H ANDBOOK OF M ATHEMATICAL F LUID DYNAMICS VOLUME IV
H ANDBOOK OF M ATHEMATICAL F LUID DYNAMICS Volume IV
Edited by
S. FRIEDLANDER University of Illinois-Chicago Chicago, Illinois, USA
D. SERRE Ecole Normale Supérieure de Lyon Lyon, France
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Printed and bound in The Netherlands
07 08 09 10 11
10 9 8 7 6 5 4 3 2 1
Contents of the Handbooks Volume I
1. 2. 3. 4. 5.
The Boltzmann equation and fluid dynamics, C. Cercignani A review of mathematical topics in collisional kinetic theory, C. Villani Viscous and/or heat conducting compressible fluids, E. Feireisl Dynamic flows with liquid/vapor phase transitions, H. Fan and M. Slemrod 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 71 307 373 421 545 653
Volume II
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Statistical hydrodynamics, R. Robert Topics on hydrodynamics and volume preserving maps, Y. Brenier Weak solutions of incompressible Euler equations, A. Shnirelman Near identity transformations for the Navier–Stokes equations, P. Constantin Planar Navier–Stokes equations: Vorticity approach, M. Ben-Artzi Attractors of Navier–Stokes equations, A.V. Babin Stability and instability in viscous fluids, M. Renardy and Y. Renardy Localized instabilities in fluids, S. Friedlander and A. Lipton-Lifschitz Dynamo theory, A.D. Gilbert Water-waves as a spatial dynamical system, F. Dias and G. Iooss Solving the Einstein equations by Lipschitz continuous metrics: Shock waves in general relativity, J. Groah, B. Temple and J. Smoller
v
1 55 87 117 143 169 223 289 355 443 501
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Contents of the Handbooks
Volume III
1. From particles to fluids, R. Esposito and M. Pulvirenti 2. Two-dimensional Euler system and the vortex patches problem, J.-Y. Chemin 3. Harmonic analysis tools for solving the incompressible Navier–Stokes equations, M. Cannone 4. Boundary layers, E. Grenier 5. Stability of large-amplitude shock waves of compressible Navier–Stokes equations, K. Zumbrun (with Appendix by H.K. Jenssen and G. Lyng) 6. Some mathematical problems in geophysical fluid dynamics, R. Temam and M. Ziane
1 83 161 245 311 535
Preface This is the fourth volume in a series of survey articles covering many aspects of mathematical fluid dynamics. Our original intention was to stop with three volumes. However, even over 2000 pages in the first three volumes do not cover a number of lively areas in the subject. We therefore decided to edit another volume, not with the unrealistic goal of being completely comprehensive, but in order to add certain topics where there is currently considerable activity and progress. We hope that the total collection of articles illustrates the breath and depth of mathematical fluid dynamics, which remains a vital source of open mathematical problems and exciting physics. Volume 4 starts with an article illustrating that topology and geometry (branches of mathematics not conventionally associated with the study of fluids) have an important role to play in examining the behaviour of the motion of an ideal fluid. This is followed by an article about inviscid compressible fluids, a field having not received attention in the second and third volumes. It is devoted to the interaction of the most elementary solutions, namely multi-dimensional shock waves. The next paper makes the important bridge between the compressible and incompressible fluid models. It is also the first of three in this volume to deal with asymptotic analysis for fluids. The remaining six articles address various topics concerned with incompressible viscous fluids and the Navier–Stokes equations. An important mathematical issue (related to a onemillion dollar prize!) is, of course, the regularity of the solutions of these equations: known results and open questions are presented in the 4th article. The next two articles discuss geophysical fluid dynamics. Mathematical models for the ocean and the atmosphere are presented and it is shown how certain models can be used to analyze mathematically some precise geophysical phenomena. The 5th has an emphasis on asymptotic analysis, with applications to either mid-latitudes or the tropics. The 6th is more concerned with periodic forcing due, for example, to circadian or seasonal rhythms. Close connections with physics continue in the next articles. The mathematical properties of the equations governing the flow of fluids with pressure and shear rate dependent viscosities is the topic of the 7th article. The 8th paper is a very comprehensive review of results concerning flows in domains with outlets at infinity. The final article explains how homogenization theory provides models for flows in porous media and other non-homogeneous situations. For instance, it gives a rigorous derivation of Darcy’s law. Again we repeat our heartfelt thanks to authors of the articles in volume 4 and also the authors of all the articles in the previous volumes. We are deeply indebted to them for the immense amount of work that writing such excellent survey articles requires. We are vii
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Preface
also very grateful to the referees who generously spent much time and thought to ensure the high quality, scholarship and enduring value of the Handbook of Mathematical Fluid Dynamics. We thank the Editors and staff at Elsevier who were unfailingly helpful and professional and who produced the excellent printing of the published books. Chicago and Lyon, September, 2006 Susan Friedlander and Denis Serre
[email protected] and
[email protected]
List of Contributors Conca, C., Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, and Centro de Modelamiento Matemático, UMI 2807 CNRS-UChile, Casilla 170/3-Correo 3, Santiago, Chile (Ch. 9) Gallagher, I., Institut de Mathématiques de Jussieu UMR 7586, Université Paris VII, 175, rue du Chevaleret, 75013 Paris, France (Ch. 5) Ghrist, R., Department of Mathematics and Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801, USA (Ch. 1) Málek, J., Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83, 186 75 Prague 8, Czech Republic (Ch. 7) Pileckas, K., Vilnius University, Faculty of Mathematics and Informatics, Naugarduko Str., 24, Vilnius, Lithuania and Institute of Mathematics and Informatics, Akademijos 4, Vilnius, Lithuania (Ch. 8) Rajagopal, K.R., Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA (Ch. 7) Saint-Raymond, L., Laboratoire J.-L. Lions UMR 7598, Université Paris VI, 175, rue du Chevaleret, 75013 Paris, France (Ch. 5) Schochet, S., Tel Aviv University, School of Mathematical Sciences, Ramat Aviv, 69978 Tel-Aviv, Israel (Ch. 3) Sell, G.R., School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA (Ch. 6) Seregin, G., Steklov Institute of Mathematics at St. Petersburg, St. Petersburg, Russia (Ch. 4) Serre, D., École Normale Supérieure de Lyon, 69364 Lyon Cedex 07, France (Ch. 2) Vanninathan, M., TIFR Center, IISc. Campus, P.O. Box 1234, Bangalore-560012, India (Ch. 9)
ix
Contents Contents of the Handbooks Preface List of Contributors
v vii ix
1. On the Contact Topology and Geometry of Ideal Fluids R. Ghrist 2. Shock Reflection in Gas Dynamics D. Serre 3. The Mathematical Theory of the Incompressible Limit in Fluid Dynamics S. Schochet 4. Local Regularity Theory of the Navier–Stokes Equations G. Seregin 5. On the Influence of the Earth’s Rotation on Geophysical Flows I. Gallagher and L. Saint-Raymond 6. The Foundations of Oceanic Dynamics and Climate Modeling G.R. Sell 7. Mathematical Properties of the Solutions to the Equations Governing the Flow of Fluids with Pressure and Shear Rate Dependent Viscosities J. Málek and K.R. Rajagopal 8. The Navier–Stokes System in Domains with Cylindrical Outlets to Infinity. Leray’s Problem K. Pileckas 9. Periodic Homogenization Problems in Incompressible Fluid Equations C. Conca and M. Vanninathan
1 39 123 159 201 331
407
445 649
Author Index
699
Subject Index
707
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CHAPTER 1
On the Contact Topology and Geometry of Ideal Fluids Robert Ghrist∗ Department of Mathematics and Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801, USA
Contents 1. Ideal fluids on Riemannian manifolds . . . . . . 1.1. The Euler equation . . . . . . . . . . . . . 1.2. An analogy . . . . . . . . . . . . . . . . . 1.3. A geometric formulation . . . . . . . . . . 2. The geometry of steady solutions . . . . . . . . 3. Basic contact topology . . . . . . . . . . . . . . 3.1. Definitions . . . . . . . . . . . . . . . . . . 3.2. Local contact topology . . . . . . . . . . . 3.3. Global contact topology . . . . . . . . . . 4. Contact structures and steady Euler fields in 3-d 4.1. Reeb fields . . . . . . . . . . . . . . . . . . 4.2. A correspondence . . . . . . . . . . . . . . 4.3. Existence on 3-manifolds . . . . . . . . . . 5. Knots and links in three-dimensional flows . . . 5.1. Unknots . . . . . . . . . . . . . . . . . . . 5.2. Knots . . . . . . . . . . . . . . . . . . . . . 6. Instability . . . . . . . . . . . . . . . . . . . . . 6.1. Instability criteria . . . . . . . . . . . . . . 6.2. Generic curl eigenfields . . . . . . . . . . . 6.3. Contact homology . . . . . . . . . . . . . . 6.4. Generic instability . . . . . . . . . . . . . . 7. Concluding unscientific postscript . . . . . . . . 7.1. Generic fluids . . . . . . . . . . . . . . . . 7.2. Closing questions . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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* RG supported in part by NSF PECASE Grant # DMS-0337713.
HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME IV Edited by S.J. Friedlander and D. Serre © 2007 Elsevier B.V. All rights reserved. 1
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R. Ghrist
Abstract We survey certain topological methods for problems in inviscid fluid dynamics in dimension three. The tools come from the topology of contact structures, or nowhere-integrable plane fields. The applications are most robust in the setting of fluids on Riemannian threemanifolds which are not necessarily Euclidean. For example, these methods can be used to construct surprising examples of inviscid flows in non-Euclidean geometries. Because of their topological basis, these methods point one toward a theory of “generic” fluids, where the geometry of the underlying domain is the genericity parameter.
On the contact topology and geometry of ideal fluids
3
1. Ideal fluids on Riemannian manifolds 1.1. The Euler equation The equation of motion of an unforced, incompressible, inviscid fluid with velocity field u(x, t) is ∂u + (u · ∇)u = −∇p, ∂t
;
div(u) = 0,
(1.1)
where p denotes a real-valued pressure function. In most instances in the literature, the domain in which the fluid resides is a Euclidean domain. On those occasions when compactness is desired and the complexities of boundary conditions are not, the fluid domain is usually taken to be a Euclidean torus T 3 given by quotienting out Euclidean space by the action of three mutually orthogonal translations. Although such a situation is of dubious physical relevance, it is nevertheless a fairly common domain in the literature on mathematical fluids. There are several unavoidable problems in the attempt to analyze Euler flows in dimension three, not the least of which is that the fundamental starting point, the global existence of solutions to the Euler equations, is unknown in dimension three and perhaps not true. And although fluid dynamics can lay claim to having inspired many ingenious and fundamental contributions to analysis and PDEs, a casual reading of the relevant literature shows that fluid dynamics in general (both viscous and inviscid fluids) is “hard” in dimension three without some type of symmetry or other reduction to a lower-dimensional setting. Despite what we teach our undergraduate students about general principles for fluid flows—e.g., Kelvin’s Circulation Theorem—there is not an abundance of theorems which hold for ideal fluid flows in dimension three. Since so little is known about the rigorous behavior of fluid flows, any methods which can be brought to bear to prove theorems about their behavior are of interest and potential use. Following the pioneering work of Arnold [3,4], Moffatt [81], and others, we propose that a more geometric and topological view of fluid dynamics can provide new tools and insights which lead to very general results. Neither geometric nor topological approaches to fluid dynamics are novel: Tait’s initial foray into knot theory was inspired by Kelvin’s interests in knotted vortex tubes in the æther [95]. The contact topological tools which we survey in this paper are of rather recent relevance to fluid dynamics.
1.2. An analogy The precise topological tools which this article discusses are those which come from a particular branch of geometric topology concerning contact structures. In a three-dimensional domain, a contact structure is a field of tangent planes which varies smoothly point-to-point and which is “nowhere integrable”, meaning that these planes, unlike scales on a fish, do not fit together to yield two-dimensional sheets. Such plane fields possess a wealth of wonderful local and global properties: see §3 for detailed information. In this article, contact
4
R. Ghrist
Fig. 1. Two types of stagnation points in a planar flow: hyperbolic [left] and elliptic [right].
structures will arise naturally as plane fields orthogonal to certain vector fields which solve the steady Euler equations. This yields a variety of novel topological methods and results which apply to arbitrary three-dimensional domains. This connection between the topology of a steady Euler field and its orthogonal plane field is not so foreign as might at first appear. Indeed, the study of steady (timeindependent) inviscid fluids on two-dimensional domains has a very topological feel to it (cf. texts such as [20]) which is not unrelated to the notion of a dual hyperplane field. We sketch out a simple analogy in dimension two which illustrates the naturality of contact topological perspectives in dimension three. Consider a steady Euler field u(x, y) = (u(x, y), v(x, y)) on the Euclidean plane R2 . ∂v Incompressibility implies that ∂u ∂y = − ∂x and so, as any student of fluid dynamics knows, there exists a stream function : R2 → R which is constant along integral curves of u. The coarse qualitative features of a steady Euler field u on R are perfectly encoded in this stream function. For example, the critical points of correspond to the stagnation points of the fluid. For an incompressible fluid in 2-d, there are two types of (nondegenerate) stagnation points: an elliptic point (or center) and a hyperbolic point (or saddle), as illustrated in Figure 1. There is a dynamical means of extracting this information from : consider the gradient field ∇. This auxiliary dynamical system tells one what is happening in the direction orthogonal to the fluid flow. The fixed points of ∇ are precisely the stagnation points of u and the Morse index of a fixed point for (the dimension of its unstable manifold) says whether the critical point of is a local max or min (an elliptic stagnation point) or is a saddle (a hyperbolic stagnation point). Upon moving to dimension three, the Bernoulli Theorem states that there is a real-valued function H which, like above, is constant along flowlines. The first general theorem for 3-d flows we survey, due to Arnold, states that as long as H is not a constant, almost all flowlines are constrained to tori which fill up the 3-d domain. For this case, the gradient of the function H gives dynamical information in the direction orthogonal to the level sets of H and likewise allows one to recapture the rough features of the flow. But three-dimensional ideal fluids admit the possibility of fully nonintegrable flowlines in certain cases (the eigenfields of the curl operator). The theme of this survey is that instead of trying to generalize the idea of a stream function, the appropriate generalization
On the contact topology and geometry of ideal fluids
5
is to consider the topological structure of what is happening orthogonal to the velocity field. In 2-d, this orthogonal structure is a line field, which corresponds to the gradient field of the stream function. In 3-d, the analogous orthogonal structure takes the form of a plane field. For those steady Euler flows with nonintegrable dynamics, this plane field likewise exhibits its own form of nonintegrability: it is a contact structure. A careful analysis of the topology and dynamics of contact structures can yield global information about the velocity field, including the existence of periodic orbits and the types of periodic orbits which arise (elliptic and hyperbolic). This analogy is not strict, but rather points to the fact that there is a relationship between the dynamics of a steady ideal fluid and the geometry of the field orthogonal to the fluid. 1.3. A geometric formulation All of the results of the previous section about the stream function in 2-d are dependent only on the fact that the domain is R2 : the precise Euclidean features of the domain geometry are not necessary for the existence of a stream function. Such is the case in higher-dimensional fluids as well. We begin by interpreting the Euler equations on more arbitrary geometric domains: see, e.g. ([1], §8.2). Let M denote a sufficiently smooth, connected, oriented differential manifold. In order to make sense of operations such as directional derivative and divergence, it behooves us to assume an underlying geometry and volume on the flow domain M. We therefore take g to be a Riemannian metric on M: a symmetric 2-tensor which defines an inner product on tangent spaces of M. We also choose a volume form μ on M, a topdimensional form which is pointwise nonvanishing. One can of course choose the precise volume form μg induced by the metric; however, for the sake of generality, we allow for arbitrary μ. This has physical significance to compressible isentropic fluids, as noted in [6]. See, e.g., [1] for a wealth of background material on the tools and language of global geometry and analysis on manifolds. The form which the Euler equations take on an oriented Riemannian manifold is the following: ∂u + ∇u u = −∇p ; Lu μ = 0, ∂t
(1.2)
where ∇u is the covariant derivative along u defined by the metric g, and Lu is the Lie derivative along u. We give an explicit example on the 3-sphere, S 3 , the points in R4 a unit distance from the origin. Via stereographic projection, this 3-manifold is equivalent to R3 with an added “point at infinity.” The round metric on S 3 is that inherited by it as a subset of Euclidean R4 , much in the same manner as we would describe the geometry of the round 2-sphere S 2 ⊂ R3 . The simplest example of a steady Euler field on the round S 3 is that given by the Hopf field, pictured in Figure 2. One may realize this field in Euclidean coordinates on R4 via x˙ = −x2 ; x˙3 = −x4 . (1.3) u= 1 x˙2 = x1 ; x˙4 = x3
6
R. Ghrist
Fig. 2. The Hopf flow on S 3 = R3 ∪ {∞}. Every flowline is a simple closed curve, each pair of which is linked.
These equations correspond to the Hamiltonian equations of two identical uncoupled simple harmonic oscillators at a fixed energy level; it therefore preserves the natural volume form on the round 3-sphere. Also, because the oscillators are identical, every orbit of the flow is periodic. This exhibits a very special type of solution to the Euler equations: ∇u u is identically zero, meaning that every flowline is a geodesic on the round S 3 . The vector-calculus identity for the directional derivative in dimension three, 1 (u · ∇)u = (∇ × u) × u + ∇ (u · u) , 2 has a more general geometric formulation [1, p. 588] by using the operation to transform vector fields to dual 1-forms: 1 (∇u u) = Lu u − d u (u) . 2 This allows one to transform (1.2) into an equation on differential forms: ∂(ιu g) + Lu ιu g − d ∂t
1 ιu ιu g = −dp ; Lu μ = 0. 2
Here, u = ιu g = g(u, ·) denotes the one-form obtained from u via contraction into the first slot of the metric. One uses the Cartan formula Lu = dιu + ιu d and absorb the energy terms into the pressure function to obtain ∂(ιu g) + ιu dιu g = −dH ; Lu μ = 0, ∂t
(1.4)
On the contact topology and geometry of ideal fluids
7
where H = p + 12 ιu ιu g. The dependence of H upon u is of no consequence—one says that u(t) is an Euler field if it satisfies the Euler equations for some function H (t). The vector calculus form of (1.4) is ∂u + u × (∇ × u) = −∇H, ∂t where ∇ × u is the vorticity of the fluid. In terms of differential forms, the vorticity corresponds to the 2-form dιu g. One correspondingly defines the curl operator for a Riemannian 3-manifold with volume (M, g, μ) to be the linear operator ∇× on divergence-free vector fields which takes a vector field u to the field ∇ × u satisfying ι∇×u μ = dιu g.
(1.5)
This operator is well-defined since μ is a nonvanishing 3-form. In the language of dual 1-forms, the curl operator is ∗d, where ∗ is the Hodge star operator [1]. With this notion of curl, the Euler equations take the particularly clean form ∂(ιu g) + ιu ι∇×u μ = −dH ; Lu μ = 0. ∂t
(1.6)
By taking the exterior derivative of (1.6), one obtains the Helmholtz equation for the evolution of the vorticity 2-form ω = dιu g: ∂ω + Lu ω = 0. ∂t
(1.7)
This equation has the advantage of discarding the exact pressure-related term dH from (1.6). For more information on the equations of motion for fluids on Riemannian manifolds, see [7,1].
2. The geometry of steady solutions There are numerous deep open problems about the solutions to (1.6), the most famous being the existence of finite-time blow-ups. We circumvent this and other delicate questions by focusing on a much simpler problem, namely, the problem of existence and classification of steady solutions to (1.6). This approach is not without precedence in the broader context of dynamical systems. It is an argument going back to the work of Poincaré and elucidated by Conley, Hale, and others, that in any dynamical system it is the bounded solutions which are most important and which should be investigated first. These, then, are stratified according to dimensions of invariant sets: first are the fixed points, then connecting orbits between fixed points along with periodic orbits, etc. We therefore consider most carefully the fixed points of the dynamical system that the Euler equation induces on the space of volume-preserving vector fields on a fixed Riemannian manifold.
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R. Ghrist
A classical result of Arnold’s on integrable Hamiltonian systems [5] has applications to the geometry and topology of fluids. It has an unfortunately strong smoothness requirement: all fields, metrics, and volumes are assumed to be C ω (real-analytic). T HEOREM 2.1 (Arnold [7]). Let u be a C ω nonvanishing Euler field on a closed oriented Riemannian three-manifold M. Then, at least one of the following is true: (1) There exists a compact analytic subset ⊂ M of codimension at least one which splits M into a finite collection of cells diffeomorphic to T 2 ×R. Each torus T 2 ×{x} is an invariant set for u having flow conjugate to linear flow. (2) The field u is a curl-eigenfield: ∇ × u = λu for some λ ∈ R. P ROOF. The Bernoulli Theorem states that H is an integral for u, namely, Lu H = ιu dH = ιu ιu ι∇×u μ = 0.
(2.1)
The Inverse Function Theorem implies that, when c is not a critical value of H : M → R, H −1 (c) is a closed 2-manifold which has a nonvanishing vector field u tangent to it. Thus it has Euler characteristic zero. Since dH = 0 here, dH orients the surface and thus H −1 (c) is a disjoint union of tori for all regular values c. Since this is a steady Euler field, the vorticity is also time-independent, and Equation (1.7) implies that the velocity and vorticity fields commute. This implies that the flow on each such torus is conjugate to linear flow, since the flowlines of the velocity field are transported by the vorticity field. Define ⊂ M to be the inverse image of the critical values of H . As all the data in the equation is assumed real-analytic, H must be C ω . If H is non-constant, real-analyticity implies that is a C ω subset of M of codimension at least one, and the complement of is composed of invariant tori with linear flow. The only other possibility is that H is constant and = M; namely, that the vorticity and velocity fields are collinear at each point. Assume ∇ × u = hu for some h : M → R. In this case h is an integral of u: 0 = d(dιu g) = d(hιu μ) = dh ∧ ιu μ + hdιu μ = dh ∧ ιu μ,
(2.2)
which implies that ιu dh = 0. Since h is an integral for u, the same argument as that for H yields the appropriate set σ ⊂ M off of which the flow consists of invariant tori. The only instance in which u is not integrable is thus when h is constant and u is a curl eigenfield. There is a surprising corollary to this proof: ‘most’ 3-manifolds do not admit a nonvanishing (analytic) integrable vector field. Any 3-manifold which admits such a decomposition into blocks of the form T 2 × R glued together appropriately can be outfitted with a round handle decomposition [14]. Those 3-manifolds which admits such a decomposition have been classified and are among the so-called graph 3-manifolds. These are relatively rare [98,99], e.g., no 3-manifold which admits a metric of constant negative curvature (a hyperbolic 3-manifold) can ever appear. This is a very powerful conclusion, and leaves us
On the contact topology and geometry of ideal fluids
9
Fig. 3. A 2-d projection of some flowlines of the ABC fields on a periodic R3 . Some orbits of this nonintegrable field are not constrained to 2-d surfaces.
with two choices: for sufficiently smooth fluid flows on a typical closed 3-manifold, the only steady ideal fluids either (1) always have stagnation points, or (2) are curl eigenfields. Arnold’s Theorem leads one naturally to examine carefully the class of fields for which the integral H degenerates. A vector field u is said to be a Beltrami field if it is parallel to its curl: i.e., ∇ × u = hu for some function h on M. A rotational Beltrami field is one for which h = 0; that is, it has nonzero curl.1 Curl eigenfields possess several interesting geometric, analytic, and dynamical features: see [7,58,22,47] for more information. A key example of a curl eigenfield is the class of ABC fields: x˙ = A sin z + C cos y y˙ = B sin x + A cos z . z˙ = C sin y + B cos x
(2.3)
Here, A, B, C 0 are constants, and the vector field is defined on the three-torus T 3 . By symmetry in the variables and parameters, we may assume without loss of generality that 1 = A B C 0. Under this convention, the vector field is nonvanishing if and only if B 2 + C 2 < 1 (see [22]). At ‘most’ parameter values, the ABC fields exhibit a so-called Lagrangian turbulence—there are apparently large regions of nonintegrability and mixing 1 There is some variation in the literature over whether Beltrami means that vorticity is parallel to velocity or, more restrictively, a scalar multiple. For purposes of this article, we use Beltrami to mean parallel and curl eigenfield to mean a scalar multiple. In [7], the term “Beltrami” means that vorticity is a scalar multiple of velocity.
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within the flow. Though this has nothing whatsoever to do with genuine turbulence in fluids, it remains a fascinating phenomenon. Although the ABC flows have been repeatedly analyzed [3,22,47,54,67,101], few rigorous results are known, other than for near-integrable examples. Beltrami fields in general are even less well understood, due to the fact that nonintegrable dynamics is both prevalent and difficult. We survey some topological tools which are especially suited to global problem in nonintegrable dynamics, with a focus on curl eigenfields. In particular, we consider the following problems: (1) Given a steady Euler flow on a Riemannian 3-manifold, are there any periodic flowlines? (2) In the case where there are periodic orbits, which knot and link types may/must occur? (3) For a ‘generic’ steady Euler flow, is this solution hydrodynamically stable or unstable? Curl eigenfields occupy an important place not only within hydrodynamics [7,22,43,80], but also within the study of magnetic fields and plasmas (where they are known as forcefree fields) [7,16,77] and in the stability of matter [75]. It is not unreasonable to hope that the techniques surveyed here have implications in these fields as well. 3. Basic contact topology This rather lengthy excursion into contact topology is justified by the later use of nearly every definition, example, and theorem in the remainder of this survey. We restrict to the setting of three-dimensional manifolds, although contact structures are defined in any odd dimension. Roughly speaking, a contact structure on a three-manifold is a distribution of smoothly varying tangent plane fields which “twist” so as to be maximally nonintegrable. Recall that any nonvanishing smooth vector field on a manifold can be integrated, or stitched together so as to fill up the space by curves tangent to the vector field. An integrable plane field likewise comes from a foliation of the space by two-dimensional sheets tangent to the plane field. The subject of whether a field of k-dimensional subspaces of the tangent space to a manifold is integrable is classical. Interestingly enough, the subject of contact transformations was long ago very closely related to topics in partial differential equations related to the method of characteristics: see, e.g., the text of Goursat from 1891 [53]. The subject of contact structures finds its origin in this time period, when Lie considered “contact elements” of curves were studied. We present a modern formulation. 3.1. Definitions The following definitions come from the proper interpretation of the Frobenius condition on the integrability of a distribution, and provides a way of saying that a plane distribution is maximally nonintegrable via the calculus of differential forms [1]. Let M denote a threedimensional manifold. A contact form on M is a differential 1-form α satisfying α ∧ dα = 0
(3.1)
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Fig. 4. The standard contact structure on R3 .
pointwise on M. A contact structure is a smooth plane field ξ on M which is (locally) the kernel of a contact form, namely ξ = ker(α)
⇔
α(v) = 0
∀ v ∈ ξ.
(3.2)
The standard contact structure on R3 is the kernel of the contact form dz + x dy. We see that its kernel consists of tangent planes in R3 which (conflating local and global coordinates) satisfy dz = −x, dy and thus correspond to the plane field illustrated in Figure 4. Along the y–z plane, all the contact planes have slope zero. As one walks along the x direction, the planes twist in a counterclockwise fashion. The reader should convince himself that such a plane field cannot arise as the tangent planes to a 2-dimensional foliation of R3 . More interesting examples are abundant. The standard contact structure on the 3-sphere S 3 ⊂ R4 is given by the kernel of the 1-form αH =
1 (x1 dx2 − x2 dx1 + x3 dx4 − x4 dx3 ) . 2
(3.3)
The contact structure ξ = ker(αH ) is the plane field orthogonal to the Hopf field on the round sphere in (1.3). Both of the above examples are coorientable contact structures—they are the kernel of a globally defined 1-form. There exist non-coorientable contact structures, just as there exist line fields on a surface which are not coorientable. However, since all the contact structures which arise in fluid dynamics come with a globally-defined contact form, we will assume for the remainder of this chapter that all relevant contact structures are cooriented.
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3.2. Local contact topology The interesting (and difficult) problems in contact geometry are all of a global nature, thanks to the following results. These results give local classifications for contact structures in terms of the natural equivalence relation. This relation is contactomorphism, or, diffeomorphism which takes one contact structure to another. T HEOREM 3.1 (The Darboux Theorem). All contact structures on a 3-manifold are locally contactomorphic to the standard structure on R3 . See, e.g., [79,2] for a proof. In polar coordinates, the standard contact structure is contactomorphic to the kernel of dz + r 2 dθ : see Figure 5[left]. There is a broad generalization of the Darboux Theorem due to Moser and Weinstein that characterizes a contact structure in a neighborhood of a surface as opposed to a neighborhood of a point. Given a surface embedded in M, the contact structure ξ on M “slices” along a (possibly singular) line field. This induces the characteristic foliation, ξ . The Moser–Weinstein Theorem implies that studying characteristic foliations is an effective means of analyzing the contact structure. T HEOREM 3.2 (The Moser–Weinstein Theorem). The contactomorphism type of a neighborhood of in M is determined by the characteristic foliation ξ . One final local result in contact topology says that deforming a contact structure on M is really the same thing as fixing the contact structure and deforming M. More precisely: T HEOREM 3.3 (Gray’s Theorem). If αt is a smooth family of contact forms, then there exists a smooth family of diffeomorphisms φt such that αt = ft φt∗ (α0 ) for some functions ft : M → R.
3.3. Global contact topology Contact structures form a rich class of plane fields for 3-manifolds. We begin with an existence result. T HEOREM 3.4. Every 3-manifold possesses a contact structure. The proof of this result for compact manifolds comes from work of Lutz [76] and Martinet [78], who used a construction called the “Lutz twist” to perform Dehn surgery on a closed loop transverse to a given contact structure. More specifically, given such a closed loop on a contact manifold (M, ξ ), one chooses a tubular neighborhood N which is homeomorphic to D 2 × S 1 . Using a parameterized Darboux Theorem, this N can be chosen so that the characteristic foliation (∂N )ξ consists of closed curves of a fixed (small) slope on the boundary torus. To obtain a new 3-manifold, one removes N from M and replaces it with a solid torus N = D 2 × S 1 sewn in by a map : ∂N → ∂N which is not homotopic
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Fig. 5. Tight [left] and overtwisted [right] contact structures in R3 with cylindrical symmetry. Pictures are the contact planes at a slice {z = 0}: the full structure translates these planes along the vertical axis.
to the identity (see, e.g., [91] for an introduction to Dehn surgery). One can explicitly construct a contact structure on N that matches that of M when glued via by controlling the characteristic foliation on ∂N and using the Moser–Weinstein Theorem to show that the contact structures match if the foliations match. The Fundamental Theorem of Surgery on 3-manifolds [91] states that any closed 3-manifold can be obtained by performing Dehn surgery along some link in S 3 ; hence, all 3-manifolds possess a contact structure. There are several other means of proving this result, just as there are several means of constructing all compact 3-manifolds: open book decompositions [96], branched covers [52], etc. The proof for non-compact 3-manifolds is more subtle, following from Gromov’s h-principle [55,28]. Contact structures on closed 3-manifolds are implicitly global objects whose properties depend crucially upon a dichotomy first explored by Bennequin [10] and Eliashberg [24]. A contact structure ξ is overtwisted if there exists an embedded disc D in M whose characteristic foliation Dξ contains a limit cycle. If ξ is not overtwisted then it is called tight. The rationale behind the definition of tight and overtwisted is difficult to comprehend unless one is familiar with some of the intricacies in classifying codimension-1 foliations on 3-manifolds [29]. It is perhaps best to point out that there is a coupling between dynamical features of characteristic foliations of surfaces in (M, ξ ) [limit cycles] and the global topological features of the plane field. An example helps justify the name. The standard contact structure in polar coordinates, ker(dz + r 2 dθ ), is tight [10]. The contact structure defined by the 1-form cos r dz + r sin r dθ is in fact overtwisted. A disc of the form {z = r 2 ; 0 r π/2} has a limit cycle in its characteristic foliation at the boundary. What this limit cycle picks up is the fact that the slope of the contact planes dz/dθ = −r tan r becomes vertical and “twists over” in a periodic manner: see Figure 5[right]. Note that, in a neighborhood of the z-axis, this overtwisted contact structure is tight (Taylor expand the form about z = 0): this is consistent with the Darboux Theorem, which implies that all contact structures are locally tight. Hence, being overtwisted is a global property. Gray’s theorem reinforces this dichotomy by implying that tight and overtwisted structures are stable up to deformation—you cannot have a smooth family of contact structures change from tight to overtwisted or vice versa.
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The rationale for the term “tight” has to do with their lack of flexibility as compared to the overtwisted structures. For example, Eliashberg [24] has completely classified overtwisted contact structures on closed 3-manifolds. In short, on a compact 3-manifold, two overtwisted contact structures are isotopic through contact structures if and only if they are homotopic as plane fields. This means that algebraic topological invariants (characteristic classes) suffice to distinguish overtwisted contact structures. Such insight into tight contact structures is slow in coming. Some 3-manifolds (such as S 3 , [24]) admit a unique tight contact structure; some 3-manifolds (such as T 3 , [51,68]) admit an infinite number of tight contact structures; and some 3-manifolds admit no tight contact structures at all [37]. This completes our brief introduction to contact topology. For a more comprehensive treatment, see [25,26,30].
4. Contact structures and steady Euler fields in 3-d We now attend to our stated goals. First, we give existence and classification results for steady Euler fields on three-dimensional Riemannian manifolds using a contact-topological approach.
4.1. Reeb fields To every contact form α is associated a unique vector field, called the Reeb field, which captures the geometry of the 1-form in the directions transverse to the contact structure. The Reeb field of α, denoted X, is defined implicitly via the two conditions: ιX dα = 0 ;
ιX α = 1.
(4.1)
These conditions are sensible. The kernel of the contact 1-form α at a point is a plane in the tangent space. The kernel of the 2-form dα at a point is a line in the tangent space. The contact condition α ∧ dα = 0 is equivalent to saying that these two subspaces are transverse. Hence, the Reeb field captures the geometry of the contact form in the “orthonormal” direction to ξ . Of course, since a contact structure is defined independent of a Riemannian metric, there is no rigid notion of orthonormality other than that induced by the geometry of the contact 1-form. We say that a vector field is Reeb-like for α if ιX dα = 0 and ιX α > 0. A Reeb-like field has the same dynamics as the Reeb field, but with a different parametrization. Note that a fixed contact structure ξ possesses many different defining 1-forms, possessing Reeb fields which are potentially very different. The dynamics of the Reeb field, together with the geometry of the contact structure, suffice to reconstruct the contact 1-form. In the case of the standard tight contact form on S 3 given in (3.3), the Reeb field is precisely the Hopf field of (1.3). Thus, in the round metric, this contact structure is orthogonal to its Reeb field. On the other hand, for the standard tight contact structure on R3 , dz + x dy, the Reeb field is ∂/∂z. This is not orthogonal with respect to the Euclidean metric.
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4.2. A correspondence It has been observed at various points in the literature [18,94,50] that a nonvanishing curl eigenfield is dual to a contact form. A more exact correspondence between Reeb-like fields for contact forms and Beltrami fields follows by adapting metrics to contact forms [17]. The following theorem gives a precise formulation. T HEOREM 4.1 ([32]). Any rotational Beltrami field on a Riemannian 3-manifold is a Reeb-like field for some contact form. Conversely, any Reeb-like field associated to a contact form on a 3-manifold is a rotational Beltrami field for some Riemannian structure. P ROOF. Assume that u is a Beltrami field where ∇ × u = f u for some f > 0. Let g denote the metric and μ a volume form on M with respect to which u is divergence-free. Let α denote the one-form ιu g. The condition ∇ × u = f u translates to dα = f ιu μ. Hence, α is a contact form since α ∧ dα = f ιu g ∧ ιu μ = 0.
(4.2)
Finally, u is Reeb-like with respect to α since ιu dα = f ιu ιu μ = 0.
(4.3)
Conversely, assume further that α is a contact form for M having Reeb field X. Assume that Y = hX for some h > 0. There is a natural geometry making Y an eigenfield of the curl operator. Let g(v, w) =
1 (α(v) ⊗ α(w)) + dα(v, J w), h
(4.4)
where J is any almost-complex structure on ξ = ker α (a bundle isomorphism J : ξ → ξ satisfying J 2 = −I D ) which is adapted to dα so that dα(·, J ·) is positive definite. Such a J is known to exist [79] since dα is nondegenerate on ξ . Then, by definition of the Reeb field, ιY g = α. Thus, dιY g = dα. Let μ be the volume form h−1 α ∧ dα on M. Then ιY μ = dα and Y is a divergence-free eigenform of curl in this geometry. We leave it as an exercise to show that Y is divergence-free under the particular ginduced volume form if and only if the scaling function h is an integral for the flow, LY h = 0, as one would expect. 4.3. Existence on 3-manifolds Knowing the basic properties of the curl operator tells us that every Riemannian 3-manifold admits a volume-preserving curl eigenfield and thus a steady solution to the Euler equations. It seems at first unclear how one would construct a nonvanishing curl eigenfield solution on an arbitrary three-manifold.
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C OROLLARY 4.2. Every closed oriented three-manifold has a nonvanishing steady solution to the Euler equations under some geometry. P ROOF. Theorem 3.4 implies the existence of a contact structure. Choose any defining 1-form, consider its Reeb field, and outfit the manifold with a geometry as in Theorem 4.1. It remains an open question whether every Riemannian 3-manifold admits admits a nonvanishing curl eigenfield, or whether there are certain geometries which are hostile to contact geometry. Another way to think of this problem is in terms of the space of contact 1-forms in the set of divergence-free 1-forms on (M, g). The eigenforms of the curl operator (or, if you like, the Laplacian) yield a basis for this space. Question: is the set of contact forms “fat” or “thin” with respect to the spectral geometry of this operator? Does the eigenbasis always pierce the set of contact forms for any geometry?
5. Knots and links in three-dimensional flows Any exploration of topological features of 3-dimensional flows prompts a discussion of knotting and linking. Any periodic flowline forms an embedded loop, and, in a 3dimensional domain, there are a countably infinite number of distinct embedding classes for loops. More formally, a knot is an embedded loop in a 3-manifold, and a link is a collection of disjoint knots. Two knots or links are said to be equivalent or (ambient) isotopic if there exists a deformation (1-parameter family of homeomorphisms starting at the identity map) of the manifold taking the one knot/link to the other. The trivial knot or unknot is any loop which bounds an embedded disc. In 3-dimensional continuous dynamics, there is an intriguing relationship between the dynamics of a flow and the types of knots and links that are present. For example, (1) Any 3-d flow which is dynamically complicated (has positive topological entropy on a bounded invariant set) must necessarily possess infinite number of distinct knot types as periodic orbits [40]. (2) One can use linking data to order and infer information about bifurcations in certain classes of flows and in suspensions of 2-d maps [65,63,64]. (3) The existence of certain knot types as periodic orbits is incompatible with a nondegenerate integrable Hamiltonian system [14,39] and hence can imply that a given Hamiltonian system is nonintegrable. (4) There is a type of renormalization theory for links of periodic orbits which mirrors the dynamical renormalization theory in that the most complex types of periodic orbit links can be self-similar on the level of knot types [49]. With regards to fluid flows, there have been numerous investigations into the knotting and linking of flowlines [82–84,88,11,41]. Perhaps the best known topological feature of fluid flows is helicity, a quantity which measures the average asymptotic linking of flowlines [4,81,7] and has applications to energy bounds.
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Fig. 6. Examples of a nontrivial knot [left], an unknot [center], and a link [right].
5.1. Unknots Recall the example of the Hopf field (1.3) on the round 3-sphere. This nonvanishing field satisfies the steady Euler equations and fills the round 3-sphere with periodic orbits. Topologists are intimately familiar with this filling up of S 3 by circles: it is the well-known Hopf fibration of S 3 . The reader may verify that each of these periodic orbits is in fact an unknot. Furthermore, any two distinct orbits of the Hopf field are linked with linking number one, giving rise to the so-called Hopf link. To what extent do these knot-theoretic features hold for more general steady Euler fields on the 3-sphere with different geometries? There is a rigidity for topological fluids on spheres which is metric-independent. Again, as in the case with Arnold’s theorem, we require a sufficient degree of smoothness. T HEOREM 5.1 ([36]). Any nonvanishing solution to the C ω steady Euler equations on a Riemannian S 3 must possess an unknotted closed flowline. It is significant both that there is a periodic orbit and that it is unknotted. The existence of a periodic orbit in a nonvanishing vector field on S 3 was the content of the Seifert Conjecture, counterexamples to which have been ingeniously constructed in the C ω class by K. Kuperberg [72] and by G. Kuperberg [71] in the C 1 volume-preserving class. (It is as yet not known if a sufficiently smooth nonvanishing volume-preserving field on S 3 must possess a periodic flowline, but the suspected answer is ‘no’.) It is known [71] that there exist arbitrarily smooth nonvanishing volume-preserving fields on S 3 with a finite number of periodic orbits, all of which have nontrivial knot types. Hence, Theorem 5.1 implies that there is something peculiar to the topology of inviscid flowlines which is not shared by more general flows which are merely volume-preserving. There are two components to the proof of Theorem 5.1, corresponding precisely to the dichotomy implicit in Arnold’s theorem. Given the smoothness requirements, it is either the case that a steady nonvanishing Euler field u has a nontrivial integral, or that u is a curl eigenfield. In the former case, the following result gives a slightly stronger conclusion.
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T HEOREM 5.2 ([36]). Any nonvanishing vector field on S 3 having a C ω integral of motion must possess a pair of unknotted closed orbits. We do not give the proof of this result here, as it involves both a very detailed examination of round handle decompositions of 3-manifolds [8,85,97] as well as a classification of all possible critical sets for said integral. It is perhaps best to think of this theorem via a 2-dimensional analogue. Given a 2-sphere S 2 and a flow on it that possesses a nontrivial C ω integral (a stream function), there must exist at least two stagnation points of elliptic type corresponding to a maximum and a minimum of the stream function. The second component to the proof of Theorem 5.1 concerns the curl-eigenfield case. If there is no nonconstant integral for u, then we know that ∇ × u = λu for some λ ∈ R. It cannot be the case that λ = 0, for this would mean that u is dual to a closed 1-form on S 3 . This closed form is not exact, since it is nonvanishing, and thus violates the fact that the first cohomology group of S 3 vanishes. Hence, we are left with the case where λ = 0 and u is, by Theorem 4.1, a Reeb-like field for the contact form α = ιu g. This is the point at which tools from contact topology show their strength. An extremely strong result of Hofer, Wysocki, and Zehnder provides the desired result. T HEOREM 5.3 (Hofer, Wysocki, and Zehnder [61]). Any Reeb field on S 3 possesses an unknotted periodic orbit. We can provide neither a proof nor a careful exposition of this proof. Instead we give an introduction to a related theorem, as this plays a role in many current ideas in contact topology and will be very important in our discussion of hydrodynamic instability in §6. T HEOREM 5.4 (Hofer [60]). Any Reeb field associated to an overtwisted contact structure on a closed orientable 3-manifold possesses a periodic orbit. The proof of this result contains the key idea for Theorem 5.3, and itself draws on the work of Gromov on pseudoholomorphic curves in symplectic manifolds [9]. The counterintuitive idea is that to analyze the three-dimensional contact manifold (M, α) with a Reeb field X, it is helpful to construct a four-dimensional symplectic manifold (W, ω) built from (M, α) and use two-dimensional techniques from complex analysis. This is perhaps not too foreign an idea to fluid dynamicists, for whom complex analysis is already an invaluable tool for planar flows. It is in the application to higher dimension fluid flows, however, that more sophisticated contact and symplectic topology arises. Given a 3-manifold M with contact form α, the symplectization of M is the 4-manifold W = M × R outfitted with the 2-form ω = d(et α), where t denotes the R-factor in W . This 2-form is a symplectic form, as it is closed (dω = 0) and nondegenerate (ω ∧ ω = 0). Nondegeneracy follows from the fact that α is contact: ω2 = [et (dα + dt ∧ α)]2 = 2e2t α ∧ dα ∧ dt = 0.
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Fig. 7. The effect of the almost-complex multiplication J on T W is to rotate in the contact plane ξ according to a symplectic basis for dα and to rotate the Reeb direction into the symplectization direction ∂/∂t .
