and D(T ) = D(0) + f (u2 ) − f (u1 ) T .
(7.21)
Suppose that there is a shock y present in u(x, t) between the characteristics x1 and x2 . Since the characteristics on either side of a shock run into the shock according to (7.19), for any given time T , there exist two characteristics y1 and y2 intersecting the shock y at exactly time T . Assume that there are no other shocks present. Then the increasing variations of u(x, t) on the intervals (x1 (t), y1 (t)) and (x2 (t), y2 (t)) are independent of t. From (7.20), u(x, t) decreases across shocks, and then the increasing variation of u(x, t) over [x1 (T ), x2 (T )] equals the sum of the increasing variations of u(x, t) over [x1 (0), y1 (0)] and over [y2(0), x2 (0)]. This sum is in general less than the increasing variation of u(x, t) over [x1 (0), x2 (0)]. Thus, if shocks are present, the total increasing variation of u(x, t) between two characteristics decreases with time. To give a quantitative estimate of this decrease, we assume for simplicity that u0 (x) is piecewise monotone. Let I0 be any interval of the x-axis. Subdivide it into subintervals [yj −1 , yj ], j = 1, . . . , N , in such a way that u(x, 0) is alternatively increasing and decreasing on the subintervals. Denote by yj (t) the characteristic issuing from the j -th
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point yj with the understanding that, if yj (t) runs into a shock, yj (t) is continued as that shock. Then, for all t > 0, u(x, t) is alternately increasing and decreasing on the intervals (yj −1 (t), yj (t)), i.e., increasing for j odd and decreasing for j even. Since f (u) is an increasing function and u(x, t) decreases across shocks, the total increasing variation TV+ (T ) of f (u(x, t)) across the interval I (T ) = [y0 (T ), yN (T )] is TV+ (T ) =
f uj (T ) − f uj −1 (T ) ,
(7.22)
j odd
where uj −1 (T ) = u(yj −1 (T )+, T ) and uj (T ) = u(yj (T )−, T ). Denote, as before, by xj −1 (t), xj (t) the characteristics starting out inside yj −1 , yj , which intersect yj −1 (t), yj (t), respectively, at t = T . Then uj (t) is the constant value of u(x, t) on xj (t). Set Dj (t) = xj (t) − xj −1 (t). Then, by (7.21), Dj (T ) = Dj (0) + f uj (T ) − f uj −1 (T ) T . Take the sum over odd j to get, from (7.22),
Dj (T ) =
j odd
Dj (0) + TV+ (T )T .
(7.23)
j odd
Since the intervals [xj −1 (T ), xj (T )] are disjoint and lie in I (T ), their total length cannot exceed the length |I (T )| of I (T ), and then TV+ (T )
|I (T )| . T
(7.24)
Suppose that the solution u(x, t) is periodic in x with period L. Take I0 to be an interval of length L, then I (t) has length L for all t > 0. From the strict convexity f (u) > c0 > 0, (7.24) implies that the increasing variation per period of u(x, t) itself does not exceed L(c0 T )−1 . Since u(x, t) is periodic, its decreasing and increasing variations are equal and serve as a bound for the oscillation of u(x, t), especially for the deviation of u(x, t) from L its mean-value over period u¯ = L1 0 u(x, t) dx. Therefore, the total variation of u(x, t) per period at time t does not exceed 2L(c0 T )−1 and |u(x, t) − u| ¯ (c0 T )−1 . To generalize this idea to a system of two conservation laws, we first note that there exist Riemann invariants w1 and w2 which are functions of u(x, t) and satisfy the following equations: ∂t wi + λj ∂x wi = 0,
i = j,
where λi , i = 1, 2, are the eigenvalues of system (1.19) with n = 2 which can be considered as functions of the Riemann invariants w = (w1 , w2 ). Along 1-characteristics: dx/dt = λ1 and along 2-characteristics: dx/dt = λ2 , w2 and w1 are constant, respectively. If xj (t), j = 1, 2, are two 1-characteristics, dxj /dt = λ1,j , then u(x, 0) along the interval [x1 (0), x2 (0)] and u(x, T ) along the interval [x1 (T ), x2 (T )] are equivariant. The
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1-characteristics are no longer straight lines, but in general they intersect if the system is genuinely nonlinear. The 1-shocks satisfy the following Lax entropy condition: λ1 (u− ) > σ > λ1 (u+ ), which is the analogue of the condition (7.19) and implies that 1-characteristics drawn in the direction of increasing time t run into 1-shocks. Thus, as before, the presence of the 1-shocks decreases the total variation of w2 . Similarly, the presence of the 2-shocks decreases the total variation of w1 . To estimate the decrease of the total variation of w1 , the effect of 1-shocks on the total variation of w1 has to be considered. It is known that, across weak 1-shocks, ∆w1 is proportional to (∆w2 )3 , where ∆wj , j = 1, 2, denote the change in wj , j = 1, 2, respectively. Then the change in total variation of w1 due to 1shocks does not exceed O(ε)TV(u0 )2 , where ε is the oscillation of the solution. The width D(t) = x2 (t) − x1 (t) of a strip bounded by 1-characteristics xj (t), j = 1, 2, satisfies D (t) = λ1,2 − λ1,1 = ∂w1 λ1 (w1,2 − w1,1 ) + ∂w2 λ1 (w2,2 − w2,1 ), according to the mean-value theorem, where λi,j := λi (w(xj (t), t)) and wi,j := wi (xj (t), t). If the oscillation ε of the solution is small, then ∂wj λ1 = O(ε), j = 1, 2. The quantities w2,j , j = 1, 2, are independent of t, but w1,j , j = 1, 2, are not. This difficulty can be overcome by measuring the width of the strip, bounded by the 1-characteristics not between points with the same t coordinates but between points which lie approximately on the same 2-characteristics. Since w1 is constant along 2-characteristics, w1,2 − w1,1 is small in the above equation on D(t). After constructing approximate characteristics, one can derive the approximate conservation laws of the increasing and decreasing variations of wj , j = 1, 2, which are formulated as a balance between the amount of shock wave and rarefaction wave of either family entering and leaving a region, the amount of rarefaction and shock wave of the same family cancelling each other in the region, and a correction term accounting for the interaction between waves belonging to different families. Finally, the inequalities for the variations of wj , j = 1, 2, can be obtained by passage to the limit. See Glimm and Lax [133] for the details of proof. N-waves. Now we consider the Glimm solutions of system (1.19) with initial data supported on a compact set, i.e., u0 (x) = 0,
if |x| > R,
(7.25)
for some constant R > 0. The solution may decay to an N-wave. In the case of scalar conservation laws, the solution with initial data of compact support approaches an N-wave in L1 as t → ∞; see Lax [177], DiPerna [101], and Dafermos [88], as well as a different proof of this result by Keyfitz [259] for piecewise smooth solutions. An N-wave consists of a rarefaction wave bracketed by two shock waves. It propagates at a constant speed while its support expands at the rate t 1/2 . The L∞ -norm of an N-wave decays at the rate t −1/2 , but its L1 -norm remains constant with time. For systems of n conservation laws, it has been conjectured by Lax [177] that, if the initial data have compact support, then the asymptotic
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form of the solution consists of n distinct N-waves, each propagating at one of the n distinct characteristic speeds λi (0) of zero state. This conjecture has been proved for the case of two conservation laws (n = 2) with initial data of large total variation (DiPerna [101]) and for the the case of n conservation laws with initial data of small total variation (Liu [204]). The primary mechanisms of decay of solutions are the spreading of rarefaction waves and the cancellation of shock and rarefaction waves of the same kind. For a genuinely nonlinear and strictly hyperbolic system of two conservation laws with the eigenvalues λi and the Riemann invariants wi , satisfying ∂wi λi = 0, i = 1, 2, define the N-waves: Ni (x, t; pi , qi ) 1 x − λ (0, 0) , −(pi ki t)1/2 < x − λi (0, 0)t < (qi ki t)1/2 , i k t = i 0, otherwise, for ki = ∂wi λi (0, 0) and some constants pi , qi > 0, i = 1, 2. One has the following decay behavior in L1 due to DiPerna [101]. T HEOREM 7.5. For the genuinely nonlinear and strictly hyperbolic system of two conservation laws (1.19) with n = 2, if the initial data u0 (x) ∈ L∞ (R) have small oscillations and compact support, then there exist positive constants pi and qi such that $ $ $wi (·, t) − Ni (·, t; pi , qi )$ 1 Ct −1/6 , i = 1, 2, L (R) for some constant C > 0. For the BV solutions constructed by the Glimm scheme to any genuinely nonlinear and strictly hyperbolic system of n conservation laws with initial data of small total variation, Liu proved in [204] that the solution also decays to the N-waves at the rate t −1/6 if the initial data have compact support; the generalization of this result to systems with linearly degenerate characteristic fields was given in Liu [205]. Decay properties for general BV solutions to systems of two conservation laws were obtained by Dafermos [88] by using the theory of generalized characteristics under the assumption that the traces of the solutions along any space-like curve are functions of locally bounded variation (see Section 8.1). See also Greenberg [143] on the decay of special solutions for a class of two conservation laws generated by a second-order wave equation, and Greenberg and Rascle [144] for an interesting example of periodic solutions in both space and time when the flux-function is C 1 but not C 2 . For periodic entropy solutions only in L∞ , an analytical framework has been established in Chen and Frid [46] (also see Section 9.6). 7.3. L1 -stability of Glimm solutions We now discuss the stability of solutions to the Cauchy problem (1.19) and (1.20). The existence proof in Theorem 7.1 based on compactness arguments does not provide
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information on this issue. By monitoring the time evolution of a certain functional, it can be shown that the Glimm solutions depend continuously on their initial data. Let u(x, t) and v(x, t) be two approximate solutions of (1.19) constructed by the Glimm scheme, with small total variation. As in Theorem 7.1, it is assumed that system (1.19) is strictly hyperbolic and each characteristic field is either genuinely nonlinear or linearly degenerate. We discuss how the distance u(·, t) − v(·, t) L1 (R) changes in time. Denote by s → Ri (s)(u− ),
s → Si (s)(u− ),
i = 1, . . . , n,
(7.26)
the i-rarefaction and i-shock curve of (1.19) through the state u− , parametrized by arclength, and set Ri (s)(u− ), s 0, Υi (s)(u− ) = (7.27) Si (s)(u− ), s < 0. For any fixed point (x, t), consider the scalar function qi (x, t), which can be regarded intuitively as the strength of the i-shock wave in the jump (u(x, t), v(x, t)), defined implicitly by (7.28) v(x, t) = Sn qn (x, t) ◦ · · · ◦ S1 q1 (x, t) u(x, t) . It is clear that n
qi (x, t) C1 u(x, t) − v(x, t) C1−1 u(x, t) − v(x, t)
(7.29)
i=1
for some constant C1 > 0. For each i = 1, . . . , n, define -
. α u(x, t) + α v(x, t) . + + Wi (x, t) = −
+
0
(7.30)
In (7.30), − sums the strengths |α(u(x, t))| (and |α(v(x, t))|) of all kα -waves xα (t) < x of u(x, t) (and v(x, t)) with i < kα n, respectively; + sums the strengths |α(u(x, t))| (and |α(v(x, t))|)of all kα -waves xα > x of u(x, t) (and v(x, t)) with 1 kα < i, respectively; and 0 sums the strengths |α(u(x, t))| (and |α(v(x, t))|) of all i-waves, here kα = i, with xα < x (and xα > x) of u(x, t) (and v(x, t)) if qi (x, t) < 0, or with xα > x (and xα < x) of u(x, t) (and v(x, t)) if qi (x, t) > 0, respectively. Define a functional, equivalent to the L1 distance of u(x, t) and v(x, t), as Φ(u, v)(t) n
qi (x, t) 1 + K1 Fu (nN∆t) + Fv (nN∆t) + K2 Wi (x, t) dx, = i=1 R
for each t ∈ (nN∆t, (n + 1)N∆t), where K1 and K2 are sufficiently large positive constants, N is a constant in the wave tracing method, Fu and Fv are the Glimm functionals
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defined in (7.9) for u(x, t) and v(x, t), respectively, valued at the end time t = nN∆t. The definition of this functional is given by Liu and Yang [215], and is similar to those used in Bressan, Liu and Yang [25] and Hu and LeFloch [155] for the solutions constructed by the wave-front tracking algorithm. The key estimate is that the functional Φ(u, v)(t) can be controlled by its initial value Φ(u, v)(0), up to a certain error term which approaches zero as the mesh size tends to zero. From Theorem 7.1, there exist subsequences of the approximate solutions which converge to the exact Glimm solutions, locally in the L1 norm. Therefore, one has the following theorem on the L1 -stability of Glimm solutions to the Cauchy problem of the genuinely nonlinear and strictly hyperbolic system (1.19) with initial data (1.20): T HEOREM 7.6. If the initial data u0 (x) and v0 (x) have sufficiently small total variation and u0 − v0 ∈ L1 (R), then, for the corresponding exact Glimm solutions u(x, t) and v(x, t) of the Cauchy problem (1.19) and (1.20), there exists a constant C > 0 such that $ $ $u(·, t) − v(·, t)$ 1 C u0 − v0 1 , L (R) L (R)
(7.31)
for all t > 0. An immediate consequence of this theorem is that the whole sequence of the approximate solutions constructed by the Glimm scheme converges to a unique entropy solution of (1.19) and (1.20) as the mesh size tends to zero. See also Bressan [19] for the uniqueness of limits of Glimm’s random choice method. The details of the proof of Theorem 7.6 can be found in Liu and Yang [215–217].
7.4. Wave-front tracking algorithm and L1 -stability Assume that system (1.19) is strictly hyperbolic, with eigenvalues λ1 (u) < · · · < λn (u), and each characteristic field is either genuinely nonlinear or linearly degenerate. The Glimm scheme has been the basic tool for the construction and analysis of entropy solutions to systems of conservation laws. An alternative method for constructing approximate solutions is the wave-front tracking algorithm, which generates entropy solutions of the Cauchy problem (1.19) and (1.20) with initial data of small total variation and provides an alternative proof of Theorem 7.1. The entropy solutions of (1.19) and (1.20) obtained by the wave-front tracking algorithm, the same as those obtained by the Glimm scheme, are L1 -stable, i.e., the solutions depend Lipschitz continuously on the initial data in the L1 norm, based on a priori estimates on the distance between two approximate solutions. Wave-front tracking algorithm. The wave-front tracking algorithm generates piecewise constant approximate solutions of the Cauchy problem (1.19) and (1.20). A wave-front tracking ε-approximate solution is, roughly speaking, a piecewise constant function u = u(x, t) whose jumps occur along finitely many segments x = xα (t) in the x–t plane and
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can be classified as shocks, rarefactions, and non-physical waves. At each time t > 0, these jumps should approximately satisfy the Rankine–Hugoniot condition:
x (t) u(xα +, t) − u(xα −, t) − f u(xα +, t) − f u(xα −, t) = O(ε), α
α
as well as the following condition:
% & q u(xα +, t) − q u(xα −, t) − xα (t) η u(xα +, t) − η u(xα −, t) α
O(ε), for any entropy–entropy flux pair (η, q) with convex η. The small parameter ε controls three types of errors: errors in the speeds of shock and rarefaction fronts, the maximum strength of rarefaction fronts, and the total strength of all non-physical waves. The notations in (7.26) and (7.27) will be adopted in this section. D EFINITION 7.1. Given ε > 0, a function u : [0, ∞) → L1 (R; Rn ) is called an ε-approximate solution of the Cauchy problem (1.19) and (1.20) if the following conditions are satisfied: (1) The function u(x, t) is piecewise constant with discontinuities along finitely many lines in the x–t plane. The step function u(x, 0) of the approximate solution u(x, t) at t = 0 approximates the initial data u0 (x) in L1 within distance ε: $ $ $u(·, 0) − u0 (·)$ 1 < ε. L
(7.32)
Only finitely many wave-front interactions occur, each involving exactly two incoming fronts. Jumps can be of three types: shocks (or contact discontinuities), rarefaction waves, and non-physical waves. (2) Along each shock (or contact discontinuity) x = xα (t), the values u± = u(xα ±, t) are related by u+ = Skα (sα )(u− ) for some kα ∈ {1, . . . , n} and some wave size sα satisfying sα < 0 if the kα -th characteristic field is genuinely nonlinear. Moreover, the speed of the shock front σ (u+ , u− ) with left and right states u± satisfies x (t) − σ (u+ , u− ) ε. α
(3) Along each rarefaction front x = xα (t), one has u+ = Rkα (sα )(u− ) with sα ∈ (0, ε] and |xα (t) − λkα (u+ )| ε, for some genuinely nonlinear field kα . ¯ where λ¯ is a fixed (4) All non-physical fronts x = xα (t) have the same speed xα (t) = λ, constant strictly greater than all characteristic speeds. The total strength of all nonphysical fronts in u(x, t) remains uniformly small in the sense:
u(xα +, t) − u(xα −, t) ε for all t 0, where the sum is taken over all non-physical fronts.
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The algorithm for constructing these wave-front tracking approximations is described below. The basic ideas were introduced in Dafermos [80] for scalar conservation laws and DiPerna [102] for 2 × 2 systems, then extended in Bressan [18] and Risebro [266] to general n × n systems. The construction starts at time t = 0 by taking a piecewise constant function u(x, 0) approximating u0 (x) satisfying (7.32) and TV(u(·, 0)) TV(u0 ). Let x1 < · · · < xN be the points where u(·, 0) is discontinuous. For each α = 1, . . . , N , the Riemann problem generated by the jump u(xα ±, 0) is approximately solved on a forward neighborhood of (xα , 0) in the x–t plane by a function of the form u(x, t) = φ((x − xα )/t) with φ : R → Rn piecewise constant. More precisely, if the exact solution of the Riemann problem contains only shocks and contact discontinuities, then we let u(x, t) be the exact solution which is piecewise constant. If centered rarefaction waves are present, they are approximated by a centered rarefaction fan containing several small jumps traveling with a speed close to the characteristic speed. Suppose that the first set of interactions between two or more wave-fronts occurs at a time t1 . Since u(·, t1 ) is still a piecewise constant function, the corresponding Riemann problems can again be approximately solved within the class of piecewise constant functions. The solution u(x, t) is then continued up to a time t2 , where the second set of wave interactions takes place, etc. However, it is observed that, at a generic interaction point, there will be two incoming fronts, while the number of outgoing fronts is n if all waves generated by the Riemann problem are shocks or contact discontinuities, or even larger if rarefaction waves are present. In turn, these outgoing wave-fronts may quickly interact with several other fronts, generating more and more lines of discontinuity. Therefore, for general n × n systems, the number of wave-fronts may approach infinity in a finite time, which causes the breakdown of the construction. To avoid this breakdown, the algorithm must be modified, which can be achieved by using two different procedures for solving a Riemann problem within the class of piecewise constant functions: (1) an accurate Riemann solver which introduces several new wave-fronts; and (2) a simplified Riemann solver which involves a minimum number of outgoing fronts. Although the number of wave-fronts could approach infinity within a finite time if all Riemann problems were solved accurately, the new fronts generated by further interactions are very small since the total variation remains small. When their size becomes smaller than a threshold parameter ν > 0, a simplified Riemann solver is used, which generates one single new non-physical front with very small amplitude and traveling with a fixed speed λ¯ strictly larger than all characteristic speeds, that is, all new waves are lumped together in a single non-physical front. The total number of fronts thus remains bounded for all times. We now describe these two procedures which will be used to solve the Riemann problem of (1.19) and (1.20) at a given point (x, ¯ t¯) with u(x, t¯) =
¯ u− , x < x, u+ , x > x. ¯
(7.33)
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The accurate Riemann solver is as follows. Given u− and u+ in (7.33), one first determines the states u0 , u1 , . . . , un and parameter values s1 , . . . , sn such that, using the notations in (7.26) and (7.27), u0 = u− ,
un = u+ ,
ui = Υi (si )(ui−1 ),
i = 1, . . . , n.
These states u0 , u1 , . . . , un are the constant states present in the exact solution of the Riemann problem. If all jumps (ui−1 , ui ) were shocks or contact discontinuities, then the Riemann problem would have a piecewise constant solution with at most n lines of discontinuity. In the general case, the exact solution of (7.33) is not piecewise constant because of the presence of rarefaction waves. These will be approximated by piecewise constant rarefaction fans, inserting additional states ui,j as follows. Let δ > 0 be a fixed small constant. If the i-th characteristic field is genuinely nonlinear and si > 0, consider the integer Ni = 1 + [si /δ],
(7.34)
where [si /δ] denotes the largest integer less than or equal to si /δ. For j = 1, . . . , Ni , define ui,j = Υi (j si /Ni )(ui−1 ),
xi,j (t) = x¯ + (t − t¯)λi (ui,j ).
If the i-th characteristic field is genuinely nonlinear and si 0, or if the i-th characteristic field is linearly degenerate (with si arbitrary), define Ni = 1 and ui,1 = ui ,
xi,1 (t) = x¯ + (t − t¯)σi (ui−1 , ui ),
with σi (ui−1 , ui ) the Rankine–Hugoniot speed of a jump connecting ui−1 with ui so that σi (ui−1 , ui )(ui − ui−1 ) = f(ui ) − f(ui−1 ). Then, define an approximate solution to the Riemann problem (7.33) as ⎧ u− , ⎪ ⎪ ⎪ ⎨u , i,j u(x, t) = ⎪ ui , ⎪ ⎪ ⎩ u+ ,
x < x1,1 (t), xi,j (t) < x < xi,j +1 (t), j = 1, . . . , Ni − 1, xi,Ni (t) < x < xi+1,1 (t), x > xn,Nn (t).
(7.35)
Thus each centered i-rarefaction wave is here divided into Ni − 1 equal parts and replaced by a rarefaction fan containing Ni wave-fronts. The strength of each one of these fronts is less than δ because of (7.34). The simplified Riemann solver is as follows. The first case is that i1 and i2 are the families of two incoming wave-fronts with i1 i2 , i1 , i2 ∈ {1, . . . , n}. In this case, let uL , uM , and uR be the left, middle, and right states before the interaction, related by uM = Υi1 (s1 )(uL ),
uR = Υi2 (s2 )(uM ).
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Define the auxiliary right state: u˜ R =
Υi1 (s1 ) ◦ Υi2 (s2 )(uL ), i1 > i2 , i1 = i2 . Υi1 (s1 + s2 )(uR ),
(7.36)
˜ Let u(x, t) be the piecewise constant solution of the Riemann problem with data uL , uR , ˜ constructed as in (7.35). Because of (7.36), the piecewise constant function u(x, t) contains exactly two wave-fronts of size s1 , s2 , if i1 > i2 , or a single wave-front of size s1 + s2 if i1 = i2 . In general, u˜ R = uR . Let the jump (u˜ R , uR ) travel with a fixed speed λ¯ strictly bigger than all characteristic speeds. In a forward neighborhood of the point (x, ¯ t¯), we thus define an approximate solution u(x, t) as u(x, t) =
˜ u(x, t), uR ,
¯ − t¯), x − x¯ < λ(t ¯ x − x¯ > λ(t − t¯).
(7.37)
This simplified Riemann solver introduces a new non-physical wave-front, traveling with constant speed λ¯ . In turn, this front may interact with other (physical) fronts. One more case of interaction thus needs to be considered, that is, a non-physical front hits a wavefront of the i-characteristic field for some i ∈ {1, . . . , n} from the left. Let uL , uM , and uR be the left, middle, and right states before the interaction. If uR = Υi (s)(uM ), define u˜ R = Υi (s)(uL ).
(7.38)
˜ Let u(x, t) be the solution to the Riemann problem with data uL and u˜ R , constructed as ˜ in (7.35). Because of (7.38), u(x, t) will contain a single wave-front belonging to the i-th field with size s. Since in general u˜ R = uR , we let the jump (u˜ R , uR ) travel with the fixed speed λ¯ . In a forward neighborhood of the point (x, ¯ t¯), the approximate solution u(x, t) is thus defined again according to (7.37). By construction, all non-physical fronts travel with the same speed λ¯ . The above cases therefore cover all possible interactions between two wave-fronts. A threshold parameter ν > 0 is used to determine which Riemann solver is used at any given interaction. The accurate method is used at time t = 0 and at every interaction where the product of the strengths of the incoming waves is |s1 s2 | ν; while the simplified method is used at every interaction involving a non-physical wave-front and also at interactions with |s1 s2 | < ν. In the above, it is assumed that only two wave-fronts interact at any given point, which can always be achieved by an arbitrarily small change in the speed of one of the interacting fronts. It should also be adopted that, in the accurate Riemann solver, rarefaction fronts of the same field of one of the incoming fronts are never partitioned (even if their strength is bigger than δ). This guarantees that every wavefront can be uniquely continued forward in time, unless it gets completely cancelled by interacting with another front of the same field and opposite sign. The above construction of an approximate solution involves three parameters: a fixed speed λ¯ strictly larger than all characteristic speeds, a small constant δ > 0 controlling the maximum strength of rarefaction fronts, and a threshold parameter ν > 0 determining whether the accurate or the simplified Riemann solver is used. This wave-front tracking
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algorithm generates alternatively an entropy solution of the Cauchy problem (1.19) and (1.20) in Theorem 7.1. T HEOREM 7.7. Let u0 (x) have small total variation over R. For any fixed small ε > 0, approximate u0 (x) by some step function uε0 (x) such that $ ε $ $u − u0 $ 1 ε, TVR uε0 TVR (u0 ). 0 L (R) Then, for the fixed speed λ¯ independent of ε and strictly larger than all characteristic speeds, small δε = ε > 0 controlling the maximum strength of rarefaction fronts, and a threshold parameter νε > 0 determining whether the accurate or the simplified Riemann solver is used and depending on ε and on the number of jumps of uε0 (x), the wave-front tracking algorithm with initial data uε0 (x) generates the global ε-approximate solutions uε (x, t) which have a subsequence converging, a.e. on R2+ , to an entropy BV solution u(x, t) of the Cauchy problem (1.19) and (1.20) with the estimates (7.3) and (7.4). To prove Theorem 7.7, the argument used in the proof of Theorem 7.1 can be applied with some modification. The proof consists of two steps. The first step is to show that the ε-approximate solution uε (x, t) is defined for all t 0, which can be achieved by showing two facts: the total variation of uε (·, t) remains uniformly bounded and the number of wave-fronts in uε (·, t) remains finite. To derive the bound of the total variation of uε (·, t), as in (7.9), introduce the total strength of waves L(t) in uε (x, t) cross the t-time line:
Luε (t) = |sα |, (7.39) α
where the summation is taken over all wave fronts of uε (x, t); and the wave interaction potential Q(t) cross the t-time line:
|sα sβ |, (7.40) Quε (t) = α,β∈A
where the summation runs over all pairs of approaching waves. For a non-physical front x = xα (t), we simply call sα = |u(xα (t)+, t) − u(xα (t)−, t)| the strength of the nonphysical front at xα (t). For convenience, non-physical fronts are regarded as belonging to a fictitious linearly degenerate (n + 1)-th characteristic field. Two fronts of the families kα , kβ ∈ {1, . . . , n + 1}, located respectively at xα , xβ with xα < xβ (kα = n + 1 if xα is non-physical), are approaching if either kα > kβ , or kα = kβ , and at least one of them is a genuinely nonlinear shock. The total strength L of waves stays constant along time intervals between consecutive collisions of fronts and only changes across points of wave interaction. The wave interaction potential Q also stays constant along time intervals between consecutive collisions. It can be proved that, for a suitably large constant M, the quantity Fuε (t) = Luε (t) + MQuε (t)
(7.41)
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analogous to the Glimm functional, bounding the total variation of uε (x, t), is nonincreasing in time. The key observation is that Q(t) is positive and decreasing after each interaction. The number of physical fronts can grow only at times t where the accurate Riemann solver is used. The set of times where the accurate solver is used can be proved finite. Thus the number of physical fronts is finite. In turn, a new non-physical front can be generated only when two physical fronts interact. Since any two physical fronts interact at most once, it follows that the number of non-physical fronts also remains finite. The second step is to show that the limit of the approximate solutions is an entropy solution. By Helly’s compactness theorem, the estimate on the total variation of uε (x, t) implies that there exists a subsequence (still denoted) uε (x, t) converging to some function u(x, t) in L1loc , as ε → 0. To prove that u(x, t) is an entropy solution of (1.19) and (1.20), one needs to verify that both the maximum size of rarefaction fronts and the total strength of non-physical fronts in uε (x, t) tend to zero as ε → 0, which follows from the construction and the interaction estimates. The approximate solutions uε (x, t) satisfy the entropy inequality with an error tending to zero as ε → 0, which shows that the solution satisfies also the entropy condition. See Bressan [18] for the details of the proof. L1 -stability. As in Theorem 7.1, the existence proof of Theorem 7.7 provides no clue on the stability of solutions of the Cauchy problem (1.19) and (1.20). By monitoring the time evolution of a certain functional, it can be shown that the ε-approximate solutions constructed by the wave-front tracking algorithm depend continuously on their initial data up to a certain error of order ε. This shows, by passing to the limit ε → 0, that the fronttracking approximations converge to a unique limit, and the solution depends Lipschitz continuously on the initial data. Suppose that system (1.19) is strictly hyperbolic and genuinely nonlinear. Let u(x, t) and v(x, t) be two ε-approximate solutions of (1.19) and (1.20) with small total variation. For any fixed point (x, t), define the scalar function qi (x, t) as in (7.28). Define the functional n
qi (x, t) 1 + K1 Qu (t) + Qv (t) + K2 Wi (x, t) dx, Ψ (u, v)(t) = i=1 R
where K1 and K2 are sufficiently large positive constants, Qu (t) and Qv (t) are the wave interaction potentials for u(x, t) and v(x, t) respectively defined in (7.40), and Wi (x, t) is defined as in (7.30). Notice that the strengths of non-physical fronts do enter in the definition of Q, but play no role in the definition of Wi . If the total variations of u(x, t) and v(x, t) are sufficiently small such that 0 K1 (Qu (t) + Qv (t)) + K2 Wi (x, t) 1 for all i, then $ $ $ $ C1−1 $u(·, t) − v(·, t)$L1 (R) Ψ (u, v)(t) 2C1 $u(·, t) − v(·, t)$L1 (R).
(7.42)
Bressan, Liu and Yang [25] indicates that, if Fu and Fv defined in (7.41) are sufficiently small, then the functional Ψ (u, v)(t) is almost decreasing in t, that is, Ψ (u, v)(t1 ) − Ψ (u, v)(t2 ) C2 ε(t1 − t2 ),
0 t2 < t1 .
(7.43)
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For small constant δ > 0, with the notation in (7.41), define the domain & % D = CL u ∈ L1 R; Rn : u(x, t) is piecewise constant, Fu (t) < δ , where CL denotes the closure in L1 (R). Estimate (7.43) implies the L1 -stability of entropy solutions obtained by the wave-front tracking method. T HEOREM 7.8. For any initial data u0 ∈ D with δ sufficiently small, as ε → 0, any subsequence of the approximate solutions uε (x, t) constructed by the wave-front tracking algorithm for the Cauchy problem (1.19) and (1.20) converges to a unique limit u(x, t). The map (u0 , t) → St (u0 ) = u(·, t) defines a uniformly Lipschitz continuous semigroup whose trajectories are entropy solutions of (1.19) and (1.20). If u(x, t) and v(x, t) are two such entropy solutions of (1.19) and (1.20) with initial data u0 (x) and v0 (x), respectively, then $ $ $u(·, t) − v(·, t)$ 1 C u0 − v0 1 , L (R) L (R)
(7.44)
for some constant C > 0. With the assumption (7.43), Theorem 7.8 can be proved as follows. For a given u0 ∈ D, consider any sequences {ul (x, t)}, l = 1, 2, . . . , and {uk (x, t)}, k = 1, 2, . . . , of the εl approximate solutions and εk -approximate solutions of (1.19) and (1.20), respectively, with $ l $ $u (·, 0) − u0 (·)$ 1 εl , $L $ k $u (·, 0) − u0 (·)$ 1 εk , L
liml→∞ εl = 0, Ful (t) < δ, limk→∞ εk = 0, Fuk (t) < δ,
for any t > 0. From (7.42) and (7.43), for any l, k 1, and t > 0, $ l $ $u (·, t) − uk (·, t)$ 1 C1 Ψ ul , uk (t) C1 Ψ ul , uk (0) + C1 C2 t max{εl , εk } L $ $ 2C12 $ul (·, 0) − uk (·, 0)$L1 + C1 C2 t max{εl , εk }. As l, k → ∞, the right-hand side tends to zero, the two sequences have the same limit, and thus any sequence of ε-approximate solutions converges to a unique limit. The semigroup property St2 (St1 u0 ) = St1 +t2 u0 follows immediately from the uniqueness. Let u0 (x) and v0 (x) be the initial data of the entropy solutions u(x, t) and v(x, t) which are the limits of the corresponding εj -approximate solutions uj (x, t) and vj (x, t) of (1.19) and (1.20), respectively, with uj (·, 0)−u0 (·) L1 εj , vj (·, 0)−v0(·) L1 εj , and limj →∞ εj = 0. From (7.42) and (7.43), one has $ $ j $u (·, t) − vj (·, t)$ 1 C1 Ψ uj , vj (t) C1 Ψ uj , vj (0) + C1 C2 tεj L $ $ 2C12 $uj (·, 0) − vj (·, 0)$L1 + C1 C2 tεj . Taking j → ∞ yields (7.44).
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Theorem 7.8 was established in Bressan, Liu and Yang [25], where the proof of (7.43) can be found, and in Hu and LeFloch [155], where Haar’s method was extended to nonlinear systems of conservation laws. A sharper version of the L1 -continuous dependence estimate, containing dissipation terms in the left-hand side of (7.44), was later established by Dafermos [88] (for scalar equations) and Goatin and LeFloch [137] (for systems). For other related results and discussions, see [20,21,88,216] and the references therein. The approach for Theorem 7.8 in [25] provides a much simpler proof of the existence of a Lipschitz semigroup, called the standard Riemann semigroup [21] generated by the n × n systems of conservation laws (1.19) and (1.20), which is defined as a continuous map S : D × [0, ∞) → D such that, for some Lipschitz constant L, denoting St (·) = S(·, t), (1) S0 u0 = u0 , St2 St1 u0 = St1 +t2 u0 ; (2) For all u0 , v0 ∈ D, t1 , t2 0, St1 u0 − St2 u0 L1 L( u0 − v0 L1 + |t1 − t2 |); (3) If u0 ∈ D is piecewise constant, then, for t > 0 sufficiently small, the function u(·, t) = St u0 coincides with the solution of (1.19) and (1.20) obtained by piecing together the standard self-similar solutions of the corresponding Riemann problems. For any initial data u0 ∈ D with δ sufficiently small, the solution u(x, t) as the limit of the ε-approximate solutions constructed by the wave-front tracking algorithm can be identified with a trajectory of the standard Riemann semigroup [21], which also indicates that the limit of the ε-approximate solutions by the wave-front tracking algorithm is unique. As discussed earlier, the results in Bressan [19] and Liu and Yang [217] also imply the uniqueness of limits of Glimm’s random choice method. For initial data u0 (x) which are small BV perturbation of a large Riemann data, some progress has been made in Lewicka [190] and Lewicka and Trivisa [191]. There are some recent important developments on uniform BV estimates for artificial viscosity approximations for hyperbolic systems of conservation laws with initial data of small total variation, as well as the L1 -stability of BV solutions constructed by the vanishing viscosity method; see Bianchini and Bressan [12,13]. This uniqueness property can be extended to any solutions satisfying certain extra regularity condition as stated in the following theorem. T HEOREM 7.9. Any solution u(x, t) of the Cauchy problem (1.19) and (1.20), with u(·, t) ∈ D for all t 0, which satisfies the following time oscillation condition: u(x±, t + h) − u(x±, t) β TV[x−λh,x+λh] u(·, t)
(7.45)
for all x ∈ R, t 0, and any h > 0, with λ and β some positive constants, coincides with the trajectory of the standard Riemann semigroup St emanating from the initial data: u(·, t) = St u0 (·). In particular, u(x, t) is uniquely determined by its initial data. The solutions constructed by either Glimm’s random choice method or the wavefront tracking algorithm satisfy the tame oscillation condition (7.45). Such a uniqueness result of entropy solutions to systems was established first by Bressan and LeFloch [23] under a stronger assumption, called the tame variation condition. By improving upon the arguments, Theorem 7.9 was established by Bressan and Goatin [22]. The tame oscillation
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condition (7.45) can be also replaced by the assumption that the trace of solutions along space-like curves has local bounded variation (see Bressan and Lewicka [24]). Also see Hu and LeFloch [155] for a different approach based on Harr’s method, and Baiti, LeFloch and Piccoli [5] for some further generalization. For other discussions about the wave-front tracking algorithm, standard Riemann semigroup, uniqueness, and related topics, we refer to Bressan [20], Dafermos [88], and LeFloch [187] which provide extensive discussions and references.
8. Global discontinuous solutions III: entropy solutions in BV In this section we focus on general global discontinuous solutions in L∞ ∩ BVloc satisfying the Lax entropy inequality and without specific reference on the method for construction of the solutions.
8.1. Generalized characteristics and decay Consider the BV entropy solutions of (1.19) having bounded variation in the sense of Tonelli and Cesari, i.e., functions whose first-order distributional derivatives are locally Borel measures (Volpert [312]). The notion of characteristics for classical solutions can be extended to generalized characteristics for BV entropy solutions. The generalized characteristics provide a powerful tool for studying the structure and behavior of BV entropy solutions. Suppose that system (1.19) is strictly hyperbolic with n real distinct eigenvalues λ1 < λ2 < · · · < λn and u(x, t) is a BV entropy solution of (1.19) for (x, t) ∈ R2+ . The domain R2+ can be written as C ∪ J ∪ I with C, J , and I pairwise disjoint, where C is the set of points of approximate continuity of u(x, t), J is the set of points of approximate jump discontinuity (shock set) of u(x, t), and I denotes the set of irregular points of u(x, t). The one-dimensional Hausdorff measure of I is zero. The shock set J is essentially the (at most) countable union of C 1 arcs. With any point (x, t) ∈ J are associated distinct one-sided approximate limits u± and a shock speed σ related by the Rankine–Hugoniot condition (6.7) and satisfying the Lax entropy condition (6.8). To handle shock waves in solutions, we employ the concept of generalized characteristics introduced by Dafermos (cf. [81]). The generalized characteristics are defined in Filippov’s sense of differential inclusion [118] as follows. D EFINITION 8.1. A generalized i-characteristic for (1.19) on an interval [t1 , t2 ], 0 t1 < t2 < ∞, associated with the solution u(x, t), is a Lipschitz function ξ : [t1 , t2 ] → R such that ξ (t) ∈ λi u ξ(t)+, t , λi u ξ(t)−, t , for almost all t ∈ [t1 , t2 ].
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Generalized characteristics propagate with either classical characteristic speeds or shock speeds, as indicated in the following proposition. P ROPOSITION 8.1. Let ξ(t) be a generalized i-characteristic on [t1 , t2 ]. Then, for almost all t ∈ [t1 , t2 ], ξ(t) propagates with classical i-characteristic speed if (ξ(t), t) ∈ C and with i-shock speed if (ξ(t), t) ∈ J . ¯ t¯) of the upper half-plane, there exists at P ROPOSITION 8.2. Given any point (x, least one generalized i-characteristic, defined on [0, ∞), passing through (x, ¯ t¯). The set of i-characteristics passing through (x, ¯ t¯) spans a funnel-shaped region bordered by a minimal i-characteristic and a maximal i-characteristic ( possibly coinciding). Furthermore, if ξ(t) denotes the minimal or the maximal backward i-characteristic issuing from (x, ¯ t¯), then u ξ(t)+, t = u ξ(t)−, t ,
ξ (t) = λi u ξ(t)±, t ,
for almost all t ∈ [0, t¯]. D EFINITION 8.2. A minimal (or maximal) i-divide, associated with the solution u(x, t), is a Lipschitz function φ : [0, ∞) → R with the property that φ(t) = limk→∞ ξk (t), uniformly on compact subsets of [0, ∞), where ξk (t) is the minimal (or maximal) backward i-characteristic emanating from some point (xk , tk ) with tk → ∞, as k → ∞. Two minimal (or maximal) i-divides φ1 (t) and φ2 (t), with φ1 (t) φ2 (t), 0 t < ∞, are disjoint if the set {(x, t): 0 t < ∞, φ1 (t) < x < φ2 (t)} does not intersect the graph of any minimal (or maximal) i-divide. The graphs of any two minimal (or maximal) i-characteristics may run into each other but they cannot cross. Then the graph of a minimal (or maximal) backward i-characteristic cannot cross the graph of any minimal (or maximal) i-divide and the graphs of any two minimal (or maximal) i-divides cannot cross. Any minimal (or maximal) i-divide divides the upper half-plane into two parts in such a way that no forward i-characteristic may cross from the left to the right (or from the right to the left). The concept of i-divide plays a central role in the investigation of the large-time behavior of solutions with periodic initial data through the approach of generalized characteristics. The set of minimal or maximal i-divides associated with a particular solution may be empty, but it is nonempty if the solution is periodic. P ROPOSITION 8.3. If φ(t) is any minimal or maximal i-divide, then u φ(t)+, t = u φ(t)−, t ,
φ (t) = λi u φ(t)±, t ,
for almost all t ∈ [0, ∞). In particular, φ(t) is a generalized i-characteristic on [0, ∞). Furthermore, if {φk (t)} is a sequence of minimal (or maximal) i-divides converging to some function φ(t) uniformly on compact subsets of [0, ∞), then φ(t) is a minimal (or maximal) i-divide.
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P ROPOSITION 8.4. The set of minimal (or maximal) i-divides associated with any solution u(x, t), periodic in x, with period P , is not empty. The union of the graphs of these i-divides is invariant under the translation by P in the x-direction. The above theory of generalized characteristics follows Dafermos [84,86,88]. The proofs of these propositions and further discussions can be found in these references. A closely related alternative definition of generalized characteristics was given in Glimm and Lax [133] which are Lipschitz curves propagating with either classical characteristic speeds or shock speeds, constructed as limits of families of approximate characteristics. The following result is due to DiPerna [104]. P ROPOSITION 8.5. Let (1.19) be an n × n strictly hyperbolic system endowed with a strictly convex entropy. Suppose u(x, t) is an L∞ ∩ BVloc entropy solution of (1.19) n (t) denote the maximal forward n-characteristic through and (6.1) for (x, t) ∈ R2+ . Let xmax 1 (0, 0). Let xmin (t) denote the minimal forward 1-characteristic passing through (0, 0). 1 (t), and u(x, t) = u , for a.e. (x, t) with Then u(x, t) = uL , for a.e. (x, t) with x < xmin R n (t). x > xmax Using the theory of generalized characteristics, Dafermos in [84,86,88] proved a series of decay properties for general BV solutions to hyperbolic systems of two conservation laws. For this purpose, the following structural condition on the BV solution u(x, t) is imposed: The traces of the Riemann invariants w1 and w2 along any space-like curve are functions of locally bounded variation. Here, a space-like curve relative to the BV solution u(x, t) is a Lipschitz curve, with graph embedded in the upper half-plane, such that, for each point (x, ¯ t¯) on the graph of the curve, the set {(x, t): 0 t < t¯, ζ(t) < x < ξ(t)} of points confined between the maximal backward 2-characteristic ζ and the minimal backward 1-characteristic ξ , emanating from the point (x, ¯ t¯), has empty intersection with the graph of the curve. Under this condition, one has the following results on the regularity and decay of the BV entropy solutions to hyperbolic systems of two conservation laws, which are due to Dafermos [81,84,86,88]. T HEOREM 8.1. Suppose that u(x, t) is a BV entropy solution of the genuinely nonlinear and strictly hyperbolic system (1.19) with n = 2. Then any point of approximate continuity is a point of continuity of u(x, t), any point of approximate jump discontinuity is a point of classical jump discontinuity of u(x, t), the set of irregular points is (at most) countable, and any irregular point is the focus of a centered compression wave of either, or both, characteristic fields, and/or a point of interaction of shocks of the same or opposite characteristic fields. If the initial data (w1 , w2 )(x, 0) belong to L1 (R) with small oscillation, then the solution (w1 , w2 )(x, t) to the genuinely nonlinear and strictly hyperbolic system (1.19) with n = 2 decays, as t → ∞, at the rate O(t −1/2 ) (Dafermos [84]), which is an analogue for the scalar conservation laws (Lax [181]). For solutions with periodic initial data, one has the following decay property (Dafermos [86]).
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T HEOREM 8.2. Suppose that u(x, t) is a BV entropy solution of the genuinely nonlinear and strictly hyperbolic system (1.19) with n = 2, and the initial data u0 (x) are periodic with period P and mean zero. Then the upper half-plane is partitioned by minimal (or maximal) divides of the first (or second) characteristic field, along which the Riemann invariant w1 (or w2 ) of the first (or second) field decays to zero, O(t −2 ), as t → ∞. If φ(t) and ψ(t) are any two adjacent 1- (or 2-) divides, then ψ(t) − φ(t) approaches a constant at the rate O(t −1 ), as t → ∞, and there is a 1- (or 2-) characteristic χ(t) between φ(t) and ψ(t) such that, as t → ∞, χ(t) = (ψ(t) + φ(t))/2 + o(1), and x−φ(t ) ∂wi λi (0, 0) wi (x, t) =
+ o 1t , t x−ψ(t ) + o 1t , t
φ(t) < x < χ(t), χ(t) < x < ψ(t),
i = 1, 2.
(8.1)
The proof of Theorem 8.2 is based on the analysis of the large-time behavior of divides. Assume φ(t) is a minimal 1-divide, say the limit of a sequence {ξk (t)} of minimal backward 1-characteristics emanating from some points {(xk , tk )} with tk → ∞ as k → ∞. Consider the traces of w1 and w2 along ξk (t): w1,k (t) := w1 (ξk (t)−, t) and w2,k (t) := w2 (ξk (t)+, t). The total variation of w2,k and the supremum of |w2,k | over any interval [t, t + 1] ⊂ [0, tk ] are O(t −1 ), uniformly in k. Then w1,k is a nonincreasing function whose oscillation over [t, t + 1] is O(t −3 ) uniformly in k since w1,k (t−) − w1,k (t+) C|w2,k (t) − w2,k (t+)|3 (see [84]). Thus, for any t ∈ [0, tk ], w1,k = O(t −2 ) + O(tk−1 ) uniformly in k, and it can be concluded that, for almost all t ∈ [0, ∞), w1 (φ(t)±, t) is a nonincreasing function which decays to zero, O(t −2 ), as t → ∞. Further analysis of divides leads to (8.1). See Dafermos [86,88] for the details. Now we consider system (1.19) with initial data of compact support (7.25). The BV entropy solution decays to an N-wave as follows (Dafermos [84,88]). T HEOREM 8.3. Suppose that u(x, t) is an entropy BV solution of the genuinely nonlinear and strictly hyperbolic system (1.19) with n = 2, and the initial data u0 (x) have compact support (7.25) and small oscillation. Then the minimal i-characteristics φi− (t) issuing from the point (−R, 0) and the maximal i-characteristics φi+ (t) issuing from the point (R, 0), i = 1, 2, satisfy, for t large, φ1− (t) = λ1 (0, 0)t − (p− t)1/2 + O(1), φ1+ (t) = λ1 (0, 0)t + (p+ t)1/2 + O(t 1/4 ), φ2− (t) = λ2 (0, 0)t − (q− t)1/2 + O t 1/4 , φ2+ (t) = λ2 (0, 0)t + (q+ t)1/2 + O(1), for some nonnegative constants p± and q± , and T V[φ − (t ),φ +(t )] (w1 , w2 )(·, t) = O t −1/2 ; 1
2
(8.2)
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and, if p+ > 0 and q− > 0, then $ $ $wi (·, t) − Ni (·, t)$ 1 = O t −1/4 , L (R)
i = 1, 2,
(8.3)
as t → ∞, with the N-waves N1 (x, t) and N2 (x, t) defined by N1 (x, t) =
− λ1 (0, 0) ,
0,
N2 (x, t) =
1 x ∂w1 λ1 (0,0) t
−(p− t)1/2 x − λ1 (0, 0)t (p+ t)1/2 , otherwise,
1 x ∂w2 λ2 (0,0) t
− λ2 (0, 0) ,
0,
−(q− t)1/2 x − λ2 (0, 0)t (q+ t)1/2 , otherwise.
The main ingredients of the proof of Theorem 8.3 include the following estimates: u(x, t) = 0 for any t > 0 and x ∈ / (φ1− (t), φ2+ (t)); for large t, λ1 (w1 (x, t), 0) = x/t + O(t −1 ) for x ∈ (φ1− (t), φ1+ (t)) and λ2 (0, w2 (x, t)) = x/t + O(t −1 ) for x ∈ (φ2− (t), φ2+ (t)); and for large t, 0 −w1 (x, t) C(x − λ1 (0, 0)t)−3/2 for x > φ1+ (t) and p+ > 0, and 0 −w2 (x, t) C(λ2 (0, 0)t − x)−3/2 for x < φ2− (t) and q− > 0. These estimates indicate that, as t → ∞, the two characteristic fields decouple and each one develops an N-wave profile, of width O(t 1/2 ) and strength O(t −1/2 ), which propagates into the rest state at the characteristic speed. See Dafermos [84,88] for the details of the proof. From Theorem 8.3, we see that the total variation of the solution decays to zero as O(t −1/2 ) (Glimm and Lax [133]). The solution decays to N-waves at the rate O(t −1/4 ) slower than the rate O(t −1/2 ) for scalar conservation laws, due to the interaction of the characteristic fields in the systems, a phenomenon which is not present in a single conservation law. However, an improvement in this uniform rate may be possible, while, in the scalar case, simple examples show that the decay rate O(t −1/2 ) cannot be improved. See also Greenberg [143] on the decay of special solutions for a class of two conservation laws generated by a second-order wave equation.
8.2. Uniqueness of Riemann solutions In this section we prove the uniqueness of Riemann solutions of the Riemann problem (1.13) and (6.1) in the class of entropy solutions in BV without extra regularity condition on the solutions. Without loss of generality, we assume that the classical Riemann solution has the following generic form: ⎧ uL , ⎪ ⎪ ⎪ ⎪ u ⎪ ⎨ M, R(x/t) = uN , ⎪ ⎪ ⎪ V3 (x/t), ⎪ ⎪ ⎩ uR ,
x/t < σ1 , σ1 < x/t < 0, 0 < x/t λ3 (uN ), λ3 (uN ) < x/t < λ3 (uR ), x/t λ3 (uR ),
(8.4)
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where σ1 = σ1 (uL , uM ) is the shock speed, determined by (6.30), and V3 (ξ ) is the solution of the boundary value problem: dV3 (ξ ) = r3 V3 (ξ ) , dξ V3 |ξ =λ3 (uR ) = uR .
ξ < λ3 (uR ),
(8.5)
The 1-shock connecting uL and uM satisfies the Lax entropy condition: λ1 (uM ) < σ1 < λ1 (uL ) < 0. The states uM and uN are also completely determined by the shock curve formula (6.34)–(6.36) and (8.5). The best way to see this fact is first to recall that S is increasing across 1-shock waves and is constant over rarefaction curves, since S is a Riemann invariant of the first and third fields (see [290,291]). Similarly, v and p are both constant over the wave curves of the second (linearly degenerate) field. Hence, in the space (v, p, S), we can project the curves S1 and R3 on the plane (v, p), find the intersection point (vM , pM ) of these projected curves, and immediately obtain the two intersection points (vM , pM , SM ), (vM , pM , SN ), of the line {(v, p, S): v = vM , p = pM } with the 1-shock curve S1 and the 3-rarefaction curve R3 in the phase space. We now state and prove the uniqueness theorem in Chen, Frid and Li [52]. 2
T HEOREM 8.4. Let u(x, t) = (τ, v, e + v2 )(x, t) be an entropy solution of (1.13) and (6.1) in ΠT := {(x, t): 0 t T } for some T ∈ (0, ∞), which belongs BVloc (ΠT ; D) with 2 D ⊂ {(τ, v, e + v2 ): τ > 0} ⊂ R3 bounded. Then u(x, t) = R(x/t), for a.e. (x, t) ∈ ΠT . P ROOF. Step 1. Consider the auxiliary function in ΠT : ⎧ x < x(t), ⎨ uL , % & x(t) < x < max x(t), σ1 t , ˜ u(x, t) = uM , % & ⎩ R(x/t), x > max x(t), σ1 t , where x(t) is the minimal 1-characteristic of u(x, t), and x = σ t is the line of 1-shock in R(x/t). One of the main ingredients in the proof is to use the state variables W = (τ, v, S) as the basic variables, rather than the conserved variables u(x, t), and we let W(x, t) denote R(x/t) in these state variables. Motivated by a procedure introduced by Dafermos (cf. [87,104]), we use the quadratic entropy-entropy flux pairs obtained from (η∗ , q∗ ): − ∇η∗ W · W−W , = η∗ (W) − η∗ W (8.6) α W, W = q∗ (W) − q∗ W − ∇η∗ W · f(W) − f(W) . β W, W (8.7) We then consider the measures t) + ∂x β W(x, t), W(x, t) , (x, t) ∈ ΠT , µ = ∂t α W(x, t), W(x, ν = ∂t η∗ W(x, t) + ∂x q∗ W(x, t) − ∂S η∗ W(x, t) ∂t S(x, t), (x, t) ∈ ΠT − { T ∪ LT }, where t = {(0, s): 0 s t} and Lt = {(x(s), s): 0 s t}.
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Then the uniqueness problem essentially reduces to analyzing the measure µ over the region, where the Riemann solution is a rarefaction wave, and over the curve (x(t), t), which for simplicity may be taken as the jump set of W(x, t). dH1 and Step 2. The first important fact is that µ{ T } = 0, since µ{ T } = T [β(W, W)] = (v − v)(p = 0, H1 -a.e. over T . The latter follows from β(W, W) ¯ − p) ¯ and [β(W, W)] over T , the fact that v, p, v, ¯ p¯ cannot change across the jump discontinuities of W and W because of the Rankine–Hugoniot relation (6.7). Let % & Ω3 := (x, t): λ3 (uN ) < x/t < λ3 (uR ), 0 < t T denote the rarefaction wave region of the classical Riemann solution. Over the region Ω3 , = W, and µ satisfies W µ = ∂t α W, W + ∂x β W, W = ν − ∇ 2 η∗ W ∂x W, Qf W, W , (8.8) where we used the fact that ∇ 2 η∗ ∇f is symmetric, and Qf(W, W) = f(W) − f(W) − ∇f(W) · (W − W) is the quadratic part of f at W. Since ˜l3 (W) = r˜ 3 (W)∇ 2 η∗ (W) is a left-eigenvector of ∇f(W) corresponding to the eigenvalue λ3 (W), and ∂W(x, t) 1 = r˜ 3 W(x, t) , ∂x t
for (x, t) ∈ Ω3 .
Then, for any Borel set E ⊂ Ω3 , we have 1˜ µ(E) = ν(E) − l3 W Qf W, W dx dt. E t
(8.9)
The fact ˜l3 (W)Qf(W, W) 0 yields µ(Ω3 ) 0. Step 3. Using the Gauss–Green formula for BV functions and the finiteness of propagation speed of the solutions yields ∞ α W(x, t), W(x, t) dx. (8.10) µ{Πt } = −∞
is a constant, On the other hand, since µ reduces to the measure ν on the open sets where W and W = W over Ω3 , % & % & µ{Πt } = µ{Lt } + µ Ω3 (t) + ν Πt − Lt ∪ t ∪ Ω3 (t) , (8.11) where we have used the fact that µ{ t } = 0. Hence, it suffices to show µ{Lt } 0.
(8.12)
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Thus, we consider the functional − β W, W . −, W + = σ α W, W D σ, W− , W+ , W Step 4. We now prove −, W + 0, D σ, W− , W+ , W
(8.13)
− and W + are conif W− and W+ are connected by a 1-shock of speed σ = x (t), W nected by a 1-shock of speed σ¯ , and also W− = W− . Using Proposition 8.5, it is then − , a careful calculation clear that (8.13) immediately implies (8.12). Thus, when W− = W shows that −, W + = d σ, W− , W+ − d σ¯ , W− , W + D σ, W− , W+ , W + σ (S− − S+ ) − σ¯ S− − S+ − ∂S η W + , − (σ − σ¯ )α W− , W (8.14) where d(σ, W− , W+ ) = σ [η(W)] − [q(W)], and (η, q) = (η∗ , q∗ ) is the energy–energy flux pair in (1.13). From the Rankine–Hugoniot relation (6.30), we may view the state W+ = (τ+ , v+ , S+ ) connected on the right by a 1-shock to a state W− = (τ− , v− , S− ) as parametrized by the shock speed σ , with σ λ1 (W− ) < 0. + = W+ (σ¯ ) in (8.14), According to the parametrization, we set W+ = W+ (σ ) and W and define h(σ ) := d σ, W− , W+ (σ ) = σ η(W) − q(W) . Then, using Proposition 6.6 and making a careful calculation yield −, W + h(σ ) − h(σ¯ ) − h( ˙ σ¯ )(σ − σ¯ ). D σ, W− , W+ , W On the other hand, (6.30) implies h(σ ) = 0 for all σ ; thereby, (8.13) holds. Step 5. Now, by (8.10), we conclude that W(x, t) = W(x, t), a.e. in ΠT . In particular, W(x, t) is an entropy solution of (1.13) and (6.1), and then the Rankine–Hugoniot rela tion (6.30) implies that W(x, t) must coincide with the classical Riemann solution W(x, t). This concludes the proof. 8.3. Large-time stability of entropy solutions In this section we follow the framework established in Chen and Frid [46] to show that the uniqueness of the classical Riemann solution R(ξ ), corresponding to the Riemann data (6.1), implies the large-time stability of entropy solutions u ∈ L∞ ∩ BVloc (R2+ ) of the Cauchy problem (1.13) and u|t =0 = R0 (x) + P0 (x),
P0 (x) ∈ L1 ∩ L∞ (R),
whose local total variation satisfies a certain natural growth condition.
(8.15)
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T HEOREM 8.5. Let S(R2+ ) denote a class of functions defined on R2+ . Assume that the Cauchy problem (1.19), (8.15), and (6.1) satisfies the following. (i) System (1.19) has a strictly convex entropy; (ii) The Riemann solution is unique in the class S(R2+ ); (iii) Given any entropy solution u ∈ S(R2+ ) of (1.19) and (8.15), the sequence uT (x, t) = u(T x, T t) is compact in L1loc (R2+ ), and any limit function of its subsequences is still in S(R2+ ). Then the Riemann solution R(x/t) is asymptotically stable in S(R2+ ) with respect to the corresponding initial perturbation P0 (x): ess lim
u(ξ t, t) − R(ξ ) dξ = 0,
L
t →∞ −L
for any L > 0.
(8.16)
2
System (1.13) has a strictly convex entropy S(τ, v, e + v2 ) in D, and hence the condition (i) follows. We choose S(R2+ ) as the class of entropy solutions in L∞ ∩ BVloc (R2+ ) satisfying a natural growth condition of local total variation: There exists c0 > 0 such that, for all c c0 , there is C > 0 depending only upon c such that TV(u | Kc,T ) CT ,
for any T > 0,
(8.17)
where Kc,T = {(x, t) ∈ R2+ : |x| ct, t ∈ (0, T )}. Such a condition is natural, since any solution obtained by the Glimm method or related methods satisfies (8.17). For such solutions and for any T > 0, uT (x, t) also satisfies (8.17) with the same constant C depending only upon c. Furthermore, the sequence uT (x, t) is compact in L1loc (R2+ ). Then the condition (iii) follows. Therefore, the uniqueness result established in Section 8.2 yields the large-time stability of entropy solutions satisfying (8.17). T HEOREM 8.6. Any Riemann solution of system (1.13), staying away from the vacuum, with large Riemann initial data (6.1) is large-time asymptotically stable in the sense of (8.16) in the class of entropy solutions in L∞ ∩ BVloc (R2+ ) of (1.13) with large initial perturbation (8.15) satisfying (8.17). R EMARK 8.1. A uniqueness theorem of Riemann solutions was first established by DiPerna [104] for 2 × 2 strictly hyperbolic and genuinely nonlinear systems in the class of entropy solutions in L∞ ∩ BVloc with small oscillation. In [48], Chen and Frid established the uniqueness and stability of Riemann solutions, with shocks of small strength, for the 3 × 3 system of Euler equations with general equation of state in the class of entropy solutions in L∞ ∩ BVloc with small oscillation. However, the uniqueness result presented here neither imposes smallness on the oscillation nor the extra regularity of the solutions, as well as does not require specific reference to any particular method for constructing the entropy solutions. In this connection, we recall that, for system (1.13) for polytropic gases, there are many existence results of solutions in L∞ ∩ BVloc via the Glimm scheme [130],
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especially when the adiabatic exponent γ > 1 is close to one (see, e.g., [212,302,253]). We also refer the reader to Dafermos [87] for the stability of Lipschitz solutions for hyperbolic systems of conservation laws. 9. Global discontinuous solutions IV: entropy solutions in L∞ In this section we extensively discuss the Cauchy problem for the one-dimensional isentropic Euler equations in (1.14) and show the existence, compactness, decay, and stability of global entropy solutions in L∞ . In the study of entropy solutions to the Euler equations, several numerical approximate schemes or methods have played an important role. As an example, we show here the convergence of the Lax–Friedrichs scheme and the Godunov scheme for the Cauchy problem. 9.1. Isentropic Euler equations Consider the Cauchy problem for the isentropic Euler equations in (1.14) with initial data: (ρ, m)|t =0 = (ρ0 , m0 )(x),
(9.1)
where ρ and m are in the physical region {(ρ, m): ρ 0, |m| C0 ρ} for some C0 > 0. For ρ > 0, v = m/ρ is the velocity. The pressure function p(ρ) is a smooth function in ρ > 0 (nonvacuum states) satisfying (6.22) when ρ > 0, and p(0) = p (0) = 0,
ρp(j +1) (ρ) = cj > 0, ρ→0 p(j ) (ρ) lim
j = 0, 1.
(9.2)
More precisely, we consider a general situation of pressure law that there exist a sequence of exponents 1 < γ := γ1 < γ2 < · · · < γJ
3γ − 1 < γJ +1 2
and a function P (ρ) such that p(ρ) =
J
κj ρ γj + ρ γJ +1 P (ρ);
j =1
P (ρ), ρ 3 P (ρ) are bounded as ρ → 0, −1) for some κj , j = 1, . . . , J, with κ1 = (γ 4γ after renormalization. For a polytropic gas obeying the γ -law (1.10), or a mixed ideal polytropic fluid, 2
p(ρ) = κ1 ρ γ1 + κ2 ρ γ2 ,
κ2 > 0,
the pressure function clearly satisfies (6.22) and (9.3).
(9.3)
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System (1.14) is strictly hyperbolic at the nonvacuum states {(ρ, v): ρ > 0, |v| C0 }, and strict hyperbolicity fails at the vacuum states {(ρ, m/ρ): ρ = 0, |m/ρ| C0 }. 9.2. Entropy–entropy flux pairs A pair of mappings (η, q) : R+ × R → R × R is called an entropy–entropy flux pair (or entropy pair for short) of system (1.14) if it satisfies the hyperbolic system: ∇q(ρ, m) = ∇η(ρ, m)∇f(ρ, m).
(9.4)
Furthermore, η(ρ, m) is called a weak entropy if = 0.
η| ρ=0
(9.5)
v=m/ρ fixed
For example, the mechanical energy (a sum of the kinetic and internal energy) and the mechanical energy flux ρ p(s) m2 η∗ (ρ, m) = +ρ ds, 2ρ s2 0 (9.6) ρ p (s) m3 ds q∗ (ρ, m) = 2 + m 2ρ s 0 form a special entropy pair; η∗ (ρ, m) is convex for any γ > 1 and strictly convex (even at the vacuum states) if γ 2, in any bounded region in ρ 0. D EFINITION 9.1. A bounded measurable function u(x, t) = (ρ, m)(x, t) is an entropy solution of (1.14), (6.22), (9.1), and (9.2) in R2+ if u(x, t) satisfies the following: (i) There exists C > 0 such that 0 ρ(x, t) C, m(x, t)/ρ(x, t) C; (ii) The entropy inequality holds in the sense of distributions in R2+ , i.e., for any weak entropy pair (η, q)(u) with convex η(u) and any nonnegative function φ ∈ C01 (R × [0, ∞)), ∞ ∞ ∞ η(u)∂t φ + q(u)∂x φ dx dt + η(u0 )(x)φ(x, 0) dx 0. (9.7) −∞
0
−∞
Notice that η(u) = ±u are both trivial convex entropy functions so that (9.7) implies that u(x, t) is a weak solution in the sense of distributions. In the coordinates (ρ, v), any weak entropy function η(ρ, v) is governed by the secondorder linear wave equation ηρρ − k (ρ)2 ηvv = 0, η|ρ=0 = 0, with k(ρ) =
ρ 0
p (s) s
ds.
ρ > 0,
(9.8)
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In the Riemann invariant coordinates w = (w1 , w2 ) defined in (6.26), any entropy function η(w) is governed by ηw1 w2 +
Λ(w1 − w2 ) (ηw1 − ηw2 ) = 0, w1 − w2
(9.9)
where
Λ(w1 − w2 ) = −k(ρ)k (ρ)
−2
k (ρ),
with ρ = k
−1
w1 − w2 . 2
(9.10)
The corresponding entropy flux function q(w) is qwj (w) = λi (w)ηwj (w),
i = j.
(9.11)
In general, any weak entropy pair (η, q) can be represented by η(ρ, v) = χ(ρ, v; s)a(s) ds, q(ρ, v) = σ (ρ, v; s)b(s) ds, R
R
(9.12)
for any continuous function a(s) and related function b(s), where the weak entropy kernel and entropy flux kernel are determined by χρρ − k (ρ)2 χvv = 0, χ(0, v; s) = 0,
χρ (0, v; s) = δv=s ,
(9.13)
and σρρ − k (ρ)2 σvv = σ (0, v; s) = 0,
p (ρ) χv , ρ σρ (0, v; s) = vδv=s ,
(9.14)
with δv=s the Delta function concentrated at the point v = s. The equations in (9.8)–(9.9) and (9.13)–(9.14) belong to the class of Euler–Poisson– Darboux-type equations. The main difficulty comes from the singular behavior of Λ(w1 − w2 ) near the vacuum. In view of (9.10), the derivative of Λ(w1 − w2 ) in the coefficients of (9.9) may blow up like (w1 − w2 )−(γ −1)/2 when w1 − w2 → 0 in general, and its higher derivatives may be more singular, for which the classical theory of Euler–Poisson–Darboux equations does not apply (cf. [11,324,325]). However, for a gas obeying the γ -law, Λ(w1 − w2 ) = λ :=
3−γ , 2(γ − 1)
the simplest case, which excludes such a difficulty. In particular, for this case, the weak entropy kernel is χ(ρ, v; s) =
λ w1 (ρ, v) − s s − w2 (ρ, v) + .
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A mathematical theory for dealing with such a difficulty for the singularities can be found in Chen and LeFloch [57–59]. Now we list several important entropy pairs and their properties. First, we have P ROPOSITION 9.1. For the general pressure law (6.22), (9.2), and (9.3), any weak entropy η(ρ, m) satisfies that, when (ρ, m) ∈ DM := {0 ρ M, |m| Mρ}, 2 ∇ η(ρ, m) CM ∇ 2 η∗ (ρ, m). ∇η(ρ, m) CM , The equations in (1.14) have several important entropy pairs from (9.9)–(9.12). As an example, we give their formulae for the case γ = 5/3. (i) Goursat entropy wave G0 = (η0 , q0 ): η0 (w) = w1 w2 X(w),
q0 (w) = λ2 η0 + τ0 ,
1 τ0 := w12 w2 X(w), 3
(9.15)
where X(w) is the characteristic function with X(w) = 1, when w1 > 0 > w2 ; and X(w) = 0, otherwise. (ii) Goursat entropy wave G1 = (η1 , q1 ): η1 (w) = (w1 + w2 )X(w), q1 (w) = λ2 η1 + τ1 ,
(9.16)
1 τ1 := w1 (w1 + 2w2 )X(w). 3
(iii) Lax entropy waves G±k = (η±k , q±k ) for k ' 1: 1 1 kw1 1/3 , qk = ηk λ2 + O , 1+O ηk (w) = e ρ k k
(9.17)
and η−k (w) = e
−kw2 1/3
ρ
1 , 1+O k
1 q−k = η−k λ1 + O . k
(9.18)
(iv) Entropy wave sequence G = (η , q ): η (w) := η(w; ψ ) = (w1 − w2 ) ψ (w1 ) + ψ (w2 ) − 2
w1
ψ (x) dx; w2
q (w) := q(w; ψ ) = λ2 η + τ , (9.19) w1 w1 2 4 τ (w; ψ ) := (x − w2 ) ψ (x) + ψ (w2 ) dx − (w1 − x)ψ (x) dx, 3 w2 3 w2 where ψ (s) = ψ( s), ψ(s) ∈ C0∞ (−1, 1), 1 − ε/2], ε < 1/4. Then we have
1
−1 ψ(s) ds
= 0, and supp ψ ⊂ [−1 + ε/2,
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P ROPOSITION 9.2. For (ρ, m) ∈ DM , there exists C = CM > 0 such that |η0 q − η q0 |
C ,
|η1 q − η q1 | C.
1 ˆ = tψ(t) + P ROPOSITION 9.3. For ψ(s) ∈ C0∞ (−1, 1) with −1 ψ(s) ds = 0, choose ψ(t) t 1 ˆ ψ(s) ds, which implies ψ(s) ds = 0. Define −1 −1 B (w; ψ) = η qˆ − ηˆ q , where (η , q ) = η(w; ψ ), q(w; ψ ) ,
(ηˆ , qˆ ) = η(w; ψˆ ), q(w; ψˆ ) .
(9.20)
Then, for (ρ, m) ∈ DM ,
B (w; ψ) =
⎧ (w1 − w2 )2 A( wj ) ⎪ ⎪ ⎨ j
+ (w1 − w2 )B (w) + ⎪ ⎪ ⎩ 1 O ,
j
C (w) ,
in Swε, j , j = 1, 2,
(9.21)
otherwise,
where A(x) =
2 3
2
x −1
ψ(s) ds
,
j B (w) C ψ( wj ) +
j C (w) C < ∞, j = 1, 2, 1−ε Swε, j = w: |wj | .
wj
−1
ψ(s) ds , (9.22)
9.3. Compactness framework We now establish the following compactness framework. T HEOREM 9.1. Consider the Euler equations (1.14) for compressible fluids under the assumptions (6.22) and (9.3). Let (ρ ε , mε )(x, t) be a sequence of functions satisfying 0 ρ ε (x, t) C, mε (x, t) Cρ ε (x, t), for a.e. (x, t), (9.23) such that, for any weak entropy pair (η, q), −1 2 ∂t η ρ ε , mε + ∂x q ρ ε , mε is compact in Hloc R+ . Then the sequence (ρ ε , mε )(x, t) is compact in L1loc (R2+ ).
(9.24)
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P ROOF. We now give a sketch of the proof for the case γ = 5/3. Step 1. First, with the aid of the div-curl lemma and the Young measure representation theorem (see Murat [238,239] and Tartar [299]; also see Chen [39]), the conditions (9.23)– (9.24) imply that there exists a family of probability measures {νx,t ∈ Prob.(R+ × R)}, uniquely determined by (ρ ε , mε )(x, t), such that supp νx,t ⊂ DM ,
(9.25)
and, for any continuous or bounded measurable weak entropy pairs (ηj , qj ), j = 1, 2, η1 ν, η2
ν, η1 ν, q1 q1 , = ν, η2 ν, q2 q2
a.e.
(9.26)
For simplicity, we often drop the index (x, t) of νx,t . Then the compactness problem reduces to the question whether the Young measures are Delta masses concentrated at u(x, t) = (ρ, m)(x, t) = w∗ -limε→0 (ρ ε , mε )(x, t), that is, νx,t = δu(x,t ) .
(9.27)
To achieve (9.27), it suffices to show supp ν ⊂ V ∪ P ,
(9.28)
where V = {w: ρ = 0}, the vacuum set, and P = (w10 , w20 ) = w(ρ 0 , v 0 ), ρ 0 > 0, is the vertex of the smallest triangle K containing supp ν − V in the w-coordinates. This can be seen as follows. If (9.28) holds, then there are only three possibilities: (i) supp ν = {P }; (ii) supp ν ⊂ V ; (iii) ν = ν|V + αδP , α = 0, 1. It is clear that (i) and (ii) imply (9.27). For (iii), we choose (η1 , q1 ) = (ρ, m) and 2 (η2 , q2 ) = (m, mρ + p(ρ)) in (9.26) to have αρ 0 p ρ 0 = α 2 ρ 0 p ρ 0 , which implies that α = α 2 since ρ 0 > 0. That is, either α = 0 or α = 1, which is a contradiction. Step 2. To achieve (9.28), it suffices to prove lim
→∞
2
' i=1
( ν|S ε, , (w1 − w2 )2 = 0. wi
This can be seen as follows. Set ν˜ = (w1 − w2 )2 ν, a weighted measure. Define ˜ | a < wi < b}, Pwi ν˜ (a, b) = ν{w
(9.29)
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an orthogonal projection of ν˜ onto the segment parallel to the wi -axis. If we can prove that the following Lebesgue derivatives of Radon measures are zero: DPwi ν˜ (wi ) = 0,
w20 < wi < w10 , i = 1, 2,
(9.30)
then we conclude ν˜ (K − P ) = 0, that is, ν K − (V ∪ P ) = 0, which implies (9.28). By Galilean invariance, it suffices for (9.30) to show that DPwi ν˜ (0) = 0,
i = 1, 2,
(9.31)
with w20 < 0 < w10 , which is equivalent to (9.29). Step 3. Claim: If supp ν ∩ (K − V ) = ∅, then P ∈ supp ν. If P ∈ / supp ν, then there exists δ > 0 such that B2δ (P ) ∩ supp ν = ∅. In (9.26), we choose (η1 , q1 ) = (ηk , qk ) and (η2 , q2 ) = (η−k , q−k ) to have ν, ηk q−k − η−k qk ν, qk ν, q−k = − . ν, ηk ν, η−k ν, ηk ν, η−k
(9.32)
Observe that, as k is sufficiently large, √ ν, ηk q−k − η−k qk Cek(w10 −w20 − 2δ) ,
and ν, ηk c0 ek(w10 − 2δ ) ,
ν, η−k c0 e−k(w20 + δ2 ) .
We conclude from (9.32) that lim
k→∞
ν, qk ν, q−k − ν, ηk ν, η−k
= 0.
2 Define the probability measures µ± k ∈ Prob.(R ):
'
( ν, hη±k µ± , k ,h = ν, η±k
h ∈ C0 R2 ,
(9.33)
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∞ as k is sufficiently large. Then µ± k M = 1, and there exists a subsequence {µkj }j =1 such that ± w∗ - lim µ± kj = µ , j →∞
and & % supp µ+ ⊂ w1 = w10 ∩ K,
& % supp µ− ⊂ w2 = w20 ∩ K.
Notice that λ1w1 = λ2w2 = 1/3 > 0. We have lim
j →∞
ν, qkj ν, ηkj
' ( ν, q−kj = µ+ , λ2 λ2 (P ) > λ1 (P ) µ− , λ1 = lim , j →∞ ν, η−kj
which is a contradiction to (9.33). Step 4. We now show that there exists C > 0, independent of , such that ν, η + ν, q C. If not, there exists a subsequence { j }∞ j =1 such that lim ν, η j = ∞,
j →∞
and/or
lim ν, q j = ∞.
j →∞
For concreteness, we assume lim
j →∞
ν, q j ν, η j
= α ∈ (−∞, ∞).
Consider the commutativity relations ν, q0 −
ν, q j ν, η j
ν, η0 =
ν, η j q0 − η0 q j ν, η j
,
and ν, q1 −
ν, q j ν, η j
ν, η1 =
ν, η j q1 − η1 q j
Let j → ∞ and use Proposition 9.2. Then ν, q0 − αν, η0 = 0, ν, q1 − αν, η1 = 0,
ν, η j
.
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which implies ν, η0 ν, q0 ( ' = ν, η0 q1 − η1 q0 = 1 ν, (w1 w2 )2 X(w) . 0 = 3 ν, η1 ν, q1 This implies that P ∈ / supp ν, which is a contradiction to Step 3. Step 5. Claim: For (η , q ) and (ηˆ , qˆ ) defined in (9.20), lim ν, η qˆ − ηˆ q = 0.
→∞
If not, there exists a subsequence such that lim ν, η j qˆ j − ηˆ j q j = 0.
j →∞
Step 4 indicates that there further exists a subsequence (still denoted) { j } such that lim ν, η j , ν, q j , ν, ηˆ j , ν, qˆ j exists. j →∞
Proposition 9.2 and the identity (9.25) imply −ν, η0 limj →∞ ν, q j + ν, q0 limj →∞ ν, η j = 0, −ν, η0 limj →∞ ν, qˆ j + ν, q0 limj →∞ ν, ηˆ j = 0. Since ν, η0 > 0 from Step 3, we have limj →∞ ν, η j limj →∞ ν, q j = lim ν, η qˆ − ηˆ q , 0= j j j j limj →∞ ν, ηˆ j limj →∞ ν, qˆ j j →∞ which is a contradiction. Step 6. Proposition 9.3 and Step 5 imply that 2
'
( ν|S ε, , (w1 − w2 )2 A( wi ) + (w1 − w2 )B i (w) → 0, wi
i=1
Choose ψ(s) = a 2−ε (s + 4
aδ (s) = δa Then
1
s , δ
2−ε (s 4 ) − a 2−ε 4
a(s) =
= 0, and 2 ψ(s) ds cε > 0,
−
2−ε 4 ),
1
e |s|2 −1 , 0,
→ ∞.
(9.34)
where
|s| 1, otherwise.
−1 ψ(s) ds
x −1
x ∈ [−1 + ε, 1 − ε].
Combining (9.34) with (9.21) and (9.35) yields (9.29). This completes the proof.
(9.35)
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R EMARK 9.1. The proof of Theorem 9.1 is taken from Chen [37] and Ding, Chen and Luo [96]. For a gas obeying the γ -law, the case γ = N+2 N , N 5 odd, was first treated by DiPerna [106], while the case 1 < γ 5/3 was first solved by Chen [37] and Ding, Chen and Luo [96]. Finally, motivated by a kinetic formulation, the cases γ 3 and 5/3 < γ < 3 were treated by Lions, Perthame and Tadmor [203] and Lions, Perthame and Souganidis [202], respectively, where their analysis applies to the whole interval 1 < γ < 3. For the general pressure law (6.22) and (9.3), Theorem 9.1 is due to Chen and LeFloch [57,59].
9.4. Convergence of the Lax–Friedrichs scheme and the Godunov scheme We now apply the compactness framework established in Theorem 9.1 to show the convergence of the Lax–Friedrichs scheme [176] for the Cauchy problem (1.14), (6.22), (9.1), and (9.3) under the assumptions: 0 ρ0 (x) C0 ,
m0 (x) C0 ρ0 (x),
for a.e. x and some C0 > 0.
(9.36)
The convergence proof for the Godunov scheme [139] is similar (see [97,40]). As every difference scheme, the Lax–Friedrichs scheme satisfies the property of propagation with finite speed, which is an advantage over the vanishing viscosity method: the convergence result applies without assumption on the decay of initial data at infinity. We now construct the family of Lax–Friedrichs approximate solutions (ρ h , mh )(x, t), similar to these in Section 7.1 for the Glimm scheme. We also set v h = mh /ρ h when ρ h > 0 and v h = 0 otherwise. The Lax–Friedrichs scheme is based on a regular partition of the half-plane t 0 defined by tk = k∆t, xj = j h for k ∈ Z+ , j ∈ Z, where ∆t and h are the sizes of time-step and space-step, respectively. It is assumed that the ratio ∆t/ h is constant and satisfies the Courant–Friedrichs–Lewy stability condition: $ ∆t $ $λj ρ h , v h $ ∞ < 1. L h In the first strip {(x, t): xj −1 < x < xj +1 , 0 t < τ, j odd}, we define (ρ h , mh )(x, t) by solving a sequence of Riemann problems for (1.14) corresponding to the Riemann data:
0 ρj −1 , m0j −1 , x < xj , ρ , m (x, 0) = 0 ρj +1 , m0j +1 , x > xj , h
h
with
ρj0+1 , m0j +1
1 = 2h
xj+2
(ρ0 , m0 )(x) dx. xj
Recall that the Riemann problem is uniquely solvable (see Section 6.3).
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If (ρ h , mh )(x, t) is known for t < tk , we set k k 1 ρj , mj = 2h
xj+1
ρ h , mh (x, tk − 0) dx.
xj−1
In the strip {(x, t): xj < x < xj +2 , tk < t < tk+1 , j + k = even}, we define (ρ h , mh )(x, t) by solving the Riemann problems with the data: h h ρ , m (x, tk ) =
ρjk , mkj , ρjk+2 , mkj +2
,
x < xj +1 , x > xj +1 .
This completes the construction of the Lax–Friedrichs approximate solutions (ρ h , mh )(x, t). T HEOREM 9.2. Let (ρ0 , m0 )(x) be the Cauchy data satisfying (9.36). Extracting a subsequence, if necessary, the Lax–Friedrichs (or Godunov) approximate solutions (ρ h , mh )(x, t) converge strongly almost everywhere to a limit (ρ, m) ∈ L∞ (R2+ ) which is an entropy solution of the Cauchy problem (1.14) and (9.1). The following two propositions will be used in the proof of Theorem 9.2. P ROPOSITION 9.4. For any w10 > w20 , the region % & Σ w10 , w20 = (ρ, m): w1 w10 , w2 w20 , w1 − w2 0 is also invariant for the Lax–Friedrichs approximate solutions, where wi , i = 1, 2, are the Riemann invariants. P ROOF. Proposition 6.5 indicates that Σ(w10 , w20 ) is an invariant region for the Riemann solutions. Since the set Σ(w10 , w20 ) is convex in the (ρ, m)-plane, it follows from Jensen’s inequality that, for any function satisfying {(ρ, m)(x): a x b} ⊂ Σ(w10 , w20 ) for some (w10 , w20 ), (ρ, ¯ m) ¯ :=
1 b−a
b a
(ρ, m)(x) dx ∈ Σ w10 , w20 .
Therefore, Σ(w10 , w20 ) is also an invariant region for the Lax–Friedrichs scheme.
In particular, Proposition 9.4 shows that the approximate density function ρ h (x, t) remains nonnegative, and both ρ h (x, t) and mh (x, t)/ρ h (x, t) are uniformly bounded so it is indeed possible to construct the approximate solutions globally, as described earlier. Consider the entropy pair (η∗ , q∗ ) defined from the kinetic and internal energy by (9.6).
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P ROPOSITION 9.5. For any weak entropy pair (η, q) and any invariant region R(w0 , z0 ), there exists a constant C > 0 such that, for any solution (ρ, m)(x, t) of the Riemann problem with initial data in R(w0 , z0 ), x (t) η(ρ, m) (t) − q(ρ, m) (t) C x (t) η∗ (ρ, m) (t) − q∗ (ρ, m) (t) , where x (t) is the speed of any shock located at x(t) in the Riemann solution (ρ, m)(x, t). The proof given in [37,96] for the γ -law case extends immediately to the general pressure law. P ROOF OF T HEOREM 9.2. Since the scheme satisfies the property of propagation with finite speed, we can assume without loss of generality that the initial data have compact support. To establish the strong convergence of the scheme, it suffices to check that the sequence uh (x, t) = (ρ h , mh )(x, t) satisfies the compactness framework in Theorem 9.1. The L∞ bound is a direct corollary of Proposition 9.4. We will prove (9.24). Consider the weak entropy dissipation measures ∂t η(uh ) + ∂x q(uh ) associated with a weak entropy pair (η, q). Using the Gauss–Green formula, for any test-function ϕ(x, t) compactly supported in R × [0, T ] with T ≡ K∆t for some integer K, one has R 0
T
η uh ∂t ϕ + q uh ∂x ϕ dx dt
= M h (ϕ) + S h (ϕ) + Lh1 (ϕ) + Lh2 (ϕ),
(9.37)
where M h (ϕ) :=
R
η uh (x, T ) ϕ(x, T ) dx −
S h (ϕ) := 0
Lh1 (ϕ) :=
j,k
Lh2 (ϕ) :=
T
R
η uh (x, 0) ϕ(x, 0) dx,
x (t)[η](t) − [q](t) ϕ x(t), t dt,
shocks x(t )
ϕjk
j,k
xj+1 xj−1
η uk− − η ukj dx,
(9.38)
xj+1 xj−1
η uk− − η ukj ϕ(x, tk ) − ϕjk dx.
Here the same notations as in Proposition 9.5 are used, and uk− (x) := uh (x, tk −) and ϕjk := ϕ(xj , tk ).
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Since each uh (x, t) has compact support, we may substitute (η, q) = (η∗ , q∗ ) and ϕ ≡ 1 in the formulas (9.37)–(9.38) to obtain
xj−1
j,k
while
xj+1
R
=
T 0
x (t)[η∗ ](t) − [q∗ ](t) dt
shocks x(t )
η∗ u0 (x) dx,
xj+1 xj−1
j,k
η∗ uk− − η∗ ukj dx +
j,k
(9.39)
η∗ uk− − η∗ ukj dx xj+1 1 0
xj−1
uk− − ukj
,
∇ 2 η∗ ukj + τ uk− − ukj
× uk− − ukj (1 − τ ) dτ dx,
(9.40)
where the summations are over all k K. In view of Proposition 9.5, the entropy inequality, x (t)[η∗ ](t) − [q∗ ](t) 0, is satisfied for the shocks. On the other hand, η∗ is convex in the conservative variables (ρ, m). Estimates (9.39)–(9.40) yield
T 0
x (t)[η∗ ](t) − [q∗ ](t) dt C,
(9.41)
shocks x(t )
j,k
xj+1 1 xj−1
0
uk− − ukj
,
∇ 2 η∗ ukj + τ uk− − ukj
× uk− − ukj (1 − τ ) dτ dx C.
(9.42)
Then we observe the following: (i) For 1 < γ 2, the entropy η∗ is uniformly convex so that the Hessian matrix ∇ 2 η∗ is bounded below by a positive constant, which implies
xj+1 uk − uk 2 dx C. (9.43) − j j,k
xj−1
(ii) For γ > 2, the estimate (9.42) implies
j,k
xj+1 ρ k mk − − k 2 ρ− xj−1
1
+ 0
−
mkj
2
ρjk
k p (ρjk + τ (ρ− − ρjk )) k ρjk + τ (ρ− − ρjk )
k 2 (1 − τ ) dτ ρ− − ρjk dx C.
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In view of the assumption (9.3), there exists C1 > 0 depending on γ such that
1
k − ρ k )) p (ρjk + τ (ρ− j
ρjk
0
k + τ (ρ−
− ρjk )
γ −2 & % k , − ρjk (1 − τ ) dτ C1 min 1, ρ−
which yields
j,k
xj+1
k ρ−
xj−1
mk− k ρ−
−
mkj
2
k k γ + ρ− − ρj dx C.
ρjk
The Cauchy–Schwarz inequality implies
j,k
xj+1 xj−1
k k m− ρ− k ρ−
mkj − k dx Ch−1/2 , ρ
(9.44)
j
and
j,k
xj+1 xj−1
ρ k − ρ k dx Ch1/γ −1 . − j
(9.45)
For any bounded set Ω ⊂ R × [0, T ] and for any weak entropy pair (η, q), we deduce from (9.37), (9.38), (9.40)–(9.45), and Propositions 9.4 and 9.5 that, for any ϕ ∈ C0 (Ω), M(ϕ) = 0, h S (ϕ) C ϕ C
0
T 0
% & x (t)[η∗ ] − [q∗ ] dt C ϕ C0 (Ω) ,
h L (ϕ) C ϕ C 1
0
j,k
xj+1
1
dx xj−1
0
uk− − ukj
,
∇ 2 η∗ ukj + τ uk− − ukj
× uk− − ukj (1 − τ ) dτ C ϕ C0 (Ω) . Hence |(M h + S h + Lh1 )(ϕ)| C ϕ C0 , which yields a uniform bound in the space M(Ω) of bounded measures for M h + S h + Lh1 , considered as a functional on the space of continuous functions: $ $ h $M + S h + Lh $ 1 M(Ω) C. compact
The embedding theorem M(Ω) &→ W −1,q0 (Ω), 1 < q0 < 2, yields that M h + S h + Lh1
is a compact sequence in W −1,q0 (Ω).
(9.46)
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G.-Q. Chen and D. Wang
It remains to treat Lh2 (ϕ). Let ϕ ∈ C0α (Ω), 1/2 < α < 1. We distinguish two cases: (i) For 1 < γ 2, we deduce from (9.43) that
h L (ϕ) hα ϕ C α 2
0
k
j
xj+1 xj−1
hα−1/2 ∇η L∞ ϕ C0α
η uk − η uk 2 dx − j
j,k
Chα−1/2 ϕ W 1,p (Ω) ,
1/2
xj+1 xj−1
uk − uk 2 dx − j
for all p >
0
1/2
2 . 1−α
(9.47)
(ii) For γ > 2, the estimates (9.44) and (9.45) yield
h L (ϕ) hα ∇η L∞ ϕ C α 2
0
j,k
mk ρ k − ρ k + ρ k m− − j dx − − k j ρ− ρjk
xj+1 xj−1
Chα+1/γ −1 ϕ C0α (Ω) .
(9.48)
The estimates (9.47) and (9.48) imply $ h$ $L $ −1,q Chα0 → 0, 0 (Ω) 2 W
when h → 0, for 1 < q0 <
2 < 2, 1+α
(9.49)
where α0 = max{α − 1/2, α − 1 + 1/γ }. Finally, we combine (9.46) with (9.49) to obtain that M h + S h + Lh1 + Lh2
is compact in W −1,q0 (Ω).
(9.50)
Since 0 ρ(x, t) C, |m(x, t)/ρ(x, t)| C, we have that M h + S h + Lh1 + Lh2
is bounded in W −1,r (Ω), r > 2.
(9.51)
The interpolation lemma in [96] (also see [39]), (9.50), and (9.51) imply that M h + S h + Lh1 + Lh2
is compact in W −1,2 (Ω),
which implies that ∂t η uh + ∂x q uh
is compact in W −1,2 (Ω).
(9.52)
In view of Theorem 9.1 and (9.52), there exists a subsequence uh (x, t) converging for almost every (x, t) to a limit function (ρ, m) ∈ L∞ .
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Now we check here that u(x, t) = (ρ, m)(x, t) is actually an entropy solution of the Cauchy problem (1.14) and (9.1). For any weak entropy pair (η, q) with convex η and for any nonnegative function ϕ ∈ C0∞ (R × [0, ∞)), we obtain from (9.37) and (9.38) that
∞ R
0
h η u ∂t ϕ + q uh ∂x ϕ dx dt +
= S h (ϕ) +
ϕjk
xj−1
j,k
+
j,k
xj+1
R
η uh (x, 0) ϕ(x, 0) dx
η uk− − η ukj dx
xj+1
ϕ(x, tk ) − ϕjk η uk− − η ukj dx.
xj−1
(9.53)
Since η is a convex function, it satisfies the entropy inequality so that S h (ϕ) 0,
(9.54)
and
ϕjk
j,k
=
xj+1
η uk− − η ukj dx
xj−1
ϕjk
j,k
xj+1 1 xj−1
uk− − ukj
0
,
∇ 2 η ukj + τ uk− − ukj
× uk− − ukj (1 − τ ) dτ dx 0.
(9.55)
Furthermore, for 1 < γ 2, one has x j+1 k k k ϕ(x, t η u − η u dx ) − ϕ k − j j j,k
xj−1
Ch1/2 ϕ C 1
0
j,k
xj+1 xj−1
uk − uk 2 dx − j
1/2 .
Thus, when h → 0, xj+1 k k k ϕ(x, tk ) − ϕj η u− − η uj dx Ch1/2 → 0. j,k
xj−1
For γ > 2, (9.43) and (9.44) imply xj+1 k k k ϕ − ϕ η u − η u dx − j j j,k
xj−1
(9.56)
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G.-Q. Chen and D. Wang
Ch ϕ C 1
0
j,k
k mk ρ k − ρ k + ρ k m− − j dx − − k j ρ− ρjk
xj+1 xj−1
C ϕ C 1 h1/γ → 0.
(9.57)
0
Since |mh (x, t)/ρ h (x, t)| C and (ρ h , mh )(x, t) → (ρ, m)(x, t) for almost every (x, t), we have 0 ρ(x, t) C and |m(x, t)|/ρ(x, t) C almost everywhere. We also conclude from (9.53)–(9.57) that u(x, t) = (ρ, m)(x, t) satisfies the entropy inequality ∞
0
R
η(u)∂t φ + q(u)∂x φ dx dt +
R
η u0 (x) φ(x, 0) dx 0,
for any nonnegative function φ ∈ C0∞ (R × [0, ∞)). This completes the proof of Theorem 9.2. R EMARK 9.2. The convergence for the γ -law case was first proved by Ding, Chen and Luo [96,97] and by Chen [37]. The proof presented above for the general pressure law basically follows [96,97,37] with some simplifications and modifications. We refer to Tadmor [296] for further discussions on various approximate solutions of nonlinear conservation laws and related equations.
9.5. Existence and compactness of entropy solutions T HEOREM 9.3 (Existence and Compactness). Assume that the initial data (ρ0 , m0 )(x) satisfy (9.36). Assume that system (1.14) satisfies (6.22) and (9.3). Then (i) There exists an entropy solution (ρ, m)(x, t) of the Cauchy problem (1.14) and (9.1), in the sense of Definition 9.1, globally defined in time. (ii) The solution operator (ρ, m)(·, t) = St (ρ0 , m0 )(·), defined in Definition 9.1, is compact in L1loc for t > 0. P ROOF. The existence is a direct corollary of Theorem 9.2. Now we prove the compactness. Consider any (oscillatory) sequence of initial data (ρ0ε , mε0 )(x), ε > 0, satisfying 0 ρ0ε (x) C0 ,
ε m (x) C0 ρ ε (x), 0
0
(9.58)
with C0 > 0 independent of ε > 0. Then there exists C > 0 independent of ε > 0 such that the corresponding sequence (ρ ε , mε )(x, t), determined by Theorem 9.2, satisfies 0 ρ ε (x, t) C,
ε m (x, t) Cρ ε (x, t).
Since (ρ ε , mε )(x, t) are entropy solutions satisfying ∂t η ρ ε , mε + ∂x q ρ ε , mε 0
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505
in the sense of distributions, for any C 2 convex weak entropy pair (η, q), we deduce from the Murat lemma (see Murat [237] or [39] for the details) that ∂t η ρ ε , mε + ∂x q ρ ε , mε
−1 2 is compact in Hloc R+ ,
for any weak entropy pair (η, q), not necessarily convex. Combining with Theorem 9.1 yields that (ρ ε , mε )(x, t) is compact in L1loc (R2+ ), which implies our conclusion. R EMARK 9.3. The existence and compactness of entropy solutions for the fluids obeying the γ -law, the case γ = (N + 2)/N, N 5 odd, was treated by DiPerna [106], the case 1 < γ 5/3 by Ding, Chen and Luo [96] and Chen [37], the case γ 3 by Lions, Perthame and Tadmor [203], and then 1 < γ < 3 by Lions, Perthame and Souganidis [202]. For the more general pressure law (6.22) and (9.3), Theorem 9.3 is due to Chen and LeFloch [57,59]. R EMARK 9.4. Notice that Greenberg and Rascle [144] found an interesting nonlinear system with only C 1 (but not C 2 ) flux function admitting time-periodic and space-periodic solutions, which indicates that the compactness and asymptotic decay of entropy solutions are sensitive with respect to the smoothness of the flux functions. However, Theorem 9.3 shows that, although the flux-function of system (1.14) is only Lipschitz continuous, the entropy solution operator is still compact in L1loc for this system.
9.6. Decay of periodic entropy solutions Now we show the large-time decay of periodic entropy solutions in L∞ in the sense of Definition 9.1, established in Chen and Frid [46]. T HEOREM 9.4 (Decay). Consider the Cauchy problem (1.14) and (9.1) satisfying (6.22) and (9.3). Let (ρ, m) ∈ L∞ (R2+ ) be its periodic entropy solution with period [0, a]. Then (ρ, m)(x, t) asymptotically decays as t → ∞: ess lim
t →∞ 0
where (ρ, ¯ m) ¯ :=
ρ(x, t) − ρ¯ + m(x, t) − m ¯ dx = 0,
a
1 a a 0 (ρ0 , m0 )(x) dx.
P ROOF. We divide the proof into four steps: Step 1. Set uε (x, t) = ρ ε , mε (x, t) := (ρ, m)(x/ε, t/ε). Then uε (x, t) is a sequence of entropy solutions with oscillatory initial data. Theorem 9.3 implies the compactness of uε (x, t) in L1loc (R2+ ). Therefore, there exists a subsequence
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G.-Q. Chen and D. Wang
¯ (still denoted) uε (x, t) converging to some function u(x, t) ∈ L∞ (R2+ ) in L1loc (R2+ ). We ε ¯ ¯ from the periodicity of u (x, t). conclude that u(x, t) = u(t) Now, writing the equation of uε (x, t) in the weak integral form and setting ε → 0, we can check that ¯ =0 ∂t u(t) ¯ = u¯ := inthe sense of distributions. This implies from the periodicity of u0 (x) that u(t) 1 a ∗ a 0 u0 (x) dx = w - limε→0 u0 (x/ε). Since the limit is unique, the whole sequence uε (x, t) strongly converges to u¯ in 1 Lloc (R2+ ) when ε → 0. Therefore, we have 1 0 |x|rt
ε u (x, t) − u¯ dx dt → 0,
when ε → 0.