Indeed, contact and symplectic structures are intimately related, the former being an odddimensional cousin of the latter. See, e.g., [79] for a more complete exploration of these ideas. The symplectic manifold W therefore captures the contact geometry of M. To utilize complex methods, we outfit W with an almost complex structure (a linear map J : T W → T W satisfying J 2 = −I D ) that respects the geometry of W . Think of each tangent space Tx W as being spanned by the Reeb field X, the t-direction ∂/∂t, and the contact plane ξ . Choose J so that it rotates the plane spanned by the X and ∂/∂t directions by a quarter-turn (like multiplication by i in C). In the plane of ξ , J is chosen so as to be adapted to dα, as in Equation (4.4). This choice of J decouples the contact directions from the Reeb field X and entwines the dynamics of this Reeb field with the t-direction: see Figure 7. Fixing such a J , one defines J -holomorphic curves in W as maps ϕ : → W from a Riemann surface (, j ) to W such that dϕ ◦ j = J ◦ dϕ. It follows easily from Stokes’ Theorem that there are no compact Riemann surfaces in W ; one must introduce punctures [60]. If is a punctured Riemann surface, the energy of is defined to be ϕ ∗ (dα). This energy measures how much area of the surface is “visible” to the contact planes (on which dα is an area form). The crucial lemma of Hofer’s: if a J -holomorphic curve ϕ = (w, h) : → M × R has finite energy then any (non-removable) puncture can be shown to possess a neighborhood parametrized by {(θ, τ ) : θ ∈ S 1 and τ ∈ [0, ∞)} such that limτ →∞ h approaches ±∞ and limτ →∞ w(θ, τ ) approaches a parametrization of a periodic orbit γ for X. The intuition behind this is that if a surface has finite energy, then in the limit as t → ±∞, the surface must be orthogonal to the contact planes, and thus tangent to the (X, ∂t∂ ) planes: see Figure 8. For more information on finite energy holomorphic curves and their asymptotics see [9]. Now the goal is to find finite energy J -holomorphic curves in W . The main result of [60] is that whenever ξ is an overtwisted contact structure on M, then there exists a finite energy J -holomorphic curve in the symplectization. (The process for constructing this is derived from a Bishop filling.) Thus, any Reeb field for any overtwisted contact structure on a closed orientable 3-manifold possesses a periodic orbit (solving the Weinstein Conjecture in these cases). On S 3 , since there is a unique tight contact structure up to contactomorphism [24], one need merely augment the above argument with a variational method that
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Fig. 8. A finite energy J -holomorphic map ϕ from a punctured surface into the symplectization W = M × R has punctures limiting to cylinders over periodic orbits of the Reeb field as t → ±∞.
works for the specific tight case [60]. With that, one has that all Reeb fields on S 3 possess a periodic flowline. This (strenuous) effort yields a single periodic orbit, with no information about its knot type. To prove the unknotting result of Theorem 5.3 requires the construction of parametrized families of finite-energy curves in the symplectization W which project down in M to a foliation branched over certain singular loops in M. See [61] for the full details. These sophisticated methods give information about fluids which are of a purely topological nature. No matter what geometry is placed on a 3-sphere, any (smooth enough) steady Euler flow is guaranteed to possess an unknotted flowline. The results are completely implicit: one has no idea where the periodic orbit lies.
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5.2. Knots The results on unknots above provide a “lower bound” on the size and structure of the periodic orbit link for a fluid flow on the 3-sphere. One naturally wonders if there is a corresponding “upper bound” for the periodic orbit link, or, indeed, any other types of restrictions on what is and is not possible. We begin by noting that integrable fields have a great deal of restrictions associated to the periodic orbit knot types. This should come as no surprise, given the constraints imposed by Arnold’s Theorem—the manifold is filled with invariant tori. It is known precisely which types of knots can appear as closed flowlines in a nonvanishing integrable field on a 3-sphere: these are the so-called zero-entropy knots [49,36]. This class consists of those knots obtainable from the unknot by iterating the operations of cabling (wrapping around a core curve) and connected sum (splicing two knots together), as in Figure 9. Such knots are a very small subclass of knots, excluding, e.g., the hyperbolic knots (knots whose complement in S 3 supports a hyperbolic geometry). Hence we conclude that the only types of (sufficiently smooth) fluid flows which may support more complex periodic orbit links are the curl eigenfields, and thus, Reeb fields. In a series of papers [82,83], Moffatt discusses knots and links in Euler flows “with arbitrarily complex topology.” What is meant by this is the construction of steady solutions to the Euler equations on Euclidean R3 which realize the same orbit topology as any given volume-preserving flow on the space. These results have the advantage of staying within the class of Euclidean metrics. However, there are two caveats associated to this work: (1) the techniques do not guarantee a continuous solution—so-called “vortex sheet” discontinuities may develop; (2) the proof itself relies crucially upon the global-time existence of solutions to the Navier–Stokes equations (with a modified viscosity term). Such an existence theorem is to date unknown. We circumvent these problems by allowing for flexibility in the geometry of the flow domain. With contact-topological techniques, it is relatively simple to use surgery (or cutand-paste) methods on contact forms: cf. the discussion after Theorem 3.4. Using such techniques one can produce all manner of complicated knotting and linking in a Reeb field, and, hence, in a steady fluid flow. In fact, the most complex forms of knotting and linking imaginable are realizable within the category of steady ideal fluids.
Fig. 9. Examples of zero-entropy knots.
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Fig. 10. An example of a flow defined by stretching, squeezing, and twisting of tubes [left] which possesses a 1-d hyperbolic invariant set and collapses to a branched surface [right]. This branched surface is a universal template: the semiflow on this object possesses all knot and link types as closed orbits.
T HEOREM 5.5 ([34]). For some C ω Riemannian structure on S 3 (or R3 if preferred), there exists a C ω nonvanishing steady Euler field possessing periodic flowlines of all knot and link types. Note that this does not merely say that any given knot or link type may be realized in some steady Euler flow; rather, all of them are realized together in a single fluid flow. It is not a priori obvious that such any vector field in 3-d can have this property, much less one which is a volume-preserving Euler field: it was shown in [48] that such vector fields do exist in general. The types of flows which admit such periodic orbits links must of necessity be very complex and chaotic. Those chaotic flows which can be best analyzed possess hyperbolic invariant sets of dimension one. Roughly speaking, this means that one can find a onedimensional invariant set and a flow-invariant splitting of the tangent bundle T = E s ⊕ E c ⊕ E u so that E c is spanned by the vector field and the flow is exponentially expanding (resp. contracting) on E u (resp. E s ). To analyze knotting and linking of orbits within , one collapses the E s direction in a neighborhood of to obtain a branched surface with a semiflow—a template, as in Figure 10—which captures the topology of a 1-dimensional hyperbolic invariant set in a 3-d vector field, much as a 2-dimensional approximation to the Lorenz attractor does [57,12]. To construct such a flow for an ideal fluid in 3-d is possible, thanks to Theorem 4.1. L EMMA 5.6. There exists a Reeb field on the 3-torus possessing a nontrivial onedimensional hyperbolic invariant set.
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P ROOF. Consider the ABC equations of §2. From Equations (2.3) and (4.1) it follows that the ABC fields lie within the kernel of the derivative of the 1-form (A sin z + C cos y)dx + (B sin x + A cos z)dy + (C sin y + B cos x)dz,
(5.1)
and that this is a contact form when the vector field is nonvanishing. In the limit where A = 1, B = 1/2, and C = 0, the vector field is nonvanishing and has the form x˙ = sin z y˙ = 12 sin x + cos z .
(5.2)
z˙ = 12 cos x For these values, there exists a pair of periodic orbits whose stable and unstable invariant manifolds intersect each other nontransversally (see, e.g., [22]). Upon perturbing C to a small nonzero value, this connection may become transverse, inducing chaotic dynamics. Indeed, a Melnikov perturbation analysis establishes this fact [19,43,47,54,67,101]. It thus follows from the Birkhoff–Smale Homoclinic Theorem [56,89] that nearby parameters force Equation (2.3) to possess a nontrivial 1-d hyperbolic invariant set as a solution. Sketch of proof of Theorem 5.5: It follows from the Birkhoff–Smale Theorem that the 1-d hyperbolic invariant set of Lemma 5.6 is a suspended Smale horseshoe (see [56,89] for background and explanation). As such, this invariant set lies within a neighborhood N , a solid handlebody of genus two. It is an unfortunate consequence that the image of the embedding of N in T 3 unknown: it may be a very thin and tangled subset. To construct a vector field on S 3 with all knots and links, N must be unwound. It suffices, via [48] to embed N in S 3 so that the handles of N are (1) unknotted, (2) unlinked, and (3) twisted in an appropriate manner as in Figure 10[right]. See [100,49] for an exposition of this result. By choosing the handles of N to be sufficiently thin (which is possible by restricting to a smaller hyperbolic invariant set), one can use an argument as in the Darboux Theorem (Theorem 3.1) to determine the characteristic foliation of the contact structure on ∂N . A surgery argument then suffices, as follows. Choose two curves, γ1 and γ2 , which are transverse to the characteristic foliation on ∂N and wrap around each handle of N once with an appropriate number of twists. One then glues two thickened discs of the form D 2 × (−, ) to N by attaching the annuli ∂D 2 × (−, ) to neighborhoods of γ1 and γ2 on ∂N . By outfitting these balls with the standard contact structure for R3 and matching the characteristic foliations along the gluing, the Moser–Weinstein Theorem (Theorem 3.2) gives a well-defined contact structure on the union, which completes N to a contact 3-ball. Contact structures on the 3-ball are classified [24] and it is known that any such structure can be completed to a contact structure on S 3 . Thus, there exists a contact form α on S 3 which agrees with the 1-form of (5.2) at the chosen parameter values on N ⊂ T 3 and whose Reeb field has a 1-d hyperbolic invariant set embedded so as to exhibit all knot and link types. This method of proof yields only a C ∞ Reeb field, since cutting and pasting arguments are used. To obtain a C ω Reeb field, one can use a classification argument in [34] to show
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that the contact form for this Reeb field is equivalent to f αH , where αH is the C ω form associated to the standard contact form on S 3 in Equation (3.3). A small perturbation to the coefficient function f yields f˜αH , a C ω form with Reeb field of the same smoothness class. Since 1-d hyperbolic invariant sets in 3-d flows are structurally stable, this perturbation does not change the link of periodic orbits in this set: hence, the all-knots property still holds. 6. Instability Recall that the theme of our investigations into the Euler equations is to begin with the dynamically simplest elements—the steady solutions—and then build up a repertoire of invariant objects. The steady solutions comprise the fixed points of the Euler equations in the space of volume-preserving vector fields. The next natural phenomenon to investigate is the local dynamics of the time-dependent Euler equations near these fixed points. This corresponds to the classical question of hydrodynamic stability. The problem of hydrodynamic stability and instability for steady Euler flows on three-dimensional domains is an important subject with a rich history. See, e.g., [15,23,74]. There are numerous notions of stability and instability for fluids, the full extent of which we do not here survey. A steady velocity field u is said to be (nonlinearly) hydrodynamically stable if, given any neighborhood of u in the space of volume-preserving vector fields, then any arbitrarily small divergence-free perturbation u + v evolves under the timedependent Euler equations in such a way that it does not leave the assigned neighborhood. Otherwise, u is said to be hydrodynamically unstable (some authors require an exponential divergence of a perturbed field from u). In all this, it is necessary to carefully specify the norm via which “smallness” is measured. We will use the L2 or energy norm exclusively: u = u · u dμ. M
Examples of stable flows are hard to come by in dimension higher than two, since longtime existence of solutions is unknown. One of the common themes to be found in the literature on hydrodynamic stability is that “almost every” steady Euler field is hydrodynamically unstable in dimension three. This begs the question of what is meant by “almost every” in the context of steady Euler fields, when, as we have seen in Theorem 2.1, steady Euler fields have some topological constraints and may or may not appear in parameterized families. We therefore turn to questions of generic fluid behavior in dimension three. The small literature on generic properties of fluid flows (primarily [38,92]) focuses on the Navier– Stokes setting and uses external forcing or Dirichlet data as a parameter, see [1, 3.6A]. In accordance with the theme of this article, we use the geometry of the domain as the genericity parameter. We outline a clear formulation of the problem and present a generic instability theorem for the curl eigenfields with respect to the underlying Riemannian metric. This approach skirts the difficulty of dealing with having steady solutions which are isolated in the space of volume-preserving vector fields (up to normalization). This approach also allows for some very fascinating topological methods to be applicable.
On the contact topology and geometry of ideal fluids
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6.1. Instability criteria The literature on hydrodynamic stability is full of various criteria, which vary in scope and effectiveness. See [15,23,74] for a classical introduction and [42,46] for more recent surveys. It suffices for our purposes to note that stability and instability criteria for 3-d Euler flows, when they exist at all, tend to be very limited in applicability. We therefore turn to a weaker notion of stability: linear stability. A steady Euler field u is said to be linearly stable if, for every sufficiently small divergence-free field v(0), the evolution of v(t) under the linearized Euler equation about u, ∂v + (u · ∇)v + (v · ∇)u = −∇p, ∂t
(6.1)
is bounded in some predetermined norm. We will use exclusively, following [42,45], the energy or L2 norm on vector fields. The solution u is thus said to be linearly unstable if, for some arbitrarily small v(0), the solution v(t) has exponential growth in energy norm. We note that, unlike in finite-dimensional dynamics, it is not necessarily the case that linear instability implies nonlinear instability. Arnold suggested that the underlying dynamics of the flowlines of the steady solution u can force instability. We now know this to be the case following the contributions of several researchers, including Bayly, Friedlander, Hameiri, Lifshitz, and Vishik (see, e.g., [42,44, 46] and references therein). The criterion we will use comes from the work of Friedlander– Vishik [45], who used techniques from geometric optics developed by Lifshitz–Hameiri [73]. This instability criterion requires some expanding dynamics within the flow, the simplest examples of which are fixed points and periodic orbits which are nondegenerate and of saddle-type. A nondegenerate fixed point is one whose eigenvalues are all nonzero. A nondegenerate periodic orbit for a volume-preserving flow is defined to be one whose Floquet multipliers (eigenvalues of the linearized return map to a cross-section of the orbit) are not equal to one. Nondegenerate periodic orbits of a 3-d volume-preserving flow are either of hyperbolic or elliptic type: see Figure 11 for an illustration, and compare with the 2-d case of Figure 1. (It is not the case that elliptic orbits are necessarily surrounded by invariant tori—this classification records only linear behavior of the return map.) Instability Criterion: [45]. The presence of a nondegenerate hyperbolic fixed point or periodic orbit in a steady Euler flow induces linear instability in the energy norm.
6.2. Generic curl eigenfields The genericity result we survey here uses the Riemannian metric as a parameter space. Theorem 2.1 gives an indication of how to approach the set of steady Euler fields. Surprisingly, it appears much easier to examine the generic behavior of curl eigenfields than that of integrable solutions, even though the dynamics of curl eigenfields can be so much more complex. In order to analyze the “typical” behavior of curl eigenfields, we use an Implicit Function Theorem argument similar to that Uhlenbeck [102] and Henry [59] used for eigenfunctions
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Fig. 11. Elliptic [left] and hyperbolic [right] periodic orbits in a volume-preserving 3-d flow are characterized by the linearization of the cross-sectional return map.
of the Laplacian. We switch to eigenforms for simplicity, under which the curl operator translates to ∗d. We work on the space of divergence-free 1-forms on M with respect to the metric g as a parameter. Denote the space of Riemannian metrics on M by G and let
E0 = (g, α) ∈ G × 1 (M) : divα = 0 ,
(6.2)
E = ker(∗d|E0 )⊥ .
(6.3)
and
Note E is a bundle over G and the curl operator ∗d is a fibrewise map. From the Hodge theorem we know that ∗d : E → E is a bundle isomorphism. Normalize E to S = {(g, α) ∈ E : α2 = 1} and consider the operator φ :S×R→E
;
φ(g, α, λ) = (g, ∗dα − λα) .
(6.4)
The φ-inverse image of the zero section gives the curl eigenforms. Thanks to the normalization, this is a fibrewise index zero Fredholm operator to which the transversality theory detailed in [102] applies. The following technical result is extremely illustrative, as it demonstrates that the curl eigenfield solutions are very well-behaved for a “typical” Riemannian geometry. We sketch enough of the proof to give an idea of what is involved. L EMMA 6.1. For each r 1, there exists a residual set in the space of C r metrics on a closed M 3 such that the eigenspaces of the curl operator (with non-zero eigenvalue) are 1-dimensional and vary smoothly with the metric. To prove this, one shows that the zero-section 0 of E is a regular value of φ; then, following [102], Q = φ −1 (0) is a manifold that fibers over G with projection π . A Gδ dense set of metrics will be regular values of π and, for these values, Qg = π −1 (g) =
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φg−1 (0) is a 0-dimensional manifold (here φg = φ|π −1 (g) ). For each point (α, λ) in Qg we have ∗dα = λα. This λ is a simple eigenvalue of curl since 0 is a regular value of φ (cf. [102, Lemma 2.3]). The eigendecompositions vary smoothly since Q is a manifold. The smoothness condition is required for the application of the Sard–Smale theorem. Showing that the zero-section of E is a regular value of φ is a fairly straightforward computation of the derivative of φ with respect to the Riemannian metric: see [35] for details. Although this result seems technical, it is really quite straightforward—one wishes to use an Implicit Function Theorem style of argument, and to do so always requires care in computing derivatives and checking smoothness. What this work gains us is the following. For such a generic metric, one can unambiguously designate the i th eigenfield of curl, for i ∈ N. These eigenfields vary smoothly with the metric parameter and allow one to work “one eigenfield at a time.” Since we are concerned with topological genericity (countable intersections of open, dense sets), we may prove a genericity result for the i th curl eigenfield and then take the intersection over all i ∈ N. This is the strategy behind the proof of the following results: L EMMA 6.2. There is a Gδ dense subset of C r metrics in G (r 2) for which all curl eigenfields with non-zero eigenvalues have all fixed points nondegenerate. The idea behind this proof is to consider ψ : Q × M → T ∗M
;
ψ(g, α, λ, x) = α(x)
(6.5)
and show that the zero-section 0 is a regular value of ψ. L EMMA 6.3. For each i ∈ Z − {0} and each positive integer T , there exists an open dense set of metrics in G so that, if the i th eigenfield of curl has no fixed points, then all of the periodic orbits of period less than T are nondegenerate. This result is more involved, and requires a few tools from contact geometry. We sketch the main steps. The idea is to consider an open set Uα of contact 1-forms near a given contact eigenform α. The goal is to argue for the existence of a dense open set in Uα of 1-forms whose Reeb fields are nondegenerate (i.e., all periodic orbits are nondegenerate). Let α be a contact 1-form in Uα . Gray’s Theorem (Theorem 3.3) says that any nearby contact form α can be deformed through a contact isotopy to the contact form f α, for some near-identity scalar function f . From the proof of Gray’s Theorem, this isotopy is smooth with respect to α —the entire neighborhood of 1-forms near α can be contactisotoped to near-identity rescalings of α. From this, one proceeds as in [62, Prop. 6.1] to examine how the Reeb of α = f α behaves for generic choice of f . This is best seen by using the symplectization of (M, α). Recall from the proof of Theorem 5.4 that this is the 4-manifold M × R with the symplectic form ω = d(et α). The contact form α is that induced on M by ω via the embedding of M to M × {0} ⊂ M × R and the Reeb field is the Hamiltonian dynamics on an energy level in the 4-manifold M × R defined by the Hamiltonian function f . A genericity result of C. Robinson [90, Thm. 1.B.iv]
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meant for Hamiltonian dynamics now implies that there is a generic set of near-identity functions f such that the Reeb field for f α has all periodic orbits nondegenerate. From Lemmas 6.1 through 6.3, we have that, for a generic geometry on M, all the curl eigenfields have either all fixed points nondegenerate, or, in the case of no fixed points, all the periodic orbits are nondegenerate. If there are any fixed points, then volumeconservation implies that they are of saddle type and hence induce linear instability. The obvious question is, then, what to do about the case where there are no fixed points? We have a guarantee that all periodic orbits must be nondegenerate, but we do not know that any periodic orbits necessarily exist. Indeed, problems associated with the existence of periodic orbits in nonvanishing vector fields on 3-manifolds are exceedingly subtle. Fortunately, we do have Theorem 5.4 for the case when the contact structure is overtwisted. However, even in that case, there is the added complication of worrying about elliptic versus hyperbolic periodic orbits. Although both types are nondegenerate, only the hyperbolic orbits yield information about hydrodynamic instability. Fortunately, this is easily overcome for overtwisted structures (see the proof of Theorem 6.5 for the argument). The question of what to do with the case of tight structures is, as always, more challenging. To address this last, most delicate, question, we turn to a very powerful tool in contact topology.
6.3. Contact homology One of the central problems in the topology of contact structures is the classification problem: given contact structures ξ and ξ on M, are they equivalent? This problem was greatly clarified by the tight versus overtwisted dichotomy and the theorem of Eliashberg [24] that the overtwisted structures are classified by the homotopy type of the plane field. That is, the plane fields are isotopic if there is a continuous 1-parameter family of smooth plane fields connecting the two. This is fairly simple to determine with standard tools from algebraic topology [characteristic classes]. This classification is much more subtle in the case for the tight structures. See [30,66] for recent progress in this area. With regards to classification, Eliashberg, Givental, and Hofer have constructed a powerful new homology theory for contact structures. Homology is a tool from algebraic topology which, in the usual category, provides an invariant of topological spaces up to homotopy type. There are numerous flavors of homology theory, each of which involves ‘counting’ special objects in a particular manner. E.g., the Euler characteristic of a triangulated surface is a particular way of counting simplices which yields a topological invariant: this invariant is homological in nature. A more relevant example is Morse homology for a finite dimensional oriented manifold M [93]. One begins with a choice of function f : M → R. The Morse homology counts the number and type of fixed points of the gradient vector field X = −∇f . It is necessary to assume that f is chosen so that these fixed points are all nondegenerate (i.e., that f is Morse). One defines the grading on these fixed points to be the Morse index, the dimension of the unstable manifold. Next define the n-chains to be the real vector space Cn (M, f ) which has basis all the fixed points of Morse index n. The tricky part of this homology theory is to define boundary operators: linear transformations ∂ : Cn → Cn−1 which satisfies ∂ ◦ ∂ = 0.
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Fig. 12. An example of Morse homology on a 2-sphere with height function f : S 2 → R as shown. The critical points and connecting orbits generate chain groups and boundary maps.
For any fixed point p ∈ Cn , ∂(p) is a linear combination of points qi ∈ Cn−1 whose stable manifolds intersect the unstable manifold of p in a heteroclinic connection. By assigning the proper orientation to the stable and unstable manifolds, one gets ∂ ◦ ∂ = 0. The homology of the resulting chain complex, MH∗ (M, f ), is well-defined and can be shown to be independent of f ; in fact, the Morse homology is isomorphic to the homology of M. See Figure 12 for an example on a 2-sphere, where the relevant portion of the chain complex is:
1 1 0 0
0 1 0 1
Za ⊕ Za −→ Zb ⊕ Zb −→ Zc ⊕ Zc .
(6.6)
This chain complex has MH2 ∼ = Z, MH1 ∼ = 0, and MH0 ∼ = Z, as expected from H∗ (S 2 ). Contact homology shares many similarities with Morse homology, though it is an infinite-dimensional theory which exploits contact topology. Briefly stated, contact homology counts periodic orbits of a Reeb field modulo the boundary map which counts pseudoholomorphic curves in the symplectization of the contact manifold.
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The details are as follows. Given the contact structure ξ , one begins with a choice of a contact form α. Any two such 1-forms are related by some f : M → R; hence the ‘choice,’ as in Morse homology, is an appropriate f . Given α, the Reeb field X is the dynamical object corresponding to the gradient field in Morse homology. It’s natural invariant sets, the periodic orbits, are what one counts. Let C be the set of periodic orbits for X. To each element c ∈ C, a grading, |c|, can be assigned using a shifted Conley–Zehnder index— an integer which is approximately equal to the number of half-twists the linearized flow performs along one cycle of the periodic orbit. We do not give a precise definition as the only feature of the grading of concern here is the following: Any nondegenerate orbit c with |c| odd is hyperbolic [60]. At this stage, the difficulty arises in defining the chains at grading n and the boundary maps between chains. A fair amount of algebraic machinery is required. One defines the graded algebra A as the free super-commutative unital algebra over Z2 with generating set C. This algebra A will be the set of all chains for contact homology. To define the boundary operator, we recall the relationship between periodic orbits of the Reeb field and finite-energy pseudoholomorphic curves, from the proof of Theorem 5.4. Unlike in the case of Theorem 5.4, we care about J -holomorphic curves which have punctures which go to +∞ and −∞, each such puncture asymptoting to a periodic orbit of the Reeb field. Given periodic orbits a, b1 , . . . , bk ∈ C, let Mab1 ...bk denote the set of finite energy holomorphic curves in the symplectization W with one positive puncture asymptotic to a and k negative punctures asymptotic to b1 , . . . , bk , (modulo holomorphic reparametrization). Since J is R-invariant on W , there is an R-action on M. One now defines ∂a =
# Mab1 ...bk /R b1 . . . bk ,
(6.7)
where the sum is taken over all b1 , . . . bk such that the dimension of Mab1 ...bk is 1. It is a deep and difficult theorem [27,13] that the differential ∂ lowers the grading by 1, and, for a generic contact 1-form (and almost complex structure) ∂ 2 = 0 and the homology of (A, ∂) is independent of the contact form chosen for ξ (and the almost complex structure). The resulting homology of (A, ∂) is called the contact homology of (M, ξ ) and is denoted CH (M, ξ ). Contact homology comes in a number of specialized flavors (by restricting the types of surfaces used) which are useful in computations: see [13]. In addition, there is an extremely broad generalization of contact homology called symplectic field theory [27] which we do not here discuss. We now have the tools available to prove a genericity result for hydrodynamic instability. For concreteness, we restrict to the case of periodic flows on R3 , or, flows on a Riemannian 3-torus T 3 = R3 /Z3 . Our aim is to show that, for a generic geometry, all of the curl eigenfields (except perhaps that with eigenvalue zero) are hydrodynamically unstable. Proving the hydrodynamic instability theorem requires knowing the existence of a hyperbolic periodic orbit for all non-degenerate Reeb fields on T 3 . The proof of Theorem 5.4 guarantees that any Reeb field for an overtwisted contact structure possesses a closed orbit of grading +1. In the nondegenerate case, such an orbit is of hyperbolic type. Thus,
On the contact topology and geometry of ideal fluids
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we need merely cover the case of the tight contact structures. For T 3 , these are classified [51,68]: there is an infinite family of isomorphism classes represented by ξk = ker (sin(kz)dx + cos(kz)dy) ,
(6.8)
for k ∈ Z − {0}. A relatively simple contact homology computation is the crucial step in the following result: L EMMA 6.4 [35]. For a nondegenerate Reeb field associated to any tight contact structure on T 3 , there is always a hyperbolic periodic orbit. The idea of this proof is, of course, to compute the contact homology of each ξk above and show that there must (either because of a nontrivial homology class or through clever counting) exist a nonzero chain in an odd grading; hence a saddle-type hyperbolic periodic orbit. See [35] for the computation.
6.4. Generic instability These ingredients combine to prove the following result. T HEOREM 6.5. For generic choice of C r metric (2 r < ∞), all of the curl-eigenfields on a three-torus T 3 (with nonzero eigenvalue) are hydrodynamically unstable [linear, L2 norm]. P ROOF. Lemma 6.1 implies that we can work one eigenfield at a time. First, use Lemmas 6.2 and 6.3 to reduce everything to either nondegenerate fixed points or periodic orbits. Given such a field u, if it possesses a fixed point, then it is immediately of saddle type due to volume conservation and satisfies the Instability Criterion. If the field is free of fixed points, then it is (after a suitable rescaling which preserves the topology of the flowlines) a Reeb field for the contact form α = ιu g. If the contact structure ξ = ker α is overtwisted, Theorem 5.4 implies the existence of a periodic orbit with grading +1. The nondegeneracy implies that the orbit is of saddle type and hence forces hydrodynamic instability. In the final case where ξ is tight, the contact homology computation of Lemma 6.4 implies instability. The advantage of a result such as Theorem 6.5 is that it is precise—there is no guessing what ‘generic’ means. That is uses some very deep ideas from contact topology is an advantage from the point of view of a topologist, but perhaps only to a topologist. To a fluid dynamicist, however, the result is by no means optimal. Genericity in the geometry means that the Euclidean geometry (a highly non-generic metric for which fluid dynamicists have a preference) is not covered by this result. In addition, the use of the Instability Criterion provides information only about linear instability, and a fairly localized high-frequency instability at that: information about nonlinear instability would be greatly preferable. It is likely that there are other approaches to a ‘generic instability’ theorem
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which would yield stronger results. Our goal was to demonstrate rigorous results about fully 3-d ideal flow using a minimal set of restrictions or assumptions.
7. Concluding unscientific postscript Flows are inherently topological. Many of the most basic and general results in inviscid fluids have a topological spirit to them—the Bernoulli theorem, the Kelvin theorem, and the Helmholtz theorem being prominent examples that all spring from very basic results of differential topology. Results which depend upon more sophisticated topological machinery abound in certain fragments of the literature. The best place to start is the fairly recent monograph of Arnold and Khesin [7] which contains a wealth of information that is complementary to the tools described in this survey. Fluid dynamics (mathematical or physical) encompasses far more than topological methods, as a brief perusal of this volume will demonstrate. However, we do argue that there is merit in exploring very basic open questions about the topological features of ideal flow to which the most current ideas in topology and geometry are applicable. This article has outlined the small role played by methods in contact topology alone—itself a very small but rapidly developing branch of topology. We neither assert nor believe that these techniques compete with current analytic methods for understanding physical fluids.
7.1. Generic fluids However, we do argue that this perspective on fluids—which is ‘unscientific’ in so far as it is of limited use to a scientist for whom fluids are wet—is not without benefit. Throughout the article we have seen theorems that either did not depend on the Riemannian structure or which applied to generic Riemannian metrics. This leads one to the question, “Which features of a fluid are generic with respect to the domain geometry?” We have already seen certain deleterious effects of Euclidean geometry on steady solutions to the Euler equations. For example: (1) Any attempt to analyze the knots and links which can arise as periodic flowlines of a steady Euler field on a Euclidean domain is doomed to languish in numerical simulations. As Theorem 5.5 demonstrates, it is relatively easy to prove that anything is possible, so long as one is not confined to the Euclidean setting. (2) We showed in Lemma 6.1 that for a generic Riemannian metric, the curl operator has a simple eigendecomposition with one-dimensional eigenspaces and smooth continuation with respect to the metric. This is certainly false in the Euclidean case: witness the ABC fields, which form a three-dimensional eigenspace of curl, reflecting the symmetry of the Euclidean 3-torus. (3) In Lemmas 6.2 and 6.3, we further demonstrated that for a generic set of metrics, all of the curl eigenfields have nondegenerate fixed points or periodic orbits, which permits the use of Morse-theoretic tools. This is certainly not the case in the Euclidean geometry.
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(4) Such results were helpful in proving the generic linear instability result—again, a result that does not apply to the Euclidean case. Could it be the case that a number of difficulties in global results about fully 3-d inviscid flow are the result of degeneracies in the Euclidean geometry? Perhaps the biggest open problem in mathematical fluid dynamics is that of finite-time blow-ups. Several controversial papers have attempted to detect blow-ups numerically [69,86,87]. Though these results are not conclusive, they do indicate the possibility of a blow-up. Furthermore, some of these very clever configurations are leveraging symmetry in order to try and force a blow-up [87]. Given the examples above in which degenerate phenomena vanish when restricting to a generic set of Riemannian geometries, one is led to the highly speculative idea that Eulerian finite-time blow-ups, if they exist, may be a function of the degeneracies implicit in Euclidean geometry. Whether this is the case or not, it might help remove some difficulties in the analysis to restrict attention to a generic Riemannian geometry on a compact manifold and use global analysis to develop a theory of “generic” inviscid fluids.
7.2. Closing questions Turning again to the contact topological methods surveyed in this article, there is an abundance of open directions for future work. (1) This article has not covered domains with boundary, for which one must modify the Euler equations to keep the boundary an invariant set for the velocity field. Such settings are very relevant in, e.g., accelerators in plasma dynamics and MHD. The methods we use are applicable to such domains. See [33,31] for examples of contact topological methods on domains with boundary. One of the possible improvements in the generic instability result of Theorem 6.5 would be to work on a solid torus M = D 2 × S 1 embedded in Euclidean R3 and use as a parameter space the space of all embeddings of the boundary ∂M into R3 . Instead of varying the metric, one varies the shape of the boundary. It is an open problem to determine if the genericity results hold for this parameter space. The recent monograph of Henry [59] gives a careful analysis of the generic behavior of the scalar Laplacian with respect to this parameter space. (2) A related generalization arises in consider flows on non-compact domains. Proofs involving hydrodynamic instability, in particular, may change their character substantially as the spectral geometry of the curl operator differs on a noncompact domain. (3) The initial reaction to the statement of the Arnold Theorem is that the integrable solutions are the ‘typical’ steady Euler fields and the curl eigenfields are the ‘exceptional’ solutions. Likewise, one is tempted to say that among the class of Beltrami fields, where ∇ × u = f u, the case of a pure curl eigenfield where f is a constant is the exceptional case. To what extent are these intuitions true? Lemma 6.1 implies that the curl eigenfields are very robust with respect to perturbations of the metric— most eigenfields can be continued smoothly. It is not clear to us that the same can be said for integrable solutions to the Euler equations.
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(4) The tight-overtwisted dichotomy in contact topology is one which has a great amount of power: in general, any problem in 3-d contact topology is ‘easy’ in the overtwisted category and ‘hard’ in the tight category, due to the rigidity of the tight structures. A prime example of this is the question of existence of periodic orbits in the Reeb field of a contact form: for overtwisted structures, this is guaranteed by Theorem 5.4; for tight structures, one needs to use case-specific methods. Does this dichotomy have a place in fluid dynamics? (5) The topology of contact structures has been intensely investigated for the past twenty years with amazing success. What has been slower in evolving is the role that geometry plays in understanding contact structures. In particular, the tight-overtwisted dichotomy has not impacted nor been effected by geometric considerations. Is there a geometric approach to tightness? It would be sublime if perspectives from fluid dynamics [an intrinsically geometric field] can return the favor and give some insight into the geometry of contact structures. It was conjectured in [32] that any curl eigenfield with smallest nonzero eigenvalue (the principal eigenfield) is, if nonvanishing, always dual to a tight contact structure. This is not the case, as shown recently by Komendarczyk [70]. His method of proof establishes some intriguing relationships between nodal curves of Laplacians on surfaces and characteristic foliations of overtwisted contact structures. Hopefully, this will lead to a better understanding of geometric tightness.
References [1] R. Abraham, J. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Second edition, Springer-Verlag, New York (1988). [2] B. Aebischer, M. Borer, M. Kählin, Ch. Leuenberger and H. Riemann, Symplectic Geometry, Birkhäuser (1994). [3] V. Arnold, Sur la géométrie differentielle des groupes de Lie de dimension infinie et ses applications à l’hydronamique des fluides parfaits, Ann. Inst. Fourier 16 (1966) 316–361. [4] V. Arnold, The asymptotic Hopf invariant and its applications, Proc. of Summer School on Diff. Eqns., 1973, Erevan, Armenian SSR Acad. Sci. (1974). Translation in Sel. Math. Sov. 5 (4) (1986) 327–345. [5] V. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, Berlin (1989). [6] V. Arnold and B. Khesin, Topological methods in hydrodynamics, Annu. Rev. Fuid Mech. 24 (1992) 145– 166. [7] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag (1998). [8] D. Asimov, Round handles and non-singular Morse-Smale flows, Ann. Math. 102 (1975) 41–54. [9] M. Audin and J. Lafontaine, eds., Holomorphic Curves in Symplectic Geometry, Birkhäuser (1994). [10] D. Bennequin, Entrelacements et équations de Pfaff, Asterisque 107–108 (1983) 87–161. [11] M. Berger, Measures of topological structure in magnetic fields, in: An Introduction to the Geometry and Topology of Fluid Flows, NATO Science Series II, vol. 47, Kluwer (2001), 239–252. [12] J. Birman and R. Williams, Knotted periodic orbits in dynamical systems—I: Lorenz’s equations, Topology 22 (1) (1983) 47–82. [13] F. Bourgeois, A Morse-Bott approach to contact homology, in: Symplectic and Contact Topology: Interactions and Perspectives, Fields Inst. Comm. 35, Amer. Math. Soc. (2003), 55–77. [14] J. Casasayas, J. Martinez Alfaro and A. Nunes, Knotted periodic orbits and integrability, in: Hamiltonian Systems and Celestial Mechanics (Guanajuato 1991), number 4 in Adv. Ser. Nonlinear Dynam., pages 35–44, World Sci. Pub., River Edge, NJ (1993). [15] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford Press (1961).
On the contact topology and geometry of ideal fluids
35
[16] S. Chandrasekhar and P. Kendall, On force-free magnetic fields, Astrophys. J. 126 (1957) 457–460. [17] S. Chern and R. Hamilton, On Riemannian metrics adapted to three-dimensional contact manifolds, in: Workshop Bonn 1984 (Bonn, 1984), volume 1111 of Lecture Notes in Math., pages 279–308, Springer, Berlin (1985). [18] C. Chicone, The topology of stationary curl parallel solutions of Euler’s equations, Israel J. Math. 39 (1981) 1–2, 161–166. [19] C. Chicone, A geometric approach to regular perturbation theory with an application to hydrodynamics, Trans. Amer. Math. Soc. 347 (12) (1996) 4559–4598. [20] A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag (1993). [21] V. Colin, E. Giroux and K. Honda, in: Topology and geometry of manifolds (Athens, GA, 2001), 109–120, Proc. Sympos. Pure Math., 71, Amer. Math. Soc., Providence, RI (2003). [22] T. Dombre, U. Frisch, J. Greene, M. Hénon, A. Mehr and A. Soward, Chaotic streamlines in the ABC flows, J. Fluid Mech. 167 (1986) 353–391. [23] P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge Press (1981). [24] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989) 623– 637. [25] Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier, Grenoble 42 (1–2) (1992) 165–192. [26] Y. Eliashberg, Invariants in contact topology, in: Proceedings of the International Congress of Mathematicians, vol. II (Berlin, 1998), number Extra vol. II, pages 327–338 (electronic) (1998). [27] Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000), Special Volume, Part II, 560–673. [28] Y. Eliashberg and N. Mishachev, Introduction to the H-Principle, Grad. Stud. in Math. 48, Amer. Math. Soc. (2002). [29] Y. Eliashberg and W. Thurston, Confoliations, volume 13 of University Lecture Series, American Mathematical Society (1998). [30] J. Etnyre, Introductory lectures on contact geometry, Proc. Sympos. Pure Math. 71 (2003) 81–107. [31] J. Etnyre and R. Ghrist, An index for closed orbits of Beltrami fields, Physica D 159 (2001) 180–189. [32] J. Etnyre and R. Ghrist, Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture, Nonlinearity 13 (2) (2000) 441–458. [33] J. Etnyre and R. Ghrist, Contact topology and hydrodynamics II. Solid tori, Ergodic Theory Dynam. Systems 22 (3) (2002) 819–833. [34] J. Etnyre and R. Ghrist, Contact topology and hydrodynamics III. Knotted flowlines, Trans. Amer. Math. Soc. 352 (2000) 5781–5794. [35] J. Etnyre and R. Ghrist, Generic hydrodynamic instability for curl eigenfields, SIAM J. Appl. Dynamical Systems 4 (2) (2005) 377–390. [36] J. Etnyre and R. Ghrist, Stratified integrals and unknots in inviscid flows, Contemp. Math. 246 (1999) 99– 112. [37] J. Etnyre and K. Honda, On the non-existence of tight contact structures, Ann. Math. 153 (3) (2001) 749– 766. [38] C. Foias and R. Temam, Structure of the set of stationary solutions of the Navier–Stokes equations, Comm. Pure & Appl. Math. 30 (1977) 149–164. [39] A. Fomenko and T.-Z. Nguyen, Topological classification of integrable nondegenerate Hamiltonians on isoenergy three-dimensional spheres, in: A. T. Fomenko, editor, Advances in Soviet Mathematics, vol. 6, pages 267–296. American Mathematical Society (1991). [40] J. Franks and R. Williams, Entropy and knots, Trans. Am. Math. Soc. 291 (1) (1985) 241–253. [41] M. Freedman and Z.-X. He, Links of tori and the energy of incompressible flow, Topology 30 (2) (1991) 283–287. [42] S. Friedlander, Lectures on stability and instability of an ideal fluid, in: Hyperbolic equations and frequency interactions (Park City, UT, 1995), 227–304, IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI (1999). [43] S. Friedlander, A. Gilbert and M. Vishik, Hydrodynamic instability for certain ABC flows, Geophys. Astrophys. Fluid Dynam. 73 (1993) 97–107.
36
R. Ghrist
[44] S. Friedlander and A. Lipton-Lifshitz, Localized instabilities in fluids, in: Handbook of Mathematical Fluid Dynamics, vol. 2, Elsevier (2003), 289–354. [45] S. Friedlander and M. Vishik, Instability criteria for steady flows of a perfect fluid, Chaos 2 (3) (1992) 455–460. [46] S. Friedlander and V. Yudovich, Instabilities in fluid motion, Notices Amer. Math. Soc. 46 (11) (1999) 1358– 1367. [47] J.-L. Gautero, Chaos lagrangien pour une classe d’écoulements de Beltrami, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 301 (15) (1985) 1095–1098. [48] R. Ghrist, Branched two-manifolds supporting all links, Topology 36 (2) (1997) 423–447. [49] R. Ghrist, P. Holmes and M. Sullivan, Knots and Links in Three-Dimensional Flows, volume 1654 of Springer Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York (1997). [50] V. Ginzburg and B. Khesin, Steady fluid flows and symplectic geometry, J. Geom. and Phys. 14 (1994) 195–210. [51] E. Giroux, Une infinité de structures de contact tendues sur une infinité de variétés, Invent. Math. 135 (3) (1999) 789–802. [52] J. Gonzalo, Branched covers and contact structures, Proc. Am. Math. Soc. 101 (1987) 347–352. [53] E. Goursat, Leçons sur l’Intégration des Équations aux Dérivées Partielles du Premier Ordre, Hermann (1891). [54] A. Goriely and M. Tabor, The singularity analysis for nearly integrable systems: homoclinic intersections and local multivaluedness, Phys. D 85 (1–2) (1995) 93–125. [55] M. Gromov, Partial Differential Relations, Springer-Verlag (1986). [56] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag (1983). [57] J. Guckenheimer and R. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979) 59–72. [58] M. Hénon, Sur la topologie des lignes de courant dans un cas particulier, C. R. Acad. Sci. Paris 262 (1976) 312–314. [59] D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, London Math. Soc. Lecture Notes, 318, Cambridge Univ. Press (2005). [60] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (3) (1993) 515–563. [61] H. Hofer, K. Wysocki and E. Zehnder, Unknotted periodic orbits for Reeb flows on the three-sphere, Topol. Methods Nonlinear Anal. 7 (2) (1996) 219–244. [62] H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2) 148 (1) (1998) 197–289. [63] P. Holmes, Knotted periodic orbits in suspensions of Smale’s horseshoe: period mutiplying and cabled knots, Physica D 21 (1986) 7–41. [64] P. Holmes, Knotted periodic orbits in suspensions of annulus maps, Proc. Roy. London Soc. A 411 (1987) 351–378. [65] P. Holmes and R. Williams, Knotted periodic orbits in suspensions of Smale’s horseshoe: torus knots and bifurcation sequences, Archive for Rational Mech. and Anal. 90 (2) (1985) 115–193. [66] K. Honda, 3-dimensional methods in contact geometry, in: Different Faces of Geometry, Int. Math. Ser. (N.Y.), Kluwer/Plenum, New York (2004), 47–86. [67] D.-B. Huang, X.-H. Zhao and H.-H. Dai, Invariant tori and chaotic streamlines in the ABC flow, Phys. Lett. A 237 (3) (1998) 136–140. [68] Y. Kanda, The classification of tight contact structures on the 3-torus, Comm. Anal. Geom. 5 (3) (1997) 413–438. [69] R. M. Kerr, Evidence for a singularity of the three-dimensional, incompressible Euler equations, Phys. Fluids A 5 (1993) 1725. [70] R. Komendarczyk, On the contact geometry of nodal sets, to appear in Trans. of Amer. Math. Soc. [71] G. Kuperberg, A volume-preserving counterexample to the Seifert conjecture, Comment. Math. Helv. 71 (1) (1996) 70–97. [72] K. Kuperberg, A smooth counterexample to the Seifert conjecture, Ann. of Math. 140 (1994) 723–732.
On the contact topology and geometry of ideal fluids
37
[73] A. Lifshitz and E. Hameiri, Localized instabilities of vortex rings with swirl, Comm. Pure Appl. Math. 46 (10) (1993) 1379–1408. [74] C. Lin, The Theory of Hydrodynamic Stability, Cambridge Press (1961). [75] M. Loss and H.-T. Yau, Stability of Coulomb systems with magnetic fields III: zero-energy bound states of the Pauli operator, Comm. Math. Phys. 104 (1986) 283–290. [76] R. Lutz, Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier 3 (1977) 1–15. [77] G. Marsh, Force-free Magnetic Fields: Solutions, Topology, and Applications, World Scientific Press (1996). [78] J. Martinet, Formes de contact sur les variétés de dimension 3, Springer Lecture Notes in Math. 209 (1971) 142–163. [79] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, New York (1995). [80] D. McLaughlin and O. Pironneau, Some notes on periodic Beltrami fields in Cartesian geometry, J. Math. Phys. 32 (3) (1991) 797–804. [81] H. Moffatt, The degree of knottedness in tangled vortex lines, J. Fluid Mech. 35 (1969) 117–129. [82] H. Moffatt, Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology: part I, J. Fluid Mech. 159 (1985) 359–378. [83] H. Moffatt, Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology: part II, J. Fluid Mech. 166 (1986) 359–378. [84] H. Moffatt, editor, Topological aspects of the dynamics of fluids and plasmas, NATO ASI Series E, vol. 218, Kluwer Academic Press (1994). [85] J. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds, Topology 18 (1978) 41–54. [86] R. Pelz, Locally self-similar finite-time collapse in a high-symmetry vortex filament model, Phys. Rev. E 55 (1997) 1617–1626. [87] R. Pelz, Symmetry and the hydrodynamic blowup problem, J. Fluid Mech. 444 (2001) 299. [88] R. Ricca, Geometric and topological aspects of vortex motion, in: An Introduction to the Geometry and Topology of Fluid Flows, NATO Science Series II, vol. 47, Kluwer (2001), 203–228. [89] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton (1995). [90] C. Robinson, Generic properties of conservative systems, Amer. J. Math. 92 (1970) 562–603. [91] D. Rolfsen, Knots and Links, Publish or Perish, Berkely, CA (1977). [92] J. Saut and R. Temam, Generic properties of Navier–Stokes equations: genericity with respect to the boundary values, Indiana Univ. Math. J. 29 (3) (1980) 427–446. [93] M. Schwarz, Morse Homology, Birkhäuser (1993). [94] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976) 225–255. [95] W. Thomson, On vortex motion, Trans. Royal Soc. Edin. 25 (1869) 217–260. [96] W. Thurston and H. Winkelnkemper, On the existence of contact forms, Proc. Am. Math. Soc. 52 (1975) 345–347. [97] M. Wada, Closed orbits of nonsingular Morse-Smale flows on S 3 , J. Math. Soc. Japan 41 (3) (1989) 405– 413. [98] F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I, Invent. Math. 2 (1967) 308–333. [99] F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten II, Invent. Math. 4 (1967) 87–117. [100] R. Williams, The universal templates of Ghrist, Bull. Amer. Math. Soc. 35 (2) (1998) 145–156. [101] X.-H. Zhao, K.-H. Kwek, J.-B. Li and K.-L. Huang, Chaotic and resonant streamlines in the ABC flow, SIAM J. Appl. Math. 53 (1) (1993) 71–77. [102] K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (4) (1976) 1059–1078.
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CHAPTER 2
Shock Reflection in Gas Dynamics Denis Serre École Normale Supérieure de Lyon∗
À ma Mère
Contents Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Models for gas dynamics . . . . . . . . . . . . . . . . . . . 1.1. Barotropic models . . . . . . . . . . . . . . . . . . . . 1.2. Irrotational models . . . . . . . . . . . . . . . . . . . 1.3. Steady flows, potential flows . . . . . . . . . . . . . . 1.4. Characteristic curves . . . . . . . . . . . . . . . . . . 1.5. Entropy inequality . . . . . . . . . . . . . . . . . . . . 1.6. Other models . . . . . . . . . . . . . . . . . . . . . . 2. Multi-dimensional shocks . . . . . . . . . . . . . . . . . . 2.1. Jump relations for a single shock . . . . . . . . . . . . 2.2. Triple shock structures . . . . . . . . . . . . . . . . . 2.3. The generation of vorticity across shocks . . . . . . . 2.4. Diffraction for the full Euler system . . . . . . . . . . 2.5. Diffraction for a barotropic gas . . . . . . . . . . . . . 3. Reflection along a planar wall . . . . . . . . . . . . . . . . 3.1. Regular Reflection . . . . . . . . . . . . . . . . . . . 3.2. Mach Reflection . . . . . . . . . . . . . . . . . . . . . 3.3. Uniqueness of the downstream flow in supersonic RR 4. Reflection at a wedge . . . . . . . . . . . . . . . . . . . . . 4.1. A 2-D Riemann problem . . . . . . . . . . . . . . . . 4.2. Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The supersonic domain behind the reflected shock . . 4.4. Mathematical difficulties . . . . . . . . . . . . . . . . 5. Reflection at a wedge: Qualitative aspects . . . . . . . . . 5.1. Weak incident shock . . . . . . . . . . . . . . . . . . 5.2. Small and large angle . . . . . . . . . . . . . . . . . . 5.3. Entropy-type inequalities . . . . . . . . . . . . . . . . 6. Regular Reflection at a wedge: Quantitative aspects . . . .
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∗ UMPA (UMR 5669 CNRS), ENS de Lyon, 46, allée d’Italie, F-69364 Lyon, cedex 07, France. The research of
the author was partially supported by the European IHP project “HYKE”, contract # HPRN-CT-2002-00282. HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME IV Edited by S.J. Friedlander and D. Serre © 2007 Elsevier B.V. All rights reserved. 39
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6.1. Minimum principle for the entropy 6.2. Minimum principle for the pressure 6.3. Estimate for the diffracted shock . . 6.4. Using the Bernoulli invariant . . . . 6.5. Conclusion; pointwise estimates . . 6.6. The vortical singularity . . . . . . . Acknowledgement . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .
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Abstract This paper is about multi-dimensional shocks and their interactions. The latter take place either between two shocks or between a shock and a boundary. Our ultimate goal is the analysis of the reflection of a shock wave along a ramp, and then at a wedge. Various models may be considered, from the full Euler equations of a compressible fluid, to the Unsteady Transonic Small Disturbance (UTSD) equation. The reflection at a wedge displays a self-similar pattern that may be viewed as a two-dimensional Riemann problem. Most of mathematical problems remain open. Regular Reflection is the simplest situation and is well-understood along an infinite ramp. More complicated reflections occur when the strength of the incident shock increases and/or the angle between the material boundary and the shock front becomes large. This is the realm of Mach Reflection. Mach Reflection involves a so-called triple shock pattern, where typically the reflection of the incident shock detaches from the boundary, and a secondary shock, the Mach stem, ties the interaction point to the wall. The triple shock pattern is pure if it is made only of the incident, reflected and secondary shocks, but of no other wave. As predicted by J. von Neumann, pure triple shock structures are impossible. A common belief was that this impossibility is of thermodynamical nature. We prove here that the obstruction is of kinematical nature, thus is independent either of an equation of state or of an admissibility condition. This holds true for all situations: Euler models, irrotational flows and UTSD, the latter case having been known for a decade. Because the Regular Reflection problem along a wedge gathers several major technical difficulties (a free boundary, a domain singularity, a solution singularity, a mixed-type system of PDEs, a type degeneracy across the sonic line), its solvability is still far from our knowledge, except in the simplest context of potential flows with small incidence, a problem solved recently by G.-Q. Chen and M. Feldman. Good though partial results have been obtained by ˇ c et al. for the UTSD model and by Y. Zheng for the Euler system. S. Cani´ As far as the Euler equations are concerned, we improve and derive with higher mathematical rigour our pointwise estimates of 1994. Our improvements concern most of the estimates: • We give a now rigorous proof of the minimum principle for the pressure, whenever the flow is piecewise smooth, • Our new bound of the size of the subsonic domain applies now to data of arbitrary strength and incidence, • This together with the observation that the entropy increases, yields much better pointwise estimates of field variables, • We prove that there must exist a vortical singularity, at least in the barotropic case: the vorticity of the flow may not be square integrable, • Last but not least, we give a rigorous justification that the flow is uniform between the ramp, the pseudo-sonic line and the reflected shock, the latter being straight.
Shock reflection in gas dynamics
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Definitions Let d = 2 or 3 be the dimension of the physical space. Although d = 3 is relevant for the real world, many phenomena are two-dimensional at the leading order and we shall often assume that d = 2. We denote by t the time variable and by x = (x1 , . . . , xd ) ∈ Rd the space variable. Gases in thermodynamical equilibrium are described by a velocity field u ∈ Rd and scalar internal variables, namely the mass density ρ, the specific internal energy e, the pressure p, the temperature T , the entropy S and a few others. These are functions of t and x. Thermodynamics tells us that the internal variables are not fully independent but are determined by two of them. It turns out that every internal variables can be computed from the primary variables (ρ, S) and the complete equation of state e = ε(ρ, S), using the partial derivatives of ε; see the discussion in Section 10 of the book by S. Benzoni-Gavage and the author [6]. However, when considering the dynamics through the Euler equations below, it is classical to make the choice of (ρ, e) as the primary variables1 and we may content ourselves with an incomplete equation of state that gives the pressure: p = P (ρ, e).
(1)
A perfect gas, also called an ideal gas2 , obeys p = (γ − 1)ρe, where the adiabatic constant γ > 1 equals 1 + 2/N , N being the number of freedom degrees (translational, rotational, vibrational,. . .) of a molecule. For instance, γ = 5/3 for a monoatomic gas (helium, argon,. . .) and γ = 7/5 for a di-atomic gas (hydrogen, oxygen, nitrogen,. . .). At moderate pressure and temperature, air may be considered as a perfect diatomic gas. The temperature and entropy then obey the identity (between differentials forms) 1 T dS = de + pd . ρ
(2)
The evolution of the field U := (ρ, u, e) obeys the Navier–Stokes equations for compressible fluids. In the absence of external forces, they consist in the conservation laws of: • mass, • momentum, • energy. For this reason, we shall call the densities of mass ρ, of momentum ρu and of mechanical energy 12 ρ|u|2 + ρe the conserved variables of the system. In suitable regimes, one may neglect the shear stress (Newton viscosity) and heat diffusion (Fourier law). Then the system reduces to first order in space and time and is called the full system of Euler equations: ∂t ρ + div(ρu) = 0,
(3)
1 Other choices may be useful in some questions, as (p, T ) or (p, τ ) where τ := 1/ρ denotes the specific
volume. 2 Some people prefer saying that a gas is perfect if pτ = RT where R is a constant. Then it is ideal if moreover e is proportional to T .
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∂t
∂t (ρu) + div(ρu ⊗ u) + ∇p = 0, 1 1 ρ|u|2 + ρe + div ρ|u|2 + ρe + p u = 0. 2 2
(4) (5)
As for general first-order systems of conservation laws, the Euler system is scaleinvariant, in that if (x, t) → U solves (3, 4, 5) and if λ is a positive real number, then U λ (x, t) := U (λx, λt) is a solution too. Together with the translation invariance and the Galilean invariance (ρ, u, e, x, t) → (ρ, u − u, ¯ e, x + t u, ¯ t) (u¯ a constant vector), this allows for a rather large set of explicit simple solutions. At the simplest level are all the constant fields. Next, there are simple centered waves, which are purely one-dimensional: planar rarefaction wave, contact discontinuities and shock waves. Beginners will find thorough descriptions of these waves in C. Dafermos’ book [30] or in the Handbook article by G.-Q. Chen and D. Wang [23]. In this paper, we investigate truly two-dimensional simple patterns: • Planar shocks reflecting along a planar wall, • Planar shocks reflecting at a wedge. These problems amount to solving a system of nonlinear partial differential equations, together with boundary conditions. The flow is steady in the first case and pseudo-steady in the second one. The latter terminology means that the governing PDEs are the same, up to lower order derivatives, as in the former case. We warn the reader however, that in pseudo-steady flows, the relevant unknown is (ρ, u − xt , e) instead of (ρ, u, e). The field u − x/t is called the pseudo-velocity. When the strength of the incident shock is moderate, or when its angle of incidence is not too small, the reflection along a planar wall is solved by means of two shocks (incident and reflected), separating three constant states. This pattern is known as a Regular Reflection and is designated by the acronym RR. Since the incident shock and the states that it separates are given data, one needs only to determine the reflected shock and the state behind. This is done through explicit algebraic manipulations. Such computations had been known as soon as 1940, and are due to J. von Neumann [58,59]. As we shall see below, the nonlinear equation to solve has generally two solutions for a moderate incident strength, yielding two types of Regular Reflection, a weak and a strong one. Physical and numerical experiments suggest that the strong RR is unstable, its instability being presumably of Hadamard type. See the analytic study by V. Teshukov [75]. When the strength of the incident shock increases, the boundary-value problem fails to admit a RR and one needs to consider a more elaborate reflection configuration, called a Mach Reflection and designated by MR. This terminology was coined by von Neumann after the very first description by Ernst Mach [53] in 1878. The common feature to all kinds of MR is the presence of a triple point where three shock waves meet, together with a slip line (vortex sheet). Numerical or physical experiments suggest several types of MR, called single (SMR), transitional (TMR), double (DMR) or complex (CMR); this list is ordered by increasing complexity. In some regimes, a Mach Reflection occurs where it is not permitted by the theory based on shock polar analysis. For some reason, the approximation of the flow by plane waves separating constant states is not valid. Numerical experiments suggest that slightly different configurations may occur, for which several scenarii have been proposed. In one of them,
Shock reflection in gas dynamics
43
called von Neumann Reflection (vNR), the reflected shock degenerates into a compression wave near the triple point (see [8,9,17]) while the incident shock and the so-called Mach stem (the third shock) seem to have a common tangent at the triple point. In another one, called Guderley Mach Reflection, an array of triple points takes place along the Mach stem immediately after the reflection point; at each of these points one observes an interaction between the Mach shock and a wave that bounces between this Mach shock and the sonic line. This wave is alternatively a shock and an expansion fan (see [45,73]). Since there is essentially no rigorous mathematics on such patterns, we shall mainly limit our study to RR, although MR is relevant in most of realistic situations. The reader interested in MR theory should consult the nice review by G. Ben-Dor [5] on the experimental side, and ˇ c and coll. given in references, on the the papers by J. Hunter and coll. and by S. Cani´ theoretical side. Ben-Dor’s book [4] displays a lot of experimental results and describes, at least on a phenomenological basis, the various strategies for transition from one kind of reflection to another. See also the review paper by H. Hornung [43]. The present paper is devoted on the one hand to the basics of the theory, like shock polars and terminology, and on the other hand to rigorous qualitative results that could be used in a strategy toward well-posedness of reflection problems. Galilean invariance. As mentioned above, the unsteady models are Galilean invariant: If (ρ, u, e) is a solution (even in the sense of distributions), and u¯ is a constant vector, then (ρ, ˆ u, ˆ e)(x, ˆ t) := (ρ(x + t u, ¯ t), u(x + t u, ¯ t) − u, ¯ e(x + t u, ¯ t)) defines another solution. This observation is particularly useful when dealing with simple flows as shock waves for instance: it will be enough to consider steady shock waves. Remark that, because of Galilean invariance, the notions of supersonic, sonic and subsonic flows do not make sense in the absence of a specified reference frame, since the sound speed is independent of the Galilean frame, while the flow velocity is not.
1. Models for gas dynamics Besides the full Euler system (3, 4, 5), there are a variety of models, each depending on some simplifying assumptions.
1.1. Barotropic models We begin by extracting from (3, 4, 5) equations that are not in conservative form. First of all, (3) and (4) imply 1 (∂t + u · ∇)u + ∇p = 0, ρ
u · ∇ :=
d
α=1
uα
∂ . ∂xα
(6)
44
D. Serre
Eliminating between (6) and (3, 5), we obtain (∂t + u · ∇)e +
p div u = 0. ρ
(7)
Finally, (3) and (7) yield the following transport equation for the entropy: (∂t + u · ∇)S = 0.
(8)
We notice that all these computations involve the chain rule and do not make sense when the field experiences a discontinuity. Therefore the conclusion that S is constant along the particle paths (these are integral curves of the equation dx/dt = u(x, t)) is not correct when such a trajectory crosses a shock. However, it is known that the jump [S] of the entropy at a shock is of the order of the cube of the shock strength. This reflects the transmission identity 1 = 0, [e] + p ρ
p :=
p + + p− , 2
(9)
which should be compared with (2). When the oscillations of the flow are small, it is therefore reasonable to assume that S does not vary across shocks. Under this simplification, and if S was constant at initial time, we conclude that S remains constant forever. The flow is then called isentropic. The constancy of S means that ρ and e are not any more independent of each other: the specific energy and therefore the pressure become functions of the density. We denote p = P (ρ). For this reason, such models are also called barotropic. For a perfect gas, one has P (ρ) = A(S)ρ γ . As in the full Euler equation, the equation of state P determines everything else. For instance, starting from (2) and expressing that dS = 0, we obtain de = −P , d(1/ρ) or equivalently e(ρ) =
ρ
p(s)
ds . s2
(10)
We check easily that for an isentropic flow, (5) is formally a consequence of (3) and (4). Therefore we feel free to drop it and retain the following shorter system, called improperly the isentropic Euler equation: ∂t ρ + div(ρu) = 0,
(11)
∂t (ρu) + div(ρu ⊗ u) + ∇P (ρ) = 0.
(12)
Another way to end with a barotropic model is to assume that heat dissipation is strong enough to drive the temperature to a constant value. The model is called isothermal. The
Shock reflection in gas dynamics
45
constancy of the temperature is again a relation between ρ and e, and thus implies an equation of the form p = P (ρ). The isothermal case of a perfect gas yields a linear P . In this situation, (5) is not any more a consequence of (3) and (4). However, it may be dropped with the following argument: the right-hand side of the conservation law of energy is not zero, but accounts for the heat diffusion. Since we assumed that heat diffusion is dominant, the convective part of this equation (the left-hand side of (5)) may be dropped. The rest just ensures that T ≡ cst. 1.2. Irrotational models Barotropic models still satisfy formally Equation (6). However, the fact that p = P (ρ) ensures that ρ −1 ∇p is curl-free. Taking the curl of (6), we thus obtain an equation for u only, which reads in dimension d = 3: (∂t + u · ∇)ω + (div u)ω = (ω · ∇)u,
ω := ∇ × u.
(13)
We notice that when d = 2 the right-hand side is absent in (13). Nevertheless, in both cases, (13) is a linear transport equation in ω. Therefore irrotationality propagates as long as the vector field u is Lipschitz continuous. Once again, Equation (13) is not valid across shocks3 ; it is therefore an approximation to claim that the flow is irrotational forever. When the flow is irrotational, one may introduce a velocity potential φ by u = ∇φ and work in terms of the unknown (ρ, φ). A clever choice of φ and an integration of (6) yield the equation 1 ∂t φ + |∇φ|2 + i(ρ) = 0, 2
i (s) = p (s)/s.
(14)
The function ρ → i(ρ) is called the enthalpy. Equation (14) is to be coupled with the conservation of mass that we rewrite now ∂t ρ + div(ρ∇φ) = 0.
(15)
We notice that i = e + ρ −1 p, with e(ρ) given by (10). 1.3. Steady flows, potential flows A flow is steady when (ρ, u, e) depend only on the space variable, but not on the time variable. For instance, the full Euler system becomes div(ρu) = 0, div(ρu ⊗ u) + ∇p = 0, 3 See Section 2.3. The jump of vorticity is proportional to the curvature of the shock front when the state is constant on one side.
46
D. Serre
div
1 ρ|u|2 + ρe + p u = 0. 2
Combining the first and the third line above, we immediately obtain u · ∇B = 0,
p 1 B := |u|2 + e + . 2 ρ
(16)
The quantity B is called the Bernoulli invariant. Equation (16) tells that B is constant along the particle paths, at least as long as the solution is smooth (see below for a better result). This constant may vary from one particle path to another. In some cases, it may be relevant to assume that this constant does not depend on the particle and therefore B ≡ cst. For instance, the flow might be uniform in some remote region, while every particle path comes from this region. This assumption allows us to eliminate e and to work with the conservation of mass and momentum only. We warn the reader that such a reduced system is not of barotropic form, as the pressure becomes a function of both ρ and |u|. In the barotropic case, the conservation of momentum reads formally (u · ∇)u + ∇i(ρ) = 0. Multiplying by uT , one obtains again u · ∇B = 0,
1 p 1 B := |u|2 + i(ρ) = |u|2 + e + . 2 2 ρ
(17)
Steady irrotational flow. The same observations as above hold true when curl u ≡ 0. The assumption that B equals a constant now yields an equation of the form ρ = F (|u|). If moreover the flow is irrotational, then the barotropic model reduces to a single second order equation in the velocity potential: div(F (|∇φ|)∇φ) = 0.
(18)
Equation (18) may be rewritten in the quasilinear form F (|∇φ|)φ +
F (|∇φ|) ∂α φ∂β φ∂α ∂β φ = 0. |∇φ|
(19)
α,β
It follows that its type depends on the local state of the fluid. Since F is positive, this equation is elliptic (respectively hyperbolic) whenever F (|u|) + |u||F (|u|)| is positive (resp. negative). From 1 i ◦ F (r) + r 2 ≡ i0 , 2 we derive F (r)i ◦ F (r) = −r. In particular F is negative. We may rewrite this as F (r)p ◦ F (r) = −rF (r), from which we find that ellipticity amounts to saying that |u|2 < p (ρ). Likewise, hyperbolicity of (18) is equivalent to |u|2 > p (ρ).
Shock reflection in gas dynamics
47
Remark. Since the conservation of energy reads div(ρBu) = 0 (for steady flows), B does not vary across steady shocks (i.e. discontinuities with non-zero mass flux). Thus the assumption that B is constant is reasonable when it is so in the far field, provided we know that the vortex sheets have small enough amplitude.
1.4. Characteristic curves The previous analysis shows that c(ρ) := p (ρ) plays a fundamental role and has the dimension of a velocity. This quantity is called the sound speed. In a non-barotropic flow (full Euler system), it is given by the expression c(ρ, e) :=
∂p p ∂p + . ∂ρ ρ 2 ∂e
(20)
For a perfect gas we obtain the well-known formula c=
γp . ρ
Coming back to Equation (19), we may eliminate F and F , and find its equivalent form c2 φ −
∂α φ∂β φ∂α ∂β φ = 0.
(21)
α,β
In unsteady models, the notion of sound speed is relevant too, and one can show that the corresponding system is hyperbolic whenever c2 is positive, that is c is a non-zero real number. Then the Cauchy problem is locally well-posed in spaces of smooth functions. See [23,30,66] for instance. Characteristic curves are well-defined in two independent variables, which may be either (t, x) if d = 1, or x = (x1 , x2 ) if d = 2. Let us begin with the time-dependent flows that depend only on one space variable. Characteristic curves are integral curves, that is solutions of differential equations dx = λ(x, t), dt
(22)
where λ, a function of U , is one of the characteristic velocities. For a general hyperbolic quasilinear system ∂t U + A(U )∂x U = 0,
(23)
the characteristic velocities are the eigenvalues of the matrix A(U ) ; hyperbolicity tells that these eigenvalues are real.
48
D. Serre
For the full Euler system, the characteristic speeds are λ− = u − c(ρ, e), λ0 = u and λ+ = u + c, whence three characteristic families. Notice that the characteristic curves for the second family are particle paths. The other ones “propagate” sound waves. For the barotropic model, we still have λ− = u − c and λ+ = u + c, and it seems that λ0 = u is dropped. This observation is correct if the conservation of momentum is restricted to momentum in the x variable, but it is not if we also take in account the momenta in the transverse directions (think of a gas in 3-d, of which the flow is planar). The situation for steady flows in 2-d is significantly different, for the evolution is governed by some quasilinear system A(U )∂x U + B(U )∂y U = 0.
(24)
If we take x (for instance) as a time-like variable, the characteristic velocities are generalized eigenvalues: det(B(U ) − λ(U )A(U )) = 0.
(25)
Characteristic curves are therefore integral curves of dy = λ(x, y), dx
(26)
or in other words and more generally det(A(U )dy − B(U )dx) = 0.
(27)
Although hyperbolicity tells that ξ1 A(U ) + ξ2 B(U ) is diagonalizable with real eigenvalues for every choice of ξ ∈ R2 , there is no reason why (25) would admit real solutions. The only general remark that can be made is that there is at least one such solution in the barotropic case because the matrices are 3 × 3 and 3 is an odd integer. To go further, let us denote λj (U ; ξ ) (j = 1, . . .) the eigenvalues of ξ1 A(U ) + ξ2 B(U ). Then λ = −ξ2 /ξ1 solves the problem (25) as soon as λj (U ; ξ ) = 0 for some j . Since there holds λ− (U ; ξ ) = u · ξ − c|ξ |,
λ− (U ; ξ ) = u · ξ,
λ+ (U ; ξ ) = u · ξ + c|ξ |, (28)
we find the following characteristic directions R(1, λ) = R(ξ2 , −ξ1 ) for the steady problem: • In all situations, Ru, the direction of the particle paths. • When |u| > c (supersonic flow), and only in this case, the directions V such that det(u, V ) = c|V |. There are two such directions, which make an angle α with the particle path, where | sin α| = c/|u|. • In the borderline case where |u| = c, these two directions coincide with the direction of the flow. This is a rather degenerate situation.
Shock reflection in gas dynamics
49
• If the flow is subsonic (that is |u| < c), we still have two such vectors V , but they are complex conjugate and cannot serve to define characteristic directions. We conclude that a steady supersonic flow is hyperbolic, in every direction that is not characteristic. For a subsonic flow, the model is of mixed type hyperbolic-elliptic: the hyperbolic mode corresponds to the sole real characteristic direction u, while the elliptic one accounts for the two complex conjugate characteristic directions V . Real characteristic directions are crucial in the study of propagation of singularities. Let us assume that U is a continuous, piecewise C 1 , solution of a first-order system in two independent variables. Then the locus of the singularities (here discontinuities of the first derivatives) is a union of characteristic curves. In particular, for a smooth solution that is constant on some open domain, the boundary of that domain consists of a union of characteristic curves that have a very simple geometry. This is an important remark when constructing explicit steady flows, for it is not hard to match local flows along characteristic lines as long as they are supersonic, but it is harder to deal with subsonic flows, because of the rigidity of solutions of elliptic problems (analyticity, maximum principle,. . .) In particular, a flow must have some regularity properties in its subsonic domain, though not in the supersonic region. Let us end this paragraph by warning the reader: when a discontinuity separates two regions in which the flow is smooth, the discontinuity does not need to be a characteristic line. The characteristic property of interfaces is only valid for discontinuities of derivatives of order one at least. Remark. In Section 4, we shall encounter systems of the form A(U )∂x U + B(U )∂y U = g(U ),
(29)
where g is some smooth function. The same theory as above applies. The characteristics of the system (24) do propagate the weak singularities (i.e. those of derivatives) of the piecewise smooth solutions of (29), in spite of the presence of an additional lower order term. These characteristics are defined by the same equation (27).
1.5. Entropy inequality Since the unsteady Euler equations (full or barotropic) form a first-order hyperbolic system of conservation laws, its solutions usually develop singularities in finite time. These singularities are most often discontinuities along moving hypersurfaces. Of course, the conservation laws have to be understood in the sense of distributions. It turns out that too many piecewise smooth solutions exist, so that the uniqueness for the Cauchy problem fails dramatically. Parallel to this mathematical difficulty, thermodynamics tells us that discontinuous solutions are not reversible, despite the formal reversibility of the conservation laws. These two remarks suggest that some of the discontinuities that solve the PDEs must be ruled out by some criterion, which has to be of mathematical nature while having a physical relevance. The most popular criterion, the entropy inequality, was introduced by
50
D. Serre
E. Jouguet [48] for gas dynamics and then generalized to hyperbolic first-order systems by S. Kruzkov [50] and P. Lax [51]. It states that for every conservation law ∂t η + div Q = 0 that is formally consistent with the model, where η and Q are given functions of the conserved variables with η convex, a weak solution is admissible if and only if it satisfies moreover ∂t η + div Q ≤ 0.
(30)
A function η as above is called an entropy by mathematicians (sometimes a convex entropy). This terminology differs from that of physicists. As a matter of fact, there is essentially one non-trivial entropy inequality for gases, where η := −ρS for the full Euler system (with Q = ηu = −ρSu), and η := 12 ρ|u|2 + ρe for the barotropic model (with Q = (η + p)u). It is worth noticing that in the latter case, the energy is not conserved across shocks, and its global decay plays the role of an entropy condition! For a perfect gas, one has S = log e − (γ − 1) log ρ. It is an interesting exercise to prove that
1 2 −ρS ρ, ρu, ρ|u| + ρe → 2
is a convex function. For a general equation of state, (30) is a necessary condition for admissibility but might not be sufficient. A wide literature exists on this topic, for which we refer to Chapter VIII of [30]. Several refined conditions are known, with various degrees of efficiency. For perfect gases however, all admissibility conditions are equivalent to (30). For this reason, we shall content ourself with the entropy inequality, assuming that the gas is perfect or close to be so. Minimum principle for the full Euler system. For the full Euler system, it has been shown by L. Tartar [72] and independently by A. Harten and coll. [38] (see also Exercise 3.18 in ([66])) that, besides the conservation laws, the mathematical entropies are all of the form ρf (S) with f : R → R. For a perfect gas, the convex ones correspond to non-increasing functions f such that s → e−s/γ f (s) is non-decreasing. For such functions, the solutions must obey the generalized entropy inequality ∂t (ρf (S)) + div(ρf (S)u) ≤ 0.
(31)
In particular, choosing for f a function of the form s → max{0, S0 − s}, we obtain (see in particular [69] and also [29,64]) a minimum principle that fits with the physical intuition that the entropy S tends to increase:
Shock reflection in gas dynamics
51
T HEOREM 1.1. For an entropy solution of the full Euler system in an isolated domain (meaning that u · ν = 0 along the boundary), the physical entropy S is globally nondecreasing: t → inf{S(x, t) ; x ∈ }
(32)
is a non-decreasing function of time. 1.6. Other models Since the Euler equations form a rather complicated system of PDEs, with strong interaction between nonlinearity and the characteristics, simplified models have been used in order to reduce the complexity, while catching the main features of the flows under consideration. The unsteady transonic disturbance equation. In the course of this article, we shall be concerned with the transition between Regular Reflection (RR) and Mach Reflection (MR) at a wedge of angle 2α. Following [44,56,45], and assuming that the strength of the incident shock is weak, this transition occurs when M − 1 = O(α 2 ),
(33)
M being the Mach number of the incident shock. This regime allows for a competition between nonlinearity and diffraction. In the sequel, the parameter α a := √ M −1
(34)
is kept fixed. Since only small disturbances of the rest state (ρ0 , 0, p0 )T are considered, we may assume that the gas is ideal for some γ > 1. With := 2(M − 1), one defines new unknowns v and w by the expansion ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ √ ρ ρ0 ρ0 0 2 2 2 ⎝u⎠=⎝ 0 ⎠+ v ⎝ c0 e x ⎠ + 3/2 w ⎝ c0 e y ⎠ + O( 2 ). (35) γ + 1 γ + 1 p γp 0 p 0
0
Rescaling now the space and time variables by X :=
x − c0 t ,
Y :=
√
2
y 1/2
,
T = c0 t,
we obtain the Unsteady Transonic Small Disturbance equation (UTSD) 1 2 ∂T v + ∂X v + ∂Y w = 0, 2 ∂Y v − ∂X w = 0.
(36)
(37) (38)
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D. Serre
This system is sometimes called the 2-D Burgers equation. We notice that, because of the anisotropic change of scales, (37, 38) is neither Galilean nor rotationally invariant. We point out that the parameter a in (34) determines the slope dX/dY of the incident shock. Since the UTSD admits shock waves, there must be a selection criterion, in order to eliminate those discontinuities that are physically irrelevant. This can be done thanks to an entropy inequality as for general first-order systems of conservation laws. For every parameter k ∈ R, classical solutions of (37, 38) satisfy 3 v kv 2 + w 2 (v − k)2 + ∂X − + ∂Y ((v − k)w) = 0, ∂T 2 3 2 and the admissible discontinuous solutions should satisfy the inequality ∂T
3 v (v − k)2 kv 2 + w 2 + ∂X − + ∂Y ((v − k)w) ≤ 0 2 3 2
(39)
in the distributional sense. On a single shock, it just tells that the jump of v is negative from the left to the right in the X-direction. The UTSD is especially useful in that it catches many of the features of the Euler equations in the presence of shock reflection. For large a, it admits a strong and a weak Regular Reflection (see Paragraph 3.3.2), while it displays a transition to Mach Reflection when a diminishes. Like the Euler models, it does not admit pure triple-point configurations (see Section 2.2). At last, fewer computational resources are required to solve it numerically. It has been studied intensively at the theoretical and numerical level as well; see for instance [8,11,12,14,16–18,32,56,68,73] and the references cited above. This series of papers culminates with the proof of existence of a local solution to the transonic (see [12]) and the supersonic Regular Reflection (see [14]). A major difficulty in these works is of course the presence of a free boundary (the reflected shock, see Section 4), for which tools were elaborated in [16]. In the transonic case, one also faces an elliptic equation whose symbol degenerates at the boundary, say along the sonic line. The pressure-gradient model. For numerical purposes (see [1]), one may be tempted to split the full Euler system into two evolution problems, one in which we drop the pressure and energy terms, and one in which we drop the convection term. The latter reads ∂t ρ = 0, ∂t (ρu) + ∇p = 0, ∂t (ρe) + p divu = 0, where ρe =: (ρ, p). Assuming that ρ was a constant ρ0 at initial time, it stays a constant forever because of the first equation. Elimination of u between the last two equations yields ∂t
ρ0 ∂t (ρ0 , p) = p. p
Shock reflection in gas dynamics
53
This is a non-linear wave equation of the form ∂t2 ψ(p) = p, with ψ =
ρ0 ∂(ρ0 , p) . p ∂p
For an ideal gas, ψ(p) is a positive constant times log p. After a rescaling, the pressuregradient system may be rewritten as ∂t u + ∇p = 0,
∂t E + div(pu) = 0,
1 E := |u|2 + p. 2
According to Y. Zheng [83], this model is a good approximation of the Euler system when the velocity is small and γ is large4 .
2. Multi-dimensional shocks 2.1. Jump relations for a single shock Beyond constant flows, we consider piecewise constant flows of the form U (x, t) =
U− , for U+ , for
x · ν < σ t, x · ν > σ t,
(40)
where ν, the direction of propagation, is a unit vector and σ ∈ R is the normal velocity of the shock. Hereabove, U stands for the set of unknowns, say (ρ, u, e) in the full system, (ρ, u) in the isentropic case and the irrotational case. Since a constant flow is obviously a solution of any of these models, U is a solution if and only if it satisfies the correct transmission conditions, called Rankine–Hugoniot relations. The first one is that associated to the conservation of mass (3), [ρ(u · ν − σ )] = 0,
(41)
where [h] denotes the jump of a quantity h accross the hyperplane x · ν = σ t. From (41), we may define the mass flux across the interface: j := ρ− (u− · ν − σ ) = ρ+ (u+ · ν − σ ). It is positive when the fluid flows from the negative side (that defined by x · ν < σ t) to the positive side, and negative in the opposite configuration. Besides (41), we write jump relations that depend on the model under consideration: 4 A large γ is not realistic when speaking of gases, of course.
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D. Serre
Full system: We write the Rankine–Hugoniot conditions for (4, 5), namely
[ρ(u · ν − σ )u] + [p]ν = 0, 1 2 ρ|u| + ρe (u · ν − σ ) + [pu · ν] = 0. 2
Because of (41), they read equivalently: j [u] + [p]ν = 0, 1 j |u|2 + e + [pu · ν] = 0. 2
(42) (43)
Barotropic case: We only write the first two jump conditions (41, 42). Irrotational flow: We write the jump condition (41), plus that associated to ∇ × u = 0, namely [u] × ν = 0.
(44)
From this, φ is given by φ(x, t) = u · (x − σ tν) + cst. We insert this formula into (14) and get our last relations 1 2 |u| − σ u · ν + [i(ρ)] = 0. (45) 2 We notice that thanks to (44), (45) is equivalent to 1 [(u · ν − σ )2 ] + [i(ρ)] = 0. 2
(46)
In the subsequent analysis, we must distinguish the contact discontinuities (CD, for which j = 0), from the shock waves (or just shocks), which correspond to j = 0. CDs do not happen in the irrotational case, because i is positive for realistic gases and therefore (j = 0) ⇒ ([i(ρ)] = 0) ⇒ ([ρ] = 0). In a CD of the full or a barotropic model, there holds [p] = 0 and u± · ν = σ . Conversely, any set (U− , U+ ; ν, σ ), satisfying these two identities, defines a CD. In the barotropic case, the constancy of the pressure implies [ρ] = 0 and therefore CDs are slip lines, also called vortex sheets. In the full Euler case, one may have [u] = 0 since [p] = 0 does not imply (ρ+ , e+ ) = (ρ− , e− ). The description of shocks is a bit more involved. On the one hand, (42) implies 2 1 + [p] = 0, (47) j ρ [u] × ν = 0.
(48)
Shock reflection in gas dynamics
55
On the other hand, combining (42, 43) and the following identities
1 2 |u| = u[u], 2
[pu] = p[u] + [p]u,
gives j [e] + p[u · ν] = 0.
(49)
This proves (9), because of [u · ν] = j [1/ρ], which is nothing but the definition of j . We show now that the jump of the entropy is of cubic order in the shock strength. It is not hard to see that (9) defines a curve in the (ρ, e)-plane when one of the states, say (ρ0 , e0 ) is fixed. Let s be a smooth, non-degenerate parameter along this Hugoniot curve. With dots standing for derivations along the curve, we have T S˙ = e˙ + p τ˙ ,
τ :=
1 . ρ
On the other hand, we have 1 ˙ ] + pτ˙ = 0, e˙ + p[τ 2 whence 2T S˙ = [p]τ˙ − [τ ]p. ˙ Differentiating once more, we have 2
d dS d 2τ d 2p T = [p] 2 − [τ ] 2 . ds ds ds ds
These identities imply dS d 2 S = 2 =0 ds ds at the origin (ρ0 , e0 ) of the curve. Whence [S] = O(s 3 ). This estimate is accurate for most of equations of state, for instance that of an ideal gas, but a degeneracy could cause the jump of the entropy to be of higher order. Entropy conditions for shocks. Besides the Rankine–Hugoniot conditions, a shock has to satisfy the jump relations associated to the entropy inequality (30). Since Lipschitz solutions of the Euler equations do satisfy the entropy equality, selecting admissible shocks
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D. Serre
among all discontinuities is precisely the role of the entropy inequality. The jump condition that is obtained is, of course, an inequality, which reads either j [S] ≥ 0,
(50)
in the full Euler case (meaning that the entropy of a small part of fluid increases when one crosses a shock), or
1 2 j |u| + e + [pu · ν] ≤ 0 2
(51)
in the barotropic case. The latter apparently violates the conservation of energy, but this is due to the fact that we do not take in account one form of the bulk energy; the inequality just tells that there is some heat release. Of course, one may eliminate u from (51) as we did in the full Euler case. We obtain the equivalent form of entropy inequality across a discontinuity: 1 ≤ 0. j [e] + p ρ
(52)
Shocks are compressive. Let us restrict to the equation of state of an ideal gas. In the full Euler case, p = (γ − 1)ρe and S = log e − (γ − 1) log ρ. Thanks to (9), we may eliminate e± and obtain exp(S+ − S− ) = X γ
1 − μ2 X , X − μ2
X :=
ρ− , ρ+
μ2 :=
γ −1 . γ +1
(53)
As a matter of fact, (9) implies also that the ratio of the densities is bounded by 1/μ2 . From (53), we immediately find the equivalent formulation of the entropy inequality: j [ρ] ≥ 0.
(54)
We examine next the barotropic case. We have p = Aρ γ where A is a positive constant. Then e = p/((γ − 1)ρ). We may assume A = 1. We eliminate again e± , now in inequality (51) and we end with the same entropy condition (54). In conclusion, shocks are compressive in ideal gases: the density is higher in the domain into which the gas flows. This domain is usually named downstream while the region from which the gas flows is named upstream. Because of the above analysis, we also impose (51) in the irrotational model. Shocks are supersonic on the front side. Given a steady shock (U− , U+ ), one may always modify the velocities by adding the same constant vector, provided the latter is parallel to the shock front. Since the tangential component of the velocity is continuous, we see that it may be cancelled simultaneously on both sides. Hence, not only there holds |u| ≥ |u · ν|, but also we may choose a reference frame in such a way that the shock be still steady,
Shock reflection in gas dynamics
57
and that it hold |u| = |u · ν|. This explains why the super-/sub-sonic property of one of the states with respect to the shock, is encoded only in the normal component of the velocity: We say that a steady shock between U− and U+ is supersonic (resp. subsonic) with respect to a neighbouring state U if |u · ν| > c(ρ, e) (resp. |u · ν| < c(ρ, e)). Accordingly, a state U = U± is said to be subsonic relatively (resp. supersonic relatively) to the shock if |u · ν| is smaller (resp. larger) than the sound speed. T HEOREM 2.1. Consider a steady shock for a perfect gas with the full Euler model. Say that p− < p+ . Then there holds ρ+ > ρ− . Furthermore, the shock is relatively supersonic on the front side (upstream) and relatively subsonic on the back side (downstream): |u− · ν| > c− ,
|u+ · ν| < c+ .
P ROOF. Inserting the√assumption ρ+ e+ > ρ− e− in (9), we obtain ρ+ > ρ− . Let us define c∗ = [p][ρ]. Eliminating j , we have (u− · ν)(u+ · ν) = c∗2 .
(55)
On the other hand, simple manipulations transform (9) into (1 − μ2 )((u · ν)2 − c2 ) = (u · ν)2 − c∗2
(56)
on both sides. From ρ− < ρ+ we have |u+ · ν| < |u− · ν|. With (55), this implies |u+ · ν| < c∗ < |u− · ν|.
(57)
Finally, (57) and (56) yield |u+ · ν| < c+ ,
c− < |u− · ν|.
(58)
We point out that the above proof works for a more general equation of state, provided that the ratio p − p0 ρ − ρ0
(p = p(ρ, e), p0 = p(ρ0 , e0 ))
increases with ρ when (ρ0 , e0 ) is kept fixed, along the Hugoniot curve defined by e − e0 +
1 p + p0 1 = 0. − 2 ρ ρ0
As a matter of fact, such a property yields c− < c∗ < c+ . Since (57) still holds true, we deduce (58).
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Let us turn towards the barotropic case, with p = Aρ γ . We still have (55) and |u+ · ν| < |u− · ν|, hence (57). On the other hand, p being convex, there holds c− < c∗ < c+ . Whence (58). Remark that this proof needs only the convexity of p. This proves T HEOREM 2.2. Consider a steady shock for a barotropic gas with p > 0 and p
> 0. Say that p− < p+ . Then there holds ρ+ > ρ− . Furthermore, the shock is relatively supersonic on the front side (upstream) and relatively subsonic on the back side (downstream): |u− · ν| > c− ,
|u+ · ν| < c+ .
For more general equations of state, it may be necessary to reinforce the admissibility condition for shock waves. The most popular admissibility condition is the Lax shock condition, which tells that if a hyperbolic system of conservation laws consists of n scalar conservation laws, there must be exactly n + 1 incoming characteristic curves in a shock. This is an obvious necessary condition, at the linearized level, for the well-determination of the solution and of the shock front, from the PDEs on each side and their associated Rankine–Hugoniot relations only (see for instance [54]). In gas dynamics, this means that with ν the unit normal to the shock front pointing towards the plus region, then either (u · ν)+ − c+ < 0 < (u · ν)− − c− , or (u · ν)+ + c+ < 0 < (u · ν)− + c− . This condition ensures that the shock is supersonic with respect to one of the states, the minus one in the first case, or the plus one in the second case. We unify these cases by saying that a steady shock is always supersonic with respect to the upstream flow. Since on the other hand the pressure is lower upstream than downstream, we have that the shock is supersonic with respect to the side of lower pressure. An equivalent statement is that if a steady shock is subsonic with respect to the state on one side, then the pressure on this side is higher than on the other one. A subtlety about subsonic states. We warn the reader that the state on the subsonic side of a steady shock may be either subsonic or supersonic in the coordinate frame! The subsonicity holds with respect to the shock, thus exactly means that |u · ν| < c, while the supersonicity with respect to the coordinate frame means that |u| > c. Obviously these two inequalities are compatible. On the other hand, the state on the supersonic side of the shock is supersonic in every sense, since we have |u| ≥ |u · ν| > c. This subtlety is of some importance, as supersonicity in the reference frame means precisely that the model, which is a system of PDEs in the (x, y)-plane, is hyperbolic in some direction, though not in every one. When we consider a 3-D steady flow, saying that the state is subsonic with respect to the steady shock means that the model is not hyperbolic in the direction normal to the shock. More generally, the steady Euler system is hyperbolic in a direction N ∈ R3 if and only if |u · N | > c|N |. For the sake of simplicity, we illustrate this
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59
claim with the barotropic case. Working with unknowns V = (ρ, u1 , u2 , u3 )T , the system reads in the quasilinear form 1 A (V )∂1 + A2 (V )∂2 + A3 (V )∂3 V = 0. Defining A(V ; ξ ) =
α α ξα A (V ),
A(V ; ξ ) = (u · ξ )I4 +
we have 0 c2 ρξ
ρξ T 0
.