(9.59)
Step 2. Define the following quadratic entropy-entropy flux pairs: ¯ = η∗ (u) − η∗ (u) ¯ − ∇η∗ (u) ¯ · (u − u), ¯ α(u, u) ¯ − ∇η∗ (u) ¯ · f(u) − f(u) ¯ , ¯ = q∗ (u) − q∗ (u) β(u, u)
(9.60)
where (η∗ , q∗ ) is the special entropy–entropy flux pair with convex η∗ , defined in (9.6). Then the periodic entropy solution u(x, t) satisfies the entropy inequality ∂t α(u, u¯ ) + ∂x β(u, u¯ ) 0
(9.61)
in the sense of distributions. It follows that there exists T ⊂ R+ with meas(T ) = 0 such that, for any q ∈ R,
q+a
α u(x, t2 ), u¯ dx
q
q+a
α u(x, t1 ), u¯ dx,
(9.62)
q
/T. for all 0 t1 < t2 , t1 , t2 ∈ Step 3. Given t > 0, t ∈ / T , we take all the rectangles given by x ∈ [q, q + a], for q integer, and s ∈ [[rt]/(2r), t], in the interior of the cone {|x| rs: 0 s t} ([a] is the largest integer less than or equal to a). The number of such rectangles is larger than [rt]. Using the periodicity of u(x, ·) in x and the inequality (9.62) with t2 = t, which holds for a.e. t1 = s ∈ (0, t) over the period [q, q + a], we obtain that there exist c0 > 0, C > 0, independent of t, such that
a
c0 0
[rt] α u(x, t), u¯ dx 2 t
[rt] t2
t [rt] 2r
t [rt] 2r
a
α u(x, t), u¯ dx ds
a
α u(x, s), u¯ dx ds
0
0
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1 t2
C
t 0 |x|rs
1 0
|x|rs
507
α u(x, s), u¯ dx ds
ε u (x, s) − u¯ dx ds → 0,
ε=
1 → 0. t
That is, a
1
ess lim
t →∞ 0
0
, (1 − τ ) u(x, t) − u¯ ∇ 2 η∗ u¯ + τ u(x, t) − u¯ × u(x, t) − u¯ dτ dx = 0.
(9.63)
Step 4. We observe the following: (a) For 1 < γ 2, the entropy η∗ is uniformly convex, and then (9.63) is equivalent to a u(x, t) − u¯ 2 dx = 0. ess lim (9.64) t →∞ 0
(b) For γ > 2, (9.63) implies that a γ ¯ 2 m(x, t) m − + ρ(x, t) − ρ¯ dx = 0. (9.65) ρ(x, t) t →∞ 0 ρ(x, t) ρ¯
ess lim
Note by Hölder’s inequality that ¯ 2 m m 2 − , + (ρ − ρ) ¯ |m − m| ¯ C ρ ρ ρ¯ β 2/γ β |ρ − ρ| ¯ 2 dx C |ρ − ρ| ¯ γ dx . 2
α
(9.66)
α
We conclude from (9.65), (9.66), and the uniform boundedness of the solution (ρ, m)(x, t) that a ρ(x, t) − ρ¯ + m(x, t) − m ¯ dx = 0. (9.67) ess lim t →∞ 0
Combining (9.64) with (9.67) leads to the completion of the proof.
R EMARK 9.5. Theorem 9.4 indicates that periodic entropy solutions asymptotically decay to the unique constant state, determined solely by the initial data. R EMARK 9.6. Although the proof above for L∞ entropy solutions only in the onedimensional case, the argument applies to any space dimension and may extend to entropy solutions in Lp for p 1. See Chen and Frid [46].
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9.7. Stability of rarefaction waves and vacuum states We now consider the global stability of rarefaction waves in a broad class of entropy solutions in L∞ containing the vacuum states for (1.14). Rarefaction waves are the only case that may produce a vacuum state in later time in the Riemann solutions when the Riemann initial data stay away from the vacuum. In Section 6.3, we have discussed the global solvability of the Riemann problem (1.14) and (6.21). Now we show the global stability of rarefaction waves in the following broader class of entropy solutions of (1.14) and (9.1) containing the vacuum states. D EFINITION 9.2. A bounded measurable function u(x, t) = (ρ, m)(x, t) is an entropy solution of (1.14) and (9.1) in R2+ if u(x, t) satisfies the following: (i) There exists a constant C > 0 such that 0 ρ(x, t) C,
m(x, t)/ρ(x, t) C;
(ii) u(x, t) satisfies the equations in (1.14) and one physical entropy inequality in the sense of distributions in R2+ , i.e., for any nonnegative function φ ∈ C01 (R2+ ), ∞ ∞
0
−∞
η(u)∂t φ + q(u)∂x φ dx dt +
∞ −∞
η(u0 )(x)φ(x, 0) dx 0,
(9.68)
2
for (η, q) = ±(ρ, m), ±(m, m 2ρ + p(ρ)), (η∗ , q∗ ), where (η∗ , q∗ ) is the mechanical energy–energy flux pair defined in (9.6). R EMARK 9.7. In Definition 9.2, we require that the entropy solutions in L∞ satisfy solely one physical entropy inequality, besides Equations (1.14), thus admitting a broader class than the usual class of entropy solutions in L∞ that satisfy all weak Lax entropy inequalities (compare with Definition 9.1). R EMARK 9.8. For the Cauchy problem (1.14), (9.1), (6.22), and (9.3), there exists a global entropy solution satisfying all weak Lax entropy inequalities (see Theorem 9.3). The following theorem is taken from Chen [42]. T HEOREM 9.5. Let R(x/t) be the Riemann solution of (1.14), (6.21), (6.22), and (9.2), consisting of one or two rarefaction waves, constant states, and possible vacuum states, as constructed in Section 6.3. Let u(x, t) be any entropy solution of (1.14), (6.22), (9.1), and (9.2) in R2+ in the sense of Definition 9.2. Then, for any L > 0,
|x|L
α(u, R)(x, t) dx
|x|L+Nt
α(u0 , R0 )(x) dx,
(9.69)
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where N > 0 depends only on C > 0 in Definition 9.2 and is independent of t, and α(u, R) ≡ (u − R)
,
1 0
∇ η∗ R + τ (u − R) dτ (u − R) > 0, 2
if u = R and both stay away from the vacuum. In particular, if u0 (x) = R0 (x) a.e., then u(x, t) = R(x/t) a.e. P ROOF. Without loss of generality, we prove the assertion only for the Riemann solution (6.29) which consists of two rarefaction waves with vacuum states as intermediate states. The other cases can be proved similarly. The proof is based on normal traces and the generalized Gauss–Green theorem for divergence-measure vector fields in L∞ , established in Chen and Frid [50,51], and the techniques developed in [47,49,104] for strictly hyperbolic systems. One of the new difficulties here is that strict hyperbolicity fails at the vacuum, yielding singular derivatives of the mechanical energy at the vacuum, which is absent in the strictly hyperbolic case. Another difficulty is that the entropy solutions are only in L∞ . Step 1. Denote u = (ρ, m) and R = (ρ, ¯ m). ¯ First we renormalize the mechanical energyenergy flux pair in (9.6) as in (9.60) and consider µ = ∂t α u(x, t), R(x/t) + ∂x β u(x, t), R(x/t) , d = ∂t η∗ u(x, t) + ∂x q∗ u(x, t) . Since u(x, t) is an entropy solution, µ 0 in any region in which R is constant and µ 0, in the sense of distributions. Then µ and d are Radon measures, and (q∗ (u), η∗ (u))(x, t) and (β(u(x, t), R(x/t)), α(u(x, t), R(x/t))) are divergence-measure vector fields on R2+ . Step 2. Let % & Ω1 := (x, t): λ1 (u− ) < x/t < vc1 , t > 0 , % & Ω2 := (x, t): vc2 < x/t < λ2 (u+ ), t > 0 , the rarefaction wave regions of the Riemann solution, and % & Ω0 := (x, t): vc1 < x/t < vc2 , t > 0
(9.70)
the vacuum region. Over the regions Ωj , j = 1, 2, µ = ∂t α(u, R) + ∂x β(u, R) = d − (∂x R), ∇ 2 η∗ (R)Qf(u, R),
(9.71)
where Qf(u, R) = f(u) − f(R) − ∇f(R)(u − R), and we used the fact that ∇ 2 η∗ ∇f is symmetric. Recall that, for (x, t) ∈ Ωj , 1 ∂x R(x/t) = rj R(x/t) , t
j = 1, 2.
(9.72)
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Then, by (9.71) and (9.72), for any Borel set E ⊂ Ωj , j = 1, 2, we have 1 rj (R), ∇ 2 η∗ (R)Qf(u, R)(x, t) dx dt. µ(E) = d(E) − t E
(9.73)
Over the vacuum region Ω0 , ρ(x, ¯ t) = 0, we may choose the velocity v(x, ¯ t) = x/t,
vc1 < x/t < vc2 .
Then a careful calculation yields 1 m 2 µ=d − + p(ρ) , ρ v¯ − t ρ which implies that, for any Borel set E ⊂ Ω0 , µ(E) = d(E) − E
1 m 2 + p(ρ) (x, t) dx dt. ρ v¯ − t ρ
Step 3. For any δ > 0, denote % & δ1 (t) = x/s = λ1 (uL ), δ < s < t , δ3 (t) = {x/s = vc2 , δ < s < t},
(9.74)
δ2 (t) = {x/s = vc1 , δ < s < t}, % & δ4 (t) = x/s = λ2 (uR ), δ < s < t .
Then % & % & µ δj (t) = d δj 0,
j = 1, 2, 3, 4.
(9.75)
δ denote the region {(x, s): |x| < L + M(t − s), 0 < δ < Step 4. For any L > 0, let ΠL,t δ δ s < t} and Ωj (t) = Ωj ∩ ΠL,t , Ωj (t) = Ωj ∩ {(x, s): 0 < s < t}, j = 0, 1, 2, where
$ $ M M0 := $β(u, R)/α(u, R)$L∞ (R2 ) . +
First, by the entropy inequality (9.68), the Gauss–Green formula for divergence-measure vector fields in [50], and the convexity of η∗ (u) in u, it is standard (cf. [63]) to deduce that any entropy solution defined in Definition 9.2 assumes its initial data u0 (x) strongly in L1loc : u(x, t) − u0 (x) dx = 0, for any K > 0. (9.76) lim t →0 |x|K
Furthermore, we apply normal traces and the Gauss–Green formula for divergencemeasure vector fields in [50] to conclude again % δ & µ Πt,L = α u(x, t), R(x/t) dx − α u(x, δ), R(x/δ) dx |x|L
|x|L+M(t −δ)
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+
δ ∂Πt,L
(β, α) · ν dσ,
where ν is the unit outward normal field and σ is the boundary measure. Then we can choose M M0 such that ∂Π δ (β, α) · ν dσ 0. Therefore, we have t,L
% δ & µ Πt,L
|x|L
α u(x, t), R(x/t) dx
−
|x|L+M(t −δ)
α u(x, δ), R(x/δ) dx.
On the other hand, since R(x/t) is constant in each component of Πt −
54
δ j =1 j (t)
(9.77) 52
δ j =0 Ωj (t) −
and d 0, we have
2
% δ & µ Πt,L − j =1
Ωjδ (t )
1 rj (R), ∇ 2 η∗ (R)Qf(u, R)(x, s) dx ds, s
(9.78)
from (9.73)–(9.75). Step 5. A careful direct calculation yields hj (x, s) := rj (R), ∇ 2 η∗ (R)Qf(u, R)(x, s) m m ¯ 2 ¯ 2p (ρ) ρ − = ρp ¯ (ρ) ¯ + 2p (ρ) ¯ ρ ρ¯
+ p(ρ) − p(ρ) ¯ − p (ρ)(ρ ¯ − ρ) ¯ (x, s) 0,
for j = 1, 2, since p(ρ) is convex in ρ 0. Also, from (6.22) and (9.2), we can see that hj (x, s), j = 1, 2, are uniformly bounded everywhere, even near the vacuum, which means that hj (x, s), j = 1, 2, are integrable in Ω1 (t) ∪ Ω2 (t) as s > 0. This fact in combination with (9.77) and (9.78) yields |x|L
α u(x, t), R(x/t) dx
|x|L+M(t −δ)
α u(x, δ), R(x/δ) dx.
Then (9.76) and (9.79) imply (9.69) as δ → 0. This completes the proof.
(9.79)
In the previous proof, the values of the divergence-measure field (β(u, R), α(u, R))(x, t) on the line segments in the (x, t)-plane should be understood in the sense of normal traces. If one wishes to forego the normalization of our solution through normal traces, then (9.69) should be considered to hold for almost all t ∈ [0, ∞). As a corollary, the following theorem in Chen [42] holds.
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T HEOREM 9.6. Let R(x/t) be the Riemann solution of (1.14), (6.21), (6.22), and (9.2), consisting of one or two rarefaction waves, constant states, and possibly vacuum states, as constructed above. Let u(x, t) be any entropy solution of (1.14), (6.22), (9.1), and (9.2) with initial data u|t =0 = R0 (x) + P0 (x), in the sense of Definition 9.2. Then R(x/t) is asymptotically stable under the initial perturbation P0 (x) ∈ L1 ∩ L∞ in the sense of lim
t →∞ |ξ |L
α u(ξ t, t), R(ξ ) dξ = 0,
for any L > 0.
R EMARK 9.9. The analysis above has also been extended in Chen [42] to the system for non-isentropic fluids, which is more complicated. This has been achieved by identifying a good Lyapunov functional and making an appropriate choice of entropy functions. Also see Chen and Frid [51] for more recent results.
9.8. Other results Further results include the following. Equations of elasticity. Consider the equations in (1.15) with p(τ ) = −σ (τ ), σ (τ ) > 0. In elasticity, genuine nonlinearity is typically precluded by the fact that the medium in question can sustain discontinuities in both the compressive and expansive phases of the motion. In the simplest model for common rubber, one postulates that the stress σ , as a function of the strain τ , switches from concave in the compressive mode τ < 0 to convex in the expansive mode τ > 0, i.e., sgn τ σ (τ ) > 0,
if τ = 0.
(9.80)
In [105], DiPerna proved the existence of global entropy solutions in L∞ of system (1.15) and (9.80). Also see Shearer [278], Lin [200], and Gripenberg [146]. As a corollary, the compactness and decay of global entropy solutions follows with the aid of the approach in Chen and Frid [46]. Euler equations for non-isentropic fluids. Consider the Euler equations for non-isentropic fluids in (1.13). Selecting (τ, v, S) as the state vector, we have the constitutive relations (e, p, θ ) = eˆ(τ, S), p(τ, ˆ S), θˆ (τ, S) satisfying the conditions p = −eˆτ ,
θ = eˆS .
(9.81)
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Under the standard assumptions pˆ v < 0 and θˆ > 0, system (1.13) is strictly hyperbolic. Consider the following class of constitutive relations w h(y) dy, p = h(τ − αS), e = βS − (9.82) θ = αh(τ − αS) + β > 0, where α, β, w = τ − αS, and h(w) is a smooth function with h (w) < 0 satisfying αh (w)2 h (w) − 4 αh(w) + β
> 0, if w < w, ˆ < 0, if w > w. ˆ
(9.83)
The model (9.82) can be regarded as a “first-order correction” to the general constitutive relations (see [43]). The existence and compactness of distributional entropy solutions for the Cauchy problem of (1.13) and (9.81)–(9.83) was established in Chen and Dafermos [43], and the decay of periodic solutions was established in Chen and Frid [46]. In particular, although the periodic solutions do not decay because of linear degeneracy of the system, several important physical quantities, including the velocity, the pressure, and the temperature, do asymptotically decay. As for the Euler equation with more general constitutive relations, including those for polytropic gases with p = (γ − 1)ρe, the problems of existence, compactness, and decay of entropy solutions with arbitrarily large initial data, beyond the BV theory, are still open.
10. Global discontinuous solutions V: the multidimensional case In this section we discuss global discontinuous solutions for the multidimensional Euler equations for compressible fluids.
10.1. Multidimensional Euler equations with geometric structure We first discuss global solutions with geometric structure for the multidimensional Euler equations for isentropic gas dynamics in (1.9) and (1.10). Consider spherically symmetric solutions outside a solid core: ρ(x, t) = ρ(r, t),
x m(x, t) = m(r, t) , r
r = |x| 1.
(10.1)
Then (ρ, m)(r, t) is determined by the equations: A (r) ∂t ρ + ∂r m = − m, 2 A(r) A (r) m2 m + p(ρ) = − , ∂t m + ∂r ρ A(r) ρ
(10.2)
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subject to the Cauchy data: (ρ, m)|t =0 = (ρ0 , m0 )(r),
r > 1,
(10.3)
with homogeneous boundary condition: m|r=1 = 0,
(10.4)
d/2
where A(r) = Γ2π(d/2) r d−1 is the surface area of d-dimensional sphere. Although system (10.2) is here presented in the context of spherically symmetric flow, the same system also describes many flows, important in physics, such as the transonic nozzle flow with variable cross-sectional area A(r) c0 > 0. The eigenvalues of (10.2) are λ± = m ρ ± c = c(M ± 1), where c = p (ρ) is the sound m speed and M = ρc is the Mach number. We notice that λ+ − λ− = 2c(ρ) = 2ρ (γ −1)/2 → 0 as ρ → 0. On the other hand, the geometric source speed is zero, and the eigenvalues λ± are also zero near M ≈ ±1, which indicates that there is also nonlinear resonance between the geometric source term and the characteristic modes. The natural issues associated with this problem are: (a) whether the solution has the same geometric structure globally; (b) whether the solution blows up to infinity in a finite time, especially the density. These issues are not easily resolved through physical experiments or numerical simulations, especially the second one, due to the limited capacity of available instruments and computers. The central difficulty of this problem in the unbounded domain lies in the reflection of waves from infinity and their amplification as they move radially inwards. Another difficulty is that the associated steady-state equations change type from elliptic to hyperbolic at the sonic point; such steady-state solutions are fundamental building blocks in our approach. Consider the steady-state solutions: A (r) m, mr = − 2 A(r) m A (r) m2 + p(ρ) = − , ρ A(r) ρ r (ρ, m)|r=r0 = (ρ0 , m0 ).
(10.5)
The first equation can be integrated directly to get A(r)m = A(r0 )m0 .
(10.6)
The second equation can be rewritten as m2 + A(r)p(ρ)r = 0. A(r) ρ r Hence, using (10.6) and θ = (γ − 1)/2, we have ρ 2θ θ M 2 + 1 = ρ02θ θ M02 + 1 .
(10.7)
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Then (10.6) and (10.7) become
ρ ρ0
θ+1 =
A(r0 )M0 , A(r)M
ρ ρ0
2θ =
θ M02 + 1 . θM2 + 1
(10.8)
Eliminating ρ in (10.8) yields F (M) =
A(r0 ) F (M0 ), A(r)
(10.9)
where
1+θ F (M) = M 1 + θM2
θ +1 2θ
satisfies F (0) = 0, F (1) = 1; F (M) → 0, when M → ∞; F (M)(1 − M) > 0, when M ∈ [0, ∞); F (M)(1 + M) > 0, when M ∈ (−∞, 0]. Thus we see that, if A(r) < A(r0 )|F (M0 )|, no smooth solution exists because the righthand side of (10.9) exceeds the maximum values of |F |. If A(r) > A(r0 )|F (M0 )|, there are two solutions of (10.9), one with |M| > 1 and the other with |M| < 1, since the line 0) F = A(r A(r) F (M0 ) intersects the graph of F (M) at two points. For A (r) = 0, the system becomes the one-dimensional isentropic Euler equations, which have been discussed in Section 9. For A (r) = 0, the existence of global solutions for the transonic nozzle flow problem was obtained in Liu [207] by first incorporating the steady-state building blocks into the random choice method [130], provided that the initial data have small total variation and are bounded away from both sonic and vacuum states. A generalized random choice method was introduced to compute transient gas flows in a Laval nozzle in [129,135]. A global entropy solution with spherical symmetry was constructed in [224] for γ = 1, and the local existence of such an entropy solution for 1 < γ 5/3 was also discussed in [225]. Also see Liu [206–208], Glaz and Liu [129], Glimm, Marshall and Plohr [135], Embid, Goodman and Majda [109], and Fok [119]. In Chen and Glimm [53], a numerical shock capturing scheme was developed and applied for constructing global solutions of (1.9) and (1.10) with geometric structure and large initial data in L∞ for 1 < γ 5/3, including both spherically symmetric flows and transonic nozzle flows. The case γ 5/3 was treated in [66]. It was proved that the solutions do not blow up to infinity in a finite time. More precisely, the following theorem due to Chen and Glimm [53] holds: T HEOREM 10.1. There exists a family of approximate solutions (ρ ε , mε )(r, t) of (10.2) such that for any T ∈ (0, ∞), there is C = C(T ) < ∞ independent of ε so that, when t ∈ [0, T ],
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(r,t ) (i) 0 ρ ε (r, t) C, | m ρ ε (r,t ) | C;
−1 (Ω) for any weak entropy pair (η, q), (ii) ∂t η(ρ ε , mε ) + ∂r q(ρ ε , mε ) is compact in Hloc 2 where Ω ⊂ R+ or Ω ⊂ (1, ∞) × R+ . Furthermore, there is a convergent subsequence (ρ ε , mε )(r, t) of approximate solutions (ρ ε , mε )(r, t) such that
ε ρ , mε (r, t) → (ρ, m)(r, t),
a.e.,
and the limit function (ρ, m)(r, t) is a global entropy solution of (10.2) with the assigned initial data in L∞ and satisfies 0 ρ(r, t) C,
m(r, t) ρ(r, t) C.
Moreover, (ρ, m)(x, t), defined in (10.1) through (ρ, m)(r, t) of (10.2)–(10.4), is a global entropy solution of (1.9) and (1.10) with spherical symmetry outside the solid core for the initial data in L∞ . The approach in Chen and Glimm [53] for constructing the family of approximate solutions in Theorem 10.1 is to merge shock capturing ideas with the fractional-step techniques in order to develop first-order Godunov shock capturing schemes, replacing the usual piecewise constant building blocks by piecewise smooth ones. The main point is to use the steady-state solutions, which incorporate the main geometric source terms, in order to modify the wave strengths in the Riemann solutions. This construction yields better approximate solutions and permits a uniform L∞ bound. There are two technical difficulties to achieve this, both due to transonic phenomena. The first one is that no smooth steady-state solution exists in each cell in general. This problem was solved by introducing a standing shock. The other is that the constructed steady-state solution in each cell must satisfy the following requirements: (a) The oscillation of the steady-state solution around the Godunov value must be of the same order as the cell length so as to obtain the L∞ estimate for the convergence arguments; (b) The difference between the average of the steady-state solution over each cell and the Godunov value must be higher than first-order in the cell length in order to ensure the consistency of the corresponding approximate solutions with the Euler equations. That is, 1 ∆r
(j + 12 )∆r (j − 12 )∆r
u(r, k∆t − 0) dr = ukj 1 + O |∆r|1+δ ,
δ > 0.
These requirements are naturally satisfied by smooth steady-state solutions that stay away from the sonic state in the cell. The general case must include the transonic steady-state solutions. The sonic difficulty was overcome, as in experimental physics, by introducing an additional standing shock with continuous mass and by
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adjusting its left state and right state in the density and its location to control the growth of the density. These requirements can yield the H −1 compactness estimates for entropy dissipation measures ∂t η ρ ε , mε + ∂r q ρ ε , mε and the strong compactness of approximate solutions (ρ ε , mε )(r, t) with the aid of the compactness framework discussed in Section 9.3. R EMARK 10.1. The above method has been applied to studying the Euler–Poisson equations for compressible fluids, which describe the dynamic behavior of many flows of physical importance including the propagation of electrons in submicron semiconductor devices and the biological transport of ions for channel proteins. See Chen and Wang [65] and the references cited therein. R EMARK 10.2. Some related results for entropy solutions with symmetric structure can be found in [40,76,289,322,336,337]. R EMARK 10.3. In the spherically symmetric problem, one of the main difficulties is from infinity because of the reflection of waves from infinity and their amplification as they move radially inwards; this difficulty has been overcome above. Another difficulty is the singularity of entropy solutions at the origin; it would be interesting to study the existence and behavior of spherically symmetric entropy solutions near the origin. 10.2. The multidimensional Riemann problem The multidimensional Riemann problem is very important, since it serves as a building block and standard test model of mathematical theories and numerical methods for solving nonlinear systems of conservation laws, especially the Euler equations for compressible fluids, in any space dimension; and its solutions may also determine the large-time behavior of general entropy solutions. See Glimm and Majda [134], Chern, Glimm, McBryan, Plohr and Yaniv [69], Glimm, Klingenberg, McBryan, Plohr, Sharp and Yaniv [132], and Chen and Frid [47–49]. The elementary waves are the building blocks out of which a Riemann solution is constructed. The Riemann solution is characterized by invariance under scale transformations: x → αx,
t → αt,
α > 0,
while the elementary wave is invariant under additional symmetry: it moves as a travelling wave with a fixed velocity. The elementary waves for the Euler equations for a polytropic fluid were classified in the following theorem by Glimm, Klingenberg, McBryan, Plohr, Sharp and Yaniv [132]. T HEOREM 10.2. Generally, the elementary waves for the Euler equations are one of the following simple types: cross, overtake, Mach triple point, diffraction, and transmission.
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Two-dimensional Riemann problems arise when one-dimensional waves cross or overtake one another or when these waves reflect from or interact with walls or boundaries. Generally, an interaction will arise when two waves meet or a single wave meets a boundary; it is such simple and generic problems which are fundamental. The following two problems have been studied extensively on the level of experiment and computation: (a) the shock-wedge problem of reflection of a shock wave by a wedge in a shock tube (e.g., [323,93,134]); (b) the shock diffraction problem of reflection and transmission of a shock wave by a contact surface (e.g., [1,134]). There are a series of topologically distinct patterns for the various reflected, transmitted and incident waves. Similar issues apply to the interior interaction of waves. Moreover, a two-dimensional Riemann problem can also be generated by the self-interactions of a single two-dimensional elementary wave. See Glimm [131] for more detailed discussions. In Chang, Chen and Yang [31,32], Kurganov and Tadmor [173], Lax and Liu [184], Schulz-Rinne, Collins and Glaz [270], and Zhang and Zheng [335], the two-dimensional Riemann problem with the following form was analyzed for gas dynamics: The initial Riemann values are constant states in each quadrant of the (x, y)-plane, and the four initial constant states satisfy that each jump in the initial data away from the origin produces exactly one of planar forward shocks, backward shocks, forward centered rarefaction waves, backward centered rarefaction waves, or slip surfaces. It was shown that all possible wave combinations can be clarified into nineteen genuine different cases, and there may be some subcases in each case. For each case, numerical solutions of each subcase were illustrated by using various shock capturing methods, and the corresponding theoretical analyses were given by the method of characteristics. In particular, in the case of the interaction of rarefaction waves propagating in the opposite direction, the numerical solutions clearly show that two compressive waves, even shock waves, appear in the solutions. This phenomenon can be explained as the effect of compression of the flow characteristics. The observation of the essential difference of two types of contact discontinuities distinguished by the sign of the vorticity yields two genuinely different cases for the interaction of four contact discontinuities. For one case, the four contact discontinuities roll up and generate a vortex, and the density monotonically decreases along the stream curves. For the other, two shock waves are formed; and, in the subsonic region between two shock waves, a new kind of nonlinear hyperbolic waves appears, called smoothed Delta-shock waves, in the compressible Euler flow, which were first observed by Chang, Chen and Yang [31,32]. The formation of Delta-shocks and the phenomena of concentration and cavitation in the vanishing pressure limit have been rigorously analyzed in Chen and Liu [54]. In general, the solution structures of the Riemann problem are extremely complicated. The following four numerical examples show the complexity of the density contour curves for different interactions of elementary waves in the Riemann problem. Figures 1 and 2 were taken from Lax and Liu [184]; and Figures 3 and 4 from Kurganov and Tadmor [173], respectively. More analytical and numerical results about the Riemann problem and shock reflection problems for the isentropic or non-isentropic Euler equations can be found in AbdElFattah,
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Fig. 1. Interaction of four rarefaction waves.
Fig. 2. Interaction of four shock waves.
Henderson and Lozzi [1], Chang and Chen [30], Chang and Hsiao [33], Deschambault and Glass [93], Gamba, Rosales and Tabak [127], Glimm and Majda [134], Harabetian [151], Hunter and Brio [156], Hunter and Keller [157], Keller and Blank [168], Li, Zhang and Yang [192], Lighthill [196], Serre [274,275], Tabak and Rosales [295], Tesdall and
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Fig. 3. Interaction of two rarefaction waves and two contact discontinuities.
Fig. 4. Interaction of two shock waves and two contact discontinuities.
Hunter [305], Woodward and Colella [323], Zakharian, Brio, Hunter and Webb [332], Zheng [338], and the references cited therein.
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For potential compressible fluid flows, recent mathematical efforts have been made to establish the existence and behavior of solutions. See Chen [67,68], Chen and Feldman [45], Gamba and Morawetz [126], Gu [147], Li [193], Lien and Liu [199], Morawetz [230,232], Zheng [333], and the references cited therein. A related model, called the unsteady transonic small disturbance (UTSD) equations, has been analyzed in Canic, Keyfitz and Lieberman [27] and Canic, Keyfitz and Kim [28]. For the Euler equations for pressureless, isentropic fluids, global Riemann solutions are now well-understood. We refer the reader to Bouchut and James [15], Chen and Liu [54], Ding and Wang [98], Grenier [145], LeFloch [186], Poupaud and Rascle [260], Sheng and Zhang [280], Tan and Zhang [298], Yang and Huang [326], and the references cited therein. Also see E, Khanin, Mazel and Sinai [107] and E, Rykov and Sinai [108] for the effects of random initial data and stochastic forcing. For the multidimensional Riemann problem for Hamilton–Jacobi equations, we refer the reader to Glimm, Kranzer, Tan and Tangerman [136], Bardi and Osher [8], and the references cited therein.
11. Euler equations for compressible fluids with source terms In Sections 2–10, we have discussed the Cauchy problem for the Euler equations for equilibrium, compressible fluids. In this section, we discuss two of the most important examples for the Euler equations for compressible fluids with source terms: relaxation and combustion.
11.1. Euler equations with relaxation The Euler equations with relaxation in (1.1), (1.21), and (1.22) fit into a general setting of hyperbolic systems of conservation laws in the form: 1 ∂t U + ∇ · F(U) + S(U) = 0, ε
x ∈ Rd ,
(11.1)
where U = U(x, t) ∈ RN represents the density vector of basic physical variables, and ε is the relaxation time, which is very short. It is assumed that the system is hyperbolic, and the relaxation term S(U) is endowed with an n × N constant matrix Q with rank n < N such that QS(U) = 0. This yields n independent conserved quantities u = QU. In addition, it is assumed that each u uniquely determines a local equilibrium value U = E(u) satisfying S(E(u)) = 0 and such that QE(u) = u, for all u. For the system in (1.1), (1.21), and (1.22), N = d + 3, n = d + 2, U = (ρ, m, E, ρq), , u = (ρ, m, E), , E(u) = (ρ, m, E, ρQ(ρ, e)), , and I(d+2)×(d+2) 0 , Q= 0 0 where I(d+2)×(d+2) is the (d + 2) × (d + 2) identity matrix.
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The simplest models of (11.1) are 2 × 2 systems: ∂t u + ∂x f (u, v) = 0, 1 ∂t v + ∂x g(u, v) + h(u, v) = 0, ε
(11.2)
where h(u, v) = a(u, v)(v − e(u)), a(u, e(u)) = 0. For such systems, d = 1, N = 2, n = 1, U = (u, v), , E(u) = (u, e(u)), , and Q = (1, 0). In particular, the p-system is a special case of (11.2): ∂t u + ∂x v = 0, 1 ∂t v + ∂x p(u) + (v − f (u)) = 0, ε
(11.3)
with Λ1 = − p (u) < Λ2 = p (u). One of the most important issues is the relaxation limit of hyperbolic systems of conservation laws with stiff relaxation terms to the corresponding local systems. This may model how the dynamic limit from the continuum and kinetic nonequilibrium processes to the equilibrium processes is attained, as the relaxation time tends to zero. Typical examples for such a process include gas flow near thermo-equilibrium, viscoelasticity with vanishing memory, kinetic theory with small Knudsen number, and phase transition with short transition time. The local equilibrium limit turns out to be highly singular because of shock and initial layers and to involve many challenging problems in nonlinear analysis and applied sciences. Roughly speaking, the relaxation time measures how far the nonequilibrium states are away from the corresponding equilibrium states; understanding its limit behavior is equivalent to understanding the stability of the equilibrium states. It connects nonlinear integral partial differential equations with nonlinear partial differential equations. This limit also involves the singular limit problem from nonlinear strictly hyperbolic systems to mixed hyperbolic-elliptic ones, or in some cases even purely elliptic (see [61]). The basic issue for such a limit is stability. Consider system (11.3). If f (u) = λu, p(u) = Λ2 u, then u satisfies ∂t u + λ∂x u + ε ∂t t u − Λ2 ∂xx u = 0.
(11.4)
The limit ε → 0 is stable if and only if the characteristic speeds satisfy −Λ < λ < Λ (cf. [322]). To understand the stability of the zero relaxation limit for the nonlinear case, we first analyze the p-system in (11.3). Notice that v ε = f (uε ) − ε(∂t v ε + ∂x p(uε )). If one can show ε ε u , v (x, t) → (u, v)(x, t),
a.e.,
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then the zero relaxation limit of (uε , v ε )(x, t) is a weak solution of the local equilibrium: v = f (u), ∂t u + ∂x f (u) = 0.
(11.5)
Consider a formal expansion of v ε (x, t) in the form: v ε ≈ f uε + εv1 uε + ε2 v2 uε + · · · . Then, in the ε0 -level, one has ∂t uε + ∂x f uε ≈ 0, ∂t f uε + ∂x p uε + v1 uε ≈ 0,
(11.6)
(11.7)
which implies 2 v1 uε ≈ − p uε − f uε ∂x uε .
(11.8)
Dropping all the higher-order terms in the expansion leads to a first-order correction to the local equilibrium approximation in the form: 2 ∂t uε + ∂x f uε ≈ ε∂x p uε − f uε ∂x uε .
(11.9)
This evolution equation will be dissipative provided the following stability criterion holds: Λ1 < λ < Λ2 , where λ = f (uε ), Λj = (−1)j p (uε ). For the general system (11.1), similar arguments yield that the first correction is −1 U = E(u) − ε ∇U S E(u) I − P(u) ∇x · F E(u) , ∂t u + ∇x · f(u) % −1 & = ε∇x · Q∇U F E(u) ∇U S E(u) I − P(u) ∇x · F E(u) ,
(11.10)
where P(u) = ∇u E(u)Q is a projection (P2 = P) onto the tangent space of the image of E(u). D EFINITION 11.1. A twice-differential function Φ(U) is called an entropy for system (11.1) provided that (i) ∇ 2 Φ(U)∇F(U) · ω is symmetric, for any ω ∈ S d−1 ; (ii) ∇Φ(U)S(U) 0; (iii) The following are equivalent:
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(a) S(U) = 0, (b) ∇Φ(U)S(U) = 0, (c) ∇Φ(U) = ν , Q, for some ν ∈ Rn . An entropy Φ is called convex if ∇ 2 Φ(U) 0.
(11.11)
If the inequality (11.11) is strict, the entropy Φ(U) is called strictly convex. Such a strictly convex entropy exists for many physical systems. For example, under certain conditions, Coquel and Perthame [75] showed that the system in (1.1), (1.21), and (1.22) has a globally defined, strictly convex entropy. In Chen, Levermore and Liu [61], the following theorem was proved. T HEOREM 11.1. Suppose that system (11.1) is endowed with a strictly convex entropy pair (Φ, Ψ ). Then (i) The local equilibrium approximation ∂t u + ∇x · f(u) = 0
(11.12)
is hyperbolic with strictly convex entropy pair: η(u), q(u) = (Φ, Ψ )|U=E (u).
(11.13)
(ii) The characteristic speeds of (11.12) associated with any wave number ω ∈ Rd are determined as the critical values of the restricted Rayleigh quotient: w→
W, ∇U2 Φ(E(u))∇U F(E(u)) · ω W W, ∇U2 Φ(E(u))W
,
(11.14)
where W = ∇u E(u)w for w ∈ Rn . The characteristic speeds of (11.12) are interlaced with the characteristic speeds of (11.1). That is, given a wave number ω ∈ Rd , for each u ∈ Rn , if the characteristic speeds Λk = Λk (E(u)) of (11.1) satisfy Λ1 · · · Λk Λk+1 · · · ΛN , while the characteristic speeds λj = λj (u) of (11.12) satisfy λ1 · · · λk λk+1 · · · λn , then λj ∈ [Λj , Λj +N−n ].
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(iii) The first correction (11.10) is locally dissipative with respect to the entropy η(u). For a 2 × 2 system in (11.2), this implies the subcharacteristic stability condition: Λ1 < λ < Λ2 ,
on v = e(u).
(11.15)
(iv) For the 2 × 2 system in (11.2) satisfying the strictly subcharacteristic stability condition (11.15), the existence of a strictly convex entropy pair (η, q) for the local equilibrium equation implies the existence of a strictly convex entropy pair (Φ, Ψ ) for system (11.2) over an open set Oη containing the local equilibrium curve v = e(u), along which (11.13) is satisfied. Theorem 11.1 indicates that a strictly convex entropy function always exists for 2 × 2 systems endowed with the strictly subcharacteristic condition (11.15) in the regions which are close to the local equilibrium curves. It would be interesting to explore an approach to construct such an entropy for hyperbolic systems of conservation laws with relaxation. Generally, the convexity of entropy could fail at the nonequailibrium states which are far away from the local equilibrium manifolds. In [310], Tzavaras considered the criteria to have such an entropy, as dictated from compatibility with the second law of thermodynamics in the form of the Clausius–Duhem inequality, and found that, roughly speaking, the existence of the entropy is equivalent to the requirement of the relaxation model to be compatible with the second law. The next issue is how the strong convergence of the zero relaxation limit to the local equilibrium equations can be achieved for systems with a strictly convex entropy. For this purpose, we consider a 2 × 2 system in (11.2). Assume that Uε (x, t) = (uε , v ε )(x, t) ⊂ K, bounded open convex set, are solutions of (11.2), which satisfy the following entropy condition: For any convex entropy pair (Φ, Ψ ), 1 ∂t Φ(Uε ) + ∂x Ψ (Uε ) + Φv (Uε )h(Uε ) 0, ε
for all ∇U2 Φ(U) 0,
in the sense of distributions. For simplicity, it is assumed that there exist two convex and dissipative entropy pairs (Φi , Ψi ), i = 1, 2, on K such that φ2 (u) − φ1 (u) = cf (u),
c = 0,
where φ(u) = Φi |v=e(u) , f (u) = f1 (u, e(u)). The existence of such entropy functions is related to the stability theory (Theorem 11.1) (see [60,61,241,46]). T HEOREM 11.2. Assume that there is no interval in which f (u) is linear. Let the Cauchy data (uε0 , v0ε )(x) satisfy $ ε $ $ u − u, ¯ v0ε − v¯ $L2 C < ∞, 0 with v¯ = e(u). ¯ Then Uε (x, t) strongly converges almost everywhere: Uε (x, t) → U(x, t),
a.e.
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The limit function U(x, t) = (u, v)(x, t) satisfies that (i) v(x, t) = e(u(x, t)) almost everywhere for t > 0; (ii) u(x, t) is the unique entropy solution of the Cauchy problem ∂t u + ∂x f (u) = 0, u|t =0 = w∗ - lim uε0 (x). Theorem 11.2 proved in [60,63] was obtained by combining the compactness theorem in [62] with the uniqueness theorem in [63], with the aid of Theorem 11.1. This limit is of compressible Euler type. Theorem 11.2 shows that, when the stability condition is satisfied, the solutions of the relaxation system indeed tend to the solutions of the local relaxation approximation, which are inviscid conservation laws. The main difficulty here is that the solutions of the full system are only measurable functions with certain boundedness. The following remarks are in order: (a) Notice that the initial data may even be far from equilibrium. The convergence result indicates that the limit function (u, v)(x, t) indeed goes into the local equilibrium instantaneously as t becomes positive. This shows that the limit is highly singular. In fact, this limit consists of two processes simultaneously: one is the initial layer limit, and the other is the shock layer limit. (b) The compactness of the zero relaxation limit indicates that the sequence Uε (x, t) is compact no matter how oscillatory the initial data are. Note that the relaxation systems are allowed to be linearly degenerate; and the initial oscillations can propagate along the linearly degenerate fields for the homogeneous systems (cf. [38]). This shows that the relaxation mechanism coupled with the nonlinearity of the equilibrium equations can kill the initial oscillations, just as the nonlinearity of the homogeneous system can kill the initial oscillations. (c) The above discussions are based on the L∞ a priori estimate. In many physical systems, such estimates can be derived. Examples include the p-system and the models in viscoelasticity, chromatography, and combustion (see [46,60,61,241,264,309,322]), which possess natural invariant regions. The technique based on the extensions of entropies has been further pursued by Serre [276] for semilinear and kinetic relaxations of systems of conservation laws. Another technique based on some strong dissipation estimates on derivatives, which are available for several semilinear systems, has been used by Tzavaras [310], Gosse and Tzavaras [140], and the references cited therein. For some special models, even uniform BV bounds of relaxation solutions (uε , v ε )(x, t) can be obtained, which ensure the convergence to the zero relaxation limit. See Natalini [241], Tveito and Winther [309], Shen, Tveito and Winther [279], and the references cited therein. We are now concerned with the weakly nonlinear limit for (11.2). Let uε = u¯ + εwε ,
v ε = v¯ + εzε ,
where (u, ¯ v¯ ) = (u, ¯ e(u¯ )) is an equilibrium state.
(11.16)
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Upon rescaling the time variable t and translating the space variable x, as the slow time variable εt and the moving space variable x − λ(u¯ )t, respectively, (x, t) → x − λ(u¯ )t, t , the flux function in system (11.2) with the stability condition satisfies λ(u¯ ) = 0,
Λ1 (u¯ )Λ2 (u¯ ) < 0.
The limit process as ε → 0 is a weakly nonlinear limit corresponding to the limit from the Boltzmann equation to the Navier–Stokes equations for incompressible fluids. The main observation is that the linearization of the local relaxation approximation about an equilibrium reduces to a simple advection dynamics with the equilibrium characteristic speed. This can be understood in a formal fashion. If one applies the same asymptotic scaling to the first correction to the local equilibrium approximation, one again arrives at the weakly nonlinear approximation. This shows that the weakly nonlinear limit is a distinguished limit of the local equilibrium limit and makes clear why it inherits the good features of the former. The advantage of the weakly nonlinear limit is that the solutions of the Burgers equation are smooth even for the case that the initial data are not smooth. Thus the solutions remain globally consistent with all the assumptions that were used to derive the weakly nonlinear approximation. In [61], the weakly nonlinear approximation was justified by using the stability theory and the energy estimate techniques. The linearized version of the limit is well understood and is related to “random walk” in Brownian motion (cf. [116,174,256]). From (11.2) and (11.16), (wε , zε )(x, t) satisfy ε2 ∂t wε + ∂x f u¯ + εwε , v¯ + εzε = 0, 1 ε2 ∂t zε + ∂x g u¯ + εwε , v¯ + εzε + h u¯ + εwε , v¯ + εzε = 0, ε ε ε w , z t =0 = w0ε , z0ε (x).
(11.17)
T HEOREM 11.3. There exist ε0 > 0 and C0 > 0 such that, if 0 < ε ε0 , and $ ε ε $ $ w ,z $ 0
0
H3
$ $ $ ε e(u¯ + εw0ε ) − e(u) ¯ $ $ C0 ε, $z − $ 2 $ 0 ε L
C0 ,
(11.18)
then there exists a unique global solution (wε , zε ) ∈ H 3 of (11.17) such that ε ε w , z (x, t) → (w, z)(x, t) ∈ L2 ,
ε → 0,
z(x, t) = e (u)w(x, ¯ t), ∂t w
+ λ ( u¯ )∂x
w2 2
¯ 2 (u) ¯ Λ1 (u)Λ hv u, ¯ e( u¯ ) ∂xx w = 0. + hv (u, ¯ e(u)) ¯
(11.19)
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O UTLINE OF THE P ROOF. Since the proof of Theorem 11.3 is technical, we list only the main steps below. Step 1. We replace (Φ, Ψ ) by (Φ∗ , Ψ∗ ), where Φ∗ (U) := Φ(U) − Φ U − ∇Φ U · U − U , Ψ∗ (U) := Ψ (U) − Ψ U − ∇Ψ U · F(U) − F U . Then Φ∗v (u, e(u)) c0 (v − e(u))2 for some c0 > 0. From ∇ 2 Φ∗ (U) > 0, one has
∞
ε −∞
Φ∗ uε (x, t), v ε (x, t) dx + c0
ε
∞
−∞
Φ∗ uε0 (x), v0ε (x) dx
t
∞
(v ε − e(uε ))2 dx dτ ε
0 −∞ ∞ ε 2 3 w0 (x) Cε −∞
+ z0ε (x)2 dx.
Therefore, $ $ $ ε e(u¯ + εwε ) − e(u) ¯ $ $z − $ Cε. $ $ 2 ε L
(11.20)
Step 2. Eliminating zε (x, t) leads to ¯ ε 2 Λ1 (u)Λ ¯ 2 (u) ¯ λ (u) ε2 ∂x w + ∂xx wε + ∂t t wε 2 hv (u, ¯ e(u)) ¯ hv (u, ¯ v(u)) ¯ = E ε x, t, D 2 wε , D 2 zε .
∂t wε +
(11.21)
Using the energy estimates yields
i,j =1
ε2(i−1)
t
∞
0 −∞
∂ i ∂xj wε , εi zε 2 (x, τ ) dx dτ C, τ
(11.22)
i+j3
where C is a constant independent of ε. Step 3. Then we prove E ε (x, t, D 2 wε , D 2 zε ) → 0, when ε → 0. Step 4. Since wε H 1 C, the Sobolev embedding theorem yields that there exists a subsequence (still denoted) wε (x, t) converging strongly in L2 . That is, wε (x, t) → w(x, t). Estimate (11.20) implies that zε (x, t) strongly converges in L2 : zε (x, t) → e (u)w(x, ¯ t). Then Theorem 10.3 follows.
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More details of the proof can be found in Chen, Levermore and Liu [61]. Some further results for hyperbolic systems of conservation laws with relaxation can be found in [39,46,159,160,241,276,309,310] and the references cited therein. Some recent ideas and approaches in attacking hyperbolic conservation laws with memory can be found in Dafermos [85], Nohel, Rogers and Tzavaras [247], and Chen and Dafermos [44] with the aid of the compensated compactness methods. For special memory kernels, these conservation laws reduce to hyperbolic systems of conservation laws with relaxation.
11.2. Euler equations for exothermically reacting fluids We now consider the Euler equations in (1.12) and (1.23) (d = 1), which govern the behavior of plane detonation waves. In a detonation wave, the effect of pressure gradient, which supports the shock wave, and the conversion of chemical energy to mechanical energy is far greater than the diffusive effect of viscosity, heat conduction, and diffusion of chemical species. This justifies the use of the Euler equations in (1.12) and (1.23), rather than the Navier–Stokes equations, in this context. The shock wave solutions in this model are jump discontinuities. This is a very good representation of the shock waves one observes experimentally, which have a width of several molecular mean free paths. The reaction zone of a detonation wave, by way of contrast, is generally hundreds of mean free paths wide. The main interest in this system of equations lies in a new type of behavior exhibited by solutions. Whereas non-reacting shock waves are known to be stable under reasonable assumptions [221], linearized stability analysis, as well as numerical and physical experiments, have shown that certain steady detonation waves are unstable [16,113,117, 185,251]. One particular kind of instability that takes place within the context of one space dimension produces pulsating detonation waves. In certain parameter regimes, steady planar detonation waves are unstable and evolve into oscillating waves. These oscillating waves generate a steady stream of waves which propagate behind the wave [64]. This implies that the exothermic reaction can increase the total variation in a number of ways. For example, in the formation of a detonation wave, a chemical reaction behind a shock wave can increase the strength of that shock wave. More subtle phenomena are also possible. In a nearly constant, unreacted state, a very small variation in temperature can cause the gas in one region to react prior to the gas in nearby regions, resulting in a large increase in total variation. Moreover, the hot spot created by such an event would generate waves, some of which would be shock or rarefaction waves. These waves could propagate away from the hot spot before the remaining reactant ignites. The theorem we discuss here from Chen and Wagner [64] is a first-step in dealing with these difficulties. It is assumed that the initial data are such that the reaction rate function φ(θ ) never vanishes. In a sense, this is a very realistic condition. Typically φ(θ ) has the Arrhenius form (1.23): φ(θ ) = Ke−θ0 /θ ,
(11.23)
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which vanishes only at absolute zero temperature. However, in a typical unburned state, φ(θ ) is very small. This assumption is made in order to obtain uniform decay of the reactant to zero. Thus, although the total variation of the solution may very well increase while the reaction is active, the reaction must eventually die out. Consequently, the increase in total variation can be estimated rigorously. Consider a one-parameter family of functions e(τ, S, ε), τ = 1/ρ, ε 0, which is C 5 and satisfies (1.5). For a polytropic gas, ε = γ − 1. It is assumed that, when ε = 0, the equation of state is that of an isothermal gas: e(τ, S, 0) = − ln τ +
S . R
(11.24)
For a polytropic gas, e(τ, S, ε) =
−ε 1 τ exp(−S/R) −1 . ε
(11.25)
One may easily check that this function is C ∞ and that, as ε → 0+, all partial derivatives converge uniformly on compact sets in τ > 0 to the corresponding derivatives of e(τ, S, 0). In particular, one may use L’Hôpital’s rule to calculate 1 S 2 , ∂ε e(τ, S, 0) = − ln τ + 2 R
(11.26)
and that ∂ε e(τ, S, ε) is continuous at ε = 0, τ > 0. The value ε = 0 is mathematically special because, at this value, system (1.12) and (1.23), in Lagrangian coordinates, has a complete set of Riemann invariants: (r, s, S, Z) = v − ln(p), v + ln(p), S, Z .
(11.27)
Moreover, all shock, rarefaction, and contact discontinuity curves in the (r, s, S, Z)-space are invariant under translation of the base point. We also use (r, s, S, Z) as the coordinates for the analysis in ε 0. Note that, since p = −∂τ e(τ, S, ε), and e(τ, S, ε) is C 5 , the transformation between (τ, v, S) and (r, s, S) is C 4 and is a diffeomorphism (e.g., [302]). T HEOREM 11.4. Let K ⊂ {(τ, v, S, Z): τ > 0} ⊂ R+ × R2 × [0, 1] be a compact set, and let N 1 be any positive constant. Then there exists a constant C0 = C0 (K, N) > 0, independent of ε > 0, such that, for every initial data (τ0 , v0 , S0 , Z0 )(x) ∈ K with TVR (τ0 , v0 , S0 , Z0 ) N , when ε TVR (τ0 , v0 , S0 , Z0 ) C0 ,
(11.28)
the Cauchy problem (1.12) and (1.23) in Lagrangian coordinates, with the initial data determined by (r0 , s0 , S0 , Z0 )(x), has a global BV entropy solution U(x, t) = (τ, v, e + v2 2 , Z)(x, t).
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There is a trade-off between the size of ε (or γ − 1) and the size of the initial data allowed. When ε is close to 0, the initial data are allowed to be of large total variation. There is also a trade-off between the minimum reaction rate and the size of the initial data allowed. If the minimum reaction rate is slow, the increase in total variation due to the reaction is potentially large so that the initial data are only allowed to be of small total variation. If, however, the minimum reaction rate is large, then larger initial data are somewhat allowed. There is an interesting common thread connecting the results with previous ones concerning balance laws (cf. [89,97,219,329,330]). While earlier results had in view lowerorder terms that exerted a damping effect, or otherwise reduced total variation, the result in Theorem 11.4 requires the decay of the lower-order term, even though total variation may increase in the process. Thus, in either case, decay of some kind seems essential.
Acknowledgments Gui-Qiang Chen’s research was supported in part by the National Science Foundation through grants DMS-0204225, DMS-9971793, INT-9987378, and INT-9726215. Dehua Wang’s research was supported in part by the National Science Foundation and the Office of Naval Research. This paper was completed when the first author visited the Institute for Pure and Applied Mathematics at the University of California at Los Angeles; the first author thanks Russ Caflisch, Tony Chan, Bjorn Engquist, Stanley Osher, and Eitan Tadmor for stimulating conversations on related topics. The authors also thank Constantine Dafermos, Jame Glimm, Peter Lax, Tai-Ping Liu, Denis Serre, and Thanos Tzavaras for valuable suggestions, comments, and remarks, which significantly improved the presentation of this paper.
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CHAPTER 6
Stability of Strong Discontinuities in Fluids and MHD Alexander Blokhin and Yuri Trakhinin Sobolev Institute of Mathematics, Russian Academy of Sciences, Koptyuga pr. 4, 630090 Novosibirsk, Russia
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. General concept of stability of strong discontinuities . . . . . . . . . . . . . . . . . . . . . 1.2. Normal modes analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Well-posedness theory for the hyperbolic stability problem . . . . . . . . . . . . . . . . . . 2. Basic steps of the stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Symmetrization of quasilinear systems of conservation laws . . . . . . . . . . . . . . . . . 2.2. Equations of strong discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Setting of the linearized stability problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Separating of instability domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Uniform linearized stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Method of dissipative energy integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Stability of gas dynamical shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. System of gas dynamics and Rankine–Hugoniot conditions . . . . . . . . . . . . . . . . . . 3.2. The LSP for gas dynamical shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Fourier–Laplace analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Construction of the dissipative energy integral for the LSP for gas dynamical shock waves 3.5. Structural stability of uniformly stable gas dynamical shocks . . . . . . . . . . . . . . . . . 4. Stability of shock waves in relativistic gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Relativistic gas dynamics equations and Rankine–Hugoniot conditions . . . . . . . . . . . 4.2. The LSP for relativistic gas dynamical shock waves . . . . . . . . . . . . . . . . . . . . . . 4.3. Instability/uniform stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Construction of the dissipative energy integral for the LSP for relativistic shock waves . . 5. Stability of MHD shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Equations of ideal MHD and strong discontinuities . . . . . . . . . . . . . . . . . . . . . . 5.2. Magnetoacoustic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Solvability of the jump conditions for MHD compressive shocks . . . . . . . . . . . . . . 5.4. The LSP for fast MHD shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. The LSP for slow MHD shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Uniform stability of fast MHD shocks under a weak magnetic field . . . . . . . . . . . . . 5.7. Uniform stability condition for the fast parallel MHD shock wave . . . . . . . . . . . . . . HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME I Edited by S.J. Friedlander and D. Serre © 2002 Elsevier Science B.V. All rights reserved 545
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5.8. Instability of slow MHD shocks under a strong magnetic field . . . . . 6. Stability of the MHD contact discontinuity . . . . . . . . . . . . . . . . . . 6.1. The LSP for the MHD contact discontinuity . . . . . . . . . . . . . . . 6.2. Uniform stability of the MHD contact discontinuity . . . . . . . . . . . 7. Rotational discontinuity in MHD . . . . . . . . . . . . . . . . . . . . . . . . 7.1. The LSP for the rotational discontinuity . . . . . . . . . . . . . . . . . 7.2. The equivalent statement of Problem 7.1 . . . . . . . . . . . . . . . . . 7.3. Instability of the rotational discontinuity under a strong magnetic field 8. Instability of the MHD tangential discontinuity . . . . . . . . . . . . . . . . 8.1. The LSP for the MHD tangential discontinuity . . . . . . . . . . . . . 8.2. Ill-posedness of Problem 8.1 . . . . . . . . . . . . . . . . . . . . . . . 9. Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract This chapter is devoted to the issue of stability of strong discontinuities in fluids and magnetohydrodynamics (MHD) and surveys main known results in this field. All the main points in the stability analysis are demonstrated on the example of shock waves in ideal models of gas dynamics, relativistic gas dynamics, and MHD. Ideal MHD is a good example containing, besides shock waves, different other types of strong discontinuities. Other MHD discontinuities include contact, tangential, and rotational discontinuities. The issue of stability for all these MHD discontinuities is also examined in this chapter. The main attention is concentrated on the linearized stability analysis and the issue of uniform stability. The issue of structural (nonlinear) stability is briefly discussed for gas dynamical shock waves. Open problems and future directions are discussed in the end of the chapter.
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1. Introduction Transitional zones of strong gradients, where parameters of a fluid (density, pressure, temperature, velocity, etc.) vary rapidly, often appear while a fluid is in motion. If dissipative mechanisms (e.g., viscosity or heat conduction) are neglected, i.e., the fluid is supposed to be ideal (inviscid etc.),1 then such thin zones (shocks) are usually viewed as surfaces of strong discontinuity. In this case, the flow parameters change step-wise with jumps on a propagating surface of strong discontinuity.2 One of the main starting points in the mathematical modelling of shocks considered as strong discontinuities is the issue of their stability.
1.1. General concept of stability of strong discontinuities The term “stability of strong discontinuity” was introduced by physicists and denotes the following. Consider a planar surface of strong discontinuity. Let it be slightly perturbed. Let constant parameters of a steady and uniform fluid flow, behind and ahead of the planar discontinuity, be also slightly perturbed. The question is the following. Are small initial perturbations bounded with time? If yes, then the strong discontinuity is stable. Otherwise, it is unstable. In fact, such a definition of stability is that of linear stability of strong discontinuities with respect to small perturbations. But, as will be noted below, such a linear stability does not always guarantee the existence (at least, short-time) of discontinuous solutions of the quasilinear hyperbolic system of conservation laws governing the fluid motion, i.e., the real existence of a strong discontinuity as a physical structure (structural stability). In this connection, the concept of stability must be determined more accurately. Actually, the intuitive concept of structural stability (we do not yet give its rigorous definition) is coupled with that of uniform linearized stability. Namely, we will call a strong discontinuity be uniformly stable if the initial perturbations decrease with time. This definition is yet not sufficiently rigorous. Mathematically, the linearized stability analysis is reduced to the study of a certain linear initial boundary value problem (linearized stability problem; see Section 2 of the present chapter), and, as we will see below, uniform stability implies the fulfilment of the so-called uniform Kreiss–Lopatinski condition [79] for the formulated stability problem. The issue of structural stability of strong discontinuities in different models of fluid dynamics is of great importance, likewise, because in the last years continuum movements with surfaces of strong discontinuities are advantageously calculated numerically. However, before performing calculations it should be sure that the surface of strong discontinuity is structurally stable. The point is that if a strong discontinuity introduced in the framework of a mathematical model is unstable, then it does not really exist, and any calculations in such a case are not adequate to the physical model of a considered phenomenon. Even if the conditions of the existence (in the first moment) of an unstable discontinuity are created artificially in a physical experiment, this discontinuity decays immediately. 1 As is known, motions of ideal fluids are usually described by hyperbolic systems of conservation laws. 2 Recall that if the flow parameters change continuously, but their functions have discontinuous derivatives on
some surface, such a surface is said to be weak discontinuity.
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It is clear that solving the problem on strong discontinuities stability is the first and necessary step in the investigation of discontinuous flows in fluid dynamics. In particular, structural stability is the necessary condition to proving the convergence of a numerical solution to the weak solution of the hyperbolic system.
1.2. Normal modes analysis First works devoted to the multidimensional stability of strong discontinuities were carried out in the 1950–60’s by D’yakov, Freeman, Kontorovich, Iordanskii, Erpenbeck, and others and relate to gas dynamics (see, especially, [49,77,52]). The approach of these works to the problem on strong discontinuities stability is based on the standard analysis by normal modes (see, e.g., [123]). This approach is as follows. One seeks particular exponential solutions to the abovementioned linearized stability problem. A conclusion about the stability or instability of a strong discontinuity is drawn by the behavior of these solutions. Namely, following [49, 77,52] (see also, e.g., [115,55,72,62,125,110] etc.), the exponential solution of a certain linearized stability problem is sought in the form % & U = U0 exp i(−ωt + kx1 + lx2 + mx3 ) , t 0, x1 > 0, (x2 , x3 ) ∈ R2 ,
(1.1)
where U0 is a constant vector, ω, k, l, m are constants, t the time, (x1 , x2 , x3 ) the Cartesian coordinates. If there exists such a solution of the stability problem that Im ω > 0,
Im k > 0,
Im l = Im m = 0,
(1.2)
the shock wave is unstable. Otherwise, it is stable with respect to exponentially growing perturbations (see, e.g., [52]). But, as was first pointed out by D’yakov [49], one should also introduce so-called neutral stable strong discontinuities. It is the case when the stability problem does not have solutions in form (1.1) with property (1.2), but it has the exponential solution (1.1) with Im ω = Im k = Im l = Im m = 0.
(1.3)
For gas dynamical shock waves such a neutral stability domain was called in [49] as the domain of spontaneous sound radiation by the discontinuity.3 Thus, the described standard approach based on the normal modes analysis is, in fact, that to investigating linear stability. Below we will give arguments which show that for the transitional case of neutral stability it is impossible to judge the existence of a strong discontinuity (as a physical structure) on the linear level of investigation. Although, one 3 Actually, in the generic case, there is also a possibility of the existence of normal modes with Im ω = 0, Im k > 0, Im l = Im m = 0 (see below Remark 2.11). Although, e.g., for gas dynamical shocks such a kind of neutral stability does not appear (see Section 3).
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should also note that a strong discontinuity which stability problem has no exponential solutions like (1.1) with properties (1.2) and (1.3) is, actually, uniformly stable (the exponential solutions decrease with time). That is, uniform stability domains can be, in principle, found by the standard normal modes analysis described above. But, to do the passage from uniform linearized stability to structural (nonlinear) stability one has to use another approach to the problem on strong discontinuities stability. It is based on the theory of hyperbolic partial differential equations and operates with such rigorous mathematical notions as uniform Lopatinski condition, well-posedness, etc. This approach dates from work of Blokhin and Majda at the end of 1970’s and the beginning of the 1980’s (see [13–15,17,92–94]).
1.3. Well-posedness theory for the hyperbolic stability problem The linearized stability analysis plays the key role in arguments of [13–15,17,92–94] and is a basis for the passage to nonlinear stability theory. Actually, the normal modes argument mentioned above can be interpreted in terms of the Fourier–Laplace transform that follows to the introduction of the Lopatinski condition (LC) and the uniform Lopatinski condition (ULC) for the linearized stability problem. This was first pointed out by Blokhin [13,14, 19] and Majda [92,94]. In particular, the concept of ULC and the extension of Kreiss’ symmetrizer techniques [79] to the linearized stability problem4 are basic elements in Majda’s linearized stability analysis [92,94]. Namely, the L2 -well-posedness theory (more exactly, L2,η ; see Section 2) developed by Kreiss (see also supplementing works [104, 105,95]) for IBVP’s for linear hyperbolic systems was extended in [92] to the linearized stability problem for strong discontinuities being Lax k-shocks (see, e.g., [75,92,94,107] and Section 2), and a priori estimates without loss of smoothness (see Section 2) for exponentially weighted square-integrable norms of solutions were derived in the domain of fulfilment of the ULC. The term “well-posedness” is used here in a classical sense. That is, a problem is said to be well-posed if its solution exists, is unique, and continuously dependent on initial data. Furthermore, let us, in the framework of this chapter, suppose that the continuous dependence means the existence of a priori estimates without loss of smoothness. Such a priori estimates for the linearized stability problem were first obtained by Blokhin [13,14] for the case of gas dynamical shock waves. Observe also that the estimates in [13,14] are, unlike above-mentioned “weighted” estimates in [79] and [92], of a “layer-wise” type with standard Sobolev norms (see Sections 2, 3). The linearized stability analysis is thus reduced to the study of well-posedness of a corresponding stability problem. On the other hand, if the stability problem is ill-posed, i.e., it admits the construction of an ill-posedness example of Hadamard type (see Section 2), then the strong discontinuity is unstable. Actually, as we will see below, such a “wellposedness” approach establishes a connection between linearized and structural (nonlinear) stability. Here the term “nonlinear stability” means the short-time well-posedness of the 4 This problem is an initial boundary value problem (IBVP) of a nonstandard type for linear hyperbolic systems (see Section 2).
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initial (nonlinear) free boundary problem for the quasilinear system of conservation laws with a propagating curvilinear boundary of a strong discontinuity. Concerning the specific case of neutral stability, we have already noted above that it does not lend itself to linear analysis, and, generally speaking, one has to analyze the initial, free boundary, stability problem (see also discussion below in Section 2). The passage from uniform linearized stability to structural (nonlinear) stability was first performed in [15,17] for shock waves in ideal gas dynamics. It is crucially based on a priori estimates without loss of smoothness obtained in [13,14] for the corresponding linearized stability problem. More exactly, these estimates were derived by the dissipative integrals techniques (DIT) (see, e.g., [48,75,66] and Section 2), and nonlinear analogues of constructions of energy integrals for the linearized stability problem are used in [15,17] to deriving a priori estimates for the nonlinear, free boundary, stability problem. In this connection, symmetrizable hyperbolicity (in the sense of Friedrichs [60]) of a quasilinear system of conservation laws governing the fluid motion plays the crucial role in such an approach utilizing the energy method. Actually, as we will see below, symmetric form can also be very useful for the derivation of the LC and the ULC. Likewise, symmetry property is essentially used in arguments of [92,93]. In view of above lines, the linearized stability analysis includes the following basic steps. I. The symmetrization of the quasilinear system of conservation laws governing the fluid motion; II. The determination of equations of strong discontinuity (jump conditions) for the system of conservation laws; III. The linearization of the quasilinear equations and the jump conditions. The formulation of the linearized stability problem; IV. The separation of ill-posedness domains (Hadamard example) and, if possible, domains of fulfilment of the ULC for the formulated stability problem; V. The derivation of a priori estimates without loss of smoothness for the stability problem by the DIT in such domains where Hadamard-type ill-posedness examples are not constructed. If we succeeded in realizing of this approach in full volume, we have the rigorous mathematical basis of the linearization method applied to the investigation of strong discontinuities stability. Note that, taking account of the results in [92], if we managed to finding the uniform stability domain (the ULC holds), then in this domain “weighted” a priori estimates without loss of smoothness are fulfilled for solutions of the linearized stability problem. But, on the other hand, standard “layer-wise” estimates are, in some sense, more preferable (in particular, for a subsequent numerical analysis; see discussion in Section 2). Moreover, we can obtain these estimates without the preliminary separation of the uniform stability domain. Concerning the passage to nonlinear stability theory, the listed basic steps can be completed by the following two ones in the domain where the point V has been realized. VI. The derivation of a priori estimates for the nonlinear, free boundary, stability problem; VII. The proving of the local (short-time) theorem of existence and uniqueness of the shock front solution to the quasilinear system of conservation laws.
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The last two steps were realized for gas dynamical shock waves in [15,17] (see also [19]), where the local theorem of existence and uniqueness of classical solutions to the quasilinear gas dynamics equations ahead and behind the curvilinear shock wave is proved (see Section 3). These classical solutions belong to a Sobolev space W23 (or W2s with s 3), and the local (short-time) theorem is valid in the domain of initial data that is determined by the domain of fulfilment of the ULC for the linearized stability problem (see Section 3). We refer also to the work [93] where, using another techniques (in particular, that of pseudodifferential operators), Majda has proved the theorem on short-time W2s existence (s 10 for 3-D) of discontinuous shock fronts solutions of a quasilinear system of conservation laws that satisfies some block structure conditions.5 In the present chapter the main attention is given to the linearized stability analysis. In the next section we describe in detail the basic steps I–V listed above and give simultaneously all the necessary definitions (e.g., the definition of the ULC, etc.) and explanations. In the subsequent sections, all the main points in the stability analysis are demonstrated on the example of shock waves in ideal models of gas dynamics, relativistic gas dynamics, and MHD. Observe also that ideal MHD is a good example containing, besides shock waves, different other types of strong discontinuities. Other MHD discontinuities include contact, tangential, and rotational discontinuities. The issue of stability for all these MHD discontinuities is also examined in this chapter. Concerning the issue of nonlinear (structural) stability, as was already noted above, the nonlinear analysis is crucially based on that of linearized stability, and we just briefly discuss it for gas dynamical shock waves. Although, there are some open problems in this point which are discussed in the last section.
2. Basic steps of the stability analysis 2.1. Symmetrization of quasilinear systems of conservation laws Let us consider a quasilinear system of partial differential equations A0 (U)Ut +
3
Ak (U)Uxk = 0,
(2.1)
k=1
where Aα are quadratic matrices of the order n, U = (u1 , . . . , un )∗ is the vector-column of unknowns. D EFINITION 2.1. System (2.1) is called symmetric t-hyperbolic (in the sense of Friedrichs [60]) if Aα = A∗α
(symmetry),
A0 > 0 (positive definiteness).
(2.2) (2.3)
5 In particular, Majda’s block structure conditions hold for the gas dynamics system (see also discussion in Section 2).
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Here and below the symbol ∗ denotes matrix transposition, the Greek indices run from 0 to 3, and the Latin ones from 1 to 3, except where stated otherwise. Observe that symmetric t-hyperbolic systems are hyperbolic, in the sense of the general definition of hyperbolic systems (see, e.g., [100,75]). Just in case, we recall here this definition. D EFINITION 2.2. The quasilinear system Ut +
3
Ak (U)Uxk = 0
k=1
is called hyperbolic if all the eigenvalues λi (i = 1, n) of the matrix P(ω) =
3
ωj Aj
j =1
are real for all ω = (ω1 , ω2 , ω3 ) ∈ R3 \{0}, and this matrix is reduced to the diagonal form (with the real diagonal elements λi = λi (U, ω)). Let the fluid motion is governed by the system of conservation laws ∂Pi0 + div P i = 0, ∂t
i = 1, n,
(2.4)
where P i = (Pi1 , Pi2 , Pi3 )∗ , i = 1, n, Piα = Piα (U). System (2.4) can be written in the matrix form (2.1), with Aα = (∂Piα /∂uj ), i, j = 1, n. In practice, the system of conservation laws (2.4) governing a fluid motion is not necessarily symmetric, and the problem now is to symmetrize it. Recently a great number of works was devoted to the symmetrization of various systems of conservation laws (see the review [21]). What is the reason of such an interest to the problem of symmetrization? The point is that the possibility to representing the system of conservation laws (2.4) in a symmetric form, provided the hyperbolicity condition (2.3) hold, means that the mathematically developed and improved theory of t-hyperbolic systems (see [48,75,19]) can be applied to a model of fluid dynamics governed by system (2.4). For example, for a symmetric t-hyperbolic system we have the local wellposedness of the Cauchy problem in a Sobolev space W2s (s 3; see [114,88,76]).6 The scheme of symmetrization of the quasilinear system of conservation laws (2.4) with the help of the additional and a priori known conservation law ∂Φ 0 + div Φ = 0, ∂t
(2.5)
6 For the linear case, with constant matrices, we have the global L -well-posedness of the Cauchy problem. 2
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553
that holds on smooth solutions of system (2.4), was first suggested by Godunov [64] (see also [61,60,45,21,43]). Consider a generalization of this scheme to systems of conservation laws which are supplemented by a set of divergent constraints (stationary conservation laws)7 div Ψ j = 0,
j = 1, m.
(2.6)
Here Φ = (Φ 1 , Φ 2 , Φ 3 )∗ , Φ α (U); Ψ j = (Ψj1 , Ψj2 , Ψj3 )∗ , Ψjα = Ψjα (U), j = 1, m. Such a generalized symmetrization scheme was first used by Godunov [65] for the symmetrization of ideal MHD. Let us now describe it in detail (see also [108,19,21]). Actually, in practice the additional conservation law is a consequence of system (2.4) and the compulsory conditions (2.6): 0
n m
∂Pi ∂Φ 0 qi rj div Ψ j = + div P i + + div Φ = 0, ∂t ∂t j =1
i=1
where qi = qi (U), i = 1, n, are so-called canonical variables (or Lagrange multipliers), and rj = rj (U), j = 1, m, are some functions which, together with the canonical variables, can be determined from the relations dΦ 0 =
n
dΦ k =
qi dPi0 ,
i=1
n
qi dPik +
m
rj dΨjk .
(2.7)
j =1
i=1
Following [65,21], let us introduce so-called productive functions: L = L(Q) =
n
qi Pi0 − Φ 0 ,
M k = M k (Q) =
n
i=1
i=1
qi Pik +
m
rj Ψjk − Φ k ,
j =1
with Q = (q1 , . . . , qn )∗ . Then, in view of (2.7), Lqi = Pi0 +
n
ql
l=1
Mqki
= Pik
+
∂Pl0 ∂Φ 0 − = Pi0 , ∂qi ∂qi
m
∂rj j =1
∂qi
Ψjk
+
n
l=1
∂rj ∂P k ∂Ψj ∂Φ k ql l + rj − = Pik + Ψ k, ∂qi ∂qi ∂qi ∂qi j m
j =1
k
m
j =1
i.e., Pi0 = Lqi ,
Pik = Mqki −
m
Ψjk (rj )qi .
(2.8)
j =1 7 The system of ideal MHD with the divergent constraint div H = 0 (H the vector of magnetic field; see Section 5) is an example of such systems.
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By virtue of (2.6), (2.8), on smooth solutions system (2.3) can be rewritten as m m
∂ Lqi + div Mqi − (rj )qi Ψ j + (rj )qi div Ψ j = 0, ∂t j =1
i = 1, n,
(2.9)
j =1
where M = (M1 , M2 , M3 )∗ . Finally, system (2.9) is written in the symmetric form 3
A0 (Q)Qt +
Ak (Q)Qxk = 0,
(2.10)
k=1
with the symmetric matrices A0 = (Lqi qj ),
Ak = Mqki qj −
m
Ψlk (rl )qi qj ,
i, j = 1, n.
l=1
In practice, we can explicitly find the symmetric matrices Aα by the following procedure suggested by Blokhin [18] (see also [21,43]). Let Lq = (Lq1 , . . . , Lqn )∗ ,
∗ Mkq = Mqk1 , . . . , Mqkn .
Then dQ = J dU,
dLq = I0 dU,
dMkq = Ik dU,
(2.11)
where J , I α are quadratic matrices. By (2.11), A = I0 J 0
−1
,
A = Ik J k
−1
−
m
Ψjk Irj .
j =1
Here Irj = ((rj )qi ql ), i, l = 1, n. Actually, I0 = A0 = (∂Pi0 /∂ul ), cf. (2.1), (2.4), (2.8), (2.11), and A0 = A0 J −1 , Ak = Ak J −1 . Moreover, the system of conservation laws (2.4) (or (2.1)) can be, likewise, rewritten in terms of the initial vector of unknowns U as the symmetric system B0 (U)Ut +
3
Bk (U)Uxk = 0,
k=1
Bα = J ∗ Aα J = J ∗ Aα . So, J ∗ is the matrix that symmetrizes system (2.1)/(2.4). Furthermore, if the matrix A0 (Q) is positive definite, the last system is hyperbolic. R EMARK 2.1. In accordance with the above described scheme of symmetrization, the equations of ideal MHD [65], MHD of Chew, Goldberger and Low [29] (for collisionless
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magnetized plasma; see [47]), relativistic fluid [31,32], superfluid [18,24], etc. were symmetrized (see the review [21]). Actually, such a scheme (in other terms) was applied also, for example, by Ruggeri and Strumia [109] for the symmetrization of the relativistic MHD equations.
2.2. Equations of strong discontinuity Now, we are interested in piecewise smooth solutions to system (2.4) with smooth parts separated by a surface of strong discontinuity. Let such a surface is given by the equation f˜(t, x) = x1 − f (t, x ) = 0
(2.12)
(x = (x1 , x ), x = (x2 , x3 )). As is known (see, e.g., [75,102,94,85]), on surface (2.12) jump conditions should hold for limit values of solutions of the system of conservation laws ahead (f˜ → −0) and behind (f˜ → +0) the discontinuity front. For system (2.4) such jump conditions (equations of strong discontinuity8) read [PiN ] = DN Pi0 ,
i = 1, n,
(2.13)
where PiN = (P i , N); |∇ f˜| = (1 + fx22 + fx23 )1/2 ; N=
1 |∇ f˜|
(1, −fx2 , −fx3 )∗ ,
DN = −
f˜t |∇ f˜|
=
ft |∇ f˜|
the unit normal to the discontinuity front and the discontinuity speed in the normal direction, [g] = g − g∞ denotes the jump for every regularly discontinuous function g with corresponding values behind (g := g|f˜→+0 ) and ahead (g∞ := g|f˜→−0 ) the discontinuity front (here and below the subindex ∞ stands for boundary values ahead the shock front). R EMARK 2.2. System (2.13) is closed, in the sense that we can uniquely determine the values ui behind the discontinuity by the values ui∞ ahead the discontinuity (i = 1, n), assuming the discontinuity speed DN is known. Observe that sometimes from physical reasons one of the equations of strong discontinuity is reduced from the additional conservation law (2.5). In that case, not all the equations from (2.13) are included into a full system of equations of strong discontinuity. Actually, jump conditions like (2.13) are deduced from corresponding initial integral conservation laws (see, e.g., [75,102,107]). So, while writing jump conditions for a system of conservation laws, we identically fix initial integral conservation laws, and, in this sense, the conservation laws (2.4) were supposed to be “proper” (see, e.g., [107]). In practice, the Rankine–Hugoniot jump conditions for the equations of matter, momentum, and energy conservation are always inserted into the full system of discontinuity equations for different models of fluid dynamics. 8 Sometimes, they are called, by analogy with gas dynamics, as Rankine–Hugoniot conditions.
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2.3. Setting of the linearized stability problem Nonlinear, free boundary, stability problem. The initial nonlinear stability problem for a strong discontinuity is the IBVP for the quasilinear system (2.4) (or (2.1)) with the boundary conditions (2.13), on a propagating surface of strong discontinuity being the free boundary x1 = f (t, x ), and initial data for the vector-function U(t, x) and the function f (t, x ). In addition, the function f (t, x ) should also be determined during the solving of such a free boundary stability problem (FBSP). Structural stability. It is clear that a strong discontinuity really exists as a physical structure if the corresponding FBSP has a unique solution at least for a small time. D EFINITION 2.3. Let a strong discontinuity exists in the first moment (initial data for the corresponding FBSP are supposed to be admissible; see Section 3). It is called structurally stable if the corresponding FBSP is locally well-posed, i.e., for a small time and admissible initial data there exist the unique surface x1 = f (t, x ) and the unique classical solution U(t, x) ahead (x1 < f (t, x )) and behind (x1 > f (t, x )) the discontinuity front. Linearization of quasilinear equations and jump conditions. We now describe the process of linearization of the quasilinear system of conservation laws (2.4) and the equations of strong discontinuity (2.13). As a result, we will obtain the linearized stability problem (LSP) mentioned above. Let the process of symmetrization of quasilinear system is already performed, i.e., without loss of generality we assume system (2.1) to be symmetric t-hyperbolic. Let us consider a planar strong discontinuity (stepshock) with the equation9 x1 = 0 and a piecewise constant solution to system (2.1), U(t, x) =
# U∞ = (uˆ 1∞ , uˆ 2∞ , . . . , uˆ n∞ )∗ , x1 < 0; # x1 > 0, U = (uˆ 1 , uˆ 2 , . . . , uˆ n )∗ ,
(2.14)
that satisfies the jump conditions (2.13) on the plane x1 = 0:
1 # #1 = P 1 # P i i U − Pi U∞ = 0,
where uˆ i∞ , uˆ i (1, n) are constants (here and below all the hat values stand for parameters of the piecewise constant solution). Linearizing system (2.4) and conditions (2.13) with respect to solution (2.14), we obtain the LSP to determining the vector of small perturbations δU and the small disturbance of discontinuity surface δf = F = F (t, x ) (in order to simplify the notation we indicate the vector δU again by U). 9 Without loss of generality we assume the planar shock with the equation x = σ t to be stationary, i.e., the 1 shock speed σ = 0.
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P ROBLEM 2.1 (Linearized stability problem). We seek the solutions of the systems A 0 Ut +
3
Ak Uxk = 0,
t > 0, x ∈ R3+ ;
(2.15)
k=1
A0∞ Ut +
3
Ak∞ Uxk = 0,
t > 0, x ∈ R3− ;
(2.16)
k=1
satisfying the boundary conditions 3
k 0 # Fxk = 0 # Ft + P −[A1 U] + P
(2.17)
k=2
at x1 = 0 (t > 0, x ∈ R2 ) and the initial data U(0, x) = U0 (x),
x ∈ R3± ,
F (0, x ) = F0 (x ),
x ∈ R2 ,
(2.18)
for t = 0. Here R3± = R± × R2 (for the case above R± = {x1 ≷ 0} ); Aα = Aα (# U) = A∗α ,
Aα∞ = Aα # U∞ = A∗α∞ ,
A0 > 0,
A0∞ > 0,
[A1 U] = A1 U|x1 →+0 − A1∞ U|x1 →−0 , ∗ α # = Pα # P α = P1α , . . . , Pnα , P U − Pα # U∞ . R EMARK 2.3. In the considered case of an abstract discontinuity the number of the boundary conditions (2.17), let us say m, coincides with that of the conservation laws (2.4), i.e., m = n. But in the case of a concrete type of strong discontinuity (e.g., shock waves, contact discontinuities, rotational discontinuities in MHD, etc.) boundary conditions can be sometimes interdependent (e.g., for MHD tangential discontinuities; see Section 8), i.e., the number of independent from each other boundary conditions can be less than n (m n). R EMARK 2.4. While solving Problem 2.1, we also determine the function F = F (t, x ). To this end, one of the boundary conditions (2.17) must be the equation to determining the function F , that describes a small disturbance of the planar discontinuity front. Thus, the question on linearized stability of strong discontinuity is reduced to the investigation of well-posedness of IBVP (2.15)–(2.18) for symmetric t-hyperbolic systems with constant real coefficients. Observe that, in view of the presence of the function F in the boundary conditions (2.17), this IBVP for linear hyperbolic equations has an unusual form in comparison with problems studied, for example, in [79].
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R EMARK 2.5. Through changing variables by x1 to −x1 in system (2.16) we can reformulate Problem 2.1 as a conventional IBVP in a half-space. Moreover, the LSP can be, in principle, reduced to a problem with homogenous initial data.10 Namely, with the notation 0 0 Aβ A1 , β = 0, 2, 3, A1 = , Aβ = 0 Aβ∞ 0 −A1∞ ∗ V = (U − U0 )∗ |x1 >0 , (U − U0 )∗ |x1 <0 , φ = F − F0 , 3 3
− k=1 Ak (U0 )xk |x1 >0 k # (F0 )xk , F= , g = [A1 U0 ] − P 3 − k=1 Ak∞ (U0 )xk |x1 <0 k=2 Problem 2.1 becomes the form A0 Vt +
3
Ak Vxk = F,
t > 0, x ∈ R3+ ;
(2.19)
k=1 3
0 k # φt + # φxk = g, −A1 V + P P
t > 0, x1 = 0, x ∈ R2 ;
(2.20)
k=2
V(0, x) = 0,
x ∈ R3+ ,
φ(0, x ) = 0,
x ∈ R2 .
(2.21)
In particular, such a reduced form of the LSP was exploited by Majda in linearized stability arguments [92]. Evolutionarity condition as the necessary one for well-posedness. Before the investigation of well-posedness of the LSP it should be sure that the Landau evolutionarity condition −1 − + − [84,85] is fulfilled, i.e., m = n+ (A−1 0 A1 ) + n (A0∞ A1∞ ) + 1, where n (n ) is the num11 ber of positive (negative) eigenvalues of a matrix. It is assumed here that the boundary conditions (2.17) are independent from each other, i.e., their number m cannot be reduced. The evolutionarity condition is, in fact, necessary but insufficient for the well-posedness of the stability problem. D EFINITION 2.4. A strong discontinuity is said evolutionary if the condition −1 − m = n+ A−1 0 A1 + n A0∞ A1∞ + 1
(2.22)
is fulfilled for Problem 2.1. R EMARK 2.6. If ahead a planar strong discontinuity there are no outgoing characteristics 3 modes, i.e., n− (A−1 0∞ A1∞ ) = 0, then in the domain t > 0, x ∈ R− the solution U(t, x) is 10 Although, such a way suffers from some shortcomings that will be discussed below. 11 Since A > 0, A ± −1 ± ± −1 ± 0 0∞ > 0, then n (A0 A1 ) = n (A1 ), n (A0∞ A1∞ ) = n (A1∞ ).
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completely determined by initial data for x1 < 0, and without loss of generality we can presume that U(t, x) ≡ 0 for x1 < 0. In the nonevolutionary case the stability problem can be underdetermined or overdetermined (corresponding strong discontinuities are said to be undercompressive and overcompressive, see, e.g., [113,91,57]). D EFINITION 2.5. Stability Problem 2.1 is underdetermined if −1 − m < n+ A−1 0 A1 + n A0∞ A1∞ + 1, and it is overdetermined if −1 − m > n+ A−1 0 A1 + n A0∞ A1∞ + 1. Important remarks on evolutionary (classical) and nonevolutionary (nonclassical) discontinuities. It is clear that an overdetermined stability problem is ill-posed, because there do not exist solutions to such a problem for arbitrary initial data.12 For example, as was shown by Akhiezer et al. [1] and Syrovatskij [116], overcompressive MHD strong discontinuities should decay into a number of evolutionary ones and simple waves. In the next subsection we prove the almost evident, but important, proposition on the ill-posedness of underdetermined stability problems (see also [107]). Thus, undercompressive strong discontinuities do not seem to exist really. Although, one should note that this conclusion is quite correct if we are in the framework of hyperbolic (inviscid ) theory. On the other hand, there is another approach to undercompressive discontinuities. Actually, the stability problem for such discontinuities lacks boundary conditions. But, sometimes lacking boundary conditions can come from a system of viscous conservation laws that is a high order regularization of the hyperbolic system and coincides with it in the limit of vanishing dissipation (e.g., viscosity or heat conduction). Namely, lacking boundary conditions are derived from the requirement that the strong discontinuity should have a structure (or viscous profile, see, e.g., [63,123, 107]) that is represented by travelling wave solutions connecting asymptotically constant states. In this connection, we refer, for example, to gas-ionizing MHD shock waves studied by Kulikovskii and Lyubimov [80] and phase transitions treated as discontinuities examined by Benzoni-Gavage [9,10]. For these undercompressive discontinuities one can set additional boundary conditions following from the viscous profile analysis. Moreover, we refer the reader to a theoretical generalization of such an approach done by Freistühler in [57–59] where the issue of the linearized stability of undercompressive shocks is also discussed. We finally mention that overcompressive shocks being considered as viscous profiles can also exist. In this connection, we especially refer to intermediate MHD shocks studied by Wu [124], that can be either undercompressive or overcompressive, and to 1-D numerical 12 Recall that the boundary conditions are assumed to be independent from each other.
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and analytical results of Freistühler [56,59] and Zumbrun and Howard [126] indicating that viscous profiles for nonclassical (undercompressive or overcompressive) shocks can be stable. The multidimensional stability of viscous planar shock fronts (for classical and as well as for nonclassical shocks) and its connection with inviscid linearized stability was analyzed by Zumbrun and Serre [127]. We note that in the framework of the present chapter we are concerned only with evolutionary (classical) strong discontinuities. Furthermore, as was mentioned in [127], the uniform inviscid stability of classical strong discontinuities that is the main point of this chapter yields the multidimensional stability of corresponding viscous shock fronts. Although, for the case of neutral (weak) stability of classical discontinuities the viscous stability analysis can give some additional useful information (see [127]), especially, if a viscous regularization is “real” (e.g., the Navier–Stokes equations as a regularization of the Euler equations).13 Classical Lax discontinuities. In the end of this section, we discuss a generalization of the linear 1-D evolutionarity condition (2.22) to the multidimensional quasilinear case. So-called k-shock conditions [75,92,94,107] can be such a generalization for special cases (e.g., for shock waves). Observe that, in turn, these conditions are a generalization of the Lax entropy conditions introduced in [87] for 1-D hyperbolic systems of conservation laws. Let the system of conservation laws (2.4) be written in the form of system (2.1) with the nonsingular matrix A0 . Then, without loss of generality, we can assume A0 to be the unit matrix. Taking this into account, we give the following definition. D EFINITION 2.6. A strong discontinuity for system (2.4) with surface (2.12) and the jump conditions (2.13) is called k-shock14 if there exists an integer number k (k = 1, n) so that the eigenvalues (see Definition 2.1.2) λ1 (U, N) · · · λn (U, N),
λ1 (U∞ , N) · · · λn (U∞ , N)
satisfy, on surface (2.12), the k-shock (Lax entropy) conditions λk (U, N) < DN < λk (U∞ , N),
λk−1 (U∞ , N) < DN < λk+1 (U, N),
with λk−1 < λk < λk+1 for U and U∞ . 15 R EMARK 2.7. Notice that in gas dynamics, the Lax entropy conditions are equivalent to the physical condition of entropy increase (S > S∞ ) under the passage through the discontinuity front (this is the cause that k-shock conditions are also called as entropy ones). But, in generic case (e.g., for MHD; see Section 5) they cannot guarantee the fulfilment of the physical entropy condition. 13 It was announced in [127] that the parabolic stability analysis could be generalized, with slight modifications, to the case of “real” (semidefinite) viscosity. 14 Sometimes k-shocks are called compressive or Lax shocks. 15 Let us say that λ (U , N) = −∞ and λ 0 ∞ n+1 (U, N) = +∞.
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561
For a planar stationary discontinuity k-shock conditions read λk (A1 ) < 0 < λk (A1∞ ),
λk−1 (A1∞ ) < 0 < λk+1 (A1 ),
(2.23)
where λi (A1 ), λi (A1∞ ) (i = 1, n) are the eigenvalues of the matrices A1 , A1∞ . If the number of the boundary conditions (2.17) is n (see Remark 2.3), the last inequalities imply the fulfilment of the evolutionarity condition (2.22). Indeed, as n+ (A1 ) = n − k, n− (A1∞ ) = k − 1, then n+ (A1 ) + n− (A1∞ ) + 1 = n = m. So, k-shock conditions are sufficient for the evolutionarity of strong discontinuities. Looking ahead, we observe that, for example, gas dynamical shock waves as well as evolutionary shock waves in MHD (fast and slow MHD shock waves; see Section 5) are k-shocks. But, on the other hand, it ought to underline that k-shock conditions are not necessary for evolutionarity. For example, rotational discontinuities in MHD are evolutionary (see Section 7) but not k-shocks. This occurs because of the existence of zero eigenvalues of the matrices A1 , A1∞ in the LSP (the stepshock x1 = 0 is a characteristic boundary). The same can be said also for contact and tangential MHD strong discontinuities (see Sections 6, 8). At last, notice that we can generalize Definition 2.6 and introduce so-called Lax discontinuities [74]. For Lax (classical) discontinuities the inequalities in Definition 2.6 (k-shock conditions) are not strict: λk (U, N) DN λk (U∞ , N),
λk−1 (U∞ , N) DN λk+1 (U, N).
In the sense of this generalized definition, all the classical MHD strong discontinuities (shock waves, rotational, contact, and tangential discontinuities) are Lax discontinuities.
2.4. Separating of instability domains Let us now describe the process of construction of an ill-posedness example of Hadamard type for LSP’s in the form of Problem 2.1. The domain of ill-posedness of Problem 2.1 is that of instability of a corresponding strong discontinuity. To show the ill-posedness of Problem 2.1 we look for the sequence of its exponential solutions in the form (0) % & U∞ exp k τ t + ξ∞ x1 + i(γ , x ) , x1 < 0; (2.24) Uk = % & x1 > 0; U(0) exp k τ t + ξ x1 + i(γ , x ) , % & Fk = F (0) exp k τ t + i(γ , x )
(2.25)
(k = 1, 2, 3, . . .). Here U∞ = (u1∞ , . . . , un∞ )∗ , U(0) = (u1 , . . . , un )∗ are real constant vectors; F (0) , τ , ξ , ξ∞ are constants; γ = (γ2 , γ3 ), Im γ2,3 = 0. (0)
(0)
(0)
(0)
(0)
R EMARK 2.8. The case γ2,3 = 0 corresponds to an ill-posedness example in 1-D form that can be constructed for stability problems violating so-called Majda’s conditions [94]
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(e.g., for underdetermined stability problems; see Proposition 2.1) that expresses, in fact, the general principle [66] of setting IBVP’s for linear hyperbolic systems. Note also that if a strong discontinuity is evolutionary, an ill-posedness example in 1-D form can be constructed (Majda’s conditions are violated) only in very specific cases when, for the 1-D LSP reduced to the canonical diagonal form of Riemann invariants (see, e.g., [66]), outgoing Riemann invariants cannot be expressed through incoming ones at the boundary x1 = 0 (a corresponding matrix in boundary conditions is singular). If a LSP does not admit ill-posedness examples in 1-D form, one can assume, without loss of generality, that |γ | = 1. If sequences (2.24), (2.25), with Re τ > 0,
Re ξ∞ > 0,
Re ξ < 0,
(2.26)
satisfy systems (2.15), (2.16) and the boundary conditions (2.17), then the solution √ U = Uk exp − k ,
√ F = Fk exp − k
is the ill-posedness example of Hadamard type for Problem Indeed, at t = 0 we have: ⎧ √ U(0) ⎪ ⎪ ∞ exp − k + kx1 Re ξ∞ −→ 0, ⎪ k→∞ ⎪ ⎪ ⎪ (0) √ ⎪ ⎪ ⎨ U∞ exp − k −→ 0, k→∞ |U|t =0| = √ (0) ⎪ ⎪ U exp − k −→ 0, ⎪ ⎪ k→∞ ⎪ ⎪ √ ⎪ ⎪ (0) ⎩ U exp − k + kx1 Re ξ −→ 0, k→∞
2.1 with special initial data.
x1 < 0; x1 → −0; x1 → +0; x1 > 0;
√ |Ft =0 | = F (0) exp − k −→ 0, k→∞
but, on the other hand, for t > 0: |U| −→ ∞, k→∞
|F | −→ ∞. k→∞
Thus, there is no the continuous dependence of solutions of Problem 2.1 on initial data in any reasonable norm, and, in that case, Problem 2.1 is ill-posed. Now, we will prove the proposition on the ill-posedness of underdetermined stability problems (see Definition 2.5). Such a proposition is almost evident, but it is important under the investigation of stability of strong discontinuities and useful for the demonstration of the procedure of the construction of Hadamard-type ill-posedness examples. P ROPOSITION 2.1 (Ill-posedness of underdetermined problems). If stability Problem 2.1 is underdetermined, then it is ill-posed.
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563
P ROOF. We will construct an ill-posedness example of Hadamard type in 1-D form, i.e., we look for exponential solutions in the form Uk =
% & (0) U∞ exp k(τ t + ξ∞ x1 ) , % & U(0) exp k(τ t + ξ x1 ) ,
x1 < 0;
(2.27)
x1 > 0;
Fk = F (0) exp(kτ t)
(2.28)
(0)
(k = 1, 2, 3, . . .), where U∞ , U(0) are constant vectors; F (0) , τ , ξ , ξ∞ constants. The proposition will be proved if we will establish the existence of solutions (2.27), (2.28) for Problem 2.1 with property (2.26). Substituting (2.27) into (2.15), (2.16), we obtain the linear algebraic systems (τ A0 + ξ A1 )U(0) = 0, (τ A0∞ + ξ∞ A1∞ )U(0) ∞ =0 (0)
for determining components of the vectors U(0) , U∞ . These systems have nontrivial solutions if det(τ A0 + ξ A1 ) = 0,
(2.29)
det(τ A0∞ + ξ∞ A1∞ ) = 0.
(2.30)
Considering (2.29), (2.30) as equations to determining ξ = ξ(τ ), ξ∞ = ξ∞ (τ ), we find the roots ξi = ξi (τ ) = −
τ −1 λ+ i (A0 A1 )
ξj ∞ = ξj ∞ (τ ) = −
,
i = 1, k+ ;
τ −1 λ− j (A0∞ A1∞ )
,
j = 1, k− ,
with the necessary property (2.26), i.e., Re τ > 0,
Re ξj ∞ > 0,
Re ξi < 0,
i = 1, k+ , j = 1, k− .
− −1 −1 Here λ+ i (λj ) are positive (negative) eigenvalues of the matrix A0 A1 (A0∞ A1∞ ); +
= n+ = n+ A−1 A ri+ , 1 0
k
i=1
−
= n− = n− A−1 A rj− ; 1∞ 0∞
k
j =1
−1 − −1 ri± (rj− ) is the multiplicity of the eigenvalue λ+ i (A0 A1 ) (λj (A0∞ A1∞ )).
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Finally, we find exponential solutions to systems (2.15), (2.16) in the form k− Uk =
% & (j ) j =1 U∞ exp k(τ t + ξj ∞ x1 ) , % & k+ (i) i=1 U exp k(τ t + ξi x1 ) ,
x1 < 0;
(2.31)
x1 > 0;
(j )
where U∞ , U(i) (j = 1, k− ; i = 1, k+ ) are constant vectors. − + − If λ+ i (λj ) is an eigenvalue with the multiplicity ri (rj ), then, in view of symmetry of the matrices Ai = τ A0 + ξi A1 and Aj ∞ = τ A0∞ + ξj ∞ A1∞ (j = 1, k− ; i = 1, k+ ), rang Ai = n − ri+ , rang Aj ∞ = n − rj− . Choosing some linearly independent equations of the systems Ai U(i) = 0,
(j )
Aj ∞ U∞ = 0, (j )
we obtain l equations for determining components of the vectors U∞ , U(i) (j = 1, k− ; i = 1, k+ ), with l=
k+ k−
(n − rj− ) = n(k+ + k− ) − n+ + n− . n − ri+ + i=1
j =1
Substituting (2.28), (2.31) into the boundary conditions (2.17) and taking into account l (j ) independent equations for U∞ , U(i) , we construct the linear algebraic system ZW = 0
(2.32)
to finding the vector ∗ ∗ ∗ ∗ (k− ) ∗ W = U(1) , U(1) , . . . , U(k+ ) , F (0) . ∞ , . . . , U∞ Here Z is a rectangular matrix of order (m + l) × (n(k+ + k− ) + 1). Since Problem 2.1 is underdetermined, m + l < n+ + n− + 1 + l = n(k+ + k− ) + 1. Thus, choosing some τ with Re τ > 0 and a nontrivial solution of system (2.32), we complete the construction of the ill-posedness example of Hadamard type. Therefore, the proposition is proved. In the generic case of constructing a 3-D ill-posedness example (γ = 0), we have to solve, on the first stage, the equations det(τ A0 + ξ A1 + iγ2 A2 + iγ3 A3 ) = 0, det(τ A0∞ + ξ∞ A1∞ + iγ2 A2∞ + iγ3 A3∞ ) = 0 to determining the roots ξ = ξ(τ ), ξ∞ = ξ∞ (τ ). Finally, we obtain a linear algebraic system in the form of system (2.32). If the strong discontinuity is evolutionary, i.e., stability Problem 2.3.1 is well formulated with respect to the number of boundary conditions
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−1 − (m = n+ (A−1 0 A1 ) + n (A0∞ A1∞ ) + 1), then the matrix Z is quadratic. Hence, an illposedness example of Hadamard type is constructed if the equation
det Z = D(τ ) = 0 for τ has a solution with Re τ > 0. The last equation is said, by physicists, to be dispersion relation.