Let P (V ; ξ ) := det(A(V ; ξ )) be the principal symbol of the operator α Aα ∂α . Let N ∈ R3 be such that P (N) = 0 (the normal plane to N is not characteristic). Then the system is hyperbolic in the direction N if the roots of the polynomial s → P (sN + ξ ) are real for every ξ . In our problem, P is written P (V ; ξ ) = (u · ξ )2 (u · ξ )2 − c2 |ξ |2 , thus there is always a double real root −(u · ξ )/(u · N ) in the fourth degree polynomial P (sN + ξ ). Taking ξ normal to N and u, we observe that the two remaining roots are real if and only if |u · N | ≥ c. Actually, these roots are the eigenvalues of some matrix and hyperbolicity requires in addition that this matrix be diagonalizable . This rules out the borderline case |u · N | = c|N |. About notations. When a steady shock is given, we always label U+ the subsonic state and U− the supersonic one. Therefore we always have ρ− < ρ+ and p− < p+ . Also, we choose the unit normal ν that points toward the “plus” zone. Since U− is upstream and U+ downstream, we have u± · ν > 0. In particular, there holds j > 0,
c− < u− · ν,
u+ · ν < c+ .
In many shock problems, a state U0 is given and a pair (U1 ; ν) is sought so that (U0 , U1 ; ν) is a steady shock. Such notations are employed as long as one does not know whether the zero state is sub- or super-sonic (mind that this depends on ν). In some circumstances, it may be necessary to employ both notations in the same paragraph.
2.2. Triple shock structures A shock interaction is the simplest pattern that is genuinely two-dimensional. At first glance, it would be a point where three shocks meet. We shall see below that it must be of even higher complexity, because of a very general obstruction result. Typically, such a configuration needs at least an additional wave: a vortex sheet or an expansion wave.
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2.2.1. Vorticity flows In this paragraph, the flow is governed by either the full Euler system, or that of a barotropic fluid. As a matter of fact, we only use the conservation of mass and momentum, without any information about the pressure. To our knowledge, the following result is new in its generality, especially because the obstruction appears to be of kinematical nature, rather than thermodynamical. T HEOREM 2.3. Consider a planar steady flow. There does not exist a pure triple shock structure, that is a piecewise constant flow with only three states separated by straight shocks. Amazingly enough, the proof involves only the kinematics, that is the conservations of mass and momentum. P ROOF. Let us denote U0 , U1 , U2 the constant states. We choose an orientation around the triple point. The unit normal vectors to the shocks are, in cyclic order, ν0 , ν1 , ν2 , meaning that the line of equation x · να = 0 separates Uα+1 from Uα+2 , the normal being oriented from the (α + 1)-zone to the (α + 2)-zone; see Figure 1. The mass fluxes across the shocks are denoted as well; for instance, j0 = ρ1 u1 · ν0 = ρ2 u2 · ν0 . We recall that jα = 0 for every α. Notice that the να ’s are pairwise distinct unit vectors. We do not exclude that two normals be parallel, for instance ν1 = −ν0 , but at least two of them are linearly independent. We make use of (42) and of its equivalent form (thanks to (47) and to j = 0) [u] = j
1 ν. ρ
Fig. 1. The (impossible) pure triple shock structure.
(59)
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In both (42) and (59), we sum circularly on the triple of shocks, in order to eliminate the [u]’s. We obtain an identity ([h]0 denotes h2 − h1 , and so on) that we may write in two forms:
1
να [p]α =0 or jα να = 0. (60) jα ρ α circ
circ
Since on the other hand we have
1
[p]α = 0, = 0, ρ α circ
circ
where the jumps of p and of 1/ρ do not vanish5 , (60) tells us that the vectors Pα := jα−1 να are collinear, and that the Qα = jα να are collinear too. Let L be the line passing through the Pα ’s. Since there is a pair of linearly independent να ’s, L does not pass through the origin. Thus its image under inversion with respect to the unit circle is a circle C passing through the origin. Since the Qα ’s belong to this circle and are collinear, two of them must be equal: say we have j1 ν1 = j2 ν2 . The ν’s being pairwise distinct, this implies ν1 = −ν2 (and thus jβ = −jα ). Now, any of the equalities in (60) tells that ν3 is collinear to the other ones, a contradiction. Remarks. • Theorem 2.3 is a folk result in multi-dimensional gas dynamics. However, it seems that all the previous proofs needed specific assumptions on the equation of state. R. Courant and K. Friedrichs ([28], paragraph 129) stated it for a polytropic gas. L. Henderson and R. Menikoff [42] considered a more general gas, but still with a restriction on the equation of state. • Henderson and Menikoff’s proof is based on the variation of the entropy across a sequence of shocks, at given final pressure. This estimate, which has its own interest, gives a rigorous basis to the following claim made several times by von Neumann [58, 59,57]: Among the wedges separated by the shocks, there must be an upstream wedge and a downstream one. As the gas flows from upstream to downstream, it passes either on one side of the triple point or on the other side. Thus some molecules cross only one shock, a strong one, while others cross two shocks that are weaker. The increase of entropy should be higher in the former case than in the latter, because the entropy jump is superlinear (typically cubic for small shocks) in the shock strength. Hence the state downstream may not be uniform. We point out that this entropy-based argument does not apply to the barotropic case. We do not exclude that a similar argument, based on the variation of total energy, might work in this latter case. • Von Neumann’s claim looks to be a kind of convexity inequality. Therefore it certainly needs some assumptions, like those of Henderson and Menikoff, in order to be proved. In particular, it applies only within the class of entropy-admissible shock waves. 5 If one of them vanished, we should find immediately [ρ] = [p] = 0 and [u] = 0, meaning that the shock was not present.
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• On the other hand, Courant and Friedrichs’ proof, though being restricted to a small class of equations of state, is valid even for non-admissible shocks, since it does not involve an inequality at all. Thus it is completely independent from von Neumann’s claim. • The proof above only assumes that the stress tensor is of the form −pId for some scalar p. This means that, given an arbitrary interface , the force applied by the fluid on one side of onto the fluid on the other side, is normal to . • It is remarkable that Theorem 2.3 holds true even if hyperbolicity fails in some region of the phase space, when an equation of state is given. For instance, it applies to a van der Waals gas. • For the same reason, it is valid both for the full Euler model and the barotropic one. • This two-dimensional result extends to three space dimensions, because the tangential velocity is constant across a shock. • Actually, the result is valid for time-dependent flows, in the following form: In 2 + 1 space-time dimensions, there does not exist a (locally defined) piecewise smooth solution U (x, t) of the (full or barotropic) Euler system, where the discontinuities are three shocks (the normal velocities are non-zero) that meet transversally along a smooth curve t → X(t). This because at a given time t0 , one may assume (thanks to ˙ 0 ) is zero. Then the field V defined by Galilean invariance) that the velocity X(t V (x, t) := lim U (rx + X(t0 ), rt + t0 ) r→0+
is a steady triple shock pattern. The most important consequence of Theorem 2.3 is the von Neumann paradox. Numerical simulations as well as laboratory experiments show that in some parameter regimes, a Mach Reflection takes place for which the flow seems to consist of three shocks separated by smooth regions, without any slip line or rarefaction fan. This is clearly in contradiction with our result. P. Collela and Henderson [27] suggested that such an irregular Mach Reflection actually contains a very small rarefaction fan that ties the diffracted shock to the now smooth curve formed by the incident shock and the Mach stem. Several authors have performed more and more accurate simulations in order to give a refined description of this pattern, called a von Neumann Reflection. An other plausible scenario is the one given by A. Tesdall and Hunter [73], after careful numerical experiments: There is a supersonic region behind the triple point, which consists of a sequence of supersonic patches formed by a sequence of expansion fans and shock waves that are reflected between the sonic line and the Mach shock. Each of the reflected shocks intersects the Mach shock, resulting in a sequence of triple points, rather than a single one. The numerical results do not indicate whether there are finitely many such triple points or not. 2.2.2. The one-dimensional case The proof of Theorem 2.3 resembles that of a wellknown result about unsteady flows: In one-space dimension, the interaction of two shocks cannot produce a single shock (see Exercise 4.12 in [66]). Here, u is scalar and each shock
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has its own velocity. Let us denote τ := 1/ρ. The Rankine–Hugoniot conditions for conservations of mass and momentum are [ρ(u − σ )] = 0, from which we define j := (ρ(u − σ ))± (notice that the shocks are not steady), and j [u] + [p] = 0. We have again j 2 [τ ] + [p] = 0, and in particular [u] = j [τ ] = ± [p][τ ]. Summing circularly, we infer
± [p][τ ] = 0. circ
The signs in this equality cannot be all equal. Therefore it reads a2 b2 = a0 b0 + a1 b1 , where in addition aj bj > 0,
aj = 0,
bj = 0.
Developing, we have (a0 + b0 )(a1 + b1 ) = a2 b2 = a0 b0 + a1 b1 + 2 a0 b0 a1 b1 , √ from which we obtain a0 b1 + a1 b0 = 2 a0 b0 a1 b1 , whence a0 b1 = a1 b0 . Coming back to the definition of the a’s and b’s, this amounts to saying p2 (τ1 − τ0 ) + p1 (τ0 − τ2 ) + p0 (τ2 − τ1 ) = 0. In other words, the three points (pj , τj ) in the (p, τ )-plane are collinear. This implies that the slopes between them be equal. Since these are the j 2 ’s, two of the j ’s must be equal, for instance j0 = j1 . But this reads ρ2 (u2 − σ0 ) = ρ2 (u2 − σ1 ). Whence σ0 = σ1 : two among the three shocks have the same velocity. It is straightforward to conclude that the configuration is trivial. P ROPOSITION 2.1. Consider unsteady flows in one space dimension. Whatever the equation of state (which could be barotropic or not), there does not exist a pure triple shock configuration satisfying the conservation of mass and momentum.
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2.2.3. Irrotational flows Slightly more difficult is the proof that a pure triple shock does not exist for potential flows. Again, it is unclear whether the following result has been stated before. The analysis in [56] (Appendix 2) proceeds only for weak shocks. It uses only approximate jump relations, thus it is not a genuine proof. T HEOREM 2.4. Consider a planar steady irrotational flow. There does not exist a pure triple shock structure, that is a piecewise constant flow with only three states separated by straight shocks. P ROOF. We proceed as in the proof of Theorem 2.3, except that the jump relations are 2 1 + 2[i] = 0 [u] × ν = 0, j ρ2 and as usual 1 [u] = j ν. ρ Once again, we do not need to know the way i is determined6 by the other parameters. We thus have
1
1 2 jα 2 = 0, jα να = 0 ρ α ρ α circ
while obviously
1 = 0, ρ α circ
circ
1 = 0. ρ2 α circ
Let us define the following nine points −1 1 Pα := jα να , ρ α
Qα := jα να ,
The following identities come from above
1
1 2 jα 2 Pα = 0, Qα = 0, ρ α ρ α circ
circ
Rα := jα
−1 1 να . ρ α
1 Rα = 0. ρ2 α circ
Since the sums of coefficients in each of these equalities vanish, we deduce that each triplet is collinear: the points Pα lie on a line LP , the points Qα belong to a line LQ and the Rα ’s 6 The context is ambiguous here. On the one hand, we do not use at all the way the enthalpy varies with the density. On the other hand, we assume an equation of state p = p(ρ), otherwise (14) would not hold.
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are on a line LR . Since the να ’s are distinct and unitary, none of these line passes through the origin. On another hand, each triplet (να , Pα , Qα , Rα ) is on a ray Dα passing through the origin. Each ray may be identified with the real line, on which we have να2 Rα = Pα Q2α . Up to a rotation we may assume that LQ has equation x1 = s with s a nonzero constant. Then the relation between the P ’s and the R’s are R = s2
P 2 P. 2 x1P
Let now ax1 + bx2 = c (c = 0) be an equation of LR . The above formula shows that in the equation Ax1 + Bx2 − C = 0 (C = 0) of LP , the form Ax1 + Bx2 − C must divide the polynomial s 2 (x12 + x22 )(ax1 + bx2 ) − cx12 . The ratio would be of degree two, and of valuation two also, hence should be a quadratic form, and more precisely x12 . This is clearly impossible. Remark. Theorem 2.4 is a bit astonishing, because there is no possibility to resolve this obstruction by inserting vortex sheets. The latter simply do not exist in potential flows. This suggests that a correct pattern in a Mach stem involves at least a centered rarefaction wave. 2.2.4. The UTSD model R. Rosales and E. Tabak [68] have shown result similar to Theorem 2.3 for the UTSD system (37, 38). We borrow our proof from Theorem 11.1 in [84]. Since (37, 38) is not Galilean invariant7 , we must consider more general travelling waves U (x − at, y − bt). T HEOREM 2.5. Consider travelling waves of the UTSD model. There does not exist a pure triple shock structure, that is a piecewise constant flow with only three states separated by straight shocks. P ROOF. We may suppose that the wave travels only in the direction y, since there is a Galilean invariance in the x direction. We assume that the shocks are non-trivial: [u]α = 0 for α = 1, 2, 3. The Rankine–Hugoniot relations read
1 2 v νx + [w − bv]νy = 0, 2
[w]νx = [v]νy .
In particular, the component νx is nonzero for each shock. Noticing that [v 2 ] = 2v[v], we obtain vνx2 − bνx νy + νy2 = 0, 7 The UTSD model is Galilean invariant in the x direction but not in the y direction.
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or equivalently v = bm − m2 , where m := νy /νx is the slope of the shock. Since [v] = −2[v], this yields8 [v]α = 2[m2 − bm]α . Using now [w] = m[v], we deduce [w]α = 2mα [m2 − bm]α . Summing up, there comes
mα [m2 − bm]α = 0,
circ
which also reads [m]α = 0. circ
Therefore at least two shocks are aligned. Since [u] × ν = 0, this means that two of the jumps [u]α are collinear. By difference, the third one is also collinear with them, meaning that the three normal vectors are aligned. This is a contradiction. Let us point out that the UTSD model does not allow any kind of contact discontinuity. Since numerical experiments strongly suggest that we do encounter triple-point configurations, we must decide which additional phenomenon prevents Theorem 2.5 to apply, in order to solve this von Neumann paradox. K. Guderley [35,36] suggested that there is a supersonic region behind the triple point, in which an additional expansion fan develops. The presence of a tiny supersonic region was validated by careful numerical experiments of Hunter and M. Brio [45]. Next, Tesdall and Hunter [73] refined the calculations and found an array of at least four supplementary triple points, next to the main one along the Mach shock. They are produced by a wave bouncing between the Mach shock and the sonic line. At each bounce, the nature of the wave flips, from shock to expansion fan, and back. The possibility of a rarefaction fan is ruled out in the transonic case because hyperbolicity of the stationary equations is lost. Then there remains the possibility that the solution be non-smooth at the triple-point. This situation is plausible, after a nice analysis by I. Gamba et al. [32]. Notice that the v component must remain bounded because of physical considerations, so that the singularity manifests itself at the leading order in the component w only. This is similar to the situation of an elliptic first-order linear system, say the Cauchy–Riemann equations, when one of the components is only piecewise continuous on the boundary. Then the other component experiences a logarithmic singularity at every point of discontinuity of its conjugate. 8 One should not confuse [m] with m . For instance, [m] = m − m . α α 2 3 1
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2.2.5. Four shocks and more Let us go back to the obstruction found in Section 2.2.1. It is interesting to know whether a pure shock interaction pattern exists with more than three shocks. This question turns out to have a simple answer, found by W. Bleakney and A. Taub [7]. Let us anticipate a little bit. We shall construct in Section 3.1 simple patterns called Regular Reflection along a planar wall, denoted by the acronym RR. This patterns is described in Figure 4. The flow is piecewise constant, varying only across two straight shocks that meet at a boundary point. Along the wall, the normal component of the velocity vanishes. Having in hands such a RR, we may built a four-shocks pure pattern in the plane, by making a reflection across the wall: ⎞ ρ(x) := ρ(x1 , −x2 ), ⎜ e(x) := e(x1 , −x2 ), ⎟ ⎟ (x2 < 0) ⇒ ⎜ ⎝ u1 (x) = u1 (x1 , −x2 ), ⎠ . u2 (x) = −u2 (x1 , −x2 ) ⎛
Hence pure four-shocks patterns do exist, contrary to triple-shock patterns. We point out that the manifold of all pure four-shocks patterns is a priori of dimension four (the four states and four shocks being described by 16 scalar parameters, while the Rankine– Hugoniot conditions giving 12 scalar constraints). On the other hand, the symmetric patterns built by Bleakney and Taub are described by four scalar parameters, say the upstream flow and the angle that it makes with the incident shock. Hence it is likely (though this claim would need a more detailed analysis) that every pure four-shock pattern has the symmetric form described above. This mandatory symmetry makes the class of pure four-shocks patterns useless. An alternative to it consists in introducing a rarefaction fan or a slip line, instead of a fourth shock. The choice of a slip line yields the so-called Mach Reflection (MR); see Section 3.2. The existence of patterns with three shocks and a rarefaction fan was proved by V. Bargmann and D. Montgomery [3]. However, this latter class does not seem as useful as the MR class. Notice that in old papers, as [3,7], a rarefaction fan is called a Prandtl–Meyer variation. It has the property that the component of the flow normal to the radius vector is always sonic (|uθ | = c(ρ)). Of course, we may consider a more general pattern organized around a center. Typically, the flow will depend only on the polar angle θ and not on the radius r. Such a flow is made of constant states, shocks, rarefaction fans and slip lines. For a barotropic flow with a reasonable equation of state p = p(ρ), say with p > 0 and p
≥ 0, one proves easily that a shock is adjacent to constant states only. Except across slip lines, the flow is irrotational and thus we can derive it from a potential that is homogeneous of degree one. Along the unit circle, the local extrema of the potential must arise in zones where the flow is uniform.
2.3. The generation of vorticity across shocks This section is devoted to the barotropic model, for which the vorticity ω := ∇ × u obeys the transport equation (13). As seen in Section 1.2, this gives ω ≡ 0, provided the initial
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vorticity vanishes and the flow remains Lipschitz continuous. In steady flows, (13) reduces to (u · ∇)ω + (divu)ω = (ω · ∇)u, again a transport equation along the flow. Thus ω vanishes downstream if it does upstream and the flow is Lipschitz continuous. Notice that in two space dimensions, ω equals ∂1 u2 − ∂2 u1 and satisfies (u · ∇)ω + (divu)ω = 0.
(61)
Notice that if we assume conversely that the flow be steady and irrotational, then we find that the fluid behaves locally like either a barotropic gas or an incompressible one, because of the identity 1 2 ⊥ 0 = curl(ρ(u · ∇)u) = curl ρ∇|u| + ρωu , 2 from which it follows, when ω ≡ 0, ∇ρ × ∇|u|2 = 0. Thus either ρ is locally constant, or |u|2 = h(ρ) for some function h, at least locally. But then 1 ∇p = −ρ∇ |u|2 , 2
(62)
whence p = H (ρ),
s H (s) = − h (s). 2
(63)
We point out however that we do not need that a barotropic equation of state be given a priori. What does happen if the flow experiences a discontinuity, while being irrotational on one side? The answer depends of course on the type of discontinuity. Along a vortex sheet V , the tangential velocity admits a nonzero jump [u × ν] (this is the only discontinuous quantity there), meaning that ω has a non-trivial singular part [u × ν] ⊗ δV . The situation is a little more interesting across a shock wave S. Our first observation is that because of (48), the vorticity does not have a singular part on S, but is an ordinary function away from slip lines, as long as the flow is piecewise smooth. In the simple case of a planar shock wave separating two constant states, the vorticity is thus identically zero, even in the distributional sense: the flow is irrotational. It turns out that this is not true once the shock is curved. For the sake of simplicity, the following result is stated in two space dimensions.
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T HEOREM 2.6. Let U be a two-dimensional steady flow, constant on one side − of a C 1 shock curve S and of class C 1 on the other side + . If ω vanishes on + , then S is a straight line. P ROOF. We proceed in two steps. We show that if ρ has a regular point (∇ρ = 0) at some point P ∈ S + , then the normal ν to S is constant in a neighbourhood of P . There remains the case where ρ is locally constant, and there we obtain the same conclusion. We begin with the easier second case. Thus let ρ ≡ ρ+ be constant in + near P ∈ S + , say in D+ . From (62), we have 1 p = p0 − ρ+ |u|2 2 in D+ (p0 a constant). Let us write the Rankine–Hugoniot conditions 2 1 . ρ+ u · ν = ρ− u− · ν, u × ν = u− × ν, [p] = −j ρ Elementary calculations yield an equation for the normal (since the flow is constant in − , p ≡ p− is constant there) 2 1 ρ 2 (u− · ν)2 = 0. − N (ν) := 2(p− − p0 ) + ρ+ (u− × ν)2 + ρ− ρ+ − Since N is a quadratic trigonometric polynomial, its roots are isolated. The only exception to this rule is when N is constant, but this means 2 1 ρ+ = ρ2 , − ρ− ρ+ − or in other words [ρ] = 0. But then we obtain [u] = 0 and there is no shock at all. Since the roots of N are isolated and ν varies continuously along S, we deduce that ν is locally constant, hence S is locally straight. We turn now to the case where ∇ρ = 0 at P . Then we may write locally |u|2 = h(ρ) and p = H (ρ) with H as in (63), and h, H are√ C 1 functions. We denote θ = h−1 . Let us introduce the sine s := (u− × ν)/|u− | (with say 1 − s 2 = (u− · ν)/|u− |). We may choose a system of coordinates in which s does not vanish at P . We write the Rankine–Hugoniot equations in the form (recall that τ = 1/ρ) ⎛ ⎞ ⎛ 2 ⎞ j (τ − τ− ) − H (ρ)+ p− j 2 M ⎝ τ ⎠ := ⎝ τ θ τ 2 j 2 + |u√ − |s − 1 ⎠ = 0. s j − ρ− |u− | 1 − s 2 A lengthy computation gives the Jacobian of M at any root: (M = 0) ⇒ det(dM) = s|u− |2 [ρ]2 (1 + 2j 2 τ 3 θ (τ 2 j 2 + |u− |s 2 )) .
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The last parenthesis vanishes precisely when the flow on the + side is sonic with respect to the shock (see Section 2.1). When this happens, we find easily that ⎛
2j [τ ] dM = ⎝ −j −1 1
⎞ 0 0 0 −sj −2 ρ 2 |u− |2 ⎠ . 0 j s/(1 − s 2 )
The rank of this matrix equals two. Its kernel is spanned by the vector (0, 1, 0)T . Therefore the derivatives of j and s along S + vanish. In particular, ν is constant, so that S is a straight line. Remarks. • If the flow is piecewise C 3 , we may continue the calculation above. Assuming that τ is non-constant along S + , we find two other equations which involve j , τ , θ and θ
. Eliminating j and τ , we conclude that θ θ
= 3θ 2 . This means exactly that the fluid behaves locally like a Chaplygin gas. In all other cases, a shock between a constant state and an irrotational flow separates two constant states. • Since we do not prescribe a priori an equation of state, the proof above indicates that the generation of vorticity across curved shocks is, at least at the qualitative level, a phenomenon of kinematical nature. We may see Theorem 2.3 as a variant of Theorem 2.6, because a wedge formed by two shocks can be viewed as a unique shock with curvature concentrated at one point. Thus the impossibility of pure triple shock configuration expresses the fact that the vorticity generation at each of the three wedges cannot cancel. The barotropic case. When a barotropic equation of state is prescribed, it becomes possible to predict in a quantitative way the generation of vorticity across a curved shock. We again assume that the flow is uniform in − . Let τ be the unit tangent to the shock and κ be the curvature. In the following calculations, the dot is the derivative along S with respect to arc length: τ˙ = κν. Denoting by B := i(ρ) + |u|2 /2 (with i (s) = s −1 p (s)) the Bernoulli invariant, we have 1 B˙ = τ · ∇B = u · (τ · ∇)u + τ · ∇p = (∇u) : (u ⊗ τ − τ ⊗ u) ρ ω = ωu × τ = j , ρ
Shock reflection in gas dynamics
71
where we drop the subscript on the plus side. Since the identity 1 [p(ρ)] =: F (ρ) [B] = [i(ρ)] − ρ holds true, because of (47), this gives F (ρ)ρ˙ = j
ω . ρ
(64)
On the other hand, the Rankine–Hugoniot conditions give u− · ν =
ρ[p(ρ)] =: G(ρ). ρ− [ρ]
Differentiating along the shock, we obtain G (ρ)ρ˙ = κ(u− × ν), from which we deduce the formula G (ρ)j ω = κ(u− × ν)ρF (ρ).
(65)
This identity shows that, given the states ρ− , ρ = ρ+ , j and the direction of the shock, the vorticity on the non-constant side is proportional to the curvature of the shock front. It is also proportional to the tangential component of the velocity, hence vanishes at points where the shock is normal. With Theorem 2.6, this shows that such points are isolated along a curved shock.
2.4. Diffraction for the full Euler system We aim to give in this section a precise description of the planar steady shocks in which the state U0 is prescribed on one side, and the unit normal ν pointing from the state U0 to the state U is given. For convenience, we shall use coordinates (ρ, u, p) in state space. The important point is that, whenever the shock front is not normal to the speed u0 (that is, ν is not parallel to u0 , one speaks of an oblique shock), the direction of the speed must change across the shock, i.e. u1 is not parallel to u0 . More precisely, the angle between the velocity and the normal to the front is larger downstream9 than upstream. This is clear from the fact [ρu · ν] = 0 while ρ+ > ρ− , hence |u+ · ν| < |u− · ν|, while [u × ν] = 0. This is the diffraction phenomenon, analogous to the one that arises in optics. Since shock reflection in presence of walls imposes a direction to the velocity in the diffracted flow (because of the boundary condition u · n = 0), it is important to describe 9 Notice that we do not specify whether U is downstream or upstream. 0
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quantitatively the diffraction angle across a shock, in terms of the shock strength and of the incidence angle. This is not too hard since an oblique shock, that is a shock where u × ν = 0, is nothing but the superposition of a normal shock (namely one with u × ν = 0) with an arbitrary (though continuous) tangential component of the velocity u − (u · ν)ν. Notice that, given U0 and the direction of the shock front, we just have to determine a normal shock, ignoring the tangential part of the velocity; this normal shock will usually be completely determined by (ρ0 , u0 · ν, p0 ) and the fact that the shock is steady. The worst situation is when these data lie beyond some threshold, so that there does not exist any such normal shock. This usually happens when the incidence angle is too large, exactly as in optics, where a light ray cannot enter a medium of higher refraction index when the incident angle exceeds a critical angle. This phenomenon is one of the ingredients that are responsible for the transition from RR to MR. The steady shock curve (full Euler system). Our first observation is that the thermodynamical variables (ρ, p(ρ, e)) on the other side of the shock front must belong to the curve (the Hugoniot curve) defined by (9), namely (recall that τ denotes the specific volume 1/ρ) e − e0 +
p(ρ, e) + p0 (τ − τ0 ) = 0. 2
(66)
On an other hand, the mass flux j across the shock is j = ρ0 u0 · ν. Hence the Rankine– Hugoniot condition for the conservation of momentum is linear in (p, τ ): j 2 (τ − τ0 ) + p − p0 = 0 (j := ρ0 u0 · ν).
(67)
Conversely, assume that (j, p, τ ) satisfies (66, 67). We define u by u · ν := j τ,
u × ν = u0 × ν.
By definition, we have [ρu · ν] = 0. From (67), we immediately have [ρ(u · ν)2 + p] = 0. With [u × ν] = 0, this gives the Rankine–Hugoniot condition for the conservation of momentum. At last, we have
2 1 2 j 2 [p][τ 2 ] |u| + e + pτ = τ + e + pτ = [e + pτ ] − 2 2 2[τ ] = [e + pτ ] − τ [p] = [e] + p[τ ] = 0,
which is the Rankine–Hugoniot equation for the conservation of energy. In summary, a solution (j, p, τ ) of (66, 67) yields a piecewise constant solution of the full Euler model, where the front is normal to ν. This discontinuity is an admissible shock if j , or equivalently u0 · ν, has the sign of τ0 − τ . In conclusion, the set of admissible shocks with the state U0 on one side is a curve parametrized by the angle of incidence. Calculations. We consider the system (66, 67) when the equation of state is that of an ideal gas, pτ = (γ − 1)e. Then (66) reads [pτ ] + (γ − 1)p[τ ] = 0. In other words,
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73
[p]τ + γ p[τ ] = 0. Eliminating [τ ] with (67), we obtain [p](j 2 τ − γ p) = 0. The case [p] = 0 corresponds to the trivial pattern where U = U0 . Hence non-trivial patterns obey the linear equation j 2 τ − γ p = 0. Using again (67), we obtain the unique value of p: p = (1 − μ2 )j 2 τ0 − μ2 p0 .
(68)
We notice that in the limit of an infinitesimal shock, that is p = p0 , one has (1 + μ2 )p0 = (1 − μ2 )j 2 τ0 , which yields |u0 · ν| = c0 . In other words, the small shocks behave like sonic waves. From (68), we derive the value of the specific volume τ = μ2 τ0 + (1 + μ2 )j −2 p0 .
(69)
Although (69) gives a positive value, this is not always true for (68). A value of p is relevant as a pressure only if being positive. Denoting by θ0 the angle of incidence on the (0) side, namely the angle between the shock normal ν and the velocity u0 , this relevance is equivalent to
γ −1 < M0 cos θ0 , 2γ
M0 :=
|u0 | , c0
(70)
where M0 is the Mach number10 on the (0) side. We point out that this inequality imposes a lower bound for the Mach number on either side of a steady shock wave:
γ −1 < M0 . 2γ
(71)
Notice that (71) is automatically satisfied when the state is supersonic (M0 > 1). For a given state U0 , the Hugoniot curve is parametrized by the incidence angle θ0 ∈ [0, θ0∗ ) where θ0∗ < π/2 is given by the equality in (70). This interval is non trivial when (71) is fulfilled, in particular when U0 is supersonic. Notice that, contrary to the optics, the maximal reflected angle that is obtained when θ0 = θ0∗ is smaller than π/2; it is given by tan θ∗ = μ2 tan θ0∗ . From [u × ν] = 0 and [ρu · ν] = 0, we get |u| sin θ = |u0 | sin θ0 and ρ|u| cos θ = ρ0 |u0 | cos θ0 . This gives [τ tan θ ] = 0. Together with (69), this yields tan θ =
μ2
tan θ0 . + (1 + μ2 )j −2 ρ0 p0
10 This is an absolute Mach number, with respect to the reference frame, rather than a Mach number with respect to the shock, √ whose value is M0 (ν) := |u0 · ν|/c0 . In terms of the latter, the positivity of p amounts to saying that M0 (ν) > (γ − 1)/(2γ ). But since we shall let θ0 vary while keeping u0 fixed, it is better to make use of M0 , which remains fixed.
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Equivalently, we obtain the relation between the incident and reflected angles: tan θ =
(γ + 1)M02 sin θ0 cos θ0 (γ − 1)M02 cos2 θ0 + 2
(72)
.
This yields a function θ0 → θ that is not necessarily monotonic at a fixed Mach number. However, it has at most one change of monotony. It is more interesting to consider the deviation angle [θ ] = θ − θ0 . From the addition formula for tangents, and from (72), we obtain tan[θ ] = 2
M02 sin θ0 cos θ0 − tan θ0 2 + (γ − 1)M02 + 2M02 sin2 θ0
[θ ] := θ − θ0 .
(73)
Notice that if we use the relative (to the shock front) Mach number m0 = |u0 · ν|/c0 = M0 cos θ0 , then Formula (73) reads tan[θ ] = 2
m20 − 1 2 + (γ + 1)M02 − 2m20
tan θ0 .
This expression is interesting in that it shows that [θ ] vanishes only for θ0 = 0 (that is for normal shocks) and for m0 = 1. The latter case necessitates M0 ≥ 1, and then is always compatible with condition (70). Variations of θ0 → [θ ]. The function θ0 → [θ ] must be studied on the interval [0, θ0∗ ). Its derivative vanishes when P (m20 ) = 0, where P is the polynomial P (X) := 2γ X 2 + (4 − (γ + 1)M02 )X − 2 − (γ + 1)M02 . This polynomial has always one positive real root X0 , which corresponds to an admissible angle θ0 if and only if X0 ∈
γ −1 , M02 . 2γ
This means precisely that
γ −1 P 2γ
< 0 ≤ P (M02 ).
These inequalities read M02 > (γ − 3)/(3γ − 1) and M02 ≥ 1 respectively, and the second one implies the first one. Hence the function θ0 → [θ ] is monotonic decreasing when M0 ≤ 1 (the state U0 is subsonic in the reference frame), but has exactly one maximum if M0 > 1 (the state U0 is supersonic). We recall that in the latter case, U0 may be either subsonic or supersonic with respect to the shock. Actually, P (1) = 2(γ + 1)(1 − M02 ) being
Shock reflection in gas dynamics
! Fig. 2. Deviation angle vs incidence angle. Left (a):
75
γ −1 2γ < M0 ≤ 1. Right (b): M0 > 1.
negative, we see that θ0 → [θ ] is monotonic increasing on the interval [0, θ0s ) for which U0 is supersonic with respect to the shock. In particular, we have [θ ] ∼ 2
M02 − 1 2 + (γ − 1)M02
θ0
for small incident angles (almost normal shock). Shock polar. We summarize the above calculations. ! −1 If M0 ≤ γ2γ , there does not exist any steady shock from U0 . ! −1 If γ2γ < M0 ≤ 1, the interval [0, θ0∗ ) is non-trivial, and θ0 → [θ ] is monotone decreasing. Both θ0∗ and the corresponding reflected angle θ∗ are less than π/2. If M0 > 1, the function θ0 → [θ ] is monotone increasing until a critical angle, and then is decreasing towards a negative value. Notice that in all non-trivial cases, the deviation is negative at the end point, meaning that θ∗ < θ0∗ . The patterns are represented in Figure 2, called shock polars. Small strength. We say that a shock has small strength if [U ] is small, but the incidence θ0 is not. In particular, [θ ] ! 1, meaning that m0 ∼ 1. Such a shock is therefore almost sonic and is close to an acoustic wave11 . We have M ∼ M0 , and since one of the states must be supersonic, we see that both states are absolutely supersonic: M > 1,
M0 > 1.
11 An acoustic wave is a continuous, piecewise C 1 , solution of which the pressure is not C 1 .
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On the other hand, both m and m0 are close to one, but they are on opposite sides of the unity: (m − 1)(m0 − 1) < 1, as it must be for every steady shock. Small vs strong shocks. Small shocks correspond to the part of the curve around its intersection with the upper semi-axis in Figure 2b. The deviation angle of such a shock is small, as well as the jump of the state; as mentioned above, the shock is approximately a sonic wave, meaning that c ∼ c0 ∼ u · ν ∼ u0 · ν. More generally, we say that a shock is small if it corresponds to an angle θ where the slope of the shock polar is negative, while it is strong if this slope is positive. With this terminology, a normal shock is always strong, although we may built a one-parameter family of normal shocks whose strength tends to zero at some value of the parameter. The flow behind a strong shock is always absolutely subsonic (see [74]).
2.5. Diffraction for a barotropic gas When the fluid is barotropic, we have only the conservation of mass and momentum, which yield the Rankine–Hugoniot conditions [ρu · ν] = 0,
[ρ(u · ν)u] + [p]ν = 0.
With the same notations as in the previous section, they read j := (ρu · ν)± ,
j [u] + [p]ν = 0.
Once again, we have [u × ν] = 0, from which we derive easily tan θ = φ −1 tan θ0 ,
φ :=
ρ . ρ0
We determine the ratio φ through j 2 [τ ] + [p] = 0, which we rewrite as φ
[p] = c02 M02 cos2 θ0 , [ρ]
M0 :=
|u0 | . c0
(74)
For a perfect gas with p(ρ) = ρ γ , this reads F (φ) = M02 cos2 θ0 ,
F (s) :=
s(s γ − 1) . γ (s − 1)
(75)
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77
As before, it is more interesting to work with the angle of deviation [θ ]: tan[θ ] =
(1 − φ) tan θ0 . φ + tan2 θ0
(76)
The same description as in the previous section holds. The deviation angle vanishes for θ0 = 0 (normal shock). It vanishes also if φ = 1 (infinitesimal shock), a property that happens, when M0 > 1, for some angle that solves cos θ0 =
1 . M0
The shock polar has the same general form (see Figure 2) as in the case of the full Euler system. Notice that the relative Mach number m0 := M0 cos θ0 is not limited here, contrary to the general case.
3. Reflection along a planar wall Let us consider the case where the physical domain is a half-plane, bounded by a rigid wall, say the horizontal axis. We give ourselves (see Figure 3) an incident steady shock I . This means that the states U0 and U1 , as well as the angle α, are given. By incident, we always mean that α ∈ (0, π/2) and U flows into the shock. With our convention, we have (U− , U+ ) = (U1 , U0 ) with respect to the incident shock. Although we shall not make use of the data U1 and α, it is worth saying that (U0 , U1 ; α) satisfies the Rankine–Hugoniot relations and the entropy condition, together with the natural boundary condition that u1 is parallel to the boundary: u1 · n = 0.
Fig. 3. Incident shock along a planar wall (the reflected pattern is to be determined).
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In general12 , U0 does not satisfy the slip boundary condition and therefore the picture given in Figure 3 is not complete. We are thus looking for a reflected shock R, that is a state U behind U0 , and an angle β, which satisfy the Rankine–Hugoniot relations and the entropy condition; naturally, we require the boundary condition that u is parallel to the boundary (see Figure 4 for the simplest pattern). Remark. The state U0 will flow into the reflected shock; thus U0 will play the role of U− with respect to the reflected shock, contrary to what held along the incident shock. We summarize this important remark in the formula U−R = U+I .
(77)
We notice that, U0 being supersonic with respect to the reflected shock, we need that M0 be larger than one13 . We have seen above that this requirement is automatically satisfied if the strength of the incident shock is weak enough.
3.1. Regular Reflection Figure 4 displays the Regular Reflection (RR), where the exact solution of our problem is piecewise constant, the only discontinuities being the straight lines where the incident and reflected shocks take place. The shocks meet at a point located along the boundary, which can be taken as the origin.
Fig. 4. Regular Reflection (RR) along a planar wall. 12 It may happen exceptionally that u · n = 0: When M > 1, there is a critical angle α for which the deviation c 0 1 angle is zero (see Figure 2b). 13 Hence U is absolutely supersonic, though subsonic relatively to the incident shock. 0
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79
The problem to solve: Since U0 is given, and in particular its angle δ with respect to the wall, we only have to select a steady shock from U0 to a state U , such that the deviation [θ ] across the shock equals δ. Coming back to Formula14 (73), we have to solve the equation 2
M02 sin θ0 cos θ0 − tan θ0 2 + (γ − 1)M02 + 2M02 sin2 θ0
= tan δ,
θ0 ∈ [0, θ0∗ (M0 )),
(78)
where θ0 is the (unknown) incidence angle of U0 in the reflected shock. This amounts to taking the intersection of the shock polar (Figure 2) with the vertical line of abscissa δ. Then the reflected angle β will be given by β=
π − θ0 − δ. 2
(79)
We point out that, since U0 is given as part of a steady shock (the incident one), (71) holds true. Therefore θ0∗ is positive and the interval [0, θ0∗ ) where the left-hand side of (78) is defined is non trivial. Since the range of θ0 → [θ ] is strictly contained in (−π/2, π/2), we see that there is a critical angle δc beyond which (78) has no solution. In this regime, there is a need for a more complicated pattern, say a Mach Reflection (see Section 3.2). Actually, experiments suggest that the transition from RR to MR happens at some angle δt strictly smaller than δc . In this situation, a Mach Reflection occurs, while a Regular Reflection is still theoretically possible. This anticipated transition must be due to some instability, but has not been explained rigorously so far. 3.1.1. Weak and strong reflections Although small values of δ do not always correspond to a weak incident shock, this will be the case in RR, according to both physical and numerical experiments. Hence we shall restrict to this case. As seen above, we have M0 > 1 and M > 1. The Figure 2b is thus relevant for both the incident and the reflected shocks. In the former, the angle of incidence is close to that at which the curve intersects the upper semi-axis. Hence α is not close to π/2: the incident shock is significantly not perpendicular to the wall. The deviation δ may be either negative or positive (the Figure 3 displays a positive δ). Concerning the reflected shock, the situation depends whether δ is positive or negative. In the former case, there are two solutions of Equation (78). One of them is small (θ0 ! 1), meaning that β is close to π/2. We warn the reader that the corresponding [U ] is not small, and we call this a strong reflection. The other solution is associated to a weak shock and is called a weak reflection. For a negative δ, there is only one solution, which is the weak one. The physical and numerical experiments suggest that a strong reflection, though perfectly defined at a mathematical level, does not happen in practice. This might be related to a Hadamard instability with respect to the evolutionary problem. Such instability is not that of one of the shock waves, these being individually stable, as shown by A. Majda [54]. It 14 For definiteness, we consider a perfect gas obeying the full Euler system. We leave the reader to carry out the calculations in the barotropic case. The results are qualitatively similar.
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must be a property of the whole pattern formed by the initial-boundary value problem and both shocks. This phenomenon has been investigated by Teshukov [75], see Section 3.3.1. We therefore restrict in the sequel to the case of weak RR. Notice that if the incident shock strength is small, both incident and reflected shocks are approximately sonic waves. In particular, we have u0 · νR ∼ c0 ∼ u0 · νI . Since δ is small, that is u0 is approximately parallel to the wall, this reads M0 sin β ∼ M0 sin α. We conclude that a weak RR with small incident strength follows approximately the law of optical (specular) reflection: β ∼ α.
(80)
Transonic vs supersonic RR. In a RR, we know that the state U2 behind the reflected shock R is subsonic with respect to R. However, it may happen that this state be either subsonic (if |u2 | < c2 ) or supersonic (if |u2 | > c2 ) with respect to the reference frame. In the first case, we say that the reflection is transonic. In the latter case, we say that it is a supersonic reflection. Teshukov [74] has shown that under a mild assumption on the equation of state, a strong RR is always transonic. 3.1.2. Normal reflection When we let the angle α tend to zero, a limit situation happens, where it is not possible any more to keep the reflection point at a finite distance: the incident shock becomes parallel to the wall and bounces against the wall at time t = 0. For this reason, the strong reflection does not happen (another insight that a strong reflection is not physical) and the reflected shock is parallel to the wall too. However, the incident shock is present only at negative times, while the reflected one is there only at positive times (Figure 5). In this problem, the state U1 is the only data. It defines a Hugoniot curve of normal shocks, parametrized by the shock velocity s. The rest states U0 and U1 are found by saying that their velocities vanish. For a reasonable equations of state, there is exactly one solution with a positive s (the incoming shock) and one with a negative s (the reflected shock). Let us see the calculations.
Fig. 5. A normal reflection. The shock bounces on the wall at time t = 0.
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Fig. 6. Normal reflection for a barotropic gas with p > 0. The rest densities at positive (ρ− ) and negative (ρ+ ) times.
Barotropic gas. Let U be the rest state and j be the net flow across the shock. We identify the velocity vectors with their normal component. We have j = ρ1 (u1 − s) = −ρs and p(ρ) = p(ρ1 ) + j u1 . Eliminating j and s, we arrive at the equation p(ρ) = p(ρ1 ) +
ρρ1 2 u . ρ − ρ1 1
(81)
Let g(ρ) denote the right-hand side of (81). Apart from a pole at ρ = ρ1 , this is a decreasing function (see Figure 6). When p is monotone increasing, Equation (81) admits precisely two solutions ρ0 and ρ2 with ρ0 < ρ1 < ρ2 . There correspond the rest states U0 and U2 , with corresponding shock speeds sI > 0 and sR < 0. Full Euler system. The identity (81) still hold true, except that now p = p(ρ, e) and p1 = p(ρ1 , e1 ). We have to solve the system made of (81) and the analogue of (66): e − e1 +
p(ρ, e) + p1 (τ − τ1 ) = 0. 2
(82)
ρρ1 2 u , which is a piecewise increasing function of ρ Let us define g(ρ, e) := p − p1 − ρ−ρ 1 1 (provided pρ > 0), with a pole at ρ = ρ1 and
g(0, e) = −p1 < 0,
g(ρ1 − 0, e) = +∞,
g(ρ1 + 0, e) = −∞,
g(+∞, e) = +∞. Equation (81) thus defines two functions ρ− (e) < ρ1 < ρ+ (e), which are decreasing under the natural assumption that pe > 0. In terms of the specific volume, we have increasing functions τ+ (e) < τ1 < τ− (e). Because of (81), solving (82) amounts to solving 1 e + p1 τ± (e) = e1 + p1 τ1 + u21 . 2
(83)
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Since the left-hand side is an increasing function of e, it is enough to verify that 1 p1 τ± (0) < e1 + p1 τ1 + u21 . 2 This is true provided e1 >
1 u21 , 2 p1
(84)
because τ− (0) = τ1 + u21 /p1 and τ+ (0) = 0. Condition (84) is precisely that which ensures the existence of an incoming normal shock with downstream state U1 . As a matter of fact, s=−
ρ1 u1 [ρ]
is of the sign opposite to that of [ρ]. Hence ρ− corresponds to s > 0, that is to the incoming shock, while ρ+ correspond to the reflected shock. In summary, for reasonable equations of state, a normal incident shock always result in a uniquely defined normal reflected shock. This one is the limit of the weak Regular Reflection as the angle α tends to zero. 3.1.3. Regular Reflection for a barotropic gas For a barotropic gas, the equation to solve is (1 − φ) tan θ0 = tan δ, φ + tan2 θ0 where φ and θ0 are related through (75). Recall that M0 and δ are given, and that θ0 is our unknown. This problem amounts, as above, to finding the intersection of a shock polar with a vertical line of abscissa δ, yielding a strong and a weak reflection. On an analytical level, tan θ0 may be eliminated, thanks to the formula 1 + tan2 θ0 =
M02 1 . = cos2 θ0 F (φ)
There remains an “algebraic” equation in φ, of degree 2(γ + 1). For a polytropic gas with D degrees of freedom, this is a genuine algebraic equation in φ 2/D , since γ = 1 + 2/D.
3.2. Mach Reflection When the incident shock is too strong, or the angle α between the front and the wall is too large, the RR does not happen, either because Equation (78) does not have a solution, or because its solutions yield unstable patterns (see Section 3.3). Experiments suggest that the reflection point P is removed from the wall. There is a reflected shock R, but the point
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Fig. 7. A Mach stem. The velocities u2 and u3 are parallel though not equal. The pressures p2 and p3 are equal, but the entropies S2 and S3 are not.
Fig. 8. A Mach stem in presence of a boundary. The vortex sheet and the shock S are curved.
P where I and R meet is not any more along the boundary. Of course, the extreme states U1 and U2 are distinct (as they were in the simpler situation of the RR) and there is a need of a third shock to match them. Thus we expect a triple shock pattern at point P . Since we have proved in Section 2.2 that a pure triple shock structure does not exist, we also need a fourth wave at P . The simplest possibility is that of a slip line V (for vortex sheet), giving rise to a Mach stem shown in Figure 7. This terminology is due to von Neumann, after the experiments by Ernst Mach [53]. At a first glance, the Mach stem is a piecewise constant steady solution, thus has the advantage of being explicit (algebraic calculations). However, its flaw is that it cannot fit the boundary condition u · n = 0. Were U3 to satisfy the boundary condition, V would be horizontal since it is parallel to the flow; but then U2 would be horizontal too so that I and R could be used to make a RR. Hence the third shock would be useless. This argument also shows that in a Mach stem, the slip line cannot be horizontal. Another difficulty in the matching of the Mach stem with the boundary condition is that since U3 and U1 are parallel, the third shock S must be normal. If S is straight, it must be vertical, but this is a severe restriction that makes the construction overdetermined. From this analysis, we must take the following conclusion. First of all, the shock reflection is not piecewise constant. Although the shock S and the slip line V may be straight
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near P when either of U2 or U3 is supersonic, they must be curved when approaching the boundary. In particular, the tangent to S at P can be determined algebraically using shock polars, while it is perpendicular to the wall. Next, the solution is unsteady. What we may expect in general is a self-similar solution generated, for instance, by the interaction of the incident shock with a very thin wedge. Then the state U (x, t) will depend only on the selfsimilar variable y = x/t. As a matter of fact, it is hard to imagine a physical experiment yielding automatically a steady shock reflection. When a RR occurs, it is only as part of a self-similar solution, and it actually travels at constant speed in the laboratory frame. 3.3. Uniqueness of the downstream flow in supersonic RR Let us consider a supersonic RR, meaning that we assume that the flow U2 behind the reflected shock is supersonic (|u2 | > c2 ). In particular, the RR is a weak one. For instance, in a near-to-normal weak RR, u2 is very large (in the reference frame where P is fixed) and therefore we do have |u2 | > c2 . Under this assumption, the steady Euler system is hyperbolic in the direction of the flow, that is in the direction of the horizontal axis. Thus we may view this system as an evolution system of PDEs, and consider the determination of the downstream state and of the reflected shock as a kind of Boundary Value problem (BVP). Actually, since R is a free boundary, this is a Free Boundary Value Problem (FBVP), with data U1 and boundary conditions. The latter are the no-flow condition along the wall, and the Rankine–Hugoniot condition across R. We constructed in Section 3.1 a solution (actually two solutions) of this FBVP by assuming a priori that R was straight and the downstream flow uniform. These assumptions were natural since the incident pattern is self-similar and the steady Euler system is invariant under space dilations. In particular, if a germ of solution defines uniquely the solution, then the latter must be self-similar, which means that R is straight and the downstream flow is uniform. The purpose of this paragraph is to get rid of these assumptions, by showing that in reasonable regimes, they are necessary. In other words, we are concerned with a uniqueness problem. The FBVP has a rather special form, since the boundaries meet at initial “time”. They form a wedge at P , so that there is no “initial data”. We assume however that the solution is piecewise smooth; in particular, the downstream state has a limit U (P ) and R has a tangent R at P . From the invariance of the Euler equations under dilation, we see that U (0) and R do provide a uniform solution of the FBVP, that is a uniform RR. Thus U2 := U (0) must be one of the states computed in Section 3.1. The behaviour at P of any piecewise smooth solution is given therefore by either the weak RR or the strong RR. Our problem here is whether such a limit value must extend downstream. This is a uniqueness question concerning the FBVP, related to the well-posedness. The general theory for that kind of problem has been developed by Li Ta-tsien and Yu Wen-ci [67]. Since the FBVP is quasilinear and first-order, we shall analyze first the linearized problem at the state (weak or strong RR) U2 = (ρ2 , u2 , e2 ). It consists of linearized PDEs. For the sake of simplicity, we restrict to the barotropic case. The PDEs are u2 · ∇ρ + ρ2 divu = 0,
ρ2 (u2 · ∇)u + c22 ∇ρ = 0.
(85)
Shock reflection in gas dynamics
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Along the ramp, we just write
cos α u· sin α
= 0,
while along R we set the linearized Rankine–Hugoniot condition. After elimination of the increment δν of the normal, this reads MU = 0 with M a 2 × 3 matrix. Using characteristic components (U− , U0 , U+ ), these boundary conditions are respectively15 U+ = c 0 U0 + c − U− along the ramp, and U 0 = b 0 U+ ,
U − = b − U+
along R. We describe now the analysis of [67]. The solution in the interior is completely determined thanks to (85), once we know the trace of U− on R and of U0 , U+ on the ramp. Denoting by t > 0 a time-like variable (t = 0 at P ), these traces f− (t), f0 (t), f+ (t) are to be determined by a kind of difference equations: f+ (t) = c0 f0 (σ0 t) + c− f− (σ− t),
f0 (t) = b0 f+ (θ0 t),
f− (t) = b− f+ (θ+ t), (86)
where the numbers 0 ≤ σ0 , . . . , θ+ < 1 involve the characteristic velocities and the slopes of the boundaries, while the coefficients b0 , . . . , c− come from the boundary conditions. There remains to check the local well-posedness of the linear problem (86). The appropriate tool is the Laplace transform in terms of the logarithm of t. A natural space is L2 (dt/t), though we may also consider Sobolev spaces associated to the same weight, and we need them when considering the non-linear well-posedness. The linear BVP is strongly wellposed in these spaces if and only if the modulus of the following Evans function is bounded away from zero in the right complex half-plane ("z ≥ 0): (z) := 1 − b0 c0 (σ0 θ0 )z − b− c− (σ− θ+ )z . It is strongly ill-posed in Hadamard’s sense (lack of uniqueness) if vanishes somewhere in the open half-space ("z > 0). Remark that (z) vanishes precisely when (86) admits a non-trivial solution homogeneous of (complex) degree z. For general coefficients b, c, σ and θ , it is not easy to determine whether vanishes in the closed right-half plane. Since we clearly have |(z)| ≥ 1 − |b0 c0 | − |b− c− |, 15 The fact that U can be taken as the output on the ramp, or the input on the shock, amounts to the normality + of the boundary conditions.
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one often contents oneself with the sufficient condition |b0 c0 | + |b− c− | < 1 for uniform well-posedness. However, in the present problem, a simplification arises, because c0 = 0, while σ− θ+ > 0. Then (z) = 1 − b− c− (σ− θ+ )z . Therefore it becomes clear that a necessary and sufficient condition for strong well-posedness is |b− c− | < 1.
(87)
Remark that although b− and c− depend on the normalization of the eigenmodes of System (85), the amplifying ratio b− c− does not. Eigenmodes. We begin by computing the eigenmodes and characteristic directions of (85). This amounts to solving (u2 · ξ )ρ + ρ2 u · ξ = 0,
ρ2 (u2 · ξ )u + c22 ρξ = 0.
The characteristics are either parallel to the flow (u2 · ξ = 0) or normal to the vectors ξ± given by u2 · ξ = c2 |ξ |. We normalize |ξ± | = 1, so that u2 · ξ± = c2 . Remark that this equation has two solutions because of our assumption c2 < |u2 |. The corresponding modes are, in (ρ, u) variables, −ρ2 0 , r0 = . r± = c2 ξ± u2 We point out that this eigenbasis becomes ill-conditioned as we approach a sonic shock, since we have r+ = r− in the limit. Calculation of c0 and c− . Let us write the boundary condition u · ν = 0 along the wall, with U = U− r− + U0 r0 + U+ r+ : c2 (U+ ξ+ + U− ξ− ) · ν + U0 u2 · ν = 0. Since we already have u2 · ν = 0, this reduces to U+ ξ+ · ν + U− ξ− · ν = 0, confirming that c0 = 0. We obtain additionally that c− = −(ξ− · ν)/(ξ+ · ν). Since ξ+ and ξ− are symmetric to each other with respect to u2 , that is to the wall, we deduce c− := 1.
(88)
Calculation of b− . We now perform the calculation of b− . We start with the linearized boundary condition ρu2 · ν + ρ2 u · ν + [ρu] · δν = 0,
(89)
ν · δν = 0,
(90)
ρ1 u1 · δν[u] + j u + [p]δν + c22 ρν = 0.
(91)
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We have used the following notations: [p] = p(ρ2 ) − p(ρ1 ) and so on, while δν is the increment of the unit normal vector ν. We point out that since δν is a tangent vector, we have [u] · δν = 0. In particular, the last term in (89) can be written [ρ]u2 · δν, while in (91), we may replace u1 by u2 . Let us investigate first the case of an infinitesimal incident shock. Recall that there are two cases: either the reflected shock is strong, or it is weak. Since the downstream flow is subsonic in the former case16 , we concentrate to the latter case. Then the reflected shock is infinitesimal too, hence is characteristic. This makes our FBVP in a corner a little bit harder to analyze. However, we can use the fact that the background state is constant everywhere. Therefore, our stability problem is equivalent to the stability of the constant state (ρ2 , u2 ) = (ρ1 , u1 ) = (ρ0 , u0 ) in a standard IBVP, where the “time variable” is −x1 and the boundary is x2 = 0. This 1-d IBVP needs exactly one boundary condition, a normal one, a fact that is easy to check. In conclusion, our FBVP is well-posed in this limit case. We now pass to the general case. To understand what is going on, it is easier to concentrate first on the problems which are transitional between the strongly well-posed ones and the strongly ill-posed. This means b− = ±1. If b− = −1, the solution (U, δν) of (89, 90, 91) satisfies U = U+ (r+ − r− ) + U0 r0 , meaning that ρ = 0. We easily obtain that u2 ν; that is, the shock is normal. This is not the case, for any weak or strong RR. We point out that this situation occurs only in the limit of a strong RR, when the incident shock strength tends to zero. If b = 1, we have U = U+ (r+ + r− ) + U0 r0 , meaning that u u2 . After some calculations that we leave to the reader, we find that a transition occurs when the following quantity vanishes G := ((u2 · ν)2 + c22 )(u2 × ν)2 + ((u2 · ν)2 − c22 )(u2 · ν)(u1 · ν).
(92)
The conclusion of this analysis is that Property (87) depends only on the sign of G. Notice that, since |u2 · ν| < c2 < |u2 |, the sign of G is not obvious. We have shown above that the FBVP is well-posed for the weak reflection of an infinitesimal incident shock. Hence we have to determine signG in this limit case, in order to decide which sign of G gives well-posedness: we have |u2 · ν| = c2 (an infinitesimal shock is characteristic) and u1 = u2 . Whence lim G = 2c22 (u2 × ν)2 is positive. Thus we obtain T HEOREM 3.1. Consider a supersonic RR for a barotropic gas. Then the FBVP (96, 97), in the wedge defined by P , the wall and the reflected shock, is strongly well-posed locally at P whenever G, defined by (92), is positive. This FBVP is strongly ill-posed if G < 0. Applications. We have seen above that at fixed angle α, the weak reflection is strongly stable as the incident strength goes to zero. Consider now the weak reflection for a quasinormal shock. In the limit α → 0, we find that P escapes to infinity, while the reflected 16 The reflected shock in a strong RR is almost normal, thus |u | ∼ |u · ν| < c . 2 2 2
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shock tends to that of the perfectly normal reflection, thus remains at finite distance of the origin. This shows that |v2 × ν| → ∞, while |v2 · ν| = O(1). Therefore G ∼ ((v2 · ν)2 + c22 )(v2 × ν)2 is positive. We conclude that, given an incident shock, the FBVP associated to the weak shock reflection is strongly well-posed when the angle α is small enough. Non-linear stability. The passage from linear stability to the nonlinear one is performed by a fixed point argument and integration along characteristics. See [67] for details. Strong linear well-posedness implies nonlinear well-posedness. The limit case when b− c− = 1 is unclear. Strong linear instability is likely to imply ill-posedness. Uniqueness and constancy of the downstream flow. From the non-linear stability, we deduce that, under the assumption that G is positive at (U2 , ν), there is only one stationary solution downstream the state U1 , achieving the limit U2 at P . By a solution, we mean a flow U of the Euler system, together with a reflected shock (a free boundary in this problem) between U and U1 . This unique solution is made of the constant state U2 and the straight reflected shock with normal ν. 3.3.1. Dynamical stability Teshukov has considered in [75] the most important dynamical stability of the RR as a stationary solution of the (full) Euler system. For this purpose, he linearizes the system about the RR. This resembles our analysis above, but with an extra ∂t U . A Laplace transform in time replaces this term by τ U , and one is reduced to the study of a stationary PDE system, where additional terms come from the fact that the shocks are free boundaries originally. The problem is to get an estimate in L2 (H ) (H the physical half-plane), uniform with respect to τ when "τ > 0. Notice that the limit case where τ = 0 yields exactly the same system as the one we studied in Section 3.3. Since the background state is constant along rays, it makes sense to use polar coordinates (r, θ ), at the expense that the new problem has variable coefficients. Teshukov chooses to perform a second Laplace transform in r (instead of log r as above), though this does not reduce the complexity of the problem, since the r-dependency of the coefficients turns into partial derivatives in the new Laplace frequency. After a rather much elaborated analysis, which involves the solvability of a Riemann– Hilbert problem, Teshukov obtains the result that for realistic gases, in particular for a perfect gas, the steady RR is dynamically stable if and only if it is a weak RR. We emphasize that this result is independent of ours, since we dealt instead with a static property, of the steady supersonic RR only. Teshukov’s analysis applies also to the transonic case, which is relevant in particular for every strong RR. 3.3.2. The analysis for the UTSD model The UTSD model is an approximation near the reflection point, valid for small strength and small angle (thin wedge), with the strength of the order of the square of the angle. With these restrictions in mind, it is interesting to consider RR for this model, as well as stability properties. The characterization of the RR regime was done in the seminal paper by Hunter [44].
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In moving coordinates associated to the sound wave of the state U0 , the UTSD reads (see Paragraph 1.6)
v2 vt + 2
+ wy = 0,
vy − wx = 0.
(93)
x
The coordinate x is parallel to the wall and y is normal. The unknowns v and w do not represent a velocity, but leading coefficients in an asymptotic expansion of U in terms of α 2 /|U1 − U0 |. System (93) is Galilean invariant with respect to (x, v), but not in terms of (y, w). Therefore, we may look for a steady state, where the incident shock is given only in terms of w0 , w1 and the jump v1 −v0 . Typically, v1 > v0 , because the shock is compressive. Since the system is invariant under the scaling (t, x, y, v, w) → (δ −1 t, δx, y, δ 2 v, δ 3 w), we may also fix this jump to [v] = 1. Then the strength of the incident shock wave is small when [w] is large. Using the Rankine–Hugoniot conditions, the slope of I is dx = a := v = dy
−
v0 + v1 . 2
Notice in particular that we need v < 0. Because of [w] = −a[v] = −a, the parameter a plays the role of an inverse of the shock strength. The boundary condition along y = 0 is written w = 0. Therefore the data satisfies w0 = 0. This implies w1 = −a < 0. Looking for a RR, we have to solve w1 = b(v1 − v2 ),
b=
−
v1 + v2 , 2
(94)
where −b is the slope of R, and we have used the boundary condition w2 = 0. This yields an algebraic equation of degree 3, P (v3 ) = 0, where P (v) := (v + v1 )(v − v1 )2 + 2w12 , and the√restriction that v1 < v2 < √ −v1 . It is easy to see that a solution exists if and only if a ≥ 2. Actually, when a > 2, there are two admissible solutions v ∗ ∈ (v1 , −v1 /3) (strong RR) and u∗ ∈ (−v1 /3, −v1 ) (weak RR). The state U2 is supersonic provided that v2 is negative and subsonic otherwise. It is immediate that subsonic in a strong RR, while in a weak RR it can be either supersonic, ! it is√ √ if a > as := 1 + 5/2, or subsonic, if a ∈ ( 2, as ). Let us assume that U2 is supersonic (in particular, the RR is weak). We perform the same stability analysis as in Section 3.3., by looking at the stationary model
v2 2
+ wy = 0,
v y − wx = 0
x
as a hyperbolic system in the direction of negative x’s. The boundary conditions along the wall (w = 0) and the reflected shock (Rankine–Hugoniot) are clearly normal. There
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remains to check an Evans condition. Once again, the latter reduces to |ρ| < 1, where ρ is an amplifying factor. We compute easily ρ=
4αw1 − P (v2 ) , 4αw1 + P (v2 )
α :=
√ −v2
where α is the sound speed. On the one hand, we have P (v2 ) < 0 because the RR is weak. On the other hand, we already know that w1 < 0. Therefore |ρ| < 1. We conclude that this stationary problem in the wedge between the wall and the reflected shock is strongly well-posed. In particular, the state remains constant and R remains straight. This remains true in the self-similar problem
v2 2
+ wy = xvx + yvy ,
vy − wx = 0,
x
until one reaches the sonic locus. Since U ≡ U2 on one side of the sonic curve, the latter is (an arc of) the parabola given by the equation y 2 = 4(v2 − x).
4. Reflection at a wedge We now consider a more complex as well as interesting geometry, presented in Figure 9. A solid wedge is surrounded by the gas. For t < 0, a planar shock wave is approaching the wedge and the state is piecewise constant, the gas being at rest near the wedge. This is clearly an admissible solution of the initial-boundary value problem. At time t = 0, the shock wave hits the wedge and something happens, since the flow downstream does not satisfy the slip boundary condition. A kind of reflection develops, which is plainly twodimensional. As a matter of fact, the incoming shock plays the role of an incident shock
Fig. 9. An incoming shock at a wedge. The upstream flow is at rest: u0 = 0. At time t = 0 the pattern is self-similar.
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on both sides of the wedge, yielding two reflected shocks, and these latter interact in a complicated way.
4.1. A 2-D Riemann problem At first glance, the problem is genuinely time dependent. However, considering the state at time t = 0 as an initial data, we observe that it is invariant under space dilations. The same is true for the spatial domain (the complement of the wedge). Since the system to solve is quasilinear of first order, we expect therefore that the solution (if there is any) be self-similar: x1 x2 , (x ∈ , t > 0). U (x1 , x2 , , t) = U˜ t t Denoting the self-similar variable by y := x/t ∈ , we may rewrite the Euler equations as a stationary-like system. Generally speaking, a fist-order system of conservation laws ∂t u +
∂α f α (u) = 0
(95)
α
yields
∂α f α (u) ˜ = (y · ∇y )u˜
α
for self-similar solutions (mind that now ∂α is the derivative with respect to yα ). Such a system has the bad feature of having variable coefficients on its right-hand side. It is a remarkable property of the Euler equations, a consequence of Galilean invariance, that these variable coefficients may be removed, at the price of the introduction of a pseudovelocity v. This vector field is defined by x v(y) := u(y) ˜ − y = u(x, t) − , t where as usual, u is the fluid velocity. The full Euler equations (3, 4, 5) in self-similar variables then can be rewritten div(ρv) + 2ρ = 0,
(96)
div(ρv ⊗ v) + 3ρv + ∇p = 0, 1 ρ|v|2 + ρe + p v + 2(ρ|v|2 + ρe + p) = 0. div 2
(97) (98)
Likewise, the entropy inequalities (31) yield div(ρf (S)v) + 2ρf (S) ≤ 0
(99)
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for the same set of functions f . Away from discontinuities, Equation (97) may be combined with (96) to give ρ(v · ∇)v + ρv + ∇p = 0,
(100)
while (98) may be rewritten as v·∇
p 1 2 |v| + e + 2 ρ
+ |v|2 = 0.
(101)
This equation generalizes (16). Recall that when v is replaced by the genuine velocity u, the expression in parenthesis above has been called the Bernoulli invariant: p 1 B := |u|2 + e + . 2 ρ In the self-similar case, we should merely speak of the pseudo-Bernoulli invariant: 1 p B := |v|2 + e + . 2 ρ Of course, in a barotropic flow (isentropic or isothermal), we have p = P (ρ). Equation (99) must be dropped, while (98) becomes an inequality: 1 2 div ρ|v| + ρe + p v + 2(ρ|v|2 + ρe + p) ≤ 0. 2
(102)
Boundary conditions. In a self-similar problem, the initial data is equivalent to a boundary condition at infinity, since t → 0+ amounts to |y| → +∞. Hence we search a solution of (96, 97, 98) (only the first two equations in the barotropic case), such that U (ry) → V (y) as r → +∞, where V is the shock data at time t = 0. However, we point out that a hyperbolic system has the property of finite speed of propagation. This means that given a (small) domain ω, our solution U must equal, at time t = 1 in ω, any other solution U such that U (y, 0) = U (y, 0) for y in a large enough domain ω0 , called the dependence domain of ω. Typically, ω0 ⊂ ω + B(0; ), where is the maximum of the propagation velocities. Of course depends on the solution itself, and it is therefore important to establish pointwise bounds on the solution, in order to have an explicit bound of , justifying a posteriori our qualitative assumptions. Since our initial data U (·, 0) and our physical domain are piecewise one-dimensional data, we first apply the above idea by choosing = R2 and extending U (·, 0) by U0 in the wedge. The corresponding solution U is just a travelling wave U (x − sI tνI ) (sI νI the velocity of the incident shock), and the equality U = U is valid away from the influence domain of the wall. Thus U is piecewise constant for d(y : ∂) > . A more involved choice allows to treat points in the strip d(y : ∂) < that are not in the influence domain of the wedge tip: |y| > . If there exists a regular reflection (in the sense of Section 3.1) when we extend infinitely one of the walls of the wedge, we
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may take for U the corresponding exact RR and the corresponding half-plane. Mind that U will be a travelling wave of constant speed parallel to the ramp, since our incident shock is moving. If its velocity is larger than , we deduce that in our true solution U , the incident shock remains straight until it meets the ramp, where it reflects according to the calculations of Section 3.1. Additionally, the reflected shock is straight and the state behind it is constant, as long as one stays away from B(0; ). This pattern is again called Regular Reflection. See Figure 10 for the case where the incident shock propagates in the direction of the symmetry axis of the wedge. Self-similar vs steady flows. It is worth to notice that the system (96, 97, 98) for selfsimilar flows differs from (3, 4, 5) for steady flows, only by low order terms 2ρ, . . . We see two important consequences. The first is that the type may be determined in exactly the same terms (though with v replacing u) as in Section 1.4. The second is that the discontinuities (shocks, contacts) obey the same set of Rankine–Hugoniot conditions and admissibility criteria. In particular, a shock must separate a supersonic state from a subsonic one (in the pseudo sense, see immediately below). Mind however that the low order terms play a crucial role in the sequel, as they allow to get a priori estimates that would be false in the stationary case. See Section 6.2. Sub- or supersonic self-similar flows. In a time-dependent problem, the notions of subsonic or supersonic flows are not well-defined, because the Galilean invariance makes us free to add a constant to the velocity. When some external relation between time and space is given, these notions become meaningful. This is the case for steady flows, as we have discussed in Section 1.4. This is also relevant in self-similar flows. In this latter case, sonicity refers to the pseudo-velocity v. We say that the flow is pseudo-subsonic (respectively pseudo-supersonic) when |v| is smaller (respectively larger) than the sound speed |c|. Finally, the flow is sonic where |v| = c. Thus it makes sense to introduce the pseudo-Mach number M :=
|v| . c
When M < 1, the self-similar system is not hyperbolic in any direction, while if M > 1, it is hyperbolic in directions ξ ∈ S 1 such that |v · ξ | > c. In the former case, the principal part defines only one family of characteristic lines through the equation y˙ = v. In the latter case, there are two other families of characteristic lines, which obey |v × y| ˙ = c, using arclength parametrization. It is worth noticing that if the real flow U is constant in some open set (as it will be in applications), v = u − y and M are non constant. In particular, the sonic line defined by M = 1 will be an arc of the circle centered at u, with radius c. For a weak incident shock, u vanishes upstream and is small downstream for weak reflection, so that the center of this circle is close to the origin. A note about figures. For a self-similar flow, U˜ is identical to U at time t = 1. Thus we always display the flow at time one. In practice, only the geometrical features are shown:
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incident (I), reflected (R) and diffracted () shocks, vortex sheet (if any), sonic line (S) and the walls. Special points are denoted by P (reflection point), O (wedge tip), and Q (pseudo-sonic point on R). We point out that α is now the angle between the ramp and the horizontal axis. Since the incident shock is vertical, it makes an angle π/2 − α with the wall, instead of α in Section 3. Flow and pseudo-flow. The pseudo-velocity turns out to have a physical meaning. For let t → x(t) be a particle trajectory of the flow: x dx = u(x, t) = u . dt t For y := x/t, we have dy 1 = v(y), dt t or in other words dy = v(y), dτ
τ := log t.
This shows that the physical flow is identical, up to the parametrization of the trajectories, to the pseudo-steady flow generated by the pseudo-velocity field.
4.2. Patterns There are several possible patterns, depending on the strength of the incident shock and on its angles with the walls of the wedge. Ordered in increasing complexity, we mention Regular Reflection (RR), the Mach Reflection (MR), the Double Mach Reflection (DMR) and the Complex Mach reflexion (CMR). Though realistic phenomena often involve a MR or more shaky patterns, we shall address mainly the RR in the sequel, since there are more mathematical results available in this case. 4.2.1. The symmetric RR We say that the reflection is symmetric when the incident shock is normal to the axis of the wedge. Then we expect (if uniqueness holds) that our solution is symmetric: ⎞ ⎞ ⎛ ρ ρ ⎟ ⎜ ⎜ v1 ⎟ ⎜ ⎟ (y1 , −y2 ) = ⎜ v1 ⎟ (y1 , y2 ). ⎝ −v2 ⎠ ⎝ v2 ⎠ e e ⎛
In particular, the vertical velocity vanishes along the vertical axis y2 = 0. This amounts to considering the half-domain + bounded below by the wedge and the horizontal axis.
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Fig. 10. The symmetric RR. Incident (I), reflected (R) and diffracted () shocks, sonic line (S) and subsonic domain D. Upstream (U0 ) and downstream (U1 ) flows are given.
The latter plays the role of another rigid wall. Figure 10 displays the symmetric regular reflection. This reduction may seem useless at first glance. Actually, it is of great importance in a priori estimates, because the reduced domain + is convex. This will allow us to establish a minimum principle for the pressure; the possibility of such a property is unclear in the non-symmetric case. Both numerical and physical experiments suggest that, if the incident shock strength := ρ/ρ0 is small, and if the ratio α −2 is small too, then the domain of influence of the wedge tip does not contain the intersection point P . Then the reflected shock is a straight line and the state (ρ, u, e) remains constant, until the sonic line at point Q (this will be discussed in Section 4.3). The interaction with the sonic line gives rise to the diffracted shock, which bends until it reaches the symmetry axis. The pattern consisting of the states U0 , U1 and U2 , separated by the incident and reflected shock waves can be computed explicitly with the help of shock polars (see Sections 3.1 and 4.1). Since the state equals the constant U2 on one side of the sonic line, this must be a circle (or an arc of the circle) with equation |y − u2 | = c2 . The present picture is characterized by a weak RR at point P , which is supersonic in the pseudo sense (|u2 − yP | > c2 ). There is another possibility for a RR, when the state U2 given by shock polar analysis gives a subsonic pseudo-state yP − u2 (transonic RR). Then we do not exclude that the incident shock still reaches the ramp and the diffracted shock begins at P . In this case, there is no sonic line, and the subsonic zone fills the domain between and the wall. See Figure 11. 4.2.2. Mach Reflections When the incident shock strength increases, or when α decays, there happens a transition from a RR to a MR. The interaction between the incident shock and the ramp has been described in Section 3.2. The rest of the picture resembles much the RR for moderate data (SMR), but becomes increasingly complex (DMR and CMR) as the angle α decreases or the shock strength increases. If the pseudo-state U2 behind P is subsonic, there occurs a simple Mach Reflection. Otherwise, the pattern depends on
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Fig. 11. The symmetric RR when |yP − u2 | ≤ c2 . Incident (I) and diffracted () shocks, subsonic domain D. Upstream (U0 ) and downstream (U1 ) flows are given.
Fig. 12. The three kinds of Mach Reflection. Left (a) simple (SMR); center (b) complex (CMR), right (c) double (DMR).
whether the slip line reflects against the ramp before crossing the sonic circle attached to U2 , or not. In one case, a fourth shock will interact again with the reflected shock, giving rise to a secondary Mach stem typical of a DMR. In the other case, the reflected shock suffers only a kink, which characterizes a CMR, intermediate between SMR and DMR. A detailed analysis based on shock polars can be found in [4,5,43] and [19], while [27,62,63] provide convincing numerical experiments. We point out however that an existence theory remains to be worked out. For the convenience of the reader, we display in Figure 12 the various kinds of Mach Reflection, although we do not intend to attack them at a mathematical level. Notice that experiments suggest that a transition occurs between every pair of types RR, SMR, CMR, DMR, but the pair (SMR, DMR).
4.3. The supersonic domain behind the reflected shock In the case of a supersonic RR, it is usually taken from granted that the state U (the physical one, but not the pseudo-state) is constant in the supersonic region D2 surrounded by the
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reflected shock R, the ramp and the pseudo-sonic line S defined by |v| = c. In particular, the sonic line obeys the equation |u2 − y| = c2 .
(103)
The sonic line is therefore an arc of the circle C of radius c2 , centered at u2 . Additionally, the reflected shock, separating two constant states U1 and U2 , remains straight until it reaches the sonic line. This constancy, though a little bit intuitive, deserves some mathematical investigation. In the sequel, we assume only that the state U is smooth in D2 and tends to U2 as y → yP within D2 . Our task is to prove that U remains constant in D2 . The idea is to apply a uniqueness result for a boundary value problem (referred to as BVP in the sequel) associated with an evolution system of PDEs. The system is of course given by the self-similar Euler equations. The assumption that the reflected state U2 be pseudo-supersonic at P tells that this system is hyperbolic in the direction of the flow. Defining a time-like variable in the direction of the ramp, from P to O, we see on the one hand that the system is (at least locally) evolutionary and hyperbolic in this direction, and on the other hand that D2 is locally in the “future” of P . These properties will remain valid as long as U stays close enough to U2 . We point out that the domain of this BVP is wedge-like. The boundary value problem is completed by the slip condition u · ν along the ramp and the Rankine–Hugoniot condition along R. Notice that R is a free boundary. A particular solution of our BVP consists of U ≡ U2 , together with the straight reflected shock given by the shock polar analysis, and the circular sonic line defined by (103). Hence a uniqueness result will solve the question. We decompose the uniqueness problem in three parts: • A local uniqueness for the BVP in the wedge-like domain at P . This is essentially the problem addressed in Section 3.3, with v instead of u. The lower order terms play no role in the strong well-posedness or in the strong instability, though we do not exclude that they be important in the marginal case of weak linear well-posedness. • Local uniqueness for standard BVPs, either along the ramp, or along the free boundary R. Following Section 3.3, we thus assume that the quantity G := ((v2 · ν)2 + c22 )(v2 × ν)2 + ((v2 · ν)2 − c22 )(v2 · ν)(v1 · ν)
(104)
is positive. For instance, this is true for the weak RR when either the incident strength is small, or the angle α is close to π/2 (near normal reflection). Since the state is pseudo-supersonic, the system is hyperbolic in D2 . It has three characteristics, a backward one, a forward one and a third one in between. The words backward and forward refer to an evolution, with the coordinate along the boundary as a pseudotime; recall that this coordinate increases from the reflection point P to the tip of the wedge. Forward characteristics emerge from the wall when the pseudo-time increases, that is when one escapes from P ; on the contrary, backward characteristics approach the ramp as the pseudo-time increases (see Figure 13). The remaining characteristics are tangent to the boundary, because of the no-flow boundary condition; they are flow line (y˙ = v), with
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Fig. 13. A forward (f) and backward (b) characteristics. The arrow shows the time-like direction in the supersonic domain D2 .
multiplicity two in the case of the full Euler system or one in the barotropic case. Because of hyperbolicity, there is a propagation property, implying that the region K where U ≡ U2 is bounded either by boundaries of D2 , or by pseudo-characteristics. Since U is smooth, these characteristics are either the tangents to the pseudo-sonic circle C associated to the state U2 : (y − w) · N = 0,
N :=
w − u2 , c2
(105)
where w is some point of the circle, or flow lines. We point out that each tangent consists in two characteristic lines, a forward one and a backward one, both separated by the point of tangency; see Figure 13. Because of local uniqueness at P , K is non void. Assume also that K is strictly contained in D2 . Then the boundary of K contains at least one straight characteristic described above, which meets either the ramp (if it is forward) or R (if it is backward or a flow line). Thus we have to verify the uniqueness property for two special BVPs, associated with the following domains. Such BVPs have been studied thoroughly by Li and Yu [67]. The domain of the first BVP is bounded by the ramp and a characteristic , which is either backward or forward. On the latter, we have a Goursat problem, on which the state is prescribed in a consistent way. Along the ramp, one characteristic is incoming, one is glancing and one is outgoing. Thus we need precisely one boundary condition. Since there are 1 + 1 dimensions, the BVP is well-posed provided the boundary condition, here u1 cos α + u2 sin α = 0 is normal and is forward. Normality means that the unique incoming mode does not satisfy the boundary condition, a fact that can be verified easily. We conclude that either K extends along the ramp up to the sonic line, or it is bounded by a backward characteristic . We point out that such a cannot meet the circle within the physical domain; the tangency point lies below the ramp. If K is bounded by a backward characteristic , we face the second BVP, which is actually a FBVP (free boundary value problem). Its domain is bounded by and the shock
Shock reflection in gas dynamics
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R. Along the former, we still have a Goursat problem and the state is prescribed in a consistent way. On the reflected shock, two characteristics17 are incoming, because the flow is pseudo-incoming to D2 . The forward characteristic is outgoing, because the flow is subsonic, relatively to the shock. Was the boundary fixed, we should need precisely two (barotropic case) or three (full Euler case) scalar boundary conditions. Since the shock is a free boundary, we actually need one more boundary condition, that is as many as the number of equations in the system. The Rankine–Hugoniot conditions give exactly the right number of boundary conditions. The verification that they form a normal boundary condition at U2 is straightforward. Therefore the FBVP is locally well-posed, whence the uniqueness property. This proves that K extends along the ramp up to the sonic line. In conclusion, K extends between the ramp and R, till S. Thus we have proved that T HEOREM 4.1. Consider a symmetric RR past a wedge. Assume that this reflection is supersonic: |u2 − P | > c2 , in particular, the reflection is a weak one. Assume in addition that G > 0 (with G defined in (104)). Then 1. the reflected shock R is straight between P and the circle C (of equation |u2 − y| = c2 ), 2. the state U remains constant, equal to U2 , within the domain bounded by the ramp, the reflected shock and the circle C. When the incident shock strength increases, G may become negative even though the reflection remains supersonic at P . In such a case, one does not expect either that R remains straight, or U remains constant, till C. In particular, C does not any more coincide with the sonic line of the pseudo-flow. The instability of the accepted pattern could explain why the RR gives way to Mach Reflection in cases where the planar RR still exists.
4.4. Mathematical difficulties The following list displays the mathematical difficulties that are encountered in the mathematical treatment of reflection past a wedge. All of them but the slip line arise already in the simplest RR pattern. Vertex at the origin. The boundary of the physical domain has a singularity at the origin. It will be responsible for a lack of smoothness of the solution. In view of Equation (118), it is unclear whether the pressure has a singularity, or the velocity vanishes. Sonic line. This difficulty arises in a supersonic RR. Across the sonic line S, the type of the Euler system changes from hyperbolic (in the supersonic zone labelled 2) to hyperbolicelliptic (in the subsonic domain D). In D, the elliptic part of the system degenerates as one approaches the boundary. When the strength of the incident shock is small, we are tempted to use linear or weakly non-linear geometrical optics (see next Paragraph), which suggest that the flow is Hölder continuous across the sonic line, with a singularity √ of order d(y; S) where d(·; S) is the distance to S in D; see for instance the acoustic approximation of J. Keller and A. Blank [49]. This picture turns out to be incorrect. It 17 One being double in the full Euler case.
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has been uncovered recently by G.-Q. Chen and M. Feldman [22] that the nonlinearity in the system yields Lipschitz continuity, at least in the potential case, where the system reduces to a nonlinear wave equation. Vertex at P . Alternatively, in a transonic RR, the subsonic zone extends till P , and we have to solve a nonlinear system of PDEs in a domain that has a vertex at P . Mixed type. In the subsonic domain D, the system is of mixed type hyperbolic-elliptic, meaning that one characteristic field is real, while the other ones are complex. This makes the analysis very hard, since it is not possible either to apply standard techniques of hyperbolic problems or to employ ideas from elliptic theory. Vortical singularity. We shall see in Section 6.6 that the vorticity ∇ × u cannot be square integrable in D, at least in the barotropic case. A singularity is expected at some point along the ramp. It is unclear whether this singularity is present in full gas dynamics. Slip line. In Mach Reflection, the interaction point P (defined as the point where the reflected and the incident shocks meet) is detached, and a secondary shock called Mach stem connects P to the ramp. Since the Rankine–Hugoniot relations are the same as in the steady case, a pure three-shocks pattern is not possible (Theorem 2.3). Hence a fourth wave must originate from P . Experiments suggest that it is a slip line, or in other words a vortex sheet. According to M. Artola and Majda [2], such jumps are √ known to be dynamically unstable unless the jump of the tangential velocity exceeds 2 2 c, which is unlikely18 . Finally, one observes that the slip line rolls up endlessly. Diffracted shock. The solution is known everywhere but in the subsonic zone D. However, the part of the boundary of D formed by the diffracted shock is a free boundary. For an incident shock of small strength, this curve is approximately a circle, the continuation of the sonic line. Triple point. The sonic line and the reflected shock form a corner at their meeting point Q. This is a singularity of the boundary of the subsonic domain. This singularity is non uniform in terms of the shock strength, as both lines tend to become tangent when the strength vanishes. We point out that the vortical singularity and the slip line are obviously not present in irrotational flows. Additionally, the system becomes purely elliptic in D. This makes the irrotational RR much more tractable, with only the difficulties of non-uniform ellipticity, boundary vertex, triple point and free boundary. In particular, one may expect that the flow be of class H 1 within D. Zheng [86], as well as Chen and Feldman [22] obtained recently an existence result in this case, when the incident shock is almost normal. Another simplification occurs in the transonic case (|u2 − P | < c2 ). Then we avoid the degeneracy problem across the sonic line and the triple point Q. In conclusion, the simplest situation for a RR is that of an irrotational flow for which the subsonic zone reaches the point P . Then the mathematical problem is to solve a scalar second order nonlinear elliptic equation in terms of the potential. The remaining difficulties are the free boundary and two geometrical singularities, at O and P . We point out on the one hand that this problem does not seem to follow from the minimisation of some action. On the other hand, it happens in some intermediate (very narrow) range of parameters and thus is not a perturbation of some trivial configuration. Therefore it cannot be attacked by pertubative tools. 18 In a three dimensional setting, vortex sheets are unconditionally unstable.
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Finally, let us mention the work by S. Chen [24], who proves the existence of a local solution for the reflexion of a shock against a smooth convex obstacle. Of course, this result is sensitive to the curvature of the boundary, and does not survive when the shape of the obstacle becomes sharp. S. Chen also proved a local stability result, near the triple point, of a Mach configuration; this result is in the spirit of Paragraph 3.3.
5. Reflection at a wedge: Qualitative aspects 5.1. Weak incident shock When the strength of the incident shock tends to zero, the initial data (at t = 0) of the evolutionary Euler system tends to a constant state, say U0 . This constant being a rest state, it is a solution of the IBVP. Since it is a smooth one, we may apply stability results (see for instance Dafermos [30], Chapter 5.2)19 . Thus the expected solution must be close20 to U0 . Although this has not been proved yet, we shall assume that this proximity holds uniformly: sup |U (y) − U0 | ! 1.
(106)
y
Assuming also that the solution has the piecewise smooth structure described in Figure 10, it should be described, at first order, by weakly nonlinear geometric optics, applied to the background state U0 . This approach has been introduced in [49] and developed further in [47]. Linear and nonlinear geometrical optics. We begin with linear geometrical optics, which gives the leading order for the evolution of the singularity locus. It was worked out by Keller and Blank [49]. There are two types of initial disturbances. First the incident shock itself, and then the vertex, where the downstream state is not compatible with the boundary condition. In general, every pointwise disturbance generates a front that travels at one of the velocities ±c0 or 0 associated to U0 . The zero speed means that a part of the singularity stays fixed, while the velocity ±c0 concerns every direction: a pointwise singularity surrounded by a constant state generates a circular front. In particular, we must have a direct wave along a (approximate) circle S whose center is located at the wedge tip. The situation is a bit different when the disturbance is localized along a line. In general, the initial front splits into three fronts, of which two move apart in opposite directions at normal velocity c0 , and one stays fixed. In our problem, the pattern is simpler since the initial disturbance is compatible with the Rankine–Hugoniot conditions: the front that originates from the incident shock remains the part of this shock that is not influenced by the vertex. Besides these direct waves, we must keep track of secondary waves (as well as ternary,. . . if any) that are generated by the interactions between them, and between one of them and 19 Recall that stability results use relative entropy estimates, and they are closely linked to weak-strong uniqueness. 20 In terms of the relative entropy, or of the relative energy in the barotropic case.