2.5. Uniform linearized stability The idea of uniform linearized stability of strong discontinuities in fluid dynamics plays the key role in the structural (nonlinear) stability analysis. Actually, from uniform stability we can conclude, with a certain degree of strictness, structural (nonlinear) stability (see Definition 2.3), i.e., the real existence of a strong discontinuity as physical structure. Let us now give the rigorous definition of this important concept. To this end, it is necessary to introduce the notations of the LC and the ULC for Problem 2.1. Uniform Lopatinski condition. Following the classical work of Kreiss [79] (see also supplementing works of Ralston, Rauch, Majda and Osher [104,105,95,92,94] and the monograph [46]), we give the definition of the LC and the ULC for stability Problem 2.1. Applying the Fourier–Laplace transform to systems (2.15), (2.16) and the boundary conditions (2.17), we obtain the following boundary value problem for systems of ordinary differential equations: d U = M(s, ω) U, dx1
x1 > 0,
(2.33)
d U = M∞ (s, ω) U, x1 < 0, dx1 g(s, ω) = 0, x1 = 0. − A1 U +F
(2.34) (2.35)
Here U= U(x1 ) = (2π)−2 = (2π)−2 F
R3
R3
exp −st − i(ω, x ) U(t, x1 , x ) dt dx ,
exp −st − i(ω, x ) F (t, x ) dt dx
(= const)
are the Fourier–Laplace transforms of the vector function U(t, x) and the function F (t, x ); s = η + iξ,
η > 0, (ξ, ω) ∈ R3 , ω = (ω2 , ω3 ), |ω|2 = ω22 + ω32 ,
M = M(s, ω) = −A−1 1 (sA0 + iω2 A2 + iω3 A3 ), M∞ = M∞ (s, ω) = −A−1 1∞ (sA0∞ + iω2 A2∞ + iω3 A3∞ ),
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# , # + g = g(s, ω) = s P iωk P
A1 U x
k=2 1
= A1 U(x1 )|x1 →+0 − A1∞ U(x1 )|x1 →−0 . =0
In applying the Fourier–Laplace transform we, as usual, assume that U(t, x) ≡ 0,
F (t, x ) ≡ 0 for t 0.
That is, for simplicity of reasoning the initial data (2.18) are supposed to be homogeneous (U0 ≡ 0, F0 ≡ 0), and det A1 = 0, det A1∞ = 0, i.e., the boundary x1 = 0 is noncharacteristic. R EMARK 2.9. The case of IBVP’s for symmetric t-hyperbolic problems with a characteristic boundary was considered by Majda and Osher in [95]. Arguments from [95] under the determination of the LC and the ULC can be easily applied as well as for our IBVP with the nonstandard boundary conditions (2.17). The same can be said also for the case of nonhomogeneous initial data that was investigated by Rauch [105] (see also discussion below). R EMARK 2.10. In practice, the function F can be usually excluded from the boundary conditions (2.17) by means of cross differentiation. As a result, the boundary conditions (2.17) are reduced to ones including the derivatives Ut , Ux2 , Ux3 . After applying the Fourier–Laplace transform we obtain, in that case, boundary conditions for U with coefficients depending on s, ω. P ROPOSITION 2.2. For all ω ∈ R2 and η > 0, n+ (A1 ) eigenvalues λ of the matrix M lie in the left half-plane (Re λ < 0), and n− (A1 ) eigenvalues lie in the right half-plane (Re λ > 0). P ROOF. It is known that n+ (A1 ) eigenvalues of the matrix A−1 0 A1 , as well as of the matrix A1 , are positive, and n− (A1 ) eigenvalues are negative. This follows that the property of the eigenvalues λ, which is being proved, is valid for ω2,3 = 0 and η > 0. On the other hand, as system (2.15) is symmetric t-hyperbolic and det A1 = 0, then the assumption Re λ = 0 yields η = 0. Hence, the location of the eigenvalues λ relative to the imaginary axis of the complex λ-plane is independent of ω and, so, coincides with that for ω2,3 = 0. Observe that the discovered property of eigenvalues of the matrix M is a general fact for matrices in the form of M that was, in particular, proved (in other terms) by Gardner and Kruskal [62]. It is clear also that for all ω ∈ R2 and η > 0 the matrix M∞ has n+ (A1∞ ) eigenvalues in the left half-plane and n− (A1∞ ) eigenvalues in the right one. Thus, we can reduce the matrices M, M∞ to the forms + + M∞ M 0 0 −1 Λ , Λ−1 M=Λ M∞ = Λ∞ ∞, 0 M− 0 M− ∞
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where Λ = Λ(s, ω), Λ∞ = Λ∞ (s, ω) are nonsingular matrices; all the eigenvalues of the − quadratic matrix M+ of order n− (A1 ) and the quadratic matrix M+ ∞ of order n (A1∞ ) − − lie in the right half-plane; M , M∞ are quadratic matrices with eigenvalues in the left half-plane. We seek the bounded solution to problem (2.33)–(2.35) in the form
0 , x1 > 0, exp(M− x1 )C− exp(M+ ∞ x1 )C+ , x < 0, U(x1 ) = Λ∞ 1 0
U(x1 ) = Λ
(2.36)
where C− , C+ are constant vectors which are found from the system, cf. (2.35),
0 −A1 Λ C−
+ A1∞ Λ∞
C+ 0
= 0. + gF
The last system can be rewritten in the form of the following linear algebraic system to )∗ : determining the constant vector C = (C∗− , C∗+ , F L(η, ξ, ω)C = 0. Here L = L(η, ξ, ω) is a quadratic matrix of order n+ (A1 ) + n− (A1∞ ) + 1 (= m). If det L(η, ξ, ω) = 0 for some η > 0, (ξ, ω) ∈ R3 , then the sequence of the vector functions Uk and the functions Fk (k = 1, 2, 3, . . .), % √ & Uk (t, x) = exp − k + k ηt + iξ t + i(ω, x ) Λ
0 exp{kM− x1 }C−
for x1 > 0, % √ & Uk (t, x) = exp − k + k ηt + iξ t + i(ω, x ) Λ∞
exp{kM+ ∞ x1 }C+ 0
for x1 < 0, % √ & exp − k + k ηt + iξ t + i(ω, x ) , Fk (t, x ) = F which are the solutions of IBVP (2.15)–(2.17) with special initial data, is the ill-posedness example of Hadamard type. Thus, following [79], we give the definitions of the LC and the ULC. D EFINITION 2.7. Boundary conditions of stability Problem 2.1 satisfy the LC if det L(η, ξ, ω) = 0 for all η > 0, (ξ, ω) ∈ R3 .
(2.37)
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Here, the determinant of the matrix L is called Lopatinski determinant. D EFINITION 2.8. Boundary conditions of stability Problem 2.1 satisfy the ULC if requirement (2.37) is fulfilled for all η 0, (ξ, ω) ∈ R3 (η2 + ξ 2 + |ω|2 = 0). Thus, the violation of the LC means the ill-posedness of the LSP and the instability of a corresponding strong discontinuity. Below we will discuss conclusions that can follow from the fulfilment of the ULC and its violation. Concept of uniform stability. We are now in a position to give the rigorous definition of the notation of linearized uniform stability of strong discontinuities. D EFINITION 2.9. A strong discontinuity is uniformly stable if the boundary conditions of the corresponding LSP satisfy the ULC. Let us consider the domain of physically admissible parameters of the LSP. Actually, this domain is determined, in general, by (i) The hyperbolicity condition, (ii) The physical condition of entropy increase (see Remark 2.7), (iii) The evolutionarity condition, and, possibly, (iv) Some additional physical conditions (as, e.g., the positiveness of the pressure and temperature in gas dynamics). Where the hyperbolicity condition is usually equivalent to such natural physical assumptions as the positiveness of the density, sound velocity, and so on. It is clear that the whole U∗ )∗ ∈ R2n of the LSP consists of the domain of physically admissible parameters (# U∗∞ , # following subdomains: I. The domain, where the LC is violated (instability domain); II. The domain of fulfilment of the ULC (uniform stability domain); III. The domain of fulfilment of the general LC and omission of the ULC (neutral stability domain). And, we will call the union of domains II and III as weak stability domain. A priori estimates for the LSP. As was already mentioned in Section 1, a priori estimates without loss of smoothness (see below) can be obtained for solutions of the LSP in the uniform stability domain. Such estimates were first derived by Blokhin [13,14] for the LSP for gas dynamical shock waves. Observe that this LSP (see Section 3) is a stability problem in the form of Problem 2.1 with the property of characteristics described in Remark 2.6. Taking account of this fact, one can assume that there are no perturbations ahead the planar shock, and for perturbations behind the gas dynamical shock wave the a priori estimates without loss of smoothness [13,14] read (we refer also to Section 3 for more details): $ $ $U(t)$ 2 3 K1 U0 2 3 , W (R ) W (R )
(2.38)
F W 3 ((0,T )×R2 ) K2 ,
(2.39)
2
2
+
2
+
Stability of strong discontinuities in fluids and MHD
569
where the a priori estimate (2.38) of “layer-wise” type is valid for all times 0 < t T < ∞; K1 > 0 is a constant depending on T ; K2 a constant depending on T , F0 W 3 (R2 ) , and 2 U0 W 2 (R3 ) . Here standard (unweighted) Sobolev norms are used: 2
+
$ $ $U(t)$2 2 3 = W (R ) 2
+
2
α 2 D U dx, x
R3+ |α|=0
F 2W 3 ((0,T )×R2 ) = 2
∂ |α| α , ∂x1α1 ∂x2α2 ∂x3 3
(s+|β|) 2 3
∂ F
T R2
0
Dxα =
s+|β|=0
β β ∂t s ∂x2 1 ∂x3 2
dx dt;
α = (α1 , α2 , α3 ), β = (β1 , β2 ) are multiindices, |α| = α1 + α2 + α3 , etc. Observe that one has managed in obtaining a priori estimates without loss smoothness like (2.38), (2.28), likewise, for shock waves in relativistic gas dynamics [32] (see Section 4), superfluid helium [30,24], MHD (for fast shocks in a special case [37]; see Section 5), radiation hydrodynamics [38,44,7], and electrohydrodynamics [41,42] (see also estimates for the MHD contact discontinuity [28] in Section 6). All these estimates were obtained by the DIT (see below). By utilizing Kreiss’ symmetrizer techniques and (generalized) pseudodifferential calculus, Majda has extended results of [79] to the LSP for Lax k-shocks and proved the equivalence of the ULC and the existence of an a priori estimate without loss of smoothness for weighted Sobolev norms of solutions.16 For the LSP in the form of problem (2.19)– (2.21) (F = F(t, x) and g = g(t, x ) are supposed to be known vector-functions) such an estimate reads [92,94]: φ 2W 1
3 2,η (R+ )
+ V|x1 =0 2L
3 2,η (R+ )
+ η V 2L
4 2,η (R++ )
1 2 2 C F L (R4 ) + g|x1 =0 L (R3 ) , 2,η 2,η ++ + η
(2.40)
where C is a constant independent of η being a sufficiently large real number [92]; V 2L (R4 ) 2,η ++
=
R3+
0
φ 2W 1 (R3 ) + 2,η
∞ ∞
= 0
R2
exp(−2ηt)|V|2 dt dx,
R4++ = R+ × R3+ ,
exp(−2ηt) η2 φ 2 + φt2 + φx22 + φx23 dt dx ,
etc. Although, one should note that if the vector-functions F and g are given as in Remark 2.5, i.e., they depend on inhomogeneous initial data for Problem 2.1, then we actually have loss of smoothness in estimate (2.40). On the other hand, if initial data for 16 We also refer the reader to the work of Métivier [99] and Mokrane [101]. They have clarified some points in Majda’s method, by using Bony’s paradifferential calculus instead of Majda’s generalized pseudodifferential calculus.
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fluid perturbations are homogeneous, U0 ≡ 0, then (2.40) implies the following estimate without loss of smoothness for Problem 2.1: F 2W 1
3 2,η (R+ )
+ η U 2L
+ U|x1 →+0 2L 4 2,η (R++ )
3 2,η (R+ )
+ η U 2L
+ U|x1 →−0 2L
4 2,η (R+− )
3 2,η (R+ )
C F0 2W 1 (R2 ) . η 2
(2.41)
Concerning an a priori estimate with inhomogeneous initial data, U0 = 0, it seems it could be obtained by using Rauch’s arguments [105] extending Kreiss’ results [79] to the case of inhomogeneous initial data. Notice that Majda’s “weighted” a priori estimates like (2.41) are valid for the LSP for an arbitrary hyperbolic symmetrizable model of continuum mechanics satisfying the block structure conditions (see also discussion below), and for a strong discontinuity being Lax shock. On the other hand, unweighted “layer-wise” estimates like (2.38) which were obtained for some models of continuum mechanics are more preferable from the practical point of view, in particular, for a possible subsequent numerical analysis of the LSP. Namely, one can so build numerical models that they tolerate the construction of a difference analog of the dissipative energy integral used to obtaining an estimate like (2.38). The presence of such an analog gives the possibility to deducing an energy estimate that implies the stability of a suggested numerical model. Different aspects of such an approach are discussed in [19,22]. Passage to nonlinear stability. As was already noted in Section 1, with the help of the DIT Blokhin [15,17] has proved the theorem of local existence and uniqueness of the classical solution U(t, x) in a Sobolev space W2s and the function f (t, x ), cf. (2.12), in a Sobolev space W2s+1 to the quasilinear gas dynamics system ahead and behind the curvilinear shock wave (see Section 3 for details), where s = 3 as in the local well-posedness theorem for the Cauchy problem proved by Kato [76]. Initial data are supposed to lie in a physically admissible domain (see Section 3), and the ULC should hold in each point of the curvilinear shock. In [16,19] it is discussed how can generalize these results to hyperbolic systems of a so-called acoustic type. Majda’s theorem [93] on local existence of discontinuous shock fronts of a quasilinear s (s = 10 for 3-D) is, in some sense, system of conservation laws in a Sobolev space W2,η quite general. Although, the hyperbolic symmetrizable system of conservation laws in Majda’s theorem must satisfy some block structure conditions [92]. This block structure assumption is valid for strictly hyperbolic systems and, as was separately checked in [92], for the Euler equations of gas dynamics where strict hyperbolicity is violated. Furthermore, Majda’s block structure conditions have later been shown by Métivier [98] to be satisfied for a class of hyperbolic symmetrizable systems with constant multiplicities. This is a large class of systems of physical interest which contains, in particular, the gas dynamics equations, the Maxwell equations, the equations of linear elasticity, etc. At the same time, for example, for the MHD system the block structure conditions seem to require separate verification.
Stability of strong discontinuities in fluids and MHD
571
One should also note that Blokhin’s and Majda’s structural stability results are for kshocks. But, as is known, there are other evolutionary strong discontinuities (see discussion above). So, in generic case, we cannot deduce, with a full mathematical strictness, structural stability from uniform stability. But, on the other hand, in the uniform stability domain an ill-posedness example of Hadamard type cannot be constructed for the LSP and as well as for all close problems, which are obtained by perturbation of the system and the boundary conditions. Thus, with a certain degree of strictness we can say that the uniform stability domain is that of structural – nonlinear – stability. Neutral stability. Concerning the transitional case of neutral stability, recall that it admits the propagation of perturbations in form (1.1), (1.3) or in a more general form
U F
=
& % U0 (x1 ) exp i τ t + (γ , x ) F0
(γ = (γ2 , γ3 ); τ , γ2,3 are real constants), i.e., it is identified by the existence of surface waves. R EMARK 2.11. As is noted in the work of Benzoni-Gavage et al. [11], a boundary between the domains of instability and uniform stability generically belongs to the class of neutral stability as well, and for this case surface waves are of finite energy (Rayleigh waves), in the sense that, roughly speaking, the corresponding normal modes (1.1) are with Im ω = 0, Im k > 0, Im l = Im m = 0. Note that, e.g., for gas dynamical shock waves such a kind of neutral stability does not appear because the boundary between the domains of instability and uniform stability corresponds to the prohibited case with the Mach number M = 1 (see Remark 3.5 in Section 3). Majda has proved [92] that for obtaining a priori estimates (like “weighted” estimates above; see (2.40), (2.41)) for a LSP in the neutral stability domain it is necessary to require more smoothness for initial data than that of solutions. Likewise, the following a priori estimate with loss of smoothness was obtained by Blokhin [19] in the neutral stability domain (D’yakov’s domain of spontaneous sound radiation by the discontinuity) for the gas dynamical shock wave: $ $ $U(t)$ 2 3 K1 U0 3 3 , W (R ) W (R ) 2
+
2
+
0 < t T < ∞,
(2.42)
cf. (2.38). It should be noted that the presence of a priori estimates with loss of smoothness does not allow us, in generic case, to carry the well-posedness result obtained for the case of constant coefficients to the quasilinear one (see [15,17,19,92,93]). On the other hand, we observe that one can so perturb (generally speaking, by complexvalued small parameters) the linear systems and the boundary conditions in Problem 2.1 that an ill-posedness example of Hadamard type can be constructed for a perturbed stability problem in the neutral stability domain, whereas it cannot be constructed in the uniform stability domain (this follows from Definitions 2.7, 2.8, and we omit corresponding simple arguments). However, it should be noted that, as was proved by Benzoni-Gavage et al. [11],
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A. Blokhin and Y. Trakhinin
this is wrong for a real perturbation of the LSP, i.e., any real perturbation of the LSP in the interior of the neutral stability domain keeps its parameters in this domain.17 It is interesting that in the neutral stability domain one can also exhibit such a “real perturbation” of the LSP that a new “perturbed” IBVP admits the existence of ill-posedness examples. As an example, one can refer to a certain LSP [19] for gas dynamical shock waves that is distinguished by the presence of nondifferential terms for U in the system (behind the shock) and for F in the boundary conditions. This LSP was obtained by a “nonstandard” way and is, in fact, a variable coefficients stability problem with “frozen” coefficients that was obtained in [19] by the linearization of the quasilinear system of gas dynamics (written in a special curvilinear moving frame of reference; see Section 3) and the boundary conditions on a shock wave with respect to a steady solution. As was shown in [19], this problem is ill-posed in the neutral stability domain.18 So, in this sense, neutral stability can be found in practice as instability. In this connection, the results of Egorushkin [50] and Majda and Rosales [96,97,106] on the possibility of the instability and the breakdown of spontaneously radiating shock waves in gas dynamics and detonation waves (in the neutral stability domain) according to the so-called weakly nonlinear analysis [106] are quite natural. But, in general, in the neutral stability domain we cannot judge the existence of a strong discontinuity (as a physical structure) on the linearized level (and as well as on the weakly nonlinear level), and one has to analyze the initial FBSP, i.e., the initial quasilinear system (2.4) and the equations of strong discontinuity (2.13). The question on structural stability of neutrally stable strong discontinuities is an open problem that is a point of essentially nonlinear analysis (see also discussion in Section 9). At the same time, for hyperbolic IBVP’s with standard (Kreiss’ type [79,46]) boundary conditions there are first works in which the loss of derivatives phenomenon (as in (2.42)) has been overcome to prove existence theorems for the initial nonlinear setting when a kind of nonlinear illposedness does not appear. In this connection, we refer, for instance, to the existence theorem (in the whole domain of the fulfilment of the LC) proved by Sablé-Tougeron [111] for a certain IBVP in nonlinear elastodynamics.
2.6. Method of dissipative energy integrals Let us now describe main points of the DIT, that allows us to deduce uniform stability a priori estimates like (2.38), (2.39) for the LSP. The main idea of the energy method is very simple, and we explain it on the example of Problem 2.1. Let us multiply systems (2.15), (2.16) scalar-wise by the vector 2U. Because of the symmetry of the matrices Aα , Aα∞ we obtain the following identities (the energy integrals 17 Although, for boundaries of the neutral stability domain one can always find a real perturbation that moves
these boundaries to the domains of instability or uniform stability. 18 The presence of nondifferential terms for F in the boundary conditions plays the crucial role in the construction of an ill-posedness example.
Stability of strong discontinuities in fluids and MHD
573
in differential form):
∂ ∂ (A0 U, U) + (Ak U, U) = 0, ∂t ∂xk
(2.43)
∂ ∂ (A0∞ U, U) + (Ak∞ U, U) = 0. ∂t ∂xk
(2.44)
3
k=1
3
k=1
Integrating (2.43) over the domain R3+ and (2.44) over the domain R3− , and summing obtained equalities, we deduce the identity of energy integral d I (t) − dt
R2
(A1 U, U) x
1 =0
dx = 0.
(2.45)
Here I (t) =
R3+
(A1 U, U) x
(A0 U, U) dx +
1 =0
R3−
(A0∞ U, U) dx,
= (A1 U, U)|x1 →+0 − (A1∞ U, U)|x1 →−0 .
When deducing (2.45) we assume that |U|2 = (U, U) → 0 for |x| → ∞. D EFINITION 2.10. The boundary conditions of the LSP are dissipative if − (A1 U, U) x
1 =0
0
(2.46)
for any nonzero vector U satisfying the boundary conditions. Let the boundary conditions (2.17) be dissipative. Then, in view of (2.46), from (2.45) we deduce the inequality d I (t) 0 dt that has as a consequence the desired a priori estimate I (t) I (0),
t > 0.
(2.47)
Because of the positive definiteness of the matrices A0 , A0∞ the inequalities λmin (A0 )|U|2 (A0 U, U) λmax (A0 )|U|2 ,
(2.48)
λmin (A0∞ )|U|2 (A0∞ U, U) λmax (A0∞ )|U|2
(2.49)
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A. Blokhin and Y. Trakhinin
are fulfilled, where λmin , λmax are minimal and maximal eigenvalues of corresponding matrices. Therefore, with regard to (2.48), (2.49), we obtain the following variant of a priori estimate without loss of smoothness for the LSP: $ $ $U(t)$ L
3 2 (R+ )
$ $ + $U(t)$L
3 2 (R− )
&1/2 % K U0 L2 (R3 ) + U0 L2 (R3 ) , +
−
(2.50)
where K > 0 is a constant depending on coefficients of the matrices A0 and A0∞ . R EMARK 2.12. An a priori estimate for the function F (t, x ) is obtained due to a specific character of the boundary conditions. The a priori estimate (2.50) points to the L2 -well-posedness of the LSP19 and, with the estimate for the function F (t, x ) (in W21 ((0, T )×R2 )), denotes the uniform linearized stability of a corresponding strong discontinuity. Observe that having the a priori L2 estimate (2.50) obtained thanks to the dissipativity of boundary conditions, for the case of noncharacteristic discontinuities (the matrices A1 , A1∞ are nonsingular) it is not difficult to deduce analogous a priori estimates without loss of smoothness in W21 , W22 (see (2.38)), W23 , etc. (for the case of characteristic discontinuities, we refer, e.g., to MHD contact discontinuities; see Section 6). However, such a simple idea of the DIT to obtaining a priori estimates, generally speaking, does not suit for the most of LSP’s because condition (2.46) is usually not fulfilled. Of course, this does not denote the ill-posedness of a LSP. In such a case for the initial systems (2.15), (2.16) one can try to construct expanded systems (systems for the vector U and its derivatives Ut , Uxk , . . .) and dissipative boundary conditions for them. Boundary conditions for expanded systems can be obtained by differentiating the initial boundary conditions with respect to t, x2 , x3 . Equations of the initial systems (2.15), (2.16) for x1 = 0 can also be used in the capacity of boundary conditions. In the present chapter we consider such expanded systems for the LSP’s for shock waves in gas dynamics, relativistic gas dynamics, and MHD. 3. Stability of gas dynamical shock waves In this section we analyze the stability of gas dynamical shock waves. The main attention is concentrated on the linearized stability analysis and the issue of uniform stability. The issue of structural (nonlinear) stability is briefly discussed in the end of the section. 3.1. System of gas dynamics and Rankine–Hugoniot conditions Euler equations. Consider the system of gas dynamics, i.e., the well-known Euler equations (see, e.g., [85]) governing the motion of an ideal fluid: ρt + div(ρv) = 0,
(3.1)
19 The existence theorem for an IBVP for a symmetric t-hyperbolic system with dissipative boundary conditions was proved, for example, in [66].
Stability of strong discontinuities in fluids and MHD
(ρv)t + div(ρv ⊗ v) + ∇p = 0, ρ E + |v|2 /2 t + div ρv E + |v|2 /2 + pV = 0.
575
(3.2) (3.3)
Here ρ denotes the density and v = (v1 , v2 , v3 )∗ the velocity of the gas, E is the internal energy, p the pressure, V = 1/ρ the specific volume of the gas (the symbol ⊗ denotes Kronecker product). If we append the state equation E = E(ρ, S) to the gas dynamics system (3.1)–(3.3), and take account of the thermodynamical equalities20
∂E p=− ∂V
=ρ
2
S
∂E ∂ρ
,
T=
S
∂E ∂S
(3.4) ρ
following from the Gibbs relation T dS = dE + p dV ,
(3.5)
where S the entropy and T the temperature of the gas, then we can view (3.1)–(3.3) as a closed system to finding components of the vector U = (p, S, v∗ )∗ . Symmetrization. We add to system (3.1)–(3.3) one more additional conservation law (entropy conservation) (ρS)t + div(ρSv) = 0
(3.6)
which is valid on smooth solutions of system (3.1)–(3.3). Observe that it is the additional conservation law (3.6) that was used in [64] for the symmetrization of the gas dynamics equations (3.1)–(3.3). On the other hand, for the system of gas dynamics it is not necessary to use the Godunov symmetrization scheme, and we symmetrize system (3.1)–(3.3) if we just rewrite it in the nondivergent form 1 dp + div v = 0, ρc2 dt
dS = 0, dt
ρ
dv + ∇p = 0 dt
(3.7)
which is, in fact, a symmetric system for the vector U: A0 (U)Ut +
3
Ak (U)Uxk = 0.
(3.8)
k=1
Here d/dt = ∂/∂t + (v, ∇); c2 = (ρ 2 Eρ )ρ the square of the sound velocity; A0 = diag(1/(ρc2 ), 1, ρ, ρ, ρ) the diagonal matrix; Ak are symmetric matrices which can easily be written out. System (3.8) is symmetric t-hyperbolic if the following natural assumptions (hyperbolicity conditions) hold: ρ > 0,
c2 > 0.
(3.9)
20 For the sake of brevity, we will below write E instead of (∂E/∂ρ) , E instead of (∂E/∂S) , and so forth. ρ ρ S S
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A. Blokhin and Y. Trakhinin
In addition, we impose on the system of gas dynamics the physical restrictions p > 0,
T > 0.
(3.10)
R EMARK 3.1. For the case of a polytropic gas, i.e., when the state equation of the gas is E=
pV , γ −1
we have S p = c1 ρ exp , cV
c2 = γpV ,
γ
T=
pV ; cV (γ − 1)
with γ > 1 the adiabat index, c1 > 0 a constant, cV > 0 the unit heat capacity of the gas; and inequalities (3.9), (3.10) are reduced to ρ > 0. Rankine–Hugoniot conditions. Following the usual procedure (see Section 2.2), from the system of conservation laws (3.1)–(3.3) we deduce the well-known Rankine–Hugoniot conditions [j ] = 0, j [vN ] + [p] = 0, j E + |v|2 /2 + [pvN ] = 0
j [vτ ] = 0,
(3.11)
which hold on surface (2.12) of a strong discontinuity. Here j = ρ(vN − DN ) is the mass transfer flux across the discontinuity surface, vτ = (vτ1 , vτ2 )∗ ,
vN = (v, N), ∗
τ 1 = (fx2 , 1, 0) ,
vτ1,2 = (v, τ 1,2 ), ∗
τ 2 = (fx3 , 0, 1) ,
[vN ] = vN − vN∞ ,
vN∞ = (v∞ , N), etc. D EFINITION 3.1. If the fluid flows through a discontinuity, i.e., j = 0, such a strong discontinuity is called shock wave. Otherwise, j = 0, the strong discontinuity is called tangential if [vτ ] = 0, and contact if [vτ ] = 0. For the case of shock waves, the last condition in (3.12) can be equivalently rewritten in the form of the Hugoniot adiabat (see, e.g., [75,102,85]) [E] +
p + p∞ [V ] = 0. 2
Without loss of generality the Hugoniot adiabat can also be given by the relation p = H(V , p∞ , V∞ ).
(3.12)
Stability of strong discontinuities in fluids and MHD
577
Lax entropy conditions. Let a shock wave be planar and stationary. Then the k-shock conditions (see Definition 2.6 and (2.23)), with k = 1, read21 0 < v1 < c,
v1∞ > c∞ ,
(3.13)
2 = (ρ 2 E ) (ρ , S ). Recall that the k-shock conditions guarantee the evoluwhere c∞ ρ ρ ∞ ∞ tionarity of a strong discontinuity (see Definition 2.4). Besides, to describe a physically admissible domain we should assume that the entropy increases under the passage through the shock front:
(3.14)
S > S∞ .
One can show (see, e.g., [107,85]) that the k-shock inequalities (3.13), the entropy increase assumption (3.14), and the compressibility conditions ρ > ρ∞ ,
p > p∞
(3.15)
are equivalent to each other if Vpp =
∂ 2V ∂p2
> 0.
(3.16)
S
Thus, for the gas dynamical shock wave the k-shock inequalities (3.13) are indeed entropy conditions, in the physical sense (see Remark 2.7), and the shock is compressive. Notice that (3.16) is one of the Bethe conditions [12] for a so-called normal gas [107].22 Observe also that, in view of the first condition in (3.12) which for a planar stationary shock becomes ρv1 = ρ∞ v1∞ , (3.15) yields v1∞ > v1 .
(3.17)
Tangential and contact discontinuities. On the surface of a tangential discontinuity the pressure is continuous, [p] = 0, and the density and the tangent components of the velocity can have arbitrary jumps, [ρ] = 0, [vτ ] = 0. On the surface of a contact discontinuity the fluid velocity is continuous, [v] = 0, and the density (and other thermodynamical values, except the pressure) can have an arbitrary jump. In the framework of this chapter, we will not consider the issue of the stability of tangential and contact discontinuities. Actually, we will be concerned with this question in a more general case of MHD tangential and contact discontinuities (see Sections 6, 8). Concerning gas dynamical tangential discontinuities, Syrovatskij [115] (see also [85]) has proved that they are always unstable. Concerning the contact discontinuity, one can easily establish that it is uniformly stable. This directly 21 Without loss of generality we suppose that the fluid flows from the left to the right, i.e., the upstream velocity v1∞ > 0. Otherwise, k = 5. 22 Actually, condition (3.16), which expresses the decrease of the adiabatic compressibility V while the pressure p increases, is not thermodynamical (see discussion in [85]) but usually holds, for instance, for a polytropic gas (in this case it reads (γ + 1)V /(γ 2 p 2 ) > 0).
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A. Blokhin and Y. Trakhinin
follows from the dissipativity of the boundary conditions for the LSP (see Definition 2.10), and we refer to Section 6 for an analogous, and more general, situation taking place for the MHD contact discontinuity.
3.2. The LSP for gas dynamical shock waves Looking ahead, we observe that the LSP for gas dynamical shock waves being obtained after linearization have the property of symmetry along to the tangent directions x2,3 (see, e.g., [49,52,14,92,85]). Therefore, the Lopatinski determinant (see Definitions 2.7, 2.8) will depend only on the magnitude ω := |ω| (to be exact, on η, ξ , and ω), and the multidimensional stability is reduced to the 2-D stability. Let us below consider the gas dynamics system and the Rankine–Hugoniot conditions in two space dimensions (x = (x1 , x2 ), v = (v1 , v2 )∗ ). For 3-D stability we refer to the original papers of Blokhin and Majda [14,92]. Moreover, in the next section we analyze the 3-D stability of gas dynamical shock waves in the more general case of an ideal relativistic fluid.23 Discontinuous solution. Let us now linearize the gas dynamics system (3.1)–(3.3) and the Rankine–Hugoniot relations (3.12) (in 2-D). To this end, consider the piecewise constant solution U(t, x) =
∗ # S∞ , vˆ1∞ , vˆ2∞ , x1 < 0; U∞ = pˆ∞ , # ∗ # x1 > 0 U = p, ˆ # S, vˆ1 , vˆ2 ,
(3.18)
to system (3.1)–(3.3) which satisfies the jump conditions (3.12) on the stepshock x1 = 0 (the strong discontinuity is supposed to be a shock wave, i.e., j = 0): ρˆ vˆ1 = ρˆ∞ vˆ1∞ = jˆ, [vˆ2 ] = 0,
jˆ[vˆ1 ] + [p], ˆ
#, pˆ ∞ , V #∞ ). pˆ = H(V
(3.19)
Here ρ, ˆ # S, vˆ1,2 , ρˆ∞ , # S∞ , vˆ1∞,2∞ are constants; pˆ = ρˆ 2 Eρ ρ, ˆ # S , cˆ2 = 2ρE ˆ # S + ρˆ 2 Eρρ ρ, ˆ # S , ˆ ρ ρ, 2 pˆ ∞ = ρˆ∞ Eρ ρˆ∞ , # S∞ , 2 2 # = 1/ρ, V ˆ cˆ∞ = 2ρˆ∞ Eρ ρˆ∞ , # S∞ + ρˆ∞ Eρρ ρˆ∞ , # S∞ ,
#∞ = 1/ρˆ∞ ; V
jˆ = 0, i.e., vˆ1 = 0, vˆ1∞ = 0. With regard to the equality [vˆ2 ] = 0, we can choose, without loss of generality, a reference frame in which vˆ2 = vˆ2∞ = 0. 23 Actually, the 3-D construction of dissipative energy integral from Section 4 was first used in [14] for usual (nonrelativistic) gas dynamical shocks.
Stability of strong discontinuities in fluids and MHD
579
Acoustic system. Linearizing system (3.8) (in 2-D) about the uniform steady solution (3.18) in the half-space x1 > 0, we obtain the well-known acoustic system. In a dimensionless form (see below) it reads A0 Ut + A1 Ux1 + A2 Ux2 = 0,
(3.20)
where A0 = diag(1, 1, M 2 , M 2 ) is the diagonal matrix (A0 > 0); ⎞ ⎛ ⎛ ⎞ 1 0 1 0 0 0 0 1 0 ⎟ ⎜0 1 0 ⎜0 0 0 0⎟ A2 = ⎝ A1 = ⎝ ⎠, ⎠; 1 0 M2 0 0 0 0 0 0 0 0 M2 1 0 0 0 U = (p, S, v∗ )∗ the vector of perturbations, M = vˆ1 /cˆ the Mach number behind the shock (M∞ = vˆ1∞ /cˆ∞ the Mach number ahead the shock). Here we use scaled values: x = x/lˆ ˆ p = p/(ρˆ cˆ2 ), S = S/# (lˆ the characteristic length), t = t vˆ1 /l, S, v = v/vˆ1 (the primes in (3.20) were removed). One has an analogous acoustic system ahead the planar shock, in the half-space x1 < 0, but in view of the evolutionarity conditions (3.13) which now read M∞ > 1, 0 < M < 1, all the characteristic modes for this system are incoming (see Remark 2.6). So, without loss of generality one can assume that there no perturbations ahead the shock wave: U ≡ 0 for x1 < 0. Taking this into account and linearizing likewise the Rankine–Hugoniot conditions (3.12) (see also (3.12)), we obtain the following LSP (in a dimensionless form) for gas dynamical shock waves. P ROBLEM 3.1 (LSP for gas dynamical shock waves). We seek the solution of system (3.20) for t > 0, x ∈ R2+ satisfying the boundary conditions v1 + d0 p = 0,
S = d1 p,
Ft = d2 p,
v2 = d3 Fx2
(3.21)
at x1 = 0 (t > 0, x2 ∈ R) and the initial data U(0, x) = U0 (x),
x ∈ R2+ ,
F (0, x2 ) = F0 (x2 ),
x2 ∈ R
(3.22)
for t = 0. Here d0 =
1+a , 2M 2
R=
vˆ1∞ ρˆ = , ρˆ∞ vˆ1
d1 = 1 −
a , M2
d2 =
(1 − a)R , 2M 2 (1 − R)
d3 = R − 1,
#, pˆ∞ , V #∞ −1 . a = −jˆ2 HV V
ˆ # S). By eliminating Moreover, we supposed, without loss of generality, that # S = ρˆ cˆ2 /pS (ρ, the function F (t, x2 ) from the boundary conditions (3.21), we rewrite them as v1 + d0 p = 0, with a0 = d2 d3 .
S = d1 p,
(v2 )t = a0 px2 ,
(3.23)
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R EMARK 3.2. For a polytropic gas (see Remark 3.1) a=
2γ M 2 − γ + 1 1 = , 2 M∞ 2 + (γ − 1)M 2
d0 =
(γ − 1)(1 − M 2 )2 , M 2 (2 + (γ − 1)M 2 )
d2 = −
d1 =
3 − γ + (3γ − 1)M 2 , 2M 2 (2 + (γ − 1)M 2 ) γ +1 , 4M 2
d3 =
2(1 − M 2 ) , (γ + 1)M 2
and the domain of physically admissible parameters, which is determined by inequalities (3.9), (3.10) and (3.13)/(3.14)/(3.15), is the following:
γ −1 < M < 1. 2γ
(3.24)
Here the inequality M 2 > (γ − 1)/(2γ ) expresses the positiveness of the pressure ahead the shock: pˆ∞ > 0. R EMARK 3.3. The acoustic system (3.20) for the case of 1-D perturbations, U = U(t, x1 ), has one incoming Riemann invariant, r1 , and other invariants, r2,3,4 , are outgoing, where p = M(r1 + r2 ), v1 = r2 − r1 , v2 = r3 , S = r4 . It is easily verified that outgoing Riemann invariants cannot be expressed through the incoming one at the boundary x1 = 0 if 1 + d0M = 0, i.e., a = −1 − 2M. That is, in this case Majda’s conditions (see Remark 2.8) are violated, and gas dynamical shock waves are 1-D unstable.
3.3. Fourier–Laplace analysis Let us now find for Problem 3.1 the domains of instability and uniform stability. The domain of instability for gas dynamical shock waves was found in 1954 by D’yakov in the work [49], and a little bit later by Erpenbeck [52]. In the same work D’yakov have first described also the domains of uniform and neutral stability.24 In the mentioned works conditions for instability and uniform stability were derived by normal modes analysis (see Section 1). Perform this in terms of the LC and the ULC (see Definitions 2.7 and 2.8) by writing and analyzing the Lopatinski determinant. For gas dynamical shocks, one can derive the LC and the ULC by direct calculations following arguments just before Definitions 2.7 and 2.8. In fact, this was done, in other terms, by D’yakov. But, for further utility, let us obtain an equivalent form (definition) for the LC and the ULC. Such an equivalent definition can be given for the special case when a LSP, like Problem 3.1, has the property that only one characteristic mode of the linear symmetric hyperbolic system is incoming, and the others are outgoing. System (3.20) has just the same property because, in view of 0 < M < 1, only one eigenvalue of the matrix A−1 0 A1 is negative (unique incoming mode). Using this equivalent definition is rather convenient for gas dynamical shocks and 24 Actually, there was some technical mistake in [49] (this was noted by Iordanskii [72]), and in [77] Kontorovich have obtained correct bounds for these domains.
Stability of strong discontinuities in fluids and MHD
581
especially useful and plays the crucial role in the stability analysis for MHD shock waves (see Section 5). Equivalent form for the LC and the ULC. Consider a general LSP in the form of Problem 2.1. Let system (2.16) has not outgoing characteristic modes, i.e., n− (A−1 0∞ A1∞ ) = 0 (see Remark 2.6), and without of generality U ≡ 0 for x1 < 0. Let, likewise, behind the discontinuity n− (A−1 0 A1 ) = 1 (unique incoming mode). Then, applying the Fourier–Laplace transform to (2.15), (2.17) gives the boundary value problem for system (2.33) with the boundary conditions U = 0, M0 (s, ω)
x1 = 0,
(3.25)
and taking into account that which are obtained from (2.35) by eliminating the constant F U(x1 )|x1 →−0 = 0. Following ideas of Gardner and Kruskal from their work [62] devoting to the stability analysis for planar MHD shocks, we derive a formula for the solution to problem (2.33), (3.25) by means of the Laplace transform (instead of representation (2.36)). This formula is 8 1 (sA0 + λA1 + iω2 A2 + iω3 A3 )−1 A1 U0 exp(λx1 ) dλ, (3.26) U(x1 ) = 2πi C where C is a contour large enough to enclose all the singularities of the integrand; U0 is a constant vector satisfying the boundary conditions (3.25): M0 U0 = M 0 U(0) = 0. Note that the singularities of the integrand are the eigenvalues λ of the matrix M and thus satisfy the equation det(sA0 + λA1 + iω2 A2 + iω3 A3 ) = 0.
(3.27)
It follows from (3.26) that U(x1 ) is a sum of residues at the poles of the integrand. Since only one (!) eigenvalue of the matrix A−1 0 A1 is negative, with regard to Proposition 2.2, there is one eigenvalue λ with Re λ > 0, i.e., for this λ: exp(λx1 ) → +∞ as x1 → +∞. Hence the residue at this value of λ must be zero. One can show (see [62]) that this is the same as the statement that for given ω ∈ R 2 there exist complex numbers s and λ, with Re s = η > 0, Re λ > 0, such that the homogeneous system (sA0 + λA1 + iω2 A2 + iω3 A3 )X = 0,
(3.28)
X∗ A 1 U0 = 0
(3.29)
has a nonzero solution X. Recall that these values of s, λ, and ω must satisfy Equation (3.27). Since λ with Re λ > 0 is a simple eigenvalue, we can choose n − 1 linearly independent equations from system (3.28). Adding Equation (3.29) to these equations, we obtain for X a linear algebraic system, and if its determinant (Lopatinski determinant) g(M0 , η, ξ, ω, λ) = 0, then the sequence of the vector functions % √ & Uk (t, x) = exp − k + k(ηt + iξ t + iω2 x2 + iω3 x3 ) U(x1 )
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(k = 1, 2, 3, . . .) is the Hadamard-type example of ill-posedness for the stability problem with special initial data. We can thus formulate the announced equivalent definitions. D EFINITION 3.2. The stability problem satisfies the LC if g(M0 , η, ξ, ω, λ) = 0 for all η > 0, (ξ, ω) ∈ R 3 , and λ being a solution of (3.27) with Re λ > 0. Let λ = λ(η, ξ, ω) with Re λ > 0 for η > 0 be a solution of (3.27), and λ0 = λ(0, ξ, ω). D EFINITION 3.3. The stability problem satisfies the ULC if g(M0 , η, ξ, ω, λ) = 0 for all η 0, (ξ, ω) ∈ R 3 (η2 + ξ 2 + |ω|2 = 0), and λ being a solution of (3.27) with Re λ 0 and λ|η=0 = λ0 . R EMARK 3.4. Observe that, generally speaking, Re λ0 0, but the case Re λ0 > 0 (for corresponding outgoing modes Re λ|η=0 < 0) corresponds to the boundary between the domains of uniform stability and instability (this was rigorously proved by BenzoniGavage et al. [11]; see Remark 2.11). However, this boundary (as well as that between the domains of instability and neutral stability) is directly found by testing the LC. So, to locate the boundary between the domains of uniform and neutral stability we should analyze the case Re λ0 = 0. Taking account of above arguments, it is clear that the following is true. L EMMA 3.1. If a LSP in the form of Problem 2.1 have the property that there are no outgoing characteristic modes ahead the discontinuity (n− (A−1 0∞ A1∞ ) = 0), and behind the discontinuity only one characteristic mode is incoming (n− (A−1 0 A1 ) = 1), then Definition 3.2 is equivalent to Definition 2.7, and Definition 3.3 to Definition 2.8. Lemma 3.1 will be used to proving stability theorems for gas dynamical shock waves (see just below and Section 4 for relativistic shocks) and for fast MHD shock waves in a special case of parallel shocks (see Section 5). Domains of instability and uniform stability. Let us now, following D’yakov’s notations in his pioneering (in multidimensional stability) work [49], and using also well-posedness results of Blokhin [13,14], formulate the main linearized stability theorem for gas dynamical shock waves for an arbitrary state equation. T HEOREM 3.1. The gas dynamical shock wave in the domains a>1
or a −1 − 2M,
RM 2 − b 2 < a < 1, RM 2 + b 2 −1 − 2M < a
RM 2 − b2 , RM 2 + b2
(3.30) (3.31) (3.32)
Stability of strong discontinuities in fluids and MHD
583
√ with b = 1 − M 2 , is, respectively, strongly unstable, uniformly stable, and neutrally stable.25 In domain (3.31) of uniform stability the a priori estimates (2.38), (2.39) without loss of smoothness hold for solutions of the LSP. In domain (3.32) of neutral stability the a priori estimate (2.42) with loss of smoothness is valid. R EMARK 3.5. Observe that a = 1 (otherwise M = 1 that is impossible), i.e., the boundary between the domains of instability and uniform stability does not belong to the domain of physically admissible parameters, and, thus, surface waves of finite energy (see Remark 2.11) do not appear. R EMARK 3.6. As was proved by Majda [92,94], we have at once a “weighted” a priori estimate like (2.41) in the uniform stability domain, and an analogous “weighted” estimate with loss of smoothness in the neutral stability domain. P ROOF. We will derive the 2-D variant (see [14] for 3-D) of the a priori estimates (2.38), (2.39) while will prove the forthcoming theorem devoted to this subject (see Theorem 3.2 below). Let us, using Definitions 3.2 and 3.3 (see Lemma 3.1), obtain the conditions for instability and uniform stability formulated in the theorem. Consider, without loss of generality, the 2-D variant of the LSP (see discussion above), i.e., Problem 3.1. So, we suppose in above arguments concerning the Fourier–Laplace analysis that ω := ω2 and ω3 = 0. For the acoustic system (3.20) Equation (3.27) is written explicitly as follows: M 2 Ω 2 M 2 Ω 2 − λ2 + ω2 = 0,
(3.33)
where Ω = s + λ. Suppose that a = −1 − 2M, i.e., 1-D instability does not appear (see Remark 3.3). In this connection, one can assume that ω = 0 (otherwise, one could construct an ill-posedness example in 1-D form; see Remark 2.8) or, without loss of generality, ω > 0. Moreover, without loss of generality we suppose that in Equation (3.33) ω = 1. Since η > 0,
Re λ > 0,
(3.34)
the second factor (the expression in parentheses) in Equation (3.33) must be equal to zero. This can be written as ζ 2 + θ 2 = 1,
(3.35)
with ζ = λ/(MΩ), θ = −i/(MΩ). Equation (3.35) has two roots θ1 and θ2 = −θ1 , and let us choose the root θ = θ1 given by √ z2 − 1 , θ = θ1 = − z 25 The correct bound (RM 2 − b2 )/(RM 2 + b2 ) was found first by Kontorovich [77].
(3.36)
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A. Blokhin and Y. Trakhinin
with z = 1/ζ . It follows from (3.34) that Im(1/θ ) > 0,
Im(ζ /θ ) > 0.
By (3.36), the last inequalities imply that the domain of z is the right half (Re z > 0) of the z-plane with the segment from 1 to +∞ removed. Observe that conditions (3.34) does not hold for θ = θ2 . From system (3.28) we obtain the relations p(1) = −Mθ
z2 (1) v , 1 − z2 2
(1)
v1 = −
θ z (1) v , −1 2
S (1) = 0
z2
(3.37)
for components of the vector X = (p(1) , S (1) , v1 , v2 )∗ . With regard to the boundary conditions (3.23), the components of the vector U0 = (p(0) , S (0) , v1(0) , v2(0) )∗ are connected by the relations (1)
v1(0) = −d0 p(0) ,
v2(0) = −a0
(1)
Mz θp(0), z−M
S (0) = d1 p(0) .
By substituting these relations into (3.29) and applying (3.37), one gets θ zb4 h(z)p(0) = 0, (z2 − 1)(z − M)
(3.38)
and since p(0) must not be equal to zero, (3.38) becomes the equality26 h(z) = 1 − R − M 2 R z2 + RMz + (R − 1)M 2 = 0,
(3.39)
where R=
M 2 (R − 1) 2(a − M 2 ) , = #p a−1 ρˆ E
#p = Ep (ρ, E ˆ p) ˆ =−
# T . #, # E V S (V S)
Suppose, as in [52,62], the thermal coefficient to be positive, Ep > 0.27 Note that the domains of instability and uniform stability were found by D’yakov [49] without this assumption. But, let us keep it for simplicity of arguments. It is easily verified that h(M) > 0. On the other hand, h(1) = (1 − M)F , and, therefore, if F = 1 + M − R < 0, 26 Exactly this equality was obtained by Gardner and Kruskal [62] for fast parallel MHD shock waves (see also Section 5). 27 Actually, it is one of the Bethe conditions [12] mentioned above.
Stability of strong discontinuities in fluids and MHD
585
then Equation (3.39) admits a real root z so that M < z < 1. Recall that if a = −1 − 2M, gas dynamical shocks are 1-D unstable (see Remark 3.3). But a = −1 −2M implies F = 0. Hence, in the domain F 0
(3.40)
the LC is violated, and the gas dynamical shock wave is unstable. The instability condition (3.40) is in the form of that given by Erpenbeck [52], and one can verify that it is totally equivalent to condition (3.30) found by D’yakov [49]. At the same time, it is easy to show that in the domain F > 0 the roots of Equation (3.39) are, in accordance with the sign of the coefficient of z2 in (3.39), either with negative real parts or 1. That is, the inequality F > 0 (or −1 − 2M < a < 1) presents the domain of weak stability, where the LC is fulfilled. To separate the uniform stability domain, in which the ULC holds, one should find a subdomain of the domain F > 0 where h(z) = 0 as well as for such z which correspond to the case η = 0, λ = λ0 (Re λ0 = 0; see Remark 3.4). It is clear that only the root z = z1 (z1 > 1) of Equation (3.39) can correspond √ to the case η = 0, λ = λ0 = iδ, δ ∈ R. In the domain z >√ 1 the function δ = δ(z) = 1/ z2 − 1 decreases, and the function ξ = ξ(z) = (z − M)/(M z2 − 1) decreases up to its minimum 1/M and increases for z > 1/M. Solving (3.33), we find λ = λ1,2 =
M 2s ±
√ M 2s 2 + 1 − M 2 . 1 − M2
It is easy to see that λ0 = λ1 |η=0 = iδ. Then we have δ=
M 2ξ +
M 2ξ 2 + M 2 − 1 . 1 − M2
(3.41)
The graph of the function ξ = ξ(z) has two points of intersection z = r1,2 (1 < r1 < 1/M < r2 ) with the line ξ = ξˆ = const for z > 1, z = 1/M.28 One of these points of intersection corresponds to the case η = 0, λ = λ0 = iδ, with δ determined by formula (3.41). By (3.41), δ (ξ ) > 0. On the other hand, δ (ξ ) = δ (z)/ξ (z). Since δ (z) < 0 for z > 1, the interval z > 1/M, which contains the point of intersection z = r2 (ξ (r2 ) > 0), determines a part of the uniform stability domain. The other part is determined by roots z of Equation (3.39) with negative real parts. Both roots have such a property if the coefficient of z2 in (3.39) is nonpositive, i.e., F M(1 + MR).
(3.42)
At the same time, omitting detailed calculations, we find that Equation (3.39) in the domain F > 0 has a real root z lying to the right of 1/M (the other root is negative) if M(1 + MR − M) < F < M(1 + MR). 28 The constant ξˆ is supposed to lie in the range of values of the function ξ = ξ(z) for z > 1.
(3.43)
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A. Blokhin and Y. Trakhinin
By combining inequalities (3.42) and (3.43), we thus obtain the uniform stability condition (the ULC holds) F > M(1 + MR − M).
(3.44)
Finally, we conclude that inequalities (3.40), (3.44), and 0 < F M(1 + MR − M)
(3.45)
present, respectively, the domains of instability, uniform stability, and neutral stability for the gas dynamical shock wave. Notice also that inequalities (3.44) and (3.45) can be equivalently rewritten as, respectively, (3.31) and (3.32). This completes the proof of Theorem 3.1. C OROLLARY 3.1. Gas dynamical shock waves in a polytropic gas are always uniformly stable. P ROOF. For a polytropic gas (see Remarks 3.1, 3.2) domain (3.42) is empty, but inequality (3.43) is always satisfied. More exactly, in this case z1 = M +
M2 +
2 , γ −1
and, in view of (3.24), z1 > 1/M.
R EMARK 3.7. As was shown in the works of D’yakov [49], Iordanskii [72], and Kontorovich [77], in the neutral stability domain (3.32)/(3.45) the shock front radiates sound waves. In other words (see [19]), Problem 3.1 in the neutral stability domain admits the propagation of perturbations in the form of plane waves near the planar shock x1 = 0 with a supersonic speed (the propagation speed of a plane wave can be greater than that of sound (scaled) 1/M). R EMARK 3.8. By simple manipulations, from the acoustic system (3.20), Lp + div v = 0,
LS = 0,
M 2 Lv + ∇p = 0,
we obtain that the pressure perturbation p satisfies the equation M 2 L2 p − p = 0
(3.46)
which is, in fact, the wave equation 2 L1 − L22 − L23 p = 0,
(3.47)
Stability of strong discontinuities in fluids and MHD
587
with the new differential operators L1,2,3 determined as L1 =
M ∂ , b2 ∂t
L2 =
∂ M2 ∂ − 2 , ∂x1 b ∂t
L3 =
1 ∂ . b ∂x2
Here L = ∂/∂t + ∂/∂x1 , = ∂ 2 /∂x12 + ∂ 2 /∂x22 . Moreover, one can obtain a boundary condition for Equation (3.46). If we apply the vector differential operator (M 2 ∂/∂t, 0, −∂/∂t, 0)∗ to system (3.20) and consider the obtained expression at x1 = 0, by making use of (3.23), we get the boundary condition M 2 (1 + d0 )pt t − b2 pt x1 + M 2 a0 px2 x2 = 0,
x1 = 0.
(3.48)
Note that conditions (3.30)–(3.32) (or (3.40), (3.44), (3.45)) for instability, uniform stability, and neutral stability can be likewise found by the Fourier–Laplace analysis of problem (3.46), (3.48), as was done by Blokhin [13,19] and Majda [92]. Problem (3.46), (3.48) for the wave equation plays the key role in the analysis of Blokhin [13,14,19] by the DIT (see just below), and the uniform stability condition appearing in this analysis is equivalently written as (cf. (3.31), (3.44)) a1 = b2 d0 + a0 M 2 > 0.
a0 < 0,
(3.49)
3.4. Construction of the dissipative energy integral for the LSP for gas dynamical shock waves Let us now derive a priori estimates without loss of smoothness for the LSP for gas dynamical shock waves in the domain of uniform stability, i.e., for the case when the state equation satisfies condition (3.31). Without loss of generality we consider, as above, 2-D perturbations, i.e., Problem 3.1, and refer to [14], where the 3-D estimates (2.38), (2.39) were obtained (see also Section 4 for 3-D estimates for relativistic shocks). T HEOREM 3.2. In the uniform stability domain (3.31) Problem 3.1 is well-posed, and its solutions satisfy the a priori estimates $ $ $U(t)$ 2 2 K1 U0 2 2 , W (R ) W (R )
(3.50)
F W 3 ((0,T )×R) K2 ,
(3.51)
2
+
2
2
+
with 0 < t T < ∞; K1 > 0 a constant depending on T ; K2 a constant depending on T , F0 W 3 (R) , and U0 W 2 (R2 ) . 2
2
+
P ROOF. The main idea of the proof is based on the DIT (see Section 2). More exactly, from the acoustic system (3.20) we will construct such an expanded system that boundary conditions for it will be dissipative (see Definition 2.10). The process of construction of this system consists of two stages.
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A. Blokhin and Y. Trakhinin
At the first stage, we just expand system (3.20) up to second-order derivatives of the vector U: A0p (Up )t + A1p(Up )x1 + A2p (Up )x2 = 0,
(3.52)
with Aαp = I10 ⊗ Aα (A0p > 0; here and below Id is the unit matrix of order d), ∗ Up = U∗ , U∗t , U∗x1 , U∗x2 , V∗ ,
∗ V = U∗t t , U∗t x1 , U∗t x2 , U∗x1 x1 , U∗x1 x2 , U∗x2 x2 .
Writing out for system (3.52) the energy integral in differential form (see Section 2), and integrating it over the domain R2+ , we obtain the energy identity d I0 (t) − dt
R
(A1p Up , Up )|x1 =0 dx2 = 0,
(3.53)
where I0 (t) =
R2+
(A0p Up , Up ) dx.
When deducing (3.53) we assume that |Up |2 = (Up , Up ) → 0 as |x| → ∞. With regard to the boundary conditions (3.23) and system (3.20) for x1 = 0, we estimate the boundary integral in (3.53) that gives the inequality d I0 (t) − C1 dt
R
2 p + v22 + pt2 + px21 + px22 + P x
1 =0
dx2 0,
(3.54)
with C1 > 0 a certain constant, P = pt2t + pt2x1 + pt2x2 + px21 x1 + px21 x2 + px22 x2 . By the property of the trace of a function in W21 (R2+ ) at the line x1 = 0 (see [100]), we reduce inequality (3.54) to the form d I0 (t) − C1 dt
R
P|x1 =0 dx2 C2 I0 (t),
(3.55)
where C2 > 0 is a constant. We now proceed to the second, more complicated, stage consisting in the construction of the expanded system. In the first place, using the symmetrization of the wave equation suggested by Gordienko [69], one can rewrite the wave operator (L21 − L22 − L23 ) in (3.47) as a vector symmetric one. Namely, if the function p satisfies Equation (3.47), then the ∇ = (L1 , L2 , L3 )∗ , satisfies vector W = (Y∗1 , Y∗2 , Y∗3 )∗ , with Yi = Li Y, i = 1, 3, Y = ∇p, the system (B0 L1 − B1 L2 − B2 L3 )W = 0.
(3.56)
Stability of strong discontinuities in fluids and MHD
589
Here B0 = B2 =
K L L K M −iN
M iN K
M iN K
K L ; M
−iN −M L
B1 =
,
L K −iN
K L M
iN M , −L
K, L, M, N are as yet arbitrary Hermitian matrices of order 3. Moreover, the matrices B0 , B1 , and B2 can be represented in the form 0 −1 ∗ ∗ B0 = T {I2 ⊗H}T , ⊗H T , B1 = T −1 0 (3.57) −1 0 ∗ B2 = T ⊗H T , 0 1 with ⎛
1 0 1 ⎜ 0 −1 T =√ ⎝ 2 0 −1 1 0
⎞ −1 0 ⎟ ⎠ ⊗I3 , 0 1
H=
K−M −L + iN
−L − iN K+M
.
By returning in (3.56) to the usual differential operators τ , ξ1,2 , one gets the system 1 DWt − B1 Wx1 − B2 Wx2 = 0, b with D = (M/b 2 )(B0 + MB1 ), and, by virtue of (3.57), M ∗ 1 −M D = 2T ⊗H T . −M 1 b
(3.58)
(3.59)
Observe that D > 0 if H > 0. Let us now obtain boundary conditions for system (3.58). By making use of (3.48) and Equation (3.47) for x1 = 0, we take the conditions L1 (L1 p) − L2 (L2 p) − L3 (L3 p) = 0, L3 (L2 p) − L2 (L3 p) = 0, L1 (L2 p) − Md0 L2 (L2 p) −
M a1 L3 (L3 p) = 0 b2
as boundary ones at x1 = 0 for system (3.58). They can be written in the matrix form AY1 + BY2 + CY3 = 0,
(3.60)
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A. Blokhin and Y. Trakhinin
with A=
0 0 , 0
1 α 0 0 0 1
⎛
B=
−α 0 0
−1 0 −Md0
⎞ −1 0 ⎟; ⎠ Ma1 − 2 b
0 0 ⎜0 1 C=⎝ 0 0
α > 1 is a certain constant. Let ΛI Λ= = T W, ΛII
ΛI =
Λ1 Λ2
0 −1 , 0
,
ΛII =
Λ3 Λ4
;
Λk (k = 1, 4) are 3-D vectors. Since √ 2 (Λ1 + Λ4 ), Y1 = 2
√ √ Y2 = − 2Λ2 = − 2Λ3 ,
√ 2 (Λ4 − Λ1 ), Y3 = 2
the vector boundary condition (3.60) can also be written as ΛI = GΛII ,
(3.61)
with G=
−G2 0
G1 I3
,
G1 = 2(A − C)−1 B, G2 = (A − C)−1 (A + C).
Assuming that |W|→0 as |x| → ∞, one obtains for system (3.58) the identity d I1 (t) + dt
R
(B1 W, W)|x1 =0 dx2 = 0,
(3.62)
with I1 (t) =
R2+
(DW, W) dx > 0.
By (3.57), (3.61), (B1 W, W)|x1 =0 = (G0 ΛII , ΛII )|x1 =0 , where −G0 = G∗ H + HG.
(3.63)
Recall that the Hermitian matrices K, L, M, N are, as yet, absolutely arbitrary. We now determine them with some arbitrariness that will be used in future. Let all the eigenvalues
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591
of the matrix G lie strictly in the left semi-plane, Re λj (G) < 0, j = 1, 6. In our case, it is easily verified that the latter is valid (!) if the uniform stability condition (3.49) (or (3.31)) holds. Consider (3.63) as the Lyapunov matrix equation (see [8]) to finding the matrix H appearing in formulae (3.57). As is known, Equation (3.63) has the unique solution (see [8]) H1 H2 > 0, H1 = H1∗ , H3 = H3∗ H= H2∗ H3 for any real symmetric positive definite matrix G0 . Therewith, the matrix H is likewise real and symmetric, and the matrices K, L, M, and N read 1 K = (H1 + H3 ), 2 1 L = − H2 + H2∗ , 2
1 M = (H3 − H1 ), 2 1 iN = H2∗ − H2 . 2
Moreover, since H > 0, then D > 0, cf. (3.59). In other words, thanks to an appropriate choice of the matrices K, L, M, N , one can suppose G0 to be an arbitrary real symmetric positive definite matrix, and D > 0. That is, in (3.62) I1 (t) > 0 and (B1 W, W)|x1 =0 > 0. Moreover, since √ 2 −Y2 ΛII = , 2 Y1 + Y3 then 2 (B1 W, W)|x1 =0 > C3 L21 p + (L1 L2 p)2 + (L1 L3 p)2 2 2 + L22 p + (L2 L3 p)2 + L23 p x =0 > C4 P|x1 =0 , (3.64) 1
where C3,4 = C3,4 (G0 ) > 0 are constants depending on the norm of the matrix G0 , and (3.62) yields the inequality d I1 (t) + C4 dt
R
P|x1 =0 dx2 0.
(3.65)
By an appropriate choice of the matrix G0 (i.e., of the matrices K, L, M, and N ), we achieve that C4 − C1 > 0. Then, adding inequalities (3.55) and (3.65), one gets d I (t) C2 I (t) dt that has as a consequence the desired a priori estimate I (t) exp(C2 t)I (0),
0 < t T < ∞,
(3.66)
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with I (t) = I0 (t) + I1 (t). Actually, the a priori estimate (3.66) shows that Problem 3.1 is well-posed, and, in view of that I (t) > 0, it can be rewritten as (3.50). Finally, let us deduce an a priori estimate for the function F . Using (3.55) and (3.65) and taking into account that C4 − C1 > 0, I (t) > 0, one can obtain the inequality T R
0
P|x1 =0 dx2 dt C5 ,
(3.67)
where C5 > 0 is a constant depending on T (below Ci , i = 6, 7, . . . , are constants). By making use of the property of the trace of a function in W21 (R2+ ) at the line x1 = 0, we have the inequality 2 p + v22 + pt2 + px21 + px22 x =0 dx2 C6 I (t) 1
R
that implies the estimate T
R1
0
p2 + v22 + pt2 + px21 + px22 x
1 =0
dx2 dt C7 .
(3.68)
Combining (3.67) and (3.68) and using the boundary conditions (3.21) gives T 0
R
(Ft )2 + (Fx2 )2 + (Ft t )2 + (Ft x2 )2
+ · · · + (Fx2 x2 x2 )2 x
1 =0
dx2 dt C8 .
(3.69)
To close estimate (3.69) and get by this (3.51) we should deduce an estimate for the function F (without its derivatives). To this end, multiplying the second boundary condition in (3.21) by 2F , integrating with respect to x2 ∈ R, and making use of the Hölder inequality, one gets $ $ $ d$ $F (t)$2 C9 $F (t)$L (R) p|x1 =0 L2 (R) . L (R) 2 2 dt The last inequality, if we use the property of the trace of a function in W21 (R2+ ) at the line x1 = 0, is rewritten as $ $ $ d$ $F (t)$ C10 $p(t)$W 1 (R2 ) . L (R) 2 + 2 dt
(3.70)
With regard to (3.50), inequality (3.70) leads us to the estimate F L2 ((0,T )×R) C11 which with (3.69) gives the desired a priori estimate (3.51) for the function F . This completes the proof of Theorem 3.2.
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593
R EMARK 3.9. Strictly speaking, to conclude the W22 -well-posedness of the LSP, Problem 3.1, we must prove more the existence of its solutions. For this purpose, in [19] it is shown the equivalence of Problem 3.1 to problem (3.46), (3.48) (with corresponding initial data for the function p). At the same time, problem (3.46), (3.48) is rewritten in the form of the IBVP for the symmetric t-hyperbolic system (3.56) with the dissipative boundary conditions (3.60). As is known, for linear symmetric t-hyperbolic systems with dissipative boundary conditions the theorem of existence of weak solutions was proved by Lax and Phillips [89] and of sufficiently smooth solutions by Godunov [66]. R EMARK 3.10. Recall that we were assuming, without loss of generality, that there are no perturbations ahead the shock wave: U ≡ 0 for x1 < 0. Actually, this natural assumption can be removed, and it is easy to obtain an a priori W22 -estimate (like the L2 -estimate (2.50)) including perturbations ahead the shock wave. Indeed, since M∞ > 1, for the acoustic system ahead of the stepshock the matrix A1∞ (cf. (2.16)) is positive definite, and the corresponding quadratic form (A1p∞ Up∞ , Up∞ )|x1 =0 > 0 (cf. (3.52), (3.53)). Using, with slight modifications, the scheme of the proof of Theorem 3.2 gives the desired a priori estimate for perturbations behind and ahead the planar gas dynamical shock wave.
3.5. Structural stability of uniformly stable gas dynamical shocks Let us now briefly discuss the passage to nonlinear stability. As we already noted, this was done for gas dynamical shocks by Blokhin [15,17] and Majda [93,94] (by different techniques). Below we will write out the local W23 -well-posedness theorem for gas 10 dynamical shock waves proved by Blokhin. We just observe that Majda’s local W2,η existence theorem [93] has an analogous form. Reduced structural stability problem. Following the works of Blokhin [15,17], one can reduce the FBSP (see Section 2) for gas dynamical shock waves to an IBVP for the quasilinear gas dynamics system written in a special curvilinear moving frame of reference in which the shock wave coincides with one of the coordinate curves. In this case, boundary conditions on a free unknown boundary are reduced to ones on a stationary plane. Consider the system of gas dynamics in 2-D.29 Let us introduce the curvilinear coordinates ξ = (ξ1 , ξ2 ) as follows30: dx1 + i dx2 = (dξ1 + µ1 dt) exp(f + iϕ) + i(dξ2 + µ2 dt) exp(g + iψ),
(3.71)
where f, g, ϕ, ψ are some functions of ξ and t; µ1 , µ2 the contravariant components of the velocity of the moving reference frame (the functions ϕ and ψ are, in fact, the inclinations of the tangents to the coordinate curves ξ2 = const and ξ1 = const to the 29 The introduction of a special curvilinear moving frame of reference for the 3-D gas dynamics equations is discussed in [19]. 30 Formula (3.71) was suggested by Godunov.
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axes x2 = 0, x1 = 0; see [15,17] for more details). The equations of gas dynamics in the moving coordinates ξ were written in a conservative form by Vinokur [121], and for a nonconservative (nondivergent) form utilized by Blokhin in the mentioned works [15,17] we refer to the monograph of Godunov et al. [68]. Let us choose a curvilinear reference frame so that the coordinate curve ξ2 = 0 coincides with the shock wave. Let, without loss of generality, ξ2 > 0 be the state ahead the shock, and let the upstream be uniform, stationary,31 and parallel to the axis x1 , i.e., 2 = p (ρˆ , # S∞ = const (cˆ∞ ρ = ρˆ∞ = const > 0, p = pˆ ∞ = const > 0, S = # ρ ∞ S∞ ) > 0) v1 = uˆ ∞ = const, v2 = 0 under ξ2 > 0. Suppose also the coordinate curves ξ1 = const to be straight lines. The integrability conditions for relation (3.71) leads us to a certain system of partial differential equations (see [15]) to finding the functions f, g, ϕ, ψ, µ1 , µ2 . It has a nonstationary solution, with (see [15] for more details) g = 0,
ψ = ϕ0 (ξ1 ) = ϕ(t, ξ )|t =0,ξ2=0 ,
µ1 = 0,
µ2 (t, ξ ) = µ(t, ξ1 ).
Taking account of this solution and omitting all the details (see [15,17]), we reduce the FBSP for the quasilinear system (3.1)–(3.3) (in 2-D) with the Rankine–Hugoniot conditions (3.12) (see also (3.12)) on a free boundary x1 = f (t, x2 ) to the following IBVP with boundary conditions on the line ξ2 = 0. P ROBLEM 3.2 (Nonlinear stability problem). We seek the solution of the system B0 Ut + B1 Uξ1 + B2 Uξ2 + F = 0,
(3.72)
for (t, ξ ) ∈ ω ⊂ {(t, ξ ) | t > 0, ξ1 ∈ R, ξ2 < 0} satisfying the boundary conditions u1 = −uˆ ∞ sin ϕ0 + ∆G(u3 )/ef ,
u2 = uˆ ∞ cos ϕ0 − Fξ1 G(u3 )/ef ,
#∞ ), u3 = H(V , pˆ ∞ , V Ft + Fξ1
uˆ ∞ cos ϕ0 − ∆
(3.73) ef G
1 (u3 ) + uˆ ∞ sin ϕ0 = 0 ∆
at ξ2 = 0 ((t, ξ1 ) ∈ Sb ⊂ {(t, ξ1 ) | t > 0, ξ1 ∈ R}) and the initial data U(0, ξ ) = U0 (ξ ),
F (0, ξ1 ) = 0,
(3.74)
for t = 0, ξ ∈ Ω(0) ⊂ R2− = {ξ | ξ2 < 0, ξ1 ∈ R}. 31 Actually, in the proof of the mentioned local well-posedness theorem [15,17] (see below) this restriction can
be easily removed by arguments like linearized ones in Remark 3.10.
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595
Here B0 = diag(ρ, ρ, 1/(ρc2 ), 1) is the diagonal matrix; ⎛ ρu2
0
0
ρu2 ∆ 1 ∆
1 ∆ u2 ρc2 ∆
0
0
∆
⎜ ⎜ 0 B1 = ⎜ ⎜ 0 ⎝ 0
0
⎟ 0 ⎟ ⎟, 0 ⎟ ⎠
⎜ ⎜ 0 B2 = ⎜ ⎜ ⎝ 1 0
1
ρu0 −
Fξ1 ∆
Fξ1 ∆ u0 ρc2
−
0
0
0
⎞
⎟ 0⎟ ⎟, ⎟ 0⎠ u0
∗
U = (u1 , u2 , u3 , u4 ) ,
F = (F1 , F2 , F3 , 0) ,
u1 = v2 cos ϕ0 − v1 sin ϕ0 , u4 = S,
0
ρu0
u2 ∆
∗
u3 = p,
⎛
⎞
u2 = v1 cos ϕ0 + v2 sin ϕ0 ,
u0 = u1 − u2 Fξ1 /∆ − Ft ;
∆ = exp(f + g) cos(ϕ − ψ) = exp(f) cos(ϕ − ϕ0 ) = exp(f0 )(1 − Kb ξ2 ) − ϕ0 F is the Jacobian of the transform D(x1 , x2 )/D(ξ1 , ξ2 ); Kb = ϕ0 / exp(f0 ) the curvature of the coordinate curve ξ2 = 0; f0 = f|t =0,ξ2=0 , ϕ0 = dϕ0 /dξ1 ,
t
F = F (t, ξ1 ) =
µ(s, ξ1 ) ds, 0
F3 = −
ϕ0 u1 ∆
,
G(u3 ) =
F1 =
ϕ0 ρu22 , ∆
#∞ − V ), (u3 − pˆ ∞ )(V
F2 = −
ϕ0 ρu1 u2 , ∆
G1 (u3 ) =
#∞ G(u3 )V #∞ − V V
(see [15,17] for more details). The natural physical domain of the thermodynamical values p, ρ, S, T is that described by conditions (3.9), (3.10). In that case, the quasilinear system (3.72) is symmetric t-hyperbolic (in the sense of Friedrichs; see Definition 2.1) because the matrices B0 = B0 (U), B1 = B1 (U, F, ξ ), B2 = B2 (U, F, Ft , Fξ1 , ξ ) are symmetric and B0 > 0. We are interested in classical solutions of Problem 3.2 in the domain ω: U(t, ξ ) ∈ C 1 (ω), ¯
F (t, ξ1 ) ∈ C 2 (S¯b ).
The domain ω is a bounded domain adjoining to the plane ξ2 = 0. Its boundary ∂ω consists of the following pieces: Ω(0) the piece of the plain t = 0; Ω(T ) the piece of the plain t = T ; Sb the piece of the plain ξ2 = 0; Slat the lateral area which is exactly determined (by usual procedure; see, e.g., [66]) while constructing a priori estimates to proving the local well-posedness theorem [15,17] for Problem 3.2 (Ω(t) is the (bounded) cross-section by the plane t = const; σ (t) the cross-section of Sb by the plane t = const. Local well-posedness theorem. Let us now write out the local well-posedness theorem (without proof) for the nonlinear stability Problem 3.2. This theorem has been proved by Blokhin [15,17] and can be rewritten in the form a local W23 -well-posedness theorem 10 -existence theorem [93]) for the corresponding FBSP. Observe (like Majda’s local W2,η that the restriction U ≡ const under ξ2 > 0 is technical and can be removed. In some sense, the analogous technical restriction that U ≡ const, ahead the shock, for |x| > const was
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A. Blokhin and Y. Trakhinin
imposed by Majda [93]. In the theorem below we remove also some additional technical restrictions which were superimposed by Blokhin in the works [15,17]. T HEOREM 3.3. Let the initial data U0 (ξ ) belong to W23 (Ω(0)) and are admissible, in the sense that they satisfy not only corresponding compatibility conditions but also (1) The hyperbolicity and physical conditions (3.9), (3.10) and, in each point of the curvilinear shock wave (i.e., at ξ2 = 0), (2) The Lax entropy conditions32 which, with regard to the above notations in Problem 3.2, have the form (cf. (3.13)) uˆ ∞ sin ϕ0 > cˆ∞ ,
0 < −u1 < c,
(3) The uniform stability condition which now reads (cf. (3.31)) (u1 )2 ρ/(ρˆ∞ c2 ) − b 2 #∞ −1 < 1, < j 2 HV V , pˆ∞ , V (u1 )2 ρ/(ρˆ∞ c2 ) + b 2 where j = ρˆ∞ uˆ ∞ sin ϕ0 = −ρu1 , b2 = 1 − u21 /c2 . Then for a sufficiently short-time, T , in the domain ω¯ Problem 3.2 has the unique classical solution U, F which satisfies the estimate $2 %$ & max $U(t)$W 3 (Ω(t )) + F 2W 3 (σ (t )) + U 2W 3 (ω) + F 2W 4 (S ) < K < ∞.
0t T
2
2
2
2
b
Main idea of the proof. The proof of Theorem 3.3 is based on a straightforward adaptation of the DIT applied in linear theory in the proof of Theorem 3.2, and we refer the reader to [15,17] for more details. Notice that from system (3.72) one can deduce a quasilinear analog of the wave equation (3.47) which plays the crucial role in the construction of an expanded system with dissipative boundary conditions. The estimate obtained by the DIT yields uniqueness, and the short-time existence of classical solutions is proved by adapting the standard proof of short-time existence for smooth solutions of the Cauchy problem (see [114,100,76,122]). R EMARK 3.11. It should be noted that Blokhin’s nonlinear analysis [15,17] based on the DIT is highly technical and more technical than Majda’s one [93] based on pseudodifferential calculus (or the nonlinear analysis of Métivier [99] using paradifferential calculus). In particular, the use of such a powerful mathematical instrument as pseudo- or paradifferential calculus is preferable to establish such general theoretical facts as Majda’s existence theorem for shock fronts [93]. On the other hand, the a priori estimate in Theorem 3.3 is like the estimate for the Cauchy problem in Kato’s local existence theorem [76]. It is also written in uniform Sobolev norms in time and requires the same 10 regularity (W23 for U) as Kato’s estimates and less regularity than Majda’s estimates (W2,η 32 See Definition 2.6. We suppose the Bethe condition (3.16) to be fulfilled, i.e., the Lax conditions guarantee the fulfilment of the physical entropy condition (3.14). Otherwise, the initial data must satisfy more the condition of S∞ . entropy increase, S0 (ξ1 , 0) > #
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597
for 3-D). Actually, it is very important for practice, especially, for numerical analysis. The issue of construction of adequate calculation models for quasilinear hyperbolic systems is a rather difficult problem (especially, for IBVP’s). Some first ideas of the use of the DIT in this direction can be found, for example, in the work of Blokhin and Sokovikov [33]. On the linearized level, different aspects of the approach to numerical analysis based on the DIT are discussed in [19,22]. In particular, in [22] Blokhin writes out a certain difference-differential model for the LSP for gas dynamical shock waves (see Problem 3.1). The energy estimate establishing the stability of this calculation model is deduced by constructing a difference analog of the dissipative energy integral used to obtaining the a priori estimate (2.38). 4. Stability of shock waves in relativistic gas dynamics The issue of the stability of relativistic shock waves is of great importance in connection with various applications in astrophysics, cosmology, plasma physics, etc. The linearized stability analysis for relativistic shocks was first carried in 1958 by Kontorovich [78]. By the normal modes analysis, he has extended the results of D’yakov’s pioneering work [49] to relativistic gas dynamics and found the domains of instability, uniform stability, and neutral stability for relativistic shock waves. Note that this was later rediscovered by Anile and Russo [5,110], and they have also obtained the conditions for instability and weak stability by a different techniques based on the concept of so-called corrugation stability [4].33 Concerning the passage to structural (nonlinear) stability, the necessary step for its realizing was performed in the work of Blokhin and Mishchenko [32] where they have deduced a priori estimates without loss of smoothness in the form of (2.38), (2.39) for the LSP for relativistic shock waves. Then, the local well-posedness theorem, like Theorem 3.3 in gas dynamics, could be proved by a straightforward adaptation of Blokhin’s techniques [15,17] applied to nonrelativistic shocks. In this section, the main attention is given to obtaining the a priori estimates (2.38), (2.39) for uniformly stable relativistic shock waves. 4.1. Relativistic gas dynamics equations and Rankine–Hugoniot conditions Relativistic gas dynamics. The field equations governing the motion of an ideal relativistic fluid (see, e.g., [85,3]) can be written in the conservative form (ρΓ )t + div(ρu) = 0,
(4.1)
(ρhΓ u)t + div(ρhu ⊗ u) + ∇p = 0, ρhΓ 2 − p t + div(ρhΓ u) = 0,
(4.2) (4.3)
33 This concept dates back to Whitham’s geometric shock dynamics [123] and, in short, corrugation stability denotes that, by perturbing the planar shock front, the shock velocity decreases where the front is expanding and increases where the front is converging. It is still an open problem to find the rigorous relationship between corrugation stability and weak linearized stability. But, it seems the condition for corrugation stability must coincide with that for linearized weak stability. At least, this is true for shock waves in classical and relativistic gas dynamics and for fast transverse MHD shocks (see [4,6]).
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where ρ and p are the rest frame density and pressure, (u0 , u) the unit 4-velocity, oriented towards the future, u0 = Γ = (1 − |v|2 )−1/2 the Lorentz factor, v = (v 1 , v 2 , v 3 )∗ the velocity of the gas, u = (u1 , u2 , u3 )∗ = Γ v, Γ 2 = 1 + |u|2 ; h = 1 + E + pV , V = 1/ρ; E(ρ, S) and S are rest frame internal energy and entropy. In our consideration the speed of light is equal to unity (|v| < 1). As in classical gas dynamics, taking account of the thermodynamical equalities (3.4) following from the Gibbs relation (3.5), we can view the relativistic gas dynamics system (4.1)–(4.3) as a closed one to finding components of the vector U = (p, S, u∗ )∗ . Symmetrization. The system of relativistic gas dynamics was symmetrized by Blokhin and Mishchenko [31] (see also [32]) by the usual scheme of symmetrization suggested by Godunov (see (2.7)–(2.9) etc., with Ψjk = 0), where the additional conservation law is that of entropy conservation: (ρΓ S)t + div(ρSu) = 0. Following [31,32], system (4.1)–(4.3) is written in the symmetric form (2.10), with the following canonical variables and productive functions: u∗ Γ ∗ h Q = S − ,− , , T T T
L = −p
Γ , T
M k = −p
uk . T
The matrices Aα are symmetric matrices presented in [31,32]. At the same time, the relativistic gas dynamics equations of the canonical form (2.10) are rewritten in terms of the initial vector U as the symmetric system A0 (U)Ut +
3
Ak (U)Uxk = 0.
(4.4)
k=1
The symmetric matrices Aα = Aα (U) for the special case u2 = u3 = 0 become the form ⎛
Γ ρc2
⎜ 0 ⎜ A0 = ⎜ v 1 ⎝ 0 0 ⎛
u1 ρc2
⎜ 0 ⎜ A1 = ⎜ ⎜ 1 ⎝ 0 0
0 1 0 0 0 0 v1 0 0 0
v1 0
0 0 0 ρhΓ 0
hρ Γ
0 0
0 0 0 0 ρhΓ
1
0
0
0 0 ρhu1 0
hρv 1 Γ
0 0
⎞
⎛
⎟ ⎟ ⎟, ⎠
0
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0
⎞ 0 0⎟ ⎟ 0⎟, ⎠ 0 0
0 ⎜0 ⎜ A3 = ⎜ 0 ⎝ 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 ⎜0 ⎜ A2 = ⎜ 0 ⎝ 1 0 ⎞
0 ⎟ ⎟ 0 ⎟ ⎟, ⎠ 0 ρhu1
⎛
⎞ 1 0⎟ ⎟ 0⎟; ⎠ 0 0
Stability of strong discontinuities in fluids and MHD
599
c2 = (ρ 2 Eρ )ρ . Suppose conditions (3.9) hold with the natural assumption that the square of the relativistic speed of sound cs2 = c2 / h is positive and less than that of light (relativistic causality): 0 < cs2 < 1.
(4.5)
Then inequalities (3.9), (4.5) guarantee A0 > 0 and are, in fact, the hyperbolicity conditions for the symmetric system (4.4). Likewise, suppose the physical restrictions (3.10) to be satisfied. Relativistic Rankine–Hugoniot conditions. By the usual procedure described in Section 2, we write out the jump conditions [j ] = 0,
j [huN ] + [p] = 0,
j [huτ ] = 0,
j [hΓ ] + DN [p] = 0
(4.6)
that hold on surface (2.12) of a strong discontinuity for piecewise smooth solutions of system (4.1)–(4.3)/(4.4). Here j = ρΓ (vN − DN ), uN = (u, N), uτ = (uτ1 , uτ2 )∗ , uτ1,2 = (u, τ 1,2 ); the vectors N, τ 1,2 are described above, etc. For the case of shock waves, j = 0, the jump relations (4.6) imply the equation of Taub adiabat [117] (the relativistic analog of the Hugoniot adiabat for classical gas dynamics) 2 h = [p](hV + h∞ V∞ ) that, with regard to (3.4), can be written in form (3.12). Lax entropy conditions. Let a relativistic shock wave be planar and stationary. Then the k-shock conditions (see Definition 2.6), with k = 1, have the form 0 < v1 < cs ,
v1∞ > cs∞ ,
(4.7)
2 = c 2 / h , c 2 = (ρ 2 E ) (ρ , S ). Observe that the k-shock inequaliwhere cs∞ ∞ ρ ρ ∞ ∞ ∞ ∞ ties (4.7), the physical entropy condition (3.14), and the compressibility conditions (3.15) are equivalent to each other under the assumption [118] (the relativistic analog of the Bethe condition (3.16) for classical gas dynamics)
∂ 2 (hV ) ∂p2
> 0.
(4.8)
S
4.2. The LSP for relativistic gas dynamical shock waves Discontinuous solution. Consider a piecewise constant solution to system (4.1)–(4.3), ∗ # S∞ , uˆ 1∞ , uˆ 2∞ , uˆ 3∞ , x1 < 0; U∞ = pˆ ∞ , # U(t, x) = (4.9) ∗ # x1 > 0, U = p, ˆ # S, uˆ 1 , uˆ 2 , uˆ 3 ,
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satisfying, on the planar shock wave (j = 0) with the equation x1 = 0, the jump conditions (4.6): 1 2 ρˆ hˆ uˆ + pˆ = 0, 3 3 uˆ /Γ# = vˆ = 0.
ρˆ uˆ 1 = ρˆ∞ uˆ 1∞ = jˆ, 2 2 uˆ /Γ# = vˆ = 0,
hˆ Γ# = 0,
(4.10) (4.11)
Here ρ, ˆ # S, uˆ k , ρˆ∞ , # S∞ , uˆ k∞ are constants; ˆ pˆ = ρˆ 2 Eρ ρ, ˆ # S , cˆ2 = 2ρE ˆ # S + ρˆ 2 Eρρ ρ, ˆ # S , cˆ2s = cˆ2 /h, ˆ ρ ρ, 2 #, # = 1/ρ, ˆ # S + pˆ V V ˆ pˆ ∞ = ρˆ∞ Eρ ρˆ∞ , # S∞ , hˆ = 1 + Eρ ρ, etc. Suppose also the fulfilment of the evolutionarity conditions, cf. (4.7), 0 < vˆ 1 < cˆs < 1,
1 0 < cˆs∞ < vˆ∞ < 1.
(4.12)
Because of (4.11) we can choose, without loss of generality, a reference frame in which the unperturbed flow behind and ahead the shock is parallel to the axis x1 : uˆ 2 = uˆ 2∞ = uˆ 3 = uˆ 3∞ = 0. Linearization. As in classical gas dynamics, if we linearize system (4.4) about the uniform steady solution (4.9) in the half-space x1 < 0, we obtain a linear system for which all the characteristic modes, by virtue of (4.12), are incoming (see Remark 2.6), and without of generality one can assume that there no perturbations ahead the shock wave: U ≡ 0 for x1 < 0. Then, linearizing system (4.4) and the jump conditions (4.6) about solution (4.9) in the half-space x1 > 0, we obtain the following LSP for shock waves in relativistic gas dynamics. P ROBLEM 4.1 (LSP for relativistic gas dynamical shock waves). We seek the solution of the system A 0 Ut +
3
A k Ux k = 0
(4.13)
k=1
for t > 0, x ∈ R3+ satisfying the boundary conditions u1 + d0 p = 0, uk = d3 Fxk
S = d1 p,
Ft = d2 p,
(4.14)
(k = 2, 3)
at x1 = 0 (t > 0, x2 ∈ R2 ) and the initial data U(0, x) = U0 (x), for t = 0.
x ∈ R3+ ,
F (0, x ) = F0 (x ),
x ∈ R2
(4.15)
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601
Here Aα = Aα (# U) (A0 > 0), the matrices Aα (U) are described above (see system (4.4)); Γ#(b1 − b2 ) , ρˆ hˆ vˆ 1 b1 d3 = −Γ# vˆ 1 , d0 =
d1 =
b2 [vˆ 1 ] , #vˆ 1 b1 b2 ρˆ T
d2 =
1 b2 vˆ∞
ρˆ hˆ Γ#2 b1 b2 vˆ 1 [vˆ 1 ]
,
ˆ # S) vˆ 1 [vˆ 1 ] 2 EV S (ρ, cˆs + − , #ρˆ b2 T 2 1 b2 = 1 − vˆ 1 vˆ∞ . b2 = cˆs2 − vˆ 1 ,
b1 = 2cˆs2
1 < 0, cf. (3.17). By eliminating the function F , the boundary Note that [vˆ 1 ] = vˆ 1 − vˆ∞ conditions (4.14) become k S = d1 p, u t = a0 pxk (k = 2, 3), (4.16) u1 + d0 p = 0,
with a0 = d2 d3 . R EMARK 4.1. Simple manipulations with system (4.13) gives the wave equation L21 p − L22 p − px2 x2 − px3 x3 = 0
(4.17)
for the function p = p/(ρˆ hˆ Γ#), with the new differential operators L1 =
1 ∂ , # Γ b ∂t
L2 =
Γ#b ∂ uˆ 1 (1 − cˆs2 ) ∂ . − cˆs ∂x1 cˆs b ∂t
As in classical gas dynamics (see Remark 3.8), one can also obtain a boundary condition for Equation (4.17). It reads cˆ2 a1 L21 + a2 L22 − s1 L1 L2 p = 0, vˆ
x1 = 0,
(4.18)
where a1 =
cˆs Γ#b
2 a0 +
cˆs2 d0 , uˆ 1 Γ#
a2 = −
cˆs Γ#b
2 a0 .
Following [19], one can show the equivalence of Problem 4.1 to problem (4.17), (4.18) (with corresponding initial data for the function p). 4.3. Instability/uniform stability conditions Following directly the scheme of the proof of Theorem 3.1, one can deduce the conditions of instability, uniform stability, and neutral stability for relativistic shock waves that
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were obtained first by Kontorovich [78]. Here we just present these conditions (without derivation) in the form obtained by Anile and Russo [5,110] (like Erpenbeck’s form [52], see also (3.40), (3.44), (3.45)). We can also refer the reader to the work of Trakhinin [119] in which the instability/uniform stability conditions for relativistic shocks are rediscovered (following the scheme of the proof of Theorem 3.1) while investigating the linearized stability of relativistic MHD shock waves. T HEOREM 4.1. The gas dynamical relativistic shock wave in the domains F 0,
(4.19)
[vˆ 1 ] vˆ 1 1 − 1 − vˆ 1 vˆ∞ , cˆs cˆs
(4.20)
[vˆ 1 ] vˆ 1 1 1 1 − vˆ vˆ∞ − , 0
(4.21)
F>
with 1 2 vˆ 1 (vˆ 1 )2 vˆ∞ vˆ 1 [vˆ 1 ] F = 1 − vˆ 1 + − + , #p cˆs2 cˆs cˆs ρˆ E
#p = Ep (ρ, E ˆ p) ˆ =−
T# , #, # E V S (V S)
is, respectively, strongly unstable, uniformly stable, and neutrally stable.34 C OROLLARY 4.1. Relativistic gas dynamical shock waves in a polytropic gas are always uniformly stable. P ROOF. Anile and Russo [110] have deduced a sufficient condition for the uniform stability of relativistic shocks that totally coincides with that obtained by Fowles [54] for nonrelativistic gas dynamical shock waves. Namely, if 0<1+
1 ρˆ cˆ2 , #p pˆ ρˆ E
(4.22)
then condition (4.20) hold, i.e., the relativistic shock wave is uniformly stable. It is easily verified that for a polytropic gas (see Remark 3.1) inequalities (4.22) are fulfilled. R EMARK 4.2. The linearized stability/instability conditions (4.19)–(4.21) can also be obtained by the normal mode analysis of problem (4.17), (4.18) (see [32]). As in classical 34 As for nonrelativistic shocks (see Remark 3.3), it is easily verified that for F = 0 relativistic shock waves are 1-D unstable.
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603
gas dynamics, the IBVP for the wave equation (see (4.17), (4.18)) plays the key role in the DIT analysis of Blokhin and Mishchenko [32] for relativistic shock waves (see just below), and in this analysis, the uniform stability condition (4.20) equivalently reads a1 > 0,
a2 > 0.
(4.23)
4.4. Construction of the dissipative energy integral for the LSP for relativistic shock waves For Problem 4.1, a priori estimates without loss of smoothness in the form of (2.38), (2.39) can be derived by applying the construction of the dissipative energy integral used by Blokhin in [14] to obtaining the a priori estimates (2.38), (2.39) for the 3-D LSP for shock waves in classical gas dynamics. Let us now describe this construction on the example of relativistic shock waves. T HEOREM 4.2. In the uniform stability domain (4.20) Problem 4.1 is well-posed, and its solutions satisfy the a priori estimates (2.38), (2.39). P ROOF. As in the proof of Theorem 3.2, the process of construction of the expanded system consists of two stages. Firstly, we expand system (4.13) up to second-order derivatives of the vector U in the following way: A0p (Up )t +
3
Akp (Up )xk = 0,
(4.24)
k=1
where ∗ Up = U∗ , U∗t , U∗x1 , U∗x2 , U∗x3 , U∗t t , U∗t x1 , . . . , U∗x3 x3 ; Aαp = diag(I5 ⊗Aα , ε(I10 ⊗Aα )) are block-diagonal matrices; ε is a positive and, as yet, arbitrary constant. Writing the energy integral for system (4.24), and omitting detailed reasoning, we obtain, as in Section 3, the inequality 2 2 2 3 2 d I0 (t) − C1 p + u + u + p2t + p2x1 2 dt R + p2x2 + p2x3 + ε(P + Q) x =0 dx 0, (4.25) 1
with C1 > 0 a certain constant, I0 (t) = (A0p Up , Up ) dx, R3+
P = p2t t + p2t x1 + p2t x2 + p2t x3 +
3
i,j =1
p2xi xj ,
Q=
3
3
i 2 ux j x k . i=2 j,k=2
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From the boundary conditions (4.16) and system (4.13) at x1 = 0 we deduce uix2 x2 + uix3 x3 = c1 pt xi + c2 px1 xi
(i = 2, 3),
x1 = 0,
(4.26)
where c1,2 are certain constants. Using the inequality [100] R2
Q|x1 =0 dx const
2
i 2 ux2 x2 + uix3 x3 x
R2 i=1
1 =0
dx ,
relation (4.26), and the property of the trace of a function in W21 (R3+ ) on the plane x1 = 0, we reduce inequality (4.25) to d I0 (t) − εC2 dt
R2
P|x1 =0 dx C3 I0 (t),
(4.27)
where C2,3 > 0 are constants. At the second stage of the construction of the expanded system, we use, as in Section 3, a symmetrization for the wave equation. By making use of the symmetrization of the 3-D wave equation suggested by Godunov and Gordienko in [67], one gets that if the function p satisfies Equation (4.17), then the vector Y = (L1 p, L2 p, px2 , px3 )∗ solves the symmetric system R0 L1 Y + R1 L2 Y + R2 Yx2 + R3 Yx3 = 0,
(4.28)
with the symmetric matrices Rα = Rα (m1 , l2 , l3 ), ⎛
1 ⎜ −m1 R0 = ⎝ −l2 −l3 ⎛ l2 ⎜ 0 R2 = ⎝ −1 0
−m1 1 0 0 0 −l2 m1 0
−l2 0 1 0
⎞ −l3 0 ⎟ ⎠, 0 1 ⎞
−1 0 m1 0 ⎟ ⎠, l2 l3 l3 −l2
⎛
⎞ m1 −1 0 0 l2 l3 ⎟ ⎜ −1 m1 R1 = ⎝ ⎠, 0 l2 −m1 0 0 l3 0 −m1 ⎞ ⎛ l3 0 0 −1 0 m1 ⎟ ⎜ 0 −l3 R3 = ⎝ ⎠; 0 0 −l3 l2 −1 m1 l2 l3
m1 , l2 , l3 are certain constants, and R0 > 0 if m21 + l22 + l32 < 1. Let us now obtain boundary conditions at x1 = 0 for system (4.28). For this purpose, we utilize the boundary condition (4.18) that can be rewritten as (L1 − qL2 )L0 p = 0,
x1 = 0,
with L0 = a1 L1 + q1 L2 . The constants q and q1 are found from the system qq1 = −a2 ,
qa1 − q1 = q0 = cˆs /vˆ 1 .
Stability of strong discontinuities in fluids and MHD
We solve this system and take q, e.g., in the form q = (q0 +
605
q02 − 4a1 a2 )/(2a1 ). In
general, q is a complex value (if q02 − 4a1 a2 > 0, then q is real). Therefore, the function Lp and the vector function Yp = (τ Y∗ , ξ1 Y∗ , ξ2 Y∗ , ξ3 Y∗ , L0 Y∗ )∗ are, generally speaking, complex-valued. The vector Yp satisfies the expanded system R0p L1 Yp + R1p L2 Yp + R2p (Yp )x2 + R3p (Yp )x3 = 0
(4.29)
constructed from (4.28). Here Rαp are block-diagonal matrices of order 20, Rαp = diag σ1 Rα(1) , σ2 Rα(2) , σ3 Rα(3) , σ4 Rα(4) , σ5 Rα(5) ,
Rα(i) = Rα (m1i , l2i , l3i );
2 + l 2 < 1. The energy integral in σi > 0, m1i , l2i , l3i (i = 1, 5) are constants, and m21i + l2i 3i the differential form for system (4.29) is
Γ#b Dp Yp , Yp t + R1p Yp , Yp x 1 cˆs + R2p Yp , Yp x + R3p Yp , Yp x = 0, 2
with Dp =
1 R Γ#b 0p
− uˆ 1 Γ#
(4.30)
3
1−cˆs2 R1p . cˆs
Let us choose
1 i = 1, 5, m11 = 0, m12 = − , 2 a1 |q|2 a2 Re q 1 2 Re q m13 = m14 = min , , m15 = , 2 2 a2 Re q a1 |q| 1 + |q|2 a1 a2 (Re q)2 a2 Re q σ3 = σ4 = σ , σ = − m 5 2 13 σ4 , (1 + |q|2 )|q|2 a1 |q|2 l2i = l3i = 0,
σ1 =
a1 σ2 , a2
with σ5 an arbitrary positive number. Such a choice is motivated by the following. Firstly, by the uniform stability condition (4.23), the constants σi are positive, and it is easy to show that Dp > 0, i.e., system (4.29) is symmetric t-hyperbolic. Secondly, thanks to such a choice of the constants mij and σi , one gets 2 1 2q0 1 − R1p Yp , Yp x =0 + (L1 L2 p)2 + σ2 L22 p 1 a2 2 2 + k0 (px3 x3 )2 +
3
% k1i (L0 px1 )2 + k2i (L1 pxi )2 i=2
& + k3i (L2 pxi )2 + k4i (px2xi )2 C4 P|x1 =0 , where kij (i = 1, 4, j = 2, 3), k0 , and C4 are positive constants.
x1 =0
(4.31)
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Integrating (4.30) over the domain R3+ , and accounting for (4.31), we obtain the energy inequality d I1 (t) + C4 P|x1 =0 0, (4.32) dt R with I1 (t) =
R3+
Dp Yp , Yp dx > 0.
Summing (4.27) and (4.32), and choosing the constant ε so that C5 − εC2 > 0, one gets the inequality d I (t) C3 I (t) dt that has as a consequence the desired a priori estimate I (t) exp(C3 t)I (0),
0 < t T < ∞,
with I (t) = I0 (t) + I1 (t) > 0. The last estimate is, in fact, an a priori estimate without loss of smoothness that can be rewritten as (2.38). In turn, as in the previous section (see the proof of Theorem 3.2), one can derive the a priori estimate (2.39) for the function F . 5. Stability of MHD shock waves In this section we are concerned with the linearized stability of shock waves in ideal MHD. Let us, at the beginning, write out the system of ideal MHD and jump conditions for MHD strong discontinuities (in passing, we will also give the classification of MHD strong discontinuities).
5.1. Equations of ideal MHD and strong discontinuities Equations of MHD for an ideal fluid. The MHD equations describe movements of dense conducting gaseous and liquid media. It is known (see, e.g., [81,84]) that the applicability condition of these equations is the smallness of the free path and time of particles (electrons and ions) as compared with characteristic lengths and times. The system of MHD governing the motion of an ideal fluid in the magnetic field can be written in the conservative form (see, e.g., [71,84,81,62]) ρt + div(ρv) = 0, |H|2 1 H⊗H +∇ p+ = 0, (ρv)t + div ρv ⊗ v − 4π 8π
(5.1) (5.2)
Stability of strong discontinuities in fluids and MHD
607
Ht − rot(v×H) = 0, |v|2 |H|2 + ρE + ρ 2 8π t 1 |v|2 + pV + H×(v×H) = 0. + div ρv E + 2 4π
(5.3)
(5.4)
Here the gas dynamical values, ρ, V , S, p, v, E, denote the same as in system (3.1)–(3.3); H = (H1 , H2 , H3 )∗ is the magnetic field. Taking account of the Gibbs relation (3.5), we obtain the thermodynamical equalities (3.4) and can regard the MHD system (5.1)–(5.4) as a closed one to finding components of the vector U = (p, S, v∗ , H∗ )∗ . Moreover, system (5.1)–(5.4) should be supplemented by the compulsory (in MHD) divergent constraint div H = 0,
(5.5)
that is, as a matter of fact, an additional requirement on initial data (with regard to Equations (5.3), condition (5.5) holds for t > 0 if it is valid at t = 0). Symmetrization. Following the symmetrization scheme described in Section 2, Godunov [65] has symmetrized the MHD system (5.1)–(5.4) supplemented by the divergent constraint (5.5). The symmetrization of the conservation laws (5.1)–(5.4) is based on the use of the additional (to (5.1)–(5.4)) conservation law (3.6) expressing the entropy conservation. Following [65] (see also [20,21]), system (5.1)–(5.4) is written in the symmetric form (2.10), with the following canonical variables, productive functions, and the function R = r1 (Ψ 1 := H, cf. (2.6), (2.7) (j = 1)): E + pV − ST − |v|2 /2 v∗ H∗ 1 ∗ Q= − ,− ,− , , T T 4πT T L=−
p + |H|2 /(8π) , T
M k = vk L,
R=−
(v, H) . 4πT
The matrices Aα are symmetric matrices described in [20]. The MHD system presented in the canonical form (2.10) can likewise be rewritten in terms of the initial vector U as the symmetric system A0 (U)Ut +
3
Ak (U)Uxk = 0
(5.6)
k=1
that also reads (cf. (3.7)) dS = 0, dt
(5.7)
1 dv + ∇p − (rot H) × H = 0, dt 4π
(5.8)
1 dp + div v = 0, ρc2 dt ρ
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v 1 1 Ht − rot(v×H) + div H = 0, 4π 4π 4π
(5.9)
where A0 = diag(1/(ρc2 ), 1, ρ, ρ, ρ, 1/(4π), 1/(4π), 1/(4π)) is the diagonal matrix; Ak are symmetric matrices which can easily be written out (see, e.g., [20]) with the help of representation (5.7)–(5.9) of system (5.6); c2 = (ρ 2 Eρ )ρ the square of the sound velocity. Observe that, with regard to the divergent constraint (5.5), subsystem (5.9) becomes (5.3). So, the symmetric system (5.6) implies the initial MHD system (5.1)–(5.4) written in the nondivergent form (5.7), (5.8), (5.3). System (5.6) is symmetric t-hyperbolic if, as in gas dynamics, the hyperbolicity conditions (3.9) are fulfilled. Moreover, as in gas dynamics, we superimpose on the MHD system the natural physical restrictions (3.10). Jump conditions for MHD strong discontinuities. By the usual procedure (see Section 2), from the conservation laws (5.1)–(5.4) one can deduce the jump conditions (see, e.g., [71, 84,81]) [HN ] = 0,
[j ] = 0, j [vτ ] =
HN [Hτ ], 4π
j [vN ] + [p] +
1 2 |H| = 0, 8π
HN [vτ ] = j [V Hτ ],
(5.10) (5.11)
HN |v|2 |H|2 |H|2 j E+ + (H, v) = 0 + p+ vN − 2 8πρ 8π 4π
(5.12)
which hold on surface (2.12) of a strong discontinuity. Here HN = (H, N), and other notations are the same as in Section 3 (see (3.12)). Let us now give the classification of strong discontinuities in MHD (see also [71,84,81]). D EFINITION 5.1. If the fluid does not flow through a MHD strong discontinuity, i.e., j = 0, such a strong discontinuity is called contact if HN = 0, and tangential if HN = 0. Otherwise, j = 0, the strong discontinuity is called rotational (or Alfvén) if [ρ] = 0, and shock wave if [ρ] = 0. MHD contact discontinuity. In the case of contact discontinuity, it follows from (5.10)– (5.12) that j = j∞ = 0,
[v] = 0,
[H] = 0,
[p] = 0.
(5.13)
In addition, generally speaking, [ρ] = 0. MHD tangential discontinuity. The jump conditions (5.10)–(5.12) imply that on the surface of tangential discontinuity there hold the following relations: j = j∞ = 0,
HN = HN∞ = 0,
[p] = −
[|H|2 ] , 8π
(5.14)
where, generally speaking, the jumps [ρ], [vτi ], and [Hτi ] (i = 1, 2) are not equal to zero.
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609
Rotational discontinuity. Assuming that HN = 0, [Hτ ] = 0, from the jump conditions (5.10)–(5.12) one can deduce the following relations (see, e.g., [84,20,34]) that hold on the surface of a rotational discontinuity: [ρ] = 0,
|H|2 = 0,
[p] = 0,
[S] = 0,
HN , j=√ 4πV
[HN ] = 0,
[H] [v] = √ . 4πρ
(5.15)
So, the density, the pressure, and the entropy are continuous on the surface of a rotational discontinuity. Moreover, the vector of the magnetic field rotates on the discontinuity whereas its absolute value remains constant. MHD shock waves. For the case of shock waves, instead of (5.12) one can use the condition (the MHD analog of the Hugoniot adiabat for gas dynamics; see, e.g., [81] for its derivation) [E] +
|[H]|2 p + p∞ [V ] + [V ] = 0 2 16π
that can be written as (cf. (3.12)) p = H(V , g, p∞ , V∞ ),
(5.16)
with g = |[H]|2/(16π). As was proved by Iordanskii [73] (see also [84]), the compulsory condition of entropy increase under the passage through the MHD shock discontinuity, cf. (3.14), is equivalent to the compressibility conditions (3.15) if inequality (3.16) holds together with the additional assumption of the positiveness of the thermal coefficient, Ep > 0 (see Section 3). That is, for a normal gas [107] (conditions (3.9), (3.10), (3.16), and Ep > 0 hold) MHD shock waves are compression waves. In MHD there are two types of k-shocks: fast and slow shock waves. Recall that kshock conditions (see Definition 2.6) guarantee the evolutionarity of strong discontinuities. Consider a planar stationary MHD shock waves. Let, without loss of generality, v1∞ > 0 and H1 0. The matrix A−1 0 A1 , cf. (5.6), has the following eigenvalues, λ1 · · · λ8 : + , λ2,7 = v1 ∓ cA , λ1,8 = v1 ∓ cM √ Here cA = H1 / 4πρ is the Alfvén velocity [2], ± cM
− λ3,6 = v1 ∓ cM ,
λ4,5 = v1 .
2 2 H12 2 1/2 1/2 1 |H|2 |H| 2 2 +c ± +c c =√ − 4πρ πρ 2 4πρ
± are the fast and slow magnetosonic velocities. It is easily verified that the velocities cA , cM − + always satisfy the inequalities cM cA cM . Then, with regard to (2.23), there are Lax shocks of the index k = 1, + v1∞ > cM∞ ,
+ cA < v1 < cM ,
(5.17)
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and k = 3, − < v1∞ < cA∞ , cM∞
− 0 < v1 < cM .
(5.18)
D EFINITION 5.2. If conditions (5.17) hold, the MHD shock wave is called fast; if (5.18), it is called slow. Notice also that, unlike gas dynamics, the k-shock conditions ((5.17) or (5.18)) do not provide, in general, the fulfilment of the physical entropy condition (3.14).
5.2. Magnetoacoustic system #∗ )∗ , to the MHD system (5.1)–(5.4)/(5.6), Consider a constant solution, # U = (p, ˆ # S, vˆ ∗ , H 2 #k certain constants, pˆ = ρˆ Eρ (ρ, with ρ, ˆ # S, vˆk , H ˆ # S), etc. Linearizing (5.6) with respect to the constant solution # U, we obtain the magnetoacoustic system
A 0 Ut +
3
Ak Uxk = 0,
(5.19)
k=1
with Aα = Aα (# U); U is the vector of small perturbations. In a dimensionless form system (5.19) reads Lp + div v = 0,
LS = 0,
Lv + ∇p − (rot H) × h = 0,
LH − rot(v×h) = 0,
(5.20)
ˆ hk = where L = ∂/∂t + (M, ∇), M = (M1 , M2 , M3 )∗ , h = (h1 , h2 , h3 )∗ , Mk = vˆk /c, ˆ p = #k /(cˆ 4π ρˆ ); we use scaled values: x = x/lˆ (lˆ the characteristic length), t = t c/ ˆ l, H 2 p/(ρˆ cˆ ), S = S/# S, v = v/c, ˆ H = H/(cˆ 4π ρˆ ) (the primes in (5.20) were removed). While setting LSP’s for MHD shock waves, it will be more convenient for us to utilize a little bit another dimensionless form for system (5.19). Namely, if we use the same scaled ˆ v = v/vˆ1 (vˆ1 = 0 values as above, except those for the time and the velocity: t = t c/ ˆ l, for the shock wave), then the magnetoacoustic system (5.19) is written, in a dimensionless form, as Lp + div v = 0,
LS = 0,
M 2 Lv + ∇p − (rot H) × h = 0,
LH − rot(v×h) = 0,
(5.21)
with L = ∂/∂t + ∂/∂x1 + (vˆ2 /vˆ1 ) ∂/∂x2 + (vˆ3 /vˆ1 ) ∂/∂x3 ; M = vˆ1 /cˆ the Mach number.
Stability of strong discontinuities in fluids and MHD
611
5.3. Solvability of the jump conditions for MHD compressive shocks Jump conditions for planar stationary shock waves. Let us consider a piecewise constant solution to the MHD system in two space dimensions.35 Such a solution, U(t, x) =
# #1∞ , H #2∞ ∗ , S∞ , vˆ1∞ , vˆ2∞ , H U∞ = pˆ ∞ , # # #1 , H #2 ∗ , U = p, ˆ # S, vˆ1 , vˆ2 , H
x1 < 0; x1 > 0,
(5.22)
should satisfy, at x1 = 0, the jump conditions (5.10)–(5.12): ρˆ vˆ1∞ = , ρˆ∞ vˆ1
#1 = H #1∞ , H
[vˆ1 ] +
# 2] [p] ˆ [|H| = 0, + 8π jˆ jˆ
#1 H #2 , #1 [vˆ2 ], #2 = H H vˆ1 H ˆ 4π j
2 # 2 # # = 0. # + |ˆv| + pˆ + |H| − H1 vˆ , H E 2 ρˆ 4π ρˆ 4π jˆ
[vˆ2 ] =
(5.23)
# = E(ρ, Here jˆ = ρˆ vˆ1 = 0 (vˆ1 > 0, vˆ1∞ > 0), E ˆ # S), etc. Consider the case of a polytropic gas. Then inequalities (3.16) and Ep > 0 are satisfied, and the compressibility conditions (3.15) provide the entropy increase (3.14). The point is that, unlike gas dynamics, in MHD (even for the case of a polytropic gas) solutions of the jump conditions for planar shocks, (5.23), are not always consistent with the compressibility conditions and the evolutionarity inequalities ((5.17) or (5.18)). The question of their consistency is, actually, rather difficult (technically) and was analyzed, in particular, by Kulikovskii and Lyubimov [81]. Here we will consider only some special cases. Fast shock waves. R=
ρˆ , ρˆ∞
q = |h|,
By introducing the dimensionless parameters P=
pˆ∞ , ρˆ cˆ2
q∞ = |h∞ |,
M0 = l=
vˆ1 +, cˆM
h1 q
#2∞ /(cˆ 4π ρˆ )), the Alfvén and magnetosonic (h = (h1 , h2 )∗ , h∞ = (h1 , h2∞ )∗ , h2∞ = H velocities read 4 c ˆ ± ˆ 1, cˆM = √ 1 + q 2 ± (1 + q 2 )2 − 4h21 , cˆA = ch 2 35 Below we will give explanations how the stability analysis extends to 3-D (see, in particular, Remarks 5.4, 5.5, 5.6).
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and the evolutionarity inequalities (5.17) become the form & % 2 + (γ P + q 2 )2 − 4l 2 q 2 γ P , 2l 2 q 2 max R1 γ P + q∞ ∞ < M02 < 1, 2 2 2 2 2 1 + q + (1 + q ) − 4l q
(5.24)
where, in view of (3.10) and the compressibility conditions (3.15), the parameters R and P should satisfy R > 1,
0
1 . γ
(5.25)
#1 > 0, H #2 = H #2∞ = 0 At the beginning, consider the special case of parallel shocks, H (l = 1, m = h2 /q = 0), i.e., the magnetic field is supposed to be parallel to the normal to #1 = H #1∞ , do the shock front. For this case the jump conditions (5.23), except the equality H not depend on the magnetic field and coincide with the gas dynamical ones (3.19) which, for a polytropic gas, imply R=
(γ − 1)M 2 + 2 , (γ + 1)M 2
P=
1 2(M 2 − 1) + . γ γ +1
(5.26)
− − + If q = h1 < 1, then cˆM = cˆA = ch ˆ 1 and M0 = M; if q < 1, then cˆM = c, ˆ cˆA = cˆM = ch ˆ 1, and the evolutionarity inequalities are violated. So, q < 1, and solution (5.26) satisfies the compulsory conditions (5.25), provided inequalities (3.24) hold. Accounting also for (5.24), we obtain the domain of physically admissible parameters for fast parallel MHD shocks: γ −1 0 < q < M, < M < 1. (5.27) 2γ
#2∞ = 0, H #2 = 0.36 Introduce the dimensionless Consider now a general case, H 2 2 # # parameter r = H2∞ /H2 (by the way, q∞ = (l + r 2 m2 )q 2 ) measuring the competition between the tangential components of the magnetic field ahead and behind the shock wave. The jump conditions (5.23) imply the relations (1 − rR)M 2 = l 2 (1 − r)q 2 , 1 + 1 − r 2 q 2, γ 2 1 1 m2 2 − P R + (1 − R) +P + (1 − r)2 q 2 = 0. γ −1 γ γ 2
P = (1 − R)M 2 +
m2
(5.28) (5.29) (5.30)
36 The specific cases H #2∞ = 0, H #2 = 0 and H #2∞ = 0, H #2 = 0 correspond to so-called switch-on and switch-off fast shock waves (see, e.g., [81]) that are both nonevolutionary (overcompressive). Indeed, by (5.23), for switchon shocks, vˆ1 = cˆA and, for switch-off ones, vˆ1 < cˆA .
Stability of strong discontinuities in fluids and MHD
613
#2 > H #2∞ ; see likewise [84]). For By (5.25) and (5.28), r ∈ (0, 1) (i.e., for fast shocks, H the special case of a weak magnetic field, q 1, system (5.28)–(5.30) has the solution (see [27,20]): r= P=
(γ + 1)M02 2 + (γ
− 1)M02
+ O q2 ,
R=
1 + O q2 , r
1 2(M02 − 1) + + O q2 γ γ +1
√ that satisfies (5.24) and (5.25) if (γ − 1)/(2γ ) < M0 < 1. At the same time, the special case of a strong magnetic field, q ' 1, cannot be realized for fast shock waves because for this case P ' 1 (see [20]) that violates (5.25). We finally observe that, for fast transverse #2 > H #2∞ > 0 (l = 0, m = 1), solutions of the jump conditions (5.28)– #1 = 0, H shocks, H (5.30) are consistent with the evolutionarity inequalities (5.24) and the compressibility conditions (5.25), for example, for the case of a weak magnetic field. Although, we cannot analytically present the full domain of admissible parameters, M0 , q, l, r, even for the special case of transverse shocks. Slow shock waves. We will use the same dimensionless parameters as above, except M0 − which is now determined as: M0 = vˆ1 /cˆM . As follows from (5.28), for slow shocks, unlike # #2∞ = 0, H #2 = 0). As was # fast ones, r > 1, i.e., H2∞ > H2 (we consider a general case, H shown by Blokhin and Druzhinin [27] (see also [20]), for the case of a strong magnetic field, system (5.28)–(5.30) has the solution: r = 1 + r0 ε2 + O ε3 ,
R =1+
r0 + O ε2 , 2 M0
P=
1 − r0 + O ε2 , γ
with ε = 1/q, r0 = 2(1 − M02 )/(γ + 1), 0 < M0 < 1; and this solution satisfies the evolutionarity inequalities (5.18) and the compressibility conditions (5.25). Moreover, one can show (see [20]) that the case of a weak magnetic field is also realized for slow shock waves. Concerning, the special cases of parallel and transverse shocks, slow MHD shock waves can be parallel but cannot be transverse (see, e.g., [84]). Slow parallel shock waves satisfy the consistency requirement, for instance, for the case of a strong magnetic field.
5.4. The LSP for fast MHD shock waves Linearization about the piecewise constant solution. Choose, without loss of generality, a reference frame in which vˆ2 = 0. Taking this into account and linearizing the MHD system (5.1)–(5.4) (in 2-D) about the piecewise constant solution (5.22), we obtain, in the halfspace x1 > 0, the magnetoacoustic system (5.21) (in 2-D and in a dimensionless form) which now reads Lp + div v = 0,
LS = 0,
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M 2 Lv1 +
∂H2 ∂H1 ∂p + h2 − h2 = 0, ∂x1 ∂x1 ∂x2 (5.31)
∂p ∂H1 ∂H2 + h1 − h1 = 0, M Lv2 + ∂x2 ∂x2 ∂x1 2
LH1 + h1
∂v2 ∂v1 − h2 = 0, ∂x2 ∂x2
LH2 + h2
∂v1 ∂v2 − h1 = 0, ∂x1 ∂x1
where L = ∂/∂t + ∂/∂x1 . Performing the linearization of the MHD system ahead the shock, x1 < 0, we obtain that the entropy perturbation solves the linear equation L∞ S = 0, with L∞ = ∂/∂t + R∂/∂x1 + w∞ ∂/∂x2 , w∞ = vˆ2∞ /vˆ1 , which has no outgoing characteristics modes (in the half-space x1 < 0). Thus, without loss of generality we can suppose the entropy perturbation ahead the shock to be identically equal to zero: S ≡ 0 for x1 > 0 (see Remark 2.6). Taking this into account, we analogously obtain the linearized MHD system (in a dimensionless form) in the half-space x1 < 0: 1 L∞ p + div v = 0, R 2 L∞ v1 + M∞
∂H2 ∂H1 c0 2 ∂p + h2∞ − h2∞ = 0, R ∂x1 ∂x1 ∂x2
2 M∞ L∞ v2 +
∂H1 ∂H2 c0 2 ∂p + h1 − h1 = 0, R ∂x2 ∂x2 ∂x1
(5.32)
∂v2 ∂v1 1 L∞ H1 + h1 − h2∞ = 0, R ∂x2 ∂x2 ∂v1 ∂v2 1 L∞ H2 + h2∞ − h1 = 0, R ∂x1 ∂x1 where c0 = cˆ∞ /c, ˆ M∞ = vˆ1∞ /cˆ∞ . By virtue of the evolutionarity conditions (5.17), for the case of fast MHD shock waves, the situation described in Remark 2.6 takes place (i.e., system (5.32) has no outgoing characteristics modes). So, without loss of generality we can presume that there are no perturbations ahead the shock: U ≡ 0 for x1 < 0. Then, linearizing likewise the jump conditions (5.10)–(5.12) (in 2-D) and accounting for (5.16), we obtain the following LSP (in a dimensionless form) for fast MHD shock waves. P ROBLEM 5.1 (LSP for fast MHD shock waves). We seek the solution of system (5.31) for t > 0, x = (x1 , x2 ) ∈ R2+ satisfying the boundary conditions v1 + b1 p + b2 H1 + b3 H2 = 0,
S = b4 p + b5 H2 ,
Ft = b6 p + b7 H1 + b8 H2 , v2 = b9 Fx2 + b10 p + b11 H1 + b12 H2 , H1 = [h2 ]Fx2 ,
H2 = [h2 ]Ft + h1 v2 − h2 v1
(5.33)
Stability of strong discontinuities in fluids and MHD
615
at x1 = 0 (t > 0, x2 ∈ R) and the initial data x ∈ R2+ ,
U(0, x) = U0 (x),
F (0, x2 ) = F0 (x2 ),
x2 ∈ R
(5.34)
for t = 0. Here U = (p, S, v1 , v2 , H1 , H2 )∗ , b1 =
1+a , 2M 2
a=
−ρˆ 2 vˆ12 , #, g, #∞ ) HV (V ˆ pˆ∞ , V
mq − M 2 b5 M2 − a , b4 = , 2 2M M2 #, g, #∞ ) mq(1 − r)aHg (V ˆ pˆ ∞ , V b5 = − , 2 2M
b2 = −
lq , 2M 2
b3 =
b7 = −b2
2−R , 1−R
b8 =
b6 =
R(mq + M 2 b5 ) , 2M 2 (1 − R)
R(1 − a) , 2M 2 (1 − R) b9 = R − 1,
b6 w∞ b7 M 2 w∞ + mq , b11 = − , R RM 2 lqR − b8 2M 2 w∞ b12 = , [h2 ] = h2 − h2∞ . RM 2
b10 = −
R EMARK 5.1. Just as the quasilinear MHD system (5.1)–(5.4), the magnetoacoustic system (5.31) should be supplemented by the divergent constraint div H = 0
(5.35)
for the perturbations vector H = (H1 , H2 )∗ . But, with regard to (5.31) and (5.33), it is easily shown that condition (5.35) holds for t > 0 if it is valid at t = 0. So, the divergent constraint (5.35) is, as a matter of fact, an additional requirement on the initial data (5.34). Observe that the same can also be proved for all the LSP’s in MHD which will be presented below, and we will omit such a remark. The case of a polytropic gas and a weak magnetic field. Consider the case of a polytropic gas and a weak magnetic field (q 1). Then, taking account of the asymptotic representations above (for r, R, and P ), after some algebra (see [27,20]) we rewrite the boundary conditions (5.33) as follows: v1 + d0 p = N1 Fx2 ,
S = d1 p + N4 Fx2 ,
Ft = d2 p + N2 Fx2 ,
v2 = d3 Fx2 + N3 p,
H2 = mq(1 − r)Ft + qvσ , with vσ = (v, σ ), σ = (−m, l)∗ ;
H1 = mq(1 − r)Fx2 ,
(5.36)
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(0) (0) (0) d1 = d1 + O q 2 , d2 = d2 + O q 2 , d0 = d0 + O q 2 , d3 = d3(0) + O q 2 , Nk = O q 2 (k = 1, 4), M 2 = M02 + O q 2 , d0(0) =
3 − γ + (3γ − 1)M02 2M02 (2 + (γ − 1)M02 ) (γ − 1)(1 − M02 )2
(0)
d1 =
M02 (2 + (γ − 1)M02 )
,
,
(0)
d2 = −
γ +1 4M02
,
(0)
d3 =
2(1 − M02 ) (γ + 1)M02
.
Fast parallel shock waves. For the case of parallel shocks (h2 = h2∞ = 0; see above), the boundary conditions (5.33) has the form v1 + d0 p = 0, v2 = d3 Fx2 ,
S = d1 p, H2 = qv2 ,
Ft = d2 p, H1 = 0,
(5.37)
where, for a polytropic gas, M = M0 , d2 = −
γ +1 , 4M 2
d0 =
3 − γ + (3γ − 1)M 2 , 2M 2 (2 + (γ − 1)M 2 ) d3 =
d1 =
(γ − 1)(1 − M 2 )2 , M 2 (2 + (γ − 1)M 2 )
2(1 − M 2 ) . (γ + 1)(M 2 − q 2 )
R EMARK 5.2. By simple manipulations, from system (5.31) we derive the equation (the wave equation with an additional term, cf. (3.46)) M 2 L2 p − p + q G = 0,
(5.38)
with G = mH2 − lH1 . R EMARK 5.3. The last two equations in system (5.31), LH1 + q(vσ )x2 = 0,
LH2 − q(vσ )x1 = 0,
with regard to (5.35), imply the existence of the function Φ = Φ(t, x) (“potential”) so that H1 = −qΦx2 ,
H2 = qΦx1 ,
LΦ = vσ .
(5.39)
By (5.39) and (5.36), the “potential” Φ satisfies, at x1 = 0, the boundary condition: Φ = −m(1 − r)F . Besides, it is easily shown that Φ satisfies the equation M 2 L2 Φ − q 2 Φ + Lσ p = 0, with Lσ = (σ , ∇).
(5.40)
Stability of strong discontinuities in fluids and MHD
617
5.5. The LSP for slow MHD shock waves Consider now the case of slow MHD shock waves, i.e., one supposes that the piecewise constant solution (5.22) satisfies the evolutionarity inequalities (5.18) and the compressibility conditions (5.25) (we consider 2-D perturbations and refer to Remark 5.6, see below). Let, without loss of generality, the entropy perturbation S ≡ 0 for x1 < 0 (see above), and we denote (with slight abuse of notation) the vector of perturbations ahead the shock (p, v1 , v2 , H1 , H2 )∗ again by U (behind the shock, U = (p, S, v1 , v2 , H1 , H2 )∗ ). Then, the LSP for slow MHD shock waves has the following form. P ROBLEM 5.2 (LSP for slow MHD shock waves). We seek the solutions of system (5.31) for t > 0, x = (x1 , x2 ) ∈ R2+ and system (5.32) for t > 0, x ∈ R2− satisfying the boundary conditions b4 p + b5 H2 = S + b5 H2∞ + b13 p∞ ,
H1 − H1∞ = [h2 ]Fx2 ,
(1 − 1/R)Ft = p − p∞ − S + v1 − v1∞ −
h1 (H1 − H1∞ ), RM 2
h1 h2 (H1 − H1∞ ) + 2 H2 2 M M h2∞ 1 1 − 2 H2∞ + 1 + 2 p − R 1 + 2 p∞ − S = 0, M M M∞
2v1 − 2Rv1∞ −
[h2 ]Ft − w∞ H1∞ − H2 + RH2∞ − h2 v1
(5.41)
+ h2∞ Rv1∞ + h1 v2 − h1 Rv2∞ = 0, 1 1 [h2 ] 1 p − p∞ − S + v1 − v1∞ − b14 H1 R R R (1 − R)M 2 +
R[h2 ] h1 H1∞ + v2 − Rv2∞ − 2 (H2 − H2∞ ) + (1 − R)Fx2 = 0 2 (1 − R)M M
at x1 = 0 (t > 0, x2 ∈ R) and the initial data U(0, x) = U0 (x),
x ∈ R2± ,
F (0, x2 ) = F0 (x2 ),
for t = 0. Here #, g, #∞ aR , b13 = Hp∞ V ˆ pˆ∞ , V a∞ a∞ = −
ρˆ 2 vˆ12
#, g, #∞ ) HV∞ (V ˆ pˆ∞ , V
;
b14 =
Rw∞ , R−1
x2 ∈ R
(5.42)
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the values b4 , b5 , a, R, etc. are written above (see the previous subsection); p∞ , vk∞ , and Hk∞ denote the limit values, under x1 → −0, for perturbations ahead the shock. The case of a polytropic gas and a strong magnetic field. Consider the case of a strong magnetic field, i.e., let ε = 1/q be a small parameter. Then, using the expansions into series in ε for r, R, and P written above (for a polytropic gas), one has the following asymptotic formulae for the constants appearing in the boundary conditions (5.41): M 2 = l 2 M02 + O ε2 , [h2 ] = − R=
b13 =
2 + (γ − 1)M02 2γ M02
2m(1 − M02 ) ε + O ε2 , γ +1
2 + (γ − 1)M02 M02 (γ
b5 = −
2 M∞ = l2
+ 1)
+ O ε2 ,
2m(γ − 1)(1 − M02 )2 M02 (γ
+ 1)(2 + (γ
− 1)M02 )
2(γ − 1)(1 − M02 )2 M02 (2 + (γ
− 1)M02 )
−γ +1
w∞ = a = l2
+ O ε2 ,
2m(1 − M02 ) lM02 (γ
+ 1)
2γ M02 − γ + 1 2 + (γ
− 1)M02
+ O ε2 ,
+ O ε2 ,
ε + O ε2 ,
+ O ε2 ,
b14 =
m(2 + (γ − 1)M02 ) lM02 (γ
+ 1)
+ O ε2 .
5.6. Uniform stability of fast MHD shocks under a weak magnetic field Above all, it should be noted that, unlike the situation in gas dynamics (see Section 3), the issue of the linearized stability of MHD shock waves has not been fully investigated as yet. After the publication of classical works of Akhiezer et al. [1] and Gardner and Kruskal [62] only a few studies of the stability of MHD shock waves have been published. Actually, the gap in our knowledge is much more than what we know about the stability of MHD shocks. The linearized stability of MHD shock waves against 1-D perturbations was studied by Akhiezer et al. [1].37 Gardner and Kruskal [62] have obtained a condition for weak linearized stability of the fast MHD shock wave for the special cases of parallel and transverse shocks. Weak stability was thereby proved for a polytropic gas with γ < 3. Observe also that, for parallel shocks, the weak stability condition derived by Gardner and Kruskal coincides totally with that for gas dynamical shock waves (F > 0 (or −1 − 2M < a < 1); see Section 3). Lessen and Deshpande [90] have numerically found some instability domains for MHD shock waves in a polytropic gas with γ = 5/3. In particular, they have shown that the slow shock wave can be unstable. An analogous, but more complete, investigation has been 37 Recall that evolutionary strong discontinuities (e.g., fast and slow MHD shock waves) can be unstable against 1-D perturbations only in very specific cases (see Remark 2.8).
Stability of strong discontinuities in fluids and MHD
619
carried out by Filippova in [53] where some instability domains were found also for fast shock waves. The linearized stability of MHD shock waves in a polytropic gas for the asymptotic cases of a weak magnetic field and a strong magnetic field (q 1 and q ' 1; see above) was analyzed by Blokhin and Druzhinin [26,27]. They have shown that fast shocks are weakly stable under a weak magnetic field whereas slow shocks are unstable under a strong magnetic field. Besides, the uniform linearized stability of fast MHD shock waves (for a polytropic gas and a weak magnetic field) as tested by obtaining a priori estimates without loss of smoothness like (3.50), (3.51) has been proved in [26,27] for the special cases of parallel and transverse shocks. The general case of an arbitrary inclination of the vector of magnetic field to the planar shock front has been studied by Blokhin and Trakhinin [37]. Namely, by deriving a priori estimates without loss of smoothness, they have established the uniform linearized stability of fast MHD shock waves in a polytropic gas under a weak magnetic field. At last, we observe that Blokhin and Trakhinin [40] have refined the results of Gardner and Kruskal in [62] and established that fast parallel MHD shock waves in a polytropic gas are always weakly stable irrespective of the adiabat index γ . Moreover, they have obtained a necessary and sufficient condition for uniform linearized stability of the fast parallel shock.38 Notice also that, as was shown by Anile and Russo [6], fast transverse MHD shocks which weak stability for a polytropic gas with γ < 3 was established by Gardner and Kruskal [62] are also always weakly stable for all γ (γ > 1). In this chapter we describe only analytical results mentioned above and we refer to the works of Lessen and Deshpande [90] and Filippova [53] where some Hadamard-type ill-posedness (instability) examples for fast and slow MHD shock waves are constructed numerically. Let us now consider fast MHD shocks and prove their uniform stability for a polytropic gas and the asymptotic case of a weak magnetic field (q 1). Because of the uniform linearized stability of gas dynamical shock waves in a polytropic gas, at first sight, this fact seems to be almost evident (q = 0 corresponds to gas dynamics). But, the definition of uniform stability is connected with the delicate concept of the ULC (see Definitions 2.7, 2.8), and the influence of the magnetic field, even rather weak,39 is not quite clear. Moreover, as we will see, the rigorous mathematical proof of the announced uniform stability result which is below formulated as a well-posedness theorem is rather complicated. T HEOREM 5.1. The fast MHD shock wave in a polytropic gas under a weak magnetic field is uniformly stable. Moreover, solutions of Problem 5.1 satisfy the a priori estimates (3.50), (3.51). P ROOF. As was suggested in [26,27,37], making relevant modifications, we will follow the scheme of construction of the a priori estimates (3.50), (3.51) used for gas dynamical shocks in the proof of Theorem 3.2. 38 The remarkable fact of the possibility of the existence of neutrally stable (see Section 2) fast parallel MHD
shocks in a polytropic gas has been first pointed out by Egorushkin and Kulikovskii [51] (recall that gas dynamical shock waves in a polytropic gas are always uniformly stable; see Corollary 3.1). 39 Although, it is not evident that the perturbations H 1,2 will be also small for all time!
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Firstly, by expanding system (5.31), one gets (3.52), with A0 = diag(1, 1, M 2, M 2 , 1, 1), ⎛
1 ⎜0 ⎜ ⎜1 A1 = ⎜ ⎜0 ⎝ 0 0 ⎛ 0 ⎜0 ⎜ ⎜0 A2 = ⎜ ⎜1 ⎝ 0 0
0 1 0 0 0 0
1 0 M2 0 0 mq
0 0 0 0 0 0
0 0 0 0 −mq 0
0 0 0 M2 0 −lq 1 0 0 0 lq 0
⎞ 0 0 0 0 ⎟ ⎟ 0 mq ⎟ ⎟, 0 −lq ⎟ ⎠ 1 0 0 1 ⎞ 0 0 0 0⎟ ⎟ −mq 0 ⎟ ⎟, lq 0⎟ ⎠ 0 0 0 0
etc. Then, we obtain the energy identity (3.53). With regard to boundary conditions (5.36) and system (5.31) for x1 = 0, identity (3.53) yields the inequality (cf. (3.54)) 2 d I0 (t) − C0 p + v22 + pt2 + px21 + px22 + P dt R 2 + N5 Ψ + Ψt2 + Ψx22 x =0 dx2 0, 1
(5.43)
where C0 > 0 is a constant independent of q, N5 = O(q 2 ), Ψ = Fx2 x2 , and the expression P is the same as in Section 3. Cumbersome manipulations with system (5.31) for x1 = 0 and boundary conditions (5.36) lead us to the equality Ψ = (a1 pt + a2 px1 + N6 px2 )|x1 =0 , with a1 = −(1 + d0 )/d3 + O(q 2 ), a2 = −1/(M 2 d3 ) + O(q 2 ), N6 = O(q 2 ). Then, as in Section 3, we reduce inequality (5.43) to form (3.55), where C1,2 > 0 are constants independent of q. At the second stage of constructing an expanded system, we use, as in Section 3, the symmetrization of the wave operator suggested in [69]. Namely, the symmetric system (3.58) becomes, in MHD, the form 1 q DWt − B1 Wx1 − B2 Wx2 + 2 b b
K =0 L ∇G M
(5.44)
which follows from Equation (5.38) (we keep the same notations as in Section 3). Let us now obtain boundary conditions for system (5.44). Following arguments in Section 3 and omitting detailed calculations (see [37] for more details), one gets the matrix form (3.60) of boundary conditions, where the matrices A, B, and C differ from
Stability of strong discontinuities in fluids and MHD
621
the corresponding ones in Section 3 by small values of order O(q 2 ): A= C=
n1 0 N8
0 0 0
αn2 0 n3
N7 0 , N9
0 −n7 1 0 1 0 − Ma b2
B=
−αn4 0 0
−n5 0 −n6 Md0
N10 −1 N11
,
;
a1 = n8 b2 d0 + a0 M 2 , a0 = d2 d3 , ni = 1 + O(q 2 ), Nk = O(q 2 ) (i = 1, 8, k = 7, 11). Moreover, for a polytropic gas and a weak magnetic field one has the representations a1 = a1(0) + O q 2 , (0)
(0)
(0)
a0 = a0(0) + O q 2 , (0)
(0) (0)
with a1 = b2 d0 + a0 M 2 > 0, a0 = d2 d3 < 0. So, for q 1 (cf. (3.49)): a0 < 0,
a1 > 0.
(5.45)
The vector boundary condition (3.60) is written as (3.61), and, by (5.45), all the eigenvalues of the matrix G lie, as in gas dynamics, strictly in the left semi-plane. Then, as in Section 3, we determine the Hermitian matrices K, L, M, and N from the Lyapunov matrix equation (3.63) (we again have that D > 0 etc.). Let us write out the energy integral in differential form for system (5.44): 1 (DW, W)t − (B1 W, W)x1 − (B2 W, W)x2 b q% + 2 2 Y1 , K ∇G + 2 Y2 , L ∇G b & = 0. + 2 Y3 , M ∇G
(5.46)
To obtain the identity of energy integral (see Section 2) for system (5.44) we would like to represent the expression in braces in (5.46) in a divergent form: {· · ·} = (· · ·)t + (· · ·)x1 + (· · ·)x2 . For this purpose, we use the function Φ introduced in Remark 5.3 as an auxiliary action. σ , then Equations (5.39), (5.40) imply the vector equations Let X = q (∇Φ), Z = ∇v Lσ Y = qX − M 2 LZ,
LX = q Z.
(5.47)
By making use of (5.47), after cumbersome calculations one gets the following divergent representation for the expression in braces in (5.46): {· · ·} = (Ω0 )t + (Ω1 )x1 + (Ω2 )x2 ,
(5.48)
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with Ω0 = −(X, K1 X) + M 2 2(L1 Z, KX) + 2(L2 Z, LX)
+ 2(L3 Z, MX) + (Zx1 , K1 Zx1 ) + (Zx2 , K1 Zx2 ) , Ω1 = −2m (Y1 , KX) + (Y2 , LX) + (Y3 , MX) − q(X, LX) + M 2 2(L1 Z, KX) + 2(L2 Z, LX) + 2(L3 Z, MX)
− 2(Zt , K1 Zx1 ) + q(Zx2 , LZx2 ) − q(Zx1 , LZx1 ) − (2q/b)(Zx2 , MZx1 ) , Ω2 = 2l (Y1 , KX) + (Y2 , LX) + (Y3 , MX) − (q/b)(X, MX) − M 2 2(Zt , K1 Zx2 ) + 2q(Zx1 , LZx2 ) + (q/b)(Zx2 , MZx2 ) − (q/b)(Zx1 , MZx1 ) ,
K1 = (Mq/b 2)(K − ML). On the other hand, in view of (5.39), one has: Mqvσ − MH2 1 2 X= 2∆ H2 − M vσ . b −bH1 So, Ω0,1,2 are quadratic forms with the vector V: Ωα = (Mα V, V),
(5.49)
where Mα are matrices with elements of order O(1) (they have no elements of order q −1 etc.). Taking account of (5.46), (5.48), (5.49), the MHD analog of the energy identity (3.62) is d q I1 (t) + (B1 W, W) − 2 (M1 V, V) dx2 = 0, (5.50) dt b R x1 =0 where q (DW, W) + 2 (M0 V, V) dx > 0 b R2+
I1 (t) =
for q 1.
As in gas dynamics, we have (3.64), and estimating the boundary integral in (5.50), we obtain the inequality (cf. (3.65)) d I1 (t) + (C4 − qC5 ) P|x1 =0 dx2 0, (5.51) dt R where C4,5 = C4,5 (G0 ) are constant independent of q and depending on the norm of the matrix G0 .40 Observe that C6 = C4 − qC5 > 0 for small q. Then, as in gas dynamics, 40 The matrix M is determined by the matrices K, L, and M which, in turn, depend on the solution 1 H = R+ exp(tG∗ )G0 exp(tG) dt of Equation (3.63).
Stability of strong discontinuities in fluids and MHD
623
by an appropriate choice of the matrix G0 , we achieve that C6 − C1 > 0, and adding inequalities (3.55) and (5.51), one gets the a priori estimate (3.66) that gives the desired a priori estimate (3.50). As in Section 3 (see the proof of Theorem 3.2), one can derive the a priori estimate (3.51) for the function F . R EMARK 5.4. Making relevant modifications in the scheme of constructing an expanded system utilized in the proof of Theorem 4.2 for 3-D relativistic gas dynamical shocks (see also [14] for usual gas dynamics), one can obtain a 3-D variant of estimates (3.50), (3.51), in the form of (2.38), (2.39), for fast MHD shock waves in a polytropic gas under a weak magnetic field. It is also clear that, following arguments above, we can prove the uniform stability of fast MHD shock waves under a weak magnetic field for an arbitrary state equation, provided the uniform stability condition (3.49) holds.
5.7. Uniform stability condition for the fast parallel MHD shock wave Consider fast MHD shock waves for the special case of parallel shocks, i.e., the magnetic field is supposed to be parallel to the normal to the shock front (see above). Following [40], for simplicity of calculations we will suppose the gas to be polytropic (see Remark 5.5 below). Moreover, as was observed in [62], for parallel MHD shocks (as well as for transverse ones), because of the symmetry of the LSP along to the tangent directions x2,3 one can consider, without loss of generality, only 2-D perturbations. Taking this into account, we will study the LSP for fast parallel MHD shock waves in the whole domain of admissible parameters that is determined, for a polytropic gas, by conditions (5.27). This LSP is the linear IBVP for system (5.31) (with h2 = 0), A0 Ut + A1 Ux1 + A2 Ux2 = 0,
(5.52)
with the boundary conditions (5.37), where the matrices Aα are written above in the proof of Theorem 5.1 (for parallel shocks l = 1, m = 0). Let us now, following [40], formulate the main stability result for fast parallel shock waves. T HEOREM 5.2. The fast parallel MHD shock wave in a polytropic gas is always weakly stable. Moreover, it is uniformly stable if and only if
g M+
M 2 + 2/(γ − 1) > 0,
(5.53)
where g(z) = (zM − 1)z4 + q 2 {(zM − 1)(z2 − 2)z2 − q 2 (z2 − 1)2 }. P ROOF. In view of the evolutionarity conditions (5.17) which, for parallel shocks and with regard to (5.25), becomes form (5.27), system (5.52) has the property described in Lemma 3.1. So, we can utilize Definitions 3.2 and 3.3 for the LC and the ULC for problem (5.52), (5.37). In other words, one can follow the scheme of the proof of Theorem 3.1. Let us, making relevant modifications and compliments, adapt this scheme for fast parallel MHD shocks.
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First, in our case the analog of the dispersion relation (3.33) reads % & Ω 2 M 2 Ω 2 M 2 Ω 2 − λ2 + ω2 + q 2 ω2 − λ2 M 2 Ω 2 − λ2 = 0
(5.54)
(here and below we keep the same notations as in Section 3). Observe that one can show that for the case of a polytropic gas Hadamard-type ill-posedness examples in 1-D form (ω = 0) cannot be constructed (Majda’s conditions are satisfied, cf. Remark 2.8, and the fast MHD shock wave in a polytropic gas is stable against 1-D perturbations). Then, without loss of generality we suppose, as in Section 3, that ω = 1. Then (5.54) yields the equation (cf. (3.35)) 1 − q 2 + 1 ζ 2 + θ 2 + q 2ζ 2 ζ 2 + θ 2 = 0 that has the following appropriate root θ (cf. (3.36)): θ = θ1 = −
1 1 (z2 − q 2 )(z2 − 1) 1/2 2 q z 1 + q2 z2 − 1+q 2
(5.55)
(θ2 = −θ1 ). Observe that q 2 /(1 + q 2 ) < q 2 < M 2 < 1, and inequalities (3.34) determine the domain of z that is the right half (Re z > 0) of the z–plane with two segments of the real axis removed: the segment from q 2 /(1 + q 2 ) to q 2 and the segment from 1 to +∞. Omitting detailed calculations (see [40] for more details), algebraic system (3.28), (3.29) implies, for problem (5.52), (5.37), the equality h(z) = −z2 + 2Mz +
2 =0 γ −1
(5.56)
that coincides with (3.39) written for a polytropic gas. Equation (5.56) has the roots z1,2 = M ±
M2 +
2 . γ −1
The root z2 < 0. The root z1 > 1 if 2M(γ − 1) + 3 − γ > 0.
(5.57)
It is obvious that (5.57) holds for γ < 3 (see also arguments in [62]). But inequality (5.57) is likewise valid for all admissible γ (γ > 1). Indeed, it can be rewritten as 2γ M 2 − (γ − 1) + 2(1 − M)(1 + γ M) > 0. With regard to (5.27), the last inequality is fulfilled for all γ > 1. Thus, the both root z1 and z2 lie outside of the domain of z described above, and, consequently, the boundary
Stability of strong discontinuities in fluids and MHD
625
conditions (5.37) satisfy the LC. It means the weak stability of fast parallel shock waves in a polytropic gas. Let us now separate the uniform stability domain in which h(z) = 0 likewise for such z which correspond to the case η = 0, λ = λ0 (Re λ0 = 0; see Remark 3.4). For the special (gas dynamical) case q = 0, the root z1 , which can correspond to the case η = 0, λ = λ0 = iδ, δ ∈ R, lies to the right of 1/M (see the proof of Theorem 3.1 and Corollary 3.1). Proceed to the general, more complicated, case q > 0 (more exactly 0 < q < M; see (5.27)). √ From (5.55) we have ξ = ξ(z, q) = ξ(z)η(z, q), where the function ξ(z) = (z − M)/(M z2 − 1) corresponds to the gas dynamical case q = 0 (see the proof of Theorem 3.1), and η = η(z, q) =
z2 (1 + q 2 ) − q 2 . z2 − q 2
It is not difficult to check that the function η (as a function of z) decreases on the interval z > 1. Hence the function ξ = ξ(z, q) for z > 1 decreases up to its minimum z∗ (z∗ > 1/M) and increases for z > z∗ . In view of the continuous dependence of η on the parameter q, the point of intersection of the graph of the function ξ = ξ(z, q) with the line ξ = ξˆ = const, which corresponds to the case of no roots with η = 0, λ = λ0 = iδ, lies to the right of z∗ for sufficiently small q (z∗ is close to 1/M), and cannot jump over the interval 1 < z < z∗ while q increases up to M (0 < q < M). Therefore, the domain determined by the condition z1 > z∗ presents that of uniform stability. To find z∗ we have to solve the equation ξz (z, q) = 0 (for z > 1) that is equivalent to the following % 2 & = 0. g(z) = (zM − 1)z4 + q 2 (zM − 1) z2 − 2 z2 − q 2 z2 − 1
(5.58)
The coefficient of z5 in (5.58) is positive. Equation (5.58) has the root z = z∗ , and other roots are either less than 1 (1 < z∗ ) or complex. Hence the polynomial g(z) is positive on the interval z > z∗ and negative for 1 < z < z∗ . Consequently, the inequality z1 > z∗ is equivalent to g(z1 ) > 0 (cf. (5.53)). Let us analyze a little condition (5.53). It is clear that it holds, for example, for the asymptotic case of a weak magnetic field (see also Theorem 5.1). Likewise, one can see that condition (5.53) is fulfilled, for instance, for the asymptotic case of maximal admissible Mach numbers (M is close to unit) or of minimal admissible adiabat indices γ (γ is close to unit). Carrying out a little bit more delicate algebraic analysis, one can show that (5.53) holds, for example, in the domain of maximal admissible q (q is close to M) if M 2 > 1/2 and γ < 3. On the other hand, it is easily verified that the asymptotic case of minimal admissible Mach numbers, M 2 − (γ − 1)/(2γ ) 1 (in this case z1 is close to 1/M), is an example of the violation of condition (5.53), i.e., the neutral stability domain is not empty (see also [51]), unlike the case of gas dynamical shock waves for a polytropic gas (see Corollary 3.1).
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R EMARK 5.5. We supposed the gas to be polytropic only for simplicity of calculations. For the general case of an arbitrary state equation, following the technique described above, it is possible to separate a subdomain of the weak stability domain (presented by Erpenbeck’s condition F 0) where the ULC is fulfilled for the LSP for fast parallel MHD shock waves.41 Moreover, this was done by Trakhinin [119] for a more general case of fast parallel shock waves in relativistic MHD. By the way, the stability analysis in [119] was performed for 3-D perturbations. Though, it is not necessary because, as for nonrelativistic fast parallel MHD shocks, the Lopatinski determinant depends on the magnitude ω = |ω|. In the framework of this chapter we do not consider fast transverse MHD shocks, which are weakly stable (in a polytropic gas), and refer to the works of Gardner and Kruskal [62] and Anile and Russo [6].
5.8. Instability of slow MHD shocks under a strong magnetic field Some instability examples for slow MHD shock waves (for a polytropic gas with γ = 5/3) have been numerically found in [90,53]. Following the work of Blokhin and Druzhinin [27] (see also [20]), let us now analytically prove (omitting, maybe, technical details) the instability of slow shock waves in a polytropic gas under a strong magnetic field (q ' 1). T HEOREM 5.3. The slow MHD shock wave in a polytropic gas under a strong magnetic field is unstable. P ROOF. We will construct an ill-posedness example of Hadamard type (see Section 2) for Problem 5.2. For this purpose, we look for the sequence of exponential solutions to systems (5.31), (5.32) and the boundary conditions (5.41) in the form Un =
% & (0) U∞ exp n(τ t + ξ∞ x1 + ix2 ) , x1 < 0; % & U(0) exp n(τ t + ξ x1 + ix2 ) , x1 > 0
(5.59)
(n = 1, 2, 3, . . .). Here U∞ = (p∞ , v1∞ , v2∞ , H1∞ , H2∞ )∗ , U(0) = (p(0) , S (0) , v1 , (0) (0) (0) v2 , H1 , H2 )∗ are real constant vectors; τ , ξ , ξ∞ are constants satisfying inequalities (2.26). Substituting (5.59) into systems (5.31), (5.32) gives linear algebraic systems for (0) components of the vectors U(0) and U(0) ∞ . As follows from the algebraic system for U , there are two possible types of exponential solutions for x1 > 0. The first one is when L = τ + ξ = 0 and S (0) = 0, and the second one when, on the contrary, L = 0 and S (0) = 0. For the fist case, (0)
(0)
(0)
(0)
(0)
(0)
ξ = ξ0 = −τ, 41 As was shown by Gardner and Kruskal [62], fast parallel MHD shocks are unstable if F < 0.
(0)
(5.60)
Stability of strong discontinuities in fluids and MHD (0)
(0)
(0)
p(0) = v1 = v2 = 0, S (0) = 0, and the constants H1 relation (0)
(0)
and H2
627
are connected by the
(0)
τ H1 + iH2 = 0.
(5.61)
For the second type of exponential solutions S (0) = 0, and other constants satisfy the algebraic system ⎛
L
⎜0 ⎜ ⎜ ⎜0 ⎜ ⎝ξ i
0
0
ξ
L 0
0 L
−ih2 h2 ξ
h2 ξ −h1 ξ
M 2L
−ih2 ih1
0
i
⎞
⎛
p(0)
⎞
⎜ (0) ⎟ ⎜H ⎟ ih1 ⎟ ⎟⎜ 1 ⎟ ⎟ ⎜ H (0) ⎟ ⎟ = 0. ⎜ −h1 ξ ⎟ ⎟⎜ 2 ⎟ (0) ⎟ ⎜ ⎠ 0 ⎝ v1 ⎠ M 2L v2(0)
(5.62)
System (5.62) has a nontrivial solution if its determinant is zero, i.e., M 2 L2 M 2 L2 − ξ 2 + 1 + q 2 1 − ξ 2 M 2 L2 − (lξ + im)2 = 0.
(5.63)
Considering (5.63) as an equation for ξ , we find its roots ξ = ξ(τ ) with the property Re ξ < 0 for Re τ > 0 (see (2.26)). Let the magnetic field be rather strong, i.e., q ' 1 or ε 1 (see above). By expanding τ and ξ into series in ε, τ = τ (0) + τ (1) ε + τ (2) ε2 + · · · ,
ξ = ξ (0) + ξ (1) ε + ξ (2) ε2 + · · · ,
we choose the appropriate roots (1)
ξ1 = −1 + ξ1 ε + · · · , (0)
(0)
(1)
ξ2 = ξ2 + ξ2 ε + · · · ,
(5.64)
(1)
with ξ2 = −(M0 τ (0) + im/ l)/(1 + M0 ), and ξ1 = 0 if τ (0) = 1 − (im − l)/(M0 l).42 Analogously, by setting the determinant of the algebraic system for U(0) ∞ equal to zero, one chooses the appropriate root (2) 2 ξ∞ = 1 + ξ∞ ε + ···
(5.65)
(2)
(ξ∞ = 0) with the property Re ξ∞ > 0 for Re τ > 0. To have more freedom for components of the constant vectors we finally look for the sequence of exponential solutions in the form Un =
% & (0) x1 < 0; U∞ exp n(τ t + ξ∞ x1 + ix2 ) , % & 2 (α) exp n(τ t + ξ x + ix ) , x > 0 α 1 2 1 α=0 U
42 As was mentioned above, the slow shock cannot be transverse, i.e., l = 0.
(5.66)
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A. Blokhin and Y. Trakhinin
(instead of (5.59)). Here ξ∞ , ξ0 , and ξ1,2 are respectively given by (5.65), (5.60), and (0) (0) (k) (k) (k) (k) (5.64); U(0) = (0, S (0), 0, 0, H1 , H2 )∗ , U(k) = (p(k) , 0, v1 , v2 , H1 , H2 )∗ (k = 1, 2), and U(0) ∞ are real constant nonzero vectors which components are connected, respectively, by (5.61) and the relations ⎛
Lk ⎜ 0 ⎜ ⎜ ⎝ 0 ξk
0 Lk
0 0
ξk −ih2
0 −ih2
Lk h2 ξk
h2 ξk M 2 Lk
⎛
(1/R)L∞ ⎜ 0 ⎜ ⎜ ⎝ 0 γ P ξ∞
⎞
⎛
p(k)
⎞
⎜ (k) ⎟ i ⎜ H1 ⎟ ⎟ ⎟ ih1 ⎟ ⎜ ⎜ (k) ⎟ ⎟ ⎜ H2 ⎟ = 0, ⎟ −h1 ξk ⎠ ⎜ ⎜ v (k) ⎟ ⎝ 1 ⎠ 0 v2(k)
0 (1/R)L∞
0 0
ξ∞ −ih2∞
0 −ih2∞
(1/R)L∞ h2∞ ξ∞
h2∞ ξ∞ M 2 L∞
⎞
⎛
(0)
p∞
⎞
(5.67)
⎜ (0) ⎟ ⎜H ⎟ ⎟ ⎜ 1∞ ⎟ ⎟ ⎜ H (0) ⎟ ⎟ ⎜ 2∞ ⎟ = 0, ⎟ −h1 ξ∞ ⎠ ⎜ ⎜ v (0) ⎟ ⎝ 1∞ ⎠ 0 i ih1
(0) v2∞
with Lk = τ + ξk , L∞ = τ + Rξ∞ + iw∞ . If by cross differentiating we eliminate the function F (t, x2 ) from the boundary conditions (5.41) and substitute (5.66) into the obtained expression, then, in view of (5.61), (0) (0) (5.67), H1 = H2 = 0 if τ = 1. Suppose τ = 1. Then, by substituting the solution representation (5.66) into the boundary conditions (5.41) (beforehand we eliminate the function F ), with regard to H1(0) = H2(0) = 0, one gets the linear algebraic system ZW = 0
(5.68)
for finding the components ω1 , ω2 , and ω3 of the vector W, where & (k) % ωk = v2 / Lk M 2 L2k − ξk2 − h22 ξk2 − 1 (k = 1, 2), 2 % & (0) 2 2 2 ω3 = −v∞ / (1/R)L∞ (1/R)M∞ L∞ − γ P ξ∞ − h22∞ ξ∞ −1 ; the matrix Z is a constant matrix which coefficients are written out in [27,20]. Finally, we obtain the dispersion relation det Z = D(τ ) = 0, that is, in fact, a polynomial equation for finding τ (see [27,20] for more details). If this equation has a root, τ , with the restrictions τ (0) = 1 − property Re τ (0) > 0 (moreover, we superimpose the additional √ (im − l)/(M0 l) and τ = 1), then the solution U = Un exp(− n ), with Un given by (5.66), is the Hadamard-type ill-posedness example for Problem 5.2 under q ' 1, and the slow shock wave under a strong magnetic field is unstable. Taking account of the asymptotic formulae under ε 1 for coefficients of the boundary conditions (5.41), we expand the polynomial D(τ ) into series in ε. Keeping a first nonzero
Stability of strong discontinuities in fluids and MHD
629
coefficient in the expansion, one gets an equation for τ (0) (see [27,20] for calculations) that has the appropriate root √ 1 im l+1− √ τ (0) = 1 + √ l+1 M0 l 2l with the necessary property Re τ (0) > 0.43 The last proves the Theorem 5.3.
R EMARK 5.6. It is clear that the instability of slow MHD shocks (for a strong magnetic field) against 2-D perturbations follows 3-D instability. Indeed, the 2-D ill-posedness example (5.66) constructed above is, of course, also the ill-posedness example for the corresponding 3-D LSP. 6. Stability of the MHD contact discontinuity The MHD contact discontinuity (see Definition 5.1) belongs to the types of strong discontinuities characterizing by the absence of a mass transfer flux across the discontinuity surface (j = 0). Moreover, on its surface the velocity, the pressure, and the magnetic field are continuous (see (5.13)), but the density can have an arbitrary jump. Perhaps, the linearized (generally speaking, weak) stability of contact discontinuities is almost evident from the intuitive physical point of view. But, as was pointed in Sections 1, 2, it is much more important to prove uniform stability. This was done by Blokhin and Druzhinin [28] (see also [20]). Moreover, the a priori estimates without loss of smoothness obtained in [28] for the LSP for MHD contact discontinuities can be very useful for the theoretical study of corresponding numerical models. Observe also that the “layerwise” L2 -estimate like (2.50) directly follows from the dissipativity of the boundary conditions for the LSP. The a priori estimates for derivatives of small perturbations and for the disturbance of discontinuity surface F (t, x ) are deduced in [28] by direct (“trivial”) expanding the magnetoacoustic systems behind and ahead the planar discontinuity. So, the structural (nonlinear) stability of the MHD contact discontinuity can be proved in a much more easier way, in comparison with gas dynamical shock waves, by utilizing the same techniques worked out in [15,17,19] (see also the end of Section 3). 6.1. The LSP for the MHD contact discontinuity Suppose the MHD planar stationary discontinuity to be contact. This means that the piecewise constant solution (like (5.22), but for 3-D) satisfies the conditions j = 0, HN = 0, and (5.13), i.e.,44 vˆ1 = vˆ1∞ = 0, [ˆv] = 0,
#1 = H #1∞ > 0, H
# = 0, [H]
[p] ˆ = 0.
43 Observe also that τ (0) = 1 − (im − l)/(M l), i.e., ξ (1) = 0 (see (5.64)). 0 1 44 Without loss of generality we suppose that H #1 > 0.
(6.1)
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Taking account of (6.1) and linearizing the MHD jump conditions (5.10)–(5.12), we obtain the following boundary conditions at x1 = 0: [δj ] = 0, [H] = 0, # + pˆ V # δj = 0, E
[p] = 0,
#1 [vˆ2,3 ] = 0, #H #2,3 δj + H V
where δj = ρ(F ˆ t + vˆ2 Fx2 + vˆ3 Fx3 − v1 ) (a small perturbation of j ). As follows from the last boundary condition, there are two possibilities for the unperturbed discontinuous flow: # + p[ #] = 0 or [E] # + p[ #] = 0. The first condition implies δj = 0, and for this case [E] ˆ V ˆ V the LSP (in a dimensionless form) is the following. P ROBLEM 6.1 (LSP for the MHD contact discontinuity). We seek the solutions of the systems
Ut +
3
Ak Uxk = 0,
t > 0, x ∈ R3+ ;
(6.2)
k=1
A0∞ Ut +
3
Ak∞ Uxk = 0,
t > 0, x ∈ R3− ,
(6.3)
k=1
satisfying the boundary conditions [U] = 0,
v1 = LF
(6.4)
at x1 = 0 (t > 0, x ∈ R2 ) and the initial data U(0, x) = U0 (x),
x ∈ R3± ,
F (0, x ) = F0 (x ),
x ∈ R2 ,
for t = 0. Here U = (p, v∗ , H∗ )∗ , A0∞ = diag(R/c0 2 , 1/R, 1/R, 1/R, 1, 1, 1), ⎛
0 ⎜1 ⎜ ⎜0 ⎜ A1 = ⎜ 0 ⎜ ⎜0 ⎝ 0 0
1 0 0 0 0 h2 −h3
0 0 0 0 0 −h1 0
0 0 0 0 0 0 −h1
0 0 0 0 0 0 0
0 h2 −h1 0 0 0 0
⎞ 0 h3 ⎟ ⎟ 0 ⎟ ⎟ −h1 ⎟ , ⎟ 0 ⎟ ⎠ 0 0
(6.5)
Stability of strong discontinuities in fluids and MHD
⎛
M2 ⎜ 0 ⎜ ⎜ 1 ⎜ A2 = ⎜ 0 ⎜ ⎜ 0 ⎝ 0 0 ⎛
M3 ⎜ 0 ⎜ ⎜ 0 ⎜ A3 = ⎜ 1 ⎜ ⎜ 0 ⎝ 0 0
0 M2 0 0 −h2 0 0
1 0 M2 0 h1 0 h3
0 0 0 M2 0 0 −h2
0 −h2 h1 0 M2 0 0
0 0 0 0 0 M2 0
⎞ 0 0 ⎟ ⎟ h3 ⎟ ⎟ −h2 ⎟ , ⎟ 0 ⎟ ⎠ 0 M2
0 M3 0 0 −h3 0 0
0 0 M3 0 0 −h3 0
1 0 0 M3 h1 h2 0
0 −h3 0 h1 M3 0 0
0 0 −h3 h2 0 M3 0
⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟, ⎟ 0 ⎟ ⎠ 0 M3
Ak∞ = Ak + Mk (A0∞ − I7 ),
L=
631
∂ ∂ ∂ + M2 + M3 ∂t ∂x2 ∂x3
(M1 = 0, i.e., A1∞ = A1 ); we use the same notations as in Section 3, 5 (R = ρ/ ˆ ρˆ∞ , Mk = vˆk /c, ˆ etc.). R EMARK 6.1. The function S(t, x) (the entropy perturbation) solves the problem LS = 0,
t > 0, x ∈ R3± ;
S(0, x) = S0 (x),
x ∈ R3± .
Therefore, without loss of generality we can suppose that S(t, x) ≡ 0, x ∈ R3± . # + p[ #] = 0,45 generally speaking, δj = 0, and the strong discontinuity For the case [E] ˆ V ˆ is not contact (j = j + δj = δj = 0). But, from the formal point of view (for a full mathematical strictness), we should consider all the possible perturbations for the unperturbed discontinuous flow being a contact discontinuity. The LSP for the case # + p[ #] = 0 is formulated as follows.46 [E] ˆ V P ROBLEM 6.2. We seek the solutions of systems (6.2), (6.3) satisfying the boundary conditions [p] = 0,
[H] = 0,
LF =
[ρv ˆ 1] , [ρ] ˆ
hk [v1 ] − h1 [vk ] = 0 (k = 2, 3)
at x1 = 0 (t > 0, x ∈ R2 ) and the initial data (6.5) for t = 0. Likewise, S(t, x) ≡ 0 for Problem 6.2 (see Remark 6.1 above). 45 For a polytropic gas the condition [E] # + p[ #] = 0 doest not hold. ˆ V 46 Looking ahead we observe that this problem is underdetermined.
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Let us now examine the fulfilment of the evolutionarity condition (2.22) for Problems 6.1 and 6.2. P ROPOSITION 6.1. Problem 6.1 satisfies the evolutionarity condition whereas Problem 6.2 is underdetermined. P ROOF. We find that n+ (A1 ) = n− (A−1 0∞ A1∞ ) = 3. Therefore, at the first sight, Problem 6.1 is overdetermined (see Definition 2.5) because it has one boundary condition more than it needs for evolutionarity. But, on the other hand, one can show that the boundary condition [H1 ]|x1 =0 = 0 is, as a matter of fact, an additional requirement to the initial data (6.5), provided other boundary conditions in (6.4) hold. Indeed, the fifth equations of systems (6.2) and (6.3) give, at x1 = 0, the equation ∂ ∂ h1 [v2 ] − h2 [v1 ] + h1 [v3 ] − h3 [v1 ] = 0 L [H1] + ∂x2 ∂x3 that, with regard to [vk ]|x1 =0 = 0 (cf. (6.4)), becomes the form L([H1 ]) = 0 (at x1 = 0). Hence, the boundary condition [H1 ]|x1 =0 = 0 holds for t > 0 if it is valid at t = 0, and without loss of generality it can be excluded from (6.4). Thus, Problem 6.1 satisfies the evolutionarity condition (2.22). Analogous arguments show that Problem 6.2 is underdetermined.
6.2. Uniform stability of the MHD contact discontinuity Formulate the main stability theorem. T HEOREM 6.1. Problem 6.1 is well-posed, and its solutions satisfy the a priori estimates $ $ $ $ $U(t)$ ˜ 1 3 + $U(t)$ ˜ 1 3 K1 Q0 , W (R ) W (R )
(6.6)
$ $ $F (t)$ L
(6.7)
2
+
2 2 (R )
2
−
exp(t/2) F0 2L
2 2 (R )
1/2 + K2 1 − exp(−t) Q20 ,
where K1,2 > 0 are constant depending on coefficients of systems (6.2), (6.3); 1/2 Q0 = U0 2W 1 (R3 ) + U0 2W 1 (R3 ) , +
2
2
$ $ $ $ $U(t)$2˜ 1 3 = $U(t)$2 W (R ) L 2
±
2
−
$ $2 + $A1 Ux1 (t)$L (R3 ) ±
2
+ (R3 ) ±
3
$ $ $Ux (t)$2 k L k=2
3 2 (R± )
.
P ROOF. By the usual procedure (see Section 2), one gets the energy identity (2.45), with I (t) = U 2L
2
+ (R3 ) +
R3−
(A0∞ U, U) dx.
Stability of strong discontinuities in fluids and MHD
633
In view of the boundary conditions (6.4), the boundary integral in (2.25) is zero (conditions (6.4) are dissipative; see Definition 2.10), and we obtain the desired a priori estimate I (t) = I (0),
t > 0,
that can be rewritten as (2.50). So, we have an a priori L2 -estimate without loss of smoothness for the vector of perturbations U. But, to deduce an estimate for F we have to estimate Ut and Uxk . Differentiate (6.2)–(6.5) with respect to x2 and x3 . Then, reasoning as above, we get the following estimates $ $ $Ux (t)$2 k L
3 2 (R+ )
$ $2 + $Uxk (t)$L
3 2 (R− )
$ $ $ & K 2 %$ $(U0 )x $2 3 + $(U0 )x $2 3 1/2 , k L2 (R ) k L2 (R ) + − 2
(6.8)
k = 2, 3. Let us now differentiate systems (6.2), (6.3) and the boundary conditions with respect to t. In this case initial data for Ut are found from systems (6.2), (6.3). Then one can obtain the estimate $ $ $ $ $Ut (t)$2 3 + $Ut (t)$2 3 C1 Q1 , (6.9) L (R ) L (R ) 2
+
2
−
that, by expressing A1 Ux1 from systems (6.2), (6.3) and making use of (6.8), gives $ $ $A1 Ux (t)$2 1 L
3 2 (R+ )
$ $2 + $A1 Ux1 (t)$L
3 2 (R− )
C2 Q1 ,
(6.10)
where Q1 =
3
$ %$ $(U0 )x $2 k
k=1
L2 (R3+ )
$ $2 + $(U0 )xk $L
& 3 2 (R− )
;
C1,2 > 0 are constants depending on coefficients of systems (6.2), (6.3). Finally, (2.50) and (6.8)–(6.10) imply the desired a priori estimate (6.6). By making use of the next boundary condition in (6.4) and the property of the trace of a function in W21 (R3+ ) on the plane x1 = 0 (see [100]), one gets the inequality $ $ $ $ $ d$ $F (t)$2 2 $F (t)$2 2 + C3 $v1 (t)$2 1 3 , (R ) (R ) L L W2 (R+ ) 2 2 dt where C3 > 0 is a constant. The last inequality and (6.6) yield the a priori estimate (6.7) for the function F , with K2 = C3 K12 . This completes the proof of Theorem 6.1 (the a priori estimates (6.6), (6.7) point to the well-posedness of Problem 6.1). By Theorem 6.1, the MHD contact discontinuity is uniformly stable.
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7. Rotational discontinuity in MHD As for shock waves, the magnetofluid flows through the surface of a rotational discontinuity (see Definition 5.1). But, unlike shock waves, the density, the pressure, and the entropy are continuous on the rotational discontinuity (cf. (5.15)). Besides, the vector of the magnetic field rotates on the discontinuity whereas its absolute value remains constant. The linearized stability of the rotational discontinuity for an incompressible fluid was established by Syrovatskij in 1953 (see [84]). Concerning the case of compressible ideal MHD, up to now there is only the work of Blokhin and Trakhinin [34] where it was shown that the rotational discontinuity can be unstable. Namely, the rotational discontinuity proves to be strongly unstable under a strong magnetic field. Observe that this physically interesting fact has a principally multidimensional character because the 1-D rotational discontinuity is linearly stable.47
7.1. The LSP for the rotational discontinuity Suppose the MHD planar stationary discontinuity to be rotational. Then parameters of the unperturbed flow satisfy the conditions (cf. (5.15)) [ρ] ˆ = 0,
#22 H
#32 +H
# S = 0,
[p] ˆ = 0,
= 0,
#1 H vˆ1 = , 4π ρˆ
#1 = 0, H
(7.1)
# [H] [ˆv] = . 4π ρˆ
#1 = H #1∞ > 0, and [H #2 ]2 + Without loss of generality we suppose that vˆ1 = vˆ1∞ > 0, H #3 ]2 = 0. [H Taking account of (7.1), the linearization of the jump conditions (5.10)–(5.12) gives the boundary conditions for perturbations at x1 = 0: [δj ] = 0,
[δHN] = 0,
# H) (H, p+ = jˆ[δvN ], 8π
#2,3ρ] [H H2,3 v2,3 − + = 0, 4π ρˆ 2ρˆ 4π ρˆ
#2,3Q = 0, H
#2 + H #2 H T#[S] = 2 2 3 [ρ]. 4π ρˆ
(7.2)
Here δj = −vˆ1 ρ + ρ(F ˆ t + δvN ), δvN = −v1 + vˆ2 Fx2 + vˆ3 Fx3 , #2 Fx2 + H #3 Fx3 , δHN = −H1 + H
Q=
δHN vˆ1 δj − ρ. + ρˆ 4π ρˆ 2ρˆ
#2 , H #3 )∗ and (H #2∞ , H #3∞ )∗ are not collinear (i.e., in view of (7.1), If the vectors (H #2 , H #3∞ = −H #3 ), then conditions (7.2) imply Q = Q∞ , [ρ] = 0, [δvN ] = 0, #2∞ = −H H 47 The same takes place also for slow MHD shocks under a strong magnetic field (see Section 5).
Stability of strong discontinuities in fluids and MHD
635
etc., and one gets [p] = 0, [S],
H1 = 0, v1 − 4π ρˆ
# H = 0, [δHN ] = 0, H, Q = 0,
H2,3 ρ #2,3 = 0. H v2,3 − + 4π ρˆ 2ρˆ 4π ρˆ
We are now in a position to formulate the LSP (in a dimensionless form) for the rotational #2∞ = −H #2 , H #3∞ = −H #3 . discontinuity for the case H P ROBLEM 7.1 (LSP for the rotational discontinuity). We seek the solutions of the system vt + (h, ∇)w + ∇P = 0, pt + div(hp + v) = 0, Ht − (h, ∇)w + h div v = 0
(7.3)
for t > 0, x ∈ R3+ and the system pt + div(h∞ p + v) = 0,
vt + (h∞ , ∇)w + ∇P∞ = 0,
Ht − (h∞ , ∇)w + h∞ div v = 0
(7.4)
for t > 0, x ∈ R3− satisfying the boundary conditions [p] = 0,
[Hh ] = 0,
[H1 ] = [h2 ]Fx2 + [h3 ]Fx3 ,
[L] = 0, Ft = L1
(7.5)
at x1 = 0 (t > 0, x ∈ R2 ) and the initial data U(0, x) = U0 (x),
x ∈ R3± ,
F (0, x ) = F0 (x ),
x ∈ R2
(7.6)
for t = 0. Here U = (p, v∗ , H)∗ , w = v − H, P = p + Hh , P∞ = p + Hh∞ , Hh = (h, H), Hh∞ = (h∞ , H), L = (L1 , L2 , L3 )∗ = w + (p/2) h. While obtaining Problem 7.1 the Halileo transform t˜ = t,
x˜ = x − (M − h)t
(7.7)
was performed (by (7.1), M1 = M1∞ = h1 , [M2 ] = [h2 ], [M3 ] = [h3 ]). After that, for example, the boundary condition Q = 0, which in a dimensionless form reads H1 − h2 Fx2 − h3 Fx3 − v1 + M2 Fx2 + M3 Fx3 + Ft −
p M1 = 0, 2
has taken the form Ft = L1 (the tildes in Problem 7.1 were removed).
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R EMARK 7.1. The entropy perturbation S(t, x) solves the problem L1 S = 0,
t > 0, x ∈ R3+ ;
L1∞ S = 0, [S] = 0,
(7.8)
t > 0, x ∈ R3− ; x1 = 0,
t > 0,
S(0, x) = S0 (x),
(7.9) x ∈ R2 ;
x ∈ R3± ;
(7.10)
where L1 = ∂/∂t + (h, ∇), L1∞ = ∂/∂t + (h∞ , ∇). That is, without loss of generality one can suppose that S(t, x) ≡ 0, x ∈ R3± . #2 , H #3∞ = −H #3 , then the boundary conditions (7.2) yield Q + Q∞ = #2∞ = −H If H 0. In this case the LSP (in a dimensionless form and after performing the Halileo transform (7.7)) for the rotational discontinuity is formulated as follows. P ROBLEM 7.2. We seek the solutions of system (7.3), (7.8) for t > 0, x ∈ R3+ and system (7.4), (7.9) for t > 0, x ∈ R3− satisfying the boundary conditions h1 [p − S] − [v1 ] = 0, [H1 ] = [h2 ]Fx2 + [h3 ]Fx3 ,
1 [L] − [S h] = 0, 2 # 2 T h2 + h23 [p − S] = [S], #ρS ρˆ E
[p + Hh ] + h1 [v1 ] = 0,
Ft = v1 + H1 − 2h2 Fx2 − 2h3 Fx3 −
h1 p∞ − S∞ − 3(p − S) = 0 4
at x1 = 0 (t > 0, x ∈ R2 ) and the initial data (7.6), (7.10) for t = 0. − < It is easy to show that Problem 7.1 satisfies the evolutionarity condition (2.22) if cˆM + cˆA < cˆM , whereas Problem 7.2 is underdetermined. Thus, the rotational discontinuity is evolutionary except when the magnetic field turns on the discontinuity by the angle 180◦ . Since for Problem 7.1 h2∞ = −h2 , h3∞ = −h3 , h1 = h1∞ > 0, |h| = |h∞ | = q, without loss of generality we suppose that
h∞ = q(cos θ, sin θ, 0)∗ ,
∗ h = q cos θ, | sin θ | cos ϕ, | sin θ | sin ϕ ,
with |θ | < π/2, 0 < ϕ < 2π , ϕ = π . R EMARK 7.2. Setting formally θ = 0,48 i.e., h2 = h2∞ = h3 = h3∞ = 0, the boundary conditions (7.5)49 become dissipative (see [28,20] for more details). As in the proof of 48 This case, when the unperturbed flow is continuous (the perturbations H 2,3 and v2,3 can have a jump), is not physically interesting for the rotational discontinuity. 49 Actually, for this case the second boundary condition in (7.5) coincides with the penultimate one, but cˆ = cˆ− A M + or cˆA = cˆM , i.e., the rotational discontinuity remains evolutionary.
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637
Theorem 6.1, one can derive a priori estimates for the solutions of Problem 7.1. They are written in [28] and are like (6.6), (6.7). In what follows we will suppose that θ = 0. 7.2. The equivalent statement of Problem 7.1 Let us, following [34], reformulate Problem 7.1 as a LSP with a lesser number of unknown values. P ROBLEM 7.3. We seek the solutions of the systems pt + div Ω = 0,
(Hh )t + div q 2 Ω − hΩh = 0,
L2 Ω − h(h, ∇p) + ∇P = 0 for t > 0, x ∈ R3+ and the system pt + div Ω ∞ = 0,
(Hh∞ )t + div q 2 Ω ∞ − h∞ Ωh∞ = 0,
L2∞ Ω ∞ − h∞ (h∞ , ∇p) + ∇P∞ = 0 for t > 0, x ∈ R3− satisfying the boundary conditions
[p] = 0,
[Hh ] = 0,
p Ω − h =0 2
at x1 = 0 (t > 0, x ∈ R2 ) and the initial data for t = 0: p(0, x) = p0 (x), x ∈ R3± ; Hh (0, x) = h, H0 (x) , x ∈ R3+ ;
Hh∞ (0, x) = h∞ , H0 (x) ,
Ω(0, x) = v0 (x) − H0 (x) + hp0 (x),
x ∈ R3+ ;
Ω ∞ (0, x) = v0 (x) − H0 (x) + h∞ p0 (x),
x ∈ R3− ;
x ∈ R3− .
Here L2 = ∂/∂t + 2(h, ∇), L2∞ = ∂/∂t + 2(h∞ , ∇), Ωh = (h, Ω), Ωh∞ = (h∞ , Ω ∞ ), P = p + Hh , P∞ = p + Hh∞ ; p0 , H0 , v0 are the functions of initial data in (7.6), and div H0 = 0, x ∈ R3± . Let us comprehend the equivalence of Problems 7.1 and 7.3 in the following sense. If U and F solve Problem 7.1, then p,
Hh = (h, H),
Hh∞ = (h∞ , H),
Ω = w + hp,
Ω ∞ = w + h∞ p is a solution of Problem 7.3. Conversely, with a knowledge of p, Hh , Hh∞ , Ω, and Ω ∞ , we can define the functions F , vk , Hk so that F and U = (p, v∗ , H∗ )∗ is a solution of
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Problem 7.1. It is easily proved that if U(t, x) be a sufficiently smooth solution of Problem 7.1, then the functions p, Hh , Hh∞ , Ω, and Ω ∞ determined as above solve Problem 7.3. To prove the equivalence in the converse direction one has to perform a little bit more complicated arguments, and we refer the reader to [34]. So, assume that we have the following theorem (see [34] for the detailed proof ). T HEOREM 7.1. Problems 7.1 and 7.3 are equivalent. For further convenience, we rewrite Problem 7.3 as follows. Introduce the new dependent and independent variables: t = qt,
Ω = qΩ ,
Hb = qHb ,
Ω ∞ = qΩ ∞ ,
Hb∞ = qHb∞ ,
where b = εh, b∞ = εh∞ , ε = 1/q, Hb = (b, H), Hb = (b, H ). Then, by omitting the primes, we reformulate Problem 7.3 in the following form. P ROBLEM 7.4. We seek the solutions of the system pt + L1 Ωb + L2 Ωσ + L3 Ωl = 0, (Hb )t + L2 Ωσ + L3 Ωl = 0, (Ωb )t + 2L1 Ωb + ε2 − 1 L1 p + L1 Hb = 0,
(7.11)
(Ωσ )t + 2L1 Ωσ + ε2 L2 p + L2 Hb = 0, (Ωl )t + 2L1 Ωl + ε2 L3 p + L3 Hb = 0 for t > 0, x ∈ R3+ and the system pt + L1∞ Ωb∞ + L2∞ Ωσ ∞ + L3∞ Ωl∞ = 0, (Hb∞ )t + L2∞ Ωσ ∞ + L3∞ Ωl∞ = 0, (Ωb∞ )t + 2L1∞ Ωb∞ + ε2 − 1 L1∞ p + L1∞ Hb∞ = 0,
(7.12)
(Ωσ ∞ )t + 2L1∞ Ωσ ∞ + ε2 L2∞ p + L2∞ Hb∞ = 0, (Ωl∞ )t + 2L1∞ Ωl∞ + ε2 L3∞ p + L3∞ Hb∞ = 0 for t > 0, x ∈ R3− satisfying the boundary conditions [p] = 0,
[Hb] = 0,
p b Ωb − + σ Ωσ + lΩl = 0 2
at x1 = 0 (t > 0, x ∈ R2 ) and corresponding initial data for t = 0. Here Ω = bΩb + σ Ωσ + lΩl ,
Ω ∞ = b∞ Ωb∞ + σ ∞ Ωσ ∞ + l∞ Ωl∞ ,
(7.13)
Stability of strong discontinuities in fluids and MHD
L1 = (b, ∇),
L2 = (σ , ∇),
L1∞ = (b∞ , ∇),
639
L3 = (l, ∇),
L2∞ = (σ ∞ , ∇),
L3∞ = (l∞ , ∇);
the sets of the vectors b, σ , l and of the vectors b∞ , σ ∞ , l∞ are orthonormal, namely, ∗ b = cos θ, | sin θ | cos ϕ, | sin θ | sin ϕ , ∗ σ = −| sin θ |, cos θ cos ϕ, cos θ sin ϕ , l = (0, − sin ϕ, cos ϕ)∗ ,
b∞ = (cos θ, sin θ, 0)∗ , σ ∞ = (− sin θ, cos θ, 0)∗ ,
l∞ = (0, 0, 1)∗.
Suppose for definiteness that 0 < θ < π/2 (for the case θ = 0 see Remark 7.2).
7.3. Instability of the rotational discontinuity under a strong magnetic field Let the magnetic field be rather strong, i.e., q ' 1 or ε 1. Taking account of Theorem 7.1, the instability of the rotational discontinuity for a strong magnetic field will follow from the theorem below. T HEOREM 7.2. Problem 7.4 is ill-posed under ε 1. P ROOF. We will construct an ill-posedness example of Hadamard type for Problem 7.4. That is to say, we look for the sequence of exponential solutions to systems (7.11), (7.12) and the boundary conditions (7.13) in form (2.24). Reasoning as in the proof of Theorem 5.3 (see [34] for more details), we find that there are four roots ξk (ξk = (0) (1) ξk(0) + ξk(1) ε + ξk(2) ε2 + · · · , k = 1, 4) and one ξ∞ (ξ∞ = ξ∞ + ξ∞ ε + ξk(2) ε2 + · · ·): (0)
ξ1,2 = −
τ (0) + iσ1 ra , b1
(1)
ξ1,2 = −
τ (1) ± τ (0) , b1
τ (2) ∓ τ (1) + τ (0) τ + 2iσ1 ra , ξ3 = − , b1 2b1 (0) − (τ˜ (0) )2 + σ 2 b τ ˜ 1 τ (1) L(0) 1 (0) (1) 1 ξ4 = , ξ = − , 4 (0) (0) σ12 τ˜ b1 − σ12 ξ4 (0) (0) (0) τ ˜ − (τ˜∞ )2 + σ12 b 1 ∞ τ (1)L1∞ (0) (1) ξ∞ = , ξ = − , ∞ (0) (0) σ12 τ˜∞ b1 − σ12 ξ∞ (2) ξ1,2 =−
(0) where b1 = cos θ , σ1 = sin θ , ra = γ2 cos ϕ + γ3 sin ϕ, τ˜ (0) = τ (0) + iσ1 ra , τ˜∞ = τ (0) + (0) (0) (0) (0) (0) (0) iσ1 γ2 , L1 = τ˜ + b1 ξ4 , L1∞ = τ˜ + b1 ξ∞ .
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We finally look for the sequence of exponential solutions in the form Un =
% & U(0) ∞ exp n τ t + ξ∞ x1 + i(γ , x ) , % & 4 (k) k=1 U exp n τ t + ξk x1 + i(γ , x ) ,
x1 < 0; x1 > 0.
(0)
Here U(k) and U∞ are real constant nonzero vectors. In particular, ∗ −irl U(3) = 0, 0, 0, (3) Ωl(3), Ωl(3) , Σ
Σ (3) =
σ1 τ + 2ira , 2b1
Ωl(3) = const,
rl = −γ2 sin ϕ + γ3 cos ϕ. Then, reasoning - as in Section 5 (see the proof.of Theorem 5.3), ∗
∗
∗
for finding the constant vector W = U(1) , U(2) , U(4) , Ωl(3), U(0) ∞ linear algebraic system Γ W = 0,
∗ ∗
we obtain the
(7.14)
where Γ = (γij )i,j =1,21 is a quadratic matrix (because of its complication, we do not present it here). System (7.14) has a nontrivial solution if det Γ = 0. By expanding the last equality into series in ε, one gets det Γ = εD (1) + O ε2 = 0,
(7.15)
the dispersion relation D (1) = 0 (it is written in details in [34]) has the root τ (0) = (0) b1 − iσ1 ra (it solves the equation L1 = 0) that has the necessary property Re τ (0) > 0 (b1 = cos θ > 0). By the smallness of ε, there exists a root τ of Equation (7.15) with Re τ > 0. This completes the proof of Theorem 7.2. Thus, we conclude the instability of the rotational discontinuity under a strong magnetic field.
8. Instability of the MHD tangential discontinuity As the MHD contact discontinuity, the tangential one is characterized by the absence of a mass transfer flux across the discontinuity surface. As for the tangential discontinuity in gas dynamics [85] (see Section 3), on the surface of the MHD tangential discontinuity the density and the tangent components vτi of the velocity (as well as of the magnetic field, Hτi ) can have arbitrary jumps (see Section 5). The linearized stability of the tangential discontinuity in gas dynamics was being studied as far back as 1944 by Landau [83]. The final conclusion on the instability of the tangential discontinuity has been drawn by Syrovatskij [115] (see also [85]). The issue of the stability of the tangential discontinuity in MHD of an incompressible fluid was examined by Polovin
Stability of strong discontinuities in fluids and MHD
641
and Demutskij [103]. It was shown that the tangential discontinuity can be unstable, and some domains of linearized stability were found (see [103]). The stability of the tangential discontinuity in an ideal compressible magnetofluid has been fully investigated by Blokhin and Druzhinin [25,20]. The MHD tangential discontinuity is found to be uniformly stable only in the specific case of “standing” discontinuity (the unperturbed fluid is immovable; see Remark 8.1). Relating to the domain of parameters of the unperturbed fluid, except the mentioned specific case, almost everywhere in this domain one can construct an ill-posedness example of Hadamard type (see this section below, and we refer to [25,20] for more details). Such an example cannot be constructed only on a hypersurface of this domain. But, as we will see below, this point is proved to be that of neutral stability. Namely, for all the near points one can construct an ill-posedness example. Thus, the MHD tangential discontinuity is almost always strongly unstable.
8.1. The LSP for the MHD tangential discontinuity For the planar and stationary MHD tangential discontinuity, the piecewise constant solution satisfies conditions (5.14) which are written in this case as vˆ1 = vˆ1∞ = 0,
#1 = H #1∞ = 0, H
# 2 |H| pˆ + = 0. 8π
(8.1)
By virtue of (8.1), after the linearization of the jump conditions (5.10)–(5.12) one gets the boundary conditions for perturbations at x1 = 0:
[δj ] = 0,
[δHN ] = 0,
[vˆ2,3 ]δj −
δHN # H2,3 = 0, 4π
p+
# H) (H, = 0, 8π
#H #2,3 δj − δHN [vˆ2,3 ] = 0, V # 2 # 2 |ˆv|2 |H| |H| # = 0, # # δj − δHN vˆ , H + E+ + pˆ + V 2 8π ρˆ 8π 4π
(8.2)
where δj and δHN are small perturbations of j and HN (see Section 7). The last five equations in (8.2) can be considered as an algebraic system for finding δj and δHN . In the case when the rank of its matrix is greater than unit, δj = 0, δHN = 0, and the LSP (in a dimensionless form) for the MHD tangential discontinuity is the following.
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P ROBLEM 8.1 (LSP for the MHD tangential discontinuity). We seek the solutions of the system (6.2) for t > 0, x ∈ R3+ and the system (6.3) for t > 0, x ∈ R3− satisfying the boundary conditions v1 = LF, v1∞ = L∞ F, H1∞ = h2∞ Fx2 + h3∞ Fx3 ,
H1 = h2 Fx2 + h3 Fx3 , [p + h2 H2 + h3 H3 ] = 0
(8.3)
at x1 = 0 (t > 0, x ∈ R2 ) and the initial data (6.5) for t = 0. Here U = (p, v∗ , H∗ )∗ ; the matrices Ak = Ak (h1 , h2 , h3 ), A0∞ , and Ak∞ = Ak∞ (h1 , h2 , h3 , M2 , M3 ) (depending on parameters hk , M2,3 ) in systems (6.2), (6.3) are written in Section 6, and for Problem 8.1, we have: Ak = Ak (0, h2 , h3 ), Ak∞ = Ak∞ (0, h2∞ , h3∞ , M2∞ , M3∞ ). The differential operator L∞ = ∂/∂t + M2∞ ∂/∂x2 + M3∞ ∂/∂x3 , and all the other notations are the same as in Section 6. Note also that, as for Problems 6.1 and 7.1 (see Remarks 6.1, 7.1), by the same arguments, we can suppose, without loss of generality, that S(t, x) ≡ 0, x ∈ R3± . For the cases when system (8.2) imply δj = 0 and/or δHN = 0, the strong discontinuity is, generally speaking, not tangential (although, the “unperturbed” planar discontinuity is tangential). For such cases more three LSP’s were obtained in [25,20]. But, as was shown in [25,20], all these LSP’s are equivalent to Problem 8.1 under some additional restrictions on the initial data (6.5). For example, two of these LSP’s are equivalent to Problem 8.1 if the initial data (6.5) are such that the first boundary condition in (8.3) holds for t = 0: (v1 − LF )|t =0 = 0,
x1 = 0,
x ∈ R2
(8.4)
(see [25,20] for more details). According to Definition 2.4, Problem 8.1 needs three boundary conditions to satisfying the evolutionarity condition (2.22) (n+ (A1 ) = n− (A1∞ ) = 1). So, at first sight, Problem 8.1 is overdetermined (see Definition 2.5). But, let us show that under some restrictions on the initial data (6.5) Problem 8.1 has exactly three independent boundary conditions (cf. Remark 2.3), i.e., the tangential discontinuity is evolutionary. For this purpose, in [25,20] one formulates a certain LSP and proves its equivalence (under some restrictions on the initial data) to Problems 8.1. This LSP is the following. P ROBLEM 8.2. We seek the solutions of systems (6.2), (6.3)50 satisfying the boundary conditions R(v1 − LF ) = v1∞ − L∞ F,
H1 = h2 Fx2 + h3 Fx3 ,
[p + h2 H2 + h3 H3 ] = 0 at x1 = 0 (t > 0, x ∈ R2 ) and the initial data (6.5) for t = 0. 50 Recall that A = A (0, h , h ), A k k k∞ = Ak∞ (0, h2∞ , h3∞ , M2∞ , M3∞ ) (see above). 2 3
(8.5)
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643
P ROPOSITION 8.1. Problems 8.1 and 8.2 are equivalent if the initial data (6.5) satisfy (8.4) and the requirement (H1∞ − h2∞ Fx2 − h3∞ Fx3 )|t =0 = 0,
x1 = 0,
x ∈ R2 .
(8.6)
P ROOF. Let there exists a sufficiently smooth solution of Problem 8.2. If we act to the second boundary condition in (8.5) by the differential operator L, then, with regard to the fifth equation of system (6.2) considered at x1 = 0, one gets the relation (h2 ∂/∂x2 + h3 ∂/∂x3 )(v1 − LF )|x1 =0 = 0. If the initial data (6.5) satisfy (8.4), then, as follows from the last relation, the first boundary condition in (8.3) is valid for all t > 0. This condition and (8.6) give the second condition in (8.3) which, with regard to the fifth equation of system (6.3) considered at x1 = 0, yields the relation L∞ (H1∞ − h2∞ Fx2 − h3∞ Fx3 )x1 =0 = 0. If the initial data (6.5) are such that the fourth boundary condition in (8.3) holds for t = 0, i.e., requirement (8.6) is satisfied, then the last relation imply that the fourth boundary condition in (8.3) is fulfilled for all t > 0. Thus, under restrictions (8.4) and (8.6) on the initial data (6.5) Problems 8.1 and 8.2 are equivalent. C OROLLARY 8.1. If the initial data for the LSP, Problem 8.1, satisfy requirements (8.4) and (8.6), then the MHD tangential discontinuity is evolutionary. conditions (8.3) of Problem 8.1 R EMARK 8.1. It is easily verified that the boundary become dissipative if we formally set M2,3 = 0, i.e., [vˆ2,3 ] = 0.51 Then, literally repeating arguments from the proof of Theorem 6.1, we obtain the a priori estimates which coincide with (6.6), (6.7), i.e., the MHD tangential discontinuity for such a specific case is uniformly stable.
8.2. Ill-posedness of Problem 8.1 The following theorem that was proved by Blokhin and Druzhinin [25,20] says that the MHD tangential discontinuity is almost always unstable (and it cannot be uniformly stable). T HEOREM 8.1. If the vectors h = (h2 , h3 ), h∞ = (h2∞ , h3∞ ), and M = (M2 , M3 ) are parallel to each other, then the MHD tangential discontinuity is neutrally stable. Otherwise, Problem 8.1 is ill-posed, i.e., the MHD tangential discontinuity is unstable. P ROOF. Without loss of generality suppose that M2∞ = M3∞ = 0. For constructing an ill-posedness example of Hadamard type for Problem 8.1 we look for the exponential solutions to systems (6.2), (6.3) and the boundary conditions (8.3) in form (2.24), (2.25). Moreover, for technical convenience and to compare the present investigations with those of [83,115,103] we set in formulae (2.24), (2.25) τ = −iω, i.e., the constants ω, ξ , and ξ∞ 51 For this case we can choose a reference frame in which vˆ = vˆ 2 2∞ = vˆ 3 = vˆ 3∞ = 0, i.e., the unperturbed fluid is immovable.
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should satisfy the inequalities Im ω > 0, Re ξ∞ > 0, Re ξ < 0. By usual arguments (see the proof of Theorem 5.3 and [25,20] for more details), we find ξ2 = 2 ξ∞
(ω − υ)4 + 1, q 2 cos2 θ0 − (1 + q 2 )(ω − υ)2
(8.7)
ω4 = + 1, 2 cos2 θ − (c 2 + Rq 2 )ω2 Rc0 2 q∞ ∞ 0 ∞
where υ = (γ , M ) = |M | cos θ ,52 q = |h |, q∞ = |h∞ |, (γ , h ) = q cos θ0 , (γ , h∞ ) = q cos θ∞ . By substituting (2.24), (2.25) into the boundary conditions (8.3) and making use of systems (6.2), (6.3) (we omit technical details and refer to [25,20]), one gets the dispersion relation 2 ξ∞ q 2 cos2 θ0 − (ω − υ)2 R = ξ Rq∞ cos2 θ∞ − ω2 ,
(8.8)
where ξ and ξ∞ are connected by (8.7). So, the problem of constructing an ill-posedness example is reduced to the algebraic problem of finding the constants ω, ξ , ξ∞ (Im ω > 0, and Re ξ∞ > 0, Re ξ < 0) which satisfy (8.7), (8.8). Let, at the beginning, the magnetic field be rather weak, i.e., q 1, q∞ = αq, α = O(1). √ We choose the values γ2 and γ3 so that υ = q. Let us seek the solution of the algebraic system (8.7), (8.8) in the form of the expansions ω=
√ q(ω0 + ω1 q + · · ·),
ξ = −1 + ξ1 q + · · · ,
ξ∞ = 1 + ξ1∞ q + · · · .
(8.9)
Substituting these expansions into system (8.7), (8.8), we easily find ω0 with the desired property Im ω0 > 0: √ R+i R ω0 = . 1+R
(8.10)
Thus, the MHD tangential discontinuity under a weak magnetic field is unstable. Consider now the case when the vectors h and h∞ are collinear, and the vector γ is orthogonal to them, i.e., cos θ0 = cos θ∞ = 0. In this case system (8.7), (8.8) becomes53
ξ2 = 1 −
(ω − υ)2 , q2 + 1
2 ξ∞ =1−
ω2 , + c0 2
2 Rq∞
ξ∞ (ω − υ)2 R = ξ ω2 . (8.11)
52 We suppose |γ | = 1 (see Remark 2.8). 53 System (8.11) under q = q = 0 coincides with the analogous one arising under the investigation of the ∞
tangential discontinuity in gas dynamics [83,115,85].