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Fig. 14. The limit pattern as the incidence strength tends to zero.
the wall. We have already seen the effect of an incident shock along a wall, which gives rise to a reflected shock with (asymptotically) specular reflection. It is immediate that, in the limit of zero strength, this reflected wave is tangent to the circular wave S. Hence the direct and secondary waves do not produce ternary waves. On an other hand, the wave S is already compatible with the boundary condition (its velocity is normal to the wall) and does not yield a secondary wave. More importantly, because of compatibility at the tangency point Q, there is no need to continue the straight reflected shock beyond this point; the proper continuation is the circle itself! One may say that the reflected shock is bent once it meets the sonic line. Finally, it is clear that S does not meet the incident shock, because the point P in Figure 10 travels at speed c0 / cos α, hence faster than S. We summarize the previous analysis: In the limit of zero incident strength, the solution exhibits three singularities along curves. Direct ones are the incident shock I moving forward and touching the wall at point P , and a circular curve S emanating from the vertex. The reflected shock is made of a part of this circle (the diffracted wave) and a secondary wave R along the tangent to this circle between P and Q. The remaining part of the circle is the sonic line. See Figure 14. At the analytical level, the linear theory yields a second-order equation of the form ¯ + r∂r ρ¯ + ∂θ2 ρ¯ = 0, r∂r ((1 − r 2 )∂r ρ)
(107)
where ρ¯ := ρ/ρ0 is the relative density disturbance, and (r, θ ) is a polar coordinate system, rescaled in such a way that the circle |y| = c0 has equation r = 1. Equation (107) is elliptic for r < 1. The boundary condition is of Dirichlet type along the circle ρ(1, ¯ θ ) = ρb (θ ) := and of Neumann type ∂ ρ/∂ν ¯ =0
1 2
along along S,
(108)
Shock reflection in gas dynamics
103
along the wall. The solution was found in explicit form by Keller and Blank, thanks to the Busemann transformation R :=
r . √ 1 + 1 − r2
√ They found that ρ − ρb behaves like h(θ ) 1 − r for some function h = 0, everywhere but in a neighbourhood of the triple point (θ = 2α). A more elaborate analysis carried out by Hunter and Keller [46,47] uses weakly nonlinear geometrical optics. It yields a refined description of the flow in a neighbourhood of the diffracted shock. There, the leading-order approximation satisfies a one-dimensional cylindrical Burgers equation. The asymptotic expansion at the triple point Q was described by E. Harabetian [37]. When α is small together with I , the sonic circle and the triple point approach the interaction point P . Then the description of the flow near P needs the more involved UTSD approximation of [44], which combines a multi-dimensional context with a nonlinearity. More about the supersonic domain. Let δ ! 1 denote the quantity supy |U (y) − U0 |. Since the wave velocities are bounded by c = c0 + O(δ), we can prove (see again [30]) that the solution U at (x, t) depends only on the data and the geometry of the domain in a ball D(x; ct). At time t = 1, this means that the solution, outside of the disk D(0; c), is not influenced by the vertex. In particular, it coincides with solutions that are known explicitly: • For |y| > c, y2 cos α − y1 sin α > c and y1 < c, one has U ≡ U1 (downstream data), • For |y| > c, y2 cos α − y1 sin α > c and y1 > 0, one has the incident shock, namely U ≡ U1 for y1 < sI (sI the incident shock speed) and U ≡ U0 for y1 > sI , • For |y| > c and 0 < x2 cos α − x1 sin α < c, we have the piecewise constant reflection described in Section 3. In particular, the reflected shock is straight near the reflection point P and the solution U ≡ U2 is constant in some wedge bounded by P , the rigid wall and the reflected shock. We warn the reader that a constant u means a non-constant v! We have seen above (Theorem 4.1) that the constancy of U and the straightness of R extend up to the sonic line. Remarks. 1. Perhaps the most important point is that, using the qualitative assumptions made above about the structure of RR, we shall be able (see Section 6) to make quantitative pointwise estimates, and therefore to justify the smallness assumption (106) made at the beginning of this paragraph. 2. Since the reflected shock is of small strength O(δ), the reflection at point P is a weak one, and the reflection angle equals approximately the angle of incidence. Such a reflection follows approximately the laws of optics. 3. If the wedge angle α is small, it may happen that the sonic line travels faster than the point P along the ramp. A sufficient condition for this to happen is " " " " sI "u2 − " < c2 . " sI tan α "
(109)
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Fig. 15. Various limits of shock reflection past a wedge. The vertical coordinate (here M := σI /c0 > 1 is the Mach number of I ) represents the strength of the incident shock. The dashed curve separates the RR regime from the MR. The limits studied by Lighthill (L) and by Hunter, Keller and Blank (HKB) occur along horizontal and vertical lines. The parabolic scaling at the origin yields the UTSD model of Brio and Hunter.
In this case, we might have either a Regular Reflection (as in Figure 11), or a Mach Reflection with interaction point remote from the wall. Of course, we do not claim that equality in (109) is the criterion for the transition from RR to MR. 4. In some regimes, a RR solution is technically possible in a neighbourhood of P , but is not observed in numerical and/or physical experiments, presumably for instability reasons. This does happen for incident shocks of moderate strength. In this case, one observes a MR instead.
5.2. Small and large angle Another asymptotic limit occurs when one keeps the incident shock strength constant, while making α → 0+. See Figure 15. It was considered by M. Lighthill [52], with the idea that in the limit problem (α = 0), there is no reflection at all since the shock travels parallel to the wall. Therefore, in spite of the fact that a Mach Reflection must occur, we expect that the reflected pattern consists only of weak waves. Hence the problem can be studied through linearization behind the incident shock. The resulting system is essentially the same as that of Keller and Blank21 . It reduces to a second order equation in the pressure, which can be solved thanks to Busemann’s transformation. Remark that the reflected pattern fills approximately the acoustic disk D of center O and radius c1 , the sound speed behind I . Since the incident shock is of Lax type, c1 is larger than the shock speed σI , so that the disk is truncated by the shock locus (as well as by the wall). This makes clear the relevance of a Mach Reflection. The opposite case, when α tends to π/2, yields in the limit the normal reflection described in Section 3.1.2. Since it has coordinates (σI , σI tan α), the point P escapes to infinity. In particular, this kind of reflection is supersonic. The rest of the pattern in the weak RR has a limit which is piecewise constant. In this limit, we have u2 = 0 and the reflected shock is straight and vertical. The subsonic domain coincides with the part of the 21 It is more chronologically correct to say that the analysis by Keller and Blank has similarities with Lighthill’s.
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disk |y| < c2 to the right of R. The velocity in D vanishes identically. Remark that σR is less than c2 , so that the sonic line is transverse to the reflected shock, contrary to the other limits considered above. This happens because R has a non zero limit strength, hence is not sonic. The pattern is nonlinear in essence at the leading order. 5.3. Entropy-type inequalities This section is not specific to gas dynamics and may be skipped in a first reading. However, we believe that it has some interest for numerical purposes. We consider self-similar solutions of a first-order system of conservation laws (95), endowed with an entropy inequality ∂t η(u) + divx q (u) ≤ 0, η being strictly convex (D2 η > 0n ). The corresponding PDEs divy f j (u) = (y · ∇y )uj ,
(j = 1, . . . , n),
divy q (u) ≤ (y · ∇y )η(u)
(110)
are understood in the distributional sense, and u(y) is locally bounded. As pointed out in [39], these equations can be combined to give inequalities in conservative form, though involving variable coefficients. Given any open subdomain ω with smooth boundary, we denote by n the outgoing unit normal to ω, by |ω| the volume and by d the space dimension (d = 2 in our reflection problems). Then every self-similar solution satisfies
1 ((n · y)u − f (u; n)) ds(y) ≤ n · (η(u)y − q (u)) ds(y), d|ω| ∂ω ∂ω (111) where f (u; n) := α nα f α (u). It is remarkable that, as ω runs over all subdomains, (111) is equivalent to (110). Therefore it can be used to give a posteriori estimates in numerical simulations. Given a finite volume method where fluxes across control volumes are practically computable, one should either make a correction on those volumes where (111) fails, or refine the grid there. This idea has not yet been implemented to our knowledge, though it looks promising. 1 η d|ω|
6. Regular Reflection at a wedge: Quantitative aspects The purpose of this section is mainly to establish pointwise estimates for Regular Reflection. Both the barotropic and the full Euler cases are considered, except in Section 6.1 (non-barotropic model only) and 6.6 (barotropic model only). We assume throughout the section that the solution (if any), is piecewise smooth and obeys the qualitative description given in Section 4 for a supersonic RR22 . The only places where U is singular are 22 However, most of our estimates are valid for either transonic RR or MR, as long as the flow is piecewise smooth.
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• the incident and reflected shocks, where we have discontinuities, • the sonic line, where U is likely to be at best Lipschitz continuous, • possible vortex sheets (slip lines) in the interior of the subsonic domain D. Notice that, according to Theorem 2.2, steady shocks are forbidden between two subsonic states, • the origin, where we shall see that the pressure gradient experiences a Dirac mass, • the likely vortical singularity, located at some point of the ramp, where the vorticity lacks square integrability. 6.1. Minimum principle for the entropy This paragraph is devoted to the case of full Euler system. The easiest pointwise estimate has been known for two decades [69] and follows directly from (31) or from (99). As stated in Theorem 1.1, the minimum of the entropy is a non-decreasing function of time. Since the initial data experiences two states U0 (upstream) and U1 (downstream), we deduce S(y) ≥ min{S0 , S1 }. It turns out that in a shock (here, the incident one), the entropy is lower upstream than downstream, so that S0 < S1 , whence the estimate S(y) ≥ S0 .
(112)
We emphasize that this is a sharp estimate, since S(y) equals S0 for y1 ≥ sI / cos α. However, the better estimate S(y) ≥ S1
(113)
holds true whenever y1 ≤ sI / cos α. This can be proved from the transport equation v · ∇S = 0, and the fact that S increases across shocks when following the pseudo-flow. 6.2. Minimum principle for the pressure The next estimate is more involved, but still sharp. We begin by defining the angle θ (y) of the flow: cos θ v = |v| . sin θ Of course, this definition makes sense only away from stagnation points, and θ can be chosen in a smooth way in every simply connected domain where v is smooth and does not vanish. The important point is that the gradients of p and θ are related by a linear identity (see [65]) ρ|v|2 ∇θ ⊥ + ρv = (I2 − c−2 v ⊗ v)∇p,
(114)
Shock reflection in gas dynamics
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where X → X ⊥ is a rotation by 90◦ . Notice that (114) is valid even across contacts, because the pressure and the angle are continuous across a vortex sheet23 . After dividing by ρ|v|2 , one eliminates θ from (114) by taking the curl. A kind of miracle happens here because the low order term involves again the gradient of the pressure: div
1 |v|2 c−2 − 2 −2 (I − c v ⊗ v)∇p + v · ∇p = 0. 2 ρ|v|2 ρ|v|4
(115)
We notice that for the moment, we do not know whether ρ may vanish, but we shall see soon that it does not, because of an explicit lower bound. This expresses the fact that the wedge has an obvious compressive effect. We interpret (115) as a linear second order equation Np = 0 in the pressure, with variable coefficients. For cosmetic reasons, our operator N is given by the left-hand side of (115), multiplied by ρ|v|2 . We have N=
2
aij ∂i ∂j +
i,=1
bi ∂i ,
i
where the second order terms are precisely
j (δi − qi qj )∂i ∂j , i,j
q :=
v . c
We point out that the first order terms bi in N have singularities at stagnation points, namely when v vanishes. There, (115) becomes a first order equation. The operator N is elliptic precisely in the subsonic domain |q| < 1, because of the inequality between symmetric matrices: v ⊗ v < c 2 I2 . We thus have an extremum principle for the pressure: it cannot achieve a local minimum or a local maximum at an interior point, unless v vanishes simultaneously. Of course, the steady problem yields an equation similar to (115) (the right-hand side being replaced by zero) and the same conclusion can be stated. Even the incompressible case is relevant, with equation 1 div ∇p = 0, |u|2 as pointed out by H. Weinberger (private communication). For the moment, we have shown that the pressure cannot achieve a local extremum in the interior of the subsonic domain, except at stagnation points. However, it is known that 23 Actually, the angle θ equals that between the tangent of the contact curve with the horizontal axis.
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Fig. 16. Steady flows with a stagnation point. Left (a): saddle point, the pressure achieves a local maximum. Right (b): a focus, the pressure achieves a local minimum.
in the steady case, p does achieve local extrema a stagnation points. Typically, p may be maximal at saddle points of the flow, and minimal at foci (see Figure 16). Thus it seems hopeless to get an a priori bound for the pressure in general. It is therefore remarkable that the lower order terms present in the self-similar problem help in establishing a minimum principle for the pressure! Stagnation points. Let y0 be a stagnation point (v(y0 ) = 0) and assume that p achieves a local minimum at y0 . Let M denote ∇v(y0 ). From (96), we have TrM = −2. Hence the spectrum of M consists in eigenvalues −1 ± λ where λ is either real or purely imaginary, because M has real entries. Differentiating (100) yields D 2 p(y0 ) = −ρ(M 2 + M). Since the spectrum of a symmetric matrix is real, we deduce that λ is real. Then one of the eigenvalues −ρλ(λ ± 1) is non-positive. Since D 2 p(y0 ) must be non-negative, we deduce that λ = 0 and therefore D 2 p(y0 ) = 02 .
(116)
Concerning M, we already have two conclusions. On the one hand, its spectrum reduces to {−1}, while on the other hand we have M 2 + M = 02 . Since the polynomial X 2 + X has simple roots, M is semi-simple and therefore equals −I2 . We conclude that24 Dv(y0 ) = −I2 .
(117)
The analysis of [65] stopped there, so that the minimum principle for the pressure remained a formal result, leaving aside the marginal case where p is flat at y0 . We complete it here with a rigorous proof, assuming only that U is locally smooth. There remains to treat the case of a stagnation point with (116, 117). To begin with, we remark that, thanks to (117), y0 is an isolated stagnation point and that v(y) = −z + O(|z|2 ) with z := y − y0 . Our key observation is that, because of (117), N has bounded coefficients at y0 ! More precisely, we have N∼
2
i,=1
aij ∂i ∂j +
bi0 ∂i ,
i
24 This equality means that u = y holds at second order at the point y . 0
Shock reflection in gas dynamics
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where bi0 = −
2 ∂ 2 vi ∂i ρ + 2 zk zl . ρ ∂yk ∂yl |z| k,l
The boundedness of the coefficients and the uniform ellipticity of N allows (see [60], Chapter 2, Theorem 5) us to apply the maximum principle at y0 . Therefore the pressure cannot reach a local minimum at some interior point. Various components of the boundary. Since we have shown that the minimum of the pressure over D must be achieved somewhere on the boundary ∂D, we now investigate on which part it occurs. The wall. This minimum cannot occur along the wall, because of the following consequence of (114) and of the boundary condition: ∂p = 0, ∂ν where the left-hand side is the normal derivative. This, together with the elliptic equation (115) tells that p does not achieve a local minimum at such a point. There are however two subtleties: • At the corner of the wedge, which is a geometrical singularity. Remark that our domain is convex and could be approximated by smooth convex domains by smoothing out the vertex at the origin. Then the boundary condition is written, in terms of the pressure, ∂p = ρ|v|2 κ, ∂ν
(118)
where κ is the curvature of the boundary, positive in this case. Hence ∂p/∂ν is positive (treat the stagnation points as above) and p cannot be minimal along the wall. • Once again, the maximum principle does not apply directly at a boundary point where v vanishes. At such a point, ∂p/∂ν vanishes because of (118). The minimality along the boundary tells also that the tangential derivative vanishes, whence ∇p(y0 ) = 0. But then the same arguments as in the case of an interior stagnation point are valid. We find that ∇v(y0 ) and the coefficients of N are locally bounded. Therefore the maximum principle holds at such boundary points. We point out that the same argument works in the steady case and has a natural interpretation. If the ramp is compressive (α > 0 as in our case), the pressure is likely to increase ahead, while if the corner is expansive (α < 0), we have the opposite situation; the pressure cannot be maximal at a boundary point and it is likely to decrease ahead. The sonic line. Along the sonic line, the solution is continuous, thus p ≡ p2 . Since U2 is the subsonic state in the reflected shock between U1 and U2 , we have p2 > p1 and therefore p > p1 along the sonic line.
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The diffracted shock. The last component of ∂D is the part of the reflected shock between the point Q and the symmetry axis. Since the flow is subsonic on the inner side and supersonic in the outer, we have that the inner trace of the pressure is higher than p1 . In conclusion, the following minimum principle is expected for the pressure: p(y) ≥ p1 ,
∀y ∈ D.
(119)
We point out that this justifies a posteriori the fact that the density does not vanish, provided the internal energy remains bounded. 6.3. Estimate for the diffracted shock We still follow [65] and assume that the reflection agrees with Figure 10, and use the fact that the state U1 is (pseudo-)supersonic relative to the reflected shock: |v1 · ν| > c1 .
(120)
Here, v1 = u1 − y and u1 , c1 are constants. Hence this reads |(y − u1 ) × y˙ ⊥ | > c1 , when parametrizing the shock by arclength. In particular, the vector product does not vanish. By continuity, it must keep a constant sign, which is positive if the arc length is measured from Q. We have therefore (y − u1 ) × y˙ ⊥ > c1 .
(121)
For the moment, let us study the curves passing through Q, which are defined by equality in (121): (Y − u1 ) × Y˙ ⊥ = c1 ,
|Y˙ | = 1.
(122)
The system (122) defines two well-posed differential equations in the domain defined by |Y − u1 | > c1 , where Y determines two values Y˙ as intersection points of a circle and a straight line. It degenerates on the circle C1 of equation |Y − u1 | = c1 , where Y˙ must be the unit tangent to C1 . If |Y − u1 | < c1 , there is no solution Y˙ . We infer that the integral curves are made of an arc of C1 , followed at each extremity by the tangents to the circle25 . One or two parts (a tangent and/or the arc) may be omitted. Since |yQ − u1 | > c1 (because the downstream flow is supersonic at Q), we see that an integral curve originating at Q must follow the tangent to C1 , the one that approaches C1 counterclockwise. Then it is free to follow C1 until it leaves it along another tangent. Several integral curves are shown on Figure 17. 25 The conclusion in [65] was erroneous, because we did not pay attention to the degeneracy of (122) along the circle.
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Fig. 17. Integral curves for (y − u1 ) × y˙ = c1 . Boldface lines are tangents to the circle C(u1 ; c1 ). One of them goes upper to the left. The other one meets the circle at T , from which every integral curve follows the dashed part of the circle, until it leaves it along a tangent. One extreme curve ends at point M− . The other one is the continuation of the tangent passing through Q; M+ might not exist.
If the inequality in (121) held in the weak sense (≥ instead of >), we could only say that the reflected shock stays below the upper extreme integral curve of (122), at least the one going towards C1 . This was the result obtained in [65], which gives a rather poor information, both in terms of sufficient condition for the boundedness of D, and in terms of estimates. But since the inequality in (121) is strict, we know that the reflected shock stays actually below every integral curve of (121) (because it is below every one near Q). Therefore, it is bounded by the extreme integral curve defined as the tangent from Q towards C1 , followed by the circle till the horizontal axis. As a conclusion, we have the following estimate of the subsonic domain, which is illustrated in Figure 18: The subsonic domain D is contained in the convex set whose boundary is made of a part of the wall (horizontal axis and ramp), a part of the circle C1 of equation |Y − u1 | = c1 , a part of the tangent to C1 passing through Q, and the part of the sonic circle |Y − u2 | = c2 between the ramp and the point Q. 6.4. Using the Bernoulli invariant We still follow [65]. We know that, away from the discontinuities, the pseudo-Bernoulli invariant 1 p 1 B := |v|2 + e + = |v|2 + γ e 2 ρ 2 satisfies the damped transport equation v · ∇B + |v|2 = 0.
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Fig. 18. The convex set D contains the subsonic domain. It is bounded by the arc SQ of the sonic circle of state U2 , the arc T M− of the sonic circle of state U1 , plus its tangent at T through Q, and finally the part of the wall M− OS.
We recall that the damping term |v|2 is due to self-similarity. Given a smooth function f , we deduce v · ∇f (B) + |v|2 f (B) = 0. Multiplying by ρ and using (96), we obtain div(ρf (B)v) + 2ρf (B) + ρ|v|2 f (B) = 0.
(123)
Restricting our attention to D, (123) is valid everywhere in the sense of distributions, because there is no shock, and the slip condition (v · ν)± = 0 along contacts is harmless. Therefore we may integrate over D and find
ρf (B)v · ν dl +
∂D
ρ(2f (B) + |v|2 f (B)) dy = 0.
(124)
D
Let us discuss the boundary integral in (124). On the one hand, the integrand ρf (B)v · ν is a trace taken from the subsonic side. On another hand, it vanishes along the wall because v · ν = 0. Hence the integral can be taken on the upper part ∂D + only, made of the sonic line SQ and the diffracted shock. Let us denote by B¯ the supremum of the inner trace of B along ∂D + . We now choose ¯ and is a non-decreasing and smooth function, such that f vanishes all along (−∞, B] positive elsewhere. Then the boundary integral in (124) vanishes, and the last term is nonnegative, whence ρf (B) dy ≤ 0, D
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from which we deduce a maximum principle: ¯ B(y) ≤ B,
∀y ∈ D.
(125)
¯ Since U is continuous across the sonic line, we We then need an explicit bound for B. have on the one hand c2 1 γ (γ + 1) e2 B = |u2 − y|2 + γ e2 ≡ 2 + γ e2 = 2 2 2 along QS. We denote this constant by B2 . On the other hand, because of (102) (which is even an equality for the full Euler system), we have j [B] ≤ 0 across a shock, say the reflected shock. However, we know that the pseudo-flow enters26 D, meaning that j < 0 when we orient the normal to the exterior of D. Therefore the inner trace is less than the outer trace, the latter being 1 B1 (y) := |y − u1 |2 + γ e1 . 2 Of course, B1 (y) is not explicit along the reflected shock, because this curve is not known with accuracy. However, the previous paragraph gives us a rather good estimate for D: the diffracted shock is bounded by the extreme integral curve QM− of Figure 17. Therefore it is not difficult to bound B1 on this part by the supremum of B1 along QM− , that is by B1 (Q) (outer trace). Remarking as before that the outer trace B1 (Q) dominates the inner trace B(Q) = B2 , we deduce at last an explicit upper bound B¯ ≤ B1 (Q). Finally, we have B(y) ≤ B1 (Q),
∀y ∈ D.
(126)
This is an accurate bound for the pseudo-Bernoulli invariant, except for the fact that B1 (Q) is the largest value of the outer trace, which is strictly larger than the corresponding inner trace; B2 is likely to be a more accurate upper bound, though we do not have a satisfactory argument for this claim.
6.5. Conclusion; pointwise estimates Let us begin with the full Euler system. We use estimates (113, 119, 126), which imply eρ 1−γ ≥ exp(S1 ),
ρe ≥ ρ1 e1 ,
γ e B1 (Q).
Altogether, these inequalities give us explicit lower and upper bounds for the density and specific energy (hence for the pressure and the temperature):
26 This might not agree with common sense, but we must keep in mind that the pseudo-flow satisfies div(ρv) < 0 and thus is somehow convergent.
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(ρ1 e1
)1−1/γ
exp
S1 γ
≤e≤
B1 (Q) γ ,
γρ1 e1 B1 (Q)
≤ρ≤
B1 (Q) γ
1/(γ −1)
S1 exp 1−γ .
(127) Additionally, (126) gives |v|2 ≤ 2B1 (Q), which, together with the estimate of the size of D, say D ⊂ B(0; R) for an explicit R, yields |u| ≤ R +
√ 2B1 (Q).
(128)
We now turn to barotropic flow. The difference with the previous case is that there is no entropy. Hence we use only estimates (119) and (126): γ ρ γ −1 B1 (Q). γ −1
γ
ρ γ ≥ ρ1 , We deduce the bounds ρ1 ≤ ρ ≤
1/(γ −1)
γ −1 γ B1 (Q)
.
(129)
Then the same estimate as in (128) holds true.
6.6. The vortical singularity We show here that the vorticity cannot be square integrable. Our computation is valid for piecewise smooth solutions in the barotropic case. More precisely, we show that if the solution is (piecewise) smooth enough, then " "p "ω" (2 − p) ρ "" "" dy ρ D
(ω := ∂1 v2 − ∂2 v1 )
admits a nonzero limit as p → 2− ; in particular, it is not smooth. We recall that D denotes the subsonic zone, the domain bounded by the wall, the reflected shock and the sonic line (actually, ω vanishes everywhere else). Up to Equation (131) below, our calculations are valid in Dr , the complement in D of discontinuities27 . We start with Equation (100), which we divide by ρ (we have seen that the density does not vanish). Remark that ρ −1 ∇p is the gradient of enthalpy ∇i(ρ), where i (s) = s −1 p (s). Taking the curl, we obtain (v · ∇)ω + (1 + divv)ω = 0.
(130)
27 We recall that these discontinuities are only slip lines, because steady shocks are forbidden between two subsonic states, according to Theorem 2.2.
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Combining with (96), we deduce the transport equation (v · ∇)
ω ω = . ρ ρ
(131)
Let f be a Lipschitz continuous function of one variable. Multiplying (131) by f (ω/ρ), we get ω ω ω = f . (v · ∇)f ρ ρ ρ
(132)
We now recombine with (96) and get ω ω div f ρv = ρg , ρ ρ
g(s) := sf (s) − 2f (s).
(133)
We emphasize that (133) is valid off vortex sheets. However, since the normal component of v along vortex sheets vanishes, the divergence in (133) does not present any singular part. Hence this identity holds in the distributional sense in D. We now integrate over D and obtain ω ω ρv · ν dl = dy, f ρg ρ ρ ∂D D where we warn the reader that the boundary integral involves inner traces. Choosing f (s) := |s|p , this yields " "p " "p "ω" "ω" " " ρv · ν dl. ρ "" "" dy = − " " ρ D ∂D ρ
(2 − p)
(134)
Let us examine the limit of the right-hand side as p → 2: " "2 "ω" " " ρv · ν dl. " " ∂D ρ
F := −
(135)
The integral over ∂D decomposes as a sum over the wall, and the rest of the boundary ∂D− . The first part vanishes because of the boundary condition. It vanishes along the sonic line too, because the irrotationality of U2 propagates across this line28 . Along the reflected shock, we know that ω is non zero29 and that the normal pseudo-velocity v · ν is negative. In conclusion the limit F is positive. We summarize our results in the following statement 28 This means somehow that the vorticity is continuous across the sonic line, in spite of the fact that the rest of
∇u is unbounded. 29 Were it to vanish, the reflected shock would remain straight, which is false. As a matter of fact, the proof of Theorem 2.6 is easily adapted, since (130) still propagates irrotationality.
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T HEOREM 6.1. Let U be a piecewise smooth symmetric RR for a barotropic gas, with qualitative features as described in Figure 10. Then the vorticity ω := ∂1 u2 − ∂2 u1 cannot be square integrable. We point out that the vortical singularity is located at a stagnation point. As a matter of fact, (131) is an ODE that transports and amplifies ω/ρ. The amplification remains finite as long as v does not vanish. Only at stagnation points can ω/ρ blow up. Of course, since ρ and 1/ρ are uniformly bounded, the singularity concerns only ω. A reasonable expectation is that the vortical singularity arises somewhere along the ramp. As a matter of fact, there must exist a stagnation point along the boundary, since v · ν = 0 holds and the pseudo-velocity field is incoming at both ends. This does not rule out the possibility that several stagnation points exist. But, assuming the generic property that they are non-degenerate, there must be N + 1 attractors and N saddle points for some N ≥ 0, since the total degree of the pseudo-flow in the subsonic zone is +1. Remark that Equation (96) forbids repellors. If N ≥ 1, then the stable manifolds of the saddle points come from the exterior of the subsonic zone and divide D into N + 1 regions, each one containing precisely one attractor. 6.6.1. A formal description of the vortical singularity We wish to describe qualitatively the behaviour of the flow near the vortical singularity located at some point y0 , using Equation (131) and the conservation of mass (96). Because of v(y0 ) = 0, this point is critical for the pressure, thus for ρ, since the equation of state is barotropic. For this reason, we make the approximation that ρ is locally constant. Therefore, we base our analysis upon the simplified system div v = −2,
v · ∇ω = ω,
ω = curl v.
(136)
Because of the evidence given by Theorem 6.1, the solution must be such that the integral of ω2 diverges precisely at y0 . We begin with the simpler case (maybe not realistic) that y0 is an interior point. Then system (136) admits a one-parameter family of rotationally invariant solution. Using polar coordinates around y0 , we have vr = −r and vθ = vθ (r). There comes ω = vθ + 1r vθ . Since ω depends only upon r, we have v · ∇ω = −r∂r ω. Thus ω must be of the form ω0 /r, meaning that (rvθ ) = ω0 . Since the pseudo-velocity is locally bounded (see Section 6.5), we conclude that v ∼ −r e r + ω0 e θ ,
ω∼
ω0 . r
(137)
We point out that ω0 is the amplitude of the singularity. This amplitude must be somehow proportional to the vorticity generated across the diffracted shock. We now turn to the situation where the singularity occurs at a boundary point. Then the description given by (137) is not satisfactory, because the trajectories of this flow are
Shock reflection in gas dynamics
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transversal to the boundary. A refined analysis seems unable to give a solution where ω is of order 1/r at the singularity. Thus we look for a slightly more singular solution of (136). We begin by introducing a potential φ such that v = −r e r + curlφ, thanks to the first of (136); we have ω = φ. Then we search for a φ of the form φ ∼ m(r)a(θ ). Several choices of m are possible a priori, with the only constraint that dr (φ(r))2 = +∞, r 0 because of Theorem 6.1. Let us first try m(r) = r α , where we need α < 1. This gives respectively # $ (138) v · ∇ω ∼ r 2α−4 αa(α 2 a + a
) + (2 − α)a (α 2 a + a
) and ω ∼ r α−2 (α 2 a + a
). Since r 2α−4 dominates r α−2 , the dominant term in (138) must vanish. This means that a γ (α 2 a + a
) = cst =: κ,
γ :=
2−α . α
This autonomous equation of the form a
= F (a) is integrable by quadrature, as is well known. We want a(0) = a(π/2) = 0, in order that the trajectories be tangent to the boundary30 . If κ = 0, then a(θ ) = cst sin(αθ ), with α an integer. With the constraint, we obtain α = 1, but then ω is smaller than 1/r and is unlikely to satisfy Theorem 6.1. If κ = 0, a remarkable phenomenon happens. The solution of a
= F (a) with a(0) = 0 is monotonic on some interval (0, θ0 ) until a point where a (0) = 0. Thus we need that π be a multiple of 2θ0 . It turns out that θ0 ≥ π/2, with equality in the case where a satisfies the integrated equation a 2 + α 2 a 2 = κ1 a 1−γ
(κ1 (1 − γ ) = κ).
(139)
Thus a must be precisely one of the solutions of (139). However, these solutions behave like θ α near θ = 0. Since α < 1 and vθ ∼ m (r)a(θ ), we find that the trajectories are still transversal to the boundary (because a is not Lipschitzian, or quasi-Lipschitz, at θ = 0). Thus let us try a slight modification of the interior case, with m(r) = r log r. We then find log r 2 (a(a + a
)) . v · ∇ω ∼ r 30 We choose the angle θ such that the boundary is given by θ ≡ 0 (modπ ).
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Once again, this dominant term must vanish, giving a(a + a
) = cst = κ. It is unlikely that κ vanishes. If it does not, a rescaling yields a
+ a = −
1 , 2a
and therefore δ a 2 + a 2 = log , a δ a positive constant. An analysis as above yields a solution for which a(0) = 0 and a (θ0 ) = 0, where
a∗
θ0 := 0
!
da
,
log aδ − a 2
and a∗ (δ) is the root of the denominator in (0, δ). It turns out that θ0 is always less than π/2, and approaches this value when δ → +∞. It also tends to zero when δ → 0+. Taking such a solution on (0, θ0 (δ)) and extending by parity, we obtain a solution with a(0) = a(2θ0 ) = 0. We think that it is reasonable to match it with a ≡ 0 on (2θ0 , π), which still cancels the dominant term in v · ∇ω. One advantage of this construction is that we do expect a zone where the vorticity vanishes identically: this is the influence domain of the sonic line; see Figure 19. An other convenience is that the corresponding vθ is quasiLipschitz at θ = 0 and therefore the streamlines all converge to the stagnation point, instead of crossing the material boundary. In conclusion, the singularity of ω may not be significantly larger than 1/r, otherwise the streamlines rotate too much and do cross the material boundary. On the other hand, it cannot be significantly smaller, because of Theorem 6.1. Thus the real singularity must be of order 1/r, up to say a logarithmic correction. Remark. At first glance, the identity (134), and the fact that the limit F given by (135) is finite, suggest a singularity of order exactly 1/r. This is in contradiction with the impossibility to build such a vortical singularity at a boundary point. However, the derivation of (134) assumed an amount of regularity that is not valid. Being more careful, we should have integrated (133) over D \ B , with B a small (half-)disk around y0 . Then there is a contribution from ∂B in (134), which does not vanish when → 0, and may even tend to infinity. With this correction, a singularity of order higher than 1/r becomes coherent. Open problems. We conclude this discussion with two important questions that remain to be solved: • Describe in more detail and with higher rigor the vortical singularity.
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Fig. 19. A vortical singularity at a boundary point. The streamlines converge to V . The dotted ones come from the sonic line. The dashed ones come from the diffracted shock. The vorticity vanishes identically in the doted zone.
• Is there a vortical singularity for the full Euler model? This is unclear. It could be that the vortical singularity is just a consequence of the barotropic assumption. Whatever the answer to this question, we keep in mind that the solution must be singular, even in the full Euler model, because the transport equation v · ∇S = 0 that follows from the self-similar equations, and the fact that the entropy may not be constant along the diffracted shock, imply a discontinuity of S at the center V of the pseudo-flow (see Figure 19). If moreover the pressure is continuous, as suggested by the elliptic equation (115), then the discontinuity of S induces a discontinuity of the density ρ.
Acknowledgement I am happy to thank the anonymous referee, whose careful reading and deep knowledge of the subject helped me to improve and fix the mathematics, and who was patient enough to correct my English.
References [1] R.K. Agarwal and D.W. Halt, A modified CUSP scheme in wave/particle split form for unstructured grid Euler flows, in: Frontiers in Computational Fluid Dynamics, D.A. Caughey and M.M. Hafez, editors, John Wiley and Sons, New York (1994), 155–168. [2] M. Artola and A. Majda, Nonlinear development of instabilities in supersonic vortex sheet I. The basic kink modes, Physica D28 (1987) 253–281. [3] V. Bargmann and D. Montgomery, Prandtl–Meyer zones in Mach reflection. OSRD 5011, Office of Scientific Research and Development (1950). [4] G. Ben-Dor, Shock Wave Reflection Phenomena, Springer-Verlag, New York (1992). [5] G. Ben-Dor, Oblique shock wave reflections, in: Handbook of Shock Waves, volume 2, G. Ben-Dor, O. Igra and T. Elperin, editors, Academic Press, San Diego (2001), 67–179.
120
D. Serre
[6] S. Benzoni-Gavage and D. Serre, Multi-Dimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications, Oxford University Press, Oxford (2007). [7] W. Bleakney and A.H. Taub, Interaction of shock waves, Rev. Mod. Phys. 21 (1949) 584–605. [8] M. Brio and J.K. Hunter, Mach reflection for the two-dimensional Burgers equation, Physica D 60 (1992) 194–207. ˇ c and B.L. Keyfitz, Oblique shock interaction and the von Neumann paradox, in: Proc. 20th Int. [9] S. Cani´ Conf. on Shock Waves, B. Sturtevant, J. Schepherd and H. Hornung, editors, World Scientific Pub, River Edge, NJ (1996). ˇ c and B.L. Keyfitz, Quasi-one-dimensional Riemann problems and their role in self-similar two[10] S. Cani´ dimensional problem, Arch. Rational Mech. Anal. 144 (1998) 233–258. ˇ c and B.L. Keyfitz, Riemann problems for the two-dimensional unsteady transonic small disturbance [11] S. Cani´ equation, SIAM J. Appl. Math. 58 (2) (1998) 636–665. ˇ c, B.L. Keyfitz and Eun Heui Kim, Free boundary problem for the unsteady transonic small distur[12] S. Cani´ bance equation: Transonic regular reflection, Methods and Appl. of Anal. 7 (2) (2000) 313–336. ˇ c, B.L. Keyfitz and Eun Heui Kim, Mixed hyperbolic-elliptic systems in self-similar flows, Boll. [13] S. Cani´ Sociedade Brasileira Mat. 32 (3) (2001) 377–399. ˇ c, B.L. Keyfitz and Eun Heui Kim, A free boundary problem for a quasi-linear degenerate elliptic [14] S. Cani´ equation: Regular reflection of weak shocks, Comm. Pure and Applied Math. LV (2002) 71–92. ˇ c, B.L. Keyfitz and Eun Heui Kim, Free boundary problems for nonlinear wave systems: Mach stems [15] S. Cani´ for interacting shocks, SIAM J. Math. Anal. 37 (2006) 1947–1977. ˇ c, B.L. Keyfitz and G. Lieberman, A proof of existence of perturbed steady transonic shocks via a [16] S. Cani´ free boundary problem, Comm. Pure and Applied Math. LIII (2000) 484–511. ˇ c, B.L. Keyfitz and D.H. Wagner, A bifurcation diagram for oblique shock interactions in the un[17] S. Cani´ steady transonic small disturbance equation, in: Hyperbolic problems, Theory, Numerics and Applications (Stony Brook, NY, 1994), J. Glimm, M. Graham, J. Grove and B. Plohr, editors, World Scientific Pub, River Edge, NJ (1996), 178–187. ˇ c and D. Mirkovi´c, A numerical study of Riemann problems for the two-dimensional unsteady [18] S. Cani´ transonic small disturbance equation, SIAM J. Appl. Math. 58 (5) (1998) 1365–1393. [19] Tong Chang and Gui-Qiang Chen, Diffraction of planar shock along compressive corner, Acta Math. Scienta 6 (3) (1986) 241–257. [20] Tong Chang and Gui-Qiang Chen, Some fundamental concepts about systems of two spatial dimensional conservation laws, Acta Math. Scienta 6 (4) (1986) 463–474. [21] Tong Chang and Ling Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, volume 41 of Pitman Surv. in Pure and Appl. Maths., Longman (1989). [22] Gui-Qiang Chen and M. Feldman, Potential theory for shock reflection by a large-angle wedge, Proc. Natl. Acad. Sci. USA 102 (43) (2005) 15368–15372 (electronic). [23] Gui-Qiang Chen and Dehua Wang, The Cauchy problem for the Euler equations for compressible fluids, in: Handbook of Mathematical Dynamics, S. Friedlander and D. Serre, editors, North-Holland, Elsevier, Amsterdam (2002), 421–543. [24] Shuxing Chen, On reflection of multidimensional shock front, J. Differential Equ. 80 (1989) 199–236. [25] Shuxing Chen, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Diff. Equ. 21 (1996) 1103–1118. [26] Shuxing Chen, Linearization of shock reflection by almost perpendicular ramp, in: Conference on Nonlinear Evolution Equations and Infinite-Dimensionnal Dynamical Systems (Shanghai, 1995), World Scientific Pub, River Edge, NJ (1997), 13–23. [27] P. Collela and L.F. Henderson, The von Neumann paradox for the diffraction of weak shock waves, J. Fluid Mech. 213 (1990) 71–94. [28] R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, New York (1976). Reprinting of the 1948 original, Applied Mathematical Sciences, vol. 21. [29] C. Dafermos, Estimates for conservation laws with little viscosity, SIAM J. Math. Anal. 18 (1987) 409–421. [30] C. Dafermos, Hyperbolic Conservation Laws in Continuum Mechanics, volume 325 of Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg (2000).
Shock reflection in gas dynamics
121
[31] Haitao Fan and M. Slemrod, Dynamic flows with liquid-/vapor phase transitions, in: Handbook of Mathematical Dynamics, S. Friedlander and D. Serre, editors, North-Holland, Elsevier, Amsterdam (2002), 373–420. [32] I.M. Gamba, R.R. Rosales and E.G. Tabak, Constraints on possible singularities for the unsteady transonic small disturbance (UTSD) equations, Comm. Pure and Appl. Math. LII (1999) 763–779. [33] I.M. Gel’fand, Some questions of analysis and differential equations, Amer. Math. Soc. Transl. (2) 26 (1963) 201–219. [34] H.M. Glaz, Self-similar shock reflection in two space dimensions, in: Multidimensional Hyperbolic Problems and Computations, J. Glimm and A. Majda, editors, volume 29 of IMA Volumes in Mathematics and Applications, Springer, New York (1991), 70–88. [35] K.G. Guderley, Considerations on the structure of mixed subsonic-supersonic flow patterns, Air Material Command technical report F-TR-2168-ND, ATI No. 22780, GS-AAF-Wright Field No. 39, U.S. Wright-Patterson Air Force Base, Dayton, OH (1947). [36] K.G. Guderley, The Theory of Transonic Flows, Pergamon Press, Oxford (1962). [37] E. Harabetian, Diffraction of a weak shock by a wedge, Comm. Pure and Appl. Math. XL (1987) 849–863. [38] A. Harten, P.D. Lax, C.D. Levermore and W. Morokoff, Convex entropies and hyperbolicity for general Euler equations, SIAM J. Numer. Anal. 35 (1998) 2117–2127. [39] A. Heibig and D. Serre, Étude variationnelle du problème de Riemann, J. Differential Equ. 96 (1992) 56–88. [40] L.F. Henderson, Regions and boundaries for diffracting shock waves, Z. Ang. Math. Mech. 67 (2) (1987) 73–86. [41] L.F. Henderson, General laws for propagation of shock waves through matter, in: Handbook of Shock Waves, volume 1, G. Ben-Dor, O. Igra and T. Elperin, editors, Academic Press, San Diego (2001). [42] L.F. Henderson and R. Menikoff, Triple-shock entropy theorem and its consequences, J. Fluid Mech. 366 (1998) 179–210. [43] H. Hornung, Regular and Mach reflection of shock waves, Ann. Rev. Fluid Mech. 18 (1986) 33–58. [44] J.K. Hunter, Nonlinear geometrical optics, in: Multidimensional Hyperbolic Problems and Computations, J. Glimm and A. Majda, editors, volume 29 of IMA Volumes in Mathematics and Applications, Springer, New York (1991), 179–197. [45] J.K. Hunter and M. Brio, Weak shock reflection, J. Fluid Mech. 410 (2000) 235–261. [46] J.K. Hunter and J.B. Keller, Weakly nonlinear high frequency waves, Comm. Pure and Applied Math. 36 (1983) 547–569. [47] J.K. Hunter and J.B. Keller, Weak shock diffraction, Wave Motion 6 (1984) 79–89. [48] E. Jouguet, Sur la propagation des discontinuités dans les fluides, C. R. Acad. Sci. Paris 132 (1901) 673–676. [49] J.B. Keller and A. Blank, Diffraction and reflection of pulses by wedges and corners, Comm. Pure and Applied Math. 4 (1951) 75–94. [50] S. Kruzkov, First-order quasilinear equations with several space variables, Math. USSR Sbornik 10 (1970) 217–273. [51] P. Lax, Shock waves and entropy, in: Contributions to Functional Analysis, E.A. Zarantonello, editor, Academic Press, New York (1971), 603–634. [52] M.J. Lighthill, The diffraction of blast. I, Proc. Roy. Soc. Lond. Ser. A 198 (1949) 454–470. [53] E. Mach, Über dev Verlauf von Funkenwellen in der Ebene und im Raume, Sitzungsbr. Akad. Wiss. Wien 77 (1878) 819–838. [54] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York (1984). [55] A. Majda, One perspective on open problems in multi-dimensional conservation laws, in: Multidimensional Hyperbolic Problems and Computations, J. Glimm and A. Majda, editors, volume 29 of IMA Volumes in Mathematics and Applications, Springer, New York (1991) 217–238. [56] C. Morawetz, Potential theory for regular and Mach reflection of a shock at a wedge, Comm. Pure and Applied Math. XLVII (1994) 593–624. [57] J. von Neumann, The main effect and height of burst, in: Collected Works of John von Neumann, volume VI, Pergamon Press, New York (1963), 309–347. [58] J. von Neumann, Oblique reflection of shocks, in: Collected Works of John von Neumann, volume VI, Pergamon Press, New York (1963), 238–299.
122
D. Serre
[59] J. von Neumann, Refraction, intersection and reflection of shock waves, in: Collected Works of John von Neumann, volume VI, Pergamon Press, New York (1963), 300–308. [60] M.H. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, New York (1967). [61] A. Sakurai, On the problem of weak Mach reflection, J. Phys. Soc. Japan 19 (1964) 1440–1450. [62] A. Sakurai, L.F. Henderson, K. Takayama, Z. Walenta and P. Collela, On the von Neumann paradox of weak Mach reflection, Fluid Dyn. Research 4 (1989) 333–345. [63] C.W. Schultz-Rinne, J.P. Collins and H.M. Glaz, Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput. 14 (6) (1993) 1394–1414. [64] D. Serre, Domaines invariants pour les systèmes hyperboliques de lois de conservation, J. Differential Equ. 69 (1987) 46–62. [65] D. Serre, Ecoulements de fluides parfaits en deux variables indépendantes de type espace. Réflexion d’un choc plan par un dièdre compressif, Arch. Rational Mech. Anal. 132 (1995) 15–36. [66] D. Serre, Systems of Conservation Laws, vol. I, Cambridge Univ. Press, Cambridge (2000). [67] Li Ta-tsien and Yu Wen-ci, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University, Durham, NC (1987). [68] E.G. Tabak and R.R. Rosales, Focusing of weak shock waves and the von Neumann paradox for oblique shock reflection, Phys. Fluids. A 6 (1994) 1874–1892. [69] E. Tadmor, A minimum entropy principle in the gas dynamics equations, Appl. Numer. Math. 2 (1986) 211–219. [70] Dechun Tan and Tong Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws, I. Four-J cases, J. Differential Equ. 111 (1994) 203–254. [71] Dechun Tan and Tong Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws, II. Initial data involving some rarefaction waves, J. Differential Equ. 111 (1994) 255–282. [72] L. Tartar, Une introduction à la théorie mathématique des systèmes de lois de conservation, Publ. Inst. Anal. Num. 682, Univ. Pavia (1989). [73] A.M. Tesdall and J.K. Hunter, Self-similar solutions for weak shock reflection, SIAM J. Appl. Math. 63 (1) (2002) 42–61. [74] V.M. Teshukov, Shock polars in a gas with general equation of state, Prikl. Mat. Mekh. 50 (4) (1986) 98–103 (in Russian). [75] V.M. Teshukov, Stability of regular shock wave reflection, J. Appl. Mech. Tech. Phys. 30 (2) (1989) 189–196. [76] F. Ting and H.F. Ludloff, Aerodynamics of blasts, J. Aero. Sci. 19 (1952) 317–328. [77] H. Weyl, Shock waves in arbitrary fluids, Comm. Pure Appl. Math. 2 (1949) 103–122. [78] G.B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons Inc., New York (1999). Reprint of the 1974 original, A Wiley-Interscience Publication. [79] Shuli Yang and Tong Zhang, The MmB difference solutions of 2-d Riemann problems for a 2 × 2 hyperbolic system of conservation laws, Impact of Computing in Science and Engineering 3 (1991) 146–180. [80] Tong Zhang, Gui-Qiang Chen and Shuli Yang, On the 2-d Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves, Disc. Cont. Dyn. Sys. 1 (4) (1995) 555–584. [81] Tong Zhang, Gui-Qiang Chen and Shuli Yang, On the 2-d Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities, Disc. Cont. Dyn. Sys. 6 (2) (2000) 419–430. [82] Tong Zhang and Yuxi Zheng, Two-dimensional Riemann problem for a single conservation law, Trans. Amer. Math. Soc. 312 (2) (1989) 589–619. [83] Yuxi Zheng, Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions, Comm. Partial Diff. Equ. 22 (11–12) (1997) 1849–1868. [84] Yuxi Zheng, Systems of Conservation Laws. Two-Dimensional Riemann Problems, volume 38 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston (2001). [85] Yuxi Zheng, A global solution of a two-dimensional Riemann problem involving shocks as free boundaries, Acta Math. Appl. Sinica 19 (2003) 559–572. [86] Yuxi Zheng, Shock reflection for the Euler system, in: Hyperbolic Problems. Theory, Numerics and Applications (Osaka 2004) (II), F. Asakura et al., editor, Yokohama Pub, Yokohama, Japan (2005), 425–432. [87] Yuxi Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin. Engl. Ser. 22 (2) (2006) 177–210.
CHAPTER 3
The Mathematical Theory of the Incompressible Limit in Fluid Dynamics Steven Schochet Tel Aviv University, School of Mathematical Sciences, Ramat Aviv, 69978 Tel-Aviv, Israel
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Scaling and formal limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Formal limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Fast limit asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Remarks on the early development of the theory of the incompressible limit . . . . . . 4. Justification of the incompressible limit for periodic barotropic flow . . . . . . . . . . 5. Survey of further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Fast decay of fast waves on Rd for d ≥ 2 . . . . . . . . . . . . . . . . . . . . . . 5.2. Boundaries: Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Nonisentropic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Multiple spatial scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Improved error estimate and asymptotic expansions . . . . . . . . . . . . . . . . 5.6. Viscous flows: weak solutions and global solutions . . . . . . . . . . . . . . . . . 6. Some open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Multiple spatial scales for fast non-isentropic flow . . . . . . . . . . . . . . . . . 6.2. Genericity of simple non-resonant spectrum for fast periodic non-isentropic flow References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The mathematical theory of the incompressible limit in fluid dynamics
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1. Introduction The equations governing incompressible fluid flow differ from those for compressible flow in that the evolution equation for the density or the pressure is replaced by the constraint that the flow be divergence-free. The equations of incompressible flow can be derived formally by taking the limit of an appropriately rescaled version of the equations for compressible flow as the scaling parameter tends to zero. In physical terms, that scaling parameter is the ratio of the fluid particle speed to the sound speed, and is called the Mach number. The limit of the rescaled compressible equations as the Mach number tends to zero is known as the incompressible limit. Since the equations of compressible fluid flow tend to the incompressible flow equations as the Mach number goes to zero, one naturally expects that solutions of the compressible fluid equations tend to solutions of the incompressible equations in that limit. This expectation has been justified in a wide variety of circumstances. Various extensions of these results have also been obtained, and analogous results have been obtained for limits of other systems. In the succeeding sections we will consider the rescaling process and the formal limit of the resulting equations, a variety of rigorous results and proofs of some of those results, and a few problems that remain open.
2. Scaling and formal limit 2.1. Scaling The scaling properties of the equations for compressible fluids can be seen most easily by considering the compressible barotropic Navier–Stokes equations (e.g. [Lio96, (1.56)]) ρT + ∇ · (ρ U) = 0 ρ UT + ρ U · ∇U + ∇P(ρ) = μ U + (μ + λ)∇(∇ · U) .
(2.1) (2.2)
Here ρ is the density of the fluid, U is its velocity, and P is the pressure, which is assumed here to be an increasing function of the density alone. The case when additional thermodynamical variables play a role will be considered later. For simplicity, the viscosity coefficients μ and λ are assumed to be constant. In addition, if they do not both vanish they must satisfy the constraints ([Lio98, §5.1], [Dan02]) μ > 0,
2μ + λ > 0,
(2.3)
which ensure the negativity of the operator appearing on the right side of (2.2); see (4.6) in Lemma 4.4 below. Rescaling the equations is accomplished by introducing non-dimensionalized independent and dependent variables via T = T t, ρ = ρ0 + R r(t, x),
X = L x, U = U0 + U u(t, x).
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Here T and X are the original time and space variables, L is a typical spatial dimension, as determined by the initial data or the domain in which the flow occurs, T is a characteristic time of the flow to be chosen shortly, and t and x are the new nondimensional independent variables. In addition, ρ0 is a positive characteristic value of the density, such as its limiting value at spatial infinity or its average over a bounded domain in which the flow occurs, R is the typical scale of density variations in the initial data, and r is the non-dimensionalized density variable. The velocity is transformed in similar fashion, although we will henceforth assume that U0 vanishes, because the Galilean invariance of the equations means that U0 can be transformed to zero by switching to the moving coordinate system x → x − TL U0 t. Substituting these changes of variables into the compressible Navier–Stokes equations (2.1)–(2.2) yields $ ρ0 UT # ∇ · (ru) + ∇ · u = 0 L R % UT RT (ρ0 + Rr) ut + P (ρ0 + Rr)∇r [u · ∇u] + L LU
rt +
=
T [μ u + (μ + λ)∇(∇ · u)] . L2
(2.4)
(2.5)
The following considerations determine appropriate choices of the scaling parameters: First, in order to see the dynamics of the fluid motion, we must choose the time scale to be long enough for a particle moving with a typical velocity O(U ) to travel a typical distance O(L). This means that ULT should be at least O(1); for simplicity, let us define the typical time T so as to make this expression exactly one, i.e., T=
L . U
(2.6)
This choice of T may be called the convective time scale, since it makes the coefficient of the convective terms u · ∇ be of order one. Second, by linearizing equations (2.1)–(2.2) with the viscosity parameters set to zero around the reference state (ρ = ρ0 , U = 0) and combining the resulting equations, we obtain the wave equation ρT T = P (ρ0 )ρ. This √ shows that the typical speed of sound is P (ρ0 ), which we assume is positive. Since we want to model situations in which the ratio of the typical fluid velocity to the typical sound speed is small, take ε := U
(2.7)
to be a small parameter. Third, if the rescaled viscosity is larger than O(1) then solutions will presumably dissipate to zero before the above-defined typical time is reached. Hence the rescaled viscosity parameters LT2 μ and LT2 (μ + λ) must be at most O(1); this is equivalent to saying that the viscous time scale must be at least as large as the convective time scale. In the inviscid case when μ = 0 = λ this condition yields no new restriction, and if
The mathematical theory of the incompressible limit in fluid dynamics
127
T L2
tends to zero with ε then the limit equations will be inviscid even when the original equations are viscous. However, let us assume that T = 1, L2
(2.8)
which allows for the possibility of viscous limit equations. Conditions (2.6)–(2.8) imply that 1 L= , ε
T=
1 . ε2
(2.9)
Plugging formulas (2.7) and (2.9) into (2.4)–(2.5) yields rt + ∇ · (ru) + (ρ0 + Rr) {ut + u · ∇u} +
ρ0 ∇ ·u=0 R
(2.10)
R P (ρ0 + Rr)∇r = μ u + (μ + λ)∇(∇ · u) . ε2 (2.11)
We still need to fix the value of R. Although equations (2.10)–(2.11) describe slightly compressible flow on the convective time scale for any choice of R, if that parameter is larger than O(ε) then the large part of the term 3 R R R 2
R P (ρ + Rr)∇r = P (ρ )∇r + P (ρ )r∇r + O 0 0 0 ε2 ε2 ε2 ε2 in (2.11) will be nonlinear. Equations containing large nonlinear terms generally do not allow uniform estimates, and so their solutions are expected to blow up in times that tend to zero with ε. Certainly this will be the case for the one-dimensional inviscid equations, since the equations can then be written as a 2 × 2 system of hyperbolic conservation laws, for which blow-up can be established by the method of characteristics ([Lax73]). We will therefore consider only the case when R = ε,
(2.12)
for which (2.10)–(2.11) become rt + ∇ · (ru) +
ρ0 ∇ ·u=0 ε
(2.13)
1 (ρ0 + εr) {ut + u · ∇u} + P (ρ0 + εr)∇r = μ u + (μ + λ)∇(∇ · u) . (2.14) ε Although the characteristic time scale for acoustic vibrations in (2.13)–(2.14) is O(ε), as can be seen from the linearization mentioned above, the fact that the only terms in those equations that are large on the O(1) convective time scale are linear with constant coefficients turns out to allow the solutions to exist and satisfy uniform bounds on that longer time scale.
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2.2. Formal limit The limit of (2.13) can be taken simply by multiplying that equation by ε and then letting ε tend to zero. Since the typical density ρ0 is assumed to be nonzero, this yields the incompressibility constraint ∇ · u = 0.
(2.15)
The limit of equation (2.14) is more complicated. Multiplying that equation by ε and letting ε tend to zero yields ∇r = 0
(2.16)
in view of the fact that P is assumed to be positive. More limit information than (2.16) can be extracted from (2.14), however, because the large term is a gradient. To obtain further information about the limit, apply the Hodge projection operator P := I − ∇−1 ∇·,
(2.17)
with the boundary condition ν · Pu = 0 on any fixed boundaries, where ν denotes the normal to the boundary. Since P annihilates gradients, applying it to (2.14) eliminates the large term, leaving the O(1) equation P [(ρ0 + εr) {ut + u · ∇u}] = P[μ u].
(2.18)
In the limit as ε → 0 this reduces to P [ρ0 {ut + u · ∇u}] = P[μ u].
(2.19)
By the Hodge decomposition theorem, anything annihilated by P must be a gradient, so for some function π(t, x), ρ0 {ut + u · ∇u} + ∇π = μ u.
(2.20)
Actually, there is an analogous way to extract additional limit information from (2.13). Since the large term is a derivative, it can be eliminated by taking the integral over the spatial variables, which yields simply d r dx = 0 (2.21) dt because the other term involving spatial derivatives is also an exact derivative. Because (2.21) is independent of ε it is also the limit equation obtained as ε → 0. Since r is also independent of x in the limit by (2.16), if the domain is unbounded then the limit r must be identically zero if the integral in (2.21) exists. When the domain is bounded then the limiting value of r reduces to a constant that is determined by the initial data.
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Since r is an O(ε) perturbation of ρ0 , to lowest order neither the additional information (2.21) nor even (2.16) is needed: Because r does not appear in (2.15) or (2.20), those equations by themselves form a closed system, which is precisely the incompressible Navier– Stokes equations for a fluid with density ρ0 . In deriving limit equations (2.15) and (2.20) from (2.13)–(2.14), we have implicitly assumed not only that the dependent variables r and u converge to limiting values as ε → 0, but also that their time derivatives converge. This assumption concerning the time derivatives is particularly open to question, since even when r and u are of order O(1) initially, as they must be in order to converge,& equations (2.13)–(2.14) themselves show that in gen' 1 eral their time derivatives will be O ε . Put differently, those equations show that in order for rt and ut to be uniformly bounded at time zero, the initial data r(0, x, ε) and u(0, x, ε) must satisfy the conditions ∇r(0, x, ε) = O(ε) = ∇ · u(0, x, ε).
(2.22)
If we assume that the initial data take the form r(0, x, ε) = r0,0 (x) + εr1,0 (x, ε), u(0, x, ε) = u0,0 (x) + εu1,0 (x, ε),
(2.23)
with r1,0 and u1,0 being uniformly bounded, then (2.22) becomes ∇r0,0 = 0 = ∇ · u0,0 .
(2.24)
Since (2.24) implies that r0,0 is a constant, it is incompatible with our earlier characterization of R as being the typical scale of density variation of the initial data. We could simply view R as a convenient scale of density variation for rescaling the equations. Alternatively, we can drop the requirement that the time derivatives of r and u be uniformly bounded. This latter option entails the presence of waves of size O(1) moving on the fast acoustic time scale, and hence also a more complicated asymptotic structure in the limit as ε → 0. Those asymptotics will be considered next.
2.3. Fast limit asymptotics 2.3.1. Multiple time scale expansions When the solution is uniformly bounded but its time derivative is not, then the solution varies on a “fast” time scale, i.e., one that is smaller than O(1). The relevant scale here is O(ε). It will therefore be convenient to introduce an additional time variable t τ= , ε
(2.25)
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S. Schochet
and to allow functions to depend on both τ and t. Of course, if a function depends on ε as well as on t and τ then the separation of the time dependence into the variables t and τ is not unique. A convenient way of determining that separation is the expansion process described below. One advantage of this dual time formulation is that a function U (τ, t, x, ε) expressed in terms of both t and τ may converge to a limit U0 (τ, t, x) as ε → 0 even though the corresponding function U ( εt , t, x, ε) expressed in terms of t alone does not converge. If the convergence of U (τ, t, x, ε) as ε → 0 is uniform in τ then it will also be true that U ( εt , t, x, ε) − U0 ( εt , t, x) → 0 as ε → 0. The limit U0 (τ, t, x) is then called the limit profile of the function U ( εt , t, x, ε). The simplest way to derive the equations satisfied by the limit profile of the solution of (2.13)–(2.14) is the following: First, consider the dependent variables to be functions of both time variables τ = εt and t as well as x; this entails expanding the time derivative ∂t appearing in (2.13)–(2.14) into 1ε ∂τ + ∂t . Second, expand the dependent variables in powers of ε:
r = r 0 (τ, t, x) + εr 1 (τ, t, x) + . . . ,
u = u0 (τ, t, x) + εu1 (τ, t, x) + . . . (2.26)
Such an expansion may not always be valid, due to small-divisor problems or other difficulties; nevertheless, as noted in [JMR98], the expansion is still useful for deriving the correct limit equations. Third, plug these expansions into the modified versions of (2.13)–(2.14), and expand the result in powers of ε as well. Finally, take the coefficients of each power of ε in the resulting equations to be valid separately. This yields rτ0 + ρ0 ∇ · u0 = 0
(2.27)
ρ0 u0τ + P (ρ0 )∇r 0 = 0
(2.28)
as the O( 1ε ) equations, and ' & rτ1 + ρ0 ∇ · u1 + rt0 + ∇ · r 0 u0 = 0 ρ0 u1τ + P (ρ0 )∇r 1 + r 0 u0τ + ρ0 u0t + u0 · ∇u0 + P
(ρ0 )r 0 ∇r 0 = μu0 + (μ + λ)∇∇ · u0
(2.29)
(2.30)
as the O(1) equations. Equations (2.27)–(2.28) determine the τ -dependence of the limit profile U0 = (r 0 , u0 ). However, as is typical of singular perturbation problems, the O(1) equations (2.29)–(2.30) involve the first-order perturbations r 1 and u1 as well as r 0 and u0 . We therefore need
The mathematical theory of the incompressible limit in fluid dynamics
131
to eliminate r 1 and u1 from those equations in order to obtain a closed set of equations that determine the t-dependence of the limit profile. The standard method to do so is the sublinear growth condition. 2.3.2. Sublinear growth condition The sublinear growth condition is the condition that the first-order perturbation terms be o(τ ), so that the ordering of the expansion (2.26) remains correct up through fast times τ of order O( 1ε ), which by (2.25) correspond to ordinary times t of order one. Even in the “slow” case when the zeroth-order solution is independent of τ , which was considered above, the first-order perturbations generally will depend on that variable, so that the sublinear growth condition remains relevant. This implies both that the expansion method is an alternative way of deriving the limit equations for the slow case, and that the equations obtained in the fast case should reduce to those for the slow case when the initial data satisfy conditions (2.24). In order to be able to calculate explicitly and easily, let us assume that the flow is periodic in the spatial variables. Flows on all of Rd and in domains with boundaries will be discussed later. Let r V := u denote the full vector of solution components, and define the “fast” operator L := −
ρ0 ∇·
0 P (ρ0 ) ρ0
∇
0
that allows us to write the O( 1ε ) equations (2.27)–(2.28) succinctly as Vτ0 = LV 0 .
(2.31)
The solution operator to this equation is
S(τ ) := eτ L = F−1 e
−iτ
0
ρ0 k·
P (ρ0 ) ρ0 k
0
F,
(2.32)
where F denotes the Fourier transform with respect to the spatial variable x with dual variable k. By (2.31) the limit profile can be expressed in terms of S as V 0 (τ, t, x) = S(τ )V 0 (0, t, x),
(2.33)
which makes explicit the τ -dependence of that profile. The O(1) equations (2.29)–(2.30) can also be written in terms of L as Vτ1 − LV 1 = f 0 ,
(2.34)
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S. Schochet
where rt0 + ∇ · r 0 u0
f := − 0
r0 0 ρ0 uτ
0) 0 + u0t + u0 · ∇u0 + P ρ(ρ r ∇r 0 − 0
μ 0 ρ0 u
−
μ+λ ρ0 ∇∇
· u0
. (2.35)
The solution V 1 to (2.34) is
τ
V 1 = S(τ )V 1 (0, t, x) +
S(τ − τ )f 0 (τ , t, x) dτ
0
= S(τ ) V (0, t, x) + 1
τ
S(−τ )f (τ , t, x) dτ 0
.
0
˜ is antisymmetric, where Now AL A˜ =
P (ρ0 )
0 ρ0 I
ρ0
0
.
The solution operator S is therefore unitary with respect to the inner product ˜ := (g, Ah)
˜ dx, g T Ah
which implies that V 1 is sublinear iff S(−τ )V 1 is. In addition, V(0, t, x) does not depend on τ and so is certainly sublinear in that variable, and hence the sublinearity condition limτ →∞ τ1 V 1 = 0 reduces to the condition that
τ
0 = lim
1 τ →∞ τ
S(−τ )f 0 (τ , t, x) dτ = Mτ [S(−τ )f 0 ],
(2.36)
0
where the averaging operator Mτ is defined by Mτ [f ] := lim τ1 τ →∞
τ
f (τ , t, x) dτ .
(2.37)
0
Using once more the fact that S is unitary, condition (2.36) can be written in the form 0 = E[f 0 ] := S(τ )Mτ [S(−τ )f 0 ],
(2.38)
which is convenient because the operator E is a projection: Specifically [Sch94], E is the projection operator onto terms of the form S(τ )f (t, x)
(2.39)
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133
with respect to the inner product g, h := lim
1 τ →∞ τ
τ
˜ dτ = lim (g, Ah)
1 τ →∞ τ
0
τ
˜ dx dτ . g T Ah
(2.40)
0
Since, as noted in (2.33), the O( 1ε ) equations (2.31) say that V 0 has the form (2.39), this alternative characterization of E shows that V 0 = EV 0 ,
(2.41)
while combining (2.38) with (2.35) yields rt0 + ∇ · r 0 u0 0 =E 0
r 0 0 u + u0 + u0 · ∇u0 + P (ρ0 ) r 0 ∇r 0 − ρ0 τ
t
ρ0
μ 0 ρ0 u
−
μ+λ ρ0 ∇∇
· u0 (2.42)
as the equation satisfied by the limit profile V 0 =
r0 . u0
2.3.3. Calculation of the projection operator In order to write equation (2.42) more explicitly, note that the zero eigenspace of L is spanned by the x-independent density components and the divergence-free velocity vectors. The restriction of S to that subspace is the identity operator, so by (2.38) the restriction of E to that subspace is simply Mτ . Since this part of E projects onto functions that do not depend on the fast time τ , it is called the slow part of E. The slow part of E is thus the projection onto functions r that are independent of x and τ and onto functions u that are independent of τ and divergence-free: Eslow =
0 , Mτ P
Mτ,x 0
(2.43)
where Mz denotes the average with respect to the variables z and P is the projection (2.17) onto divergence-free vector fields. In order to calculate the fast part of E that projects onto functions that do depend on τ , return to the Fourier-space representation of the solution operator S in (2.32), and note that for each nonzero k the matrix ( L(k) := FLF−1 has exactly two nonzero eigenvalues. The union of the corresponding eigenvectors over all k spans a “fast” space on which the solution to (2.31) truly depends on τ . Those nonzero eigenvalues and corresponding (±) eigenvectors w(k) are (±) ( Lw(k)
=
0
− P ρ(ρ0 0 ) ik
√ = ±i P (ρ0 )|k|
−ρ0 ik·
∓ √iρ0 |k|
0 ∓ √iρ0 |k| P (ρ0 ) ik
P (ρ0 )
ik
.
(2.44)
134
S. Schochet
Restricted to the span of those eigenvectors, the projection E singles out those functions of the form √ (±) c(±) (t, k)e±i P (ρ0 ) |k|τ w(k) . (2.45) (±)
Note that for each k the two vectors w(k) are orthogonal with respect to the inner ˜ Also, for functions r 0 having the τ -dependence indicated in (2.45), product g T Ah. iρ0 √ ∓ |k|r 0 = − P ρ(ρ0 0 ) ∂τ r 0 . Hence the fast part of the projection E is
P (ρ0 )
Efast
⎧ √ ⎨ w (±) e±i P (ρ0 ) |k|τ (k)
= F−1 ⎩
±
(±) T ˜ (±) (w(k) ) Aw(k)
√
(±) ˜ Mτ e∓i P (ρ0 ) |k|τ (w(k) )T AF(I − Mx ) ·
ρ P (ρ0 ) − P (ρ0 0 ) ∂τ 1 E∂τ2 −P (ρ0 ) ( − P (ρ = ∂τ P ρ(ρ0 0 ) ∇· ) 0) 2ρ0 ∇ ρ0 1 − P (ρ0 ) ∂τ −1 E∂τ2 −P (ρ0 ) ( − ρ10 ∂τ ∇· ) , =2 ∇
⎫ ⎬ ⎭
(2.46)
where E∂τ2 −P (ρ0 ) denotes the projection onto solutions of the scalar wave equation # 2 $ ∂τ − P (ρ0 ) w = 0, i.e., FE∂τ2 −P (ρ0 ) F−1 =
√ √ e±i P (ρ0 ) |k|τ Mτ e∓i P (ρ0 )|k|τ · .
(2.47)
±
Note that the projection I − Mx is not needed in the later formulas in (2.46) since applying the operators there in the order indicated annihilates the x-independent functions. To verify formula (2.46), note that it annihilates constant densities r and divergence-free velocities (±) u, and when restricted to the span of the w(k) it reproduces functions of the form (2.45) but annihilates functions having different τ -dependence. 2.3.4. Explicit limit equations: separation into fast and slow parts The formulas just derived allow us to write the limit equations (2.42) more explicitly. Since the slow and fast projections are mutually orthogonal, (2.42) remains valid when the full projection E is replaced by either its fast or slow part. Similarly, (2.41) implies that V 0 equals the sum 0 + V0 Vfast slow of its fast and slow projections, i.e.
r0 u0
=
0 rfast
u0fast
+
0 rslow
u0slow
By (2.46), the fast part satisfies 0 rfast 0 Mτ = 0 0 ufast
.
(2.48)
(2.49)
The mathematical theory of the incompressible limit in fluid dynamics
135
and
0 rfast
u0fast
=
− P ρ(ρ0 0 ) ∂τ ∇
φ0,
(2.50)
where φ 0 := 12 −1 (∇ · u0 −
1 0 ρ0 rτ )
(2.51)
satisfies the wave equation
∂τ2 − P (ρ0 ) φ 0 = 0.
(2.52)
By (2.43), the slow part satisfies 0 0 0 rslow = Mx rslow = rslow (t),
u0slow = u0slow (t, x),
∇ · u0slow = 0.
(2.53)
In addition, (2.50) implies that ' & P u0fast · ∇u0fast = P ∇φ 0 · ∇ ∇φ 0 = 12 P ∇|∇φ 0 |2 = 0,
(2.54)
and (2.49) and (2.53) imply that Mτ [wfast wslow ] = 0
(2.55)
for any fast wfast and slow wslow . 2.3.5. The slow equations Replacing the full projection E in (2.42) by its slow part (2.43) and using (2.28), (2.48), (2.49), (2.50), (2.53), (2.54), and (2.55) yields 0 0
rt0 + u0 · ∇r 0 + ∇ · r 0 u0
= Eslow
0 0) 0 + u0t + u0 · ∇u0 + P ρ(ρ r ∇r 0 − ρμ0 u0 − μ+λ ρ0 ∇∇ · u 0 # $ ⎛ ⎞ 0 Mτ,x ∇ · r 0 u0 rslow % ⎠ = ∂t +⎝
P (ρ0 ) 0 2 u0slow Mτ P ∇ ρ0 P (ρ0 )− (r ) + u0 · ∇u0 − ρμ0 u0 2
r0 0 ρ0 uτ
0 rslow
2ρ0
0 0 $ # 0 = ∂t + 0 0 Mτ P uslow + ∇φ · ∇ uslow + ∇φ 0 − uslow 0 rslow 0 $ # + . = ∂t P u0slow · ∇u0slow − ρμ0 u0slow u0slow
μ 0 ρ0 uslow
(2.56)
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S. Schochet
The first part of (2.53) and the first part of (2.56) are equivalent to (2.16) and (2.21). The second equation of (2.56) plus the last part of (2.53) yield the standard incompressible Navier–Stokes equations (2.15) and (2.20). Thus, the slow part of the limit profile is the same as the “slow” limit obtained when rt and ut are uniformly bounded, and so in particular is independent of the fast part. This special feature of the equations for barotropic flow is a consequence of identity (2.54), which ensures that the interaction of fast waves in the convective term contributes only to the fast part of the equations. One of the difficulties in non-barotropic flow involving fast waves of order one is that this independence of the slow part of the limit from the fast part no longer holds. 2.3.6. The fast equation As noted in (2.50), the fast part of the zeroth-order solution is determined uniquely by φ 0 , so it suffices to derive an equation for the latter. A calculation ([Sch94,Sch05]) leads to
(ρ ) 0 0 0 0 = φt0 − P P (ρ0 ) rslow φτ −
2μ+λ 0 ρ0 φ
0 2 0 2 0 )ρ0 + 14 E∂τ2 −P (ρ0 ) (1 − Mx ) PP (ρ (φ ) + |∇φ | τ (ρ0 )2 & ' + E∂τ2 −P (ρ0 ) ∇ · 12 φ 0 ∇φ 0 + u0slow · ∇ ∇φ 0 .
(2.57)
3. Remarks on the early development of the theory of the incompressible limit The first theory of low Mach number flow, due to Janzen and Rayleigh (see [Sch60, §47], [VD64]) for expositions and references) dealt with steady irrotational flow. Their expansion in powers of the Mach number was used both as a computational tool and as a method for proving the existence of compressible flow. Sirovich [Sir67] extended the use of such expansions to non-steady flows, albeit on the shorter time scale of the fast acoustic waves rather than the convective time scale considered here. The effect of slight compressibility on that longer time scale was first considered in the context of the numerical method of artificial compressibility [Cho67,Tem77], in which the true equation(s) for the evolution of the thermodynamical variables are replaced by a simpler linear model equation for P . The first general proof of the convergence of compressible fluid flow to incompressible flow was given by Ebin [Ebi77], using a differential-geometric formulation that models constraints as a limit of large potentials in dynamical systems. This incompressible limit and other singular limits were formulated directly in terms of partial differential equations by Kreiss [BKK80,Kre80] using the bounded derivative method, which employs transformations to normal forms and places severe restrictions on the initial data, although in certain cases those restrictions were later relaxed [BK82]. Finally, Klainerman and Majda [KM81] proved the convergence of compressible to incompressible flow by directly obtaining uniform estimates for the scaled form of the partial differential equations. Their approach has been followed in most subsequent work, including that described here.
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4. Justification of the incompressible limit for periodic barotropic flow In the slow case, in which (2.24) holds, the same result holds whether the flow occurs on a torus (Td ) or on the whole space (Rd ), or even on some combination of the two. T HEOREM 4.1. ([KM81]) Let k be an integer greater than d2 + 1, where d is the spatial dimension, and assume that P belongs to C k+2 in a neighborhood of ρ0 . Assume further that ρ0 > 0
and
P (ρ) > 0 for ρ > 0,
(4.1)
and that either both viscosity coefficients μ and λ vanish or else (2.3) holds. Let the domain X of the spatial variables be either periodic (i.e., a circle T) or R in each variable. Suppose that the initial data for (2.13)–(2.14) have the form (2.23), where r0,0 is a constant and u0,0 (x) belongs to H k (X) and satisfies (2.24), and r1,0 and u1,0 are uniformly bounded in H k (X). Then for positive ε less than or equal to some positive ε0 , the unique H k (X) solution (rε (t, x), uε (t, x)) to (2.13)–(2.14) with initial data (2.23) exists for a time T independent of ε and is bounded uniformly in C 0 ([0, T ]; H k (X)). Furthermore, as ε → 0, k−δ (rε (t, x), uε (t, x)) converges in C 0 ([0, T ]; Hloc (X)) for every δ > 0 to (r0,0 , u), where u is the unique solution of the limit equations (2.15), (2.20) having initial data u0,0 . In the fast case, for which (2.24) does not hold, the geometry of the domain does affect the asymptotics of solutions. The following theorem covers the periodic case; the case of flow on all Rd will be described later. T HEOREM 4.2. (cf. [Sch94,Gal00]) Let k, ρ0 , P, μ and λ be as in Theorem 4.1. Suppose that the initial data (r0,ε (x), u0,ε (x)) for (2.13)–(2.14) are periodic in x, are uniformly bounded in H k (Td ), and converge in H k (Td ) as ε → 0 to (r0,0 (x), u0,0 (x)). Then for positive ε less than or equal to some positive ε0 , the unique H k (Td ) solution (rε (t, x), uε (t, x)) to (2.13)–(2.14) with initial data (r0,ε (x), u0,ε (x)) exists for a time T independent of ε and is bounded uniformly in C 0 ([0, T ]; H k (Td )). Furthermore, 0 r (τ, t, x) rε (t, x) " − →0 (4.2) uε (t, x) u0 (τ, t, x) "τ = t ε
r0 is the sum as ε → 0 for any δ > 0, where the limit profile in u0 (2.48) of slow and fast parts that are the solutions of (2.56) and (2.50), (2.52), and (2.57), respectively, with initial data
C 0 ([0, T ], H k−δ )
u0slow (0, x) = Pu0,0 (x),
0 rslow (0, x) = Mx r0,0 (x)
and φ 0 (0, 0, x) = ∇ · u0,0 (x),
φτ0 (0, 0, x) = − P ρ(ρ0 0 ) (I − Mx )r0,0 (x).
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S. Schochet
R EMARKS 4.3. 1. The uniform existence part of the theorems remains valid when ε belongs to a bounded interval (0, ε1 ] with ε1 arbitrarily large, provided that ρ0 + εr is bounded away from zero initially for such ε. In view of Lemma 4.4 below, this extension follows from the local existence theory for viscous perturbations of symmetric hyperbolic systems, which yields a uniform time of existence for ε in the compact interval [ε0 , ε1 ]. 2. A more complicated argument ([BdV95,BdV94]) shows that the convergence of the k (X)), which corresponds to solutions in Theorem 4.1 holds also in C 0 ([0, T ]; Hloc taking δ in that theorem equal to zero. 3. Theorems 4.1 and 4.2 can be expressed as results about solutions of the original unscaled equations (2.1)–(2.2) by reversing the changes of variables that lead from those equations to (2.13)–(2.14). 4. By making more extensive use of the regularizing effect of the diffusive terms, it is possible to reduce the smoothness required in the viscous case for both theorems [Dan02]. See also the discussion of weak solutions below. The proof of Theorem 4.1 is based on the fact that the equations for r and u can be written in symmetric form with the large terms having constant coefficients and the secondorder terms being definite with the helpful sign: r . u There exists a function f that is positive for small values of its argument, such that after multiplying the first equation of (2.13)–(2.14) by f (εv), those equations can be written in the form L EMMA 4.4. Let k, ρ0 , P, μ and λ be as in Theorem 4.1, and let v denote the vector
A(0) (εv)vt +
d
A(j ) (v, ε)vxj +
j =1
d d
1 (j ) C vxj = K (i,j ) ∂xi ∂xj v, ε j =1
(4.3)
i,j =1
where d is the spatial dimension, and the matrices A(0) , A(j ) , C (j ) , and K (i,j ) are all symmetric. In addition, there exists a positive m such that the following properties hold for |εv| ≤ m: 1. There exists a positive constant c0 such that A(0) (εv) ≥ c0 I.
(4.4)
2. The matrices C (j ) are constant. 3. The matrices A(0) and A(j ) are C k functions of their arguments. 4. The matrices K (i,j ) either all vanish or else are all constant and have the form K
(i,j )
=
0 0 (i,j ˜ 0 K )
,
(4.5)
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139
where
(∂xi u) K˜ (i,j ) (∂xj u) dx ≥ c1
T
|∇u|2 dx
(4.6)
i,j
for some positive constant c1 . Here and later, the superscript T denotes the transpose.
0 +εr) and P ROOF. Define f (εv) = P ρ(ρ0 +εr
A(0) (εv) :=
P (ρ0 +εr)
0 (ρ0 + εr) I
ρ0 +εr
0
(4.7)
.
Since ρ0 is assumed to be positive, and P is assumed to be positive for positive arguments, f will be positive and A(0) will satisfy (4.4) as long as |εr| ≤ m := 12 ρ0 .
(4.8)
After multiplying (2.13) by f (εv) and writing the result together with (2.14) in terms of the vector v, the coefficient of vt is A(0) (εv). Since that matrix is diagonal it is certainly symmetric. In addition, A(0) is C k+1 at least as long as (4.8) holds, because P has been assumed to be C k+2 for positive arguments. The equation for v may be written as 1 0 ∇· v A(0) (εv) [vt + u · ∇v] + P (ρ0 + εr) ∇ 0 ε 0 = . (4.9) μ u + (μ + λ)∇(∇ · u) Although the large terms in (4.9) appear to have variable coefficients, the fact that P (ρ0 + εr) − P (ρ0 ) ε is bounded allows us to assimilate the non-constant part of the term involving A(j ) . Doing so allows us to write (4.9) in the form (4.3) with C j := P (ρ0 )
0 ej
ejT 0
and P (ρ0 + εr) − P (ρ0 ) 0 ejT ε ej 0 r 0 ejT d P (ρ0 + εs) 0 = uj A (εv) + ds ε ej 0 0 ds
A(j ) (v, ε) := uj A0 (εv) +
1 ε
into the
140
S. Schochet
r
= uj A0 (εv) +
P
(ρ0 + εs) ds
0
0 ej
ejT 0
,
where ej denotes the vector whose j th component equals one and whose other components vanish. Both sets of matrices are symmetric, the C (j ) are constant, and the last expression for the A(j ) shows that those matrices are C k functions of ε and v at least as long as (4.8) holds, in view of the assumed smoothness of P. Although the explicit formula for the K (i,j ) is easily deduced from (4.9), it is more convenient to prove (4.6) directly from the right side of (4.9). We therefore note that (4.5) follows from the fact that only derivatives of the u components appear on the right of (4.9), and only in the equations for the u components. In addition, the symmetry of the operator on the right side of (4.9) implies that the K (i,j ) may be taken to be symmetric. The bound (4.6) is based on the estimate
(∇ · u) dx ≤
|∇u|2 dx,
2
(4.10)
which can be shown by integrating by parts twice and applying the Cauchy–Schwartz inequality: (∇ · u)2 dx =
u(j ) u(i) xi
i,j
xj
dx = −
u(i)
xi xj
i,j
u(j ) dx
%2 (i) (j ) u(i) u = dx ≤ dx u xj
i,j
xi
=
xj
i,j
|∇u|2 dx.
To obtain (4.6) in the case when μ and λ are not both zero, we express the left side of that estimate as the integral of −uT times right side of (2.14), integrate by parts, and apply (4.10):
(∂xi u)T K˜ (i,j ) (∂xj u) dx = −
i,j
uT
=−
K˜ (i,j ) ∂xi ∂xj u dx
i,j
uT [μ u + (μ + λ)∇(∇ · u)]
=μ
|∇u|2 dx + (μ + λ)
(∇ · u)2 dx
≥μ
|∇u| dx + min(0, μ + λ) 2
= min(μ, 2μ + λ)
|∇u|2 dx.
|∇u|2 dx
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141
Hence (4.6) holds with c1 := min(μ, 2μ + λ), which is positive by assumption (2.3) on μ and λ. R EMARKS 4.5. 1. There are two general approaches for proving convergence results like (4.2) for singular perturbation problems: In the direct approach one first obtains estimates for solutions v 0 of the limit problem and/or uniform estimates for the solutions v ε of the full problem, and then estimates the difference between the two [JMR95]. Indeed, Theorem 4.2 follows from the general convergence theorem for singular perturbations in [JMR95]. Alternatively, in the compactness method one obtains uniform estimates for both v ε and ∂t v ε , then applies Ascoli’s theorem or some extension thereof to obtain convergence to a limit for a sequence of values of ε, and finally shows uniqueness of the limit v 0 to obtain convergence without restriction to a sequence. Following [KM81], the latter approach will be used here. 2. Although Theorems 4.1 and 4.2, concern the specific system (2.13)–(2.14), the proofs in essence treat the more general system (4.3), which is either a symmetric hyperbolic system or a parabolic perturbation of such a system. The method of those proofs can therefore be applied to other singular limit problems that can be written in the form (4.3), such as the quasigeostrophic limit of the shallow water equations [Sch87, Gre97]. P ROOF OF T HEOREM 4.1. (cf. [Maj84, §2.1&2.4]) The key observation of [KM81] is that the constancy of the matrices C (j ) occurring in the large terms of (4.3) ensures that the ordinary energy estimates for symmetric hyperbolic systems yield estimates for (4.3) that are uniform in ε. Standard approximation arguments [Maj84, Chapter 2], [Tay96, §16.2] show that it suffices to derive such estimates formally, i.e., assuming as much smoothness, and decay at infinity where relevant, as required. In addition, we may assume that (4.4) holds, because the bound to be derived for v allows us to reduce condition (4.8) for (4.4) to hold to a restriction on the maximum value ε0 of ε. The basic energy estimate is obtained by multiplying (4.3) by v T , integrating over the spatial variables, integrating by parts where appropriate, and simplifying the result. To see that a uniform estimate is obtained we will consider each of the terms in (4.3). The contribution to the basic energy estimate arising from the time derivative term in (4.3) is v T A(0) (εv)vt dx, which can be written as d dt
1 2
v A (εv)v dx − T
(0)
& ' v T εvt · ∇A(0) (εv) v dx
(4.11)
on account of the symmetry of A(0) . Since vt appears in (4.11) only in the combination εvt , using (4.3) to express that time derivative in terms of spatial derivatives yields an expression that is of order one, despite the fact that terms of order 1ε appear in (4.3). Furthermore, since A(0) depends only on r, the time derivative appearing in (4.11) can be expressed in terms of first derivatives only, because no viscous terms appear in the equation for r.
142
S. Schochet
When viscosity is present then an integration by parts plus (4.6) show that the contribution V of the second-order derivative terms on the right side of (4.3) satisfies
V := v T K (i,j ) ∂xi ∂xj v dx = − (∂xi u)T K˜ (i,j ) (∂xj u) dx
i,j
≤ −c1
i,j
|∇u|2 dx.
(4.12)
As mentioned above, the main point is that the large terms in (4.3) make no contribution to the energy estimate, because they yield j 1ε v T C (j ) vxj dx, which equals T (j ) 1 j ∂xj v C v dx and so vanishes identically by periodicity in x or decay at in2ε finity. The order one spatial-derivative terms make the same contribution as in the ordinary estimate for symmetric hyperbolic systems, namely
v T A(j ) (v, ε)vxj dx j
=
1 2
∂xj v T A(j ) (v, ε)v dx −
j
⎤ ⎡
vxj · ∇A(j ) (v, ε)⎦ v dx = − vT ⎣
⎡ vT ⎣
⎤ vxj · ∇A(j ) (v, ε)⎦ v dx
j
(4.13)
j
since the exact derivative term vanishes just as for the large terms. We therefore obtain a basic energy identity d v T A(0) (εv)v dx dt ⎫ ⎧ ⎬ ⎨ = V + vT vxj · ∇A(j ) (v, ε) + εvt · ∇A(0) (εv) v dx ⎭ ⎩ j
=V +
vT
vxj · ∇A(j ) (v, ε)
j
−1 & ' (0) (j ) (j ) − A (εv) C + εA (v, ε) vxj % · ∇A(0) (εv) v dx =V +
v T M(v, ∇v, ε)v dx,
(4.14)
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143
where V was defined in (4.12) and the matrix M is defined by the formula on the preceding lines of (4.14). By the smoothness of the matrices appearing in (4.3), M satisfies M(v, ∇v, ε)L∞ ≤ G(vC 1 )
(4.15)
for some differentiable function G independent of ε. By using (4.12), (4.15), and the positive-definiteness of A(0) , we obtain from (4.14) the basic energy estimate d dt
T
(0)
v A (εv)v dx
≤ c0−1 G(vC 1 )
v T A(0) (εv)v dx,
(4.16)
where we have simply dropped the helpful term that arises when viscosity is present in order to treat the two cases in the same fashion. A fundamental theorem for ODEs states that for any differentiable function H , any function X(t) satisfying the differential inequality d X(t) ≤ H (X(t)) dt
(4.17)
and the initial bound X(0) ≤ y0 will obey the estimate X(t) ≤ Y (t)
(4.18)
for t ≥ 0, where Y (t) is the solution of the differential equation Y (t) = H (Y (t)) with initial data Y (0) = y0 . Differential inequalities of the form (4.17), in which the right side is a function only of the quantity whose derivative appears on the left side, are called closed. However, the basic energy estimate (4.16) is not closed, so it does not by itself imply an integrated estimate of the form (4.18). In order to obtain a closed estimate we need to estimate derivatives as well. The situation here is exactly the same as in the standard theory of quasilinear symmetric hyperbolic systems [Maj84, Chapter 2], [Tay96, §16.2]. Let α be a multi-index, i.e., a vector (α1 , . . . , αd ) whose components are nonnegative integers, and define ∂xα := ∂xα11 · · · ∂xαdd . The result of applying ∂xα to (4.3) can be written as
A(0) (εv)vtα +
A(j ) (v, ε)vxαj +
j
1 (j ) α C vxj = K (i,j ) ∂xi ∂xj v α + Fα,ε , ε j
i,j
(4.19) where v α := ∂xα v and Fα,ε := [A(0) (εv), ∂xα ]vt +
[A(j ) (v, ε), ∂xα ]vxj j
= [A(0) (εv), ∂xα ]
144
S. Schochet
⎧ ⎡ ⎤⎫ ⎬ ⎨
1 K (i,j ) ∂xi ∂xj v − A(j ) (v, ε)vxj − C (j ) vxj ⎦ A(0) (εv)−1 ⎣ ⎭ ⎩ ε i,j
j
+ [A(j ) (v, ε), ∂xα ]vxj ,
j
(4.20)
j
where [L1 , L2 ]w denotes the commutator L1 L2 w − L2 L1 w. Equation (4.19) has the same form as (4.3) except that v has been replaced by v α where differentiated, and a forcing term Fα,ε has been added to the right side. The basic energy estimate therefore yields d −1 α T (0) α α T (0) α (v ) A (εv)v dx ≤ c0 G(vC 1 ) (v ) A (εv)v dx + v α Fα,ε dt ≤ c0−1 G(vC 1 ) + 1 (v α )T A(0) (εv)v α dx + Fα,ε 2L2 .