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Then, if υ = 0, i.e., the vectors h , h∞ , and M are parallel to each other, one cannot construct an ill-posedness example. Let υ = ε > 0 with ε being a small value (ε 1). We look for the solution to system (8.11) in the form of the following expansions: ω = ε(ω0 + ω1 ε + · · ·),
ξ = −1 + ξ1 ε + · · · ,
ξ∞ = 1 + ξ1∞ ε + · · · .
By substituting these expansions into system (8.11), one gets ω0 in the form of (8.10). That is, the ill-posedness example is constructed under ε 1, whereas it cannot be constructed for ε = 0. Therefore, the hypersurface υ = 0 belongs to the class of neutral stability (see Section 2), i.e., the ULC is violated. Let us now consider a more general situation. Suppose the vectors h∞ and M are not √ collinear. We choose the vector γ being orthogonal to h∞ (θ∞ = ±π/2) and υ = q. Since the vectors h∞ and M are not collinear, this can be done and does not superimpose any requirements on the vectors h , h∞ , and M . If the magnetic field under x1 > 0 is rather small (q 1), the solution of problem (8.7), (8.8) is found in the form of expansions (8.9), with ω0 given by (8.10). By excluding the values ξ and ξ∞ from (8.7), (8.8), one gets R R2 − 2 ω4 (q∞ + c0 2 )ω2 =
(ω − υ)4 − (q 2 + 1)(ω − υ)2 + q 2 cos2 θ0 . {q 2 cos2 θ0 − (q 2 + 1)(ω − υ)2 }{q 2 cos2 θ0 − (ω − υ)2 }2
The last equation is the algebraic one of order 8 with respect to ω. On the other hand, the real roots of this equation are the points of intersection of two curves, one of which is moved to υ from the origin. Moreover, by an appropriate choice of υ, the number of intersections remains constant while the value q changes. Since for the special case q 1 examined above the number of intersections is, at least, not greater than 6 (there is a pair of complex conjugate roots, one of which has the desired property Im ω > 0), the illposedness example of Hadamard type is constructed for the case when the vectors h∞ and M are not collinear. For the remaining case when the vectors h and M are not collinear, one can also apply arguments as above, that gives an ill-posedness example. So, if we are outside of the situation when the vectors h , h∞ , and M are parallel to each other,54 then the MHD tangential discontinuity is always unstable. R EMARK 8.2. In [103] the stability of the MHD tangential discontinuity was examined for an incompressible fluid. In this case the first equations of systems (6.2) and (6.3) are replaced by div v = 0. The investigation of the tangential discontinuity is strongly simplified because we at ones obtain that ξ = −1 and ξ∞ = 1. The further analysis is performed analogously as above, and one obtains the quadratic equation for ω: (1/R)ω2 + (ω − υ)2 = a 2 , 54 Recall that the ULC is violated for this special case.
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with a 2 = (γ , h )2 +(γ , h∞ )2 . All the roots ω of this equation are real if (1+R)a 2 −υ 2 > 0. As note in [103], if we are outside of the situation when the vectors h , h∞ , and M are 2 ) > |M |2 an ill-posedness parallel to each other, then under the condition (1 + R)(q 2 + q∞ example cannot be constructed. However, we observe that one succeeds in constructing an ill-posedness example if the vectors h and h∞ are collinear and the vector M is not parallel to them. Indeed, in that case we choose the vector γ being orthogonal to h and h∞ . Then a = 0, and the quadratic equation for ω has complex conjugate roots. 9. Open problems The linearized stability analysis is of decisive importance in studying the stability of strong discontinuities in ideal fluids. But there are still a lot of, basically practical, open problems even for the linear analysis. As was noted in Section 5, the problem of finding the domains of instability and uniform/neutral stability for MHD shock waves (both fast and slow shocks) has been solved only for several special cases and remains open. We know about the MHD rotational discontinuity only that it is unstable under a strong magnetic field (see Section 7). Likewise, there are many other models of continuum mechanics in which the linearized stability of strong discontinuities is of great (both theoretical and practical) importance but has not been fully investigated as yet. This is, for example, relativistic MHD and anisotropic MHD of Chew, Goldberger and Low (some special cases for shock waves and rotational discontinuities have been analyzed in [35,36,39,119]). The main difficulty in the investigation of linearized stability lies in writing out the Lopatinski determinant. Actually, often (or even as a rule) it is technically impossible to obtain and test analytically the LC and the ULC for a concrete model of fluid dynamics and a concrete type of strong discontinuity. In this connection, it would be extremely important to create a computation software package for finding the domains of fulfilment/omission of the LC/ULC for LSP’s. This is not so simply and not just a technical matter as may seem at the first sight. First attempts in this direction were made for MHD shock waves in [90,53] where Hadamard-type ill-posedness (instability) examples were constructed numerically.55 It seems in certain cases one can numerically show the violation of the LC, but it would be much more interesting and important to be able to establish, by numerical analysis, the fulfilment of the LC, and especially the ULC. The creation of such a software is of great importance for applications. On the other hand, there are also some theoretical open problems in the linearized stability analysis. For example, it would be quite interesting to know whether “layerwise” a priori W22 -estimates like (2.38) are valid for the LSP for an arbitrary uniformly stable strong discontinuity. As is known, uniform stability implies the existence of Kreiss’ “weighted” a priori estimates. But, as was already pointed, estimates of a “layer-wise” type with standard Sobolev norms are more preferable, in particular, for a possible subsequent numerical analysis of the LSP. 55 We also refer to completely new results that are not included in this chapter and were obtained Trakhinin [120] during the publication process. In [120] an algorithm of numerical testing the ULC for LSP’s for 1-shocks was suggested. By means of this algorithm a complete 2-D stability analysis of fast MHD shocks in a polytropic gas was first carried out.
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At a theoretical level, it would be of interest to prove a local well-posedness theorem, like those in [15,17,19,92–94], for the nonlinear, free boundary, stability problem for an arbitrary uniformly stable strong discontinuity (for a quite arbitrary hyperbolic system of conservation laws and not only for k-shocks). But, an extremely significant direction is to understand what happens with neutrally stable discontinuities on the initial nonlinear level. This is a very difficult problem, and, at present, it even seems to be resistant to solving because, in this connection, one has to study the initial nonlinear, free boundary and multidimensional, stability problem. On the other hand, it is clear that a part of the neutral stability domain should be that of a kind of nonlinear instability,56 and the rest of the domain is that of structural (possibly, in some sense weakened) stability. That is, for “structurally stable” part of the neutral stability domain the loss of derivatives phenomenon is just a technical property appearing on the linearized level.57 However, an open problem is to understand how separate such domains of real stability for neutrally stable discontinuities. Finally, we observe that, perhaps, the viscous multidimensional stability analysis (see, e.g., [127]) could give a certain information for understanding the physical character of neutrally stable strong discontinuities.
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are analyzed according to the weakly nonlinear analysis, and their complex physical behavior such as Mach stem formation is shown. 57 We refer once more to the work of Sablé-Tougeron [111] in which the loss of derivatives phenomenon has been overcome in the nonlinear analysis (the nonlinear well-posedness domain in [111] seems to coincide with that of fulfilment of the LC, and a kind of nonlinear ill-posedness does not appear.
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[12] H.A. Bethe, On the theory of shock waves for an arbitrary equation of state, Office of Scientific Research and Development, Report No. 545 (1942), Classic Papers in Shock Compression Science, Springer, New York (1982), 421–492. [13] A.M. Blokhin, A mixed problem for a system of equations of acoustics with boundary conditions on a shock wave, Izv. Sibirsk. Otdel. Akad. Nauk SSSR Ser. Tekhn. Nauk 13 (1979), 25–33 (Russian). [14] A.M. Blokhin, A mixed problem for a three-dimensional system of equations of acoustics with boundary conditions on the shock wave, Dinamika Sploshn. Sredy 46 (1980), 3–13 (Russian). [15] A.M. Blokhin, Estimation of the energy integral of a mixed problem for gas dynamics equations with boundary conditions on the shock wave, Sibirsk. Mat. Zh. 22 (4) (1981), 23–51 (Russian); English translation: Siberian Math. J. 22 (4) (1981), 501–523. [16] A.M. Blokhin, A mixed problem for symmetric t-hyperbolic systems of acoustic type, Dinamika Sploshn. Sredy 52 (1981), 11–29 (Russian). [17] A.M. Blokhin, Uniqueness of the classical solution of a mixed problem for equations of gas dynamics with boundary conditions on a shock wave, Sibirsk. Mat. Zh. 23 (5) (1982), 17–30 (Russian); English translation: Siberian Math. J. 23 (5) (1982), 604–615. [18] A.M. Blokhin, Symmetrization of Landau equations in the theory of superfluidity of helium II, Dinamika Sploshn. Sredy 68 (1984), 13–34 (Russian). [19] A.M. Blokhin, Energy Integrals and their Applications in Problems of Gas Dynamics, Nauka, Sibirsk. Otdel., Novosibirsk (1986) (Russian). [20] A.M. Blokhin, Strong Discontinuities in Magnetohydrodynamics, Nova Science Publ., New York (1994). [21] A.M. Blokhin, Symmetrization of continuum mechanics equations, Sib. J. Differential Equations 2 (1995), 3–47. [22] A.M. Blokhin, A new concept of construction of adaptive calculation models for hyperbolic problems, NATO ASI Ser., Ser. C, Math. Phys. Sci. 536 (1999), 23–64. [23] A.M. Blokhin and R.D. Alaev, Construction of adequate difference models for gas dynamics equations, Siberian J. Comput. Math. 1 (2) (1992), 169–189. [24] A.M. Blokhin and V.N. Dorovsky, Mathematical Modelling in the Theory of Multivelocity Continuum, Nova Science Publ., New York (1995). [25] A.M. Blokhin and I.Yu. Druzhinin, Formulation of problems on the stability of discontinuities in magnetohydrodynamics, Boundary Value Problems for Partial Differential Equations, Collect. Sci. Works, Novosibirsk (1988), 16–38 (Russian). [26] A.M. Blokhin and I.Yu. Druzhinin, On the stability of a fast magnetohydrodynamic shock wave for a weak magnetic field, Partial Differential Equations, Collect. Sci. Works, Novosibirsk (1989), 15–32 (Russian). [27] A.M. Blokhin and I.Yu. Druzhinin, Stability of shock waves in magnetohydrodynamics, Sibirsk. Mat. Zh. 30 (4) (1989), 13–29 (Russian); English translation: Siberian Math. J. 30 (4) (1989), 511–524. [28] A.M. Blokhin and I.Yu. Druzhinin, Well-posedness of some linear problems on the stability of strong discontinuities in magnetohydrodynamics, Sibirsk. Mat. Zh. 31 (2) (1990), 3–8 (Russian); English translation: Siberian Math. J. 31 (2) (1990), 187–191. [29] A.M. Blokhin and D.A. Krymskikh, Symmetrization of equations of magnetohydrodynamics with anisotropic pressure, Boundary Value Problems for Partial Differential Equations, Collect. Sci. Works, Novosibirsk (1990), 3–19 (Russian). [30] A.M. Blokhin and D.A. Krymskikh, Strong discontinuities in superliquid helium, Proc. Inst. Math. Novosibirsk 24 (1994), 20–62. [31] A.M. Blokhin and E.V. Mishchenko, Symmetrization of relativistic equations of gas dynamics, Dinamika Sploshn. Sredy 88 (1988), 13–22 (Russian). [32] A.M. Blokhin and E.V. Mishchenko, Investigation on shock waves stability in relativistic gas dynamics, Matematiche (Catania) 48 (1993), 53–75. [33] A.M. Blokhin and I.G. Sokovikov, On a certain approach to constructing difference schemes for quasilinear equations of gas dynamics, Sibirsk. Mat. Zh. 40 (6) (1999), 1236–1243 (Russian); English translation: Siberian Math. J. 40 (6) (1999), 1044–1050. [34] A.M. Blokhin and Yu.L. Trakhinin, A rotational discontinuity in magnetohydrodynamics, Sibirsk. Mat. Zh. 34 (3) (1993), 3–18 (Russian); English translation: Siberian Math. J. 34 (3) (1993), 395–411.
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[61] K.O. Friedrichs and P.D. Lax, System of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 1686–1688. [62] C.S. Gardner and M.D. Kruskal, Stability of plane magnetohydrodynamic shocks, Phys. Fluids 7 (1964), 700–706. [63] D. Gilbarg, The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math. 73 (1951), 256–274. [64] S.K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521–523 (Russian). [65] S.K. Godunov, Symmetrization of magnetohydrodynamics equations, Chislennye Metody Mekhaniki Sploshnoi Sredy, Novosibirsk 3 (1972), 26–34 (Russian). [66] S.K. Godunov, Equations of mathematical physics, Nauka, Moscow (1979) (Russian); French translation: Equations de la Physique Mathematique, traduit du russe par Edouard Gloukhian, Mir, Moscou (1973). [67] S.K. Godunov and V.M. Gordienko, A mixed problem for the wave equation, Trudy Sem. S.L. Soboleva 2 (1977), 5–31 (Russian). [68] S.K. Godunov, A.V. Zabrodin, M.Ya. Ivanov, A. Krajko and G.P. Prokopov, Numerical Solving of Multidimensional Gas Dynamics Problems, Nauka, Moscow (1976) (Russian); French translation: Resolution Numerique des Problemes Multidimensionnels de la Dynamique des Gaz, sous la redaction de S. Godounov, traduit du russe par Valeri Platonov, Mir, Moscou (1979). [69] V.M. Gordienko, Symmetrization of a mixed problem for a hyperbolic equation of second order with two spatial variables, Siberian Math. J. 22 (1981), 231–248. [70] A. Harten, On the symmetric form of systems of conservation laws with entropy, J. Comp. Phys. 49 (1) (1983), 151–164. [71] F. Hoffman and E. Teller, Magnetohydrodynamic shocks, Phys. Rev. 80 (4) (1950), 696–703. [72] S.V. Iordanskii, On the stability of a planar steady shock wave, Prikl. Mat. Mekh. 21 (1957), 465–472 (Russian). [73] S.V. Iordanskii, On compression waves in magnetohydrodynamics, Dokl. Akad. Nauk SSSR 121 (1958), 610–612 (Russian); English translation: Soviet Phys. Dokl. 3 (1959), 736–738. [74] E.L. Isaacson, D. Marchesin and B.J. Plohr, Transitional waves for conservation laws, SIAM J. Math. Anal. 21 (1990), 837–866. [75] A. Jeffrey, Quasilinear Hyperbolic Systems and Waves, Pitman, New York (1976). [76] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), 181–205. [77] V.M. Kontorovich, On the shock waves stability, Zh. Eksp. Teor. Fiz. 33 (1957), 1525–1526 (Russian); English translation: Soviet Phys. JETP 33 (6) (1959), 1179–1180. [78] V.M. Kontorovich, Stability of shock waves in relativistic hydrodynamics, Zh. Eksp. Teor. Fiz. 34 (1958), 186–194 (Russian); English translation: Soviet Phys. JETP 34 (7) (1960), 127–132. [79] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–296. [80] A.G. Kulikovskii and G.A. Lyubimov, Gas-ionizing magnetohydrodynamic shock waves, Dokl. Akad. Nauk SSSR 129 (1959), 52–55 (Russian); English translation: Soviet Phys. Dokl. 4 (1960), 1185–1188. [81] A.G. Kulikovskii and G.A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, MA (1965). [82] O.A. Ladyzhenskaja, Boundary-Value Problems of Mathematical Physics, Nauka, Moscow (1973) (Russian). [83] L.D. Landau, On the stability of tangential discontinuities in a compressible fluid, Dokl. Akad. Nauk SSSR 44 (4) (1944), 151–153 (Russian). [84] L.D. Landau and E.M. Lifshiz, Electrodynamics of Continuous Media, Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford (1960). [85] L.D. Landau and E.M. Lifshiz, Fluid Mechanics, Course of Theoretical Physics, Vol. 6, Pergamon Press, New York (1997). [86] P. Lankaster, Theory of Matrices, Academic Press, New York (1969). [87] P.D. Lax, Hyperbolic systems of conservation laws (II), Comm. Pure Appl. Math. 10 (1957), 537–566. [88] P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM Reg. Conf., No. 11, Philadelphia (1973).
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[89] P.D. Lax and P.S. Philips, Local boundary conditions for dissipative symmetric linear operators, Comm. Pure Appl. Math. 13 (1960), 427–455. [90] M. Lessen and M.V. Deshpande, Stability of magnetohydrodynamic shocks waves, J. Plasma Physics 1 (4) (1967), 463–472. [91] T.P. Liu, Nonlinear stability and instability of overcompressive shock waves, IMA Vol. Math. Appl. 52 (1993), 159–167. [92] A. Majda, The stability of multi-dimensional shock fronts – a new problem for linear hyperbolic equations, Mem. Amer. Math. Soc. 41 (275) (1983). [93] A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 43 (281) (1983). [94] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, New York (1984). [95] A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math. 28 (1975), 607–675. [96] A. Majda and R. Rosales, A theory for spontaneous Mach stem formation in reacting shock fronts. I, The basic perturbation analysis, SIAM J. Appl. Math. 43 (1983), 1310–1334. [97] A. Majda and R. Rosales, A theory for spontaneous Mach-stem formation in reacting shock fronts. II. Steady-wave bifurcations and the evidence for breakdown, Stud. Appl. Math. 71 (1984), 117–148. [98] G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc. 32 (2000), 689–702. [99] G. Métivier, Stability of Multidimensional Shocks, Recent Advances in the Theory of Shock Waves, Progress in Nonlinear Differential Equations, Birkhäuser, to appear. [100] S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, New York (1973). [101] A. Mokrane, Problem es mixtes hyperboliques non lineaires, Thèse, Université de Rennes 1 (1987). [102] L.V. Ovsyannikov, Lectures on the Fundamentals of Gas Dynamics, Nauka, Moscow (1981) (Russian). [103] R.V. Polovin and V.P. Demutskij, Fundamentals of Magnetohydrodynamics, Atomizdat, Moscow (1987) (Russian). [104] F.V. Ralston, Note on a paper of Kreiss, Comm. Pure Appl. Math. 24 (1971), 759–762. [105] J. Rauch, L2 is a continuable initial condition for Kreiss mixed problems, Comm. Pure Appl. Math. 25 (1971), 265–285. [106] R. Rosales and A. Majda, Weakly nonlinear detonation waves, SIAM J. Appl. Math. 43 (1983), 1086– 1118. [107] B.L. Rozhdestvenskii and N.N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics, Transl. Math. Monogr., Vol. 55, Amer. Math. Soc., Providence (1983). [108] T. Ruggeri and A. Strumia, Main field and convex covariant density for quasilinear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. H. Poincaré Sect. A (N.S.) 34 (1) (1981), 65–84. [109] T. Ruggeri and A. Strumia, Convex covariant entropy density, symmetric conservative form, and shock waves in relativistic magnetohydrodynamics, J. Math. Phys. 22 (1981), 1824–1827. [110] G. Russo and A.M. Anile, Stability properties of relativistic shock waves: basic results, Phys. Fluids 30 (1987), 2406–2413. [111] M. Sablé-Tougeron, Existence pour un probleme de l’elastodynamique Neumann non lineaire en dimension 2, Arch. Rational Mech. Anal. 101 (1988), 261–292. [112] L.I. Sedov, Continuum Mechanics, Vol. 1, World Scientific, River Edge, NJ (1997). [113] M. Shearer, D.G. Schaeffer, D. Marchesin and P. Paes-Leme, Solution of the Riemann problem for a prototype 2 × 2 system of non-strictly hyperbolic consevation laws, Arch. Rational Mech. Anal. 97 (1987), 299–320. [114] S.L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Transl. Math. Monogr., Vol. 7, Amer. Math. Soc., Providence (1963). [115] S.I. Syrovatskij, The instability of tangential discontinuities in a compressible fluid, Zh. Eksp. Teor. Fiz. 27 (1954), 121–123 (Russian). [116] S.I. Syrovatskij, The stability of shock waves in magnetohydrodynamics, Zh. Eksp. Teor. Fiz. 35 (1958), 1466–1470 (Russian); English translation: SovietPhys. JETP 35 (8) (1959), 1024–1027. [117] A.H. Taub, Relativistic fluid mechanics, Ann. Rev. Fluid Mech. 10 (1978), 301–332. [118] K.S. Thorne, Astrophys. J. 179 (1973), 897.
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[119] Yu.L. Trakhinin, On stability of shock waves in relativistic magnetohydrodynamics, Quart. Appl. Math. 59 (2001), 25–45. [120] Yu.L. Trakhinin, A complete 2D stability analysis of fast MHD shocks in an ideal gas, Preprint 14, Novosibirsk State University (2002) (Russian); English translation: submitted for publication. [121] M. Vinokur, Conservation equations of gasdynamics in curvilinear coordinate systems, J. Comput. Phys. 14 (1974), 105–125. [122] A.I. Vol’pert and S.I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb. 87 (4), 504–528 (Russian); English translation: Math. USSR Sb. 16 (1973), 517–544. [123] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974). [124] C.C. Wu, New theory of MHD shock waves, Viscous Profiles and Numerical Methods for Shock Waves (Proc. Workshop, Raleigh/NC (USA) 1990), SIAM (1991), 209–236. [125] R.M. Zaidel’, On the stability of planar shock waves, Prikl. Mekh. Tekhn. Fiz. 8 (4) (1967), 30–39 (Russian). [126] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (1998), 741–748. [127] K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar viscous shock waves, Indiana Univ. Math. J. 48 (1999), 937–992.
CHAPTER 7
On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications Giovanni P. Galdi Department of Mechanical Engineering, University of Pittsburgh, 15261 Pittsburgh, USA
Dedicated to Professor Patrick J. Rabier on the occasion of his 50th birthday
Contents Introduction . . . . . . . . . . . . . . . . . . . . . Particle orientation . . . . . . . . . . . . . . . Self-propelled bodies . . . . . . . . . . . . . Outline of the paper . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . 1. Mathematical formulation . . . . . . . . . . . 2. The liquid models . . . . . . . . . . . . . . . 2.1. Navier–Stokes liquid . . . . . . . . . . . 2.2. Second-order liquid . . . . . . . . . . . Part I. Particle sedimentation . . . . . . . . . . . 3. The free fall problem . . . . . . . . . . . . . 4. Free fall in a Navier–Stokes liquid . . . . . . 4.1. Stokes approximation . . . . . . . . . . 4.2. The full nonlinear case . . . . . . . . . . 5. Free fall in a second-order liquid . . . . . . . 5.1. Steady free fall at zero Reynolds number 5.2. Steady fall at nonzero Reynolds number Part II. Self-propelled bodies . . . . . . . . . . . 6. The self-propelled body equations . . . . . . 6.1. Stokes approximation . . . . . . . . . . 6.2. The full nonlinear case . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
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On the motion of a rigid body in a viscous liquid
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Introduction Over the last 40 years the study of the motion of small particles in a viscous liquid has become one of the main focuses of applied research. The presence of the particles affects the flow of the liquid, and this, in turn, affects the motion of the particles, so that the problem of determining the flow characteristics is highly coupled. It is just this latter feature that makes any fundamental mathematical problem related to liquid-particle interaction a particularly challenging one. Interestingly enough, even though the mathematical theory of the motion of rigid particles in a liquid is one of the oldest and most classical problems in fluid mechanics, owed to the seminal contributions of Stokes [109], Kirchhoff [77], Thomson (Lord Kelvin) and Tait [112], and Jeffery [66], only recently have mathematicians become interested in a systematic study of the basic problems related to liquid-particle interaction [101,121,102, 53,38,34,39,45,22,58,61,62,55,23,90].1 The present article concentrates on the mathematical analysis of two of the several important and still not completely understood aspects of this fascinating subject, and is divided into two parts. Part I is devoted to the problem of the orientation of symmetric particles sedimenting in Newtonian and viscoelastic liquids, while Part II is dedicated to the motion of a self-propelled body in a Newtonian liquid. Before describing the main results, we would like to introduce some basic problems of practical interest, and experimental facts that motivated our study.
Particle orientation The orientation of long bodies2 in liquids of different nature is a fundamental issue in many problems of practical interest. (i) Composite materials. The addition of short fiber-like particles to a polymer matrix is well-known to enhance the mechanical properties of the composite material; see, e.g., [1]. Typical sizes of a fiber are a hundred micrometers in diameter and a centimeter in length [1]. The degree of enhancement depends strongly on the orientation of the fibers and the fiber orientation is in turn caused by the flow occurring in the mold; see [84]. Therefore, a better knowledge of the motion of fibers in polymer liquids (solutions and melts) with viscoelastic properties is important for the design of molding equipment and determining the optimal processing conditions. (ii) Separation of macromolecules by electrophoresis. Electrophoresis is a dominant analytical separation technique in the biological sciences [54]. Modern applications include weight determination of proteins [56], DNA sequencing [114], and diagnosis of genetic disease [8]. Electrophoresis involves the motion of charged particles (macromolecules) in 1 Of course, there is a fairly rich engineering literature dedicated to the theoretical analysis of particle-liquid
interaction; see, e.g., the review papers [83,82,14], and the references cited therein. These results, however, are not rigorous and they are all based on formal expansions of the velocity and pressure fields, like “inner-outer expansion” in the case of Navier–Stokes liquids [21], and expansion in the Weissenberg number, in the case of viscoelastic liquid [81,13,76]. 2 Loosely, a “long body” is a body where one dimension is much prevailing upon the other two.
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Fig. 1. Aggregation of spheres in Newtonian liquids (A), and viscoelastic liquids (B). When two spheres touch (“kissing”; see iii in (A) and iv in (B)) they momentarily form a “long” vertical body, that is unstable in the Newtonian case, and stable in the viscoelastic case [70]. (Courtesy of D.D. Joseph.)
solution, under the influence of an electric field. Certain types of macromolecules have a symmetric and rigid straight-rod shape (tropomyosin, fibrinogen, tobacco mosaic virus) and are several hundreds nanometers in length [54]. The orientation of the molecules plays an important role, since it is responsible for the loss of separability during steady-field gel electrophoresis [113,54]. (iii) Flow-induced microstructures. Particle pair interactions are a fundamental mechanism that enter strongly in all practical applications of particulate flows [70,100]. They are due to inertia and normal stresses and are maximally different in Newtonian and viscoelastic liquids [73]. In the most well-studied case of fluidized spheres, the principal interaction between a neighboring pair is described by the mechanism of drafting, kissing, and tumbling in Newtonian liquids, and of drafting, kissing, and chaining in viscoelastic liquids [68,69]; see Figure 1. A key to understanding microstructure in flowing suspensions of spherical bodies is the stable orientation of long bodies, since two spheres in momentary contact can be viewed as a rigid, symmetric, long body [29,63]; see Figure 1. A first, fundamental step in modeling the motion and the orientation of long bodies in liquids is to investigate experimentally their free fall behavior (sedimentation), both in Newtonian and viscoelastic liquids [81,15,17,86]. It is a well-established experimental fact that homogeneous bodies of revolution around an axis a (say) with fore-and-aft symmetry3 (like cylinders, round ellipsoids, etc., of constant density), when dropped in a quiescent viscous liquid will eventually reach a steady state that is purely translatory (no spin), and with a forming an angle with respect to the gravity g, that depends on the weight of the body, on its geometric properties (like being prolate or oblate in shape), and on the physical properties of the liquid (viscosity, inertia, non-Newtonian characteristics, etc.). In particular, if the liquid is viscous and Newtonian, 3 By this latter we mean that there is a plane Π orthogonal to a that is of symmetry for B.
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Fig. 2. Orientation of cylinders with round ends in a Newtonian liquid at nonzero Reynolds number (A), and in a purely viscoelastic liquid at vanishingly small Reynolds number (B). In the Newtonian case (A) the initial configuration is with the major axis parallel to the gravity (unstable) and the terminal configuration is with the major axis perpendicular to the gravity (stable). In the purely viscoelastic case (B) the situation is reversed [70]. One can see the similarity between the sedimentation of long bodies and the aggregation of spheres reported in Figure 1. (Courtesy of D.D. Joseph.)
(homogeneous) cylinders or prolate spheroids will always reach an equilibrium orientation with a orthogonal to the gravity, no matter what their initial orientation; see [95,4]; see Figure 2(A). Thus, these bodies take up an orientation that makes their resistance to motion a maximum (namely, an orientation that makes their speed a minimum). It is important to observe that, in these experiments, the Reynolds number Re = U d/ν can be very small. This is due to the fact that, typically, the product of the terminal speed U of the body and its characteristic length d is small compared to the kinematical viscosity ν of the liquid. For example, in a 85% aqueous solution of glycerine we have, at room temperature (20 ◦ C), ν = 1.13 cm2 /sec. Thus, for a body of diameter d = 0.5 cm falling with a terminal speed U = 1 cm/sec the corresponding Reynolds number is Re = 0.44.4 However, despite the smallness of the Reynolds number involved, these phenomena are genuinely nonlinear, and originate from the inertia of the liquid. In fact, if we make the liquid very viscous (99.9% of glycerine, ν = 15 cm2 /sec), so that the Reynolds number reduces approximately to zero and inertial effects can be neglected, then it is observed that the body will always keep its initial orientation with g [111]. In other words, all orientations are admissible at Re = 0; see Figure 3.5 If a small amount of polymer is added to the Newtonian liquid (typically, a 0.5%–2% aqueous solution), the situation changes dramatically and the final orientation may be completely different than that observed for a Newtonian liquid at nonzero Re. Detailed experimental studies were performed on slender cylinders sedimenting in aqueous solution polyacrylamide of different concentration; see [81,15]. In these experiments Re is much smaller than the corresponding dimensionless elasticity parameter, so that the effect due 4 For reference, the Reynolds number for a car speeding at 30 mls/hr is order of 106 . 5 The conservation of the initial orientation depends, of course, on the elapsed observation time. This means
that, if we wait a sufficiently long time (depending on the viscosity), inertia will eventually prevail and the body will turn with a perpendicular to g. In practice, we would need a sufficiently tall liquid container in which to drop the body, in order to observe a significant deviation from its initial orientation.
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Fig. 3. Multiple-image photograph of a slender particle sedimenting through 99.5% glycerine [81]. The particle keeps its initial orientation at all subsequent times. Notice that it also moves sideways while sedimenting. (Reprinted with the permission of Cambridge University Press.)
to the inertia of the liquid can be neglected. This typically happens when particles are very light, as in a fiber suspension or electrophoresis. The final orientation of all particles is observed to be with their broadside parallel to gravity; see Figure 2(B). This is quite remarkable, since it is in sharp contrast with the Newtonian case where, as we described before, a long particle will reach an equilibrium configuration with its broadside perpendicular to the gravity; see Figure 2(A). In fact, a recent experimental study on the orientation of long particles sedimenting in viscoelastic liquid by Liu and Joseph [86], shows another remarkable feature. Let us call tilt angle the angle formed by the long axis of symmetry a of the particle with the horizontal, when the body reaches its final equilibrium orientation. Liu and Joseph have found that for squared-off cylinders the tilt angle may vary continuously from 0 ◦ to 90 ◦ , depending on the physical properties of the cylinder and on the concentration of polymeric liquid. The tilt angle is very stable and it is reached no matter how and where the cylinder is released. It is important to emphasize that the dimensionless numbers involved in all the above experiments may be very small. For example, for cylinders made of plastic, Teflon, aluminum and titanium, with length ∼2 cm and diameter in the range 0.25 ∼ 1 cm, it is found that Re varies from 0.016 to ∼5, while the Weissenberg number We = λU/d 6 ranges between 0.048 and ∼0.3 [86], p. 580. Therefore, the phenomenon of particle orientation can be definitely considered a first order effect in Re and We.
Self-propelled bodies Self-propulsion is a common means of locomotion of macroscopic objects. Typical examples are motions performed by birds, fishes, airplanes, rockets and submarines. In 6 λ is the relaxation time.
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Fig. 4. Electromicrograph and corresponding drawing of Paramecium. The dense hairlike structures, cilia, are visible. (With permission of BIODIDAC, University of Ottawa.)
the microscopic world, many minute organisms, like flagellates and ciliates, also move by self-propulsion. Even though the hydrodynamical mechanism of self-propulsion may be different for macroscopic and microscopic bodies [110], the self-propelled motion of a body B into a viscous liquid L is essentially due to the interaction between the boundary Σ of B and L. Hence, Σ serves as the driver of B and the distribution v∗ of velocity on Σ, as its thrust. The thrust can be generated by muscular action, as in animal locomotion [51], or by a mechanical device, as in an airplane [92]. We are interested in problems where the shape of B does not change with time.7 In this case, there are two relevant mechanisms of self-propulsion: the body may generate a nonzero momentum flux through its boundary, or it may tangentially move portions of its boundary (or it may use a combination of both mechanisms). On a macroscopic scale, these types of motion can occur, for example, by inserting a mechanical pump into B, that sucks liquid into a porous front portion of Σ and expels it at the rear, or by locating on Σ a system of moving belts. On a microscopic scale, a remarkable example is furnished by ciliated micro-organisms (Ciliata), see, e.g., [11,6,9,10,75,7,16]. These micro-organisms consist of a body that is covered by a great number of hair-like organelles called cilia (see Figure 4), which move in a rather complicated way, and whose size is considerably smaller than any characteristic length of the body [11,7]. Ciliata can be found in almost every environment with liquid water. Because individual ciliate species vary greatly in their tolerance of pollution, the ciliates found in a body of water can be used to gauge the degree of pollution quickly [64,52]. Many ciliates, like those belonging to the Eukaryote family, have the shape of prolate spheroids, whose eccentricity has a very wide range of variability: from the Spirostomum Ambiguum – almost as slender as a “needle” – to the Didinium Nasutum – almost as round as a sphere; see [11], p. 376 and the references cited therein; see also Figure 5. 7 We refer the reader to [103,25,26], where the case of non-constant shape is considered.
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Fig. 5. Electromicrographs of: (A) Spirostomum Ambiguum [24] (Courtesy of Stephen Durr), and of (B) Didinium Nasutum. (With permission of BIODIDAC, University of Ottawa.)
Fig. 6. Sketch of the mechanical fish of G.I. Taylor [111].
Since the characteristic dimension and velocity of micro-organisms are extremely small, the corresponding Reynolds number is essentially zero, so that the inertial effects of the liquid can be legitimately ignored. Actually, it is just this latter circumstance that makes the problem challenging, as emphasized by G.I. Taylor [110]: “How can a body propel itself when the inertia forces, which are the essential element in selfpropulsion of all large living or mechanical bodies, are small compared with forces due to viscosity?”
As shown in a famous experiment by Taylor [111], a mechanical fish can happily swim in water but makes no progress in a very viscous liquid like corn syrup. The fish consists of a cylindrical body with a plane tail which flaps to and fro (see Figure 6). Roughly speaking, due to the reversibility of flow in a liquid with no inertia or, mathematically, due to the linearity of the equations, whatever the fish achieves by one flap of the tail, he/she will immediately lose with the next flap. In a commonly accepted model of Ciliata, the layer model, the motion of the cilia produces a distribution of velocity on a surface enclosing the layer of cilia, which serves to propel the animal [6,75]. So, several interesting questions can be posed like, for instance, which type of velocity distribution can effectively propel the micro-organism; how is this velocity related to the velocity of propulsion of the animal, and which velocity distributions can generate a purely translational motion. Also, how is the velocity of propulsion related to the shape of the micro-organism.
Outline of the paper The objective of Part I is to give a mathematical analysis of certain aspects of particle sedimentation, like those described above. We assume that the liquid fills the whole space,
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in accordance with the fact that, as established by the experiments, “wall effects” play no role on the preferred orientation of the particle. (For example, in [17], the ratio of the length of the cylinders to the diameter of the container is of the order of 10−2 .) This study is divided into two main sections. The first one (Section 4) is dedicated to the problem of sedimentation of a rigid body in a Navier–Stokes (Newtonian) liquid, and the second one (Section 5) dedicated to the same problem in a second-order (viscoelastic) liquid. Both studies are based on the concept of free fall [121], that we introduce in Section 3. In the Navier–Stokes case, we begin to consider (Section 4.1) the case of Re = 0 (Stokes approximation). In this situation we are able to decouple the equations of the particle from those of the liquid. This allows us to furnish a complete treatment of particle sedimentation (Sections 4.1.1, 4.1.2, and 4.1.3), from both steady and unsteady point of view. In the special case of a homogeneous body of revolution with fore-and-aft symmetry, we find predictions in a complete agreement with the experiment. In particular, we show that such a body, when dropped from rest, will always keep its initial orientation and will eventually reach a steady state (terminal state) with a corresponding limiting velocity that depends only on the geometric properties of the body, on its “effective mass”8 and on its initial orientation. In Section 4.2 we study the full nonlinear case. We begin to rederive (Section 4.2.1) the remarkable result of Serre [102] (see also Weinberger [121]), that ensures that the set of steady falls (terminal states) is always non-empty, for any body B, and for any Reynolds number. A fundamental question that naturally arises is that of which among the possible steady falls is stable and/or attainable. As is well-known, a basic tool in answering a question of this type is the knowledge of the asymptotic structure of the the velocity field of the steady solution [42,104]. In the case of a steady fall, this study is particularly difficult, due to the fact that, unlike the “classical” exterior problem, [36], Chapter IX, the body can also rotate. In Section 4.2.2 we give some contributions in this direction, by showing that the velocity field in any steady fall tends to zero at large distances, uniformly pointwise. This result can be generalized, by showing the same property for derivatives of arbitrary order, also for the pressure field [41]. As we emphasized previously, the problem of the orientation of certain symmetric bodies is of the utmost interest in particle sedimentation. In Sections 4.2.3, 4.2.4, and 4.2.5 we therefore concentrate on translational steady falls of a homogeneous body of revolution around a (say), with fore-and-aft symmetry, see [48]. We begin to show that such a body can execute at least two types of translational falls, namely, with a either parallel or perpendicular to g. We aim to prove that, in fact, these are the only two possible translational motions that the body can perform at small and nonzero Re, or, in other words, that these are the only two possible orientations of the body with respect to g. The physical quantity responsible for the possible orientations is the component of the torque exerted by the liquid on the body that is orthogonal to the plane containing a and the translational velocity U. Denoting by M this component (with respect to the center of mass of the body), we provide the following two-side estimate of M: 3 1 Re U1 U2 |GI | |M| Re U1 U2 |GI |. 2 2 8 Namely, the mass of the body minus the mass of the displaced liquid, which is assumed to be positive for sedimentation.
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Here U1 , U2 are the components of U along a and orthogonal to a, respectively,9 and GI is a scalar quantity depending only on the geometric properties of the body, such as size or shape, but otherwise independent of its orientation, and of the properties of the liquid, that we assume to be nonzero. We call GI the inertial torque coefficient. From the above estimates it then follows that every body with the mentioned symmetry and having GI = 0, must orient itself, at small Reynolds number, with a either orthogonal or perpendicular to g. This is in sharp contrast with the linear case (Re = 0) where all orientations are allowed. We also perform a “quasi-steady” stability analysis, at first order in Re which shows that if GI < 0 [respectively, GI > 0] the configuration with a orthogonal to g [respectively, parallel to g] is stable to small disorientations. A full stability analysis (in the sense of Liapounov) requires a great technical effort, and will be considered elsewhere. In Section 4.2.6, we specialize our results when B is a prolate spheroid of eccentricity e. In particular, we show that the torque coefficient is only a function of e and, by a numerical integration, that it is negative for all e ∈ (0, 1). Therefore, unless the body is a ball or a needle, only two possible orientations are allowed at small and nonzero Re. Moreover, in this case, the configuration with a perpendicular to g is stable to small disorientations, while the other is unstable, in agreement with experimental observation; see Figure 2. Finally, in Section 4.2.7, we briefly collect the known results concerning the unsteady fall [102]. Unfortunately, not much is known in such a case. Existence is known only for weak solution a la Leray–Hopf, for all times. Fundamental questions such as existence of strong solutions (even for small times) and asymptotic behavior in time (even for small Reynolds number) remain open. In Section 5 we begin the study of the free fall of a body in a second-order liquid. The choice of such a viscoelastic model is made for the following reasons. On one hand, as is well-known, it exhibits the “normal stress effect” that seems to play an important role in particle orientation [71,72]. On the other hand, it is a model that has been broadly investigated from the mathematical point of view [19,32,20,44,46,12,94,93,96,97,117,98]. However, compared to the Navier–Stokes case, the situation is here complicated by the fact that the equations are highly nonlinear. This reflects in the fact that one is not able, so far, to give such a general result for a steady free fall as that furnished by Serre and Weinberger in the Navier–Stokes case. In fact, one can only show [116] that the set of steady falls for a body of arbitrary shape is not empty only when Re = 0, the Weissenberg number We is sufficiently small, and the ratio ε of certain material constants (the “quadratic constants”) is −1; see Section 5.1. These results are then specialized to the case of a homogeneous body of revolution with fore-and-aft symmetry (Section 5.1.1). We are able to give a rigorous formula for the torque M exerted by the liquid on the body. An explicit calculation of M is made for a prolate spheroid (Section 5.1.2), and it is found that there are two possible orientations allowed, with the major axis of symmetry either parallel or perpendicular to the direction of the gravity. The stability to small disorientation of these configurations is found to be related to the sign of the first normal stress Ψ1 . In particular, the orientation with a parallel to g will be stable (as observed experimentally; see Figure 2(B)) if and only if Ψ1 > 0. As we mentioned previously, one of the important aspects of the experiments on flow-induced microstructures is the competition between the inertial and normal-stress 9 Without loss, we may assume U , U 0. 1 2
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torques. Actually, in [71,72], it is conjectured that this competition is responsible for the “tilt angle phenomenon” in the orientation of “heavy” long particles. This because, roughly speaking, inertia tends to rotate the axis of the body orthogonal to g, while normal stresses tend to rotate the axis in the direction of g. Therefore, the tilt angle could be the equilibrium configuration arising from this competition. Motivated by these considerations in [49] we have developed a mathematical analysis of the orientation of a homogeneous body of revolution with fore-and-aft symmetry translating in a second-order liquid, based on the evaluation of the torque acting on the body at first order in Re and We, and for arbitrary values of ε. The main features of this analysis are reported in Sections 5.2.1, 5.2.2, and 5.2.3, and results are applied to the case of a prolate spheroid in Section 5.2.4. In particular, it is shown that, perhaps at odds with intuition, there is no tilt angle phenomenon, and that, again, only two orientations are practically allowed, namely, when the major axis of symmetry a is either orthogonal or parallel to g. However, the competition between inertia and normal stress is responsible for the stability (to small disorientation) of these configurations. In particular, if inertia prevails on normal stresses in a well defined sense, then the configuration with a perpendicular to g is stable, the other configuration being stable otherwise. In Part II we perform a mathematical analysis of self-propelled motions of a rigid body in a Navier–Stokes liquid. The choice of the liquid model is made for the sake of simplicity. Among the several problems that can be addressed, we will mainly treat the following ones [37,33,38,106,107]. The first one, where the velocity v∗ at the boundary Σ of the body B is prescribed, and one has to find the corresponding velocity U of B, and the way in which it depends on v∗ . In particular, we wish to find (and, possibly, characterize) the velocities v∗ that ensure a non-zero translational velocity. A second one, where U is given, and one has to find the boundary velocity distributions v∗ that are able to propel B with velocity U. Notice that both problems may have10 an infinite number of solutions. For this reason, we would like to characterize, if possible, a class of boundary velocity and of velocity of the body that can be related by a one-to-one correspondence. A third problem regards the attainability of steady motions, that is, what are the boundary velocity distributions that can propel B from rest, until it reaches a prescribed constant velocity U. All these problems have a complete answer only at zero Reynolds number. In particular, we find that for a steady self-propelled motion to occur with a nonzero velocity, it is necessary and sufficient that the propelling boundary velocity distribution has a nonzero projection on a suitable 6-dimensional space, T (B), that depends only on the geometric properties of B (see Section 6.1.1). The corresponding velocity of B is uniquely determined. We will call T (B) the “control space”. Explicit applications of these results are given in Section 6.1.2 in the case of rotationally symmetric bodies. In particular, we find the form of the most general boundary velocity distribution that can move the body with a (nonzero) translational velocity ξ , parallel to the axis of rotation. Also, we find a relation between ξ and the (uniquely determined) boundary velocity in the control space. In the case of a prolate spheroid of eccentricity e, we find the variation of the (nondimensional) speed ξ with e. It is found that this variation is very small, from ξ = 2/3 for e = 0 to ξ = π/2 for e = 1. This is in agreement with the observation previously made 10 As in fact they do have; see Section 6.1.
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that a wide variety of prolate-spheroidal-shaped ciliata occur in nature; see also Figure 5. In Section 6.1.3 we shall report some of the recent results obtained in [107], and show that all these steady solutions are attainable from rest. In the general nonlinear case, only partial answers are available, with the exception of a symmetric body, where, more or less, the same results obtained in the linearized case can be proved, at least if the size of the Reynolds number is suitably restricted. Specifically, we show in Section 6.2.1 that, if Re is sufficiently small, for any (sufficiently regular) distribution of velocity v∗ on Σ having zero total flux through Σ, there exists at least one steady solution. However, the corresponding propulsion velocity U of B need not be nonzero. Actually, by means of an example, we show that there is a wide class of boundary velocities v∗ which do not set B into motion (i.e., U = 0). We then show in Section 6.2.2 that U is not zero whenever v∗ has a nonzero projection11 on the control space T (B). Moreover, in such a case, we furnish explicit lower and upper bounds for the translational and angular velocity of B, in terms of the projection of v∗ on T (B), and we prove that velocity distributions having different projections on T (B) will generate different motions for B. Finally, in Section 6.2.3 we specialize these results to the case of a symmetric body around an axis a ≡ y1 (say). By this we mean the following (y1 , y2 , y3 ) ∈ Σ →
(y1 , −y2 , y3 ) ∈ Σ, (y1 , y2 , −y3 ) ∈ Σ.
The main result states that every steady self-propelled motion of B with a nonzero translational velocity along a is attainable from rest, provided Re is not too large [106].
Notation12 N is the set of positive integers. Rn is the Euclidean n-dimensional space and {e1 , e2 , e3 , . . . , en } ≡ {ei } the associated canonical basis. S n−1 denotes the unit sphere in Rn . Given a second-order tensor A and a vector a, of components {Aij } and {ai }, respectively, in the basis {ei }, by a · A [respectively, A · a] we mean the vector whose components are given by Aij ai [respectively, Aij aj ]. Moreover, if B = {Bij } is another second-order tensor, by the symbol A · B we mean the second-order tensor whose components are given by Ail Blj . √ We also set A : B = trace(A · BT ), where the superscript T “ ” denotes transpose, and |A| = A : A. Given a vector field h(z) ≡ {hi (z)}, by grad h we denote the second-order tensor field whose components {grad h}ij in the given basis are given by {∂hj /∂zi }. For any domain A, C k (A), k 0, Lq (A), W m,q (A), m 0, 1 < q < ∞, denote the usual space of functions of class C k on A, and Lebesgue and Sobolev spaces, respectively. Norms in Lq (A) and W m,q (A) are denoted by · q,A , · m,q,A . Unless confusion arises, 11 In the sense of L2 (Σ). 12 As a rule, in this paper we shall use the notation of [35]. However, for the reader’s convenience, we collect
here the most frequently used symbols.
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we shall usually drop the subscript “A” in these norms. The trace space on ∂A for functions from W m,q (A) will be denoted by W m−1/q,q (∂A) and its norm by · m−1/q,q,∂ A . By D k,q (A), k 1, 1 < q < ∞, we indicate the homogeneous Sobolev space of order (m, q) on A, [108,35], that is, the class of functions u that are (Lebesgue) locally integrable in Ω and with D β u ∈ Lq (A), |β| = k, where Dβ =
∂ |β| β
β
β
∂x1 1 ∂x2 2 ∂x3 3
,
|β| = β1 + β2 + β3 .
For u ∈ D k,q (A), we set13 |u|k,q,A =
|β|=k A
β q D u
1/q ,
where, again, the subscript “A” will be generally omitted. Given a Banach space X, and an open real interval (a, b), we denote by Lq (a, b; X) the linear space of (equivalence classes of) functions f : (a, b) → X whose X-norm is in Lq (a, b). Likewise, for r a non-negative integer and I a real interval, we denote by C r (I ; X) the class of continuous functions from I to X, which are differentiable in I up to the order r included. If X = Rn , we shall simply write Lq (a, b), C r (I ), etc. Let X be any space of real functions. As a rule, we shall use the same symbol X to denote the corresponding space of vector and tensor-valued functions. Finally, we denote by R the set of all velocity fields in a rigid motion, namely, & % R = U ∈ C ∞ R3 : U = U0 + U1 × x, U0 , U1 ∈ R3 .
1. Mathematical formulation Let us consider a rigid body B moving through a liquid L that fills the whole space. B is assumed to be an open, connected and bounded set (namely, B is a bounded domain). We indicate by V = V(x, t) the velocity field associated with the motion of B with respect to an inertial frame I. Thus, denoting by C the center of mass of B and by O the origin of I, we have V(x, t) = η(t) + Ω(t) × (x − xC ), •
(1.1)
where η = xC (t), and Ω is the angular velocity of B. The Eulerian velocity and pressure fields associated to the motion of L in I, are denoted by v = v(x, t) and p = p(x, t). The 13 Typically, we shall omit in the integrals the infinitesimal volume or surface of integration.
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equations of conservation of linear momentum and mass of L, with respect to I, are then given by ⎫ dv ⎬ 9 = div T (v, p) + ρF (x, t), ρ × {t}, (x, t) ∈ D(t) (1.2) dt ⎭ div v = 0, t >0 is the where ρ is the constant density of L, d/dt is the material (total) derivative, D(t) region occupied by L at time t, F is the body force acting on L, and T is the Cauchy stress tensor. We assume that the liquid is at rest at infinity, so that we impose lim v(x, t) = 0.
(1.3)
|x|→∞
of the body B we require the following condition Moreover, at the boundary surface Σ(t) v(x, t) = v∗ (x, t) + V(x, t),
(x, t) ∈
9
× {t}, Σ(t)
(1.4)
t >0
where v∗ is a velocity distribution that takes into account the possibility that B may generate a nonzero momentum flux through its boundary, or it may tangentially move portions of its boundary (or it may use a combination of both mechanisms). The equations of motion of B are obtained by requiring the balance of linear and angular momentum. In this regard, we notice that the forces acting on B are of two different types: those due to the interaction liquid-body, internal forces, and those that are not due to this directed interaction, external forces, like gravity. Denoting by N the unit normal to Σ toward B, the internal forces can be expressed as the sum of forces exerted by L on B, like drag and/or lift: T (v, p) · N − ) Σ(t
: and of forces due to a momentum flux through Σ ρv(v − V) · N. ) Σ(t
Likewise, the total torque due internal forces with respect to C is given by − (x − xC ) × T (v, p) · N − ρv(v − V) · N ) Σ(t
Denote by m the mass of B, by ρB its density, and by J its inertia tensor with respect to C, defined by the relation ρB a × (x − xC ) · b × (x − xC ) , for all a, b ∈ R3 . a·J·b= B
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The equations of motion of B in the frame I are then given by dη m =F− dt
) Σ(t
T (v, p) · N − ρv(v − V) · N ,
d(J · Ω) = MC − dt
) Σ(t
(x − xC ) × T (v, p) · N − ρv(v − V) · N ,
(1.5)
where F and MC are total external force and external torque with respect to C, acting on B. The motion of B and L will be determined by solving the problem (1.2), (1.3), (1.4), and (1.5), once the initial conditions on v and V are prescribed. However, this formulation has an undesired feature, namely, the region occupied by L is an unknown function of time. One therefore prefers to reformulate the problem in a frame S attached to B, where this region remains the same at all times. To this end, without loss, we take the origin of coordinates of S coinciding with C, and assume I ≡ S at time t = 0. Thus, if y denotes the position vector of a point P in S and x the position vector of the same point in I, we have x = Q(t) · y + xC (t),
Q(0) = 1,
xC (0) = 0
(1.6)
with Q orthogonal linear transformation: Q(t) · QT (t) = QT (t) · Q(t) = 1.
(1.7)
From (1.1) and (1.6) we deduce, in particular, that the angular velocity Ω is related to Q by the equation A · a = Ω × a,
for all a ∈ R3 ,
•
A(t) ≡ Q(t) · QT (t).
(1.8)
In order to write the equation of L in S, we introduce the following transformed fields for B: ξ (t) = QT (t) · η(t),
ω(t) = QT (t) · Ω(t),
(1.9)
and for L: w(y, t) = QT (t) · v Q(t) · y + xC (t), t , p(y, t) = p Q(t) · y + xC (t), t ,
(1.10)
T(w, p) = Q · T (Q · w, p) · Q. T
Notice that from (1.8) and (1.9) it easily follows that B · a = ω × a,
for all a ∈ R3 ,
•
B(t) ≡ QT (t)· Q (t).
(1.11)
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We have • dv d(Q · w) ∂w • = =Q· w+Q· + y · grad w , dt dt ∂t
(1.12)
where the gradient operator is acting on the y-variable. Differentiating (1.6), and taking into account (1.10)1, (1.9) and (1.11) we get •
•
y = QT · (v − η) − QT · Q · y = w − ξ − ω × y. Thus, from this latter equation and from (1.12) we deduce QT ·
dv ∂w =ω×w+ + (w − ξ − ω × y) · grad w. dt ∂t
(1.13)
Furthermore, from (1.10), one obtains the following identities divx v = divy w, QT · divx T (v, p) = divy T(w, p). Consequently, collecting (1.2), (1.13) and (1.14) we find ⎫ ∂w ⎪ + (w − U) · grad w + ω × w ⎪ ρ ⎪ ⎪ ⎬ ∂t in D × (0, ∞), T = div T + ρQ · F , ⎪ ⎪ ⎪ ⎪ ⎭ div w = 0,
(1.14)
(1.15)
where U(y, t) = ξ (t) + ω(t) × y
(1.16)
and D is the fixed region occupied by L in S. Moreover, in view of (1.10)1, (1.9) and (1.11), the side conditions (1.3) and (1.4) become lim w(y, t) = 0
(1.17)
|y|→∞
and w(y, t) = w∗ (y, t) + U(y, t),
(y, t) ∈ Σ × (0, ∞),
(1.18)
respectively, where Σ = ∂B and w∗ = QT · v∗ . We shall now write the equation of B in the frame S. Using (1.9) and (1.11) we find •
mη = m
• • • d(Q · ξ ) = m Q · ξ + Q · ξ = mQ · ξ + ω × ξ . dt
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Moreover, denoting by n the unit normal to Σ directed toward B, we have N = Q · n, and so we obtain T (v, p) · N = Q · T(w, p) · n, ) Σ(t
Σ
and
) Σ(t
v · (v − V) · N = Q ·
w(w − U) · n. Σ
Therefore, the equation of linear momentum (1.5)1 becomes
•
m ξ + mω × ξ = QT · F −
T(w, p) · n − ρw(w − U) · n .
(1.19)
Σ
In a similar way, using the identity (Q · a) × (Q · b) = Q · (a × b),
for all a, b ∈ R3 ,
and setting I = QT · J · Q,
(1.20)
one shows that the equation of angular momentum (1.5)2 becomes •
I · ω + ω × (I · ω) T y × T(w, p) · n − ρw(w − U) · n . = Q · MC −
(1.21)
Σ
Notice that I is independent of time, since a·I·b=
B
ρB (a × y) · (b × y),
for all a, b ∈ R3 .
We may then conclude that the motion of the system body-liquid with respect to the frame S is governed by the system of equations (1.15)–(1.21).
2. The liquid models As mentioned in the Introduction, in this paper we shall consider two types of liquid models: Navier–Stokes and second-order. The relation between the Cauchy stress tensor T and the transformed tensor T will be considered separately.
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2.1. Navier–Stokes liquid In this case, the Cauchy stress tensor is given by T (v, p) ≡ T NS (v, p) = −p1 + 2µD(v),
(2.1)
where µ is the shear viscosity coefficient, and D(v) =
1 gradx v + (gradx v)T , 2
(2.2)
is the stretching tensor. From (1.6) and (1.10)1, it easily follows that gradx v + (gradx v)T = Q · grady w + (grady w)T · QT , which, in turn, with the help of (1.10)2, furnishes T NS (v, p) = −p1 + Q · grady w + (grady w)T · QT . Using (1.10)3, we thus conclude that T NS and TNS have the same functional form, that is, T NS (v, p) = TNS (w, p).
(2.3)
2.2. Second-order liquid Set A1 (h) = grad h + (grad h)T ,
L(h) = grad h.
(2.4)
The Cauchy stress tensor for a second-order liquid model can then be written as [67] T (v, p) = T NS (v, p) + S(v), where T NS is defined in (2.1) and the viscoelastic extra-stress tensor S is given by
S(v) = α1
dA1 (v) + A1 (v) · LT (v) + L(v) · A1 (v) + α2 A1 (v) · A1 (v), dt
where α1 , α2 are the so-called “quadratic constants”. They are related to the normal stress coefficients Ψ1 and Ψ2 by the formulas α1 = − 12 Ψ1 , α2 = Ψ1 + Ψ2 ; see [67], Chapter 17. Define S(w) ≡ QT · S(Q · w) · Q.
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By a direct calculation, we show that S(w) = α1 B · A1 (w) − A1 (w) · B
∂A1 (w) + α1 + (w − U) · grad A1 (w) ∂t
+ A1 (w) · LT (w) + L(w) · A1 (w) + α2 A1 (w) · A1 (w),
(2.5)
where the tensor B is given in (1.11). Therefore S(w) = S(v), that is, S and S don’t have the same functional form. So, unlike the purely Newtonian Navier–Stokes case, we find T (v, p) = T(w, p). However, introducing the relative velocity: u = w − U, and observing that A1 (U) = 0,
(2.6)
and that L(U) = BT = −B, we find B · A1 (u) − A1 (u) · B = −A1 (u) · LT (U) − L(U) · A1 (u), and from (4.8) we conclude
S(u) = α1
∂A1 (u) + u · grad A1 (u) + A1 (u) · LT (u) + L(u) · A1 (u) ∂t
+ α2 A1 (u) · A1 (u), that is, S(u) = S(v). Because of (2.6) and of (2.3), this latter property in turn implies T (v, p) = T(u, p). Such a result is, of course, expected, as a consequence of the frameinvariance condition satisfied by the Cauchy stress tensor [115].
Part I. Particle sedimentation Suppose that a rigid body B is released from rest in an otherwise quiescent liquid L, under the action of the force of gravity (sedimenting particle). We assume that “wall effects” are negligible, that is, we assume that L fills the whole space. After a certain interval of time, B will eventually execute a motion where its angular velocity and the velocity of its center of mass will be constant. We shall call this motion terminal state. Regarding this simple and familiar phenomenon, several interesting mathematical questions can be formulated. For example, is the set of terminal states always non-empty, no matter what the shape and
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the density of B, and the property of L. How many terminal states there exist for given B, and L, and which are those that can be attained or which are the stable ones. Even though these questions are very simply and spontaneously formulated, their answer is far from being trivial, and several of them remain still open, even for a “classical” liquid model like Navier–Stokes. Besides these fundamental problems, there are other interesting issues coming from experimental evidence, regarding the orientation of certain symmetric particles, such as those we largely described in the Introduction. The objective of this Part I is to give a mathematical analysis of certain aspects of particle sedimentation. Specifically, we shall present the results available and shall point out the several open questions that remain to be answered. This analysis will be subdivided into two main sections, the first dedicated to purely Newtonian, Navier–Stokes liquids (Sections 4), and the second to viscoelastic liquids described by the second-order model (Section 5). In both situations, we shall reserve particular attention to the case when B is a homogeneous body of revolution with fore-and-aft symmetry. We shall also analyze in details the limiting case of zero Reynolds number, where more complete results are available.
3. The free fall problem The mathematical analysis of particle sedimentation is based on the concept of free fall of a body B in a liquid L. D EFINITION 3.1. We shall say that B executes a free fall in L if and only if: (1) The bounding surface Σ of B is impermeable and fixed, so that w∗ ≡ 0. (2) The force of gravity is the only external force acting on B and L. (3) B is dropped from rest in an otherwise quiescent liquid L. We wish to give a mathematical formulation of free fall. To this end, we observe that, since the motion of B is not known, the direction of the vector QT · g, is not known, and we have to provide a suitable equation describing its variation. Set G(t) = QT (t) · g. •
Differentiating this expression and taking into account that g = 0, we find that •
•
•
G = Q T · g = Q T · Q · G. •
•
However, from (1.7) we have Q T · Q = −QT · Q, and so from (1.11) we find dG = G × ω. dt
(3.1)
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From (1.15)–(1.21), and (3.1) we then conclude that the relevant equations describing free fall are furnished by ⎫ ∂w ⎪ + (w − U) · grad w + ω × w = div T(w, p) + ρG, ⎬ ρ ∂t ⎪ ⎭ div w = 0,
lim w(y, t) = 0,
(3.2)
|y|→∞
w(y, t) = U(y, t), m I·
in D × (0, ∞),
(y, t) ∈ Σ × (0, ∞),
dξ + mω × ξ = mG − dt dω + ω × (I · ω) = − dt
T(w, p) · n,
(3.3)
y × T(w, p) · n,
(3.4)
Σ
Σ
dG = G × ω, dt
(3.5)
where U = ξ + ω × y. To (3.2)–(3.5) we have to append the initial conditions. Since B is dropped from rest, and L is initially quiescent, we have w(x, 0) = ξ (0) = ω(0) = 0.
(3.6)
Moreover, since Q(0) = 1, we have G(0) = g
(3.7)
which represents the initial orientation of B. Therefore, the problem of free fall can be stated as follows. Given T = T(w, p), ρ, B, m, I, and g, that is, given the liquid, the body and its initial orientation, find {w, p, ξ , ω, G} satisfying (3.2)–(3.7). For reasons given at the beginning of Section 3, of particular practical interest is the steady counterpart of problem (3.2)–(3.7). In fact, steady solutions describe the possible terminal states that B can eventually reach in a free fall, when time goes to infinity. These states are thus obtained by requiring that w, p are functions of y only and that ξ , ω, and G ≡ g are independent of time. The problem of steady free fall is then formulated as follows. Given T(w, p), ρ, m and I, find {w, p, ξ , ω, g} such that : ρ (w− U) · grad w + ω × w = div T(w, p) + ρg, div w = 0, lim w(y) = 0,
(3.8)
|y|→∞
w(y) = U(y),
in D,
y ∈ Σ,
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G.P. Galdi
mω × ξ +
T(w, p) · n = mg,
(3.9)
Σ
ω × (I · ω) +
y × T(w, p) · n = 0,
(3.10)
Σ
ω × g = 0,
(3.11)
where U(y) = ξ + ω × y. R EMARK 3.1. The special feature of the steady free fall problem is that the direction of the acceleration of gravity g is not prescribed; rather, it is an unknown to be determined. The directions of g for which the problem has a solution, will furnish the orientations of B in its steady free fall. R EMARK 3.2. The steady free fall problem is well formulated, in the sense that the number of unknowns equals that of the equations. Actually, since the magnitude of g is given and ω is parallel to g, we have a total of 10 scalar unknowns, and a total of 10 scalar equations, namely, (3.8)1,2, (3.9), (3.10), (3.11). R EMARK 3.3. Problem (3.8)–(3.11) may have, in general, more than one solution, or even an infinite number of solutions. Which one among these steady solutions is effectively realized, is related to the problem of attainability and stability, which, in turn, is governed by the asymptotic behavior in time of solutions to (3.2)–(3.7). R EMARK 3.4. If we think of g and U as prescribed quantities, and, moreover, we set ω = 0, then problem (3.8) is a “classical” exterior boundary-value problem. In the next sections we shall investigate the problem of free fall and of particle orientation in Navier–Stokes and second-order liquids.
4. Free fall in a Navier–Stokes liquid For a Navier–Stokes liquid, the Cauchy stress tensor is given in (2.1). In view of (2.3) we find div T(w, p) = µ w − grad p. We wish to write the free fall equations in a suitable non-dimensional form. To this end, we denote by W and d suitable scale velocity and length, and introduce dimensionless time t ∗ = tµ/(ρd 2 ) and mass m∗ = m/(ρd 3 ). The free fall equations (3.2)–(3.7) then become in non-dimensional form
On the motion of a rigid body in a viscous liquid
⎫ ∂w + Re (w − U) · grad w + ω × w = w − grad p + G, ⎬ ∂t ⎭ div w = 0, lim w(y, t) = 0,
|y|→∞
(y, t) ∈ Σ × (0, ∞), dξ T(w, p) · n, m + Re mω × ξ = mG − dt Σ dω I· + Re ω × (I · ω) = − y × T(w, p) · n, dt Σ
675
in D × (0, ∞), (4.1)
w(y, t) = U(y, t),
dG = Re G × ω dt
(4.2)
(4.3)
(4.4)
with initial conditions w(y, 0) = ξ (0) = ω(0) = 0,
(4.5)
G(0) = g.
In these equations Re = ρW d/µ is the Reynolds number, and all the variables are nondimensional. Moreover, by a suitable choice of the dimensional scale quantities, we can take |G(t)| = 1, at all times t 0. 4.1. Stokes approximation In the present section we will be interested in the case when B moves in L with a small velocity and/or the viscosity of L is very large. Under these circumstances, it is reasonable, in a first analysis, to assume that Re = 0 in (4.1)–(4.3), and to study the free fall in the Stokes approximation [119,120]. In this case, Equations (4.1)–(4.4), after a suitable rescaling of space, time and mass, become [120] ⎫ ∂w = w − grad p + G, ⎬ ∂t in D × (0, ∞), ⎭ div w = 0, (4.6) lim w(y, t) = 0, |y|→∞
w(y, t) = U(y, t) ≡ ξ (t) + ω(t) × y, dξ m = mG − T(w, p) · n, dt Σ dω I· =− y × T(w, p) · n, dt Σ
y ∈ Σ, (4.7)
(4.8)
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G.P. Galdi
dG = G × ω, dt
(4.9)
with initial conditions w(x, 0) = ξ (0) = ω(0) = 0,
G(0) = g.
(4.10)
Our first goal is to investigate the problem of steady fall, which will be the object of the first part of this subsection. In the second part, we shall study the unsteady case. 4.1.1. Steady free fall In the case of steady fall, the fields w, p, U and G do not depend on time, so that (4.6)–(4.9) reduce to the following ones w − grad p + g = 0,
: in D,
div w = 0.
(4.11)
lim w(y) = 0,
|y|→∞
w(y) = U(y) ≡ ξ + ω × y,
y ∈ Σ,
mg =
T(w, p) · n,
(4.12)
Σ
y × T(w, p) · n = 0,
(4.13)
Σ
g × ω = 0.
(4.14)
Following [119], it is easy to show that problem (4.11)–(4.14) has at least one solution, for any B. To this end, we introduce the auxiliary fields (h(i) , p(i) ), and (H(i) , P (i) ), i = 1, 2, 3, satisfying the following boundary value problems [57] h(i) = grad p(i) ,
: in D,
div h(i) = 0, h(i) (y) = ei ,
(4.15)
y ∈ Σ,
lim|y|→∞ h(i) (y) = 0 and H(i) = grad P (i) ,
: in D,
div H(i) = 0, H(i) (y) = ei × y,
y ∈ Σ,
lim|y|→∞ H(i) (y) = 0,
(4.16)
On the motion of a rigid body in a viscous liquid
677
where {ei } is the canonical basis in R3 . The fields (h(i) , p(i) ) [respectively, (H(i) , P (i) )] are velocity and pressure fields of L when B is translating [respectively, rotating] in L along three orthogonal directions. It is evident that the auxiliary fields depend only on geometric properties of B such as size, shape, symmetry, etc. Existence of (h(i) , p(i) ), and (H(i) , P (i) ) is well-known [35], Chapter V, even without smoothness for B, provided the boundary conditions (4.15)3 and (4.16)3 are interpreted appropriately.14 The pairs (h(i) , p(i) ) and (H(i) , P (i) ) are infinitely differentiable in D and, moreover, (h(i) , p(i) ), (H(i) , P (i) ) ∈ 1,2 Wloc (D) × L2loc (D). Set ξ = ξi ei ,
ω = ωi ei ,
and consider the following fields w ≡ ξi h(i) + ωi H(i) ,
p ≡ ξi p(i) + ωi P (i) + g · y.
(4.17)
Of course, w, p satisfy (4.11). Moreover, if B is (locally) Lipschitz, the total force and total torque exerted by L on B are well-defined.15 Since
g · yn = −
Σ
B
(g · y)y × n = Σ
grad(g · y) = −|B|g,
B
curl(yg · y) =
(4.18) B
g × y = |B|g × R,
where |B| is the volume of B, and R = C − C, with C centroid of B, we deduce, in particular, the following formulas −
T(w, p) · n = −K · ξ − C · ω − |B|g,
(4.19)
y × T(w, p) · n = −S · ξ − Θ · ω + |B|g × R,
(4.20)
Σ
− Σ
14 For example, the boundary condition (4.15) is satisfied in the sense that ψ(y)(h(i) (y) − e ) ∈ W 1,2 (D), i 3 0
where ψ(y) is a non-increasing smooth function that is equal to 1 in a neighborhood of Σ and is zero at large distances. Condition (4.16)3 is satisfied in a similar way. 15 Since div T = 0 and T ∈ L2 (D), the trace T · n| is well-defined as an element of W −1/2,2 (Σ) (the dual Σ loc space of W 1/2,2 (Σ)); see, e.g., [35], Section III.2.
678
G.P. Galdi
where Kj i = Σ
Θj i =
Σ
Σ
Cj i = Sj i = Σ
(i) (i) T h ,p · n j, y × T H(i) , P (i) · n j , y × T h(i) , p(i) · n j ,
(4.21)
(i) (i) T H ,P · n j.
The matrices K, Θ, C, and S will play an important role in the sequel and, therefore, we wish to recall here some of their main properties. First of all, we have, clearly, that they depend only on geometric properties of B such as size, shape, symmetry, etc., but they are otherwise independent of the orientation of B and of the physical properties of L. Moreover, we have the following result [57]. L EMMA 4.1. Let B be Lipschitz.16 The matrices K and Θ are symmetric and positive definite, and S = CT . Also, the 6 × 6 matrix A=
K CT
C Θ
is positive definite. The form of the matrix A can be highly simplified, depending on the (geometric) symmetry properties of B. In this paper we shall devote particular attention to homogeneous bodies of revolution around an axis a (say), that possess fore-and-aft symmetry. By this latter we mean that there is a plane Π orthogonal to a that is of symmetry for B. Typical examples are cylinders and prolate and oblate spheroids of constant density. Concerning this kind of bodies, the following result can be proved [57]. L EMMA 4.2. Let B as in the previous lemma. Assume, moreover that it is a homogeneous body of revolution around a with fore-and-aft symmetry. Then C = 0.17 Moreover, taking a ≡ y1 , we have that K and Θ are diagonal and that K22 = K33 , Θ22 = Θ33 . 16 We can alternatively require that B has positive capacity [119]. We recall that the capacity C of B is defined
as C=
∂Φ 1 , 4π Σ ∂n
where Φ is the harmonic function that is 1 at Σ and vanishes at infinity. 17 From its very definition, the matrix C depends on the point O (say) with respect to which the moment of the stress T(h(i) , p (i) ) · n at Σ is evaluated (see (4.21)3 ). So, we have, in general, C = CO . Therefore, the result in the lemma states that CC = 0.
On the motion of a rigid body in a viscous liquid
679
Taking into account that, by (4.14), ω = λg, for some λ ∈ R, in view of (4.19), (4.20), and Lemma 4.1, we find that conditions (4.12), (4.13) can be written as follows K · ξ + λC · g = me g,
(4.22)
CT · ξ + λΘ · g = |B|g × R,
where me = m − |B| is the effective mass of B, namely, in dimensionless form, the mass of the body minus the mass of the displaced liquid. Since we are only interested in sedimentation phenomena, we shall tacitly understand that the effective mass is always positive.18 Equation (4.22) characterizes the possible steady falls of B, with velocity and pressure fields in a suitable regularity class. To see this, let us set % & 1,s C s = w ∈ Wloc (D), p ∈ Lsloc (D) ,
s > 1.
Notice that, by the trace theorem, conditions (4.12) and (4.13) are meaningful for w, p ∈ C s . The following result holds. L EMMA 4.3. Let B be a Lipschitz domain. Problem (4.11)1,2,4–(4.14) has at least one (distributional) solution {w, p, ξ , ω, g} with w, p ∈ C s , some s > 1, and satisfying (4.11)3 uniformly pointwise,19 if and only if the algebraic system (4.22) has a solution {ξ , λ, g}. Moreover, if such a solution exists, the fields w, p are infinitely differentiable in D and they admit the representation (4.17). P ROOF. If (w, p) is a solution to (4.11) in the class C s , for some s > 1, then, by classical regularity and uniqueness results (see, e.g., [35], Sections V.2, V.3) it follows that the fields w, p are infinitely differentiable in D, and are of the form (4.17). Consequently, we obtain (4.22). Conversely, if (4.22) has a solution {ξ¯ , λ, g¯ }, we get that w and p defined in (4.17), with ξ = ξ¯ , g = g¯ , and ω = λ¯g, are in the class C 2 , are infinitely differentiable in D, and that, further, they solve (4.11). Clearly, (4.14) is satisfied. Since w, p verify (4.19), (4.20) and {ξ¯ , λ, g¯ } solve (4.22), we then conclude that w, p satisfy also (4.12)–(4.13), and the proof is completed. The existence of a steady free fall is thus reduced to solving (4.22). Notice that this latter equation involves quantities related only to the motion of B, that is thus decoupled from the motion of L. 18 For the sake of completeness, we wish to remark that all the results of Part I continue to hold also for m < 0. e The only difference, from the physical point of view, is that the particle will be rising instead of sedimenting. The case me = 0 is of no interest, since the particle will not move. 19 This latter condition can be weakened in many ways. For instance, we may require S2 |w(x)| → 0, as |x| → ∞; see [35], Theorem V.3.2.
680
G.P. Galdi
T HEOREM 4.1. Let B be Lipschitz.20 Then, problem (4.11)–(4.14) has at least one solution {w, p, ξ , ω, g}, with (w, p) ∈ C 2 ∩ C ∞ (D). Moreover, the motion of B is purely translatory (namely, ω = 0) if and only if |B|1 × R − me CT · K−1 · g = 0.
(4.23)
P ROOF. By Lemma 4.3, the problem is equivalent to study the solvability of (4.22). From (4.22)1 we find ξ = K−1 · (me g − λC · g).
(4.24)
Replacing this into (4.22)2 we obtain A · g = λg
(4.25)
−1 A = Θ − CT · K−1 · C · |B|1 × R − me CT · K−1 .
(4.26)
with
By Lemma 4.1, A is a well-defined 3×3 matrix and, therefore, it has at least one real eigenvalue. Thus, problem (4.22) is solvable with λ and g real eigenvalue and corresponding eigenvector of the matrix A, respectively, and ξ given by (4.24), which shows the first part of the theorem. We next observe that λ = 0 (namely, ω = 0) if and only if A · g = 0.
(4.27)
Since, by Lemma 4.1, it is (Θ − CT · K−1 · C)−1 = 0, condition (4.27) occurs if and only if (4.23) is satisfied. The proof of the theorem is completed. An important corollary of this result is furnished in the next theorem. T HEOREM 4.2. Let B be a Lipschitz homogeneous body of revolution with fore-and-aft symmetry. Then all possible solutions to problem (4.11)–(4.14) with (w, p) ∈ C s , some s > 1, are of the type {w, p, ξ , 0, g} where the direction of g is arbitrary, and ξ = me K−1 · g. P ROOF. If B is homogeneous, then R = 0, and if it is also a body of revolution with foreand-aft symmetry, by Lemma 4.2 it is C = 0. Therefore, if B possesses both properties, condition (4.23) is identically satisfied for all g. The theorem follows from this and from Equation (4.24). 20 This assumption is due to the particular simple approach followed here. It can be removed, if we are interested in weak solutions to (4.11)–(4.14). In fact, as we shall see directly in the full nonlinear context (Theorem 4.5), existence of weak solutions can be proved for an arbitrary domain B, but by a much less elementary approach (Lemma 4.13).
On the motion of a rigid body in a viscous liquid
681
4.1.2. Unsteady free fall We shall now turn to the resolution of the general initialboundary value problem (4.6)–(4.9). In particular, we shall show that, as in the steady case (see (4.22)), also in the case at hand the motion of the body can be decoupled from that of the liquid. To this end, we begin to apply the Helmholtz decomposition to the fields h(i) and H(i) to get (i)
(i)
(i)
h(i) = h0 + grad p0 ,
(i)
H(i) = H0 + grad P0 ,
i = 1, 2, 3,
(4.28)
where (i)
p0 = 0, ∂p0(i) (y) = ei · n, ∂n
y ∈ Σ,
(4.29)
(i)
lim grad p0 (y) = 0
|y|→∞
and P0(i) = 0, (i)
∂P0 (y) = ei × y · n, ∂n
y ∈ Σ,
lim grad P0(i) (y) = 0.
(4.30)
|y|→∞
q
(i) It is well-known, [35], Section III.1, that, in general, h(i) 0 , H0 ∈ Lσ (D), q > 3, where
% & Lqσ (D) = ψ ∈ Lq (D): div ψ = 0 in D, ψ · n = 0 at Σ . For i = 1, 2, 3, we define the auxiliary fields (γ (i) , p(i) ), and (Γ (i) , P(i) ), as solutions to the following initial-boundary value problems ⎫ ∂γ (i) ⎬ = γ (i) − grad p(i) , ∂t ⎭ div γ (i) = 0, γ (i) (z, t) = 0,
z ∈ Σ,
lim γ (i) (y, t) = 0,
|y|→∞
γ (i) (y, 0) = h(i) 0 (y)
in D × (0, ∞), (4.31)
682
G.P. Galdi
and ⎫ ∂Γ (i) (i) (i) ⎬ = Γ − grad P , ∂t ⎭ div Γ (i) = 0, Γ (i) (z, t) = 0,
in D × (0, ∞),
z ∈ Σ,
(4.32)
lim Γ (i) (y, t) = 0,
|y|→∞
(i)
Γ (i) (y, 0) = H0 (y). The pairs (γ (i) , p(i) ), and (Γ (i) , P(i) ), i = 1, 2, 3, are the unsteady counterpart of (h(i) , p(i) ), and (H(i) , P (i) ), respectively, defined in (4.15), (4.16). Their existence and other relevant properties are collected in the following lemma. L EMMA 4.4. Let B be of class C 2 . Then, there exists one and only one solution to (4.31) such that γ (i) ∈ C [0, T ], Lqσ (D) ∩ Lr *, T ; W 2,q (D) ∩ W 1,r *, T ; Lq (D) for arbitrary positive * and T , and arbitrary r > 1 and q > 3. Moreover, the following estimates hold
grad γ (i) (t) ct −1/2 1 + t −1/2q h0 q , Σ
(i) p (t) c t −1/2 1 + t −1/2q 2 + t −1/2q 1/q h0 q ,
all t ∈ (0, T ),
Σ
(4.33)
where the constant c depends on T , q, and D. The same conclusions are valid for problem (4.32), if we replace γ (i) and p(i) with Γ (i) and P(i) , respectively. P ROOF. The first part of the lemma follows from the work of Maremonti and Solonnikov [89]. The second part is proved in [34], Section 3. Our next goal is to establish the unsteady counterpart of (4.22). To this end, let ξ (t) and ω(t) be two vector functions such that ξ (t), ω(t) ∈ W 1,r (0, T ),
some r 1,
Set ξ (t) = ξi (t)ei ,
ω = ωi (t)ei ,
ξ (0) = ω(0) = 0.
On the motion of a rigid body in a viscous liquid
683
and consider the following fields w(y, t) ≡ ξi (t)h(i) (y) + ωi (t)H(i) (y) t dωi (i) dξi (i) γ (y, t − s) + Γ (y, t − s) ds, − ds ds 0 dξi (i) dωi (i) p0 (y) − P (y) + G · y dt dt 0 t dωi (i) dξi (i) − p (y, t − s) + P (y, t − s) ds. (4.34) ds ds 0
p(y, t) ≡ ξi (t)p(i) (y) + ωi (t)P (i) (y) −
Clearly, w satisfies the initial and boundary conditions in (4.6). Moreover, by a direct calculation that uses (4.31), (4.32), we show that ∂w dξi (i) − w + grad p − G = − γ 0 (y, 0) + grad p0(i) (y) − h(i) (y) ∂t dt dωi (i) (i) Γ 0 (y, 0) + grad P0 (y) − H(i) (y) = 0, − dt where, in the last step, we have employed (4.31)5, (4.32)5 and (4.28). We can now furnish an expression for the total force and the total torque exerted by L on B at each time t in terms of solution (4.34). Specifically, recalling (4.21) and using (4.18), we have −
T(w, p) · n = −K · ξ (t) − C · ω(t) Σ
−
t 0
dω dξ + C(t − s) · K(t − s) · ds ds ds
− M1 ·
dξ dω − I1 · − |B|G dt dt
(4.35)
and −
y × T(w, p) · n = −CT · ξ (t) − Θ · ω(t) Σ
t dω dξ + Υ (t − s) · ds − H(t − s) · ds ds 0 − I2 ·
dξ dω − M2 · + |B|G × R, dt dt
(4.36)
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G.P. Galdi
where Kj i (t) =
Σ
Σ
Υj i (t) = Cj i (t) = Σ
Hj i = Σ
(i) (i) T γ ,p · n j, y × T Γ (i) , P(i) · n j ,
y × T γ (i) , p(i) · n j ,
(4.37)
(i) (i) T Γ ,P · n j
and
(i)
M1j i =
Σ
M2j i = Σ
I1j i =
Σ
p0 nj , P0(i) (y × ·n)j , (4.38)
(i) P0 nj , (i)
I2j i = Σ
p0 (y × n)j .
The matrices (4.37) are the unsteady counterpart of the matrices (4.21). Like the latter, they depend only on geometric properties of B such as size, shape, symmetry, etc., but they are otherwise independent of the orientation of B and of the physical properties of L. As function of t, they are expected to become singular at t = 0, since the tangential component of the initial data (4.32)5, (4.31)5 does not, in general, match the tangential component of the boundary data at t = 0. Some of the properties of (4.37) are collected in the following lemma. L EMMA 4.5. Set K K= H
C Υ
,
and assume that B is of class C 2 . The following properties hold. (a) For all T , η > 0, there is cη,T > 0 such that K(t) cη,T t −1/2−η ,
for all t ∈ (0, T ].
(b) K ∈ C(0, T ] for all T > 0. (c) If B is a homogeneous body of revolution with fore-and-aft symmetry, then C = H = 0.21 21 See footnote 17.
On the motion of a rigid body in a viscous liquid
685
P ROOF. Since
3
K(t) c
grad γ (i) + grad Γ (i) + p(i) + P(i) ,
i=1 Σ
part (a) follows from Lemma 4.4, by taking q as large as we please. Part (b) is a consequence of the fact that, again from Lemma 4.4 and classical embedding theorems, (γ (i) , p(i) ), and (Γ (i) , P(i) ) are continuous in (0, T ] with values in W 2,r (D) × Lr (D). Finally, the proof of part (c) is given in [34], Theorem 5.2. R EMARK 4.1. The estimate furnished in part (a) of Lemma 4.5 is “essentially” sharp. Actually, for B a sphere, from the work of Basset [3] (see also [118]), it follows that Kij (t) = cij t −1/2 , i, j = 1, 2, 3, with cij a suitable negative definite constant matrix. Concerning the matrices (4.38), we also have that they are independent of the orientation of the body and of the physical properties of the liquid. In particular, the 6 × 6 matrix M=
M1 I2
I1 M2
is called the added mass matrix, and possesses the following properties [122]. L EMMA 4.6. Let B be Lipschitz. Then, the matrix M is positive semi-definite. If, moreover, B is a homogeneous body of revolution with fore-and-aft symmetry, then I 1 = I 2 = 0.22 Set I=
m1 0
0 . I
From (4.35), (4.36), we deduce that the fields (4.34) will be a solution to the free fall problem (4.6)–(4.10) if and only if ξ , ω, and G satisfy the following problem B·
dχ =− dt
0
dG = G × ω, dt 0 χ (0) = , 0 22 See footnote 17.
t
H(t − s) ·
dχ ds + L(G), ds (4.39)
G(0) = g,
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G.P. Galdi
where ξ , B = I + M, ω me G L(G) = . |B|G × R
χ=
H = K + A, (4.40)
Problem (4.39)–(4.40) is the unsteady counterpart of (4.22) and completely characterizes unsteady free falls of B, in a very general class of velocity and pressure fields. To show this, let % & 1,s (D) , p ∈ Lr 0, T ; Lsloc (D) ; ξ , ω ∈ W 1,r (0, T ) , CTr,s = w ∈ Lr 0, T ; Wloc r, s > 1, T > 0. The following lemma holds. L EMMA 4.7. Let B be of class C 2 , and let {w, p, ξ , ω} ∈ CTr,s , for some r, s > 1, T > 0, be a solution (in the sense of distributions) to (4.6)1,2,4, with w(y, 0) = 0, and satisfying (4.6)3 uniformly pointwise for all t ∈ (0, T ]. Then, the following statements hold. (a) The fields w, p verify the properties w ∈ Lr 0, T ; W 2,q (D) ,
dw , grad(p − G · y) ∈ Lr 0, T ; Lq (D) , dt
for all q > 1;
(4.41)
(b) The fields w and p satisfy the representation (4.34), for all t ∈ (0, T ]. Moreover, if ξ , ω ∈ W 2,2 (ε, T ), for some ε 0, then dw ∈ L∞ ε, T ; L2 (D) ∩ L2 ε, T ; W 1,2 (D) ∩ L2 η, T ; W 2,2 (D) , (c) dt η > ε.
for all
P ROOF. The proof of (a) is established in [43]. Property (b) follows from (a). Concerning (c), we notice that, denoting by V(y, t) a smooth, solenoidal extension of U(y, t) of bounded support in D (see Equation (4.81) below), we have that the field v = w − V satisfies the following Stokes problem ⎫ ∂v ⎪ = v − grad(p − G · y) + f1 ξ (t) ⎪ ⎪ ⎪ ∂t ⎬ dξ dω + f2 + f3 × ω + f4 × , ⎪ ⎪ dt dt ⎪ ⎪ ⎭ div v = 0, lim v(y, t) = 0,
|y|→∞
in D × (0, ∞),
On the motion of a rigid body in a viscous liquid
v(y, t) = 0,
y ∈ Σ,
v(y, 0) = 0,
687
y ∈ D,
where the f ’s are smooth functions of y only, with bounded support in D. Property (c) can then be proved by the same arguments used in [40], Lemma 5.4 and Lemma 5.5. The following result establishes the equivalence of the solvability of (4.39)–(4.40) and of (4.6)–(4.10) in the class C r,s . L EMMA 4.8. Let B be of class C 2 . If (w, p, ξ , ω) ∈ CTr,s is a solution (in the sense of distributions) to (4.6)1,2,4–(4.10), satisfying (4.6)3 uniformly pointwise for all t ∈ (0, T ], then (4.39)–(4.40) hold. Conversely, assume that (4.39)–(4.40) has a solution G, χ , with χ ∈ W 1,r (0, T ), r > 1,23 then problem (4.6)–(4.10) has a solution in the class CTr,s , for all s > 1. If such a solution exists, it admits the representation (4.34) and, moreover, it satisfies the properties (4.41). P ROOF. If (4.6)–(4.10) has a solution in the sense specified in the statement of the lemma, we have, by Lemma 4.7 and by the reasoning leading to (4.39)–(4.40), that it also satisfies (4.39)–(4.40). Conversely, if (4.39)–(4.40) has a solution G, χ , with χ satisfying the properties specified in the statement of the lemma, then the fields given in (4.17) are solutions of (4.6)–(4.10), and the conclusions of Lemma 4.7 apply. We are now in a position to prove the existence of solutions to the unsteady free fall problem. T HEOREM 4.3. Let B be of class C 2 . Then, for any g ∈ S 2 , problem (4.6)–(4.10) has one and only one solution {w, p, ξ , ω, G}, with {w, p, ξ , ω} ∈ CTr,s , for all r, s > 1, all T > 0 and with w, p satisfying (4.41) for all r > 1, all T > 0. Moreover, ξ , ω ∈ C 1 ([0, T ]) ∩ C 2 ((0, T ]), G ∈ C 2 ([0, T ]), for all T > 0. P ROOF. We begin to show existence. In view of Lemma 4.8, to prove the first part, it is enough to show that (4.39)–(4.40) has a solution χ , G, with χ ∈ W 1,r (0, T ), for all r > 1. To this end, we write this problem in the following equivalent form
t
X(t) =
N(t − s) · X(s) ds + F(G),
0
dχ = X, dt dG = G × ω, dt 0 χ (0) = , 0
(4.42)
G(0) = g,
23 Notice that if χ ∈ W 1,r (0, T ), r > 1, then G ∈ C 1 ([0, T ]).
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G.P. Galdi
where N = −B−1 · H,
F = B−1 · L.
The solvability of (4.42) can be obtained by combining Schauder fixed point theorem with a boot-strap argument. Let Hτ = C([0, τ ]), τ T . For any given ∈ Hτ , we solve (4.42)3 the corresponding solution. Replace G for G in the with in place of ω and denote by G right-hand side of (4.42)1, that thus becomes a Volterra integral equation of the second type with a continuous right-hand side and a weakly singular kernel. By well-known results on this type of equation, [85], Theorem 3.13 and Lemma 3.2, we find a solution X ∈ Hτ , and, consequently a corresponding χ ≡ (ξ , ω) ∈ W 1,r (0, τ ), all r > 1, with ω ∈ Hτ . We may in this way define a map N : ∈ Hτ → ω ∈ Hτ . Next, let J be an integer such that
N(ζ ) dζ < 1 , 2
τ 0
τ = T /J.
(4.43)
The number J certainly exists in view of condition (a) in Lemma 4.5. Setting []a[] = maxt ∈[0,τ ](|a(t)| + |da(t)/dt|), from (4.43) and (4.42), we easily obtain []ω[] + []ξ [] T ,
(4.44)
where T is a positive quantity depending only on g and on T . Therefore, choosing δ = T , we find that N transforms a ball of Hτ of radius δ into itself. Moreover, by the Ascoli– Arzelà theorem, the map N is compact and, by the Schauder theorem, it has a fixed point in Hτ , with τ satisfying (4.43). Therefore (4.42) has a solution χ 0 (t), G0 (t), t ∈ [0, τ ], that, in view of (4.44), belongs, in particular, to W 1,r (0, τ ), for all r > 1. For t ∈ [0, τ ], set χ 1 (t) ≡
ξ 1 (t) ω1 (t)
= χ (t + τ ),
G1 (t) = G(t + τ ),
X1 (t) = X(t + τ ).
From (4.42) we then have that χ 1 , G1 , and X1 formally satisfy the following problem
t
X1 (t) =
N(t − s) · X(s) ds + F(G1 ) + F1 (t),
0
dχ 1 = X1 , dt dG1 = G1 × ω 1 , dt χ 1 (0) = χ 0 (τ ),
(4.45)
G1 (0) = G0 (τ ),
where
τ
F1 (t) ≡ 0
N(t + τ − s) · χ 0 (s).
On the motion of a rigid body in a viscous liquid
689
Since F1 is continuous (see Lemma 4.5(b)), we repeat the same reasoning as before and find a solution χ 1 (t), G1 (t), X1 (t) to (4.45) for t ∈ [0, τ ], with χ 1 , G1 ∈ W 1,r (0, τ ), for all r > 1. Iterating this procedure we find solutions χ j (t), Gj (t), Xj (t), j = 0, . . . , J − 1, with χ j , Gj ∈ W 1,r (0, τ ), for all r > 1, and verifying χ j (τ ) = χ j +1 (0), Gj (τ ) = Gj +1 (0). Therefore, the functions χ (t), G(t), and X(t) defined by χ (t + j τ ) = χ j (t),
G(t + j τ ) ≡ Gj (t),
X(t + j τ ) = Xj (t),
t ∈ [0, τ ], j = 0, . . . , J − 1, solve (4.42) for all t ∈ [0, T ], and verify χ , G ∈ W 1,r (0, T ), for all r > 1. Using this information in (4.42)1, together with the property of the kernel N (see Lemma 4.5), we easily obtain that χ ∈ C 1 ([0, T ]), for all T > 0 which, by (4.42)3, in turn furnishes G ∈ C 2 ([0, T ]). Moreover, again from (4.42)1 and well-known regularity result for Volterra integral equations [91], Theorem 1, we have X ∈ C 1 ((0, T ]), and the proof of existence is therefore completed. To show the uniqueness part, we observe that, in view of wellknown results for the exterior Stokes initial-value problem [89], it is enough to prove that problem (4.42) has at most one solution corresponding to the given g, in the class of solutions {ξ , ω, G}, with ξ , ω ∈ W 1,r (0, T ), and G ∈ C 1 ([0, T ]), T > 0. Denoting by {ξ 1 , ω 1 , G1 }, {ξ 2 , ω2 , G2 } two such solutions, from (4.42) we find that ξ ≡ ξ 1 − ξ 2 , ω ≡ ω1 − ω2 , and γ = G1 − G2 satisfy the following problem (with the obvious meaning of symbols) X(t) = tN(t − s) · X(s) ds + F(γ ), 0
dχ = X, dt dγ = γ × ω1 + G2 × ω, dt 0 χ (0) = , γ (0) = 0. 0
(4.46)
Applying Young’s inequality for convolutions in the first of these equations we obtain, for any r > 1 X r,(0,t ) N 2r/(1+r),(0,t ) X 2r/(1+r),(0,t ) + M1 γ r,(0,t ),
(4.47)
where M1 is a positive constant. Since 2r/(1 + r) < 2, by Lemma 4.5(a) we find that N 2r/(1+r),(0,T ) ≡ M2 < ∞. Thus, by Hölder inequality and (4.47), it follows that X r,(0,t ) M2 t (r−1)/2r X r,(0,t ) + M1 γ r,(0,t ). From (4.46)3 it easily follows that t t γ (t) M3 γ (s) ds + |g| ω(s) ds, 0
0
(4.48)
(4.49)
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G.P. Galdi
where M3 ≡ maxt ∈[0,T ] |ω1 (t)|, and where we used the fact that |G2 (t)| = |g|, for all t > 0. Integrating over (0, t) and applying Hölder inequality, from (4.49) we obtain (1 − M3 t) γ r,(0,t ) |g|t 1/r
ω(s) ds.
t
(4.50)
0
Further, since ω(0) = 0, we have ω(s)
$ $ t dω $ $ ds t 1−1/r $ dω $ . ds $ dt $ 0 r,(0,t )
Thus, for t < 1/(2M3), from (4.50) it follows that $ $ $ dω $ $ 2|g|t 2 N r,(0,t ) . γ r,(0,t ) 2|g|t 2 $ $ dt $ r,(0,t ) We now use this inequality on the right-hand side of (4.48) to obtain 1 − M2 t (r−1)/2r − 2M1 |g|t 2 X r,(0,t ) 0. Thus, for all t less than a fixed, suitable positive constant t0 (say), we get that X(t) is identically zero in [0, t0 ], which in turn, since χ (0) is zero, implies that χ (t) vanishes for all t ∈ [0, t0 ]. Iterating this procedure a finite number of times, we eventually obtain that χ (t) vanishes for all t ∈ [0, T ], for arbitrarily fixed T > 0, which concludes the proof of the uniqueness part. The theorem is therefore completely proved. We wish to specialize the previous result to the case of homogeneous bodies of revolution with fore-and-aft symmetry. T HEOREM 4.4. Let B be a homogeneous body of revolution of class C 2 , possessing the fore-and-aft symmetry. Then, all solutions to (4.6)–(4.10) with {w, p, ξ , ω} ∈ CTr,s , for all T > 0, and satisfying (4.6)3 uniformly pointwise, must have ω(t) = 0 and, consequently, G(t) = g, for all times t > 0. Therefore, these bodies keep their initial orientation at all subsequent times. P ROOF. By Lemma 4.8, {ξ , ω, G} satisfy (4.39). Moreover, by Lemma 4.5(c) and Lemma 4.6, the matrices H and M are diagonal and, further, R ≡ 0.24 Thus, from (4.39) we get, in particular, dω =− dt
0
t (I + M2 )−1 · Υ (t − s) ·
dω ds, ds
ω(0) = 0. 24 If B is homogeneous, its center of mass and its centroid coincide.
(4.51)
On the motion of a rigid body in a viscous liquid
From the Young inequality for convolutions, from (4.51)1 we deduce $ $ $ $ $ dω $ $ dω $ $ $ $ $ $ $ $(I + M2 )−1 · Υ $ . $ dt $ 2r/(1+r)$ dt $ r 2r/(1+r)
691
(4.52)
Since 2r/(1 + r) < min{2, r} we have, by Lemma 4.5(a), that (I + M2 )−1 · Υ 2r/(1+r) ≡ M < ∞, and, by Hölder inequality and (4.52), that $ $ $ $ $ $ $ dω $ $ $ Mt (r−1)/2r $ dω $ . $ dt $ $ dt $ r r Therefore, we obtain dω/dt ≡ 0, for all t < M −2r/(r−1), that is, since ω(0) = 0, ω ≡ 0, for all t < M −2r/(r−1). Iterating this procedure, we conclude ω(t) = 0 for all times t > 0. The proof of the theorem is thus completed. R EMARK 4.2. The results proved in Theorem 4.2 and Theorem 4.4 for the case of homogeneous body of revolutions with fore-and-aft symmetry, give a rigorous interpretation of the sedimentation experiments performed in a Newtonian liquid at exceedingly small Reynolds number; see [111] and Figure 3. 4.1.3. Attainability of steady free falls An interesting question that deserves attention, already in the case of Stokes approximation, is that of the attainability, as t → ∞, of the possible steady motions that B can perform. In other words, if we drop B from rest, which steady motion (terminal state) will it eventually achieve? According to experimental evidence, it can happen that, for a given B, one or more of these motions can be effectively achieved. For instance, if B is a non-homogeneous cylinder with its center of mass C belonging to the axis of the cylinder, it can be shown from Theorem 4.1 that there are two possible purely translational steady motions, namely, those with the vector R = C − C (see (4.18)) having either the same or the opposite orientation with g. However, only the latter is experimentally observed; see Figure 7. On the other hand, if the cylinder is homogeneous, as we know from Theorem 4.2, there is an infinite number of steady motions, and they are all experimentally observed; see [111] and Figure 3. We would like to emphasize that the problems of attainability and stability are completely different in this setting, from both mathematical and physical points of view. For instance, consider the case of the non-homogeneous cylinder of Figure 7. If we drop the cylinder from rest with its heavier part on top and R parallel to g (case (B)), by using Theorem 4.3 it is not hard to show that R and g will stay parallel for all times, and that eventually, the cylinder will reach the steady state corresponding to the case (B). Therefore, this steady state is attainable from certain initial conditions. However, the same steady state is not stable with respect to small disorientations of R with respect to g, as expected on physical ground, and as a simple “quasi-static” mathematical analysis proves [119], §4. From the mathematical point of view, the problem of attainability of a terminal state (and of its stability) can be addressed by investigating the asymptotic behavior of solutions χ , G to (4.39). This study may present some difficulty, in general, due to the particular integro-differential structure of (4.39), and to the lack of accurate information about the
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G.P. Galdi
Fig. 7. Orientation of a non-homogeneous cylinder. The grey part is heavier than the white part. C is the centroid and C the center of mass. Both configurations in (A) and (B) correspond to purely translational steady motions of the cylinder. However, the configuration in (A) is stable, while that in (B) is unstable to a small disorientation of R with respect to g.
kernel H. In fact, even in the case of the sphere, where H is known explicitly (see also Remark 4.1), the study of the long-time behavior of χ and G requires some effort, see [3], Chapter XXII, and [118], pp. 218–224. Here, by means of a different approach that relies on the study of the full set of equations (4.6)–(4.10), we shall show that every homogeneous body of revolution with fore-and-aft symmetry, that is dropped from rest in a quiescent liquid will eventually reach the (translational) velocity corresponding to the steady state uniquely determined by its initial orientation. To reach our goal, we notice, in the first place, that from Theorem 4.4 we have that all possible orientations are attained since t = 0. Denote now by g0 the direction of the acceleration of gravity in a given orientation, and by {w0 , p0 , ξ 0 , g0 } the corresponding steady solution given in Theorem 4.2. Recalling that, by Theorem 4.3, we have ω(t) ≡ 0, G(t) ≡ g0 at all times, from (4.6)–(4.9) and (4.11)–(4.14), we find that the fields u = w − w0 , p = p − p0 , ζ = ξ − ξ 0 satisfy the following problem ⎫ ∂u = u − grad p, ⎬ ∂t ⎭ div u = 0, lim u(y, t) = 0,
in D × (0, ∞), (4.53)
|y|→∞
u(y, t) = ζ (t), dζ m =− dt
y ∈ Σ,
T(u, p) · n
(4.54)
Σ
with initial conditions u(y, 0) = w0 (y). We multiply both sides of (4.53)1 by u − grad p ≡ div T(u, p) and integrate over DR ≡ {y ∈ D: |y| < R}. Using (4.53)4 and (4.54), we find
On the motion of a rigid body in a viscous liquid
1 d D(u) : D(u) 2 dt DR 2 dζ ∂w div T(u, p)2 + · T(u, p) · n. = −m − dt DR |y|=R ∂t
693
(4.55)
We next observe that, from known results on the steady exterior Stokes problem (see [35], Chapter V), we have grad w0 , and p0 − g0 · y = O |y|−2
as |y| → ∞.
(4.56)
Moreover, by Lemma 4.7 we also have dw ∈ Lr 0, T ; Lq (D) , dt
all r, q > 1, all T > 0,
(4.57)
and, by Lemma 4.7 and by Sobolev inequality, grad w, grad(p − g0 · y) ∈ Lr 0, T ; Ls (D) , all s > 3/2, r > 1, all T > 0.
(4.58)
Consequently, from (4.56)–(4.58) and Fubini’s theorem we deduce that the function of F defined by F (R, t) ≡
t 0
|y|=R
∂w · T(w, p − g0 · y) − T(w0 , p0 − g0 · y) · n, ∂t
satisfies F (·, t) ∈ L1 (R, ∞), for sufficiently large R, and any fixed t 0. Therefore, for any t 0 there is a diverging sequence {Rk }k∈N such that lim F (Rk , t) = 0.
k→∞
Since t 0
|y|=R
∂w · T(u, p) · n = F (R, t), ∂t
integrating (4.55) from 0 to t and passing to the limit along the sequence {Rk }k∈N , we conclude D u(t) : D u(t) D
t 2 t 2 dζ div T(u, p) = + 2m + 2 D(w0 ) : D(w0 ), 0 ds 0 D D
(4.59)
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G.P. Galdi
for all t 0. Notice that the right-hand side of this equation is finite in view of (4.56). From (4.59) we obtain, in particular, ∞ dζ 2
$ $2 $ $ + $ ∂u $ < ∞ $ ∂t $ dt 2
0
(4.60)
and, $ $ $ $ $D u(t) $ $D(w0 )$ . 2
(4.61)
With the help of some results that will be shown later in Section 4.2.1 (see (4.75), (4.74), and Lemma 4.9), from (4.61) we further obtain $ $ $ $ $ $ $ $ ζ (t) κ $D(w0 )$ , $u(t)$ + $grad u(t)$ c$D(w0 )$ 2 6 2 2 for all t > 0,
(4.62)
where κ and c are positive constants depending only on B. We next differentiate (4.6)1 with respect to t, multiply the resulting equation by ∂w/∂t, and integrate by parts over D. Since, by Lemma 4.7, ξ ∈ W 2,2 (ε, T ), all ε > 0, and all T > ε, we may use (4.7) along with the regularity results of Lemma 4.7 to obtain the following relation 2 2 ∂w ∂D(w) 2 1 d + 1 m d dξ = − ∂t , 2 dt D ∂t 2 dt dt D
t > 0.
(4.63)
Since ∂w ∂u = , ∂t ∂t
dξ dζ = , dt dt
∂D(w) ∂D(u) = ∂t ∂t
from (4.63) we get 1 d 2 dt
2 2 ∂u + m dζ 0, ∂t dt D
which, together with (4.60), implies $ $ $ ∂u $ $ = lim dζ = 0. lim $ $ $ t →∞ ∂t t →∞ dt
t > 0,
(4.64)
Let Q denote the class of functions ϕ = ϕ(y) that are infinitely differentiable, solenoidal, with bounded support in D, and that equal a constant vector aϕ , depending on ϕ, in a neighborhood of Σ. Multiplying (4.53) by ϕ, integrating over D, and using (4.54), we obtain dζ ∂u ·ϕ+m · aϕ = − D(u) : D(ϕ). (4.65) dt D ∂t D
On the motion of a rigid body in a viscous liquid
695
We now let t → ∞ in this relation through an arbitrary sequence {tk }k∈N . By (4.62) we find 1,2 that there is u0 ∈ Wloc (D) and ζ 0 ∈ R3 (depending a priori on the particular sequence) and a subsequence {tk }k ∈N such that u0 ∈ L6 (D), lim
tk →∞ D
D(u0 ) ∈ L2 (D),
D u(tk ) : D(ϕ) =
u0 |Σ = ζ 0 ,
D
D(u0 ) : D(ϕ),
for all ϕ ∈ Q
lim ζ (tk ) = ζ 0 .
tk →∞
Thus, from these latter relations, and from (4.65) and (4.64) we conclude, by classical regularity results on the Stokes problem, that u0 , ζ 0 solve the following Stokes problem u0 − grad p0 = 0,
:
div u0 = 0,
in D,
lim u0 (y) = 0,
|y|→∞
u0 (y) = ζ 0 ,
y ∈ Σ,
0=
T(u0 , p0 ) · n, Σ
for some “pressure” p0 . However, as is well-known [35], Chapter V, this Stokes problem admits only the zero solution u0 ≡ grad p0 ≡ ζ 0 ≡ 0. Therefore, since the sequence {tk }k∈N is arbitrary, we conclude lim ζ (t) = 0,
t →∞
(4.66)
that is what we wanted to prove. R EMARK 4.3. Since B does not change its orientation in time, ξ (t) will have the same direction as ξ 0 , for all times t > 0. This is in complete agreement with the experiments [111]; see also Figure 3. Once (4.66) has been established, using the representation (4.34) one can prove that u tends to zero as t → ∞ in several different norms. Details will be given elsewhere [41].
4.2. The full nonlinear case As it might be expected, the free fall problem in a Navier–Stokes liquid is much more complicated, and presents several important unanswered questions, mainly in the unsteady case. In fact, in the case of free steady fall the picture is more complete. Specifically,
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G.P. Galdi
one can show the existence of a solution for arbitrary body B and arbitrary Re [102]; furthermore, one can furnish a complete characterization of translational steady falls for bodies of revolution with fore-and-aft-symmetry, at least for small Re, that is in complete agreement with the experiment [48]. However, in the unsteady case, while it is relatively trivial to show that (4.1)–(4.4) admits at least one global weak solution (in the sense of Leray and Hopf) for any data [102], it is an open question to prove existence of strong solution even for small times. Moreover, the study of the asymptotic behavior in time of unsteady falls, and the corresponding problem of attainability (or stability) of steady falls is also completely open, even for homogeneous bodies with fore-and-aft symmetry. The reason why this study is complicated is easily explained. As is known, an important role in establishing attainability (or stability) of a steady solution in an exterior domain is played by a sharp knowledge of the functional properties of the linearized operator around the “trivial” solution; see, e.g., [42,88,80,65,104]. It is also known that the Hilbert space L2 does not furnish a good setting for the study of these properties, and one has to resort to the more difficult Lq case, q = 2 [78]. Now, if the body is merely translating in the direction x1 , say, the linearization is the classical Oseen operator: ∂u LT (u, p) = − u − Re − grad p, div u , ∂y1 and these properties are well-known [35]. If, however, the body is translating and rotating, the linearized operator becomes ∂u LT R (u, p) = − u − Re + ω × y · grad u − ω × v ∂y1 − grad p, div u .
(4.67)
The study of the functional properties of the operator LT R seems to be particularly complicated. The only results available for a linearization that takes into account rotation (but no translation), have been obtained recently by Hishida [59,60], and they are, unfortunately, not as general as to be applied to the present situation. In our study of the asymptotic (spatial) behavior of the velocity field of the liquid in a steady fall (see Theorem 4.6), we shall show a number of properties for the full operator LT R , including translation (see Lemma 4.14), that are more detailed than those obtained by Hishida, but that however are still far from furnishing the necessary information for an appropriate study of stability and/or attainability. In the following sections, we shall be concerned mainly with the steady problem. The known results for the unsteady case will be briefly described in Section 4.2.7.
On the motion of a rigid body in a viscous liquid
697
4.2.1. Steady free fall: existence In the case of steady free fall, (4.1)–(4.4) reduce to the following ones : Re (w − U) · grad w + ω × w = w − grad p + g, div w = 0,
in D, (4.68)
lim w(y) = 0,
|y|→∞
w(y) = U(y) ≡ ξ + ω × y,
y ∈ Σ,
T(w, p) · n,
Re mω × ξ = mg −
(4.69)
Σ
y × T(w, p) · n,
Re ω × (I · ω) = −
(4.70)
Σ
g × ω = 0.
(4.71)
Following [121,102], we shall now give a variational formulation of problem (4.68)–(4.71). To this end, we denote by C(D) the class of test functions ϕ such that: (1) ϕ ∈ C0∞ D ; (2)
div ϕ = 0 in D;
(4.72)
(3) ϕ = ϕ¯ ≡ Φ 1 + Φ 2 × y,
for some Φ i ∈ R3 , i = 1, 2, in a neighborhood of Σ.
Extending ϕ by ϕ¯ in B and using the identity ϕ = 2 div D(ϕ) − grad div ϕ,
(4.73)
with D stretching tensor (see (2.2)), we find grad ϕ 2,R3 =
√ $ $ 2$D(ϕ)$2,D .
(4.74)
Therefore, by the Sobolev inequality, it follows that $ $ ϕ 6,R3 γ0 $D(ϕ)$2 ,
(4.75)
with γ0 absolute constant, and where the norm on the right-hand side is taken over D. Using this inequality, we prove the following (see [121], Lemma 3.1 and [102], Lemme 3.1) L EMMA 4.9. Let B be an arbitrary domain. There is κ = κ(B) such that, for all ϕ ∈ C(D): $ $ |Φ 1 | + |Φ 2 | κ $D(ϕ)$2 .
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G.P. Galdi
P ROOF. Extending ϕ by ϕ¯ in B, from the identity ϕ = − curl curl ϕ + grad div ϕ, we obtain
R3
|curl ϕ|2 =
R3
|grad ϕ|2 .
(4.76)
Since 4|B| |Φ 2 |2 =
B
curl(Φ 1 + Φ 2 × y)2 =
B
|curl ϕ|2
R3
|curl ϕ|2 ,
from (4.74) and (4.76) we deduce $ $ |Φ 2 | 2−1/2 |B|−1/2$D(ϕ)$2 .
(4.77)
Also, 1/6
|B|
1/6
|Φ 1 |
B
|Φ 1 + Φ 2 × y|
R3
1/6 |ϕ|6
6
1/6
+
B
|Φ 2 × y|
6
+ δ(B)|B|1/6|Φ 2 |,
with δ(B) the diameter of B, and the lemma follows from this inequality, (4.75) and (4.77). Let $ $ H(D) = completion of C(D) in the norm $D(·)$2 .
(4.78)
From (4.74) and (4.75) it follows that, for any B, the following (set-theoretic) inclusion holds 1,2 H(D) ⊂ Wloc D .
(4.79)
We also have the following. L EMMA 4.10. For any B, there exists a bounded linear operator T : u ∈ H(D) → u¯ ∈ R. If B is Lipschitz, then u¯ coincides with the trace of u on Σ. P ROOF. The proof is a consequence of Lemma 4.9, of (4.79), and of well-known trace theorems.
On the motion of a rigid body in a viscous liquid
699
If B has a little regularity, we can give a characterization of the space H(D). To this end, set % 1,2 D : u ∈ L6 (D), D(u) ∈ L2 (D); div u = 0 in D; Y(D) ≡ u ∈ Wloc & u = u1 + u2 × y, y ∈ Σ, ui ∈ R3 , i = 1, 2 , where, if B has no regularity, the trace of u on Σ is meant in the sense of Lemma 4.10. The following properties hold. L EMMA 4.11. For any B, we have H(D) ⊂ Y(D). Moreover, if B is Lipschitz, then H(D) = Y(D). P ROOF. From Lemma 4.10 and (4.79) we obtain at once H(D) ⊂ Y(D), for any B. Assume B Lipschitz. We have to show that, for any ε > 0, there is ϕ ε ∈ C(D) such that $ $ $D(u) − D(ϕ ε )$ < ε. (4.80) 2 We observe that the extension of u to R3 obtained by setting u = u1 + u2 × y, y ∈ B, 1,2 belongs to Wloc (R3 ). We continue to denote by u such an extension. By a simple cut-off argument that uses (4.73), it follows that grad u ∈ L2 (D), and that $ $ grad u 2 $D(u)$2 . Set 1 V = − curl curl ζ u1 y22 + ζ |y|2u2 , 2
(4.81)
where ζ is an arbitrary function from C0∞ (D) that is one near Σ and zero far from Σ. The function v = u − V is solenoidal, has a finite Dirichlet integral, vanishes at Σ in the trace sense, and belongs to L6 (D). From known results, [35], Theorem II.6.1, Lemma III.5.1, it follows that for any ε > 0 there exists a solenoidal function ψ ε ∈ C0∞ (D) satisfying $ $ $grad(v − ψ ε )$ < ε. 2 The lemma then follows by taking ϕ ε = ψ ε + V.
Multiplying (4.68)1 by ϕ ∈ C(D) and integrating over D we obtain Re (w − U) · grad w · ϕ + ω × w · ϕ D
= −2
D
D(w) : D(ϕ) +
+ Φ2 ·
D
g · ϕ + Φ1 ·
y × T(w, p) · n. Σ
T(w, p) · n Σ
(4.82)
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G.P. Galdi
Since, by (4.18), g · ϕ = −|B|g · Φ 1 − |B|Φ 2 × R · g, D
(4.83)
from (4.82) and (4.69), (4.70) we find: Re (w − U) · grad w · ϕ + ω × w · ϕ D
= −2
D
D(w) : D(ϕ) + me g · Φ 1 − |B|Φ 2 × R · g
− m Re Φ 1 · ω × ξ − Re Φ 2 · ω × (I · ω).
(4.84)
The above argument shows that every (sufficiently smooth) solution to (4.68)–(4.70) is a solution to (4.84), for any ϕ ∈ C(D). Conversely, if {w, ξ , ω, g} solves (4.84) for any ϕ ∈ C(D), and 1,2 D , div w = 0, w ∈ Wloc then w ∈ C ∞ (D),
(4.85)
and (4.68)1 is satisfied for a suitable p ∈ C ∞ (D). In fact, taking in (4.84), in particular, ϕ (solenoidal and) from C0∞ (D), we get
D
D(w) : D(ϕ) =
D
Z · ϕ,
1 Z ≡ Re (w − U) · grad w + ω × w . 2
(4.86)
Since Z ∈ L2loc (D), the property (4.85) follows from classical interior regularity results for the Stokes problem, along with a standard boot-strap argument; see, e.g., [35], Chapter V. Integrating (4.86) by parts, and using the fact that ϕ is arbitrary, we find (4.87) Re (w − U) · grad w + ω × w − div 2D(w) = −grad(p − g · y) for some p ∈ C ∞ (D). If Σ is sufficiently smooth (e.g., C 2 ), then {w, p, ξ , ω, g} also satisfies (4.69) and (4.70). Actually, using integration by parts for the integral on the righthand side of (4.84), together with (4.87) and (4.83), we get (Φ 1 + Φ 2 × y) · (2D(w) − p1) · n Σ
= mg · Φ 1 − |B|Φ 2 × R · g − m Re Φ 1 · ω × ξ − Re Φ 2 · ω × (I · ω). Since Φ 1 and Φ 2 are arbitrary, from this relation we deduce (4.69), (4.70). In view of the above considerations, we give the following definition of weak solution to the steady free fall problem [121,102].
On the motion of a rigid body in a viscous liquid
701
D EFINITION 4.1. A quadruple {w, ξ , ω, g}, with g ∈ S 2 , is called a weak solution to (4.68)–(4.71) if and only if (i) w ∈ H(D); (ii) U ≡ ξ + ω × x is the trace of w on Σ, in the sense of Lemma 4.10; (iii) Identity (4.84) is satisfied for all ϕ ∈ C(D); (iv) g × ω = 0. R EMARK 4.4. The determination of a field w satisfying (i), (ii), and (iii) of Definition 4.1 with g given could be obtained by standard methods. The difficulty of the steady free fall problem is that g is not given and we have to find it in such a way as to satisfy condition (iv). Our next objective is to prove the existence of a weak solution. To this end we propose the following two lemmas. The proof of the first is given in [38], Lemma 5.2, while the proof of the second is due to P. Rabier [99]; see also [102], Lemme 4.4. L EMMA 4.12. There exists {ϕ k }k∈N ⊂ C(D) whose linear hull is dense in H(D), and such that (i) D D(ϕ k ) : D(ϕ j ) = δkj , for all k, j ∈ N; (ii) Given ϕ ∈ C(D) and ε > 0, there exist m = m(ε) ∈ N and β1 , . . . , βm ∈ R such that $ $ m $ $
$ $ βi ϕ i $ $ϕ − $ $ i=1
< ε.
C 1 (D )
L EMMA 4.13. Let % & BR = y ∈ Rn : |y| < R ,
R > 0,
and let Π : B R × S 2 → Rn , τ : B R × S 2 → R3 be continuous maps. Suppose that Π(c, g) · c > 0,
∀(c, g) ∈ ∂BR × S 2 ,
τ (c, g) · g = 0,
∀(c, g) ∈ BR × S 2 .
and
Then, there is (c, g) ∈ BR × S 2 such that Π(c, g) = 0 and τ (c, g) = 0.
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G.P. Galdi
In what follows, we shall frequently use the following identity, that is immediately proved via integration by parts: D
(ϕ − ξ − ω × y) · grad ϕ · ϕ = 0,
for all ϕ ∈ C(D), ξ , ω ∈ R3 .
(4.88)
With these premises in hand, we have. T HEOREM 4.5. For any given B,25 m, and I , and for any Reynolds number Re there is at least one weak solution to the steady free fall problem. Namely, every body B can, in principle, execute at least one steady free fall in a Navier–Stokes liquid. P ROOF. We shall use Galerkin method with the special basis introduced in Lemma 4.12. Step 1. Construction of approximating solution. Set wn =
n
cin ϕ i ,
i=1
where {ϕ i }i∈N is the basis of Lemma 4.12. We wish to determine the coefficients (c1n , . . . , cnn ), and a vector gn ∈ S 2 in such a way that Re
D
(wn − ξ n − ωn × y) · grad wn · ϕ k + ωn × wn · ϕ k
= −Re mΦ 1k · ω n × ξ n + Φ 2k · ωn × (I · ωn ) −2 D(wn ) : D(ϕ k ) + me gn · Φ 1k − |B|Φ 2k × R · gn ,
(4.89)
D
gn × ωn = 0, for all k = 1, . . . , n, where Φ 1k + Φ 2k × y, is the rigid motion associated to ϕ k and ξn =
n
cin Φ 1i ,
ωn =
i=1
n
cin Φ 2i .
i=1
Set c = (c1n , . . . , cnn ), g = gn and consider the maps Π : B R × S 2 → Rn ,
τ : B R × S 2 → R3 ,
where, for k = 1, . . . , n,
Π(c, g)
k
= Re
D
(wn − ξ n − ωn × x) · grad wn · ϕ k + ωn × wn · ϕ k
25 That is, B is any bounded domain in R3 .
On the motion of a rigid body in a viscous liquid
703
+ 2Re mΦ 1k · ωn × ξ n + Φ 2k · ωn × (I · ωn ) +2 D(wn ) : D(ϕ k ) − me g · Φ 1k − |B|Φ 2k × R · g, D
τ (c, g) = g × ωn . Using (4.88) along with Lemma 4.9 and Lemma 4.12(i), we deduce, for all (c, g) ∈ BR × S 2 , τ (c, g) · g = 0, and, for all (c, g) ∈ ∂BR × S 2 , R > me κ/2, D(wn ) : D(wn ) − me g · ξ n 2R 2 − me κR > 0. Π(c, g) · c = 2 D
Existence for (4.89), for all n, then follows from Lemma 4.13. Step 2. Convergence of the approximating solution. Multiplying both sides of (4.89)1 by ckn , summing over k from 1 to n, and using (4.89)2 and (4.88) we deduce D(wn ) : D(wn ) = me g · ξ n , D
and so, by Lemma 4.9, (4.74) we obtain that there exists M > 0 independent of n such that $ $ |ξ n | + |ωn | + wn 6 + $grad(wn )$2 M. 1,2 (D) ∩ L6 (D), with D(w) ∈ L2 (D), vectors ξ , ω ∈ Thus, we can find a field w ∈ Wloc 3 2 R , g ∈ S , and sequences {wn , ξ n , ωn , gn }, such that
wn → w
1,2 weakly in Wloc (D),
strongly in L2loc (D),
ξn → ξ,
ωn → ω,
(4.90)
gn → g.
Letting n → ∞ in (4.89) and using (4.90) we then obtain that w, ξ , ω and g verify the following equation (w − ξ − ω × x) · grad w · ϕ k + ω × w · ϕ k Re D
= −Re mΦ 1k · ω × ξ + Φ 2k · ω × (I · ω) −2 D(w) : D(ϕ k ) + me g · Φ 1k − |B|Φ 2k × R · g, D
g × ω = 0.
(4.91)
704
G.P. Galdi
Taking suitable linear combinations of (4.91)1 and using Lemma 4.12(ii), we show that {w, ξ , ω, g} is a weak solution to (4.68)–(4.71). 4.2.2. Steady free fall: asymptotic spatial behavior The objective of this section is to investigate the asymptotic behavior (in space) of weak solutions to (4.68)–(4.71). The knowledge of the asymptotic structure of steady solutions is of fundamental importance, since it is strictly related to the study of their stability and attainability [104,42]. If ω = 0, the structure of the solution at large distance is well-known, thanks to the work of Finn [28], Babenko [2], and Galdi [31]; see [36], Chapter VIII. If, however, ω = 0, the problem seems to be much more complicated and only partial results are available. Specifically, we can prove [41] that w, p and all their derivatives of arbitrary order tend to zero uniformly pointwise, but their rate of decay is a matter that deserves further investigation. Very probably, w(y) = O(|y|−1),
uniformly as |y| → ∞,
but no proof is available. In this section we shall show that w converges uniformly pointwise to zero at large distances. The proof of this result in the case ω = 0 is straightforward and well-known since the late 50’s [27]. We begin to investigate some properties of the linearized operator associated to (4.68)1, in the case D = R3 . Specifically, we begin to prove the following. L EMMA 4.14. Consider the problem in R3 : ∂u − ω × y · grad u + ω × u = grad τ + f, ∂y1 div u = 0.
u − U
(4.92)
Set µ=
ω , |ω|
% & BR = |y| < R .
The following properties hold. (i) Given f ∈ L2 (R3 ) there exists at least one solution u, p to (4.92) such that 2,2 3 u ∈ Wloc R ∩ D 2,2 R3 ∩ D 1,6 R3 , 1,2 3 R ∩ D 1,2 R3 ∩ L6 R3 . τ ∈ Wloc
(4.93)
The solution satisfies the estimate: u q,BR + |u|1,6 + |u|2,2 + τ 6 + |τ |1,2 c(q, R) f 2 ,
1 q < 2.
(4.94)
On the motion of a rigid body in a viscous liquid
705
Moreover, if u1 , τ1 is another solution to (4.92), corresponding to the same f, with 2,2 3 R ∩ D 1,σ R3 , some σ ∈ [4, 9] u1 ∈ Wloc 1,2 3 R ∩ Lr R3 , some r ∈ [1, ∞], τ1 ∈ Wloc then u1 = u + λµ,and τ1 = τ , for some λ ∈ R. (ii) Given f ∈ L3/2 R3 , there exists at least one solution u, τ to (4.92) such that 2,3/2 3
∩ D 1,3 R3 , 1,3/2 τ ∈ Wloc R3 ∩ D 1,3/2 R3 ∩ L3 R3 . R
u ∈ Wloc
(4.95)
The solution satisfies the estimate: u s,BR + |u|1,3 + τ 3 + |τ |1,3/2 c(s, R) f 3/2 ,
1 s < 3.
(4.96)
Moreover, if u1 , τ1 is another solution to (4.92), corresponding to the same f, with 2,3/2 3
∩ D 1,σ R3 , some σ ∈ [2, 9/2], 1,3/2 τ1 ∈ Wloc R3 ∩ Lr R3 , some r ∈ [1, ∞],
u1 ∈ Wloc
R
then u1 = u + λµ, and τ1 = p, for some λ ∈ R. (iii) Given f ∈ L2 (R3 ) ∩ L3/2 (R3 ), there exists at least one solution u, τ to (4.92) satisfying (4.93) and (4.95). Moreover, this solution verifies the estimates (4.94) and (4.96). Finally, if u1 , τ1 is another solution to (4.92), corresponding to the same f, with 2,2 3 R ∩ D 1,σ R3 , some σ ∈ [2, 9], u1 ∈ Wloc 1,2 3 R ∩ Lr R3 , some r ∈ [1, ∞], τ1 ∈ Wloc then u1 = u + λµ, and τ1 = p, for some λ ∈ R. P ROOF. For f ∈ Ls (R3 ), 1 < s < ∞, we may write f = F + grad p, where div F = 0,
(4.97)
in the sense of distributions, and F s + |p|1,s c f s .
(4.98)
Now, let u be a solution to the problem u − U
∂u − ω × y · grad u + ω × u = F, ∂y1
(4.99)
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G.P. Galdi
satisfying either (4.93)1 or (4.95)1. (Notice that u need not be solenoidal.) Then, since div(−ω × y · grad u + ω × u) = ω × y · grad(div u),
(4.100)
from (4.97) and (4.99) we find that h ≡ div u satisfies h − U
∂h − ω × y · grad h = 0, ∂y1
(4.101)
1,r (R3 ), for suitable r > 1, by elliptic regularity in the sense of distributions. Since h ∈ Wloc ∞ 3 we deduce h ∈ C (R ). Let ψρ (r), r = |y|, be a non-negative, non-increasing smooth “cut-off” function such that 1 if r < ρ, ψρ (r) = 0 if r > 2ρ, (4.102) 2 D ψρ (r) Mρ −2 grad ψρ (r) Mρ −1 ,
with M independent of r and ρ. Since div(ω × y) = 0
(4.103)
and grad φ · (ω × y) = 0,
for all differentiable functions φ = φ |y| ,
(4.104)
we find 1 ψρ h|h|s−2 ω × y · grad h = div ψρ ω × y · grad |h|s , s
s 1.
(4.105)
Thus, multiplying both sides of (4.101) by ψρ h|h|s−2 , s 2, integrating by parts over R3 , and using (4.105) we obtain R3
|h|
s−2
1 | grad h| = s(s − 1) 2
∂ψρ ψρ + |h|s . ∂y1 R3
Letting ρ → ∞ in this relation, using (4.102) and the fact that h belongs to Lq (R3 ), for some q > 1, we conclude h ≡ div u ≡ 0. Therefore, a solution to (4.101) in the class (4.92) or (4.94) is necessarily solenoidal. As a consequence, since f = F + grad p, u will satisfy (4.92) with τ = −p. Moreover for 1 s < 3, by Sobolev’s inequality and (4.98), τ 26 obeys the following estimate τ 3s/(3−s) + |τ |1,s c f s . 26 Possibly modified by the addition of a constant.
(4.106)
On the motion of a rigid body in a viscous liquid
707
In the light of these observations, to prove the lemma it is enough to show existence of a solution u to (4.99), in the classes (4.93)–(4.96), with F satisfying the same assumptions as f. To reach this goal, we assume at first F ∈ C0∞ R3 , and formally take the Fourier transform of both sides of (4.99) to get F, − |ξ |2 + iU ξ1 uˆ − ω × y · grad u + ω × uˆ = #
(4.107)
where 1 aˆ (ξ ) = (2π)3/2
R3
a(y)eiy·ξ dy.
Since ˆ ω × y · grad u = ω × ξ · gradξ u, Equation (4.107) becomes − |ξ |2 + iU ξ1 uˆ − ω × ξ · grad uˆ + ω × uˆ = # F,
(4.108)
where it is understood that “grad” operates on the ξ -variables. We wish now to show the existence of a solution to (4.108), satisfying suitable estimates. This will be obtained by elliptic regularization of (4.108). For ε > 0, consider the following family of problems − |ξ |2 + iU ξ1 w − ω × ξ · grad w + ω × w = # F − ε w.
(4.109)
By using standard procedures (Galerkin method, for instance) we show the existence of a solution to (4.109) with a finite Dirichlet integral: |grad w|2 < ∞, R3
that, in addition, satisfies the following estimate 2 |ξ |2 |w|2 + ε |grad w|2 |ξ |−2 # F . R3
R3
R3
(4.110)
(Notice that the right-hand side of (4.110) is finite in view of Hardy’s inequality.) From (4.110) we get, in particular, that |w| must vanish (in a suitable sense) at large distances. Therefore, by Sobolev’s inequality we find 1/3
ε R3
|w|
6
c0
R3
2 |ξ |−2 # F ,
(4.111)
where c0 is an absolute constant. By standard elliptic regularity theory, it follows that the solution w is of class C ∞ (R3 ). We next multiply both sides of (4.109) by ψρ |ξ |2 w∗ , where
708
G.P. Galdi
ψρ is the “cut-off” function introduced previously, and “∗ ” means complex conjugate. Integrating by parts over R3 the relation so obtained, and adding it to its complex conjugate produces the following result
ψρ |ξ | |w| = 4
2
R3
2
R3
ψρ |ξ |2 (ω × w · w∗ + ω × w∗ · w)
−
R3
−
R3
ψρ |ξ |2 ω × ξ · (grad w · w∗ + grad w∗ · w) ∗ ψρ |ξ |2 # F · w∗ F · w +#
− 2ε
ψρ |ξ | |grad w| + ε 2
R3
2
R3
ψρ |ξ |2 |w|2 .
(4.112)
By the property of the triple scalar product we have that the first integral on the righthand side of (4.112) is zero. Moreover, employing (4.103) and (4.104), and reasoning as in (4.105) we show that also the second integral on the right of (4.112) is zero. Therefore, from (4.112) and the Schwarz inequality we find 2ε
ψρ |ξ |2 |grad w|2 +
R3
R3
2 # F + ε
R3
R3
ψρ |ξ |4 |w|2
ψρ |ξ |2 |w|2 .
(4.113)
We next observe that ψρ |ξ |2 |w|2 R3
=
ψρ |ξ |2 + 4 grad ψρ · ξ |w|2 + 6
Bρ,2ρ
R3
ψρ |w|2 ,
(4.114)
where % & Bρ,2ρ = ρ < |ξ | < 2ρ . Since ψρ |ξ |2 + 4 grad ψρ · ξ c, with c independent of |ξ | and ρ, we have for all ρ 1
Bρ,2ρ
ψρ |ξ |2 + 4 grad ψρ · ξ |w|2 c
|w|2 c Bρ,2ρ
|ξ |2 |w|2 . Bρ,2ρ
On the motion of a rigid body in a viscous liquid
709
Therefore, by (4.110), we deduce lim
ρ→∞ B ρ,2ρ
ψρ |ξ |2 + 4 grad ψρ · ξ |w|2 = 0.
(4.115)
Concerning the second integral on the right-hand side of (4.114), we have, for all η > 0,
1 ψρ |w| ψρ |w| + 4 ψρ |ξ |4 |w|2 η |ξ |η R3 |ξ |η 1/3 2/3 4 1 η2 |w|6 + 4 ψρ |ξ |4 |w|2 . 3 η |ξ |η R3
2
2
In view of (4.111), this last inequality leads to the following one
ε R3
ψρ |w|2 cη2
2 ε |ξ |−2 # F + 4 η R3
R3
ψρ |ξ |4 |w|2 .
(4.116)
Collecting (4.113)–(4.116) we thus obtain for large ρ ε ψρ |ξ | |grad w| + 1 − 4 ψρ |ξ |4 |w|2 2ε η R3 R3 2 2 # |ξ |−2 # F + cη2 F + o(1).
2
2
R3
Choosing η2 =
R3
√ 2ε, and letting ρ → ∞, we finally find
2ε
R3
|ξ |2 |grad w|2 +
R3
√ 2 # F + 2c 2ε
R3
R3
|ξ |4 |w|2 2 |ξ |−2 # F .
(4.117)
We now multiply both sides of (4.109) by ψρ |ξ |4 |w|w∗ and perform the same type of manipulation as before. Thus, instead of (4.113), we get, in particular, the following one
R3
ψρ |ξ |6 |w|3
R3
3 # F + ε
As before, we have 3 4 ψρ |ξ | |w| = R3
Bρ,2ρ
+ 16
R3
ψρ |ξ |4 |w|3 .
(4.118)
ψρ |ξ |4 + 8 grad ψρ · ξ |ξ |2 |w|3
R3
ψρ |ξ |2 |w|3 .
(4.119)
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G.P. Galdi
Since ψρ |ξ |4 + 8 grad ψρ · ξ |ξ |2 c|ξ |2 , for some c independent of ρ and |ξ |, we find that the first integral at the right-hand side of (4.119) I, say, can be estimated as follows
1/2
|ξ |2 |w|3 c
|I| c
1/2
|ξ |4 |w|2
Bρ,2ρ
|w|4
Bρ,2ρ
.
Bρ,2ρ
However, by Sobolev’s embedding theorems and by (4.110) it follows that
1/2 |ξ |>1
|w|4
c
|ξ |>1
c
|w|2 +
|ξ |>1
|ξ |>1
|grad w|2
|ξ |2 |w|2 +
|ξ |>1
|grad w|2 < ∞.
Thus, from (4.117) we conclude lim
ρ→∞ R3
ψρ |ξ |4 + 8 grad ψρ · ξ |ξ |2 |w|3 = 0.
(4.120)
Moreover, with the help of (4.111), for all η > 0 we find
1 ψρ |ξ | |w| |ξ | |w| + 4 ψρ |ξ |6 |w|3 η R3 R3 |ξ |η √ 1/2 2 3 1 η |w|6 + 4 ψρ |ξ |6 |w|3 3 η R3 R3 2 3/2 η3 1 c 3/2 |ξ |−2 # + 4 ψρ |ξ |6 |w|3 . F 3 3 ε η R R 2
3
2
3
(4.121)
From (4.118)–(4.121) we find for large ρ 3 2 3/2 ε η3 # 1− 4 ψρ |ξ |6 |w|3 |ξ |−2 # + o(1). F + c 3/2 F η ε R3 R3 R3 Therefore, choosing again η2 = ρ → ∞ we obtain
|ξ | |w| 2 6
R3
3
R3
√ 2ε, from the previous inequality passing to the limit
3 # F + cε1/4
R3
|ξ |
F
−2 #2
3/2 .
(4.122)
On the motion of a rigid body in a viscous liquid
711
We next multiply both sides of (4.109) by ϕ ∗ , ϕ ∈ C0∞ (R3 ), and integrate by parts to obtain (with w(ε) ≡ w) −
%
R3
=
R3
& |ξ |2 + iU ξ1 w(ε) − ω × w(ε) · ϕ ∗ − ω × ξ · grad ϕ ∗ · w(ε)
# F · ϕ∗ + ε
R3
grad w(ε) : grad ϕ ∗ .
(4.123)
We now let ε → 0 in (4.123). In view of (4.110) we obtain at once lim ε
ε→0
R3
grad w(ε) : grad ϕ ∗ = 0.
(4.124)
Moreover, again from (4.110) we get R3
2 (ε) 2 ξ | w | c,
where c is independent of ε. Therefore, setting L2 R3 = u ∈ L1loc R3 :
R3
|ξ |2 |u|2 < ∞
we find W ∈ L2 R3 such that lim
ε→0 R3
|ξ |2 w(ε) · Φ ∗ =
R3
|ξ |2 W · Φ ∗ ,
for all Φ ∈ L2 R3 .
(4.125)
Let us show that lim
ε→0 R3
w(ε) · ϕ ∗ =
R3
W · ϕ∗,
for all ϕ ∈ C0∞ R3 .
(4.126)
Actually, since
∗
R3
W·ϕ =
R3
|ξ |2 W · ϕ ∗ |ξ |−2 ,
(4.127)
and since, by the Hardy inequality, ϕ ∗ |ξ |−2 ∈ L2 (R3 ), we get that (4.126) follows from (4.125) and (4.127). Likewise, we show lim
ε→0 R3
ξ1 w
(ε)
∗
·ϕ =
R3
ξ1 W · ϕ ∗
for all ϕ ∈ C0∞ R3 .
(4.128)
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G.P. Galdi
Taking the limit ε → 0 in (4.123) and using (4.125), (4.127) and (4.128), we conclude that the field W solves (4.123) in the sense of distributions. From (4.117), (4.122) and (4.110), by means of standard arguments, we show that W obeys the following inequalities: 2 # |ξ |4 |W|2 2 (4.129) F ,
R3
R3
R3
R3
|ξ |6 |W|3 2
R3
|ξ |2 |W|2
3 # F ,
(4.130)
2 F . |ξ |−2 #
R3
(4.131)
Setting 1 u(y) = (2π)3/2
R3
W(ξ )e−iy·ξ dy,
∞ 3 from (4.129), (4.131) and the Schwarz 3 inequality it follows that u ∈ L (R ). Moreover, 2 2 by (4.129), we have that D u ∈ L R and that, by Plancherel theorem,
R3
2 2 D u 4
R3
|F|2 .
(4.132)
Since i = ξi W, ∂u/∂y
(4.133)
and ∂# F/∂ξi = y; i F, from (4.131) and Hardy inequality we have R3
u|2 c |grad
R3
3
2 grad # F = c k,l
R3
2 y; k Fl .
Therefore, by Plancherel theorem, |grad u|2 c |y|2|F|2 . R3
R3
From this latter relation we see that grad u must tends to zero suitably at large distances. Therefore, from (4.132) and the Sobolev inequality we find
1/3 R3
|grad u|
6
c
R3
|F|2.
(4.134)
On the motion of a rigid body in a viscous liquid
713
Again from (4.133) and from (4.130), it follows that R3
u|3 2 |ξ |3 |grad
R3
3 # F .
(4.135)
By Hausdorff–Young inequality and Pitt theorem [105], and [79], p. 251, we have R3
3 # F c
2 |F|
R3
u|3 c |ξ | |grad
3/2
3
R3
R3
|grad u|3 ,
and therefore, by (4.135) we conclude
1/3
R3
|grad u|
3
c
2/3 R3
|F|
3/2
(4.136)
.
Again by Pitt’s theorem, from (4.129) and (4.130) we get the following inequalities R3
|u|2 c |y|4
R3
|u|3 c |y|3
R3
|F|2 ,
2 R3
|F|3/2
,
that furnish, in particular, for all R > 0
1/q |y|
|y|
|u|
q
|u|
r
c(q, R) 1/r
1/2
c(r, R)
R3
R3
|F|
|F|
2
1 q < 2,
,
(4.137)
2/3 3/2
,
1 r < 3.
Setting BR = {|y| < R}, From (4.132), (4.134), (4.136) and (4.137) we deduce that the solution u obeys the following estimates $ $ u q,BR + |u|1,6 + $D 2 u$2 c1 (q, R) F 2 , u r,BR + |u|1,3 c1 (r, R) F 3/2 ,
1 q < 2,
1 r < 3.
(4.138)
These relations hold for F ∈ C0∞ (R3 ). However, assuming F ∈ L2 (R3 ), say, we can find a sequence {Fk }k∈N ⊂ C0∞ (R3 ) approximating F in L2 (R3 ). Then, the corresponding solutions will obey (4.138)1. From this it follows that there is a field u in the class (4.93)1 which satisfies (4.138)1 and it is also a solution to (4.99). Therefore, in view of (4.106),
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G.P. Galdi
the existence proof in the part (i) of the lemma is accomplished. By an entirely analogous reasoning, we prove the existence results of parts (ii) and (iii). Let us now consider the uniqueness of the solutions just obtained. It is reduced to showing that the following problem u − U
∂u − ω × y · grad u + ω × u = grad p, ∂y1
(4.139)
div u = 0 has only the zero solution, in a suitable class. Notice that, by elliptic regularity theory, we may assume that the solution u, p to (4.139) is of class C ∞ (R3 ). Operating with “div” at both sides of (4.139)1 and using (4.139)2 and (4.26), we find that p is harmonic in R3 . Since, by assumption, p can be written as the difference of two functions, each of which belongs to some Lebesgue space, this is enough to ensure p = 0 in R3 . Thus, (4.139)1 furnishes u − U
∂u − ω × y · grad u + ω × u = 0. ∂y1
(4.140)
We take the derivative with respect to yk of both sides of (4.140), which delivers (with ∂k ≡ ∂/∂yk ) (∂k ui ) − U ∂1 (∂k ui ) −*lrk ωr ∂l ui − *lrs ωr xs ∂l ∂k ui + *ims ωm ∂k us = 0,
(4.141)
where *ij k is the alternating symbol. We observe that *lrk ωr ∂l ui ∂k ui = *ims ωm ∂k us ∂k ui = 0.
(4.142)
Moreover, for a 1, ψρ *lrs ωr ys (∂l ∂k ui )∂k ui |grad u|a−2 a = ψρ ω × y · grad |grad u|a 2
a a y a = div ψρ ω × y · |grad u| − ψρ |grad u|a ω × y · 2 2 |y|
a = div ψρ ω × y · |grad u|a , 2 where ψρ is the function defined in (4.102). We also have −[ (∂k ui )]∂k ui |grad u|a−2 ψρ = − div grad(∂k ui )∂k ui |grad u|a−2 ψρ
(4.143)
On the motion of a rigid body in a viscous liquid
715
+ ψρ |grad u|a−2 grad(∂k ui ) · grad(∂k ui ) +
2 (a − 2) ψρ |grad u|t −4 ∂s |grad u|2 4
− |grad u|a−2 grad ψρ · grad(∂k ui )∂k ui div grad(∂k ui )∂k ui |grad u|a−2 ψρ + αψρ |grad u|a−2 grad(∂k ui ) · grad(∂k ui ) − β|grad ψρ |2 |grad u|a ,
(4.144)
where α and β are positive quantities depending on a only. Finally, a ψρ (∂1 ∂k ui )∂k ui |grad u|a−2 = − ∂1 ψρ |grad u|a . 2
(4.145)
Thus, multiplying both sides of (4.141) by ψρ |grad u|a−2 ∂k ui , a 2, and using (4.142)– (4.145) we get R3
2 ψρ |grad u|a−2 grad(grad u)
c
R3
|grad ψρ |2 + |grad ψρ | |grad u|a .
(4.146)
By the assumption of part (i) in the lemma, we may write u = u1 − u2 where u1 ∈ D 1,σ R3 ,
u2 ∈ D 1,6 R3 .
some σ ∈ [4, 9];
Choosing a = 6 in (4.146), we then have, by Hölder inequality, that, if σ ∈ [6, 9], the integral at its right-hand side can be increased as follows:
|grad ψρ |2 + |grad ψρ | |grad u1 |6
Bρ,2ρ
+
|grad ψρ | + |grad ψρ | 2
σ/(σ −6)
(σ −6)/σ
Bρ,2ρ
×
6/σ |grad u|σ
.
(4.147)
Bρ,2ρ
Since σ/(σ − 6) 3 and |grad ψρ (y)| |y|−1 , we deduce that the integrals in (4.147) go to zero in the limit ρ → ∞, that is, the right-hand side of (4.146) tends to zero in this limit. We reach the same conclusion by a similar argument if σ ∈ [4, 6). Thus, from (4.146) we infer grad u ≡ 0, and, consequently, from (4.140) we conclude u = λµ, for some λ ∈ R. The proof of uniqueness part in (i) is completed. By an entirely similar argument we prove
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G.P. Galdi
uniqueness under the assumptions stated in (ii) and (iii). The lemma is therefore completely proved. R EMARK 4.5. The result just proven can be generalized in several directions. For instance, existence, uniqueness and corresponding estimates can be shown when f belongs to D0−1,r (R3 ), for suitable values of r. Analogous results hold for f ∈ W m,2 (R3 ), m 0. 5m 5 k+2,2 (R3 ) × k+1,2 (R3 ). Details In this latter case, the solution u, p is in m k=0 D k=0 D will be given elsewhere [41]. With the help of the above lemma, we are now in a position to show the main result of this section. T HEOREM 4.6. Let {w, ξ , ω, g} be a weak solution to (4.68)–(4.71). Then lim w(y) = 0,
|y|→∞
uniformly pointwise.
P ROOF. Set v = ψρ w,
τ = ψρ p,
with ψρ is the “cut-off” function introduced in (4.102), and p is the pressure associated to w. Take ρ sufficiently large so that v and τ are zero in a neighborhood of ∂D. Without loss, we may assume that U is directed along the y1 -axis. Thus, from (4.68)1,2 and (4.104) we have that v and τ satisfy the following problem in R3 :27 v − U
∂v − ω × y · grad v + ω × v = grad τ + S, ∂y1
(4.148)
div w = Q, where S = ψρ w · grad w − ψρ w − 2 grad ψρ · grad w − wU
∂ψρ − p grad ψρ , ∂y1
Q = w · grad ψρ . We next denote by V a function such that div V = −Q,
V ∈ C0∞ (Dρ,2ρ ).
Since Q ∈ C0∞ (Dρ,2ρ )28 and Dρ,2ρ Q = 0, such a function certainly exists; see [35], Theorem III.3.2. Setting u = v + V, from (4.148) we then conclude that u, τ obey 27 Recall that a weak solution to the free steady fall problem is infinitely differentiable in D. 28 See footnote 27.
On the motion of a rigid body in a viscous liquid
717
Equation (4.92) with f ≡ S − V + U
∂V + ω × y · grad V − ω × V. ∂y1
Since w ∈ H(D) we deduce, on one hand, that u ∈ D 1,2 (R3 ) and, on the other hand, that f ∈ L3/2 (R3 ). Furthermore, since ∂u div u − U − ω × y · grad u + ω × u = 0, ∂y1 we find τ = div f. Therefore, from well-known results on the Poisson equation in the whole space, it follows that τ ∈ D 1,3/2 (R3 ), which in turn, by the Sobolev inequality, furnishes τ ∈ L3 (R3 ), where, possibly, τ has been modified by the addition of a constant. Then, from Lemma 4.14(ii), it follows that u ∈ D 1,3 (R3 ), that is, v ∈ D 1,3 (Dρ ). Since, by Hölder inequality, w · grad w 2 w 6 grad w 3 , we find that f ∈ L3/2 (R3 ) ∩ L2 (R3 ). Using Lemma 4.14(iii), we thus deduce, in particular, u ∈ D 1,6 (R3 ), namely, w ∈ D 1,6 (Dρ ). Therefore, w ∈ W 1,6 (Dρ ), and the theorem is a consequence of the well-known inequality (see, e.g., [35], Theorem II.2.4) w(y) c1 w 1,6,B with c1 independent of y.
1 (y)
,
R EMARK 4.6. By using the more general results stated in Remark 4.5, we can show that any derivative of arbitrary order of v and p tends to zero uniformly pointwise at large distances. Thus, in particular, from (4.68)1 we have that also ω × y · grad w must tend to zero uniformly pointwise. Details will be given elsewhere. 4.2.3. Translational steady free fall of symmetric bodies As discussed in the introduction to Part I, it is experimentally observed that long homogeneous bodies of revolution around an axis a (say), with fore-and-aft symmetry, when dropped in a quiescent Navier–Stokes liquid, will eventually reach a terminal state with no spin and with a perpendicular to g; see Figure 2. In Sections 4.1.1 and 4.1.2, we have proved that this phenomenon is strictly nonlinear, that is, it requires nonzero Reynolds number Re. The objective of the present
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G.P. Galdi
section and of the following three sections, is to investigate this question. In particular, we shall show that if Re is not zero and not “too large” there are only two possible orientations for B, that is, with a either perpendicular or parallel to g. By a “quasi-steady” stability analysis, we shall then show that, in the case when B is a prolate spheroid, the stable orientation will be with a perpendicular to g. When B is falling in L with zero angular velocity, that is, by a purely translational motion, problem (4.68)–(4.71) becomes w = Re(w − ξ ) · grad w + grad p + g,
:
div w = 0, w=ξ
in D,
on Σ,
lim w(y) = 0,
(4.149)
|y|→∞
T(w, p) · n = mg,
Σ
y × T(w, p) · n = 0. Σ
D EFINITION 4.2. A triple {w, ξ , g} is a weak solution to (4.149) if and only if {w, ξ , 0, g} is a weak solution to (4.68)–(4.71). We shall call {w, ξ , g} translational steady free fall, or, more simply, translational steady fall. Notice that, unlike (4.68)–(4.71), the unknowns in problem (4.149) reduce now to w, p, ξ and g, that is, nine scalar quantities, while the number of scalar equations is still ten.
The problem is thus overdetermined. The reason for this is very clear from the physical point of view, because not every B can execute a translational motion, unless we add some other conditions that prevent it from rotating. In the linearized Stokes approximation, the necessary and sufficient condition is furnished by Equation (4.27). In the full nonlinear case, a characterization is not yet available. However, if B is homogeneous and if it has a certain type of symmetry, one can show the existence of purely translational solutions. To this end, we give the definition of symmetric body around an axis. D EFINITION 4.3. B is said to be symmetric around the axis a ≡ y1 (say) of the frame S, if and only if:29,30 (y1 , y2 , y3 ) ∈ Σ →
(y1 , −y2 , y3 ) ∈ Σ, (y1 , y2 , −y3 ) ∈ Σ.
29 In the literature, this type of symmetry is sometimes referred to as rotational symmetry of order 4 around a; see [121]. 30 Notice that if B is homogeneous, its center of mass belongs to the axis of symmetry.
On the motion of a rigid body in a viscous liquid
719
Denote by Pi , i = 1, 2, 3, the operators: P1 f (y1 , y2 , y3 ) = f (−y1 , y2 , y3 ), P2 f (y1 , y2 , y3 ) = f (y1 , −y2 , y3 ),
(4.150)
P3 f (y1 , y2 , y3 ) = f (y1 , y2 , −y3 ). We shall say that a vector field v is in the class C1 if and only if v1 = P2 v1 = P3 v1 ,
v2 = −P2 v2 = P3 v2 ,
v3 = P2 v3 = −P3 v3 .
(4.151)
Likewise, we shall say that a scalar field Φ is in the class C1 if and only if Φ = P2 Φ = P3 Φ.
(4.152)
Our next objective is to show that, if B is a homogeneous, symmetric body, for a given g directed along the y1 -axis, problem (4.149) has at least one weak solution with w and corresponding pressure p in C1 . We shall call this weak solution symmetric translational steady fall. R EMARK 4.7. In the case of symmetric steady fall, and for a sufficiently smooth B, we have T(w, p) · n = ηe1 , η ∈ R, Σ
y × T(w, p) · n = 0. Σ
Therefore, the number of equations in (4.149) reduces to five, and the number of unknowns also reduces to five, if g is given in the direction of a, and ξ is directed along a. Let Cs (D) be the subclass of C(D) of those functions that are in the class C1 , and denote by Hs (D) the completion of Cs (D) in the norm D(·) 2 . Clearly, any ϕ ∈ Cs (D) must be of the form αϕ e1 , for some αϕ ∈ R in a neighborhood of Σ. Since Hs (D) is a subspace of H(D), Lemma 4.11 applies. Therefore, any u ∈ Hs (D) verifies u(y) = αu e1 , y ∈ Σ, for some αu ∈ R. Moreover, let C1⊥ be the class of vector functions v satisfying at least one parity condition opposite to those defining the class C1 , namely, v1 = v3 =
either −P2 v1 , or
−P3 v1 ,
either −P2 v3 , or
P3 v3 .
v2 =
either P2 v2 , or
P3 v2 ,
(4.153)
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G.P. Galdi
Since elements from C1 and C1⊥ have at least one opposite parity, we show by direct inspection that the following relations hold
D
v·u=
D
D(v) : D(u) = 0,
v ∈ C1 , u ∈ C1⊥ .
(4.154)
We have the following. L EMMA 4.15. Let B be symmetric. Then, any ϕ ∈ C0∞ (D) can be decomposed as follows: ϕ = ϕ (1) + ϕ (2) ,
(4.155)
where ϕ (1), ϕ (2) ∈ C0∞ (D) and, moreover, ϕ (1) ∈ C1 and ϕ (2) = 3i=1 ϕ (2,i) , with ϕ (2,i) ∈ C1⊥ , i = 1, 2, 3. So, in particular (4.154) holds with v = ϕ (1) , u = ϕ (2) . P ROOF. We write ϕ1 as follows: 1 1 (1) (2) ϕ1 = (ϕ1 + P2 ϕ1 ) + (ϕ1 − P2 ϕ1 ) ≡ φ1 + φ1 . 2 2 We then further decompose φ1 and φ2 : 1 1 φ1 = (φ1 + P3 φ1 ) + (φ1 − P3 φ1 ) ≡ ϕ1(1) + ϕ1(2,1), 2 2 1 1 (2,2) (2,3) φ2 = (φ2 + P3 φ2 ) + (φ2 − P3 φ2 ) ≡ ϕ1 + ϕ1 . 2 2 Clearly, ϕ1(1) = P2 ϕ1(1) = P3 ϕ1(1) , while ϕ1(2,i) , i = 1, 2, 3, satisfy the parity property of the first component of a vector in the class C1⊥ . In a similar way, we can decompose the other two components of ϕ: (1)
ϕ2 = ϕ2 +
3
(2,i)
ϕ2
,
i=1
ϕ3 = ϕ3(1) +
3
ϕ3(2,i) ,
i=1 (1)
(1)
where ϕ2 , ϕ3 verify the parity property (4.151)2, and (4.151)3, respectively, while ϕ2(2,i) , ϕ3(2,i) , i = 1, 2, 3, verify those in (4.153)2 and (4.153)3, respectively. The lemma is therefore proved.
On the motion of a rigid body in a viscous liquid
721
Since (w − ξ ) · grad w ∈ C1 for w ∈ C1 , ξ = ξ e1 , from Lemma 4.15 and Definition 4.2, we deduce that for {w, ξ , g} to be a symmetric steady fall it is sufficient that the following identity is satisfied 2
D
D(w) : D(ϕ) − Re
D
(w − ξ ) · grad w · ϕ = me αϕ g · e1 ,
(4.156)
for all ϕ ∈ Cs (D), where ϕ = αϕ e1 in a neighborhood of Σ. For any given g directed along y1 , the existence of a symmetric translational steady fall is then obtained by the usual Galerkin method, see, e.g., [48], on the basis of a formal a priori estimate obtained as follows. We replace ϕ with u into (4.156) to get (formally) 2
D
D(w) : D(w) = me ξ · g,
(4.157)
where we used the (formal) condition D
(w − ξ ) · grad w · w =
1 div (w − ξ )|w|2 = 0. 2 D
From (4.157) and from Lemma 4.9(iii) we find 2
D
$ $ D(w) : D(w) me κ $D(w)$2 ,
and so, by (4.74) we conclude me κ grad w 2 √ , 2 which gives the desired a priori estimate. Moreover, by standard arguments, [39], Theorem IX.4.1, it can be shown that the solution constructed in this way obeys the following energy inequality: 2
D
D(w) : D(w) me g · ξ ,
from which it follows that ξ · g > 0. We finally observe that if ξ is directed along the axis of symmetry of B, the corresponding velocity and pressure field of L must belong to the class C1 , provided Re is not “too large”. In fact, if w1 , p1 , ξ , g1 is another solution to (4.149) with w1 having a finite Dirichlet integral, where w1 , p1 are not necessarily assumed in the class C1 , it is well-known that, [36], Theorem IX.5.3, there is a positive constant c depending only on B such that if Re < c we have w1 = w, p1 = p, which in turn, in view of (4.149)5, implies g1 = g. We thus have the following.
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G.P. Galdi
T HEOREM 4.7. Let B be a homogeneous,31 symmetric body around a. Then there exists at least one symmetric translational steady fall {w, ξ , g} with g parallel to a, and ξ · g > 0. Moreover, there exists a positive constant c depending only on B such that, provided Re < c, the following property holds: if {w1 , ξ , g1 } is another translational steady fall corresponding to the same velocity ξ – that is, a priori w1 need not belong to the class C1 – then w1 = w, and g1 = g. A subclass of homogeneous, symmetric bodies are bodies of constant density (homogeneous) that are of revolution around an axis a (say), and that possess fore-and-aft symmetry. As we know, this implies that there exists a plane Π orthogonal to a, Π ≡ {y1 = 0}, say, such that (y1 , y2 , y3 ) ∈ S "⇒ (−y1 , y2 , y3 ) ∈ S. Since such bodies are symmetric with respect to a and to any other axis belonging to Π , by Theorem 4.7 we deduce the following general result. T HEOREM 4.8. Let B be a homogeneous body of revolution around an axis a, possessing fore-and-aft symmetry. Then, there exist two and only two classes of symmetric steady falls, determined by the following directions of the acceleration of gravity g: (a) g is parallel to a; (b) g is orthogonal to a. In both cases, ξ is parallel to g, and ξ · g > 0. Our next objective is to show that, provided the Reynolds number is nonzero and less than a constant depending only on B, symmetric steady falls are the only possible translational steady falls that a homogeneous body of revolution with fore-and-aft symmetry can execute. To reach this goal, we will calculate the relevant component of the torque exerted by L on B. This will be the content of the next section. 4.2.4. On the torque exerted by the liquid on a body of revolution with fore-and-aft symmetry Throughout this section we shall suppose that B is a body of revolution around a, with fore-and-aft symmetry, of class C 2 . We shall also assume that B is moving through L with a given velocity U. We fix, without loss, the axes of the frame S in such a way that a ≡ y1 , and U = (U1 , U2 , 0), with U1 and U2 non-negative. Moreover, we nondimensionalize the relevant equation by using |U|, and the diameter d of B, as velocity and length scale, respectively. Denoting by u the relative velocity of the particles of L in S, we have that u and the corresponding pressure p of L obey the following 31 This assumption can be replaced by the more general one that the center of mass of B belongs to the axis of
symmetry.
On the motion of a rigid body in a viscous liquid
723
problem div T (u, p) = Re u · grad u,
: in D,
div u = 0, u=0
(4.158)
on Σ,
lim u(y) = −U.
|y|→∞
Notice that, by the choice of the velocity scale, we have |U| = 1. D EFINITION 4.4. A triple {u, p, U} belongs to the class F , if it is a solution to (4.158), with u ∈ D 1,2 (D). R EMARK 4.8. As is known, any solution in the class F is infinitely differentiable in D and satisfies the following summability properties [36], Section IX.7, u + U ∈ Lq (D),
for all q > 2,
grad u ∈ L (D),
for all r > 4/3,
r
p ∈ Ls (D),
(4.159)
for all s > 3/2.
Set M ≡ −e3 ·
y × T(u, p) · n.
(4.160)
S
From the physical point of view, M represents the component of the total torque exerted by L on B, in the direction perpendicular to the plane containing U and a, evaluated with respect to the center of mass of B. It is clear that M is responsible for the possible orientations of B. We want to give a two-side estimate for |M|, for small and nonzero Re. We begin to introduce some suitable symmetry classes. To this end, we shall say that a vector field w belongs to the class C2 if and only if (the operators Pi are defined in (4.150)) w1 = −P1 w1 = P3 w1 ,
w2 = P1 w2 = P3 w2 , (4.161)
w3 = P1 w3 = −P3 w3 , while, a scalar field Φ is in the class C2 if and only if Φ = P1 Φ = P3 Φ.
(4.162)
Likewise, w, Φ belong to the class C3 if and only if w1 = P1 w1 = −P2 w1 , w3 = −P1 w3 = −P2 w3 ,
w2 = −P1 w2 = P2 w2 , (4.163)
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G.P. Galdi
Φ = −P1 Φ = −P2 Φ.
(4.164)
Using the symmetry property of B along with (4.151), (4.161), and (4.163) we immediately deduce that w(j ) · w(3) = 0, w(j ) ∈ Cj , j = 1, 2, w(3) ∈ C3 . (4.165) D
L EMMA 4.16. Let H, P be the solution to Stokes problem (4.16) corresponding i = 3. Then, H, P ∈ C3 . Furthermore, for all s > 3/2 and all r > 1, there is a positive constant c = c(B, s, r), such that H s + |H|1,r c. P ROOF. Consider the following fields: 1 = 1 (H1 + P1 H1 − P2 H1 − P1 P2 H1 ), H 4 2 = 1 (H2 − P1 H2 + P2 H2 − P1 P2 H2 ), H 4 3 = 1 (H3 − P1 H2 − P2 H2 − P1 P2 H2 ), H 4 1 P = (P − P1 P − P2 P − P1 P2 P ). 4 P belong to the class C3 . Moreover, since B is symmetric around the y3 The fields H, P is a solution to (4.16) and axis, in the sense of Definition 4.3, we also have that H, so, by uniqueness, it has to coincide with H, P . Concerning the second part, as is wellknown [35], Theorem V.3.2, H(y) admits the following asymptotic representation for all sufficiently large |y|: T(H, P ) · n + H1 (y), (4.166) H(y) = U (y) · Σ
where U (y) is the Stokes fundamental tensor solution, and (j = 1, 2, 3) H1j (y) = (e3 × z)i Til (uj , qj )(y − z)nl (z) dΣz Σ
Uij (y − z) − Uij (y) Til (H, P )(z)nl (z) dΣz .
− Σ
In (4.167), we have set uj = (U1j , U2j , U3j ),
(4.167)
On the motion of a rigid body in a viscous liquid
725
and we have denoted by q the “pressure” associated to U . Using the asymptotic properties of the Stokes fundamental solution (U , q) [35], §IV.2, along with the trace theorem, from (4.167) we find that H1 (y) can be pointwise increased by a constant times |y|−2 times H 2,2,N + P 1,2,N , where N is an arbitrary subdomain of D with ∂N ⊃ Σ. Likewise, grad H1 (y) can be pointwise increased by a constant times |y|−3 times H 2,2,N + P 1,2,N . Since (H, P ) ∈ C3 , it is easy to show that (see also [57], §§ 5-5, 5-7) T (H, P ) · n = 0,
(4.168)
Σ
and from (4.166) we then get H(y) = H1 (y). From local estimates for the Stokes problem [35], Theorems IV.4.1, IV.5.1, the quantity H 2,2,N + P 1,2,N can be bounded in terms of |H|1,2 + P 2 which, in turn, is bounded by a constant depending only on D, [35], Theorem V.2.1. The lemma is proved. Using the same type of argument employed in the first part of the proof of Lemma 4.16 we can show the following. L EMMA 4.17. Let h(i) , p(i) , i = 1, 2, be the solutions to (4.15). Then, h(i) , p(i) ∈ Ci . Furthermore, for all q > 3/2 there is a positive constant c = c(B, q) such that $ (i) $ $h $ + h(i) c, ∞ 1,q
i = 1, 2.
Finally, we have the following result. L EMMA 4.18. Let w(i) be sufficiently smooth vector fields belonging to the class Ci , i = 1, 2, 3. Then, w(j ) · grad w(j ) · w(3) = 0, j = 1, 2. D
P ROOF. Since w(j ) · grad w(j ) ∈ Cj , j = 1, 2, the lemma follows from (4.165).
We next introduce two suitable splittings of the solution u, p to (4.158). The first one: u = u1 + u2 ,
p = p1 + p2 ,
(4.169)
where div T(u1 , p1 ) = Re u1 · grad u1 , div u1 = 0, u1 = 0
on Σ,
lim u1 (y) = −U1 e1
|y|→∞
: in D, (4.170)
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G.P. Galdi
and div T(u2 , p2 ) = Re(u · grad u2 + u2 · grad u1 )
: in D,
div u2 = 0,
(4.171)
u2 = 0 on Σ, lim u2 (y) = −U2 e2 ,
|y|→∞
and the second one: u = u1 + u2 ,
p = p1 + p2 ,
where div T(u2 , p2 ) = Re u2 · grad u2 , div u2 = 0,
: in D, (4.172)
u2 = 0 on Σ, lim u2 (y) = −U2 e2
|y|→∞
and div T(u1 , p1 ) = Re(u · grad u1 + u1 · grad u2 ), div u1 = 0,
: in D,
u1 = 0 on Σ,
(4.173)
lim u1 (y) = −U1 e1 .
|y|→∞
Notice that, since u1 , p ∈ C1 "⇒ div T(u1 , p1 ) − Reu1 · grad u1 , div u1 ∈ C1 , by known methods, for any Re > 0 and any given U1 , we may construct a solution u1 , p1 to (4.170) such that u1 , p1 ∈ C1 .
(4.174)
Likewise, we may construct a solution u2 , p2 to (4.170) such that u2 , p2 ∈ C2 . We need the following preparatory result on the torque M given in (4.160).
(4.175)
On the motion of a rigid body in a viscous liquid
727
L EMMA 4.19. Let {u, p, U} be in the class F . Then, the following representation holds M = −Re u · grad u · H. (4.176) D
Moreover, the following equivalent alternative representations for M hold: M = −Re (u1 · grad u2 + u2 · grad u1 + u2 · grad u2 ) · H, D
M = −Re
D
(u1 · grad u2 + u2 · grad u1 + u1 · grad u1 ) · H,
(4.177)
(4.178)
where H is given in Lemma 4.16. P ROOF. Taking the scalar product of both sides of (4.158)1 by H, integrating by parts over D, and using the asymptotic properties of u, p (Equation (4.159)) and of H (Lemma 4.16), we obtain y × T(u, p) · n = 2 D(u) : D(H) + R u · grad u · H. (4.179) e3 · D
Σ
D
Likewise, taking the scalar product of both sides of (4.15)1 (with i = 3) by u + U and integrating by parts over D, we find T(H, P ) · n = 2 D(u) : D(H). (4.180) U· D
Σ
Combining (4.160), (4.168), (4.179) and (4.180), we deduce (4.176). The proof of (4.177) then follows from (4.176), (4.169), Lemma 4.16, (4.174), and Lemma 4.18. The proof of (4.178) is completely analogous and, therefore, it will be omitted. Finally, we recall two results, whose proof can be found in [48], Lemmas 2.1 and 2.2, respectively. L EMMA 4.20. Let u0 , b0 ∈ R3 , and let F be a second-order tensor field such that div F ∈ Lq (D),
1 < q < 3/2.
Then the problem u + u0 · grad u = div F + grad p, div u = 0, u = b0
on Σ,
lim u(y) = 0
|y|→∞
: in D, (4.181)
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G.P. Galdi
admits one and only one solution u, p such that u ∈ L2q/(2−q)(D) ∩ D 1,4q/(4−q) (D) ∩ D 1,3q/(3−q)(D) ∩ D 2,q (D), p ∈ L3q/(3−q) (D) ∩ D 1,q (D). This solution satisfies the following estimate u0 · grad u q + |u0 |1/2 u 2q/(2−q) +|u0|1/4 |u|1,4q/(4−q) + |u|1,3q/(3−q) + u 3q/(3−2q) + |u|2,q + p 3q/(3−q) + |p|1,q c1 div F q + |b0 | , where the positive constant c1 depends on q and u0 . However, if |u0 | B, for some positive B, then c1 depends only q and B. L EMMA 4.21. Let v, p be a solution to the following problem v = Re v · grad v + grad p,
:
div v = 0,
in D,
v = 0 on Σ, lim v(y) = ξ ,
|y|→∞
with v ∈ D 1,2 (D), and let 1 < q < 3/2. There exists c0 = c0 (B, q) > 0 such that if Re |ξ | c0 ,
(4.182)
then v, p satisfies the following properties (v − ξ ) ∈ L2q/(2−q)(D) ∩ D 1,4q/(4−q)(D) ∩ D 1,σ (D) ∩ D 2,q (D), p ∈ L3q/(3−q) (D) ∩ D 1,q (D),
(4.183)
with σ ∈ (3/2, ∞). Moreover, (v − ξ )(1 + |y|) ∞ < ∞ and the following estimate holds $ $ $(v − ξ ) 1 + |y| $ + |v|1,σ + Re ξ · grad v q + Re|ξ | 1/2 v − ξ 2q/(2−q) ∞ 1/4 + Re|ξ | |v|1,4q/(4−q) + |v|2,q + p 3q/(3−q) + |p|1,q c|ξ |, (4.184) with c = c(B, q, σ ). We are now in a position to prove the main result of this section. Set GI ≡ −
D
h(1) − e1 · grad h(2) + h(2) − e2 · grad h(1) · H.
(4.185)
On the motion of a rigid body in a viscous liquid
729
Clearly, GI depends only on the geometric properties of B, such as size or shape, but it is otherwise independent of the orientation of B and of the properties of L. We shall call GI the inertial torque coefficient. T HEOREM 4.9. Let {u, p, U} be in the class F , and suppose GI = 0.
(4.186)
Then, there exists a positive number c0 , depending only on B, such that for 0 < Re < c0 , we have32 1 3 Re U1 U2 |GI | |M| Re U1 U2 |GI |. 2 2
(4.187)
P ROOF. Set v1 = U1 h(1) − e1 , V1 = u1 − v1 ,
v2 = U2 h(2) − e2 ,
(4.188)
V2 = u2 − v2 .
Using (4.188) in (4.177), taking into account (4.165) and the symmetry properties of the fields u1 , h(1) , and h(2) , after an integration by parts we obtain 1 M = U1 U2 GI − v2 · grad H · V2 − V2 · grad H · v2 Re D D − v2 · grad H · V1 − −
D D D
V1 · grad H · v2 −
D
v1 · grad H · V2 −
V2 · grad H · V1 −
≡ U1 U2 GI +
9
D
D
V2 · grad H · v1
V1 · grad H · V2 −
Ii .
D
V2 · grad H · V2 (4.189)
i=1
Since grad H ∈ Lr (D) for any r > 1 (Lemma 4.16) and, moreover, vi ∞ ci Ui , i = 1, 2 (Lemma 4.17), by the Hölder inequality we find 9 Ii c U2 V1 s1 + V2 s2 i=1
+ U1 V2 s2 + V1 s1 V2 s2 + V2 2s2 ,
(4.190)
32 The numbers 1/2 and 3/2, in (4.187) can be replaced by 1 − ε and 1 + ε, respectively, 0 < ε < 1, in which case the constant c0 in the statement of the theorem will depend also on ε.
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G.P. Galdi
where s1 , s2 are arbitrary numbers in (1, ∞), and c is a constant depending on B, s1 , and s2 . Our next task is to give suitable estimates for V1 and V2 . To this end, we begin to observe that V1 obeys the following boundary-value problem : V1 = Re u1 · grad u1 + grad Φ, in D, div V1 = 0, (4.191) V1 = 0 on Σ, lim V1 (y) = 0.
|y|→∞
Since u1 · grad u1 = (u1 + U1 e1 ) · grad u1 − U1 e1 · grad u1 = div (u1 + U1 e1 ) ⊗ u1 , from Lemma 4.20 we find V1 3q/(3−2q) c3 Re u1 + U1 e1 2q/(2−q) |u1 |1,2 + U1 |u1 |1,q ,
1 < q < 3/2.
(4.192)
Choosing q > 4/3, we have 2q/(2 − q) ∈ (4, 6) therefore, applying Lemma 4.21 (Equation (4.184)) to u1 we obtain u1 + U1 e1 2q/(2−q) |u1 |1,2 c4 U12 .
(4.193)
Moreover, for any q > 4/3 we can always find q1 sufficiently close to 1 such that 4q1/(4 − q1 ) < q. Therefore, with this choice of q1 , by the convexity inequality, and Lemma 4.21, we get −θ/4 U1 |u1 |q |u1 |θ1,4q1/(4−q1 ) |u1 |1−θ 1,2 c5 Re
1−θ/4
,
0 < θ < 1.
(4.194)
Collecting (4.191)–(4.194), and recalling that U1 1, we find V1 s1 c6 Reγ1 U1 ,
for some s1 > 1, γ1 > 0.
(4.195)
We wish to find an analogous estimate for V2 . To this end, we shall first establish an estimate for u2 . Setting w2 = u2 + U2 e2 , from (4.169) and (4.171) we obtain ⎫ w2 + Re U · grad w2 = Re w2 · grad u ⎪ ⎪ ⎪ ⎪ ⎬ + (u1 + U1 e1 ) · grad w2 in D, − U2 e2 · grad u1 + grad p2 ,⎪ ⎪ ⎪ ⎪ ⎭ (4.196) div w2 = 0, w2 = −U2 e2
on Σ,
lim w2 (y) = 0.
|y|→∞
On the motion of a rigid body in a viscous liquid
731
Applying Lemma 4.20 to (4.196) we find for all s ∈ (1, 3/2) Re U · grad u2 s + Re1/2 u2 + U2 e2 2s/(2−s) + Re1/4 |u2 |1,4s/(4−s) + |u2 |1,3s/(3−s) % c7 Re U2 |u1 |1,s + u2 + U2 e2 2s/(2−s)|u|1,2 & + u1 + U1 e1 4 |u2 |1,4s/(4−s) + U2 .
(4.197)
From Lemma 4.21 we get for s > 4/3 3/4
Re1/4 |u1 |1,s c8 U1 ,
u1 + U1 e1 4 c9 U1 ,
|u|1,2 c10 ,
(4.198)
and so, using (4.198) in (4.197) and choosing Re less then a suitable constant depending only on B, we deduce, for all s ∈ (4/3, 3/2), Re U · grad u2 s + Re1/2 u2 + U2 e2 2s/(2−s) +Re1/4 |u2 |1,4s/(4−s) + |u2 |1,3s/(3−s) c11 U2 .
(4.199)
We are now in a position to furnish an estimate for V2 . From (4.171) we find that V2 satisfies the following Stokes-like problem ⎫ % V2 = Re (u + U) · grad u2 − U · grad u2 ⎪ ⎪ ⎪ ⎪ ⎬ + (u2 + U2 e2 ) · grad u1 & − U2 e2 · grad u1 + grad Φ, ⎪ ⎪ ⎪ ⎪ ⎭ div V2 = 0, V2 = 0
in D, (4.200)
on Σ,
lim V2 (y) = 0.
|y|→∞
From Lemma 4.20 and the Hölder inequality we find for q ∈ (1, 3/2) and r > q V2 3q/(3−2q) c12 Re u + U rq/(r−q)|u2 |1,r + U · grad u2 q
+ u + U2 e2 2q/(2−q)|u1 |1,2 + U2 |u1 |1,q .
(4.201)
We choose r ∈ (4/3, 2q/(2 − q)) and restrict the range of q to the interval (4/3, 3/2), so that rq/(r − q) > 2,
2q/(2 − q) > 2.
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G.P. Galdi
Then, from (4.199), and from (4.184) applied to u and to u1 , respectively, we deduce |u2 |1,r c13 U2 Re−1/4 ,
u2 + U2 e2 2q/(2−q) c14 U2 Re−1/2 ,
u + U rq/(r−q) c15 Re−1/2 ,
|u1 |1,q c16 Re−1/4 ,
(4.202)
|u1 |1,2 c17 . Pick q1 ∈ (4/3, q) and q2 ∈ (2, 12/5). Recalling that |U| = 1, from (4.199), we thus find U · grad u2 q1 c18 Re−1 U2 ,
U · grad u2 q2 |u2 |1,q2 c19 Re−1/4 U2 .
Using these latter relations in the following convexity inequality U · grad u2 q U · grad u2 αq1 U · grad u2 1−α q2 ,
0 < α < 1,
we conclude U · grad u2 q c20 U2 Re−
1+3α 4
.
(4.203)
Substituting (4.202) and (4.203) into (4.201) we obtain V2 s2 c21 Reγ2 U2 ,
for some s2 > 1, γ2 > 0.
(4.204)
Employing (4.195), (4.204) in (4.190), and observing that & % max Reγ1 , Reγ2 = Reγ , where γ=
max{γ1 , γ2 }
if Re > 1,
min{γ1 , γ2 }
if Re < 1,
it follows that 9 Ii c22 Reγ U1 U2 + U22 .
(4.205)
i=1
Coupling this latter inequality with (4.189) we have |M/Re − U1 U2 GI | c23 Reγ U1 U2 + U22 .
(4.206)
However, using (4.172), (4.173), (4.178), in place of (4.170), (4.171), (4.177), by a completely analogous reasoning we show that |M/Re − U1 U2 GI | c24 Reγ U1 U2 + U12 .
(4.207)
On the motion of a rigid body in a viscous liquid
733
Now, if U2 U1 , from (4.206) we have |M/Re − U1 U2 GI | c25 Reγ U1 U2 ,
(4.208)
and, if U1 U2 , from (4.207) we find again (4.208). Thus, (4.208) holds for arbitrary U1 , U2 . The proof of (4.187) then follows by choosing in (4.208) c25 Reγ 12 |GI |. From the representation (4.189) and the estimate (4.205) we also have the following. T HEOREM 4.10. Let {u, p, U} be in the class F , and suppose GI = 0. Then, there exist positive numbers c0 , C, and γ , depending only on B, such that for all 0 < Re < c0 , we have M = Re U1 U2 GI + M0 , where |M0 | C Re1+γ . 4.2.5. Orientation of homogeneous bodies of revolution with fore-and-aft symmetry Let B be a homogeneous body of revolution with fore-and-aft symmetry, and let {w, ξ , g} be a corresponding translational steady fall. Setting u = w − ξ,
p = p + g · y,
and using (4.18),33 we have that u, p satisfies (4.158) with U = ξ , that {u, p, ξ } is in the class F , and that, moreover y × T(u, p) · n = 0. Σ
Thus, in particular, M is zero (see (4.160)), and by Theorem 4.9 we deduce that, at small nonzero Re, either ξ1 or ξ2 must vanish. The translational velocity must be then directed either parallel or orthogonal to a. Therefore, from the uniqueness part of Theorem 4.7 there exists c1 > 0, such that if Re < c1 , the translational steady fall must be symmetric. We have thus proved the following. T HEOREM 4.11. Let B be a body of revolution around a of class C 2 , possessing foreand-aft symmetry and satisfying condition (4.186). Then, there is c∗ > 0, depending only on the geometric properties of B, such that for all 0 < Re < c∗ the only possible translational steady falls are symmetric. This implies, in particular, that a is either parallel or perpendicular to g. In either case, the translational velocity ξ is parallel to g with ξ · g > 0. 33 Recall that, for a homogeneous B, it is R = 0.
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G.P. Galdi
Fig. 8. Definition of the angle θ .
R EMARK 4.9. The result in Theorem 4.11 should be contrasted with that obtained in the case of zero Reynolds number in Theorem 4.2. In this latter case, in fact, every orientation is allowed in a steady fall of a homogeneous body of revolution with foreand-aft symmetry, while at nonzero and small Reynolds number only two orientations are allowed, provided B satisfies (4.186). To study the stability of these two possible orientations, we take a coinciding with the y1 -axis, and denote by θ the angle formed by the velocity ξ with a, clockwise oriented. See Figure 8. Thus, ξ1 = |ξ | cos θ,
ξ2 = −|ξ | sin θ.
From Theorem 4.10 we know that the relevant torque acting on B, at first order in Re, will be M ≡ M(θ ) = Re GI ξ1 ξ2 = −Re |ξ |2 GI sin θ cos θ. Thus, if we limit ourselves to perturbations in the form of infinitesimal disorientations of a with respect to g of the type δθ e3 , for a configuration to be stable [respectively, unstable] the variation of M(θ ) from its value at the equilibrium configuration, should have a sign opposite to δθ [respectively, the same sign]. Therefore, denoting by θ0 the equilibrium configuration (that is, θ0 is either 0 or π/2), we have dM < 0 "⇒ stability, dθ θ=θ0 dM > 0 "⇒ instability. dθ θ=θ0 Consequently, we conclude, stable θ =0 unstable stable π θ= 2 unstable
if GI > 0, if GI < 0, if GI < 0, if GI > 0.
On the motion of a rigid body in a viscous liquid
735
Table 1 Tabulations of computed inertial torque coefficient GI versus eccentricity e in the case of a prolate spheroid e
GI
0 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.98
0.00 −0.005 −0.02 −0.08 −0.19 −0.33 −0.50 −0.67 −0.83 −0.94 −0.93 −0.81 −0.64
Fig. 9. Absolute value of the inertial torque coefficient GI versus eccentricity e, in the case of a prolate spheroid.
R EMARK 4.10. The stability results just obtained, though in agreement with the experimental data (see next section), are not in the classical sense of Liapounov stability. As mentioned at the beginning of Section 4.2, this latter would require a careful analysis of the asymptotic properties of the solution to the equation LT R (u, p) = 0 (see (4.67)), but such an analysis is not yet available.
4.2.6. Orientation of homogeneous prolate spheroids We shall now specialize the results obtained in the previous section to the situation when B is a homogeneous prolate spheroid of eccentricity e. In this case, the fields h(1) , h(2) , and H ≡ H(3) are well-known [18,57], and therefore, we can explicitly evaluate the inertial torque coefficient GI given in (4.185).
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G.P. Galdi
This was done in [48], and here we summarize the results. If B is a prolate spheroid, it is found that GI is always negative for 0 < e < 1. Furthermore, GI is shown to be zero at e = 0 and 1. Table 1 summarizes the findings. Figure 9 shows a comparative plot of the absolute value of the torque coefficient. It is zero for B a sphere (e = 0) and increases monotonically for increasing e until approximately e = 0.8. It then turns around and falls very rapidly to zero for B a needle (e = 1). Therefore, the prolate spheroid must orient itself with its major axis a either parallel or orthogonal to g. However, since GI < 0 for e ∈ (0, 1), from the result of Section 4.2.5 we find that in this range of eccentricities the orientation with a orthogonal to g is stable, while the other is unstable, in accordance with experiment; see also Figure 2(A). 4.2.7. Unsteady free fall In this section we shall briefly discuss the problem of free fall in the unsteady case. In this case, one can only prove existence of weak solutions a la Leray– Hopf. Whether strong solutions exist, even for small times, is a question that deserves appropriate further investigation. In order to state results for weak solutions, we introduce some notation. Let T > 0 and let DT ≡ D × [0, T ). The space of test functions C(DT ), is then defined as the class of vector functions ϕ such that (i) ϕ ∈ C ∞ (D T ), (ii) div ϕ(y, t) = 0, for all t ∈ [0, T ), ¯ t) = −(ϕ 1 (t) + (iii) There exist ϕ 1 , ϕ 2 ∈ C0∞ ([0, T )) such that ϕ(y, t) = ϕ(y, ϕ 2 (t) × y), in a neighborhood of Σ and for all t ∈ [0, T ). Multiplying formally (4.1)1 by ϕ ∈ C(DT ), integrating by parts over DT , and taking into account (4.5)1 and (4.18), we find −
T 0
∂ϕ ·w= D ∂t
T 0
0
Σ
+ Re
T
D
0
−2
T
ϕ¯ · T(w, p) · n −
T
D
0
|B|G · ϕ 1 + |B|ϕ 2 × R · G
(w − U) · grad ϕ · w − ϕ · ω × w
D(ϕ) : D(w).
Imposing conditions (4.2), (4.3), we then get −
T 0
∂ϕ ·w= D ∂t
T
m 0
dϕ dϕ 1 · ξ + 2 · I · ω − mω × ξ · ϕ 1 dt dt
− ω × (I · ω) · ϕ 2 + me G · ϕ 1 + |B|ϕ2 × R · G + Re
T 0
−2
T 0
D
D
(w − U) · grad ϕ · w − ϕ · ω × w
D(ϕ) : D(w).
(4.209)
On the motion of a rigid body in a viscous liquid
737
Moreover, we can give the following weak form to (4.4)
0
for all ψ
· G + Re ω × G · ψ = 0,
T dψ
ψ(0) · g +
dt [0, T ) .
∈ C0∞
(4.210)
Following [102], we shall say that the quadruple {w, ξ , ω, G} is a weak solution to (4.1)– (4.5) if and only if, for all T > 0, the following conditions hold: (i) w ∈ L2 (0, T ; H(D)) (see (4.78)); (ii) w = U in Σ × (0, T ) (in the sense of Lemma 4.10), where U ≡ ξ (t) + ω(t) × y, t ∈ [0, T ]; (iii) {w, ξ , ω, G} verifies (4.209), for all ϕ ∈ C(DT ), and (4.210). The following theorem can be proved by standard methods. T HEOREM 4.12. Let B be an arbitrary domain. Then, there exists at least one weak solution to (4.1)–(4.5). We end this section by recalling, one more time, the two fundamental open problems: the existence of strong solutions (even for small times), and the asymptotic behavior in time of solutions (even for small Re). The resolutions of both problems can be obtained by careful estimates of the linearized operator (4.67).
5. Free fall in a second-order liquid We shall now consider the problem of free fall in the case of a second-order liquid. As in the Navier–Stokes case, we shall be focused only on the steady case, the unsteady case being a completely open question. From Section 2.2, we find that the equations of a steady free fall can be written in the following nondimensional form Re(u · grad u + 2ω × u + ω × U) = u − grad p − We div S(u) + g, div u = 0, lim u(y) + U(y) = 0,
⎫ ⎪ ⎬ ⎪ ⎭
in D, (5.1)
|y|→∞
u(y) = 0,
y ∈ Σ, T(u, p) · n,
Re mω × ξ = mg −
(5.2)
Σ
y × T(u, p) · n,
Re ω × (I · ω) = − Σ
(5.3)
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G.P. Galdi
g × ω = 0,
(5.4)
where u ≡ w − U is the relative velocity, We =
−α1 W dµ
(Weissenberg number)
and S(u) = u · grad A1 + A1 · LT + L · A1 + εA1 · A1 ,
(5.5)
where A1 is defined in (2.4)1, and ε = α2 /α1 . It is worth emphasizing that, even in the steady case, the situation is much less clear than the Navier–Stokes counterpart, and existence of free falls for an arbitrary body B is only known at zero Reynolds number, and under the assumption that the Weissenberg number is sufficiently small and that ε = −1. Nevertheless, if B is a homogeneous body of revolution with fore-and-aft symmetry, we are still able to characterize the translational steady falls, at least at small Reynolds and Weissenberg numbers, and to provide results that give a rigorous interpretation of certain typical sedimentation experiments performed in polymeric liquids. 5.1. Steady free fall at zero Reynolds number In most sedimentation experiments in polymeric liquid, the Reynolds number is much less than the Weissenberg number. For instance, in the experiment of Liu and Joseph [86], for plastic cylinders of diameter 0.1 in and length 0.4 in sedimenting in a 2% aqueous polyacrylamide solution, we have Re = 0.016, We = 0.048. In this circumstances, the elastic properties of the liquid dominate over its inertia, and, therefore, one may assume, in a first analysis, Re = 0. Following Giesekus [50], we shall also assume ε = −1. Typically, experiments and theoretical considerations suggest ε in the range −1.6 ∼ −2 [86,67], §17.11. The influence of the value of ε on sedimentation will be considered in Section 5.2. Under the assumptions Re = 0, ε = −1, one is able to prove existence of steady fall for bodies of arbitrary shape and density distribution, provided We is not too large [116]. If ε = −1, by a direct calculation one shows that 1 2 div S(u) = (curl u) × u + grad u · u + |A1 | . 4 As a consequence, (5.1)–(5.4) with Re = 0 and ε = −1 become : u = grad P + We (curl u) × u, in D, div u = 0, u=0
at Σ,
lim u(y) + U(y) = 0,
|y|→∞
(5.6)
On the motion of a rigid body in a viscous liquid
739
T(u, p) · n = mg,
(5.7)
y × T(u, p) · n = 0,
(5.8)
Σ
Σ
g × ω = 0,
(5.9)
where P = p − g · y + We u · u +
1 |A1 |2 . 4
(5.10)
Using the fact that if v = grad τ then (curl v) = 0, it is immediately seen that, for any ¯ p¯ ) ξ = ξi ei , ω = ωi ei , a solution to (5.6), (5.10) is given by the following pair (u, u¯ = ξi h(i) − ξ + ωi H(i) − ω × y , 2 1 (i) (i) ¯ , p¯ = ξi p + ωi P + g · y − We u¯ · u¯ + A1 (u) 4
(5.11)
where (h(i) , p(i) ), (H(i) , P (i) ), i = 1, 2, 3, are the auxiliary fields defined in (4.15), (4.16), respectively. In fact, if the Weissenberger number is sufficiently small, in a suitable sense, the only solutions to (5.6) possessing an appropriate asymptotic behavior are of the form (5.11). To this end, denote by A the class of pairs (u, p) such that (i) Regularity: 2,3 u ∈ C 2 D ∩ Wloc (D),
p ∈ C 0 D ∩ Wloc (D), 1,2
where D is any bounded subset of D. (ii) Asymptotic behavior: D β u(y) + U(y) = O |y|−1−|β| , p − g · y = p0 + O |y|−2 ,
0 |β| 3,
as |y| → ∞,
where p0 ∈ R. ¯ p¯ ) certainly satisfies condition (ii). It also satisfies (iii) if B is sufficiently Notice that (u, smooth, of class C 3 (say). Therefore, the class A is not empty if B is regular enough. The following result holds. L EMMA 5.22. Let B be of class C 3 . There is a positive constant c = c(B) such that, if |We| |ξ | + |ω| < c, ¯ p¯ ) is the only solution in the class A. (u,
(5.12)
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G.P. Galdi
¯ Φ = P − P, where P is P ROOF. Let (u, p) be another solution in A, and set v = u − u, ¯ From (5.6) we find given by (5.10) with p = p¯ , and u = u. :
¯ v = grad Φ + We[ ζ × v + ζ × u],
in D,
div v = 0,
(5.13)
v = 0 at Σ, lim v(y) = 0,
|x|→∞
where ζ = curl v. Multiplying (5.13)1 by v, and integrating by parts, we find
D
|grad v|2 = We
D
ζ × u¯ · v.
(5.14)
Integrating several times by parts and using the fact that v ≡ u¯ ≡ 0 at Σ, we find (with ∂/∂k = ∂k )
D
ζ × u¯ · v =
D
∂l vi ∂l (∂i vj − ∂j vi )u¯ j +
D
vi ∂l (∂i vj − ∂j vi )∂l u¯ j
1 2 ¯ u · grad (grad v) − = ∂l vi ∂l vj ∂i u¯ j 2 D D − ∂l vi ∂l (∂i vj − ∂j vi )u¯ j + − +
D D D D
vi vj ∂i u¯ j +
D
u¯ · grad |v|2
∂l vi ∂l vj ∂i u¯ j −
D
∂l vi ∂l (∂i vj − ∂j vi )u¯ j
vi vj ∂i u¯ j .
Therefore, from Hölder and Sobolev inequalities (see (4.75)) we obtain ζ × u¯ · v c1 grad u ¯ ∞ grad v 22 + |u| ¯ 3,3/2 v 26 D ¯ ∞ + |u| ¯ 3,3/2 grad v 22 . c2 grad u
(5.15)
From well-known results on the Stokes problem [35], Chapter V, we have ¯ ∞ + |u| ¯ 3,3/2 c3 |ξ | + |ω| . grad u The result then follows from this latter inequality, from (5.15) and (5.14).
On the motion of a rigid body in a viscous liquid
741
By means of (5.11) we can give an explicit formula for the total force and torque acting on B. Actually, recalling (4.21) and (4.18), we obtain ¯ p¯ ) · n T(u,
− Σ
= −K · ξ − C · ω − We Σ
2 1 ¯ n − S(u) ¯ · n − |B|g, A1 (u) 4
(5.16)
¯ p¯ ) · n y × T(u,
− Σ
= −D · ξ − Θ · ω 1 2 ¯ y × n − y × S(u) ¯ · n + |B|g × R. |A1 (u)| − We Σ 4
(5.17)
We can rewrite the integrals on the right-hand side of these equations in terms of the vorticity ζ¯ ≡ curl u¯ only. Actually, since u¯ vanishes at Σ we have, on one hand (see, e.g., [38], Lemma 2.1), ¯ · n = τ · A1 (u) ¯ ·τ =0 n · A1 (u)
at Σ,
with τ any unit vector tangent at Σ, and, on the other hand [5], ¯ · n = ζ¯ × n, A1 (u)
ζ¯ · n = 0
at Σ.
Therefore, 2 A1 (u) ¯ = 2|ζ¯ |2
at Σ.
(5.18)
Furthermore, we recall that the traction vector S · n at the wall Σ is given by [5] ¯ · n = |ζ¯ |2 n at Σ. S(u)
(5.19)
Using (5.18) and (5.19), we can rewrite (5.17), (5.16) as follows ¯ p¯ ) · n = −K · ξ − C · ω + T(u,
− Σ
We 2
|ζ¯ |2 n − |B|g,
(5.20)
Σ
¯ p¯ ) · n = −D · ξ − Θ · ω y × T(u,
− Σ
+
We 2
|ζ¯ |2 y × n + |B|g × R. Σ
(5.21)
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G.P. Galdi
R EMARK 5.1. A comparison between (4.19), (4.20) and (5.20), (5.21) immediately reveals that the viscoelastic contribution to the force and to the torque is given by We 2
|ζ¯ |2 n
(5.22)
|ζ¯ |2 y × n,
(5.23)
Σ
and We 2
Σ
respectively. In particular, it follows that the viscoelastic force per unit area acting on B, that is 12 We|ζ¯ |2 n, is compressive if and only if We > 0, or, equivalently, if and only if α1 < 0. Because of (5.11)1, the quantities (5.22) and (5.23) are (quadratic) functions of ξ and ω, with coefficients depending only on the properties of B. Specifically, setting (i) Z(i) 1 = curl h ,
(i) Z(i) 2 = curl H
and (i,j )
AT
=
1 2
(j )
(i)
Σ
Z1 · Z1 n,
(i) (j ) 1 (i,j ) (i) BT = Z2 · Z2 − 4Z2j n, 2 Σ (i) (j ) (i,j ) (i) Z1 · Z2 − 2Z1j n, CT = Σ
(i,j )
AR
=
1 2
(5.24) (j )
Σ
Z(i) 1 · Z1 y × n,
(i) (j ) 1 (i,j ) (i) BR = Z2 · Z2 − 4Z2j y × n, 2 Σ (i) (j ) (i,j ) (i) CR = Z1 · Z2 − 2Z1j y × n, Σ
we may write 1 2 1 2
Σ
Σ
|ζ¯ |2 n = ξi ξj AT
(i,j )
(i,j )
+ ωi ωj BT
(i,j )
+ ξi ωj CT
≡ F (ξ , ω), (5.25)
(i,j ) (i,j ) (i,j ) |ζ¯ |2 y × n = ξi ξj AR + ωi ωj BR + ξi ωj CR ≡ T (ξ , ω).
On the motion of a rigid body in a viscous liquid
743
As observed previously, all quantities defined in (5.24) depend only on geometric properties of B, such as size, shape, symmetry, etc. but are otherwise independent of its orientation. We need one more preparatory result. L EMMA 5.23. Assume that the following problem K · ξ + λC · g = me g + We F (ξ , λg), CT · ξ + λΘ · g = |B|g × R + We T (ξ , λg)
(5.26)
with F , T defined in (5.25), has at least one solution {ξ , λ, g}. Then, the quadruple ¯ p¯ defined in (5.11) is a solution to (5.6)–(5.9). ¯ p¯ , ξ , ω, g} with ω = λg, and u, {u, Conversely, if {u, p, ξ , ω ≡ λg, g} is a solution to (5.6)–(5.9) with (u, p) ∈ A and satisfying (5.12), then {ξ , λ, g} necessarily obeys (5.26). P ROOF. We begin to notice that, in view of (5.20) and (5.21), the system (5.26) is equivalent to (5.7), (5.8), with ω = λg, whenever u ≡ u¯ and p ≡ p¯ . Now, if (5.26) has ¯ p¯ , ξ , ω ≡ λg, g} is a solution to (5.6)–(5.9). a solution, then it follows at once that {u, Suppose, conversely, that {u, p, ξ , ω ≡ λg, g} is a solution to (5.6)–(5.9) with (u, p) ∈ A ¯ p ≡ p¯ and the lemma and satisfying (5.12). Then, by Lemma 5.22 we have u ≡ u, follows. We are now in a position to prove existence of steady falls [116]. T HEOREM 5.13. Assume B of class C 3 . Then there is a positive We0 = We0 (B), such that if |We| < We0 , the steady free fall problem (5.6)–(5.9) has at least one solution. P ROOF. A proof based on Lemma 4.13 is given in [116]. Here we shall give a much more elementary proof, under, however, some extra assumptions. Actually, by Theorem 4.1 we know that (5.26) has a solution for We = 0. Let us denote this solution by {ξ 0 , λ0 , g0 }. We recall that λ0 is an eigenvalue to (4.25), and its multiplicity is either 1, that is, λ0 is a simple eigenvalue, or three. We shall prove the theorem when λ0 is a simple eigenvalue, referring to [116] for the proof in the general case. We observe that (5.26) can be formally rewritten as ξ = K−1 · (me g − λC · g) + We F 1 , A · g = λg + We T 1 , where F 1 = K−1 · F ,
−1 T T 1 = − Θ − CT · K−1 · C · C · K−1 · F + T
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G.P. Galdi
and A is defined in (4.100). Set ξ X = λ ∈ R7 g and consider the map Ψ : (We, X) ∈ [0, We0 ] × R7 → R7 , with Ψ (We, X) =
ξ − K−1 · (me g − λC · g) − We F 1 A · g − λg − We T 1 g·g−1
.
- ξ0 . Since, by Theorem 4.1 the equation Ψ (0, X) = 0 has a solution X0 = gλ0 , say, the 0 existence of a solution to (5.26) will follow from the implicit function theorem, if we prove that the Fréchet derivative of Ψ evaluated at We = 0, X = X0 is a bijection. In other words, we have to show that for any given P, Q ∈ R3 and G ∈ R the problem ξ − K−1 · me g − C · (λ0 g + λg0 ) = P, (A − λ0 1) · g − λg0 = Q,
(5.27)
2g0 · g = G has a unique solution ξ , g, λ. Let us first prove uniqueness, that is, that the homogeneous system ξ − K−1 · me g − C · (λ0 g + λg0 ) = 0, (A − λ0 1) · g − λg0 = 0,
(5.28)
g0 · g = 0 has only the solution ξ = g = 0, λ = 0. Clearly, λ0 is also an eigenvalue for the transpose matrix AT . Denote by g∗0 the corresponding eigenvector, and let us show that g∗0 · g0 = 0.
(5.29)
Assuming the contrary would imply that the equation (A − λ0 1) · g¯ = g0 has at least one solution. Applying (A − λ0 1) on both sides of (5.30) we find (A − λ0 1)2 · g¯ = 0,
(5.30)
On the motion of a rigid body in a viscous liquid
745
which in turn, since λ0 is simple, implies g¯ = αg0 , for some α ∈ R. But this latter condition would contradict (5.30), and, therefore, (5.29) is proved. Since, from (5.28)2, we find λg∗0 · g0 = 0, in view of (5.29) we deduce λ = 0, and therefore, again from (5.28)2 and from the fact that λ is simple, we also deduce g = βg0 , for some β ∈ R. However, because of (5.28)3, we obtain g = 0, and so, from (5.28)1, we conclude also ξ = 0, and the uniqueness proof is completed. To show existence, set 1 γ = g − Gg0 = g − (g0 · g)g0 , 2 so that (5.27)2 can be rewritten as follows (A − λ0 1) · γ = Q + λg0 ≡ Q.
(5.31)
We next choose λ=−
Q · g∗0 . g∗0 · g0
(5.32)
Notice that, in view of (5.29), λ is well-defined, and that, moreover, Q · g∗0 = 0. Therefore, (5.31) is solvable and we have γ = (A − λ0 1)−1 1 Q, or, equivalently,
Q · g∗0 1 g = Gg0 + (A − λ0 1)−1 g Q − , 0 1 2 g0 · g∗0
(5.33)
where the subscript 1 means that the operator A − λ0 1 is restricted to the space orthogonal to g0 . (Notice that γ · g0 = 0.) Once λ and g have been determined by (5.32) and (5.33), we get ξ from (5.27)1. This proves that the Fréchet derivative of Ψ (We, X) at (0, X0 ) is a bijection, and that is enough to ensure that (5.26) has at least one solution for sufficiently small We. The theorem is thus completely proved. 5.1.1. Steady free fall of homogeneous bodies of revolution with fore-and-aft symmetry We shall now assume that B is a homogeneous body of revolution around a, say, possessing fore-and-aft symmetry. We shall show that, if We is small in a suitable sense, the only steady free falls that B can execute are with ω = 0, that is, they are purely translational.
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G.P. Galdi
Moreover, a must be either parallel or orthogonal to g. The stability of these configurations will be also investigated. To reach this goal, we premise the following result, whose proof is given in [116]. L EMMA 5.24. Assume that the coordinate planes {y1 = 0}, {y2 = 0}, and {y3 = 0} are symmetry planes for B. Then, the quantities defined in (5.24) satisfy the following (i,j ) (i,j ) (i,j ) (i,j ) (i,j ) properties: AT = BT = CR = 0, for all i, j = 1, 2, 3, and CT l = 0, ARl = 0, (i,j ) BRl = 0 if at least two of the indices l, i, j coincide. If, in addition, B is of revolution around the y1 axis, say, then we have (1,3) (1,2) (2,3) (2,3) (3,2) (3,2) + AR3 = AR1 = BR1 + BR1 = CT(2,3) AR2 1 + CT 1 = 0.
With the help of this lemma and of Theorem 5.13, we can now characterize the steady free falls of B. To this end, set (1,2) GV = 2AR3 ≡
Σ
(2) Z(1) 1 · Z1 (y1 n2 − y2 n1 ).
(5.34)
We shall call GV the viscoelastic torque coefficient. T HEOREM 5.14. Let B be a C 3 homogeneous body of revolution around a ≡ y1 , possessing fore-and-aft symmetry. Then, the class of purely translational solutions {u, p, ξ , 0, g} to (5.6)–(5.9) is not empty. Moreover, assume that (u, p) ∈ A and that (5.12) is satisfied. Then, the following properties hold. (i) If GV = 0, all possible purely translational solutions are with g either parallel or perpendicular to a. Moreover, ξ = me K −1 g, where K is one of the (positive) eigenvalues of the matrix K. (ii) If GV = 0, all possible purely translational solutions are with g of arbitrary direction, and ξ = me K−1 · g. Finally, set
a1 =
GV CT(2,3) 1 . Θ33
Then, if |We||ξ | |a1 | < K11 ,
(5.35)
the translational solutions are the only possible steady free falls in the class of solutions {u, p, ξ , ω, g} with (u, p) ∈ A.
On the motion of a rigid body in a viscous liquid
747
P ROOF. By Lemma 5.24, Equation (5.26) becomes (with ω = λg) K11 ξ1 = me g1 + We CT(2,3) 1 [ξ2 ω3 − ξ3 ω2 ], (3,1) K33 ξ2 = me g2 + We ξ1 ω3 CT(1,3) , 2 + ξ3 ω1 CT 2 (2,1) K33 ξ3 = me g3 + We ξ1 ω2 CT(1,2) , 3 + ξ2 ω1 CT 3 Θ11 ω1 = 0,
(5.36)
(1,3) (3,1) , Θ33 ω2 = We −ξ1 ξ3 GV + ω1 ω3 BR2 + BR2 (1,2) (2,1) Θ33 ω3 = We ξ1 ξ2 GV + ω1 ω2 BR3 + BR3 . According to Lemma 5.23, the solvability of (5.36) is equivalent to the existence of a solution {u, p, ξ , ω, g} to (5.6)–(5.9), with (u, p) ∈ A and satisfying (5.12). Imposing ω = 0, the above equations reduce to ξ = K−1 · g and ξ1 ξ3 GV = ξ1 ξ2 GV = 0. Now, if GV = 0, the assertion in the theorem is obvious. So, supposing GV = 0, we get either (a) ξ2 = ξ3 = 0, or (b) ξ1 = 0. In case (a), ξ is directed along a, in case (b) ξ is orthogonal to −1 −1 a. Moreover, in case (a) we find ξ = me K11 g, while in case (b) we deduce ξ = me K33 g. Next, we show that purely translational falls are the only possible solutions, under the stated assumptions. We observe that (5.36)4 furnishes ω1 = 0. If GV = 0, from (5.36)5,6 we at once obtain ω2 = ω3 = 0, and the theorem follows. Suppose, then, GV = 0. Since ω = λg, the condition ω1 = 0 implies either λ = 0, in which case the proof is completed, or g1 = 0. Assuming the latter, Equation (5.36)1 delivers K11 ξ1 = We CT(2,3) 1 [ξ2 ω3 − ξ3 ω2 ]. Replacing ω2 and ω3 from (5.36)5,6 (with ω1 = 0) into this equation, we obtain
K11 − We2 a1 ξ22 + ξ32 ξ1 = 0.
(5.37)
Now, if ξ1 = 0, from (5.36)5,6 (with ω1 = 0) we infer ω2 = ω3 = 0, and the result is achieved. If, however, ξ1 = 0, from (5.37) we get K11 − We2 a1 ξ22 + ξ32 = 0,
(5.38)
which is impossible if We2 |ξ |2 |a1 | < K11 . Therefore, it is ξ1 = 0, and the proof of the theorem is completed.
Theorem 5.14 asserts, in particular, that a (sufficiently smooth) body B having GV = 0, can orient itself in a translational fall only with its rotational symmetry axis a either parallel or perpendicular to g. This result is the same as that we have proved for a Navier–Stokes
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G.P. Galdi
liquid. However, by a simple stability analysis we can show that the stable orientation depends crucially on the sign of the Weissenberg number, that is, on the sign of the quadratic constant α1 . Our stability analysis here parallels that performed in Section 4.2.5. Specifically, assuming (without loss) that B is translating with a velocity ξ = (ξ1 , ξ2 , 0), from (5.36) and Figure 8, we find that the third component M of the torque acting on B, is given by M ≡ M(θ ) = We GV ξ1 ξ2 = −We GV |ξ |2 sin θ cos θ. Thus, limiting ourselves to perturbations in the form of infinitesimal disorientations of a with respect to g, of the type δθ e3 , and denoting by θ0 the equilibrium configuration (that is, θ0 is either 0 or π/2), we have dM < 0 "⇒ stability, dθ θ=θ0 dM > 0 "⇒ instability. dθ θ=θ0 Consequently, we conclude, θ =0 π θ= 2
stable
if We GV > 0,
unstable if We GV < 0, stable if We GV < 0,
(5.39)
unstable if We GV > 0.
5.1.2. Orientation of homogeneous prolate spheroids We wish to specialize the results found in the previous section to the case when B is a homogeneous prolate spheroid of eccentricity e. As we noticed in Section 4.2.6, in this situation the fields h(i) are explicitly known and the viscoelastic torque coefficient GV given in (5.34) can be analytically computed. This calculation has been performed in [39], where it is found the following expression for GV GV = 16π 2 A1 (e)A2 (e)A3 (e), where34 −1
1+e − 2e A1 = e2 1 + e2 log , 1−e −1
1+e + 2e A2 = 2e2 3e2 − 1 log , 1−e 34 We use here, as length scale, the semi-major axis of the spheroid.
On the motion of a rigid body in a viscous liquid
749
Fig. 10. Viscoelastic torque coefficient GV versus eccentricity e, in the case of a prolate spheroid.
A3 (e) = e
−1
1+e 2 − 6e . 3 − e log 1−e
By a straightforward calculation we show that GV is zero for e = 0 (sphere) and e = 1 (needle). Otherwise, GV is always positive, as shown in Figure 10. The evaluation of GV allows us to draw some interesting consequences concerning the stability of steady falls of a prolate spheroid. In fact, we notice that, from Theorem 5.14, the velocity ξ must be always parallel to g. Therefore, since GV > 0 when e ∈ (0, 1), in this range of eccentricities, from (5.39) we find the following stability properties of the two possible orientations of the axis of revolution a with the gravity g: a parallel to g
stable
unstable if We < 0,
a orthogonal to g
if We > 0,
stable
if We < 0,
unstable if We > 0.
These theoretical predictions are in agreement with the experiments [81,86,17], if and only if We > 0, namely, if and only if α1 < 0. We also wish to notice that, unlike the purely Newtonian case, in the case at hand the ellipsoid will orient itself in such a way that its speed is a maximum, that is, its resistance is a minimum. This follows from Theorem 5.14(i), and from the fact that K33 > K11 (see [57]).
5.2. Steady fall at nonzero Reynolds number As we already remarked, the problem of existence of steady free fall in the case of nonzero Reynolds number is completely open for bodies of arbitrary shape and density. However, if B is a homogeneous body of revolution with fore-and-aft symmetry, it is still possible to show existence of purely translational steady falls, and to find all corresponding possible orientations for B, at least at first order in Reynolds and Weissenberg numbers, and to study their stability. This problem, as we already emphasized several times, is very important in understanding the nature of the viscoelastic forces on “long” particles. Actually, in a
750
G.P. Galdi
Navier–Stokes liquid (like water), the inertia of the liquid produces a torque on the cylinder that makes it turn with its broadside horizontal, so that the stable orientation is θ = π/2 (see Figure 2(A), Figure 8). This was mathematically established in Section 4.2.5. However, if a suitable concentration of polymer is added to the liquid, so that the effect of inertia is negligible with respect to the viscoelastic one, the angle changes dramatically to θ = 0, that thus becomes the new stable orientation (see Figure 2(B)). In Section 5.1.1 we showed that this phenomenon can be explained by using the second-order liquid model. However, in liquids where inertia and viscoelasticity are of the same order of magnitude it is observed that the stable orientation of long homogeneous particles, like homogeneous cylinders, occurs at an angle ranging between θ = 0 and θ = π/2, [86]. This phenomenon is called the “tilt angle phenomenon”. In [71] a qualitative analysis is performed, according to which the tilt angle is hypothesized to arise from the balance of the inertial torque and of the viscoelastic torque generated by normal stress effects; see also [72,74]. Therefore, liquid models like second-order or Oldroyd-B, where normal stress effects are taken into account, could quantitatively explain this phenomenon. In the following sections we shall prove that the above conjecture about the tilt angle phenomenon, even though very plausible, is not correct, and that this phenomenon can not be attributed to the competition of inertia and normal stresses alone. In fact, at first order in Re and We, we shall see that only two orientations are allowed, and that this competition is only responsible for their stability. The main idea behind the proof of these results is the evaluation of the torque M exerted by the liquid on the body. In the next section, we shall outline a general method for the evaluation of M, in the case of a body possessing the above mentioned symmetry, and moving by translational motion in a generic non-Newtonian liquid, with Cauchy stress tensor of the form TNS +λS, where λ is a real parameter and S is an “extra” non-Newtonian stress. This method, introduced in [49], generalizes the one that we have used in the case of a Navier–Stokes liquid in Section 4.2.4. In particular, we shall see that if S satisfies certain general conditions, then M, at first order in Re and λ, can be simply expressed in terms of the auxiliary fields h(i) , H(i) (see (4.15), (4.16)). In subsequent sections we shall then show that these conditions are certainly satisfied in the case of a second-order liquid, and, consequently, we are able to characterize all possible orientations of B at first order in Re and We, and to ascertain their stability. Most of the results presented here are taken from the paper [49]. We shall limit ourselves to give the main ideas, referring to that paper for all technical details. 5.2.1. Evaluation of the torque Assume that a body B is moving in a viscous liquid L, with a constant translational velocity V. We suppose that the Cauchy stress tensor for L has the form T = TNS + λS, where TNS denotes the (nondimensional) Newtonian stress tensor, namely, TNS = −pI + 2D(u), λ is a (nondimensional) parameter related to the non-Newtonian character of L, and S is the non-Newtonian contribution to the stress tensor. Without loss, we may assume λ 0.
On the motion of a rigid body in a viscous liquid
751
The appropriate equations of motion, with respect to a frame attached to B, can be written in a nondimensional form as follows : Re u · grad u = div TNS (u, p) + λ div S(u), in D, div u = 0, (5.40) u = 0 at Σ, lim u(y) = −U,
|y|→∞
where u is the relative velocity, and U = V/V . As we know, the total torque M exerted by L on B is given by y × T · n. (5.41) M≡− Σ
Our objective is to compute M at first order in Re and λ. Multiplying (5.40)1 by H(i) , integrating by parts over Ω and using (4.16)2,3,4 we find Mi = 2 D(u) : D H(i) D
+λ
D
S : D H(i) + Re
D
u · grad u · H(i) .
(5.42)
The first integral on the right-hand side of this relation can be evaluated by multiplying (4.16)1 by u + U and integrating by parts over D. We get (5.43) 2 D(u) : D H(i) = U · T H(i) · n. D
Σ
From (5.41), (5.42), and (5.43) we thus obtain M = MS + Re MI + λMNN ,
(5.44)
where (i = 1, 2, 3) MSi
T H(i) · n,
= −U · Σ
MIi = −
D
u · grad u · H(i) ,
MNN =− i
D
(5.45)
S(u) : D H(i)
are the torque in the Stokes approximation (i.e., Re = λ = 0), the torque due to inertia, and the torque due to the non-Newtonian character of L, respectively.
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G.P. Galdi
We now denote by (uS , pS ) and by (uNS , pNS ) the solutions to (5.40) with Re = λ = 0 and λ = 0, respectively. We also set v = uNS − uS
z = u − uNS , and M0,I i =−
D
uS · grad uS · H(i) ,
M0,NN i
=−
D
S(uS ) : D H .
(5.46)
(i)
From (5.44) we thus get M = MS + Re M0,I + λM0,NN + N ,
(5.47)
where N = Re MI − M0,I + λ MNN − M0,NN ≡ Re N 1 + λN 2 . By a straightforward calculation we find for i = 1, 2, 3, (z + v) · grad u + uS · grad(z + v) · H(i) , N1i = − D
N2i = −
D
−
S(u) − S(uNS ) : D H(i)
D
(5.48)
S(uNS ) − S(uS ) : D H(i) .
From (5.48) it is expected that both N 1 and N 2 should vanish as Re , λ → 0, that is, N = o(Re ) + o(λ) as Re, λ → 0.
(5.49)
If this is the case, from (5.47) we deduce that, at first order in Re, λ M = M0,S + Re M0,I + λM0,NN .
(5.50)
The above considerations apply to any body B (and to any liquid L). We would like now to consider the special case when B is a homogeneous body of revolution around an axis a (say), with fore-and-aft symmetry. In such a case, from Lemma 4.1 and Lemma 4.2 we find T H(i) · n = 0, i = 1, 2, 3. Σ
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This fact has two main consequences. The first (obvious) is: MS = 0,
(5.51)
and the second is (see (4.166) and [35], Chapter V): $ $ $grad H(i) $ < ∞, for all s ∈ (1, ∞), i = 1, 2, 3. s
(5.52)
From (5.48)1, by an integration by parts we find N1i = (z + v) · grad H(i) · (u + U) + uS · grad H(i) · (z + v) .
(5.53)
D
It is well-known that vS ∞ c,
(5.54)
where c is a positive constant depending only on B. Assume now that there are Re0 , λ0 > 0 such that for all 0 < Re < Re0 , and 0 < λ < λ0 the following conditions hold (H1) u + U ∞ c1 , (H2) z q1 c2 λβ1 , for some q1 ∈ (1, ∞), β1 > 0, (H3) v q2 c2 Reγ1 , for some q2 ∈ (1, ∞), γ1 > 0, where c1 , c2 , c3 are (positive) constants depending only (at most) on B, Re0 , λ0 and q. Then, using Hölder’s inequality and (5.52) in (5.53), we find |N 1 | c4 Reγ1 + λβ1 , with a constant c4 independent of Re and λ. Likewise, assume that for all 0 < Re < Re0 , and 0 < λ < λ0 the following conditions hold (H4) S(u) − S(uNS ) q3 c2 λβ2 , for some q3 ∈ (1, ∞), β2 > 0, (H5) S(uNS ) − S(uS ) q4 c3 Re γ2 , for some q4 ∈ (1, ∞), γ2 > 0, with c2 , c3 independent of Re , λ. Then, using again Hölder’s inequality and (5.52) in (5.48)2, we find |N 2 | c4 Reγ2 + λβ2 . The results just described are summarized in the following. L EMMA 5.25. Let B be a homogeneous body of revolution with fore-and-aft symmetry. Assume that conditions (H1)–(H5) hold. Then, there are positive Re0 and λ0 such that for all 0 < Re Re0 , and 0 < λ λ0 the total torque (5.41) exerted by the liquid L on B is given by M = Re M0,I + λM0,NN + N ,
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where M0,I and M0,NN are defined in (5.46), while |N | C Re1+γ + λ1+β , with C, γ and β positive constants independent of Re and λ. 5.2.2. Torque exerted by a second-order liquid at first order in Re and We We shall now verify that conditions (H1)–(H5) of Lemma 5.25 are indeed satisfied for a secondorder liquid. We recall that, in this case, the “extra-stress” S is given in (5.5). Moreover, in view of the results of Section 5.1.2, we take α1 < 0, so that λ ≡ We .35 The equations of motion (5.40) then become : Re u · grad u = div −p1 + 2D − We (A2 + εA1 · A1 ) , div u = 0, u=0
in D, (5.55)
at Σ,
lim u(y) = −U,
|y|→∞
where A2 = A2 (u) ≡ u · grad A1 + A1 · LT + L · A1 . In the Navier–Stokes case, i.e., We = 0, the above problem specializes to the following one : Re uNS · grad uNS = div −pNS 1 + 2D(uNS ) , in D, div uNS = 0, uNS = 0 at Σ, lim uNS (y) = −U.
|y|→∞
Moreover, by taking Re = 0, this problem reduces, in turn, to the Stokes problem : div −pS 1 + 2D(uS ) = 0, div uS = 0, uS = 0
in D,
at Σ,
lim uS (y) = −U.
|y|→∞
35 We are assuming α 0, so that We 0. This assumption is not needed from the mathematical point of 1 view. However, as we already pointed out in Section 5.1.2, assuming α1 > 0, would produce results at odds with experiments.
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D EFINITION 5.1. For a given C > 0, we shall say that a solution (u, p) to (5.55) belongs to the class AC if and only if Re1/2 u + U
2q 2−q
+ Re1/4 grad u
4q 4−q
$ $ $ $ + $D 2 u$1,q + $D 2 u$1,t
+ grad p q + grad p t C. The key results of this section are collected in the following Theorem 5.15 and Theorem 5.16, while the main result is stated in Theorem 5.17. The proof of Theorems 5.15 and 5.16 is rather technical and we shall omit it here. For details, we refer the reader to Theorem 2.1 and Theorem 2.2 of [49], respectively. T HEOREM 5.15. Let B be of class C 3 . There exist positive numbers Re0 = Re0 (Ω, ε), C1 = C1 (B, Re0 , q) and C2 = C2 (B, Re0 , ε, q) such that for any 0 < Re Re0 , and 1 < q < 3/2 we have (i) uNS − uS 3q C1 Re1−η , 3−2q
(ii) S(uNS ) − S(vS ) q C2 Re1−η , where η can be taken arbitrarily close to zero, by choosing q arbitrarily close to 3/2.36 T HEOREM 5.16. Let B be of class C 3 . Let v, p ∈ AC for some C > 0. Then, there exist positive numbers We 0 = We 0 (Ω, ε, C), Re 0 = Re0 (Ω, ε, C), and C3 = C3 (Ω, We 0 , Re 0 , ε, q) such that for any 0 < Re Re0 , 0 < We We 0 , and 1 < q < 3/2 we have (i) u + U ∞ C3 , (ii) u − uNS 3q C3 We , 3−2q
(iii) S(u) − S(uNS ) q C3 We . From Lemma 5.25, Theorem 5.15 and Theorem 5.16 we immediately obtain the main result of this section. T HEOREM 5.17. Let B be a homogeneous body of revolution of class C 3 , with fore-andaft symmetry, moving in a second-order liquid L by constant translational motion. Let u, p ∈ AC , some C > 0. Then, there are positive Re0 and We 0 depending on B, ε and C, such that for all 0 < Re Re0 , and 0 < We We 0 the total torque (5.41) exerted by L on B is given by M = Re M0,I + We M0,NN + N , with M0,I and M0,NN defined in (5.46), while |N | K Re2−η + We 2 , 36 Notice, however, that C , C → ∞ as q → 3/2. 1 2
(5.56)
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G.P. Galdi
where K and η are positive constants independent of Re and We , and where η can be taken arbitrarily close to zero.37 5.2.3. Orientation of a symmetric body in a translational steady fall In analogy with the corresponding Navier–Stokes case treated in Section 4.2.3, we shall say that {u, p, ξ , g} is a translational steady fall, if it satisfies (5.1)–(5.4) with ω = 0, namely, Re u · grad u = u − grad p − We div S(u) + g, div u = 0, lim u(y) = −ξ ,
: in D, (5.57)
|y|→∞
u(y) = 0,
y ∈ Σ,
T(u, p) · n,
mg =
(5.58)
Σ
y × T(u, p) · n = 0.
(5.59)
Σ
As already noticed in the Newtonian case, some further properties for B are needed for the existence of a translational solution. By using a more complicated technique than that employed for the Navier–Stokes case, one can establish the following result, for whose proof we refer to Theorem 3.1 and Theorem 3.2 of [49]. T HEOREM 5.18. Let B be a homogeneous body of revolution around an axis a, of class C 3 and possessing fore-and-aft symmetry. Then, there exist Re0 , We0 , C > 0 depending only on B and ε such that, for Re < Re0 , We < We0 , there are at least two types of translational steady falls {u, p, ξ , g} with (u, p) ∈ AC , and they are determined by the following directions of the acceleration of gravity g: (a) g is parallel to a; (b) g is orthogonal to a. In both cases, g is parallel to ξ , with ξ · g > 0. Moreover, if {u1 , p1 , ξ , g1 } is another translational steady fall corresponding to the same velocity ξ , and with (u1 , p1 ) ∈ AC , there exist Re1 , We1 > 0 depending only on B, ε, and C such that, for Re < Re1 , We < We1 , we have u ≡ u1 , p ≡ p1 , and g = g1 . Our next objective is to show that, at first order in Re and We , these are the only possible translational falls. In other words, the only possible orientations for B are with a either parallel or perpendicular to g. A fundamental role in proving this result is played by the evaluation of the torque furnished in Theorem 5.17. Without loss of generality, we take the y1 -axis of a frame attached to B coinciding with the axis of revolution a of B, and assume the translational velocity ξ contained in the plane 37 K → ∞ as η → 0.
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y1 , y2 ; see Figure 8. With these choices, one can show the following results, for whose proof we refer again to Section 3 of [49]: 0,I 0,NN = M0,NN =0 M0,I 1 = M2 = M1 2
and M0,I 3 = ξ1 ξ2 GI ,
M0,NN = ξ1 ξ2 GV ,ε , 3
where (see (4.185)) (1) h − e1 · grad h(2) + h(2) − e2 · grad h(1) · H(3) GI = − D
(5.60)
(5.61)
and GV ,ε = −
D
(1) T h − e1 · grad A1 h(2) + grad h(1) · A1 h(2)
+ A1 h(1) · grad h(2) + h(2) − e2 · grad A1 h(1) T + grad h(2) · A1 h(1) + A1 h(2) · grad h(1) + 2εA1 h(1) · A1 h(2) : D H(3) .
(5.62)
Clearly, for a fixed ε, GV ,ε (similarly to GI ) depends only on the geometric properties of B, such as size or shape, but is otherwise independent of the orientation of B and of the properties of the liquid. Moreover, for ε = −1, GV ,ε reduces to GV given in (5.34).38 We continue to call GV ,ε the viscoelastic torque coefficient. Therefore, at first order in Re and We , from Theorem 5.17 we obtain that the torque M acting on B is given by M = (Re GI + We GV ,ε )ξ1 ξ2 e3 .
(5.63)
In the case of a steady fall, the torque must vanish (see (5.59)), and from (5.63) we deduce that, provided the term in bracket is not zero, this can happen only if ξ is either directed along the axis of revolution a of B or it is perpendicular to it. From Theorem 5.18 it then follows that ξ has the same orientation as g and so we conclude that provided the term in bracket in (5.63) is not zero, the only possible orientations of B at first order in the Reynolds and Weissenberg numbers are with a either parallel or perpendicular to g.39 Let us now consider the stability of such orientations. Since ξ1 = |ξ | cos θ , ξ2 = −|ξ | sin θ (see Figure 1), Equation (5.63) can be also written as follows M = −|ξ |2 (Re GI + We GV ) sin θ cos θ e3 .
(5.64)
38 This might not seem so obvious at a first glance, but one can prove it in a rather straightforward way, by means
of several integrations by parts. 39 Notice that if Re G + We G I V ,ε = 0 all orientations are allowed (at first order in Re and We). In a real experiment, however, this vanishing condition is practically unattainable.
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Thus, if we limit ourselves to perturbations in the form of infinitesimal disorientations of a with respect to g, of the type δθ e3 , denoting by θ0 the equilibrium configuration (that is, θ0 is either 0 or π/2), we have d(M · e3 ) < 0 "⇒ stability, dθ θ=θ0 d(M · e3 ) > 0 "⇒ instability. dθ θ=θ0 Consequently, we obtain θ =0 θ=
π 2
stable
if Re GI > −We GV ,
unstable if Re GI < −We GV ,
stable
if Re GI < −We GV ,
unstable if Re GI > −We GV .
From this we see that, perhaps at odds with intuition, the competition between the inertial torque and viscoelastic torque due to normal stress not produce an “intermediate” equilibrium configuration corresponding to an angle θ = 0, π/2, as conjectured in [71, 72,74]. Rather, it is only responsible for the stability/instability of the configurations θ = 0, π/2. 5.2.4. Orientation of homogeneous prolate spheroids This section aims to discuss the nature of the torque in the case when B is a prolate spheroid of eccentricity e. In [49] an evaluation of the viscoelastic torque GV ,ε (5.62) is performed, and the results are reported here. Graphs of the variation of the viscoelastic torque coefficient GV ,ε with eccentricity are shown in Figures 11 and 12. They depict also the variation of GV ,ε with the parameter ε. The essential profile of the curve stays remarkably consistent for each value of the parameter ε (see Figure 11), changing slightly when ε > −1 (see Figure 12). Also, GV ,ε increases with decreasing ε. It is also interesting to note in Figure 12 that GV ,ε is always positive for each e if ε is less than approximately −1. As ε becomes less than one, the torque coefficient changes sign for e close to one. Let us analyze the two situations ε −1 and ε −1 separately. The case ε −1. In this case we have that GI and GV ,ε have opposite sign; see Figures 9 and 11. In view of the results of the previous section, this means that for ε −1 the stable orientation of the prolate spheroid is with its major axis a perpendicular to the gravity g if −Re GI > We GV ,ε (inertia prevails on viscoelasticity) while the stable orientation is with a parallel to g if −Re GI < We GV ,ε (viscoelasticity prevails on inertia). The case ε −1. For values of eccentricities in the range (0, ∼ 0.9) the stability of the equilibrium configuration is the same as in the case ε −1. However, for very slender spheroids (e ∼ 1) GV ,ε becomes negative. Therefore, if ε −1, sedimenting slender spheroids experience inertial and viscoelastic torques acting in the same direction, and
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759
Fig. 11. Viscoelastic torque coefficient versus eccentricity e for different values of ε.
Fig. 12. Viscoelastic torque coefficient versus e, for ε = −0.7, −0.8. The torque coefficient changes sign for e at approximately −0.9. Note also that the curves achieve their peaks at decreasing values of e as ε increases.
the configuration with a perpendicular to g is always stable, as in the case of a purely Newtonian liquid. Since slender bodies in a viscoelastic liquid orient themselves with a parallel to g [81], our result confirms that the predicted value of a lower bound of ∼ −1.6 for ε is appropriate [86]. In Figure 12 we plot GV ,ε versus e for ε = −0.7, −0.8, since the dramatic turn to negative values is more prominent in these cases. Another important feature is that the viscoelastic torque coefficient is several times bigger than the absolute value of the inertial torque coefficient, mainly for eccentricities close to 1. Figure 13 compares the inertial torque coefficient to the viscoelastic torque coefficient for two different values of ε. We have chosen ε = −1.8 which is the value recommended in the experiments of Liu and Joseph [86], see also [67], §17.11, and ε = −1, that is the value for winch the viscoelastic torque coefficient can be computed analytically [39]. The viscoelastic effects seem to outweigh the inertial ones.
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G.P. Galdi
Fig. 13. Comparison of the inertial torque coefficient to the viscoelastic torque coefficient, for different values of ε. The viscoelastic torque coefficient is almost five times bigger than the inertial one for e around 1 (slender ellipsoids) and ε = −1.8.
Fig. 14. Viscoelastic torque coefficient versus e, for the case of an Oldroyd-B model [47].
R EMARK 5.2. Results similar to those just described are obtained in [47] for another classical model of viscoelastic liquid (Oldroyd-B), that takes only into account normal stress effects. In Figure 14 we report a graph of the corresponding viscoelastic torque coefficient versus eccentricity, and its comparison to the inertial torque. It turns out that the viscoelastic torque coefficient has the same qualitative features as in the second-order liquid case. In particular, as in the second-order model, the viscoelastic and the inertial torque coefficients have opposite sign.
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Part II. Self-propelled bodies In this second part we shall be interested in the motion of a rigid body that propels itself in a viscous liquid. In contrast to the case analyzed in Part I, here the body does not move under the action of an external force (towed body problem), but, rather, through the use of an “internal” mechanism. Our objective is to provide a mathematical analysis of the motion of self-propelled bodies of constant shape in a Navier–Stokes liquid. This choice of the liquid model is made for the sake of simplicity. A similar analysis may be performed, in principle, for more complicated non-Newtonian models, with all the corresponding troubles that we have already encountered in the case of particle sedimentation. While the performed analysis is essentially complete in the linearized approximation of zero Reynolds number, in the full nonlinear case, only partial answers are available, with the exception of a symmetric body, where, more or less, the same results obtained in the linearized case can be proved, at least if the size of the Reynolds number is suitably restricted. After giving in Section 6 the mathematical formulation of self-propelled motion, in Section 7 and corresponding subsections, we shall furnish a complete theory in the Stokes approximation. In Section 8 and corresponding subsection, we shall treat the nonlinear case where, however, results are not equally satisfactory, more or less due to the same technical difficulties encountered in Part I.
6. The self-propelled body equations In a purely self-propelled motion, a body B moves into the liquid L only by a mechanism produced by the body itself. Therefore, we shall give the following definition. D EFINITION 6.1. We shall say that B executes a self-propelled motion in L if and only if: (1) The total external force and external torque acting on B are identically zero. (2) The total external force acting on L is identically zero. From (1.15)–(1.21), we thus find that the equation describing the self-propelled motion of B, in the case of a Navier–Stokes liquid and in a nondimensional form, are given by ⎫ ∂w + Re (w − U) · grad w + ω × w ⎪ ⎪ ⎬ ∂t = div T ≡ w − grad p, ⎪ ⎪ ⎭ div w = 0,
in D × (0, ∞),
lim w(y, t) = 0,
(6.2)
|y|→∞
w(y, t) = w∗ (y, t) + U(y, t),
(6.1)
(y, t) ∈ Σ × (0, ∞),
(6.3)
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m
I·
dξ + Re mω × ξ = − dt
T(w, p) · n − Re w(w − U) · n ,
(6.4)
Σ
dω + Re ω × (I · ω) = − dt
y × T(w, p) · n − Re w(w − U) · n .
(6.5)
Σ
To these equations we should append the initial conditions. For the sake of simplicity, and also for the type of applications we have in mind, we shall assume that the body is initially at rest in a quiescent liquid. Therefore, the initial conditions become w(y, 0) = ξ (0) = ω(0) = 0.
(6.6)
6.1. Stokes approximation We shall begin to consider the limiting situation of vanishingly small Reynolds number, that is, the stress due to viscosity is predominant on that due to inertia. This happens, when the characteristic velocity is small (slow motion) and/or when the size of B is small (microscopic objects). In the case of ciliates, for example, a characteristic length is 10−4 m, and a characteristic velocity is 10−4 m/sec. Therefore, a typical value of the Reynolds number in a liquid like water is Re ∼ 10−2 . If we take the limit Re → 0 into Equations (6.1)–(6.5), we formally get the following problem ⎫ ∂w = div T(w, p), ⎬ ∂t ⎭ div w = 0,
in D,
w(y, t) = w∗ (y, t) + U(y, t), lim w(y, t) = 0,
|y|→∞
(y, t) ∈ Σ × (0, ∞), (6.7)
dξ =− T(w, p) · n, dt Σ dω =− y × T(w, p) · n, I· dt Σ
m
with initial conditions given in (6.6). As in the case of particle sedimentation, we first consider the case of steady selfpropelled motions, whose study will be the object of the next few sections.
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6.1.1. Steady self-propelled motion In the case of a steady motion, Equations (6.7) reduce to the following ones div T(w, p) = 0,
:
div w = 0,
in D,
w(y) = w∗ (y) + U(y),
y ∈ Σ,
lim w(y) = 0,
(6.8)
|y|→∞
T(w, p) · n = 0,
Σ
y × T(w, p) · n = 0. Σ
Following [38], we shall show that for any (sufficiently regular) velocity distribution w∗ with nonzero orthogonal projection40 P(w∗ ) in a suitable “control” space, see (6.22), there exists one and only one solution w, p and U to (6.8), with U = 0. Moreover, the velocity U is completely determined by P(w∗ ) and by certain geometric properties of B. To prove this, we introduce the following vector fields g(i) := T h(i) , p(i) · nΣ , G(i) := T H(i) , P (i) · nΣ ,
i = 1, 2, 3, i = 1, 2, 3,
where (h(i) , p(i) ) and (H(i) , P (i) ) are the auxiliary fields defined in (4.15) and (4.16). The vector functions g(i) = g(i) (y) and G(i) = G(i) (y) depend only on the geometric properties of B such as size or shape. In particular, they do not depend on the orientation of B and on the liquid property. For example, for B a ball of radius a, the solutions to (4.15), (4.16) are well-known, see, e.g., [57], pp. 163, 169, and we have g(i) (y) =
3 ei , 2a
G(i) (y) = 3ei × y,
i = 1, 2, 3,
(6.9)
with the origin at the center of the ball. The vector functions g(i) = g(i) (y) and G(i) = G(i) (y) will play an important role and, in particular, we are interested in their linear independence properties. In this regard, the following result holds, for whose proof we refer to [38], Lemma 2.1. L EMMA 6.26. If B is Lipschitz, the system of vector functions %
&
S1 = g(i) , G(i) , 40 In the sense of L2 (Σ).
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G.P. Galdi
is linearly independent. Moreover, if Σ is of class C 2 , also the system S2 =
%
& g(i) × n × n, G(i) × n × n
is linearly independent. R EMARK 6.1. Notice that the system S3 = {(g(i) · n)n, (G(i) · n)n} is not always linearly independent. For instance, when B is a ball of radius a, from (6.9) we get (G(i) · n) ≡ 0, i = 1, 2, 3. This implies – as we shall see later on – that, as it is intuitive, a sphere can not perform a rotation by a purely normal distribution of velocity at its boundary; see (6.9) and (6.33). We shall next furnish necessary and sufficient conditions in order that B performs a steady self-propelled motion within the Stokes approximation. To this end, we multiply (6.8)1 by h(i) and integrate by parts over D to find
ei ·
T(w, p) · n = 2
D
Σ
D h(i) : D(w),
i = 1, 2, 3.
Likewise, multiplying (4.15)1 by w and integrating by parts over D, we obtain
Σ
(w∗ + U) · g(i) = 2
D
D h(i) : D(w),
i = 1, 2, 3.
(6.10)
These two displayed relations then imply
Σ
(w∗ + U) · g(i) = ei ·
T(w, p) · n.
(6.11)
Σ
In a similar fashion, multiplying (6.8)1 by H(i) and (4.16) by w, respectively, and integrating by parts over D we find
ei ·
y × T(w, p) · n = 2
D
Σ
D H(i) : D(w),
i = 1, 2, 3,
and
Σ
(w∗ + U) · G
(i)
=2
D
D H(i) : D(w),
i = 1, 2, 3,
(6.12)
which in turn give
Σ
(w∗ + U) · G
(i)
= ei ·
y × T(w, p) · n. Σ
(6.13)
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Consequently, the self-propelling conditions (6.8)5,6 are equivalent to the following ones
Σ
(w∗ + U) · g(i) =
Σ
(w∗ + U) · G(i) = 0,
i = 1, 2, 3.
(6.14)
We wish to put (6.14) in a different form. Let us define the vectors V and W as follows: (i) Vi = − w∗ · g , Wi = − w∗ · G(i) , i = 1, 2, 3. (6.15) Σ
Σ
From (6.14) we find that w∗ generates a steady, self-propelled motion if and only if the following condition holds V = K · ξ + C · ω, W = CT · ξ + Θ · ω,
(6.16)
where the matrices K, C, and Θ are defined in (4.21) and Lemma 4.1. Since the 6 × 6 matrix K C (6.17) CT Θ is positive definite, and so are the matrices K and Θ, see Lemma 4.1, we may solve (uniquely) for ξ and ω in (6.16) to obtain ξ = A · V − C · Θ −1 · W , ω = B · W − CT · K−1 · V ,
(6.18)
−1 A = K − C · Θ −1 · CT , −1 B = Θ − CT · K−1 · C .
(6.19)
where
Notice that the translational velocity ξ of B is not zero if and only if the velocity distribution at Σ satisfies the following condition V = C · Θ −1 · W,
(6.20)
whereas the angular velocity ω is nonzero if and only if W = CT · K−1 · V.
(6.21)
In this context, it is interesting to mention the case when B is “non-screw” [57], p. 192. Orthotropic bodies (that is, those having three mutually perpendicular symmetry planes)
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G.P. Galdi
are an example of non-screw bodies. In such a case, [57], p. 174, for a suitable choice of the origin (the “center of reaction” of B), one has C = 0, so that from (6.16) it follows that the motion of B is purely translational (ω = 0) or purely rotational (ξ = 0) if and only if W = 0 or V = 0, respectively. Set Mij = g(i) · g(j ) , Nij = g(i) · G(j ) , Oij = G(i) · G(j ) . Σ
Σ
Σ
In view of the linear independence of the system S1 = {g(i), G(i) }, see Lemma 6.26, the 6 × 6 matrix M N NT O is invertible. Therefore, for any ξ , ω ∈ R3 there exists a vector field w∗ = αi g(i) + βi G(i) with uniquely determined α, β ∈ R3 satisfying (6.16). Moreover, in view of the independence of the system S2 , constituted by the tangential components of the vectors g(i) , G(i) , we may prescribe the normal component ψ (say) of the velocity field at Σ and, for any given ξ , ω ∈ R3 we can solve (6.16) with w∗ = γi (g(i) × n) × n + δi (G(i) × n) × n + ψ with uniquely determined γ , δ ∈ R3 . To describe the results obtained above, it is convenient to introduce the following 6-dimensional subspaces of L2 (Σ) % & T (B) = u ∈ L2 (Σ): u = αi g(i) + βi G(i) , for some α, β ∈ R3
(6.22)
% Tτ (B) = u ∈ L2 (Σ): u = γi g(i) × n × n + δi G(i) × n × n, & for some γ , δ ∈ R3 .
(6.23)
and
As we noticed, T (B) and Tτ (B) depend only on the geometric properties of B such as size or shape. In particular, they are independent of the orientation of B and on the liquid property. We denote by P the orthogonal projection of L2 (Σ) onto T (B). Taking into account classical existence and uniqueness theorems for the exterior Stokes problem, see [35], Chapter V, we may then summarize the results obtained thus far in the following. T HEOREM 6.19. Let B be Lipschitz. Then, for any w∗ ∈ W 1/2,2 (Σ) satisfying P(w∗ ) = 0, there exists a unique solution w, p, U to problem (6.8) with U ∈ R − {0}. Conversely, for any U ∈ R − {0}, there exists one and only one solution w, p to (6.8)1,2,4,5,6 such that the trace w∗ of w to Σ belongs to T (B). Moreover, assume Σ of class C 2 and that w · n(y) = ψ(y),
y ∈ Σ,
(6.24)
On the motion of a rigid body in a viscous liquid
767
Fig. 15. Coordinate system for a rotationally symmetric body.
where ψ ∈ W 1/2,2 (Σ) is prescribed. Then, for any U ∈ R − {0}, there exists one and only one solution to (6.8)1,2,4,5,6–(6.24) with (w × n) × n ∈ Tτ (B). Finally, The translational velocity ξ of B is not zero if and only if (6.20) is satisfied, while its angular velocity ω is nonzero if and only if (6.21) holds.
6.1.2. Application to rotationally symmetric bodies We wish now to apply the results of the previous section to some particular interesting cases. We shall do this for B possessing rotational symmetry. The case of other symmetries (e.g., helicoidal symmetry) could be treated in a similar way with the help of the results of [57], §5-5. Supposing y1 is the axis of symmetry, we denote by {N, s, eϕ } a basis for a system of orthogonal coordinates at Σ, where N is the unit outer normal, s a unit tangent vector to a meridian curve on Σ and eϕ the azimuthal unit vector. The sense of s is such that {N, s, eϕ } is right-handed (see Figure 15). T HEOREM 6.20. Let B be a Lipschitz body of revolution around the y1 -axis. Then, the most general boundary velocity distribution w∗ which can move B with nonzero translational velocity directed along y1 , is given by w∗ = αg(1) + V∗ ,
(6.25)
where α ∈ R − {0}, and V∗ ∈ (T (B))⊥ . Moreover, assume Σ of class C 2 and let ψ ∈ W 1/2,2 (Σ), ψ = P2 ψ = P3 ψ, where Pi , i = 2, 3, are defined in (4.150). Then, if w∗ · n = ψ at Σ, the most general distribution is w∗ = γ g(1) · s s + ψn + W∗ ,
(6.26)
where γ ∈ R − {0}, and W∗ ∈ (T (B))⊥ with W∗ · n ≡ 0. The corresponding translational velocity ξ of B is given by ξ = Le1 ,
(6.27)
where L=k Σ
w∗ · g(1),
(6.28)
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G.P. Galdi
and k is a negative constant, depending only on B, and representing the inverse of the force exerted by the liquid on B, when B moves with unitary velocity in the y1 direction.41 P ROOF. In the case at hand, the matrices K, C and Θ take the form, see [57], §5-5, k1 0 0 K = 0 k2 0 , 0 0 k2 θ1 0 0 Θ = 0 θ2 0 , 0 0 θ2
C=
0 0 0
0 0 −c
0 c , 0
(6.29)
where ki ( = 0), c and θi ( = 0), i = 1, 2, are constants depending only on B. Moreover, it is easy to show [37] that the vectors g(i) satisfy the following properties g1(1) = P2 g1(1) = P3 g1(1) ,
g2(1) = −P2 g2(1) = P3 g2(1) ,
g3(1) = P2 g3(1) = −P3 g3(1) ,
g1(2) = −P2 g1(2) = P3 g1(2) ,
(2)
(2)
(2)
(2)
(2)
(2)
g2 = P2 g2 = P3 g2 ,
g3 = −P2 g3 = −P3 g3 ,
g1(3) = P2 g1(3) = −P3 g1(3) ,
g2(3) = −P2 g2(3) = −P3 g2(3) ,
(3)
(3)
(6.30)
(3)
g3 = P2 g3 = P3 g3 while the vectors {G(i) } satisfy
(1) (1) G(1) 1 = −P2 G1 = −P3 G1 , (1)
(1)
(1) (1) G(1) 2 = P2 G2 = −P3 G2 ,
(1)
(2)
(2)
(2)
G3 = −P2 G3 = P3 G3 ,
G1 = P2 G1 = −P3 G1 ,
(2) (2) G(2) 2 = −P2 G2 = −P3 G2 ,
(2) (2) G(2) 3 = P 2 G3 = P 3 G3 ,
(3)
(3)
(3)
(3)
(3)
(3)
G1 = −P2 G1 = P3 G1 ,
(3)
(6.31)
(3)
G2 = P 2 G2 = P 3 G2 ,
(3)
G3 = −P2 G3 = −P3 G3 . These relations imply, in particular, the following orthogonality conditions
g(1) · g(i) = Σ
g(1) · G(j ) = 0,
for all i = 2, 3, j = 1, 2, 3.
Σ
Since any vector field w∗ at Σ can be decomposed in the following way w∗ = αi g(i) + βi G(i) + V∗ ,
⊥ αi , βi ∈ R, V∗ ∈ T (B) ,
41 Since B is symmetric along the y -axis, this force coincides with the drag. 1
(6.32)
On the motion of a rigid body in a viscous liquid
769
Equation (6.25) follows from (6.32). It is easy to show that k1 is the opposite of the force (d, say) exerted by the liquid on B, when B is moving with velocity e1 . In fact, by (4.15), we have g(1). d= Σ
However, from (6.30)1,2,3 and (4.21), we obtain d = k1 e1 , which is what we claimed. We next observe that, in view of (4.112), system (6.16) is equivalent to the following one V1 = k1 ξ1 ,
V2 = k2 ξ2 + cω3 ,
W1 = θ1 ω1 ,
V3 = k2 ξ3 − cω2 ,
W2 = θ2 ω2 − cξ3 ,
W3 = θ2 ω3 + cξ2 .
(6.33)
Now, assume that (6.25) holds. Because of (6.32) and of the invertibility of the matrix (6.17), we find that ξ given in (6.27) is the only solution to (6.33). Conversely, if B moves with a translational velocity of the type (6.27), in view of Lemma 6.26 there exists a field of the type (6.25) satisfying (6.33) for a uniquely determined α, and relating to ξ by (6.27) and (6.28). The proof in the case (6.26) is the same, and the result is proved. If B has rotational symmetry, the Stokes problem (6.8) admits a noteworthy class of solutions, namely, the potential-like solutions. Actually, let w = grad Φ + ξ , with Φ harmonic function such that lim grad Φ(y) = −ξ ,
|y|→∞
(6.34)
Φ = P2 Φ = P3 Φ, where Pi , i = 2, 3, are defined in (4.150). Clearly, w satisfies (6.8)1,2,4 with p = const. In view of (6.34)2, it is readily seen that also (6.8)6 is satisfied. Moreover, denoting by ΣR the surface of a sphere centered in B and of radius R sufficiently large, we find
T(w, p) · n = Σ
T(w, p) · n.
(6.35)
ΣR
Since any harmonic function Φ in D, having bounded first derivatives satisfies D σ Φ = O |y|−3 ,
|σ | = 2,
as |y| → ∞,
letting R → ∞ into (6.35), we then derive also the validity of (6.8)5. Potential-like solutions play an important role in the self-propulsion of ciliated microorganisms of prolate-spheroidal shape, and, in fact, they are used to calculate fundamental parameters related to this type of motions, such as velocity of propulsion and rate of energy dissipation [6,9–11,75]. It is of some interest, therefore, to compare potential-like solutions with solutions uniquely determined by boundary data in the control spaces T (B) and Tτ (B) (Theorem 6.19). For B a sphere, these two types of solutions coincide if either
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G.P. Galdi
no prescriptions are given for w∗ or if w∗ · n = 0. To show this, we denote by {r, θ, ϕ} a system of spherical coordinates and by {er , eθ , eϕ } the corresponding unit vectors. We have the following general result. T HEOREM 6.21. Let B be a sphere of radius 1. Then, the most general boundary velocity distribution w∗ which can move B with nonzero translational velocity directed along y1 , is given by w∗ = αe1 + V∗ ,
(6.36)
where α ∈ R − {0}, and where V∗ satisfies
Σ
V∗ =
Σ
V∗ × er = 0.
(6.37)
Moreover, if w∗ · n ≡ 0, the most general distribution is w∗ = β sin θ eθ + W∗ ,
(6.38)
where β ∈ R − {0}, and where W∗ satisfies W∗ · n ≡ 0 and (6.37). The corresponding translational velocity ξ = ξ e1 of B is given by ξ=
−α
in case (6.36),
2 3β
in case (6.38).
Finally, the solutions to (6.8), corresponding to the velocity distributions (6.36) and (6.38) with V∗ = W∗ = 0 are potential-like. P ROOF. The first part of the theorem follows from (6.9) and from the well-known fact that the drag on a sphere of radius a, translating with a unit velocity is given by −6πa. Let us next consider the field w = grad Φ. With the choice (a) Φ = ξy1 ;
1 a 3 (b) Φ = ξy1 1 + , 2 |y| we verify at once that (as we already noticed) w satisfies (6.8)1,2,4,5,6 with p = const, and that, moreover, w|Σ = ξ e1
(in case (a))
and 3 w|Σ = ξ sin θ eθ 2
(in case (b)).
The result then follows from Theorem 6.20.
On the motion of a rigid body in a viscous liquid
771
The case of B a sphere is a very special one. Actually, if B is a prolate spheroid with eccentricity e > 0, the following result holds. T HEOREM 6.22. Let B be a prolate spheroid with eccentricity e > 0, and unit semi-major axis. Assume that w∗ · n = 0 at Σ. Then, solutions of Theorem 6.19 are not potential-like. P ROOF. To verify these assertions, we need to evaluate g(1) for the prolate spheroid. It can be shown [38] that 1 A(e) cos θ n − sin θ s , (6.39) g(1) = √ 1 − e2 cos2 θ 1 − e2 where θ ∈ [0, π] and A(e) =
4e3 (1 + e2 ) ln 1+e 1−e − 2e
(6.40)
.
Let us now consider the potential-like velocity field w(p) whose normal component vanishes at the boundary Σ of the prolate spheroid. Such a solution is well-known, see, e.g., [87], pp. 422–423, and we have sin θ s w(p) |Σ = B(e)ξ0 √ 1 − e2 cos2 θ
(6.41)
with B(e) =
2e3 (e2 − 1) ln 1+e 1−e + 2e
(6.42)
.
Since w(p) |Σ is not proportional to (g(1) · s)s, we conclude that the solutions of Theorem 6.19 are not potential-like if B is a prolate spheroid with e > 0. In view of the use of potential-like solutions in the study of self-propulsion of ciliated micro-organism, it is of some interest to compare the propulsion velocity ξ (c) generated by a distribution of velocity w∗ belonging to the control space and that ξ (p) generated by a (p) (p) potential-like distribution w∗ . We shall do this in the case when both w∗ and w∗ have zero normal component at Σ. From (6.39) and (6.41) we find w∗ = C
sin θ s , 1 − e2 cos2 θ
(p)
w∗ = D √
sin θ s 1 − e2 cos2 θ
,
(6.43)
where C and D are constants that we fix in such a way that both ξ (c) and ξ (p) coincide with the value 2/3 for B a sphere of unit radius (see Theorem 6.21). For a prolate spheroid of unit semi-major axis we have dΣ = 2π 1 − e2 1 − e2 cos2 θ sin θ
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G.P. Galdi
Fig. 16. Variation of ξ (c) and ξ (p) with the eccentricity e.
Fig. 17. Variation of ξ (c) /ξ (p) with the eccentricity e.
and, moreover, see [57], p. 155, k = 4πA(e) with A(e) defined in (6.40). Thus, setting ξ (c) = ξ (c) e1 and ξ (p) = ξ (p) e1 , from (6.27), (6.39) and (6.43) we deduce ξ
(c)
C = 2
π 0
sin3 θ dθ, (1 − e2 cos2 θ )3/2
ξ
(p)
D = 2
π 0
sin3 θ dθ. 1 − e2 cos2 θ
Therefore, fixing C and D in such a way that ξ (c) , ξ (p) → 2/3 as e → 0, we find ξ (c) = 1/G(e),
ξ (p) = 1/B(e),
where G(e) =
e3 √ √ π/2 − e 1 − e2 − tan−1 ( 1 − e2 /e)
and B(e) is defined in (6.42). From Figures 16 and17 we see that the two propulsion velocities become different for e close to 1. In any case, the range of variability for each of them is not so large, since ξ (c) varies from 2/3 (e = 0, sphere) to π/2 (e = 1, needle), while ξ (p) varies from 2/3 (sphere) to 1 (needle). Therefore, the velocity of propulsion is not greatly altered by the shape of the prolate spheroid. This is in agreement with
On the motion of a rigid body in a viscous liquid
773
the observation that a wide variety of prolate-spheroidal-shaped ciliated micro-organisms occur in nature, see [75,11]; see also Figure 5. 6.1.3. Unsteady self-propelled motion and attainability of steady motion The problem of existence and uniqueness of self-propelled unsteady motions, that is, the solvability of (6.7)–(6.6) has been recently studied and solved in [107]. In particular, the author shows that every (sufficiently regular) velocity distribution on Σ generates a self-propelled unsteady motion. We shall limit ourselves here to quote the main result, referring to that paper for the elegant proof. To this end, let Hloc (0, T ; X), X a Banach space, be the space of functions from (0, T ) into X that are Hölder continuous on each compact set of (0, T ). Moreover, let 1 (0, T ; X) be the subspace of H (0, T ; X) constituted by functions having also first Hloc loc derivative in Hloc (0, T ; X). We have the following main result [107]. T HEOREM 6.23. Let B be of class C 2 and let 1 0, ∞; W 1/2,2(Σ) . w∗ ∈ Hloc 0, ∞; W 3/2,2(Σ) ∩ Hloc Then (6.7)–(6.6) has a unique solution {w, p, U} such that for all T > 0 ξ , ω ∈ C [0, T ] ∩ C 1 (0, T ] , w ∈ C [0, T ]; L2 (D) ∩ C 1 (0, T ]; L2 (D) ∩ C (0, T ]; W 2,2 (D) , grad p ∈ C (0, T ]; W 2,2 (D) . This result is quite general. However, it does not ensure that the body does move, that is, it does not ensure that U ≡ 0. We encountered (and solved) the same type of question in the steady case, where we showed that U ≡ 0 if and only if the velocity distribution at Σ has a non-zero component on the “control space”. In the remaining part of this section we would like to address this question within the framework of attainability of steady solutions. Specifically, let w0∗ (y) be a (sufficiently smooth) boundary velocity field with P(w∗ ) = 0. By Theorem 6.19 we know that there is a unique, corresponding steady self-propelled solution s0 ≡ {w0 , p0 , U0 }, with U0 ∈ R − 0. Let ψ = ψ(t) be a “ramping function”, that is, (i) ψ is a smooth, nondecreasing function of time only, 0, t 0, (ii) ψ(t) = 1, t t0 .
(6.44)
By Theorem 6.23 there is a unique unsteady self-propelled motion corresponding to the boundary data ψ(t)w0∗ . Denote by s ≡ {w, p, U} such a motion. One can then show that
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G.P. Galdi
s → s0 , as t → ∞, in suitable norms. Here we shall give a proof of this statement, referring to [107] for more detailed results. Actually, setting u = w − ψw0 , = ω − ψω 0 ,
p = p − ψ p0 ,
µ = ξ − ψξ 0 ,
V = U − ψU0 ,
we find that {u, p, V} satisfy the following initial-boundary value problem ⎫ ∂u = div T(u, p) − ψ w0 , ⎬ ∂t in D, ⎭ div u = 0, u(y, t) = 1 − ψ(t) w0∗ + V(y, t), (y, t) ∈ Σ × (0, ∞), lim u(y, t) = 0,
|y|→∞
(6.45)
dµ =− T(u, p) · n + mψ ξ 0 , dt Σ d =− y × T(u, p) · n + ψ I · ω0 , I· dt Σ m
with initial conditions u(y, 0) = µ(0) = (0) = 0.
(6.46)
We now recall that the velocity field w0 belongs to W 2,2 (D) and satisfies the following estimate (see [107], §4.2) w0 2,2 c |ξ 0 | + |ω0 | . In fact, this property is characteristic of steady self-propelled motions of bodies (see [35], §V.6), and does not hold in case when the body is moved by an external force, like in particle sedimentation, where w ∈ Lq (D) only for q > 3. With this in mind, we deduce that u and the corresponding pressure p satisfy the same regularity property satisfied by w, p in Theorem 6.23. Multiplying (6.45)1 by u, integrating over D and using (6.45)5,6 we find 1 d u(y, t)2 + mµ(t)2 + (t) · I · (t)2 = − D(u)2 , t > t0 . 2 dt D D Likewise, multiplying both sides of (6.45)1 by div T(u, p) and integrating over D we obtain 2 1 d D(u)2 = −m dµ − d · I · d − div T(u, p)2 , dt 2 dt D dt dt D
t > t0 .
On the motion of a rigid body in a viscous liquid
775
These two latter displayed equations imply, in particular, that D(t) 22 and m|µ(t)|2 + | (t) · I · (t)|2 are decreasing and summable in (t0 , ∞). Therefore, $ $ $D(t)$2 + mµ(t)2 + (t) · I · (t)2 2
$ 1 $ $D(t0 )$2 + mµ(t0 )2 + (t0 ) · I · (t0 )2 , 2 t
t > t0 ,
and the proof of attainability is completed.
6.2. The full nonlinear case In this and the next sections we would like to furnish a nonlinear counterpart of some of the results obtained for the Stokes approximation. In particular, we shall show, in the steady case, that if the boundary velocities have a nonzero projection in the control space T (B), and a vanishing total flux through Σ, then B moves by self-propelled motion, provided the Reynolds number is not too large. Moreover, different projections will produce different velocities for B. Concerning the unsteady case, the same fundamental remarks apply as in the problem of particle sedimentation. Actually, one can show that every (sufficiently regular) boundary velocity distribution produces a global solution to the problem. However, this solution is weak, in the sense of Leray and Hopf. In fact, it is not known if more regular solutions exist, even for small data. In the special situation of symmetric bodies (in the sense of Definition 4.3) one can prove that the weak solution is strong, and that it exists for all times, at least for small Reynolds number. Moreover, in such a case, all symmetric steady solutions are attainable. 6.2.1. Steady self-propelled motion: existence and asymptotic behavior In the case of a steady motion, Equations (6.1)–(6.5) reduce to the following ones : Re (w − U) · grad w + ω × w = w − grad p, div w = 0,
in D,
lim w(y) = 0,
(6.47) (6.48)
|y|→∞
w(y) = w∗ (y) + U(y), Re mω × ξ = −
y ∈ Σ,
T(w, p) · n − Re w(w − U) · n ,
(6.49) (6.50)
Σ
Re ω × (I · ω) = − Σ
y × T(w, p) · n − Re w(w − U) · n .
(6.51)
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G.P. Galdi
In this section we shall be interested in the resolution of the following general Problem P. Given w∗ in a suitable class of functions, determine a solution w, p, U to the system of Equations (6.47)–(6.51). We begin to give a weak formulation of Problem P. To this end, let ϕ be an arbitrary element from C(D) (see (4.72)). Multiplying, formally, both sides of (6.47)1 by ϕ, integrating by parts over D we obtain
ϕ¯ · T(w, p) · n − Re w(w − U) · n Σ
=2
D
D(w) : D(ϕ) − Re
D
(w − U) · grad ϕ · w + Re
D
ω × w · ϕ,
where ϕ¯ ≡ Φ 1 + Φ 2 × y is the rigid motion to which ϕ reduces on Σ. If we impose the self-propelling conditions (6.50), (6.51), the preceding relation reduces to 2
D
D(w) : D(ϕ)
= Re
D
(w − U) · grad ϕ · w − Re
D
ω×w·ϕ
+ Re Φ 1 · ξ × ω + Φ 2 · (I · ω) × ω for all ϕ ∈ C(D). We are thus led to the following definition. D EFINITION 6.2. A triple {w, ξ , ω}, is a weak solution to Problem P, if and only if (i) w ∈ H(D) (see (4.78)); (ii) w = w∗ + U at Σ (in the trace sense), where U ≡ ξ + ω × y; (iii) w, U satisfy (6.52). In order to construct a weak solution, we need a preparatory result. L EMMA 6.27. Let B be Lipschitz and let w∗ ∈ W 1/2,2 (Σ) with Φ≡ Σ
w∗ · n = 0.
Then, there exists a solenoidal extension V of w∗ to D such that: (i) V ∈ W 1,2 (D); (ii) There is δ > 0 such that V(y) = 0 for all |y| > δ; (iii) There is a positive constant c = c(B) such that V 1,2 c w∗ 1/2,2,Σ . Furthermore, for all u ∈ C(D), we have
(6.52)
On the motion of a rigid body in a viscous liquid
D
777
¯ · grad u · V − u2 × V · u V · grad u · u + (u − u)
$ $2 c1 w∗ 1/2,2,Σ $D(u)$2 ,
(6.53)
where u¯ ≡ u1 + u2 × y, and c1 = c1 (B). P ROOF. Since Φ = 0, we may find a solenoidal extension V of w∗ to D, satisfying (i), (ii) and (iii); see [35], Exercise III.3.5. Denote by I the left-hand side of (6.53). From Hölder inequality and inequality (4.74), we have $ $ I V 4,∆ 2$D(u)$2 u 4,∆ + |u1 | + |u2 | + |u2 | u 4/3,∆ , where ∆ is the bounded support of V. From (4.75) we have $ $ u 4/3,∆ + u 4,∆ C $D(u)$2
(6.54)
(6.55)
while, by the trace theorem and (iii) we have V 4,∆ c w∗ 1/2,2,Σ .
(6.56)
Therefore, using (6.54)–(6.56) and Lemma 3.4 we conclude $2 $ I c w∗ 1/2,2,Σ $D(u)$2 and the proof of the lemma is completed.
We are now in a position to show the following existence result for steady self-propelled motions. T HEOREM 6.24. Suppose B Lipschitz. Let w∗ ∈ W 1/2,2 (Σ) with w∗ · n = 0. Σ
There exists c = c(B) > 0 such that if Re w∗ 1/2,2,Σ < c, then Problem P admits at least one weak solution {w, ξ , ω}. Moreover, there exists C = C(B) > 0 such that $ $ $D(w)$ C w∗ 1/2,2,Σ Re w∗ 1/2,2,Σ + 1 . 2
(6.57)
P ROOF. The solution is constructed by the Galerkin method. Let {ϕ k }k∈N be the orthonormal set introduced in Lemma 4.12. We look for an “approximating” solution
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G.P. Galdi
wm = um + V, m ∈ N, where V is the solenoidal extension of Lemma 6.27, and um is defined by um =
m
cim ϕ i ,
i=1
with cim satisfying 2
D
D(um ) : D(ϕ k ) = −2
D
D(V) : D(ϕ k ) + Re
D
(um − Um ) · grad ϕ k · um
+ V · grad ϕ k · um + (um − Um ) · grad ϕ k · V + V · grad ϕ k · V − Re ωm × um · ϕ k + ω m × V · ϕ k D
+ Re Φ 1k · ξ m × ωm + Φ 2k · (I · ωm ) × ωm ,
(6.58)
where Um ≡ ξ m + ω m × y =
m
cim ϕ¯ i
i=1
and ϕ¯ i ≡ Φ 1i + Φ2i × y is the rigid motion associated to ϕ i in a neighborhood of Σ. To show the existence of a solution cim for all m ∈ N, it is enough to prove a uniform bound on the norm D(um ) 2 , see, e.g., [36], Lemma VIII.3.2. To this end, multiplying both sides of (6.58) by ckm and summing over k, we obtain $2 $ $ $ 2 D(um ) 2 = Re V · grad um · um + (um − Um ) · grad um · V D
+ Re −2
D
D
[V · grad um · V − ωm × V · um ]
D(um ) : D(V).
(6.59)
Employing the Schwarz inequality in this relation along with Lemma 6.27, we derive $ $ $ $2 $2 $ 2$D(um )$2 Re c w∗ 1/2,2,Σ $D(um )$2 + Re V 24 $D(um )$2 $ $ $ $ + 2$D(um )$2 $D(V)$2 .
(6.60)
By Lemma 6.27 and the embedding theorem, we have $ $ Re V 24 + $D(V)$2 c w∗ 1/2,2,Σ Re w∗ 1/2,2,Σ + 1 .
(6.61)
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Thus, by choosing Re c w∗ 1/2,2,Σ < 1, say, inequality (6.60) furnishes the desired bound on um , that is, $ $ $D(um )$ c v∗ 1/2,2,Σ Re w∗ 1/2,2,Σ + 1 . 2
(6.62)
Let D be any compact subset of D containing Σ. Then, using (4.74) and (4.75) we find $ $ um 1,2,D c$D(um )$2 .
(6.63)
Inequalities (6.62)–(6.63) imply in particular the existence of u ∈ H(D) such that (along a subsequence again denoted by {um }) D(um ) → D(u), um → u,
weakly in L2 (D),
(6.64)
strongly in Lq (D ), for all q ∈ [2, 6).
Also, by Lemma 4.9, there is U ∈ R such that Um → U = ξ + ω × y, $ $ u 6 c$D(u)$2 .
(6.65)
In view of (6.64), (6.65), we may pass to the limit m → ∞ in (6.58) for fixed k, and use standard arguments to show (u − U) · grad ϕ k · u D(u) : D(ϕ k ) = −2 D(V) : D(ϕ k ) + Re 2 D
D
D
+ V · grad ϕ k · u + u · grad ϕ k · V + V · grad ϕ k · V − Re [ ω × u · ϕ k + ω × V · ϕ k ]
D
+ Re Φ 1k · ξ × ω + Φ 2k · (I · ω) × ω .
(6.66)
However, from Lemma 4.12 we know that every ϕ ∈ C(D) can be approximated in C 1 (D) by linear combinations of the functions ϕ k . Therefore, using this fact and the properties of u, we may replace ϕ k by ϕ in (6.66), thus showing that the triple {w ≡ u + V, ξ , ω} is a weak solution to Problem P. Finally, estimate (6.57) follows from (6.62) and (6.61). The problem of asymptotic behavior of the velocity field of a weak solution presents exactly the same difficulty encountered in the case of free fall. However, by using the same methods employed in the proof of Theorem 4.6, we can show the following result. T HEOREM 6.25. Let {w, ξ , ω} be a weak solution to (6.47)–(6.51). Then lim w(y) = 0,
|y|→∞
uniformly pointwise.
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G.P. Galdi
6.2.2. Steady self-propelled motion: propelling boundary conditions The result proved in Theorem 6.24 does not ensure U = 0, that is, it does not ensure that B really moves. The fact that not every boundary data w∗ is able to “propel” B seems to be clear from an intuitive point of view, and can also be seen by the following example. Take Φ such that Φ = 0
in D,
∂Φ = f (y), ∂n
y ∈ Σ,
lim Φ(y) = 0,
|y|→∞
for a given f with Σ f = 0. We at once recognize that w = grad Φ, p = 12 (grad Φ)2 , and U = 0 satisfy (6.47)–(6.51) with w∗ = grad Φ|Σ . Actually, the pair {grad Φ, 12 (grad Φ)2 } verifies (6.47). Moreover, since, for any R > δ, T(w, p) · n − Re w∗ w∗ · n = T(w, p) · n − Re ww · n , (6.67) Σ
ΣR
and Σ
y × T(w, p) · n − Re w∗ w∗ · n
y × T(w, p) · n − Re ww · n ,
=
(6.68)
ΣR
taking into account that D σ Φ(y) = O(|y|−2−|σ | ), |σ | = 1, 2, we may let R → ∞ into (6.67), (6.68) to deduce that also conditions (6.50), (6.51) with ξ = ω = 0 are satisfied. In this section we shall show that a self-propelled motion does occur (U = 0) whenever w∗ has a nonzero orthogonal projection in the control space T (B) and Re is not “too large”. Moreover, boundary velocities having different projections on the space T (B) will generate different rigid motions for B. These results imply, in particular, the following one. Consider the map Q which to every element of T (B) assigns the corresponding nonzero rigid motion of B, and denote by R ⊆ R the range of Q. Then, our result implies, in particular, that for any U in R there exists one and only one w∗ ∈ T (B) which propels B with the velocity U, for sufficiently small λ. T HEOREM 6.26. Let the assumptions of Theorem 6.24 be satisfied. Suppose, in addition, that P(w∗ ) = 0, where P is the projection operator of L2 (Σ) onto the space T (B) defined in (6.22). Then, there exists C = C(B, w∗ ) > 0 such that if Re < C
(6.69)
the corresponding weak solution {w, ξ , ω} determined in Theorem 6.24 has either ξ or ω nonzero. In particular, if V = C · Θ −1 · W, we have 1 3 A · V − C · Θ −1 · W |ξ | A · V − C · Θ −1 · W 2 2
(6.70)
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781
while, if W = CT · K−1 · V, we have 1 B · W − CT · K−1 · V |ω| 2
3 B · W − CT · K−1 · V . 2
(6.71)
Here, V and W are the projections of w∗ into T (B) defined in (6.15), while the matrices A, B, C, K and Θ are given in (6.19) and (4.21). P ROOF. Let ΣR the surface of a ball centered at the origin and which contains Σ, and let DR be the intersection of D with such a ball. Multiplying (4.15) by w and integrating by parts over DR , we find
Σ
(w∗ + U) · g
(i)
=2
DR
D h(i) : D(w)
w · T h(i) , p(i) · n,
−
i = 1, 2, 3.
(6.72)
ΣR
As is well-known, [35], Chapter V, for sufficiently large |y| we have (i) (i) T h , p c|y|−2 and so with the help of Theorem 6.25, we find lim R→∞
ΣR
(i) (i) w · T h , p ·n c lim w(y) = 0. |y|→∞
From (6.72) we thus get
Σ
(w∗ + U) · g(i) = 2
D
D h(i) : D(w),
i = 1, 2, 3.
Denote by w0 , p0 , U0 ≡ ξ 0 + ω 0 × y the solution to (6.8) corresponding to the same data w∗ . From this latter displayed equation and from (6.10), we obtain
(U − U0 ) · g(i) = 2 Σ
D
D h(i) : D(w − w0 ),
i = 1, 2, 3.
(6.73)
By an entirely analogous argument which makes use of (4.16) and (6.12), we also obtain
(U − U0 ) · G(i) = 2 Σ
D
D H(i) : D(w − w0 ),
Setting ξ˜ = ξ − ξ 0 ,
ω = ω − ω0 ,
i = 1, 2, 3.
(6.74)
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G.P. Galdi
i = 2 F
D
D h(i) : D(w − w0 ),
Gi = 2
D
D H(i) : D(w − w0 ),
from (6.73), (6.74) we find − C · Θ −1 · ξ˜ = A · F G , . ω=B· G − CT · K−1 · F
(6.75)
We shall show the following property ∀η > 0, ∃Re0 = Re0 (η, B, w∗ ) > 0: Re < Re0 implies F + G < η.
(6.76)
If we temporarily give for granted the validity of (6.76), we can prove the theorem. Actually, denoting by σ = σ (B) an upper bound for the entries of the matrices A, B, CT · Θ −1 and C · K−1 , from (6.75) we obtain |ξ 0 | − cη |ξ | |ξ 0 | + cη, |ω0 | − cη |ω| |ω0 | + cη, where c = c(σ ). Thus, choosing cη = 12 min{|ξ 0 |, |ω0 |} the preceding relation furnishes 3 1 |ξ 0 | |ξ | |ξ 0 |, 2 2 1 3 |ω0 | |ω| |ω0 |. 2 2
(6.77)
Notice that η depends only on B and v∗ . Since, by assumption, P(w∗ ) = 0, from Theorem 6.19 we know that at least one of the two vectors ξ 0 , ω0 is nonzero and so, by (6.77), the same property holds for ξ and ω, which shows U = 0. Moreover, taking into account (6.18), inequalities (6.70) and (6.71) follow at once from (6.77). To show the theorem completely, it remains to prove (6.76). To this end, let {Ren }n∈N be a sequence of values of Re tending to zero and bounded above by a certain positive quantity Re. Let {wn }n∈N be the sequence of corresponding weak solutions constructed in Theorem 6.24. From (6.57), we know that there exists a positive constant C, depending only on B, w∗ , and Re such that $ $ $D(wn )$ C. 2
(6.78)
From Lemma 4.9, (4.74) and (4.75), we show that (6.78) implies |ξ n | + |ωn | + wn 1,2,D C
(6.79)
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783
for another constant C depending on B, w∗ , and Re. From (6.78), (6.79), and a well-known compactness theorem we deduce the existence of a field w ∈ H(D) such that (along a subsequence, at least) D(wn ) → D(w), wn → w,
weakly in L2 (D),
strongly in Lq (D ), for all q ∈ [2, 6).
(6.80)
We know write (6.52) for wn , Un and Ren , pass to the limit n → ∞ and use (6.79), (6.80) to show that w satisfies the following relation D(w) : D(ϕ) = 0 for all ϕ ∈ C(D). (6.81) D
We then conclude that w is a weak solution to the steady self-propelled motion in the Stokes approximation, corresponding to the data w∗ . It is easy to show that w = w0 . Actually, let u = w − w0 . From (6.8) and (6.81) we obtain D(u) : D(ϕ) = 0 for all ϕ ∈ C(D). (6.82) D
Since w0 ∈ H(D), we have u ∈ H(D) and, moreover, u = V ≡ V1 + V2 × y at Σ. From (6.16) it then follows that 0 = K · U1 + C · U 2 ,
(6.83)
0 = C T · U1 + Θ · U2
and, by Lemma 4.1, we conclude U ≡ 0. Thus, u ≡ 0 and uniqueness is proved. As a consequence, (6.80)1 is satisfied not only along a subsequence, but as long as Re → 0. This implies, in particular D h(i) : D(w − w0 ) → 0 D H(i) : D(w − w0 ) → 0 as Re → 0 D
D
and (6.76) is proved.
We shall finally show that boundary velocities having different projections on T (B) generate different rigid motions, at least for Re not too large. (1)
(2)
T HEOREM 6.27. Let w∗ , w∗ ∈ W 1/2,2 (Σ) and let {w(1), ξ (1) , ω(1)}, {w(2) , ξ (2), ω(2) } be the corresponding weak solutions constructed in Theorem 6.24. Then, there exists (2) Re0 (B, w(1) ∗ , w∗ ) > 0 such that if Re < Re0 = P w(2) implies U(1) = U(2) , P w(1) ∗ ∗ where U(i) = ξ (i) + ω(i) × y, i = 1, 2.
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G.P. Galdi (1)
(2)
P ROOF. Denote by U0 and U0 the rigid motions performed by B in the Stokes (1) (2) (1) (2) approximation and corresponding to w∗ and w∗ , respectively. We have U0 = U0 . Reasoning as in the proof of (6.73), (6.74) we prove for i = 1, 2, 3, k = 1, 2, (k) (i) U − U(k) · g = 2 D h(i) : D w(k) − w0 , 0 Σ D (6.84) (i) (k) (k) (i) = 2 D H − w U − U(k) · G : D w . 0 0 D
Σ
Assume U(1) = U(2) . Then, from (6.84) we deduce for i = 1, 2, 3 (1) (2) (i) U0 − U0 · g = 2 D h(i) : D w(2) − w0 − D w(1) − w0 , D
Σ
Σ
(1) (i) U0 − U(2) ·G =2 0
D
D H(i) : D w(2) − w0 − D w(1) − w0 ,
which, in view of Lemma 6.26 and (6.76), gives a contradiction.
6.2.3. Steady self-propelled motion: symmetric bodies We would like to specialize the results obtained in the previous two sections to the case when B is symmetric, in the sense of Definition 4.3, around the y1 -axis. This problem has been treated at some length in [37]. Here we shall limit ourselves to state some of the main results, referring the reader to that paper for proofs and technical details. Notice that, unlike Theorem 6.24, part (i) of the following one does not require zero total flux of the boundary velocity. We recall that the class C1 has been defined in (4.151). T HEOREM 6.28. Let B be symmetric around y1 , and of class C 2 . Then, the following conclusion holds (i) There is a constant C = C(B) > 0 such that, for any w∗ ∈ W 1/2,2 (Σ) ∩ C1 satisfying Re w∗ · n < C, Σ
Problem P admits at least one corresponding weak solution {w, ξ , 0}. If ξ = 0, then ξ is directed along y1 . (ii) If w∗ ∈ W 2−1/q,q (Σ) ∩ C1 for some q 3/2, P(w∗ ) = 0 and Re w∗ 2−1/q,q(Σ) < C then ξ = me1 , m = 0. In the case of a symmetric body we are also able to solve a problem, that seems difficult to address in the general case. Specifically, as in the linear case, we are able to show, at least for small R, that there is a kind of one-to-one correspondence between (translational)
On the motion of a rigid body in a viscous liquid
785
velocities of B and elements in the control space T (B) (see (6.22)). This result follows from the previous theorem (part (ii)) and from the following one. T HEOREM 6.29. Let B satisfy the assumptions of Theorem 6.28, and let ξ = me1 , m = 0, be given. Then, there exists a constant C = C(B) > 0, such that if Re |ξ | < C, the selfpropelled problem (6.47)–(6.51) has at least one solution with w∗ ∈ T . Moreover, let Sξ be the class of weak solutions to (6.47)–(6.51) corresponding to the given ξ , and such that (i) w∗ ∈ T(B); (i) maxi | Σ w∗ · g(i) | c0 |ξ |, for some c0 > 0. Then, there exists C1 = C1 (B, c0 ) > 0 such that if Re |ξ | < C1 , Sξ is constituted by only one element. 6.2.4. Unsteady self-propelled motion and attainability of steady motion As we already mentioned, the problem of unsteady self-propelled motions presents some general basic difficulties. Actually, only weak solutions are available, and also under the assumption that the boundary velocity has zero flux through the boundary Σ. It is not known if strong solutions exist, even for data of restricted size. The problem of existence of global weak solutions has been recently considered in [106]. In order to describe these results, in analogy with the free fall problem, we give a definition of weak solution. Multiplying formally (6.1) by the test function ϕ ∈ C(DT ) (see Section 4.2.7), integrating by parts over DT , and taking into account (6.6), we find −
T 0
∂ϕ ·w= D ∂t
T 0
Σ
+ Re
ϕ¯ · T(w, p) · n − Re w∗ (w∗ − U) · n
T D
0
−2
T
D
0
(w − U) · grad ϕ · w − ϕ · ω × w
D(ϕ) : D(w).
Imposing the self-propelling conditions, we then get −
T 0
∂ϕ ·w= D ∂t
T
m 0
dϕ dϕ ·ξ + ·I·ω dt dt
+ Re mϕ 1 · ξ × ω + ϕ 2 · (I · ω) × ω + Re
T 0
−2
T 0
D
D
(w − U) · grad ϕ · w − ϕ · ω × w
D(ϕ) : D(w).
(6.85)
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G.P. Galdi
Following [106], we shall say that the triple {w, ξ , ω} is a weak solution to (6.1)–(6.6) if and only if, for all T > 0, the following conditions hold: (i) w ∈ L2 (0, T ; H(D)) (see (4.78)); (ii) w = w∗ + U in Σ × (0, T ) (in the trace sense), where U ≡ ξ (t) + ω(t) × y, t ∈ [0, T ]; (iii) {w, ξ , ω} verifies (6.85), for all ϕ ∈ C(DT ). The following theorem holds, for whose proof we refer to [106], Theorem 4.3. 1
T HEOREM 6.30. Let B be of class C 2 . For any given w∗ ∈ W 1,2 (0, T ; W 2 ,2 (Σ)), all T > 0, verifying Σ
w∗ (y, t) · n dσ = 0,
for all t > 0,
there exists at least one weak solution to (6.1)–(6.6). The study of the regularity of weak solutions and of their asymptotic behavior in time is open in the general case. However, if B is symmetric (in the sense of Definition 4.3), and moves with a purely translational motion, both questions can be answered, at least in a certain class of boundary data and for small Reynolds number. In fact, we have the following result, for whose proof we refer to [106], Theorem 5.2. T HEOREM 6.31. Let B be a symmetric body of class C 2 . Let w∗ ∈ W 2−1/q,q (Σ) ∩ C1 , q > 3, with P(w∗ ) = 0, satisfy the assumptions of Theorem 6.28, and let {w0 , ξ 0 , 0} (ξ 0 = 0) be the corresponding solution. Finally, let ψ = ψ(t) be a ramping function (see (6.44)). Then, there is a positive constant K = K(B, q) such that if Re w∗ 2−1/q,q(Σ) < K all weak solution {w0 , ξ 0 , 0} to (6.1)–(6.6) with w in the class C1 , and corresponding to boundary value ψ(t)w∗ (y) satisfy the following regularity properties w ∈ L∞ 0, ∞; W 2,2(D) ,
dw ∈ L∞ 0, ∞; L2(D) . dt
Moreover, the following estimates hold for all t t0 $ $ $ ∂w $ dξ $ $ , (t) Ct −1/2 , (t) $ ∂t $ dt 2 $ $ $ $ $grad w(t)$ , ξ (t), $(w − w0 )(t)$ , (w − w0 )(t) Ct −1/4 , 2 ∞ 2,2 where C = C(B, w∗ ).
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Acknowledgments It is my pleasure to thank Professor D.D. Joseph for introducing me to the problem of sedimentation of particles and for showing me the many interesting related experiments performed in his laboratory. I also would like to thank Professor D. Serre and an anonymous reviewer for useful comments. Part of this work is the content of a Summer Course I gave at the International School “Navier–Stokes Equations and Related Topics”, held at the Instituto Superior Tecnico in Lisbon (Portugal), in the period June 29–July 2, 1999. I would like to take this opportunity to thank the Organizers of the School and, in particular, Professor Adelia Sequeira, for her wonderful hospitality and for the stimulating scientific atmosphere that she was able to create around the participants. I also would like to thank C.I.M. and Foundation C. Gulbenkian, for their generous financial support. This work was partially supported by the NSF grant DMS-0103970.
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[20] V. Coscia and G.P. Galdi, Existence of steady-state solutions of the equations of a fluid of second grade with non-homogeneous boundary conditions, Zap. Nauchn. Sem. POMI 243 (1997), 117–130. [21] R.G. Cox, The steady motion of a particle of arbitrary shape at small Reynolds numbers, J. Fluid Mech. 23 (1965), 625–643. [22] B. Desjardins and M. Esteban, Existence of weak solutions for the motion of rigid bodies in viscous fluid, Arch. Rational Mech. Anal. 46 (1999), 59–71. [23] B. Desjardins and M.J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models, Comm. Partial Differential Equations 25 (2000), 1399–1413. [24] S. Durr, Introduction to photmicrography, http://www.durr.demon.co.uk/index.html (2000). [25] B.U. Felderhof and R.B. Jones, Inertial effects in small-amplitude swimming of a finite body, Physica A 202 (1994), 94–118. [26] B.U. Felderhof and R.B. Jones, Small-amplitude swimming of a sphere, Physica A 202 (1994), 119–144. [27] R. Finn, On steady-state solutions of the Navier–Stokes partial differential equations, Arch. Rational Mech. Anal. 3 (1959), 381–396. [28] R. Finn, On the exterior stationary problem for the Navier–Stokes equations and associated perturbation problems, Arch. Rational Mech. Anal. 19 (1965), 363–406. [29] A. Fortes, D.D. Joseph and T.S. Lundgren, Nonlinear mechanics of fluidization of beds of spherical particles, J. Fluid Mech. 177 (1987), 467–483. [30] G.P. Galdi, On the Oseen boundary-value problem in exterior domains, Navier–Stokes Equations: Theory and Numerical Methods, J.G. Heywood, K. Masuda, R. Rautmann and V.A. Solonnikov, eds, Lecture Notes in Math., Vol. 1530, Springer (1991), 111–131. [31] G.P. Galdi, On the asymptotic structure of D-solutions to steady Navier–Stokes equations in exterior domains, Mathematical Problems Related to the Navier–Stokes Equation, G.P. Galdi, ed., Adv. Math. Appl. Sci., Vol. 11, World Scientific (1992), 81–105. [32] G.P. Galdi, Mathematical theory of second grade fluids, Stability and Wave Propagation in Fluids and Solids, G.P. Galdi, ed., CISM Courses and Lectures, No. 344, Springer (1995), 67–104. [33] G.P. Galdi, On the steady translational self-propelled motion of a body in a Navier–Stokes fluid, Proc. Conference Equadiff 9, Brno (Czech Republic), Electronic Publishing House, Stony Brook, NY (1998), 63–81. [34] G.P. Galdi, Slow motion of a body in a viscous incompressible fluid with application to particle sedimentation, Developments in Partial Differential Equations, Vol. 2, V.A. Solonnikov, ed., Quaderni di Matematica della II Università di Napoli (1998), 2–50. [35] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Linearised Steady Problems, Springer Tracts in Natural Philosophy, Vol. 38, 2nd corrected ed., Springer (1998). [36] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, Vol. 39, 2nd corrected ed., Springer (1998). [37] G.P. Galdi, On the steady, translational self-propelled motion of a symmetric body in a Navier–Stokes fluid, Classical Problems in Mechanics, Vol. 1, R. Russo, ed., Quaderni di Matematica della II Università di Napoli (1997), 98–173. [38] G.P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Rational Mech. Anal. 148 (1999), 53–88. [39] G.P. Galdi, Slow steady fall of a rigid body in a second-order fluid, J. Non-Newtonian Fluid Mech. 93 (2000), 169–177. [40] G.P. Galdi, An introduction to the Navier–Stokes initial-boundary value problem, Fundamental Directions in Mathematical Fluid Mechanics, Vol. 1, G.P. Galdi, J.G. Heywood and R. Rannacher, eds, Advances in Mathematical Fluid Mechanics, Birkhäuser (2000), 1–98. [41] G.P. Galdi, Asymptotic behavior of solutions to an exterior Navier–Stokes problem with rotation, in preparation (2001). [42] G.P. Galdi, J.G. Heywood and Y. Shibata, On the global existence and convergence to steady state of Navier–Stokes flow past an obstacle that is started from the rest, Arch. Rational Mech. Anal. 138 (1997), 307–319. [43] G.P. Galdi and P. Maremonti, Asymptotic spatial properties of weak solutions to the Stokes initial-value problem in exterior domains, in preparation.
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Author Index Roman numbers refer to pages on which the author (or his/her work) is mentioned. Italic numbers refer to reference pages. Numbers between brackets are the reference numbers. No distinction is made between first and co-author(s).
289 [26]; 289 [27]; 289 [28]; 289 [29]; 289 [30]; 289 [31]; 289 [32]; 289 [33]; 289 [34]; 289 [35]; 289 [36]; 289 [37]; 289 [38] Arnold, A. 198, 233, 289 [39] Arnold, V.I. 229, 289 [40] Arsen’ev, A. 178, 181, 289 [41] Arthur, M.D. 135, 289 [42] Asakura, F. 389, 417 [4] Asano, K. 119, 135, 201, 289 [43]; 289 [44]; 303 [404]; 304 [437]; 304 [438]
AbdElFattah, A.M. 518, 519, 531 [1] Abeyaratne, R. 388, 389, 417 [1]; 417 [2] Abrahamsson, F. 227, 244, 288 [1] Adams, R.A. 316, 317, 327, 329, 367 [1] Advani, A.S. 655, 787 [1] Aifantis, E.C. 378, 417 [3] Aizicovici, S. 355, 368 [30] Akhiezer, A.I. 559, 618, 647 [1] Alaev, R.D. 648 [23] Alberga, A.H. 52, 68 [141] Alexandre, R. 121, 122, 129, 133, 137, 138, 151, 165, 167, 169, 171, 173, 175, 176, 178–180, 182–184, 266, 288 [2]; 288 [3]; 288 [4]; 288 [5]; 288 [6]; 288 [7]; 288 [8]; 288 [9]; 288 [10]; 288 [11]; 288 [12]; 288 [13] Alfvén, H. 609, 647 [2] Alinhac, S. 441, 442, 531 [2]; 531 [3] Allegre, J. 62, 63 [1] Alsmeyer, H. 61, 63 [2] Amosov, A.A. 361, 367 [2]; 367 [3] Amundsen, N.R. 526, 541 [264] Andréasson, H. 86, 157, 288 [14]; 288 [15] Andries, P. 20, 52, 63 [3] Ané, C. 107, 207, 232, 286, 288 [16] Anile, A.M. 548, 569, 597, 602, 619, 626, 647 [3]; 647 [4]; 647 [5]; 647 [6]; 647 [7]; 651 [110] Antontsev, S.N. 312, 313, 361, 367 [4] Anzellotti, G. 531 [4] Aoki, K. 32, 33, 37, 63, 68 [139]; 68 [155]; 69 [163]; 113, 303 [408] Aris, R. 526, 541 [264] Aristov, V. 57, 63 [4] Arkeryd, L. 12, 63 [5]; 63 [6]; 85, 88, 89, 101, 103, 118, 133–135, 138, 139, 143, 150, 154, 161, 166, 201, 202, 235, 276, 288 [17]; 288 [18]; 288 [19]; 288 [20]; 288 [21]; 288 [22]; 289 [23]; 289 [24]; 289 [25];
Babenko, K.I. 704, 787 [2] Babin, A.V. 356, 367 [5] Babovsky, H. 60, 63 [7]; 63 [8] Bachynski, M. 121, 303 [405] Baiti, P. 479, 531 [5] Bakry, D. 207, 222, 231, 232, 289 [45] Balescu, R. 89, 93, 94, 113, 286, 289 [46]; 289 [47]; 289 [48] Balian, R. 286, 289 [49] Ball, J.M. 357, 367 [6]; 531 [6]; 531 [7] Bancel, D. 86, 289 [50] Bardi, M. 521, 531 [8] Bardos, C. 27, 28, 63 [9]; 112, 135, 138, 266, 290 [51]; 290 [52]; 290 [53]; 290 [54]; 290 [55]; 290 [56]; 290 [57]; 290 [58]; 531 [9] Barron, A.R. 255, 290 [59] Bärwinkel, K. 24, 63 [10] Bassanini, P. 42, 65 [59] Basset, A.B. 685, 692, 787 [3] Batchelor, G.K. 313, 367 [7] Beale, T. 88, 139, 290 [60]; 290 [61] Becker, H.A. 657, 787 [4] Bedjaoui, N. 389, 417 [5]; 417 [6] Beenakker, J.J.M. 49, 52, 67 [122]; 68 [141] Bellman, R. 591, 647 [8] 793
794
Author Index
Bellomo, N. 87, 136, 164, 290 [62]; 290 [63]; 290 [64]; 304 [425] Belotserkovskii, O.M. 59, 63 [11] Ben Abdallah, N. 86, 290 [65]; 290 [66] Benedetto, D. 109, 194, 195, 228, 233, 234, 273, 276–279, 290 [67]; 290 [68]; 290 [69]; 290 [70]; 290 [71]; 290 [72]; 290 [73] Benettin, G. 50, 63 [12]; 63 [13] Benzoni-Gavage, S. 383, 384, 417 [7]; 417 [8]; 418 [9]; 559, 571, 582, 647 [9]; 647 [10]; 647 [11] Berdichevskii, V. 378, 418 [10] Bereux, F. 532 [10] Berker, R. 741, 787 [5] Berry, C.J. 62, 67 [125] Bers, A. 84, 89, 93, 121, 294 [164] Berthelin, F. 206, 290 [74] Bethe, H.A. 577, 584, 648 [12] Betsadze, A.V. 490, 532 [11] Beylich, A.E. 62, 66 [92] Bhatnagar, P.L. 19, 64 [14] Bianchini, S. 381, 418 [11]; 478, 532 [12]; 532 [13] Bird, G.A. 50, 53, 57, 58, 61, 62, 64 [15]; 64 [16]; 64 [17]; 64 [18]; 64 [19]; 64 [20]; 64 [21]; 68 [128]; 69 [168] Bird, R.B. 32, 66 [86] Bisch, D. 62, 63 [1] Blachère, S. 107, 207, 232, 286, 288 [16] Blachman, N. 255, 290 [75] Blake, J.R. 659, 660, 769, 787 [6]; 787 [7] Blanchard, D. 138, 176, 290 [76]; 291 [77] Blanchard, R.C. 24, 64 [22] Blank, A. 519, 537 [168] Blokhin, A.M. 445, 532 [14]; 549–555, 568–572, 578, 582, 583, 586, 587, 593–598, 601–603, 607–609, 613, 615, 619, 620, 623, 624, 626, 628, 629, 634, 636–644, 646, 647, 647 [7]; 648 [13]; 648 [14]; 648 [15]; 648 [16]; 648 [17]; 648 [18]; 648 [19]; 648 [20]; 648 [21]; 648 [22]; 648 [23]; 648 [24]; 648 [25]; 648 [26]; 648 [27]; 648 [28]; 648 [29]; 648 [30]; 648 [31]; 648 [32]; 648 [33]; 648 [34]; 649 [35]; 649 [36]; 649 [37]; 649 [38]; 649 [39]; 649 [40]; 649 [41]; 649 [42]; 649 [43]; 649 [44] Bobylev, A. 32, 37, 68 [155]; 83, 87, 88, 101, 113, 115, 117, 128, 134, 135, 148, 150, 153, 192, 199, 211, 247, 252, 256, 259, 260, 268, 271, 274, 275, 277–279, 291 [78]; 291 [79]; 291 [80]; 291 [81]; 291 [82]; 291 [83]; 291 [84]; 291 [85]; 291 [86]; 291 [87]; 291 [88]; 291 [89]; 298 [279]; 301 [367]; 303 [408] Bogdonoff, S.M. 61, 66 [95]; 67 [123]
Bogoljubov, N. 93, 291 [90] Bogovskii, M.E. 337, 367 [8] Boillat, G. 553, 649 [45] Boldrighini, C. 99, 291 [91] Boltzmann, L. 47, 50, 64 [23]; 64 [24]; 64 [25]; 64 [26]; 81, 82, 84, 104, 107, 108, 209, 286, 291 [92]; 291 [93] Bonnetier, E. 532 [10] Bony, J. 87, 139, 277, 291 [94]; 291 [95] Borchers, W. 338, 367 [9] Borgnakke, C. 50, 64 [27] Borst, M. 655, 787 [8] Bouchut, F. 90, 137, 138, 144, 145, 157, 206, 237, 286, 290 [74]; 291 [96]; 291 [97]; 291 [98]; 291 [99]; 291 [100]; 521, 532 [15] Boudin, L. 137, 146, 157, 162, 291 [101] Bourgain, J. 99, 292 [102] Bourlioux, A. 529, 532 [16] Brauer, U. 443, 532 [17] Brenier, Y. 230, 292 [103] Brennen, C. 659, 769, 773, 787 [9]; 787 [10]; 787 [11] Brenner, H. 676, 678, 725, 735, 749, 763, 765–768, 772, 789 [57] Bresch, D. 662, 787 [12] Bressan, A. 381, 418 [11]; 470, 472, 476, 478, 479, 532 [12]; 532 [13]; 532 [18]; 532 [19]; 532 [20]; 532 [21]; 532 [22]; 532 [23]; 532 [24]; 532 [25] Brey, J. 278, 292 [104] Brio, M. 519, 520, 537 [156]; 543 [332] Broadwell, J.E. 62, 69 [168] Brunn, P. 655, 787 [13] Bryan, G.H. 48, 64 [28] Buet, C. 93, 115, 257, 292 [105]; 292 [106]; 292 [107] Bunimovich, L.A. 99, 291 [91] Buryak, O. 178, 181, 289 [41] Cabannes, H. 88, 117, 292 [108]; 292 [109]; 292 [110] Caflisch, R. 118, 135, 199, 202, 282, 284, 290 [51]; 290 [52]; 292 [111]; 292 [112]; 292 [113]; 292 [114]; 292 [115]; 532 [26] Caglioti, E. 109, 194, 195, 228, 233, 234, 273, 276–279, 290 [67]; 290 [68]; 290 [69]; 290 [70]; 290 [71]; 290 [72]; 292 [116] ˇ c, S. 521, 532 [27]; 532 [28] Cani´ Caprino, S. 88, 292 [117] Carleman, T. 103, 127, 133, 186, 201, 286, 287, 292 [118]; 292 [119] Carlen, E. 206, 207, 213–215, 218, 248, 252–257, 259, 270, 287, 292 [120]; 292 [121]; 292 [122];
Author Index 292 [123]; 292 [124]; 292 [125]; 292 [126]; 292 [127]; 292 [128] Carr, J. 378, 418 [12] Carrillo, J. 195, 198, 205, 228, 232, 233, 268, 273, 274, 275, 277–279, 290 [68]; 291 [86]; 293 [129]; 293 [130]; 293 [131] Carvalho, M. 213–215, 248, 254, 256, 257, 259, 270, 287, 292 [121]; 292 [122]; 292 [123]; 292 [126]; 292 [127] Casal, P. 378, 418 [13]; 418 [14] Castella, F. 110, 145, 281, 293 [132]; 293 [133]; 293 [134]; 293 [135] Caswell, B. 655, 787 [14] Cauchy, A.-L. v, vii [1] Celenligil, M.C. 62, 64 [29] Cercignani, C. 3, 9–12, 15, 18–20, 22–25, 27, 28, 31, 33–35, 37–39, 42–44, 47, 50–52, 57, 60, 62, 63, 63 [6]; 64 [30]; 64 [31]; 64 [32]; 64 [33]; 64 [34]; 64 [35]; 64 [36]; 64 [37]; 64 [38]; 64 [39]; 64 [40]; 64 [41]; 64 [42]; 64 [43]; 64 [44]; 65 [45]; 65 [46]; 65 [47]; 65 [48]; 65 [49]; 65 [50]; 65 [51]; 65 [52]; 65 [53]; 65 [54]; 65 [55]; 65 [56]; 65 [57]; 65 [58]; 65 [59]; 65 [60]; 65 [61]; 65 [62]; 69 [156]; 69 [157]; 69 [158]; 69 [159]; 69 [160]; 69 [163]; 78, 82, 85–89, 94–98, 100, 101, 103, 105, 107, 109, 113, 114, 117, 118, 120, 122, 132, 135, 137–139, 143, 172, 192, 196, 197, 211, 241, 259, 265, 268, 271, 273, 274, 277, 278, 286, 289 [27]; 289 [28]; 289 [29]; 289 [30]; 289 [31]; 289 [42]; 291 [80]; 291 [81]; 291 [87]; 293 [136]; 293 [137]; 293 [138]; 293 [139]; 293 [140]; 293 [141]; 293 [142]; 293 [143]; 293 [144]; 293 [145]; 293 [146]; 293 [147]; 293 [148]; 293 [149]; 293 [150]; 293 [151]; 293 [152]; 532 [29] Chafaï, D. 107, 207, 232, 286, 288 [16] Chandrasekhar, S. 87, 293 [153] Chanetz, B. 62, 65 [63] Chang, T. 446, 453, 458, 518, 519, 532 [30]; 532 [31]; 532 [32]; 532 [33] Chapman, J.S. 532 [34] Chapman, S. 32, 65 [64]; 78, 113, 280–282, 286, 293 [154] Chazarain, J. 565, 572, 649 [46] Chemin, J.-Y. 432, 443, 532 [35]; 532 [36] Chen, G.-Q. 381, 385, 418 [23]; 441, 442, 456, 458, 461, 468, 484, 486, 487, 491, 493, 497, 499, 502, 504, 505, 507–513, 515–519, 521, 522, 524–527, 529, 531, 532 [30]; 532 [31]; 532 [32]; 532 [37]; 533 [38]; 533 [39]; 533 [40]; 533 [41]; 533 [42]; 533 [43]; 533 [44]; 533 [45]; 533 [46]; 533 [47]; 533 [48]; 533 [49]; 533 [50]; 533 [51]; 533 [52]; 533 [53];
795
533 [54]; 533 [55]; 533 [56]; 533 [57]; 533 [58]; 533 [59]; 533 [60]; 533 [61]; 533 [62]; 533 [63]; 534 [64]; 534 [65]; 534 [66]; 535 [96]; 535 [97]; 542 [320] Chen, S.-X. 521, 534 [67]; 534 [68] Chen, S.Y. 61, 65 [65] Chen, X.-F. 385, 418 [15] Cheng, H.K. 61, 65 [65] Chepyzhov, V.V. 355, 367 [10] Chern, I.-L. 517, 534 [69] Chew, G.F. 555, 649 [47] Chiba, K. 656, 657, 787 [15] Childress, S. 659, 787 [16] Cho, K. 656, 661, 749, 787 [17] Cho, Y.I. 656, 661, 749, 787 [17] Choe, H.J. 320, 367 [11] Choquet-Bruhat, Y. 86, 289 [50] Chorin, A.J. 313, 367 [12]; 534 [70] Chow, W.L. 61, 65 [66]; 65 [67] Christiansen, E.B. 657, 790 [95] Chueh, K. 534 [71] Chun, C.H. 62, 68 [129] Chwang, A.T. 735, 787 [18] Coifman, R. 331, 367 [13]; 534 [72] Colella, P. 518, 520, 543 [323] Collins, J.P. 518, 541 [270] Colombo, R.M. 389, 418 [16] Conley, C. 534 [71] Constantin, P. 534 [73] Conway, E. 534 [74] Coquel, F. 524, 534 [75] Cordier, S. 93, 115, 257, 292 [105]; 292 [106]; 292 [107] Corli, A. 389, 418 [16]; 418 [17]; 418 [18]; 418 [19] Coron, F. 135, 293 [155] Coscia, V. 662, 787 [19]; 788 [20] Courant, R. 432, 446, 517, 534 [76]; 534 [77]; 550, 552, 649 [48] Cover, T. 107, 108, 207, 255, 260, 286, 293 [156]; 294 [165] Cowling, T. 24, 32, 65 [64]; 65 [68]; 78, 113, 280–282, 286, 293 [154]; 532 [34] Cox, R.G. 655, 788 [21] Crandall, M.G. 432, 534 [78] Crasta, G. 478, 532 [21] Curtiss, C.F. 32, 48, 66 [86]; 68 [132] Dacorogna, B. 534 [79] Dafermos, C.M. 332, 367 [14]; 385, 389–391, 399, 406, 414, 418 [20]; 418 [21]; 418 [22]; 436, 441, 442, 446, 453, 461, 463, 467, 468, 472, 478, 479, 481–484, 488, 513, 529, 531,
796
Author Index
533 [43]; 533 [44]; 534 [80]; 534 [81]; 534 [82]; 534 [83]; 534 [84]; 534 [85]; 534 [86]; 534 [87]; 534 [88]; 534 [89]; 534 [90]; 534 [91] Dahler, J.S. 48, 65 [69] Dahler, N.F. 48, 68 [146] Danchin, R. 318, 319, 321, 367 [15]; 367 [16] Daneri, A. 39, 42, 64 [44] Darboux, G. 535 [92] Darrozès, J.-S. 25, 35, 65 [70] Dautray, R. 86, 293 [157] de Boer, J. 135, 305 [454] de Groot, S. 86, 294 [158] De Masi, A. 27, 28, 66 [71]; 88, 135, 292 [117]; 294 [159] de Saint-Venant, B. v, vii [11] Deckelnick, K. 363, 367 [17] Decoster, A. 89, 94, 286, 294 [160] Degond, P. 86, 93, 115, 121, 135, 180, 266, 268, 281, 290 [65]; 292 [106]; 293 [133]; 294 [161]; 294 [162] Del Pino, M. 198, 233, 294 [163] Delcroix, J. 84, 89, 93, 121, 294 [164] Dembo, A. 107, 207, 255, 294 [165] Demutskij, V.P. 641, 643, 645, 646, 651 [103] Deschambault, R.L. 518, 519, 535 [93] Deshpande, M.V. 618, 619, 626, 646, 651 [90] Deshpande, S.M. 59, 66 [72] Desjardins, B. 348, 359, 365, 366, 367 [18]; 367 [19]; 368 [20]; 655, 788 [22]; 788 [23] Desvillettes, L. 84, 86, 99, 110, 114, 115, 118, 122, 124, 129, 133, 137, 138, 144, 146, 147, 149, 151, 153, 157, 162, 165, 169, 171, 173, 175, 178, 180, 183, 187, 197, 209, 210, 212, 213, 224, 225, 237, 238, 240–243, 268, 278, 288 [10]; 290 [66]; 291 [98]; 291 [99]; 291 [101]; 294 [166]; 294 [167]; 294 [168]; 294 [169]; 294 [170]; 294 [171]; 294 [172]; 294 [173]; 294 [174]; 294 [175]; 294 [176]; 294 [177]; 294 [178]; 294 [179]; 294 [180]; 294 [181]; 294 [182]; 295 [183]; 295 [184]; 295 [185] Di Blasio, G. 133, 295 [186]; 295 [187] Di Meo, M. 135, 295 [188] Diaconis, P. 270, 295 [189] Ding, X. 381, 385, 418 [23]; 497, 499, 502, 504, 505, 521, 531, 535 [94]; 535 [95]; 535 [96]; 535 [97]; 535 [98] DiPerna, R. 90, 137, 138, 144, 154, 159, 160, 164, 176, 287, 295 [190]; 295 [191]; 295 [192]; 295 [193]; 295 [194]; 295 [195]; 320–324, 327, 332, 333, 368 [21]; 368 [22]; 368 [23]; 368 [24]; 381, 385, 418 [24]; 463, 464, 467, 468, 472, 481, 484, 487, 497, 505, 509, 512,
535 [99]; 535 [100]; 535 [101]; 535 [102]; 535 [103]; 535 [104]; 535 [105]; 535 [106] Dogra, V.K. 62, 68 [130]; 68 [131] Dolbeault, J. 86, 197, 198, 233, 237, 282, 285, 291 [100]; 294 [163]; 295 [196]; 295 [197] Dorovsky, V.N. 555, 569, 648 [24] Druzhinin, I.Yu. 569, 613, 615, 619, 626, 628, 629, 636, 637, 641–644, 648 [25]; 648 [26]; 648 [27]; 648 [28] Du, Y. 277, 295 [198] Dudy´nski, M. 86, 295 [199]; 295 [200] Dufty, J. 278, 292 [104] Dukes, P. 83, 88, 291 [88]; 298 [279] Dunn, J.E. 378–380, 418 [25] Durr, S. 660, 788 [24] Duvaut, G. 244, 295 [201] D’yakov, S.P. 548, 578, 580, 582, 584–586, 597, 649 [49] E, W. 521, 535 [107]; 535 [108]; 535 [111] Ebin, D.B. 312, 368 [25] Eden, A. 391, 419 [58] Egorushkin, S.A. 572, 619, 625, 647, 649 [50]; 649 [51] Ehrenfest, P. 105, 106, 287, 295 [202] Ehrenfest, T. 105, 106, 287, 295 [202] Ekiel-Je˙zewska, M.L. 86, 295 [199]; 295 [200] Ellis, R.S. 135, 295 [203] Elmroth, T. 133, 147–149, 295 [204] Emanuel, G. 52, 66 [73] Embid, P. 515, 535 [109] Emery, M. 207, 222, 231, 232, 289 [45] Engelberg, S. 442, 535 [110] England, J. 107, 300 [331] Engquist, B. 535 [111]; 535 [112] Enskog, D. 25, 31, 66 [74] Erd˝os, L. 86, 281, 293 [134]; 295 [205]; 295 [206] Ernst, M.H. 101, 295 [207] Erpenbeck, J.J. 529, 535 [113]; 548, 578, 580, 584, 585, 602, 649 [52] Erwin, D.A. 61, 68 [140] Escobedo, M. 86, 118, 123, 279, 281, 282, 284, 295 [208]; 296 [209]; 296 [210]; 296 [211] Esposito, R. 27, 28, 66 [71]; 133–135, 206, 235, 276, 289 [32]; 292 [124]; 293 [151]; 294 [159]; 295 [188]; 296 [212]; 296 [213]; 296 [214] Esteban, M. 655, 788 [22]; 788 [23] Euler, L. v, vii [2]; 535 [114] Evans, L.C. 332, 368 [26]; 527, 535 [115]; 535 [116] Fan, H.-T. 382, 383, 385, 386, 389–392, 418 [26]; 418 [27]; 418 [28]; 418 [29]; 418 [30]; 418 [31]
Author Index Feireisl, E. 327, 333, 335, 337, 339, 340, 342–344, 346, 347, 349, 351, 352, 354–357, 361, 368 [27]; 368 [28]; 368 [29]; 368 [30]; 368 [31]; 368 [32]; 368 [33]; 368 [34]; 368 [35]; 368 [36]; 368 [37]; 368 [38]; 368 [39] Felderhof, B.U. 379, 418 [32]; 659, 788 [25]; 788 [26] Feldman, M. 521, 533 [45] Feldstein, A. 689, 790 [91] Feng, J. 656, 662, 663, 750, 758, 789 [71]; 790 [73] Ferziger, J.H. 32, 66 [75] Fickett, W. 529, 535 [117] Filbet, F. 115, 292 [107]; 296 [215] Filippov, A.F. 479, 535 [118] Filippova, O.L. 619, 626, 646, 649 [53] Finn, R. 704, 788 [27]; 788 [28] Firsov, A.N. 135, 300 [333] Fisher, R. 254, 296 [216] Foias, C. 356, 368 [40] Fok, S.K. 515, 536 [119] Ford, G.W. 78, 286, 304 [433] Fortes, A. 656, 788 [29] Fougères, P. 107, 207, 232, 286, 288 [16] Fournier, N. 177, 178, 187, 296 [217]; 296 [218]; 296 [219]; 296 [220]; 296 [221]; 296 [222]; 296 [223] Fowles, G.R. 602, 649 [54] Francheteau, J. 424, 536 [120] Freeman, N.C. 548, 649 [55] Freistühler, H. 559, 560, 649 [56]; 649 [57]; 649 [58]; 649 [59] Frezzotti, A. 57, 63, 65 [45]; 65 [46] Frid, H. 456, 458, 468, 484, 486, 487, 505, 507, 509, 510, 512, 513, 517, 525, 526, 529, 533 [46]; 533 [47]; 533 [48]; 533 [49]; 533 [50]; 533 [51]; 533 [52] Fridlender, O.G. 32, 37, 67 [101] Friedland, S. 536 [121] Friedrichs, K. 432, 442, 446, 517, 534 [76]; 536 [122]; 536 [123]; 536 [124]; 550, 551, 553, 649 [60]; 650 [61] Frisch, H.L. 20, 67 [114] Frommlet, F. 281, 293 [134] Fujita, H. 318, 368 [41] Gabetta, E. 248, 252–254, 259, 261, 292 [126]; 292 [128]; 296 [224]; 296 [225] Galdi, G.P. 338, 368 [42]; 655, 661–665, 677, 679, 681, 682, 685–687, 693, 695, 696, 699–701, 704, 716, 717, 721, 723–725, 727, 736, 740, 741, 748, 750, 753, 755–760, 763, 766, 768, 771, 774, 777, 778, 781, 784, 787 [19];
797
788 [20]; 788 [30]; 788 [31]; 788 [32]; 788 [33]; 788 [34]; 788 [35]; 788 [36]; 788 [37]; 788 [38]; 788 [39]; 788 [40]; 788 [41]; 788 [42]; 788 [43]; 789 [44]; 789 [45]; 789 [46]; 789 [47]; 789 [48]; 789 [49] Galgani, L. 50, 63 [12]; 63 [13] Galkin, V.S. 32, 37, 67 [101] Gallavotti, G. 99, 286, 296 [226]; 296 [227] Gamba, I. 274, 275, 277–279, 291 [86]; 296 [228]; 519, 521, 536 [125]; 536 [126]; 536 [127] Garcia, A. 60, 62, 66 [76]; 66 [77]; 66 [78] Gardener, R. 385, 418 [33] Gardner, C.S. 548, 566, 581, 584, 606, 618, 619, 623, 624, 626, 650 [62] Gasser, I. 138, 296 [229] Gatignol, R. 87, 88, 296 [230] Génieys, S. 86, 290 [65]; 290 [66] Gentil, I. 107, 207, 232, 286, 288 [16] Gérard, P. 137, 287, 296 [231]; 296 [232] Giesekus, H. 738, 789 [50] Gilbarg, D. 559, 650 [63] Gimelshein, S.F. 62, 66 [92]; 66 [93] Giorgilli, A. 50, 63 [12]; 63 [13] Giurin, M.C. 62, 69 [160] Glass, I.I. 518, 519, 535 [93] Glassey, R.T. 86, 90, 286, 296 [233]; 297 [234]; 297 [235]; 442, 536 [128] Glaz, H. 515, 518, 536 [129]; 541 [270] Glimm, J. 427, 428, 458, 461, 464, 467, 481, 483, 487, 515–519, 521, 533 [53]; 534 [69]; 536 [130]; 536 [131]; 536 [132]; 536 [133]; 536 [134]; 536 [135]; 536 [136] Goatin, P. 478, 532 [22]; 536 [137] Godlewski, E. 428, 536 [138] Godunov, S. 497, 536 [139]; 550, 553, 554, 562, 574, 575, 593–595, 604, 607, 650 [64]; 650 [65]; 650 [66]; 650 [67]; 650 [68] Goldberger, M.L. 555, 649 [47] Goldhirsch, I. 273, 275, 276, 297 [236]; 297 [237] Goldshtein, A. 278, 297 [238] Golse, F. 86, 99, 112, 135, 137, 138, 144–146, 180, 266, 273, 276, 287, 290 [53]; 290 [54]; 290 [55]; 290 [56]; 290 [57]; 290 [69]; 292 [102]; 293 [155]; 294 [176]; 297 [239]; 297 [240]; 297 [241]; 297 [242]; 297 [243]; 297 [244]; 297 [245]; 531 [9] Goodman, J. 381, 418 [34]; 515, 535 [109] Gordienko, V.M. 588, 604, 620, 650 [67]; 650 [69] Gosse, L. 526, 536 [140] Goudon, T. 119, 136, 166, 181, 297 [246]; 297 [247]; 297 [248] Gouin, H. 378, 418 [13]; 418 [14]; 418 [35] Grabacka, E. 659, 789 [52]
798
Author Index
Grad, H. 63, 66 [79]; 78, 95, 108, 110, 113, 120, 135, 156, 197, 236, 243, 244, 286, 287, 297 [249]; 297 [250]; 297 [251]; 297 [252]; 297 [253]; 297 [254]; 297 [255] Graham, C. 100, 114, 178, 269, 294 [177]; 297 [256]; 297 [257] Grassin, M. 436, 536 [141]; 536 [142] Gray, J. 659, 789 [51] Green, A.E. 364, 368 [43] Greenberg, J. 468, 483, 505, 536 [143]; 536 [144] Greenberg, W. 135, 297 [258] Grenier, E. 359, 368 [20]; 521, 536 [145] Grinfeld, M. 383, 384, 418 [36]; 418 [37] Gripenberg, G. 512, 537 [146] Grobbelaar-Van Dalsen, M. 655, 789 [53] Gronwall, T. 103, 297 [259]; 297 [260] Gropengießer, F. 60, 66 [80] Grosfils, P. 63, 65 [46] Gross, E.P. 19, 64 [14] Gross, L. 207, 297 [261] Grossman, P.D. 655, 656, 789 [54] Grünbaum, F.A. 253, 297 [262] Gu, C. 445, 446, 521, 537 [147]; 537 [148] Guiraud, J.-P. 25, 35, 65 [70] Gunzburger, M. 655, 789 [55] Guo, M.Z. 287, 298 [263] Guo, Y. 90, 298 [264]; 298 [265]; 298 [266]; 298 [267]; 298 [268]; 442, 537 [149]; 537 [150]; 543 [334] Gurtin, M. 378, 387, 418 [12]; 419 [38] Gustafsson, T. 87, 133, 150, 156, 161, 201, 290 [62]; 298 [269]; 298 [270]
Heywood, J.G. 661, 696, 704, 788 [42] Hicks, B. 57, 68 [136]; 69 [177] Hilbert, D. 25, 31, 35, 66 [85]; 432, 534 [77]; 550, 552, 649 [48] Hirschfelder, J.O. 32, 66 [86] Hishida, T. 696, 789 [59]; 789 [60] Hoff, D. 317, 318, 346, 357, 361, 363, 365, 367, 368 [45]; 369 [46]; 369 [47]; 369 [48]; 369 [49]; 369 [50]; 369 [51]; 369 [52]; 384, 419 [47] Hoffman, F. 606, 608, 650 [71] Hoffmann, K.H. 655, 789 [61]; 789 [62] Holway, L.H., Jr. 20, 66 [87] Horikawa, A. 656, 657, 787 [15] Hörmander, L. 537 [152] Hou, Z. 445, 446, 537 [148] Howard, P. 385, 420 [86]; 560, 652 [126] Howell, J.R. 57, 68 [149] Hrusa, W.J. 441, 537 [153] Hsiao, L. 361, 369 [53]; 385, 419 [48]; 441, 442, 446, 453, 458, 461, 519, 531, 532 [33]; 534 [89]; 534 [90]; 535 [95]; 537 [154] Hu, H. 656, 750, 758, 789 [63]; 790 [74] Hu, J. 470, 478, 479, 537 [155] Huang, A.B. 61, 66 [88]; 66 [89]; 66 [90] Huang, F. 521, 543 [326] Huber, C. 61, 65 [65] Hudjaev, S.I. 596, 652 [122] Hul, M. 659, 789 [64] Hunter, J. 519, 520, 537 [156]; 537 [157]; 542 [305]; 543 [332] Hurlbut, F. 24, 66 [91] Hwang, P.F. 61, 66 [89]; 66 [90] Hyung-Chun Lee 655, 789 [55]
Hagan, R. 382, 383, 419 [39]; 419 [40] Hale, J.K. 356, 368 [44] Hamdache, K. 132, 138, 298 [271] Hames, B.D. 655, 789 [56] Hanson, F.B. 20, 35, 44, 66 [81] Happel, V. 676, 678, 725, 735, 749, 763, 765–768, 772, 789 [57] Harabetian, E. 519, 537 [151] Harbour, P.J. 61, 66 [82] Harten, A. 650 [70] Harvey, J.K. 62, 66 [83] Hash, D.B. 58, 62, 66 [84]; 68 [131] Hassan, H.A. 58, 66 [84] Hattori, H. 390, 391, 419 [41]; 419 [42]; 419 [43]; 419 [44]; 419 [45]; 419 [46] Heintz, A. 85, 289 [33]; 298 [272]; 298 [273] Helfand, E. 20, 67 [114] Henderson, L.F. 518, 519, 531 [1] Herrero, M.A. 282, 295 [208] Hesla, T. 655, 789 [58]
Ikenberry, E. 113, 134, 148, 153, 201, 204, 298 [274] Illner, R. 9, 18, 24, 60, 63 [8]; 65 [47]; 78, 82, 83, 87, 88, 96–98, 101, 103, 105, 107, 109, 110, 114, 132, 136, 143, 162, 286, 289 [31]; 291 [88]; 293 [149]; 298 [275]; 298 [276]; 298 [277]; 298 [278]; 298 [279]; 302 [384] Imai, K. 135, 301 [359] Iordanskii, S.V. 548, 580, 586, 609, 650 [72]; 650 [73] Isaacson, E.L. 561, 650 [74] Ivanov, M.S. 62, 66 [92]; 66 [93] Ivanov, M.Ya. 594, 650 [68] Iwashita, H. 696, 789 [65] Jabin, P.E. 138, 296 [229] James, F. 138, 298 [280]; 521, 532 [15] James, R.D. 385, 419 [49] Janenko, N.N. 549, 555, 559, 560, 577, 609, 651 [107]
Author Index Janvresse, E. 270, 298 [281] Jeans, J.H. 48, 66 [94] Jeffery, G.B. 655, 789 [66] Jeffrey, A. 549, 550, 552, 555, 560, 576, 650 [75] Jenssen, H.K. 537 [158] Jerome, J.W. 533 [55] Jiang, S. 346, 361, 367, 369 [54]; 369 [55] Jin, B.J. 320, 367 [11] Jin, S. 113, 298 [282]; 529, 537 [159]; 537 [160] John, F. 436, 438, 439, 441, 442, 537 [161]; 537 [162] Johnston, T. 121, 303 [405] Jones, R.B. 659, 788 [25]; 788 [26] Joseph, D.D. 656–658, 662, 663, 670, 738, 749, 750, 758, 759, 788 [29]; 789 [67]; 789 [68]; 789 [69]; 789 [70]; 789 [71]; 790 [72]; 790 [73]; 790 [74]; 790 [86] Joss, W.W. 61, 66 [95] Jüngel, A. 198, 233, 293 [129] Kac, M. 82, 87, 100, 105–107, 265, 269, 270, 287, 298 [283]; 298 [284] Kadanoff, L. 277, 295 [198] Kan, P.-T. 533 [56] Kaniel, S. 136, 298 [285] Kaper, H.G. 32, 66 [75] Kato, T. 318, 368 [41]; 432, 442, 537 [163]; 537 [164]; 537 [165]; 552, 570, 596, 650 [76] Kawashima, S. 135, 201, 298 [286]; 432, 442, 537 [166]; 539 [226] Kazhikhov, A.V. 312, 313, 344, 345, 361, 367 [4]; 369 [56]; 371 [107] Keizer, J. 537 [167] Keller, J. 441, 519, 537 [157]; 537 [168]; 537 [169] Keller, S.R. 659, 660, 769, 773, 790 [75] Kersch, A. 57, 58, 66 [96]; 66 [97] Keyfitz, B.L. 385, 391, 419 [50]; 419 [51]; 419 [53]; 467, 521, 532 [27]; 532 [28]; 540 [259] Khanin, K. 521, 535 [107] Khesin, B.A. 229, 289 [40] Khodja, M. 384, 419 [47] Kim, E.H. 521, 532 [28] Kim, S. 655, 790 [76] Kipnis, C. 287, 298 [287] Kirchhoff, G. 655, 790 [77] Klainerman, S. 436, 441, 537 [170] Klaus, M. 118, 135, 298 [288] Klingenberg, C. 517, 536 [132] Knowles, J. 388, 389, 417 [1]; 417 [2] Knudsen, M. 24, 42, 52, 66 [98]; 66 [99] Ko, J. 655, 790 [84]
799
Kobayashi, T. 363, 369 [57]; 369 [58]; 696, 790 [78] Kogan, M. 24, 32, 37, 67 [100]; 67 [101]; 78, 101, 298 [289] Köhler, W.E. 49, 67 [122] Kompaneets, A. 282, 298 [290] Kontorovich, V.M. 548, 580, 583, 586, 597, 602, 650 [77]; 650 [78] Korteweg, D.J. 378, 379, 419 [52] Kosinski, W. 443, 537 [171] Kot, S.S. 61, 67 [102] Koura, K. 58–60, 62, 67 [103]; 67 [104]; 67 [105]; 67 [106]; 67 [107] Kozono, H. 696, 790 [80] Krajko, A. 594, 650 [68] Kranzer, H.C. 391, 419 [53]; 521, 536 [136] Kreiss, H.-O. 547, 549, 557, 565, 567, 569, 570, 572, 650 [79] Krizanic, F. 24, 67 [112] Krook, M. 19, 64 [14] Kruger, C.H., Jr. 542 [311] Kruskal, M.D. 548, 566, 581, 584, 606, 618, 619, 623, 624, 626, 650 [62] Kruzhkov, S. 537 [172] Krymskikh, D.A. 554, 569, 648 [29]; 648 [30] Kufner, A. 324, 370 [85] Kulikovskii, A.G. 559, 606, 608, 609, 611, 612, 619, 625, 649 [51]; 650 [80]; 650 [81] Kurganov, A. 518, 537 [173] Kurtz, D.S. 713, 790 [79] Kurtz, T.G. 527, 537 [174] Kušˇcer, I. 22–24, 49, 50, 67 [108]; 67 [109]; 67 [110]; 67 [111]; 67 [112]; 67 [122] Ladyzhenskaja, O.A. 650 [82] Lampis, M.C. 23, 24, 34, 47, 51, 52, 60, 65 [48]; 65 [49]; 65 [50]; 65 [51]; 65 [52]; 65 [53]; 65 [54]; 65 [55]; 65 [56]; 65 [57]; 105, 293 [150] Landau, L. 91, 116, 299 [291]; 555, 558, 574, 576–578, 597, 606, 608, 609, 613, 634, 640, 643, 644, 650 [83]; 650 [84]; 650 [85] Landim, C. 287, 298 [287] Lanford, O., III 9, 19, 67 [113]; 96, 287, 299 [292] Lankaster, P. 650 [86] Larsen, P.S. 50, 64 [27] Lax, P. 432, 433, 436, 438, 441, 448, 464, 467, 481, 483, 497, 518, 536 [124]; 536 [133]; 537 [175]; 538 [176]; 538 [177]; 538 [178]; 538 [179]; 538 [180]; 538 [181]; 538 [182]; 538 [183]; 538 [184]; 552, 553, 560, 593, 650 [61]; 650 [87]; 650 [88]; 651 [89] Le Tallec, P. 20, 52, 63 [3]
800
Author Index
Leal, L.G. 655–658, 749, 759, 790 [81]; 790 [82]; 790 [83] Lebowitz, J.L. 20, 27, 28, 66 [71]; 67 [114]; 82, 105, 135, 206, 286, 292 [124]; 294 [159]; 296 [212]; 296 [213]; 296 [214]; 299 [293] Ledoux, M. 222, 261, 299 [294]; 299 [295] Lee, C.-H. 391, 392, 419 [54] Lee, H.I. 529, 538 [185] Lee, S.C. 655, 790 [84] LeFloch, P. 388, 389, 417 [5]; 417 [6]; 419 [55]; 470, 478, 479, 491, 497, 505, 521, 531 [5]; 532 [10]; 532 [23]; 533 [57]; 533 [58]; 533 [59]; 536 [137]; 537 [155]; 538 [186]; 538 [187] Legge, H. 62, 67 [115]; 67 [116] Lemoine, J. 662, 787 [12] Lemou, M. 93, 115, 135, 268, 282, 292 [106]; 294 [161]; 299 [296]; 299 [297]; 299 [298] Lentati, A. 24, 65 [54] Leray, J. v, vii [3]; vii [4]; 112, 299 [299]; 299 [300]; 299 [301] Lessen, M. 618, 619, 626, 646, 651 [90] LeVeque, R.J. 428, 538 [189] Levermore, C.D. 138, 266, 282, 284, 290 [53]; 290 [54]; 292 [114]; 299 [302]; 522, 524–527, 529, 531 [9]; 533 [61]; 537 [159]; 538 [188] Levermore, D. 112, 138, 266, 290 [55]; 290 [56]; 290 [57]; 297 [240]; 297 [241] Lewicka, M. 478, 479, 532 [24]; 538 [190]; 538 [191] Lewis, J.H. 61, 66 [82] Li, D. 391, 419 [45]; 445, 446, 537 [148] Li, H. 277, 295 [198] Li, J. 519, 538 [192] Li, T.-C. 535 [95] Li, T.-T. 432, 436, 445, 446, 521, 538 [193]; 538 [194]; 538 [195] Li, Y. 456, 484, 533 [52] Lichtenstein, L. v, vii [5] Lieb, E. 260, 299 [303] Lieberman, G.M. 521, 532 [27] Lien, W.-C. 521, 538 [199] Lifshitz, E.M. 78, 90, 93, 299 [304]; 555, 558, 574, 576–578, 597, 606, 608, 609, 613, 634, 640, 644, 650 [84]; 650 [85] Lighthill, M.J. 519, 538 [196] Lillicrap, D.C. 62, 67 [125] Lin, L.W. 436, 438, 441, 538 [197]; 538 [198] Lin, P.X. 512, 538 [200] Linnik, Y.V. 255, 299 [305] Linz, P. 688, 790 [85] Lions, J.-L. 86, 244, 293 [157]; 295 [201]; 327, 369 [59] Lions, P.-L. v, vii [6]; 90, 103, 109, 112, 131, 136–138, 144, 146, 147, 154, 156, 157, 159,
160, 164, 170, 176, 179, 180, 195, 201, 261, 285, 287, 295 [190]; 295 [191]; 295 [192]; 295 [193]; 295 [194]; 295 [195]; 297 [242]; 299 [306]; 299 [307]; 299 [308]; 299 [309]; 299 [310]; 299 [311]; 299 [312]; 299 [313]; 299 [314]; 300 [315]; 300 [316]; 300 [317]; 300 [318]; 300 [319]; 300 [320]; 300 [321]; 300 [322]; 300 [323]; 313, 320–324, 330–333, 337, 338, 340, 344, 346, 349, 350, 357–360, 368 [20]; 368 [23]; 369 [60]; 369 [61]; 369 [62]; 369 [63]; 369 [64]; 369 [65]; 428, 497, 505, 534 [72]; 538 [201]; 538 [202]; 538 [203] Liu, H. 442, 518, 521, 533 [54]; 535 [110] Liu, T.-P. 366, 369 [66]; 390, 419 [56]; 436, 438, 441, 461, 463, 464, 468, 470, 476, 478, 488, 515, 521, 522, 524–527, 529, 532 [25]; 533 [60]; 533 [61]; 536 [129]; 538 [199]; 538 [204]; 539 [205]; 539 [206]; 539 [207]; 539 [208]; 539 [209]; 539 [210]; 539 [211]; 539 [212]; 539 [213]; 539 [214]; 539 [215]; 539 [216]; 539 [217]; 559, 560, 649 [56]; 651 [91] Liu, X.-D. 518, 538 [184] Liu, Y.J. 656, 658, 662, 663, 738, 749, 750, 758, 759, 790 [72]; 790 [73]; 790 [74]; 790 [86] Liubarskii, G.Ia. 559, 618, 647 [1] Loitsyanskii, L.G. 771, 790 [87] Lord, R.G. 52, 61, 67 [117]; 68 [150] Lordi, J.A. 48, 67 [118] Lorentz, H.A. 47, 67 [119] Loss, M. 270, 292 [123] Low, F.E. 555, 649 [47] Loyalka, S.K. 4, 69 [175] Lozzi, A. 518, 519, 531 [1] Lu, X. 86, 110, 138, 148, 157, 164, 205, 279, 280, 284, 285, 300 [324]; 300 [325]; 300 [326]; 300 [327]; 300 [328] Lu, Y.-G. 526, 533 [62] Lucquin-Desreux, B. 121, 180, 266, 294 [162] Lundgren, T.S. 656, 788 [29] Luo, P. 381, 385, 418 [23]; 497, 499, 502, 504, 505, 531, 535 [96]; 535 [97] Luo, T. 361, 369 [53]; 539 [218] Luskin, M. 531, 535 [112]; 539 [219] Lyubimov, G.A. 559, 606, 608, 609, 611, 612, 650 [80]; 650 [81] MacCamy, R.C. 442, 539 [220] Majda, A. 27, 67 [120]; 327, 333, 368 [24]; 428, 430, 432, 436, 438, 441, 442, 444, 445, 515, 517–519, 529, 532 [16]; 535 [109]; 535 [112]; 536 [134]; 537 [170]; 539 [221]; 539 [222]; 539 [223]; 549–551, 555, 558, 560, 561, 565,
Author Index 566, 569–572, 578, 583, 587, 593, 595, 596, 647, 651 [92]; 651 [93]; 651 [94]; 651 [95]; 651 [96]; 651 [97]; 651 [106] Makino, T. 354, 370 [76]; 432, 442, 463, 515, 539 [224]; 539 [225]; 539 [226]; 540 [248] Málek, J. 313, 320, 351, 352, 369 [67]; 369 [68] Malrieu, F. 107, 207, 232, 234, 286, 288 [16]; 300 [329] Mamontov, A.E. 364, 369 [69]; 369 [70] Marcati, P. 539 [227] Marchesin, D. 559, 561, 650 [74]; 651 [113] Maremonti, P. 682, 686, 689, 696, 788 [43]; 790 [88]; 790 [89] Markelov, G.N. 62, 66 [93] Markowich, P. 198, 207, 232, 233, 289 [39]; 300 [330] Markowich, P.A. 89, 94, 281, 286, 293 [134]; 294 [160]; 539 [228] Markowich, P.P. 198, 233, 293 [129] Marra, R. 135, 206, 292 [124]; 293 [151]; 296 [212]; 296 [213]; 296 [214] Marsden, J.E. 313, 367 [12]; 534 [70] Marshall, G. 515, 536 [135] Martin, H. 655, 790 [90] Martin, N. 107, 300 [331] Maslova, N.B. 135, 300 [332]; 300 [333]; 300 [334] Masmoudi, N. 138, 300 [317]; 357–360, 368 [20]; 369 [64]; 369 [65] Mates, R.E. 48, 67 [118] Matsumoto, H. 58, 67 [107] Matsumura, A. 315, 316, 361, 369 [71]; 369 [72]; 370 [73]; 370 [74]; 537 [166] Matuš˚u-Neˇcasová, Š. 347, 354, 364, 368 [31]; 370 [75]; 370 [76] Mawhin, J. 395, 419 [57] Maxwell, J. 24, 37, 61, 67 [121]; 78, 82, 102, 286, 300 [335]; 300 [336]; 300 [337] Mazel, A. 521, 535 [107] Mc Court, F.R.W. 49, 67 [122] McBryan, O. 517, 534 [69]; 536 [132] McCann, R.J. 205, 230, 232, 233, 268, 293 [130]; 300 [338] McCann, R.J. 230, 231, 300 [339]; 300 [340] McCroskey, W.J. 61, 67 [123] McDougall, J.G. 61, 67 [123] McKean, H.J. 104, 222, 223, 232, 248, 253–255, 260, 287, 300 [341]; 300 [342] McNamara, S. 276, 301 [343] Meistermann, L. 656, 791 [113] Méléard, S. 100, 114, 177, 178, 269, 294 [177]; 296 [219]; 296 [220]; 296 [221]; 296 [222]; 296 [223]; 297 [256]; 297 [257]; 301 [344] Merazhov, I.Z. 569, 649 [41]; 649 [42]
801
Messaoudi, S.A. 441, 537 [153] Messiter, A.F. 61, 67 [124] Metcalf, S.C. 62, 67 [125] Métivier, G. 424, 445, 536 [120]; 539 [229]; 569, 570, 596, 651 [98]; 651 [99] Meyer, R.E. 313, 370 [77] Meyer, Y. 137, 144, 295 [195]; 331, 367 [13]; 534 [72] Miclo, L. 216, 301 [345] Milani, A.J. 391, 419 [58] Miller, D.M. 655, 787 [8] Miller, R.K. 689, 790 [91] Millikan, R.A. 42, 67 [126] Milne-Thomson, L.M. 659, 790 [92] Mischaikow, K. 384, 390, 419 [46]; 419 [59] Mischler, S. 86, 114, 118, 123, 132, 133, 136, 138, 147, 149, 150, 153, 157, 158, 161, 162, 279, 281, 282, 284, 294 [178]; 296 [209]; 296 [210]; 296 [211]; 301 [346]; 301 [347]; 301 [348]; 301 [349] Mishchenko, E.V. 555, 569, 597, 598, 602, 603, 648 [31]; 648 [32] Mizohata, K. 463, 515, 539 [224] Mizohata, S. 552, 588, 596, 604, 633, 651 [100] Mobly, R. 61, 65 [65] Mokrane, A. 569, 651 [101] Monakhov, V.N. 312, 313, 361, 367 [4] Morawetz, C.S. 521, 536 [126]; 539 [230]; 539 [231]; 539 [232]; 539 [233] Morgenstern, D. 129, 134, 301 [350]; 301 [351] Morokoff, W.J. 57, 58, 66 [96]; 66 [97] Morrey, C. 287, 301 [352]; 539 [234] Morse, T.F. 20, 35, 44, 66 [81]; 67 [127] Moser, J. 540 [235] Moss, J.N. 62, 64 [29]; 68 [128]; 68 [129]; 68 [130]; 68 [131] Možina, J. 24, 67 [112] Muckenfuss, C. 48, 68 [132] Müller, I. 540 [236] Muncaster, R. 78, 95, 101, 113, 117, 236, 286, 304 [430] Muntz, E.P. 61, 68 [140] Murat, F. 138, 176, 290 [76]; 291 [77]; 332, 370 [78]; 493, 505, 540 [237]; 540 [238]; 540 [239]; 540 [240] Murata, H. 249, 301 [353] Mustieles, F.-J. 86, 301 [354]; 301 [355] Nanbu, K. 59, 60, 62, 68 [133]; 68 [134]; 68 [144] Nash, J. vii [7]; 319, 370 [79] Natalini, R. 389, 419 [60]; 525, 526, 529, 539 [227]; 540 [241] Navier, C.L.M.H. v, vii [8]
802
Author Index
Neˇcas, J. 313, 320, 351, 352, 364, 369 [67]; 369 [68]; 370 [80]; 370 [81] Neˇcas, M. 364, 370 [82] Nelson, J. 750, 758, 790 [74] Neunzert, H. 90, 301 [356] Neunzert, N. 60, 66 [80] Nicolaenko, B. 135, 290 [51]; 290 [52]; 292 [115]; 301 [357]; 391, 419 [58]; 419 [61] Nishida, T. 135, 301 [358]; 301 [359]; 315, 316, 369 [72]; 370 [73]; 442, 461, 463, 537 [166]; 540 [242]; 540 [243]; 540 [244]; 540 [245]; 540 [246] Nocilla, S. 24, 68 [135] Nohel, J.A. 442, 529, 534 [91]; 540 [247] Noll, W. 671, 791 [115] Nordsiek, A. 57, 68 [136] Nouri, A. 86, 101, 135, 201, 289 [34]; 289 [35]; 289 [36]; 289 [37]; 289 [38]; 301 [360] Novotný, A. 343, 344, 346, 347, 354, 364, 368 [32]; 370 [75]; 370 [80]; 370 [81]; 370 [83]; 370 [84]; 662, 790 [93]; 790 [94] Ogawa, T. 696, 790 [80] Oguchi, H. 61, 68 [137] Ohwada, T. 32, 33, 68 [138]; 68 [139] Okada, M. 354, 370 [76]; 540 [248] Olaussen, K. 260, 301 [361] Oleinik, O. 540 [249] Olla, S. 113, 236, 301 [362]; 301 [363] Opic, B. 324, 370 [85] Oppenheim, A.K. 529, 540 [251] Osher, S. 521, 531 [8]; 540 [250]; 549, 565, 566, 651 [95] Osteen, R.M. 57, 69 [177] Otto, F. 194, 198, 204, 207, 229–233, 301 [364]; 301 [365]; 301 [366] Otto, S.R. 659, 787 [7] Ovsyannikov, L.V. 555, 576, 651 [102] Padula, M. 355, 363, 370 [86]; 370 [87] Paes-Leme, P. 559, 651 [113] Pagani, C.D. 39, 42, 65 [58]; 65 [59] Palczewski, A. 87, 136, 164, 290 [63]; 301 [367]; 302 [368] Panferov, V. 277, 296 [228] Pao, Y.P. 135, 302 [369] Papanicolaou, G.C. 287, 298 [263]; 532 [26] Pareschi, L. 115, 129, 130, 302 [370]; 302 [371]; 302 [372]; 302 [373]; 302 [374] Park, N.A. 656, 661, 749, 787 [17] Pedlosky, J. 540 [252] Pedregal, P. 327, 370 [88] Pego, R. 389, 391, 419 [62]; 419 [63]
Pence, T.J. 390, 420 [64] Peng, Y.-J. 138, 298 [280]; 463, 488, 540 [253] Penland, C. 62, 66 [78] Perlat, J.P. 20, 52, 63 [3] Perthame, B.T. 20, 52, 63 [3]; 89, 90, 94, 109, 110, 115, 132, 136–138, 144, 145, 162, 286, 293 [135]; 294 [160]; 296 [229]; 297 [242]; 297 [243]; 298 [280]; 300 [318]; 300 [319]; 300 [320]; 301 [348]; 302 [370]; 302 [375]; 302 [376]; 302 [377]; 302 [378]; 302 [379]; 302 [380]; 302 [381]; 428, 442, 497, 505, 524, 534 [75]; 538 [202]; 538 [203]; 540 [254]; 540 [255] Pettyjohn, E.S. 657, 790 [95] Petzeltová, H. 337, 340, 343, 344, 346, 347, 349, 351, 354, 355, 361, 368 [31]; 368 [32]; 368 [33]; 368 [34]; 368 [35]; 368 [36]; 368 [37]; 368 [38]; 368 [39] Pfaffelmoser, K. 90, 302 [382] Pham-Van-Diep, G.C. 61, 68 [140] Philips, P.S. 593, 651 [89] Piccoli, B. 478, 479, 531 [5]; 532 [21] Pileckas, K. 662, 790 [96]; 790 [97] Pinsky, M.A. 135, 295 [203]; 527, 540 [256] Piriou, A. 565, 572, 649 [46] Pitaevski˘ı, L.P. 78, 90, 93, 299 [304] Pitteri, M. 110, 302 [383] Płatkowski, T. 87, 88, 302 [384] Plohr, B.J. 515, 517, 534 [69]; 536 [132]; 536 [135]; 561, 650 [74] Poisson, S.D. v, vii [9]; 540 [257] Pokorný, M. 662, 663, 750, 755–758, 789 [49]; 790 [93]; 790 [98] Poletto, M. 656, 790 [73] Polewczak, J. 136, 302 [385] Polovin, R.V. 559, 618, 641, 643, 645, 646, 647 [1]; 651 [103] Poupaud, F. 86, 297 [244]; 301 [360]; 302 [386]; 302 [387]; 302 [388]; 451, 463, 521, 540 [258]; 540 [260] Povzner, A.J. 88, 132, 133, 147, 148, 302 [389] Pozio, A. 178, 288 [11] Prangsma, G.J. 52, 68 [141] Presutti, E. 88, 292 [117] Price, J.M. 62, 68 [129]; 68 [130]; 68 [131] Probstein, R.F. 61, 68 [148] Prokopov, G.P. 594, 650 [68] Proutière, A. 175, 302 [390] Pulvirenti, A. 133, 157, 186, 187, 250, 302 [391]; 302 [392]; 302 [393] Pulvirenti, M. 9, 18, 24, 60, 65 [47]; 68 [142]; 78, 82, 84, 88, 89, 96–100, 103, 105, 107, 109, 114, 132–134, 136, 143, 194, 195, 228, 233–235, 273, 276–279, 286, 289 [32]; 290 [68];
Author Index 290 [69]; 290 [70]; 290 [71]; 290 [72]; 290 [73]; 292 [117]; 293 [149]; 294 [179]; 298 [275]; 298 [276]; 302 [378]; 303 [394]; 303 [395]; 303 [396] Qin, Y. 357, 371 [112] Quastel, J. 113, 303 [397] Rabier, P. 701, 790 [99] Rajagopal, K.R. 309, 313, 371 [105]; 662, 789 [44] Ralston, F.V. 549, 565, 651 [104] Rammaha, M.A. 442, 540 [261]; 540 [262] Rascle, M. 451, 463, 468, 505, 510, 521, 526, 533 [63]; 536 [144]; 540 [258]; 540 [260] Rauch, J. 549, 565, 566, 570, 651 [105] Rault, D.F.G. 61, 62, 68 [143]; 69 [176] Raviart, P.-A. 428, 536 [138] Reddy, B.D. 655, 789 [45] Redwane, H. 138, 176, 291 [77] Reichelman, D. 62, 68 [144] Rein, G. 110, 298 [277] Rendall, A.D. 442, 541 [263] Rhee, H.K. 526, 541 [264] Rickwood, D. 655, 789 [56] Riemann, B. v, vii [10]; 541 [265] Ringeisen, E. 132, 303 [398] Ringhofer, C.A. 539 [228] Risebro, N.H. 472, 541 [266] Risken, H. 198, 241, 303 [399] Rivlin, R.S. 364, 368 [43] Rjasanow, S. 115, 291 [82]; 291 [89] Robbin, J.W. 536 [121] Roberto, C. 107, 207, 232, 286, 288 [16] Roco, M.C. 656, 791 [100] Rogers, R.C. 529, 540 [247] Rokhlenko, A. 206, 292 [124] Rokyta, M. 313, 320, 369 [68] Romano, V. 553, 554, 569, 649 [43]; 649 [44] Romanovski˘ı, Y.R. 135, 300 [334] Rosales, R. 519, 536 [127]; 542 [295]; 572, 647, 651 [96]; 651 [97]; 651 [106] Rousset, F. 571, 582, 647 [11] Roussinov, V. 62, 69 [160] Roytburd, V. 420 [65]; 529, 532 [16] Rozhdestvenskii, B.L. 549, 555, 559, 560, 577, 609, 651 [107] Rubin, S.G. 61, 68 [145] Rudman, S. 61, 68 [145] Ruggeri, T. 553, 555, 651 [108]; 651 [109] Ruggeri, Y. 540 [236] Russo, G. 115, 129, 302 [371]; 302 [372]; 302 [373]; 548, 597, 602, 619, 626, 647 [4]; 647 [5]; 647 [6]; 651 [110]
803
R˚užiˇcka, M. 313, 320, 369 [68] Rykov, Y. 521, 535 [108] Sablé-Tougeron, M. 389, 418 [19]; 572, 647, 651 [111] Saint-Raymond, L. 138, 144, 145, 206, 266, 297 [241]; 297 [245]; 303 [400]; 303 [401] Saloff-Coste, L. 270, 295 [189] Salvarani, F. 115, 294 [180] Sandler, S.I. 48, 68 [146] Santos, A. 278, 292 [104] Sather, N.F. 48, 65 [69] Sauer, N. 655, 789 [53]; 791 [101] Schaeffer, D.G. 541 [267]; 541 [268]; 559, 651 [113] Schaeffer, J. 90, 303 [402] Schamel, H. 94, 305 [465] Schauder, J. v, vii [4] Scheffer, G. 107, 207, 232, 286, 288 [16] Schippers, S. 24, 63 [10] Schmeiser, C. 86, 302 [388]; 539 [228] Schmidt, B. 61, 68 [147] Schneider, J. 87, 301 [367] Schochet, S. 463, 541 [269] Schulz-Rinne, C.W. 518, 541 [270] Schwartz, L. 541 [271] Sedov, L.I. 651 [112] Sell, G.R. 352, 370 [89] Semmes, S. 534 [72] Sentis, R. 137, 144, 297 [242]; 297 [243] Sequeira, A. 662, 760, 789 [46]; 789 [47]; 790 [94]; 790 [96]; 790 [97] Seregin, G.A. 655, 789 [55] Serre, D. 340, 361, 367, 369 [48]; 370 [90]; 370 [91]; 370 [92]; 389, 391, 419 [63]; 420 [68]; 436, 446, 453, 461, 463, 519, 526, 529, 536 [142]; 539 [218]; 541 [272]; 541 [273]; 541 [274]; 541 [275]; 541 [276]; 541 [277]; 560, 571, 582, 647, 647 [11]; 652 [127]; 655, 661, 662, 696, 697, 700, 701, 737, 791 [102] Serrin, J. 313, 370 [93]; 378–380, 382, 417 [3]; 418 [25]; 419 [39]; 420 [66]; 420 [67] Shapere, A. 659, 791 [103] Shapiro, A.H. 313, 370 [94] Shapiro, M. 278, 297 [238] Sharp, D. 517, 536 [132] Shearer, J. 512, 541 [278] Shearer, M. 382, 383, 385, 386, 420 [69]; 420 [70]; 420 [71]; 420 [72]; 541 [267]; 541 [268]; 559, 651 [113] Shelukhin, V.V. 361, 370 [95] Shen, W. 526, 541 [279]
804
Author Index
Sheng, W.-C. 521, 541 [280] Sherman, F.S. 24, 66 [91] Shibata, Y. 363, 369 [58]; 661, 696, 704, 788 [42]; 790 [78]; 791 [104] Shinbrot, M. 136, 162, 298 [278]; 298 [285] Shizuta, Y. 135, 201, 303 [403]; 303 [404] Shkarofsky, I. 121, 303 [405] Shorenstein, M. 61, 68 [148] Shu, C.-W. 387, 390, 420 [73]; 533 [55] Sideris, T.C. 436, 439–442, 541 [281]; 541 [282]; 541 [283]; 541 [284]; 541 [285]; 541 [286] Siegel, R. 57, 68 [149] Šilhavý, J. 364, 370 [82] Šilhavý, M. 313, 364, 370 [80]; 370 [81]; 370 [96] Silvestre, A.L. 663, 664, 773, 774, 785, 786, 791 [106]; 791 [107] Simader, C.G. 665, 791 [108] Simmons, R.S. 61, 68 [150] Simon, A. 178, 288 [11] Sinai, Y.G. 99, 291 [91]; 521, 535 [107]; 535 [108] Sirovich, L. 20, 44, 68 [151] Slemrod, M. 113, 298 [282]; 378, 382, 383, 385, 391–393, 400, 418 [12]; 418 [31]; 419 [40]; 420 [65]; 420 [74]; 420 [75]; 420 [76]; 420 [77]; 441, 442, 517, 541 [287]; 541 [288]; 541 [289] Smith, J. 458, 484, 541 [290] Smoller, J. 367, 369 [49]; 446, 453, 458, 461, 463, 484, 534 [71]; 534 [74]; 540 [245]; 540 [246]; 541 [291]; 542 [292] Snider, R.F. 47, 68 [152] Soane, D.S. 655, 656, 789 [54] Sobolev, S.L. 552, 596, 651 [114] Soffer, A. 218, 255, 292 [125] Sogge, C.D. 156, 303 [406] Sohr, H. 338, 367 [9]; 665, 791 [108] Sokovikov, I.G. 597, 648 [33] Solonnikov, V.A. 321, 370 [97]; 682, 689, 790 [89] Soloukhin, R.I. 529, 540 [251] Sone, Y. 32, 33, 37, 68 [138]; 68 [139]; 68 [153]; 68 [154]; 68 [155]; 69 [162]; 113, 286, 287, 303 [407]; 303 [408] Song, K. 656, 657, 787 [15] Souganidis, P.E. 137, 138, 144, 300 [321]; 300 [322]; 302 [379]; 432, 497, 505, 534 [78]; 538 [202] Spohn, H. 82, 90, 96, 99, 286, 303 [409]; 303 [410] Stam, A. 207, 217, 255, 303 [411] Starovoitov, J.S. 655, 790 [90] Starovoitov, V.N. 655, 789 [61]; 789 [62] Stefanov, S. 62, 65 [60]; 69 [156]; 69 [157]; 69 [158]; 69 [159]; 69 [160] Stein, E.M. 156, 303 [406]; 713, 791 [105] Stewart, D.S. 529, 538 [185]
Stewartson, K.O. 61, 69 [161] Stoke, J.J. 542 [293] Stokes, G. v, vii [12]; 655, 791 [109] Straškraba, I. 347, 354, 361, 368 [31]; 370 [83]; 370 [84]; 370 [98]; 370 [99]; 370 [100] Strauss, W.A. 86, 90, 297 [234]; 297 [235]; 298 [265]; 298 [266]; 298 [267]; 298 [268]; 442, 542 [294] Struckmeier, J. 52, 60, 62, 65 [55]; 65 [56]; 65 [57]; 66 [80]; 69 [160] Strumia, A. 553, 555, 651 [108]; 651 [109] Sugimoto, H. 32, 37, 68 [155]; 113, 303 [408] Sulem, C. 135, 293 [155] Sylvester, J. 536 [121] Syrovatskij, S.I. 548, 559, 577, 640, 643, 644, 651 [115]; 651 [116] Sznitman, A. 100, 150, 269, 303 [412] Tabak, E. 519, 536 [127]; 542 [295] Tadmor, E. 138, 300 [319]; 300 [320]; 424, 428, 442, 497, 504, 505, 518, 535 [110]; 537 [173]; 538 [203]; 542 [296]; 542 [297] Tahvildar-Zadeh, A.S. 442, 537 [150] Tait, P.G. 655, 791 [112] Takata, S. 33, 69 [162]; 113, 303 [408] Takeno, S. 515, 539 [225] Talay, D. 237, 240, 303 [413] Tan, D. 521, 536 [136]; 542 [298] Tanaka, H. 128, 177, 201, 249, 250, 259, 301 [353]; 303 [414]; 303 [415] Tanaka, S. 32, 37, 63, 68 [155]; 69 [163] Tang, S.Q. 389, 419 [60] Tangerman, F.M. 521, 536 [136] Tani, A. 321, 370 [101] Tartar, L. 87, 303 [416]; 303 [417]; 327, 332, 371 [102]; 371 [103]; 493, 542 [299]; 542 [300]; 542 [301] Taub, A.H. 599, 651 [117] Taylor, G.I. 657, 659, 660, 691, 695, 791 [110]; 791 [111] Tcheremissine, F.G. 57, 63 [4]; 69 [164]; 69 [165] Teller, E. 606, 608, 650 [71] Temam, R. 356, 368 [40]; 371 [104] Temple, B. 463, 488, 530, 531, 539 [219]; 542 [292]; 542 [302]; 542 [303]; 542 [304] Teng, Z. 543 [328] Tesdall, A. 520, 542 [305] Thomas, J. 107, 108, 207, 255, 260, 286, 293 [156]; 294 [165] Thomson, W. 655, 791 [112] Thorne, K.S. 599, 651 [118] Ting, L. 441, 537 [169] Tinland, B. 656, 791 [113]
Author Index Tironi, G. 20, 42, 65 [61]; 65 [62] Tolman, R.C. 47, 69 [166] Toro, E.F. 428, 542 [306] Toscani, G. 110, 115, 120, 129, 130, 133, 136, 151, 153, 157, 164, 198, 206, 210, 215, 217–219, 223–227, 232, 233, 243, 250–253, 255, 256, 259–261, 275, 289 [39]; 290 [63]; 290 [64]; 291 [83]; 292 [128]; 293 [129]; 293 [131]; 296 [225]; 300 [323]; 302 [368]; 302 [372]; 302 [373]; 302 [374]; 302 [391]; 303 [418]; 304 [419]; 304 [420]; 304 [421]; 304 [422]; 304 [423]; 304 [424]; 304 [425]; 304 [426]; 304 [427]; 304 [428]; 304 [429] Trainor, G.L. 655, 791 [114] Trakhinin, Yu.L. 445, 532 [14]; 553, 554, 569, 602, 609, 619, 620, 623, 624, 626, 634, 637–640, 646, 647 [7]; 648 [34]; 649 [35]; 649 [36]; 649 [37]; 649 [38]; 649 [39]; 649 [40]; 649 [41]; 649 [42]; 649 [43]; 649 [44]; 652 [119]; 652 [120] Trivisa, K. 478, 538 [191]; 542 [307] Truesdell, C. 78, 95, 101, 113, 117, 134, 148, 153, 201, 204, 236, 286, 298 [274]; 304 [430]; 309, 313, 371 [105]; 542 [308]; 671, 791 [115] Truskinovskii, L. 378, 384, 387, 418 [10]; 420 [78]; 420 [79]; 420 [80] Tucsnak, M. 655, 790 [90] Tupciev, V.A. 385, 391, 420 [81] Turcotte, D.L. 61, 67 [102] Tveito, A. 526, 529, 541 [279]; 542 [309] Tzavaras, A. 112, 138, 302 [380]; 304 [431]; 391, 420 [82]; 525, 526, 529, 536 [140]; 540 [247]; 542 [310] Uchiyama, K. 88, 304 [432] Uhlenbeck, G.E. 46, 47, 69 [167]; 69 [172]; 78, 135, 286, 304 [433]; 305 [454] Ukai, S. 27, 28, 63 [9]; 119, 135, 289 [44]; 290 [58]; 304 [434]; 304 [435]; 304 [436]; 304 [437]; 304 [438]; 432, 442, 463, 515, 539 [224]; 539 [226] Unterreiter, A. 198, 233, 289 [39]; 293 [129] Vaidya, A. 661–663, 696, 721, 727, 736, 738, 743, 746, 750, 755–758, 760, 789 [47]; 789 [48]; 789 [49]; 791 [116] Vaigant, V.A. 344, 345, 348, 371 [106]; 371 [107] Valli, A. 316, 321, 361, 370 [100]; 371 [108]; 371 [109] van der Mee, C.V.M. 135, 297 [258] van Leuween, W. 86, 294 [158] van Weert, C. 86, 294 [158] Varadhan, S.R.S. 113, 236, 287, 298 [263]; 301 [362]; 301 [363]; 304 [439]
805
Vas, I.E. 61, 66 [95] Velazquez, J.J.L. 282, 295 [208] Victory, H.D. 83, 88, 298 [279] Victory, H.D., Jr. 83, 88, 291 [88] Videman, J.H. 662, 789 [46]; 790 [94]; 790 [96]; 790 [97]; 791 [117] Vila, J.-P. 451, 463, 540 [258] Villani, C. 90, 93, 113, 116, 117, 119–122, 125, 129, 130, 133, 137, 138, 151, 153, 157, 160, 166, 167, 169, 171, 173, 175–180, 182–184, 187, 194, 198, 204–207, 209, 210, 215, 217–219, 223–227, 229–233, 237, 238, 240–243, 251, 256, 257, 259–261, 266–268, 271, 275, 277–279, 287, 288 [10]; 288 [12]; 288 [13]; 292 [116]; 293 [130]; 293 [152]; 294 [181]; 294 [182]; 295 [183]; 295 [184]; 296 [228]; 300 [330]; 301 [365]; 301 [366]; 302 [374]; 304 [426]; 304 [427]; 304 [428]; 304 [429]; 304 [440]; 304 [441]; 304 [442]; 304 [443]; 304 [444]; 304 [445]; 305 [446]; 305 [447]; 305 [448]; 305 [449]; 305 [450]; 305 [451]; 305 [452] Villat, H. 685, 692, 791 [118] Vincenti, W.G. 542 [311] Vinokur, M. 594, 652 [121] Vishik, M.I. 355, 356, 367 [5]; 367 [10] Vogenitz, F.W. 62, 69 [168] Volpert, A. 479, 542 [312]; 596, 652 [122] Volterra, V. 542 [313] Wagner, D. 458, 461, 529, 534 [64]; 542 [314] Wagner, W. 60, 68 [142]; 69 [169]; 89, 100, 303 [396]; 305 [453] Waldmann, L. 47, 69 [170]; 69 [171] Wang, C.-H. 531, 535 [95]; 543 [329]; 543 [330] Wang, D. 441–443, 515, 517, 533 [55]; 534 [65]; 534 [66]; 541 [286]; 542 [315]; 542 [316]; 542 [317]; 542 [318]; 542 [319]; 542 [320] Wang, W.-C. 543 [321] Wang, X.P. 385, 418 [15] Wang, Z. 521, 535 [98] Wang Chang, C.S. 46, 69 [172]; 135, 305 [454] Webb, G.M. 520, 543 [332] Weigant, V.A. 420 [83] Weill, G. 656, 791 [113] Weinberger, H.F. 655, 661, 675, 676, 678, 691, 697, 700, 718, 791 [119]; 791 [120]; 791 [121] Welander, P. 19, 69 [173] Wennberg, B. 99, 129, 133, 147–151, 153, 157, 158, 161, 171, 173, 178, 183, 186, 187, 202, 211, 213, 252, 259, 288 [10]; 292 [102]; 292 [127]; 295 [185]; 296 [225]; 301 [349]; 302 [392]; 302 [393]; 305 [455]; 305 [456];
806 305 [457]; 305 [458]; 305 [459]; 305 [460]; 305 [461]; 305 [462]; 305 [463] Werner, C. 58, 66 [97] Whitham, G.B. 391, 420 [84]; 517, 522, 526, 543 [322]; 548, 559, 597, 652 [123] Wilczek, F. 659, 791 [103] Wild, E. 134, 248, 305 [464] Williams, M.M.R. 4, 24, 69 [174]; 69 [175] Winet, H. 659, 769, 773, 787 [11] Winther, R. 526, 529, 541 [279]; 542 [309] Wolf, U. 94, 305 [465] Wolibner, W. v, vii [13] Wood, W.W. 529, 535 [117] Woodward, P. 518, 520, 543 [323] Woronowicz, M.S. 61, 69 [176] Wu, C.C. 559, 652 [124] Wu, T.Y. 659, 660, 735, 769, 773, 787 [18]; 790 [75] Wu, X.-M. 490, 543 [324] Xin, Z. 319, 366, 369 [66]; 371 [110]; 381, 418 [34]; 529, 537 [160]; 543 [321] Yanagi, S. 361, 370 [74] Yang, D.Y. 655, 790 [84] Yang, G.-J. 490, 543 [325] Yang, S. 518, 519, 532 [31]; 532 [32]; 538 [192] Yang, T. 366, 369 [66]; 470, 476, 478, 532 [25]; 539 [215]; 539 [216]; 539 [217]; 543 [327] Yang, X. 521, 543 [326] Yanitskii, V. 59, 63 [11] Yaniv, S. 517, 534 [69]; 536 [132] Yau, H.-T. 86, 112, 113, 281, 287, 295 [205]; 295 [206]; 301 [363]; 303 [397]; 305 [466]; 305 [467]
Author Index Yen, S.M. 57, 69 [177] Yi, Z. 332, 371 [111] Yih, C.-S. 685, 791 [122] Ying, L. 531, 543 [328]; 543 [329]; 543 [330] Yoshizawa, Y. 56, 69 [178] You, J.R. 655, 790 [84] Young, L.C. 543 [331] Young, R. 463, 542 [303]; 542 [304] Young, W. 276, 301 [343] Yu, W. 432, 445, 446, 537 [148]; 538 [195]
Zaag, H. 117, 267, 305 [468] Zabrodin, A.V. 594, 650 [68] Zaidel’, R.M. 548, 652 [125] Zajaczkowski, M. 316, 371 [109] Zakharian, A.R. 520, 543 [332] Zanetti, G. 273, 297 [237] Zavelani, M.B. 60, 68 [142] Zavelani Rossi, M.B. 89, 100, 303 [396] Zhang, P. 346, 369 [55] Zhang, T. 517–519, 521, 535 [95]; 538 [192]; 541 [280]; 542 [298]; 543 [334]; 543 [335]; 543 [336]; 543 [337] Zhang, Y.-Q. 521, 543 [333] Zheng, S. 357, 371 [112] Zheng, Y. 517, 518, 520, 543 [335]; 543 [336]; 543 [337]; 543 [338] Ziane, M. 357, 365, 369 [50]; 369 [51] Zlotnik, A.A. 361, 367 [3]; 371 [113] Zumbrun, K. 361, 363, 369 [52]; 385, 418 [33]; 420 [85]; 420 [86]; 560, 571, 582, 647, 647 [11]; 652 [126]; 652 [127]
Subject Index
a priori estimate, 316, 321, 326, 328, 335, 336, 343, 345, 365 abbreviations – DIT, 550 – FBSP, 556 – IBVP, 549 – LC, 549 – LSP, 556 – ULC, 549 absorption event, 57 absorption of radiation, 57 accommodation coefficients, 24 acoustic system, 579 activation energy, 53–55 added mass matrix, 685 admissibility criteria, 380, 381 adsorption, 23 adsorption time, 22 aerodynamic forces, 3 aerosol particles, 3, 4 aerosol reactors, 3 aerosol science, 4 aerosols, 4 aerospace, 3, 4 angle cutoff, 10 angular momentum conservation, 55 angular velocity, 47 Arrhenius formula, 53 artificial satellites, 33 asymmetric point of view, 125, 160, 172, 175 atomic oxygen, 53 attainability of steady free falls, 691 attractor, 352, 356, 357, 365 auxiliary fields, 676 Avogadro’s number, 53
bimolecular reaction, 53, 54 binary collisions, 79 Blachman–Stam inequality, 218, 255 – for Boltzmann’s operator, 255–256 blow-up, 267 blunt bodies, 62 Bobylev’s identity, 128 Bobylev’s lemma, 129, 257 bodies with fore-and-aft symmetry, 678 body force, 9, 33 Boltzmann constant, 17 Boltzmann equation, 3, 5, 8–10, 12, 15–18, 20, 22, 25–33, 36, 48, 54, 56–58, 60–62 Boltzmann gas, 31 Boltzmann inequality, 12, 17, 19 Boltzmann’s theorem, 212, 223 Boltzmann, L.E., 47, 50 Boltzmann–Bose equation, 281 Boltzmann–Compton equation, 281 Boltzmann–Fermi equation, 280 Boltzmann–Grad limit, 7, 30, 96 Boltzmann–Plancherel formula, 172–173 Bose condensation, 283, 285 Bose–Einstein distribution, 283 bosons, 281 bounce-back condition, 85 boundary, 15, 22, 40, 42 boundary conditions, 22, 34, 37, 38, 40, 52, 57, 60, 84–86, 138 boundary interactions, 58 boundary source, 41 boundary value problem, 37 Brenier’s theorem, 230 Broadwell model, 88 bulk velocity, 5, 14, 16, 19, 21, 26, 31, 34, 37 Burnett equation, 113
bacteria, 3 Balescu–Lenard equation, 93 barotropic, 311, 316, 321, 326, 328, 330, 333, 338, 344, 346, 349, 361 BGK model, 19, 20, 44, 48, 49, 61, 87, 132 bilinear Boltzmann operator, 125
cancellation lemma, 168–170 carbon whiskers, 3 Carleman equation, 88 Carleman’s representation, 127, 154, 157, 171 Carleman’s theorem, 133 807
808
Subject Index
Carlen and Carvalho’s theorem, 214, 222, 257 catalysts, 3 Cauchy problem, 100, 130–139, 141–187, 265– 423 Cauchy stress tensor, 666 center-of-mass system, 49, 53, 54 central force, 9, 47 central limit theorem, 251, 253–255, 261 ceramics, 3 Cercignani’s conjecture, 210–211, 215, 268 Cercignani–Lampis (CL) model, 24, 50, 52, 60 Cercignani–Lampis–Lord (CLL) model, 61 chaos, 80, 96–100 – one-sided, 99 Chapman–Enskog expansion, 31, 113, 135, 276 characteristics, 433 chemical bonds, 22 chemical plants, 3 chemical reactions, 4, 10, 43, 52, 53, 61, 62 chemically reacting flows, 57 chemistry, 53 classical mechanics, 5, 47, 50 classical solutions, 344 clouds, 3 clustering, 276 collision, 6, 7, 10, 43, 44, 46–49, 54, 58–60 collision energy, 53 collision frequency, 19, 20 collision integral, 11, 12, 18, 19, 48, 57 collision invariants, 12, 16, 40, 103 collision kernel, 81–84 – angular, 120 – for quantum models, 282 – kinetic, 119 – qualitative influence, 123–124, 148 collision mechanics, 58 collision model, 18, 19, 48 collision operator, 11, 36 – Balescu–Lenard, 93 – Boltzmann, 79–82, 124–130, 192 – Boltzmann–Bose, 281 – Boltzmann–Compton, 281 – Boltzmann–Fermi, 280 – Fokker–Planck, 86, 193, 194 – Fokker–Planck–Coulomb, 94 – for Maxwellian collision kernel, 247 – inelastic Boltzmann, 274 – Kac, 87 – Landau, 91, 121, 180, 193 – linearized Boltzmann, 134 – Rostoker, 94 collision pair, 59 collision term, 10–12, 17, 18, 28, 31, 44, 47, 48 collision-free dynamics, 58
combustion, 427 compact, 340, 352 compactness, 326, 328–332, 335, 336, 338, 343, 346, 504 compactness framework, 492 components of air, 43, 48, 52 compressibility conditions, 577 compressible and/or heat conducting fluids, 364 compressible flows, 37 compressible fluids, 318, 361, 423 compressible Navier–Stokes equations, 20, 31, 32, 319, 357 compressible viscous fluid, 311 compressiblescaling, 25 concentration, 53 concentration gradient, 3 confinement, 88, 195–196 conservation equations, 30 conservation form, 30 conservation laws, 103 constants – M, 579, 610 – M0 , 611, 613 – Mi , 610 – M∞ , 579, 614 – R, 579, 611 – γ , 576 – ρ, ˆ p, ˆ . . ., ρˆ∞ , pˆ∞ , . . . – – uˆ i , uˆ i∞ , 556 – a, 579, 615 – b, 583, 601 – c0 , 614 – hi , 610 – q, 611 – q∞ , 611 – r, 612 – w∞ , 614 – vectors – – M, 610 – – h, 610 constitutive relations, 309 contact discontinuities, 448 contact discontinuity in magnetohydrodynamics, 608 contact strong discontinuity in gas dynamics, 576 continuity equation, 27, 38 continuum, 43 continuum equations, 28, 29 continuum gas dynamics, 13 continuum limit, 25, 26, 30, 54 continuum mechanics, 15, 37, 61 continuum model, 3, 61 continuum regime, 31
Subject Index contracting metrics, 201, 249–254, 258 convection, 31 convexity, 126 correlation function, 274 corrugation stability of a strong discontinuity, 597 Couette flow, 37 Coulomb interaction, 83, 90, 114, 184, 266, 267 Coulomb logarithm, 121 Cramér–Rao inequality, 255 cross-section, 47, 58, 59, 62, 81, 82 – for momentum transfer, 120, 167 Csiszár–Kullback–Pinsker inequality, 204 cut-off (Grad’s angular), 84, 120, 124, 138, 153– 164 damping, 442 Debye length, 91, 114, 121, 180 Debye potential, 84, 90, 114, 179, 180 decay, 463, 479, 505 defect measure, 326, 327, 333, 341 deflection angle, 58 degrees of freedom, 10 delocalized collisions, 89 delta function, 25, 44 delta wing, 62 dense gas, 30 density, 3, 5, 10, 13, 16, 19, 20, 27, 31, 34, 37, 41 density parameter, 31 detonation wave, 529 diatomic gases, 43, 48 diatomic oxygen, 53 differentiation of Boltzmann’s operator, 125 diffuse reflection, 60 diffusion, 31 diffusion coefficient, 53, 58 diffusive models, 86, 199 DiPerna–Lions’ theorem, 137–139, 154, 160, 163 Direct Simulation Monte Carlo (DSMC), 48, 57, 58, 60–62 discontinuous solutions, 443, 446, 458, 479, 488, 513 discrete ordinate methods, 61 discrete velocities, 87, 139 discrete velocity models, 58 discrete vibrational levels, 50 dispersion, 110 displacement convexity, 230 dissipation, 28 dissipative boundary conditions, 573 dissipative solutions, 138, 164 dissociating molecule, 54 dissociation, 54 dissociation energy, 54 dissociation-recombination reaction, 54
809
distribution function, 5, 6, 10, 18, 22, 34, 43, 47, 52, 53, 61, 77–78 DIT (dissipative integrals techniques), 574 divergence formulation, 116 drag, 22, 34, 39, 42 dual estimates, 155–156 Duhamel formula, 157–159 dust, 3 effective mass, 661, 679 effective viscous flux, 318, 330, 338 elastic collisions, 48–50, 79 elastic cross-section, 62 elasticity, 512 electric power plants, 3 electronic states, 56 electrons, 53 elementary mechanics, 4 ellipsoidal statistical (ES) model, 20 emission coefficient, 57 emission of radiation, 57 empirical measure, 78, 97 endothermic reaction, 54 energy, 15, 18, 19, 34, 53, 60, 310, 316, 317, 328, 329, 334–336, 340, 344, 346, 349, 353, 357, 363 energy balance, 46, 54 energy conservation, 30, 43–45, 55, 60 energy density, 14 energy equation, 38 energy flow, 14, 15, 25 energy integral, 573 energy release, 54 energy states, 54 energy transfer, 50 Enskog equation, 89 – for granular media, 273–274 entropy, 18, 25, 97, 104–109, 130, 191 – for quantum models, 283–284 entropy dissipation, 104, 126, 130, 146, 154, 160, 164, 167, 170–172, 178, 183, 191–244 – for Kac’s master equation, 271 entropy dissipation methods, 203–244 entropy inequality, 426 entropy power, 207 entropy rate criteria, 389 entropy solution, 444, 479, 486, 488, 489 entropy–entropy dissipation inequality, 205 – Boltzmann equation, 210–223 – Fokker–Planck equation, 206–207 – Kac’s master equation, 271–272 – Landau equation, 208–210 entropy–entropy flux pair, 426, 489
810
Subject Index
environmental problems, 3 eternal solutions, 117, 251 Euler equations, 3, 25–28, 30–32, 37, 112, 423, 521 Euler–Poisson equations, 442, 517 evolutionarity condition, 558 evolutionary strong discontinuity, 558 excited levels, 56 exothermic reaction, 54 explicit solutions, 101, 134
free fall, unsteady – Navier–Stokes, 736 – Stokes approximation, 681 free transport, 79 free-molecular conditions, 37 free-molecular flow, 31 free-molecule flow, 33 free-molecule regime, 4 frontal collisions, 125
fast and slow magnetosonic velocities, 609 fast magnetohydrodynamical shock wave, 610 Fermi–Dirac distribution, 283 fermions, 280 finite energy weak solution, 334, 335, 336, 339, 340, 343, 344, 346–348, 349, 350, 351, 354, 356, 358, 360 finite ordinate schemes, 39 Finite-Pointset method, 60 Fisher information, 193, 207, 222, 253–258 flow domain, 60 flow field, 57, 58, 62 flow past a body, 27, 34 flow past a cylinder, 62 fluctuations, 60, 61 fluorescence, 57 flying vehicle, 3 fog, 3 Fokker–Planck equation, 20, 86, 94, 196, 197, 202, 237 Fourier transform, 128–130, 171–173, 175, 247, 251–253 Fourier–Laplace transforms , 565 –F – U, 565 frame-invariance, 671 free energy, 193, 197 free fall, 672 free fall, Navier–Stokes liquid, 674 free fall, second-order liquid, 737 free fall, steady, 673 – Navier–Stokes, asymptotic behavior, 704, 716 – Navier–Stokes, existence, 697 – second-order liquid at non-zero Reynolds number, 749 – second-order liquid at zero Reynolds number, 738 – second-order liquid, existence, 743 – second-order liquid, symmetric, 745 – Stokes approximation, 676 – weak solutions, 700 free fall, Stokes approximation, 675 free fall, translational steady – Navier–Stokes liquid, 717
Γ formula, 175–176 gain term, 81, 155, 157 gas, 4, 5, 9, 10, 14, 15, 18, 19, 22, 25 gas constant, 17 gas dynamical shock wave, 576 gas dynamics, 574 gas-surface interaction, 4, 22, 24, 33, 50, 54, 57 Gaussian tails, 150 generalized characteristics, 479 geometric structure, 513 Glimm functional, 461 Glimm scheme, 458 Glimm solutions, 458, 463, 468 global well-posedness, 432 Godunov scheme, 497 Grad’s number, 243 gradient flows, 109, 194, 198, 204, 228–235 granular materials, 87, 194, 197, 206, 233, 272– 279 gravity, 5, 9 grazing collisions, 10, 83, 121–123, 130, 181, 268 – for quantum models, 282 H theorem, 18, 20, 47, 104–109, 189–244, 258 H -functional, 104, 105, 107, 108 H -solutions, 160, 167 hard potentials, 119, 122, 123, 133, 148–150, 201, 209, 216, 268 hard spheres, 5, 9, 43, 58, 59, 83 heat conductivity, 20, 32, 52, 58 heat flow, 37, 38, 42, 46 heat fluxes, 3 heat of reaction, 55 heat transfer, 22, 34, 42 heat-flow, 5, 15, 16, 25 Heisenberg inequality, 255 helium, 61 Hermite polynomials, 135, 252 high altitude flight, 4, 52 high temperatures, 52 high-temperature air, 54 Hilbert expansion, 28, 31, 113 Hilbert method, 36
Subject Index Hilbert space, 21, 22 Hilbert’s sixth problem, 114 Hilbert, D., 31 homogeneous cooling states, 278–279 Hugoniot adiabat, 576 hydrodynamic limit, 111–114, 135, 138, 206 – for granular media, 276 – for quantum models, 282 hyperbolic–elliptic mixed type, 378–380 hypersonic flow, 61, 62 hypersonic speed, 52 hypoellipticity, 180 ideal fluid, 37 ill-posedness example of Hadamard type, 561 impact parameter, 58, 80 incomplete accommodation, 61 incompressible fluid, 27 industrial emissions, 3 inelastic Boltzmann operator, 274 inelastic collisions, 49, 50, 87, 273–279 inelastic scattering, 57 inertia tensor, 666 inertial motion, 58 inertial torque coefficient, 729, 735, 760 infinite energy, 117, 192 infinite entropy, 227 infinite mass, 131, 195 information theory, 254–257 initial conditions, 27, 36 initial data, 60 initiation criteria, 387 instability of a strong discontinuity, 561, 568 interaction potential, 22 intermolecular force, 10, 58 internal degrees of freedom, 48, 50, 53 internal energy, 10, 15–17, 30, 31, 46, 47, 49, 53, 55 internal state, 46, 47, 58 internal variables, 47 invariant regions, 453 inverse collision, 47 ionization, 53, 56 ionization phenomena, 4 ionized flows, 57 ions, 3, 53 irreversibility, 82, 105–109, 117 isentropic, 333, 336, 346, 349, 355, 357, 365 isentropic Euler equations, 432, 451, 488 isentropic gas dynamics, 513 isothermal Euler equations, 449 isothermal gas dynamics, 461
811
jump conditions, 555 jump conditions for MHD (magnetohydrodynamical) strong discontinuities, 608 jump processes, 199, 216 k-shock, 560 k-shock conditions, 560 Kac model, 87, 165, 223, 253, 260 Kac’s master equation, 269–270 Kac’s spectral gap problem, 269–271 kinematic viscosity, 28 kinetic energy, 5, 10, 46 kinetic equation, 47 kinetic layer, 94, 135 kinetic model, 18, 44, 57 kinetic relation admissibility criteria, 387 kinetic theory, 3, 4, 13, 17, 22, 53, 54, 61 Knudsen gas, 31 Knudsen layers, 37 Knudsen minimum effect, 94 Knudsen number, 33, 61, 111 Kompaneets equation, 282 Korn inequality, 243 Kronecker delta, 16 Kullback entropy, 192, 204, 206 L1 -stability, 468, 470, 476 laminar regime, 62 Landau approximation, 114, 167, 180–184 Landau equation, 91–94, 105, 114, 121–123, 151, 178, 180–184, 256–257, 267 Landau length, 91, 180 Landau operator, 180 Lanford’s theorem, 96, 107, 136, 266 Larsen–Borgnakke model, 50, 51 law of interaction, 9 Lax discontinuities, 561 Lax entropy conditions, see aso k-shock conditions, 560 Lax shock, see aso k-shock, 560 Lax’s theorem, 448 Lax–Friedrichs scheme, 497 LC (Lopatinski condition), 567 leading edge, 61 lift, 22, 34 linear Boltzmann equation, 86, 99, 281 linearization, 191, 202–203 linearized Boltzmann equation, 36, 37, 39, 40, 134, 138 linearized collision operator, 21, 36 linearized collision term, 21 linearized stability problem for a strong discontinuity, 557 Linnik functional (Fisher information), 255
812
Subject Index
Lions’ theorem, 156, 157 loaded-sphere model, 48 local Maxwellian, 19 local well-posedness, 428, 443 localization of collisions, 79 localization of the distribution function, 123, 147– 153 logarithmic Sobolev inequality, 207, 209, 216, 222 long-time behavior, 109–111, 189–244, 258 – for granular media, 277 – for quantum models, 283–284 Lopatinski determinant, 568 Lorentz gas, 99 Lorentz, H.A., 47 Loschmidt’s paradox, 106 loss term, 81 low-density limit, 96 lower bounds, 185–226 LSP (linearized stability problem) for fast MHD shock waves, 614 LSP (linearized stability problem) for gas dynamical shock waves, 579 LSP (linearized stability problem) for relativistic gas dynamical shock waves, 600 LSP (linearized stability problem) for slow MHD shock waves, 617 LSP (linearized stability problem) for the MHD contact discontinuity, 630 LSP (linearized stability problem) for the MHD tangential discontinuity, 642 LSP (linearized stability problem) for the rotational discontinuity, 635 Lyapunov functionals, 104–109, 191–198, 255, 260 Mach number, 27, 61 macroscopic density, 78 macroscopic velocity, 78 magnetic field, 47, 49 magnetoacoustic system, 610 magnetohydrodynamical shock wave, 608 Manev interaction, 88 mass, 15, 19 mass conservation, 30 mass density in phase space, 13 mass flow, 14, 42 maximum principle, 186–187 Maxwell molecules, 10, 60 Maxwell’s model, 24, 50, 60 Maxwellian collision kernels, 119, 122, 128, 134, 148, 201, 214, 245–261, 275 Maxwellian diffusion (boundary condition), 85, 201
Maxwellian distribution, 27, 29, 31, 32, 34, 36, 40, 41, 44, 109 Maxwellian interaction, 84, 92 Maxwellian state – global, 110, 134, 138 – local, 109, 110, 112, 235 Maxwellians, 13, 53 McKean’s conjectures, 260–261 mean free path, 3, 4, 33, 37 mean-field, 88, 197, 276 merged layer, 61 metastability, 377 MHD (magnetohydrodynamics) for an ideal fluid, 606 micromachines, 4 microreversibility, 80, 101 microscopic variables, 77 mixture, 10, 43, 44, 46, 53, 58, 105 model molecules, 58 molar density, 53 molecular chaos, 8 molecular collisions, 4, 33, 57, 58 molecular diameter, 3, 58 molecular interaction, 9, 39, 43, 55 molecular levels, 62 molecular mass, 17 molecular model, 9, 60 molecules, 3–5, 9, 10, 22, 24, 31, 33, 34, 43, 44, 46–49, 53, 54, 58, 61, 62 moment estimates, 147–153, 226 moment method (Grad’s), 113, 135 moments, 29 momentum, 5, 14, 15, 19, 34, 60 momentum conservation, 30, 44, 45, 60 momentum density, 14 momentum flow, 14, 15 momentum transfer, 28 monatomic gas, 20, 43, 47, 48, 57, 58 monodimensional problems, 139 Monte Carlo quadrature method, 57 Monte Carlo simulation, 57 multidimensional Euler equations, 442, 443, 513 multiple collisions, 6 N-waves, 467 Navier–Stokes equations, 3, 4, 25, 27, 28, 31, 37, 61, 62, 112, 138 Navier–Stokes liquid, 670 nearly free-molecule flow, 33 nearly free-molecule regime, 4 neutral stability of a strong discontinuity, 548, 568, 571 neutrons, 86 nitrogen, 43, 53, 62
Subject Index no-slip boundary condition, 312, 324, 334, 359 no-stick boundary conditions, 347 non-equilibrium, 54 non-isentropic Euler equations, 454 non-isentropic fluids, 512 nondegenerate levels, 47 nondrifting Maxwellians, 13 nonstandard analysis, 143 nonuniqueness of solutions of Riemann problem, 386, 389 normal gas, 577 norms – · (t) W k (Ω) , 569 2
– · L2,η (Ω) , 569 – · W k (Ω) , 569 2
– · W k (Ω) , 569 2,η nuclear reactor, 3 null collision technique, 59, 62 number density, 10, 53, 54 numerical simulations, 114–115, 257 – deterministic, 114 – spectral schemes, 115, 129 – stochastic, 114, 178 ω-representation, 127 oblique shock, 62 one-particle probability density, 5 optical fibers, 3 orientation of a symmetric bodies – second-order liquid, 756 orientation of prolate spheroids – Navier–Stokes liquid, 735 – second-order liquid at non-zero Reynolds number, 758 – second-order liquid at zero Reynolds number, 748 orientation of symmetric bodies – Navier–Stokes liquid, 733 Ornstein–Uhlenbeck, 214, 218, 219 Otto’s Riemannian structure, 229 outgassing, 22 over-Maxwellian collision kernel, 208, 215 overcompressive strong discontinuity, 559 oxygen, 43 parallel magnetohydrodynamical shocks, 612 parity operator, 39 particle orientation, 655 particle sedimentation, 671 perfect gas, 18, 26, 27, 30, 31 periodic solutions, 464 periodicity boundary conditions, 27 perturbation methods, 57
813
perturbations of equilibria, 36 perturbative regime, 134–136, 147 phase space, 5, 10, 14 phenomenological derivation, 115 photon emission, 57 photons, 53, 56, 57, 86, 280, 281 physical values – E, 575, 598, 607 – Hi , 607 – S, 575, 598, 607 – T , 575, 598, 607 – V , 575, 598, 607 – ρ, 575, 598, 607 – Γ , 598 – c, 575, 599, 608 – cA , 609 + – cM , 609 − – cM , 609 – cs , 599 – p, 575, 598, 607 – vi , 575, 598, 607 – vectors – – H, 607 – – v, 575, 598, 607 Planck distribution, 57, 284 plasmas, 89–94 Poincaré’s lemma, 270 Poincaré’s recurrence theorem, 105 point masses, 6, 9, 46 pollen, 3 polyatomic gas, 10, 20, 43, 44, 46, 48, 50, 52, 54, 57, 62 polyatomic molecules, 5, 47, 52, 58 polytropic gas, 423, 576 porous medium, 198 potential energy, 54, 55 potential flow, 61 potential-like solutions, 769 Povzner equation, 88 Povzner inequalities, 148 – reverse, 150 power-law potentials, 10 Prandtl number, 20 pre-postcollisional change of variables, 126 pressure, 5, 7, 16, 20, 30, 37 pressure gradient, 3 probability density, 5–8, 10, 14, 22, 23, 44, 47 probability distribution, 59 propagation of smoothness, 124, 137, 138, 146– 147, 157, 159, 160 propelling boundary conditions, 780 pseudo-differential operators, 165, 175
814
Subject Index
quantum Boltzmann equation, 86 quantum kinetic theory, 279–285 quantum mechanics, 10, 46, 47, 54 radiation, 3, 53, 56, 57 radiation frequency, 57 radiative transfer, 56 radioactivity, 3 Radon transform, 156–157 Rankine–Hugoniot conditions, 576 Rankine–Hugoniot conditions, see aso jump conditions, 555 rarefaction curves, 447 rarefaction waves, 457, 508 rate coefficients, 53 Rayleigh waves, 571 re-entry calculations, 62 reactants, 53 reacting collision, 10 reaction cross-section, 53, 54 reaction model, 54 reaction rates, 56 reactive collision, 46 reactive cross-section, 62 reciprocity, 23, 24, 40, 49 recombination process, 54 recombination reaction, 54 reduced mass, 43 reflection, 57 regularity of the gain operator, 156–157, 171, 201, 226 regularization, 123, 138, 146–147, 165, 170–175, 177–180, 183 relative speed, 10, 55, 58 relative velocity, 5, 9 relativistic Boltzmann equation, 86 relativistic gas dynamics, 597 relativistic Rankine–Hugoniot conditions, 599 relativistic shock wave, 599 relaxation, 521 relaxation time, 48, 50 renormalization, 137, 159–160, 175–176 renormalized, 323 renormalized formulation, 182 renormalized solution, 137–139, 145–147, 159– 160, 163–164, 178–180, 321–324, 326, 328, 331, 335, 340, 341 rescaled convolution, 247 resonance, 426 Reynolds number, 61, 675 Riemann invariants, 433 Riemann problem, 380, 385, 446, 517 Riemann semigroup, 478 Riemann solutions, 483
Rostoker collision operator, 94 rotational cross-section, 62 rotational discontinuity in magnetohydrodynamics, 608 rotational energy, 48 rough sphere model, 48 Rutherford’s formula, 83, 266 scaled Boltzmann equation, 28 scalings, 25, 26 scattering amplitude, 47 scattering event, 57 scattering formulas (Maxwell’s), 82 scattering kernel, 23, 24 scattering law, 58 scattering probability, 57 screening (Debye), 90, 180 second-order liquid, 670 sedimentation, 656 self-propelled bodies, 658, 761 self-propelled motion, 761 self-propelled motion, steady – attainability, 773, 785 – Navier–Stokes, 775 – Stokes approximation, 763 – symmetric bodies, 767, 784 – weak solution, 776 self-propelled motion, unsteady – Navier–Stokes, 785 – Stokes approximation, 773 self-similar solutions, 101, 117 semi-classical limit, 282 Senftleben–Beenakker effects, 49 separated flows, 62 Shannon’s entropy, 107 Shannon–Stam inequality, 108, 217, 255 – for Boltzmann’s operator, 255–256 shear stress, 38 shock capturing scheme, 515 shock curves, 447, 454 shock front solutions, 443 shock layers, 37 shock profile, 61 shock wave, 4, 37, 61, 135, 425 shock wave structure, 61 Shuttle Orbiter, 62 σ -representation, 80 silicon chips, 3 simulation schemes, 57 singular perturbation, 31 singularity, 424, 436, 437, 439 – angular, 83, 84, 90, 91, 120, 133, 135, 165–180, 251, 268
Subject Index – kinetic, 124, 133, 165, 173, 266 slip regime, 37 slow magnetohydrodynamical shock wave, 610 small solutions, 136–137, 161 Smoluchowski equation, 198 smooth solutions, 428, 432, 436 soft potentials, 119, 122, 123, 135, 150–151, 201, 202, 209, 216, 224 solid state physics, 22 solid wall, 5, 18, 22 space shuttle, 4 space-homogeneous, 53 space-homogeneous solutions, 18 spacecraft, 3 spaces – Rn+ , 557 – Rn− , 557 spatially homogeneous theory, 133–134, 161, 192– 195 special notations – DN , 555 – F , 556 – [·], 555 – x , 555 – ρ∞ , p∞ , v∞ , . . . – – g∞ , 555 588 – ∇, – ∗ , 552 species, 43, 44, 53, 54, 56 specific heat, 20, 43 spectral analysis, 134, 135 spectral gap, 253 specular reflection, 6, 24, 85, 242 speed of light, 57 speed of sound, 27 spherically symmetric solutions, 513 sphero-cylinder model, 48 spin, 47, 49 spinodal region, 376 splitting, 114, 133, 154, 165 spontaneous emission, 56 spontaneous sound radiation by the discontinuity, see aso neutral stability of a strong discontinuity, 548 stability, 131, 138, 146–147, 160, 486, 508 stability of orientation – second-order liquid, 748, 757 – Navier–Stokes liquid, 734 Stam’s regularization argument, 217–218 stationary Boltzmann equation, 101 stationary states – for granular media, 277 – inhomogeneous setting, 110, 197 – spatially homogeneous setting, 133
815
statistics, 5, 254 steady fall, symmetric translational, 719 steady-state solutions, 514 stepshock, 556 stochastic systems, 100, 178 Stokes paradox, 27 stress tensor, 15, 16 stresses, 5, 15, 25 Strichartz estimates, 145 strong discontinuity, 547, 555 strong solution, 314, 345 structural stability of a strong discontinuity, 547, 556 sub-additivity, 260 super-H theorem, 260 supersonic flow, 61 surface layers, 22 surfaces of discontinuity, 37 symmetric t-hyperbolic system, 551 symmetric body, 718 symmetric point of view, 125 symmetric translational steady fall, 719 symmetric translational steady falls, existence, 722 symmetrization (collision kernel), 120, 124 symmetrization of systems of conservation laws, 555 Talagrand inequality, 204 tame oscillation condition, 478 Tanaka’s representation, 128 Tanaka’s theorem, 201, 250 tangential discontinuity in magnetohydrodynamics, 608 tangential strong discontinuity in gas dynamics, 576 Taub adiabat, 599 temperature, 5, 7, 17, 19, 20, 22, 24, 27, 30, 34, 37, 41, 46, 50, 52, 53, 58, 78 temperature gradient, 3, 38 temperature jump, 37, 38 terminal state, 691 termolecular reaction, 54 the self-propelled body equations, 761 thermal energy, 46 thermally radiating flows, 57 thermodynamical equilibrium, 112, 191 thermodynamics, 18, 25 tilt angle, 658 tilt angle phenomenon, 663 time reversible, 23 torque, 723, 750, 754 Toscani’s distance, 251, 258 total cross-section, 53
816
Subject Index
total energy, 48, 49 trailing edge, 61 translational degrees of freedom, 48 translational energy, 48, 53 transonic nozzle flow, 514, 515 transport operator, 78 transverse magnetohydrodynamical shocks, 613 traveling wave criterion, 382 traveling waves, 381 ULC (uniform Lopatinski condition), 568 undercompressive strong discontinuity, 559 uniform stability of a strong discontinuity, 547, 568 uniqueness, 138, 150, 178, 478, 483 upper atmosphere, 3 validation problem, 95–100, 137, 266 – Grad’s approach, 95 – Kac’s approach, 100 van der Waals, 376 vanishing similarity viscosity, 385, 391 vanishing viscosity criterion, 381 variable hard sphere model, 10 variable hard spheres, 84 variable hard-sphere (VHS) model, 58 variable soft sphere (VSS) model, 58 variational method, 38, 39 variational principle, 38, 40, 41 vehicles, 3 velocity, 41 velocity slip, 37, 38 velocity space, 12, 13, 39 velocity-averaging, 137, 143, 147, 160 vibrational cross-section, 62 vibrational energy, 50 vibrational relaxation, 62 viscoelastic torque coefficient, 746, 757, 759, 760
viscosity, 20, 28, 32, 50, 58 viscous boundary layer, 4, 61 viscous boundary layer reattachment, 62 viscous boundary layer separation, 62 Vlasov equation – linear, 79 – nonlinear, 88 – Vlasov–Poisson, 90–94 Vlasov–Fokker–Planck, 196, 197, 237 Vlasov–Maxwell system, 90 wake flows, 62 wall Maxwellian, 25 Wasserstein distance, 204, 233, 249–251 wave-front tracking, 470 weak compactness, 131 weak convergence, 160, 200 weak formulation – Boltzmann’s, 104 – for Landau’s operator, 181 – Maxwell’s, 101, 130, 147, 166, 181 weak or distributional solutions, 313 weak solution, 317, 318, 321, 323, 324, 345, 347, 348, 366, 786 (weak) solutions, 320 weak stability of a strong discontinuity, 568 weak-strong uniqueness theorem, 138, 163 weakly inhomogeneous solutions, 133, 276 Weissenberg number, 738 Weyl’s theorem, 202 Wild representation, 258 Wild sums, 248–249 Wild tree, 248 Young measures, 493 Zermelo’s paradox, 105