(4.21)
By Sobolev’s theorem and the definition of k, vC 1 ≤ cvH k . In view of the positivity of A(0) , adding the estimates (4.21) over 0 ≤ |α| := αj ≤ k will therefore yield a closed estimate of the form (4.17), with
(4.22) (v α )T A(0) (εv)v α dx, X := 0≤|α|≤k
provided that
& ' Fα,ε 2L2 ≤ H1 v2H k .
(4.23)
0≤|α|≤k
In the inviscid case, (4.23) indeed holds. To see this, note that since the time derivative gets replaced by a single space derivative, each term in Fα,ε has at most k + 1 derivatives. The presence of the commutators in (4.20) ensures that no one instance of v is differentiated more than k times, and that the factor of 1ε in (4.20) will be compensated by a factor of ε arising from the differentiation of A(0) (εv). Each term in Fα,ε can therefore be written as an application of at most k − 1 derivatives to an O(1) expression involving v and ∇v. Estimate (4.23) therefore follows from the facts that (e.g. [Maj84, Proposition 2.1] or [Tay96, Propositions 13.3.7–13.3.9]) vwH r ≤ cvH r wH r
and
F (v)H r ≤ G(vH r )
for r > d2 . (4.24)
In the viscous case, Fα,ε also contains the commutator [A(0) , ∂xα ] applied to terms involving second derivatives. Since the commutator ensures that the factor A(0) (εv) will
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be differentiated, these extra terms have the form of at most k − 1 derivatives applied to O(ε)f (v, ∇v)uxi xj . By (4.24), such terms are bounded by εH2 (vH k )∇uH k . Since this expression does not have the form F (X) with X as in (4.22), we must eliminate it by using the fact that viscosity adds helpful terms to the energy estimates. If we do not drop the term V when going from (4.14) to (4.16), then (4.21) will include the term −c1 |∇uα |2 dx on the right side. The sum of (4.21) over 0 ≤ |α| ≤ k will therefore have have the extra term inviscid denote the expression obtained for F −c1 ∇u2H k . Let Fα,ε α,ε in the inviscid case, and viscous denote the extra terms present in the viscous case. Then let Fα,ε
Fα,ε 2L2 =
0≤|α|≤k
inviscid viscous 2 Fα,ε + Fα,ε L2
0≤|α|≤k
≤2
0≤|α|≤k
&
inviscid 2 Fα,ε L2 + 2
'
viscous 2 Fα,ε L2
0≤|α|≤k
≤ 2H1 v2H k + 2ε 2 H2 (vH k )2 ∇u2H k .
(4.25)
Since we will obtain a bound for the H k norm of v, after restricting the maximum value ε0 of ε once more the term involving k + 1 derivatives on the right side of (4.25) will be smaller in absolute value than the helpful viscous term. Hence a closed estimate (4.17) with X as in (4.22) is again obtained. As noted above, estimate (4.17) for the X defined in (4.22) yields a uniform bound for the H k norm of v, for some time T independent of ε. Sobolev’s Theorem shows that this H k norm dominates the C 1 norm, so the latter is also bounded uniformly for t ∈ [0, T ] and ε ∈ (0, ε0 ] for some sufficiently small ε0 . The above estimates do not make use of the boundedness of vt at time zero; that is, they apply not only to the slow case under discussion here but also to the fast case of Theorem 4.2. In similar fashion, taking one time derivative and up to k − 1 spatial derivatives, and using the uniform H k bound for v, yields a uniform-in-ε closed estimate (4.17) for '
& ∂xα vt , A(0) (εv)∂xα vt . (4.26) X := |α|≤k−1
In obtaining that estimate, we do not use (4.3) to eliminate vt , but do use the time derivative of that equation to eliminate vtt . In the “slow” case currently under consideration, vt is bounded uniformly in ε at time zero. Estimate (4.17) for (4.26) therefore shows that vt remains bounded in H k−1 uniformly in ε for a time T independent of ε. When the spatial domain X is periodic then the inclusion of H k in H k−1 is compact. Ascoli’s theorem therefore implies that every sequence εj → 0 has a subsequence for which u = u(ε) converges in C 0 ([0, T ], H k−1 ) to a limit u0 . The uniform boundedness of u(ε) in L∞ ([0, T ], H k ) implies that every such subsequence in turn has a subsequence converging weak-∗ in that space. Since each type of convergence implies weak convergence in L2 ([0, T ] × X) and such limits are unique, the limits in both senses are the same. This shows in particular that u0 belongs to L∞ ([0, T ], H k ). Interpolation inequalities for
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S. Schochet
Sobolev spaces then show that the converge also takes place in C 0 ([0, T ], H k−δ ) for any δ > 0. Similarly, the uniform boundedness of vt in C 0 ([0, T ], H k−1 ) implies that, for a further subsequence, vt converges weak-∗ in L∞ ([0, T ], H k−1 ). When the spatial domain X r . In paris unbounded then the same convergences hold provided that we replace H r by Hloc 1 ([0, T ] × X). ticular, we always have by Sobolev’s theorem that convergence occurs in Cloc Given this convergence, the formal computation of the limit in §2.2 is easily justified: 1 convergence allows us to take the limit of the coefficient matrices and the first The Cloc derivatives with respect to spatial variables. Since the time derivatives and second derivatives occur linearly in (4.3), and the product of a strongly convergent sequence with a weakly convergent sequence converges weakly to the product of the corresponding limits, we can also take the weak limit of the terms containing such derivatives. This suffices to obtain the limit equations. Finally, an L2 energy estimate for the difference of two solutions of the limit equations shows that the solution of those equations with given initial data is unique. This shows that the limit obtained above is independent of the choice of sequence εj , which implies that converge holds as ε → 0 without restricting to a sequence. R EMARK 4.6. Estimate (4.23) for the inviscid case can be improved to
Fα,ε 2L2 ≤ H2 vC 1 v2H k .
(4.27)
0≤|α|≤k
As a result, the closed estimate (4.17) for (4.22) can be written as d X(t) ≤ H (v(t)C 1 )X(t). dt
(4.28)
Since X dominates the C 1 norm of v, this estimate is still closed. In addition, it can be integrated to yield X(t) ≤ X(0)e
t 0
H (v(s)C 1 ) ds
.
(4.29)
Estimate (4.29) shows that X(t), and hence also the H k norm of v, remains finite as long as the C 1 norm of v is finite. This shows that the only mechanisms leading to blow-up of solutions to (4.3) are the familiar maximum norm blow-up typical of ODEs and gradient blow-up typical of a single quasilinear first-order PDE like ut + uux = 0. P ROOF OF T HEOREM 4.2. As noted in the proof of Theorem 4.1, the uniform H k bound for v obtained there remains valid under the assumptions here. However, vt can no longer be uniformly bounded in general, since it is not generally uniformly bounded at time zero. Fortunately, the formal calculation of the limit suggests a remedy: Since we have formally separated the fast and slow evolutions into distinct equations (2.27)–(2.28) and (2.29)– (2.30), let us also separate the time dependence of the solution v to (4.3) into a fast variable τ and a slow variable t. To do this, replace v by V(τ, t, x, ε) := S(τ − εt )v(t, x, ε),
(4.30)
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where, as before, S denotes the solution operator (2.32) to the fast equation A(0) (0)Vτ +
C (j ) Vxj = 0
(4.31)
j
that is equivalent, after replacing V by its limit V 0 , to (2.31). Since v has no τ -dependence, (4.30) implies that V indeed satisfies (4.31). Equations (4.30)–(4.31) imply that the time derivative of v can be expressed as vt = S( εt − τ )V t = S( εt − τ )Vt + 1ε S( εt − τ )Vτ .
(4.32)
Definition (4.30) and equations (4.32) and (4.31) allow us to rewrite (4.3) in terms of V as A(0) (εS( εt − τ )V) − A(0) (0) t S( ε − τ )Vτ + A(0) (εS( εt − τ )V)S( εt − τ )Vt ε
+ A(j ) (S( εt − τ )V, ε)S( εt − τ )Vxj = K (i,j ) ∂xi ∂xj S( εt − τ )V. (4.33) j
i,j
Thus, this change of variables cancels the large O( 1ε ) term appearing in (4.3), so a uniform H k−1 estimate (or H k−2 in the viscous case) for Vt is obtained directly from the transformed PDE. Actually, in order to obtain this cancellation it would suffice to replace v by S(− εt )v; the τ -dependence has been included for convenience, so that v can be recovered by evaluating V at τ = εt , as in (2.25) of the formal calculation. By the unitarity of S with respect to the inner product defined by A(0) (0), the uniform k H estimate for v implies the same bound for V, which together with the uniform bound for Vt again yields the compactness necessary to obtain convergence for a subsequence εj → 0. Furthermore, the unitarity of S ensures that the above bounds are also uniform in r0 0 implies that τ , which shows that the convergence of V(τ, t, x, ε) to V (τ, t, x) := u0 (4.2) holds. Taking the limit of (4.30) shows that V 0 has the form S(τ ) times an expression independent of τ , i.e., (2.33) holds. As shown in the formal derivation, this may be expressed equivalently as (2.41). Since, as discussed above, the existence of a corrector V 1 remains in doubt, the justification of the limit profile equation (2.42) must differ from the formal derivation. This justification then serves as an alternative derivation of that equation. We cannot take the limit of (4.33) directly on account on the numerous occurrences of the operator S( εt − τ ), which does not converge as ε → 0. We therefore solve that equation for Vt and integrate with respect to time, obtaining V(T , τ, x, ε) = V(0, τ, x, ε) T S(τ − εt )A(0) (εS( εt − τ )V)−1 + 0
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S. Schochet
K (i,j ) ∂xi ∂xj S( εt − τ )V −
i,j
−
A(0) (εS( εt − τ )V) − A(0) (0) t S( ε − τ )Vτ ε
A(j ) (S( εt − τ )V, ε)S( εt − τ )Vxj dt.
(4.34)
j
Wherever ε appears in (4.34) outside of S and V we may expand in powers of ε and retain only the terms of order one to obtain V(T , τ, x, ε) = V(0, τ, x, ε) T + S(τ − εt )A(0) (0)−1 0
K (i,j ) ∂xi ∂xj S( εt − τ )V − S( εt − τ )V · ∇u A(0) (0) S( εt − τ )Vτ
i,j
−
(j )
A
(S( εt
− τ )V, 0)S( εt
− τ )Vxj dt
j
+ o(1).
(4.35)
Taking the limit of both sides of (4.35) along a sequence εj → 0 for which V → V 0 then yields V 0 (T , τ, x) = V 0 (0, τ, x) T S(τ − εt )A(0) (0)−1 + lim ε→0 0
K (i,j ) ∂xi ∂xj S( εt − τ )V 0 − S( εt − τ )V 0 · ∇u A(0) (0) S( εt − τ )Vτ0
i,j
−
(j )
A
(S( εt
− τ )V
0
, 0)S( εt
− τ )Vx0j
dt.
(4.36)
j
Note that although we have not yet calculated the limit on the right side of (4.36), the fact that V converges to V 0 justifies our having replaced the former with the latter inside that limit. To calculate the limit in (4.36), note that by (2.33), " S( εt − τ )V 0 = S( εt − τ )S(τ )V 0 (0, t, x) = S( εt )V 0 (0, t, x) = V 0 (τ, t, x)"τ = t . ε
(4.37)
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Since the effect of ∂τ on V 0 can be expressed via (4.31) in terms of spatial derivatives, and the solution operator S commutes with that spatial derivative operator, it is also true that " S( εt − τ )Vτ0 = Vτ0 (τ, t, x)"τ = t .
(4.38)
ε
Formulas (4.37)–(4.38) allow us to write (4.36) as V 0 (T , τ, x) = V 0 (0, τ, x) + S(τ ) lim
T
ε→0 0
" f (τ, t, x)"τ = t dt,
(4.39)
ε
where f (τ, t, x) := S(−τ )A(0) (0)−1 ⎡ ⎤
⎣ K (i,j ) ∂xi ∂xj V 0 − V 0 · ∇u A(0) (0) Vτ0 − A(j ) (V 0 , 0)Vx0j ⎦ . (4.40) i,j
j
The limit in (4.39) can be evaluated by using a version of a special case of the Bogoliubov– Mitropolsky averaging theorem. The basic idea is as follows: We want to determine the limit as ε → 0 of an integral of the form
T
z1 (T ) := 0
" f (τ, t, x)"τ = t dt;
here x is merely a parameter. After defining T := d dT
z0 z1
(4.41)
ε
=ε
T ε
and z0 = εT , we find that
1 f (T , z0 , x)
.
(4.42)
The Bogoliubov–Mitropolsky averaging theorem (e.g. [SV85, Theorem 3.3.3]) says that if f is C 1 with respect to t, and the limit 1 [MT f ](z, x) := lim T →∞ T
T
f (T1 , z, x) T1
(4.43)
0
exists, then the solution of (4.42) with given initial data tends as ε → 0 to the solution of the averaged system d dT
z˜ 0 z˜ 1
=ε
1 [MT f ](˜z0 , x)
(4.44)
having the same initial data, uniformly for 0 ≤ T ≤ O( 1ε ). Transforming back to the independent variable T and using (4.41) therefore yields
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S. Schochet
T
lim
ε→0 0
" f (τ, t, x)"τ = t dt = ε
T
[Mτ f ](t, x) dt,
(4.45)
0
where the averaging operator Mτ was defined in (2.37). The classical Bogoliubov–Mitropolsky theory can be applied pointwise if k > d2 + 2 in the inviscid case or k > d2 + 4 in the viscous case, which we temporarily assume to hold. Since the τ -derivative appearing in (4.39) can be expressed in terms of spatial derivatives of order one, the above extra conditions on k ensure that the integrand f appearing in (4.39)–(4.40) is continuously differentiable with respect to t. In addition, f is also almostperiodic with respect to τ , which ensures that the average Mτ f exists. To see this, note that since V has the form S(τ )g(t, x, ε), its limit V 0 has the form S(τ )g(t, x). Because the restriction of S(τ ) to each Fourier mode is periodic, and the Fourier series of V 0 and its spatial derivatives through order two converge absolutely by the proof of Sobolev’s theorem, V 0 and those spatial derivatives are almost periodic. Composing V 0 with C 1 functions and multiplying the result by first derivatives of V 0 still yields an almost periodic function. Because the Fourier series of the result still converges absolutely, applying S(τ ) once more yields another almost-periodic function. In view of definition (2.38), applying (4.45) to (4.39) yields V 0 (T , τ, x) = V 0 (0, τ, x) T 3 + E A(0) (0)−1 K (i,j ) ∂xi ∂xj V 0 0
i,j
4
0 (j ) 0 0 − V · ∇u A (0) Vτ − A (V , 0)Vxj dt.
0
(0)
(4.46)
j
Differentiating (4.46) with respect to t then yields the limit equation 3 Vt0
−1
= E A (0) (0)
−
K (i,j ) ∂xi ∂xj V 0
i,j
4
0 (0) 0 (j ) 0 0 − V · ∇u A (0) Vτ − A (V , 0)Vxj .
(4.47)
j
Equation (4.47) is equivalent to (2.42). The derivation of the more explicit formulas (2.56)– (2.57) then follows as in the formal calculation. Even when the additional assumption on k does not hold, the first-derivative terms in the integrand f occurring in (4.39)–(4.40) still belong to H s with s > d2 , so the proof of Sobolev’s theorem shows that their Fourier series converge absolutely. In similar fashion to the above, this ensures that those terms are almost-periodic. In addition, the time derivative of those terms is at worst in H −1 , which by interpolation ensures that those terms are Hölder continuous in L2 . The special case (4.45) of the Bogoliubov–Mitropolsky theory has been extended to such settings [Sch94b], so the above calculation of the limit of those
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151
terms remains valid. Since the second-derivative term, when present, has constant coefficients, we can consider it in the distribution sense. This effectively removes the derivatives, which makes the above argument valid for that term as well. Since we are merely trying to determine the equation satisfied by V 0 , it suffices to take the limit in the distribution sense. Alternatively, one could use the additional smoothness induced by that viscous terms to obtain the limit in similar fashion to the first-derivative terms. Finally, an L2 energy estimate with respect to the inner product (2.40) for the difference of two solutions of the limit profile equations shows that the solution of those equations with given initial data is unique. To see this note that since EV 0 = V 0 as noted in (2.41), once we multiply the difference of (4.47) for two solutions by A(0) (0) times the difference of those solutions, we can drop the projection E. The L2 energy estimate for the difference is therefore is the same as the standard estimate that holds for the PDE obtained by dropping E from (4.47). As in the proof of Theorem 4.1, this shows that converge holds as ε → 0 without restricting to a sequence.
5. Survey of further results 5.1. Fast decay of fast waves on Rd for d ≥ 2 In spatial dimensions larger than one, solutions of the wave equation (2.52) having initial data in H k ∩ L1 decay in L∞ . The potential φ 0 in (2.51), which determines the fast components of the limit via (2.50), satisfies the wave equation (2.52) on the fast acoustic time scale. As a consequence, the fast part of the solution decays on that time scale, and so is negligible on the convective time scale. Thus, even in the fast case the full solution v ε converges on compact sets to the slow limit, albeit nonuniformly in time for t near zero. This result was proven for solutions in Rd in [Uka86,Iso87b], and has been extended to exterior domains [Iso87a], including the case when there is a nonvanishing steady flow at infinity [Iso89], and also to a half-space Rd+ [Igu97]. 5.2. Boundaries: Euler equations For slow barotropic flow, results analogous to Theorem 4.1 have been proven for inviscid flow in bounded domains in [Ebi82,Sch86,Asa87]. Exterior domains can be treated in similar fashion [Iso87a]. The fast case was treated in [Sec00]. For moving domains, the formal asymptotics of solutions have been calculated [Alì03], while for domains with open boundaries uniform estimates have been derived for the linearized system [GS91].
5.3. Nonisentropic flow In non-barotropic flow, two thermodynamical variables are needed to determine the state of the fluid. It will be convenient to take the pressure and either entropy or temperature as
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S. Schochet
the basic variables, and consider the density as a function of these. In particular, such flows involve variations of entropy, i.e., they are nonisentropic. Instead of considering a typical density, we consider a typical pressure P0 , and assume that ∂ρ ∂ P > 0. 5.3.1. The inviscid case In the inviscid case we will consider the density ρ as a function of pressure P and entropy S. For such flows, rescaling leads to # $ a10 (P0 + εp, s) pt + u · ∇p + 1ε ∇ · u = 0 a20 (P0 + εp, s) [ut + (u · ∇) u] + 1ε ∇p = 0 st + u · ∇s = 0, where a10 :=
1 ∂ρ ρ ∂P ,
a20 := ρ.
(5.1)
These equations almost have the form with (4.3) the additional spatial variable and viscous 0 a10 (0) terms omitted, except that A := now depends on v as well as on εv, i.e., 0 a20 I (4.3) is replaced by A(0) (εv, v)vt +
Aj (v, ε)vxj +
j
1 (j ) C vxj = 0. ε
(5.2)
j
In general, solutions of (5.2) having uniformly bounded initial data do not exist for a time independent of ε, as can be seen from the explicitly-solvable equation (1+v 2 )vt + 1ε vx = 0. In terms of energy estimates, the difficulty with (5.2) is that the equation for vxj has the form A(0) (εv, v)vxj t + vxj · ∇v A(0) (εv, v) vt + ε vxj · ∇εv A(0) (εv, v) vt + · · · = 0, in which the O( 1ε ) term vt is not always multiplied by a compensating factor of ε, as holds for (4.3). Of course, in the slow case for which vt is bounded initially this problem does not arise, so that uniform estimates can be obtained [Sch86,Sch88]. The Euler equations have a special structure beyond that of (5.2). Among other features, A(0) depends on v rather than εv only through its dependence on s, whose time-derivative equation contains no large O( 1ε ) terms. In other words, although the fast operator is nonlinear it depends only on a slow variable. Taking into account certain other special features of the equations as well then permits a complicated set of estimates to be obtained, which yield uniform bounds for # $ p, u, s, st , and P a20 u t .
(5.3)
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In particular, even in the fast case solutions of the Euler equations with uniformly bounded initial data exist for a time independent of ε. Nevertheless, the equations exhibit nonuniform stability in that small changes in s may cause an O(1) change in u. Furthermore, on account of the nonlinearity of the fast operator, explicit Fourier-space computations like those of §2.3.2 are not possible. The uniform bounds for the quantities in (5.3) imply that, after restricting to a sequence of values of ε, uε converges weakly to some u0slow , p ε converges weakly to some $ # 0 , s ε converges strongly to some s 0 , and P a 0 (P + εp ε , s ε )uε converges strongly to pslow 0 2 # 0 $ P a2 (P0 , s 0 )u0slow . In all Rd , if the initial data decay sufficiently rapidly at infinity then the fast waves still decay quickly, so that the limit satisfies the “stratified” incompressible Euler equations in which the entropy, and hence also the density, remain non-constant [MS01]. An extension of this result to exterior domains has recently been obtained [Ala05]. In the periodic case, the above convergence results suffice to obtain the equations 0 ∇pslow = 0 = ∇ · u0slow
st0 + u0slow · ∇s 0 = 0 0 ∂t pslow =0
for the limit variables [MS03]. However, in order to obtain a closed set of equations we still need to find the equation satisfied by u0slow , which turns out not to be just the incompressible Euler equation in general. Rather, a formal calculation of the equation for u0slow shows that that equation includes an extra term involving the limit of quadratic expressions in the fast part of the solution, which tends weakly to zero. In the very special case of only one spatial dimension, the limit can be both calculated completely and justified [MS03]. The limit equation for u0slow turns out to be simply ∂t u0slow = 0. However, this result depends on the fact that a10 =
1 ∂a20 , a20 ∂P
(5.4)
as follows from (5.1). If we consider general positive functions aj0 depending on (P0 + ε p 0 ±iα t/ε j , where εp, s) then complicated equations link uslow to the weak limits of e uε the αj2 are the nonzero eigenvalues of the operator −
1 a10 (P0 , s(0, x))
5 ∂x
1 a20 (P0 , s(0, x))
6 ∂x .
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S. Schochet
The key point allowing the formula so obtained to be justified is that, after an appropriate transformation, the spectral decomposition of the corresponding operator for non-zero ε is independent of ε and t. In the multi-dimensional case the formal calculation of the extra term in the limit, which once again involves the spectral decomposition of the fast operator, assumes that the spectrum of that fast operator is simple and non-resonant. For certain finite-dimensional truncations of the equations those assumptions can be shown to be generic and to ensure convergence to the limit equations [MS03]. 5.3.2. The full fluid equations A uniform existence result, and a convergence result for flow in the whole space, has recently been proven for the rescaled full Navier–Stokes equations [Ala06], including viscous terms not only in the velocity equation but in the temperature equation as well. One striking new feature of this case is that the divergence of the limit velocity is no longer zero but instead depends on the limit temperature.
5.4. Multiple spatial scales More complicated asymptotics are obtained when the initial data involve multiple spatial scales, i.e. are H k -functions of the 2d variables x and y := εx. The formal asymptotics for both isentropic and nonisentropic flow involving multiple spatial scales have been developed in [Kle95,KBS+ 01,Mei00]. In the isentropic case the small scale flow does not affect the large scale motion. The asymptotics for that case were justified in [Sch05], simply by noting that when ordinary estimates work for one scale then they work for multiple spatial scales as well. However, the special estimates needed for the nonisentropic case do not extend to multiple spatial scales.
5.5. Improved error estimate and asymptotic expansions When can the o(1) error estimate in Theorem 4.2 or its variants be improved to v ε − v 0 + O(ε) or even to an asymptotic expansion v ε = v 0 + εv 1 + · · · + ε k v k + O(ε k+1 )? Such results cannot be expected to hold true in general because of small divisor problems in the periodic case or slow decay of the fast part or the pressure in the whole-space case. Nevertheless, such results have been proven when in the slow case [KM82,Sch88], i.e., 0 = 0, and for generic values of the ratios of the spatial periods in the periodic when vfast case [JMR95,Sch94,Gal98].
5.6. Viscous flows: weak solutions and global solutions In the energy estimates derived above for (4.3), we only made use of the fact that the viscous terms do not hinder the essentially hyperbolic estimates that are valid when viscosity is absent. Retaining the contribution of the viscous terms adds very helpful terms to those estimates. Complicated estimates make it possible to take advantage of those helpful terms
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so as to allow the initial data to be less smooth, including cases in which the equations must be understood in the weak sense, and to obtain global existence of weak solutions for small ε when the limit solution exists for all time [Lin95,Hof98,LM98,LM99,DG99, HL98,HL02,Dan02,BDGL02]. Surprisingly, when both fast waves and boundaries are present then the interaction of fast waves with a boundary layer usually makes them decay fast in bounded domains, so that v ε converges to the solution of the slow equations [DGLM99]. Even in the inviscid case the limit solution sometimes exists for all time. Without viscosity one cannot expect the full solution v ε to also exist for all time, but its time of existence has been shown in various cases to tend to infinity as ε → 0 [Sid91,DH03]. 6. Some open problems 6.1. Multiple spatial scales for fast non-isentropic flow The estimates of [MS01] for fast nonisentropic flow do not work with multiple spatial scales, because the condition (1.3) there no longer holds. However, multiple spatial scales are especially interesting for nonisentropic flows because the formal asymptotics calculated in [KBS+ 01] show that the small-scale flow does affect the large-scale one. Do fast nonisentropic solutions with multiple spatial scales exist for a time independent of ε and satisfy uniform bounds?
6.2. Genericity of simple non-resonant spectrum for fast periodic non-isentropic flow Do the genericity and convergence results proven in [MS03] for finite-dimensional truncations of the non-isentropic Euler equations also hold for the full Euler equations?
References [Ala05] T. Alazard, Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions, Adv. Differential Equations 10 (1) (2005) 19–44. [Ala06] T. Alazard, Low mach number limit of the full Navier–Stokes equations, Arch. Rational Mech. Anal. (2006), to appear. [Alì03] G. Alì, Low Mach number flows in time-dependent domains, SIAM J. Appl. Math. 63 (6) (2003) 2020–2041 (electronic). [Asa87] K. Asano, On the incompressible limit of the compressible Euler equation, Japan J. Appl. Math. 4 (1987) 455–488. [BDGL02] D. Bresch, B. Desjardins, E. Grenier and C.-K. Lin, Low Mach number limit of viscous polytropic flows: formal asymptotics in the periodic case, Stud. Appl. Math. 109 (2) (2002) 125–149. [BdV94] H. Beirão da Veiga, Singular limits in compressible fluid dynamics, Arch. Rational Mech. Anal. 128 (4) (1994) 313–327. [BdV95] H. Beirão da Veiga, On the sharp singular limit for slightly compressible fluids, Math. Methods Appl. Sci. 18 (4) (1995) 295–306. [BK82] G. Browning and H.-O. Kreiss, Problems with different time scales for nonlinear partial differential equations, SIAM J. Appl. Math. 42 (4) (1982) 704–718.
156
S. Schochet
[BKK80] G. Browning, A. Kasahara and H.-O. Kreiss, Initialization of the primitive equations by the bounded derivative method, J. Atmospheric Sci. 37 (7) (1980) 1424–1436. [Cho67] A. Chorin, A numerical method for solving incompressible viscous flow problems, J. Comp. Phys. 2 (1967) 12–26. [Dan02] R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions, Amer. J. Math. 124 (6) (2002) 1153–1219. [DG99] B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999) 2271–2279. [DGLM99] B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier–Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. 78 (1999) 461–471. [DH03] A. Dutrifoy and T. Hmidi, The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data, C. R. Math. Acad. Sci. Paris 336 (2003) 471–474. [Ebi77] D. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. of Math. 105 (1977) 141–200. [Ebi82] D. Ebin, Motion of slightly compressible fluids in a bounded domain I, Commun. Pure Appl. Math. 35 (1982) 451–485. [Gal98] I. Gallagher, Asymptotic of the solutions of hyperbolic equations with a skew-symmetric perturbation, J. Differential Equations 150 (2) (1998) 363–384. [Gal00] I. Gallagher, A remark on smooth solutions of the weakly compressible periodic Navier–Stokes equations, J. Math. Kyoto Univ. 40 (3) (2000) 525–540. [Gre97] E. Grenier, Pseudo-differential energy estimates of singular perturbations, Comm. Pure Appl. Math. 50 (9) (1997) 821–865. [GS91] B. Gustafsson and H. Stoor, Navier–Stokes equations for almost incompressible flow, SIAM J. Numer. Anal. 28 (6) (1991) 1523–1547. [HL98] T. Hagstrom and J. Lorenz, All-time existence of classical solutions for slightly compressible flows, SIAM J. Math. Anal. 29 (1998) 652–672. [HL02] T. Hagstrom and J. Lorenz, On the stability of approximate solutions of hyperbolic-parabolic systems and the all-time existence of smooth, slightly compressible flows, Indiana Univ. Math. J. 51 (6) (2002) 1339–1387. [Hof98] D. Hoff, The zero-Mach limit of compressible flows, Comm. Math. Phys. 192 (1998) 543–554. n, [Igu97] T. Iguchi, The incompressible limit and the initial layer of the compressible Euler equation in R+ Math. Methods Appl. Sci. 20 (1997) 945–958. [Iso87a] H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain, J. Reine Angew. Math. 381 (1987) 1–36. [Iso87b] H. Isozaki, Wave operators and the incompressible limit of the compressible Euler equation, Comm. Math. Phys. 110 (1987) 519–524. [Iso89] H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain. II. Bodies in a uniform flow, Osaka J. Math. 26 (2) (1989) 399–410. [JMR95] J.-L. Joly, G. Métivier and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics, Ann. Sci. École Norm. Sup. (4) 28 (1) (1995) 51–113. [JMR98] J. L. Joly, G. Metivier and J. Rauch, Dense oscillations for the compressible 2-d Euler equations, in: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. XIII (Paris, 1994/1996), volume 391 of Pitman Res. Notes Math. Ser., Longman, Harlow (1998), 134–166. [KBS+ 01] R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics, J. Engrg. Math. 39 (1–4) (2001) 261–343. [Kle95] R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics. I. One-dimensional flow, J. Comput. Phys. 121 (2) (1995) 213–237. [KM81] S. Klainerman and A. Majda, Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math 34 (1981) 481–524.
The mathematical theory of the incompressible limit in fluid dynamics
157
[KM82] S. Klainerman and A. Majda, Compressible and incompressible fluids, Commun. Pure Appl. Math. 35 (1982) 629–653. [Kre80] H.-O. Kreiss, Problems with different time scales for partial differential equations, Comm. Pure Appl. Math. 33 (3) (1980) 399–439. [Lax73] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM (1973). [Lin95] C. K. Lin, On the incompressible limit of the compressible navier–stokes equations, Comm. Partial Differential Equations 20 (1995) 677–707. [Lio96] P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1, volume 3 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York (1996). [Lio98] P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2, volume 10 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York (1998). [LM98] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. 77 (1998) 585–627. [LM99] P.-L. Lions and N. Masmoudi, Une approche locale de la limite incompressible, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 387–392. [Maj84] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag (1984). [Mei00] A. Meister, Asymptotic single and multiple scale expansions in the low Mach number limit, SIAM J. Appl. Math. 60 (1) (2000) 256–271 (electronic). [MS01] G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal. 158 (2001) 61–90. [MS03] G. Métivier and S. Schochet, Averaging theorems for conservative systems and the weakly compressible Euler equations, J. Differential Equations 187 (2003) 106–183. [Sch60] M. Schiffer, Analytical theory of subsonic and supersonic flows, in: Handbuch der Physik, vol. 9, part 3, Springer-Verlag, Berlin (1960), 1–161. [Sch86] S. Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Commun. Math. Phys. 104 (1986) 49–75. [Sch87] S. Schochet, Singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation, J. Differential Equations 68 (3) (1987) 400–428. [Sch88] S. Schochet, Asymptotics for symmetric hyperbolic systems with a large parameter, J. Differential Equations 75 (1988) 1–27. [Sch94] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations 114 (1994) 476–512. [Sch94b] S. Schochet, Resonant nonlinear geometric optics for weak solutions of conservation laws, J. Differential Equations 113 (1994) 473–504. [Sch05] S. Schochet, The mathematical theory of low Mach number flows, Math. Model. Numer. Anal. 39 (2005) 441–458. [Sec00] P. Secchi, On the singular incompressible limit of inviscid compressible fluids, J. Math. Fluid Mech. 2 (2000) 107–125. [Sid91] T. Sideris, The lifespan of smooth solutions to the three-dimensional compressible Euler equations and the incompressible limit, Indiana U. Math J. 40 (1991) 535–550. [Sir67] L. Sirovich, Initial and boundary value problems in dissipative gas dynamics, Phys. Fluids 10 (1967) 24–34. [SV85] J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag (1985). [Tay96] M. Taylor, Partial Differential Equations III, Springer-Verlag, New York (1996). Applied Mathematical Sciences 117. [Tem77] R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam (1977). [Uka86] S. Ukai, The incompressible limit and initial layer of the compressible Euler equation, J. Math. Kyoto U. 26 (1986) 323–331. [VD64] M. Van Dyke, Perturbation Methods in Fluid Mechanics, Applied Mathematics and Mechanics, vol. 8, Academic Press, New York (1964).
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CHAPTER 4
Local Regularity Theory of the Navier–Stokes Equations G. Seregin Steklov Institute of Mathematics at St. Petersburg, St. Peterburg, Russia
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2. Local regularity theory for the Stokes system . . . . . . 3. Interior case: Ladyzhenskaya–Prodi–Serrin condition . 4. Boundary case: Ladyzhenskaya–Prodi–Serrin condition 5. Interior case: ε-regularity theory . . . . . . . . . . . . . 6. Boundary case: ε-regularity theory . . . . . . . . . . . 7. Local L3,∞ -solutions . . . . . . . . . . . . . . . . . . 8. Other local regularity results . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract This is a survey paper on the local regularity theory of the non-stationary three-dimensional Navier–Stokes equations.
Keywords: Navier–Stokes equations, weak Leray–Hopf solutions, suitable weak solutions, interior regularity, boundary regularity MSC: 35K, 76D
HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME IV Edited by S.J. Friedlander and D. Serre © 2007 Elsevier B.V. All rights reserved. 159
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1. Introduction In this paper we would like to systemize data, available for the so-called local regularity theory of the Navier–Stokes equations. We restrict ourselves to the non-stationary threedimensional case. Let us enumerate typical problems of that theory. P ROBLEM 1 (Interior regularity) Consider the classical Navier–Stokes system ∂t v + v · ∇v − v = f − ∇p,
div v = 0
(1.1)
in the canonical domain Q = B×] − 1, 0[⊂ R3 × R1 . Here, B is the unit ball of R3 centered at the space origin x = 0, v and p stand for the velocity and the pressure, respectively, and f is a given force. What are minimal conditions on v and p that imply regularity of v at the space-time origin z = (x, t) = 0 = (0, 0)? Notions minimal and regularity given in the above formulation must be explained. The most popular definition of regularity is due to Caffarelli–Kohn–Nirenberg [4]. It says z = 0 is a regular point of v if there exists a positive number r ≤ 1 such that v ∈ L∞ (Q(r)), where Q(r) = B(r)×] − r 2 , 0[ is a standard parabolic cylinder and B(r) is the ball of radius r with the center at the space origin. However, we would not recommed fixing this definition for ever. In fact, the class L∞ (Q(r)) may be replaced with C(Q(r)), see [41–44], and even with C α (Q(r)) for some positive α, see [26] and also [49]. In most cases all these definitions are equivalent. By minimality of assumptions we mean conditions that naturally arise from global existence theorems for initial-boundary value problems. Usually they include boundedness of the energy. P ROBLEM 2 (Boundary regularity) Consider the same Navier–Stokes system (1.1) but in the half-cylinder Q+ = B + ×] − 1, 0[, where B + = {x = (x1 , x2 , x3 )|x| < 1, x3 > 0}, under the homogeneous Dirichlet boundary condition v|x3 =0 = 0.
(1.2)
The question is the same: to find reasonable conditions for the space-time origin z = 0 to be a regular point of v. Now, z = 0 is regular if there exists r ∈ ]0, 1] such that v ∈ L∞ (Q+ (r)) + + (C(Q (r)) or C α (Q (r))). Here, Q+ (r) = B + (r)×] − r 2 , 0[, B + (r) = {|x| < r, x3 > 0}. From the local regularity theory of quasilinear elliptic and parabolic equations and systems it is known that basically, for solutions with finite “energy”, we can have the following issues: full regularity of solutions to equations containing one unknown function, and partial regularity of solutions to systems. In turn, partial regularity is a consequence of socalled ε-regularity theory. It should be noted that for many interesting quasilinear systems in 3D the problem as to whether energy solutions are smooth or not is still open. The Navier–Stokes equations form a semilinear system and we could expect to have something better than what takes place in the general theory of regularity for quasilinear systems. In addition, they are invariant under special scaling which allows us to significantly improve regularity results for the general theory. However the incompressibility condition brings new challenges for the analysis of local regularity. To demonstrate these
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challenges we can ignore the non-linear term, i.e., consider the Stokes system. Such a consideration shows the limitations of local regularity analysis for the original Navier–Stokes equations. The reader will find corresponding discussions in Section 2. In Sections 3 and 4 a local version of the classical Ladyzhenskaya–Prodi–Serrin condition is discussed. Our version differs from known statements of J. Serrin [55] and M. Struwe [59] for the interior case. In the boundary case, we use a combination of techniques developed by the author [46] and by V. A. Solonnikov [58]. Sections 5 and 6 contain general facts concerning the Caffarelli–Kohn–Nirenberg type theory. Most of them have the form of so-called ε-regularity conditions. The famous Caffarelli–Kohn–Nirenberg condition, see [4], is formulated as Proposition 5.8 for the interior case and as Proposition 6.8 for the boundary case. It gives the best estimate for the Hausdorff dimension of the singular set. In our approach we follow papers [26] and [48–50]. In Section 7 we discuss recent results on so-called local L3,∞ -solutions, see [54], [12], and [15] for the interior case and [51] for the boundary case. It is a particular case of the Ladyzhenskaya–Prodi–Serrin condition. To prove that such solutions are in fact smooth, one needs to develop a unique continuation theory. This was done in [12–15] where new backward uniqueness theorems for the heat operator were proved. The last section is devoted to selected topics concerning local theory. Among them, the reader can find a local version of Constantin’s result on differentiability properties of the vorticity and a short discussion of recent results on self-similar solutions due to J. Neˇcas, M. Ruziˇcka, and V. Šverák, see [37], and T.-P. Tsai, see [65]. Finally we note that for systems describing non-Newtonian fluids the local regularity theory is still poorly understood. Some partial results in that directions can be founded in [45], see also [24], [25], [1], [19], [31], and [34].
2. Local regularity theory for the Stokes system Here we will distinguish between interior and boundary cases. Let us start with the simplest one, i.e. the interior case. Consider the Stokes system ∂t v − v = f − ∇p,
div v = 0
(2.1)
in the space-time cylinder Q. To understand difference between interior and boundary cases, we first assume that f = 0. There are two ways to investigate regularity problem. In the first way one can introduce vorticity ω = ∇ ∧ v and get the heat equation for it in Q: ∂t ω − ω = 0. Local regularity theory of the heat equation is well developed, see [27]. In the second way, we suppose that p ∈ Ll (Q)
Local regularity theory of the Navier–Stokes equations
163
for some l > 1. Since p(·, t) is a harmonic function in B, it is smooth in spatial variables inside B. We may interpret the first equation in (2.1) as the heat equation with smooth in space right hand side and conclude that the velocity field is also smooth in spatial variables. However, looking at the boundary value problem ∂t v − v = f − ∇p,
div v = 0
(2.2)
in Q+ and v|x3 =0 = 0,
(2.3)
we observe that ω|x3 =0 = 0 in the first case. Thus it is not clear how to apply the local regularity theory of the heat equation to our situation. In the second case, two assumptions p(·, t) ∈ Ll (B + ) and p(·, t) = 0
in
B+
do not imply, in general, that p(·, t) is smooth in the closure of B + (r) for some positive r. In addition, keeping in mind the non-linear case, we should admit that the force f is not zero and it is not smooth. Nevertheless, there is a possibility to circumvent this problem: we can use the coercive Ls,n -estimates of solutions to the initial boundary value problems for the Stokes system. Let us formulate the simplest statement of that kind. T HEOREM 2.1. Let be a bounded domain in R3 with sufficiently smooth boundary and T is a given positive number. Assume that f ∈ Ls,n (QT )
(2.4)
for some 1 < s, n < +∞. Here, Ls,n (QT ) = Ls (0, T ; Ln ()) and QT = ×]0, T [. The initial boundary value problem ∂t v − v = f − ∇p, v=0
div v = 0
in QT ,
on ∂ QT ,
(2.5) (2.6)
where ∂ QT is the parabolic boundary of the space-time cylinder QT , has a unique solu2,1 1,0 tion v ∈ Ws,n (QT ) and p ∈ Ws,n (QT ), satisfying the coercive estimate ∂t vs,n,QT + ∇ 2 vs,n,QT + ∇ps,n,QT ≤ C(, T )f s,n,QT .
(2.7)
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Here and below, the following notation is used: f s,n,QT = f Ls,n (QT ) , 1,0 Ws,n (QT ) = {v ∈ Ls,n (QT ),
∇v ∈ Ls,n (QT )},
2,1 1,0 (QT ) = {v ∈ Ws,n (QT )∂t v ∈ Ls,n (QT ), Ws,n
∇ 2 v ∈ Ls,n (QT )}.
The reader can find a proof of Theorem 2.1 in [21] and [35]. Now we demonstrate a simple and important consequence of Theorem 2.1. P ROPOSITION 2.2. Assume that condition (2.4) holds in the space-time cylinder Q and 1,0 (Q) and p ∈ Lm,n (Q) satisfy (2.1). m ∈ ]1, s]. Let a pair of functions v ∈ Wm,n 2,1 1,0 Then v ∈ Ws,n (Q(1/2)) and p ∈ Ws,n (Q(1/2)) and we have the estimate ∂t vs,n,Q(1/2) + ∇ 2 vs,n,Q(1/2) + ∇ps,n,Q(1/2) ≤ c(f s,n,Q + vm,n,Q + ∇vm,n,Q + pm,n,Q ).
(2.8)
P ROOF. It is sufficient to prove this proposition for case s = m. The general case can be deduced from it by embedding theorems and bootstrap arguments. Fix a non-negative cut-off function ϕ ∈ C0∞ (B×] − 1, 1[) so that ϕ = 1 in B(1/2)×] − (1/2)2 , (1/2)2 [. For any t ∈ ] − 1, 0[, we determine a function w(·, t) as a unique solution to the boundary value problem w(·, t) − ∇q(·, t) = 0, div w(·, t) = v(·, t) · ∇ϕ(·, t) q(x, t)dx = 0, w(·, t) = 0 on ∂B.
in
B,
B
It satisfies the estimate ∇ 2 w(·, t)s,B + q(·, t)s,B + ∇q(·, t)s,B ≤ c∇(v(·, t) · ∇ϕ(·, t))s,B .
(2.9)
Letting V = ϕv − w,
P = ϕp − q,
F = ϕf + v∂t ϕ − 2∇v∇ϕ − vϕ + p∇ϕ − ∂t w, we observe that new functions V and P form a unique solution to the following initial boundary value problem ∂t V − V = F − ∇P ,
div V = 0
in Q,
Local regularity theory of the Navier–Stokes equations
V =0
165
on ∂ Q.
Taking into account the statement of Theorem 2.1 and estimate (2.9), we find ∂t vs,n,Q(1/2) + ∇ 2 vs,n,Q(1/2) + ∇ps,n,Q(1/2) ≤ cA + c∂t ws,n,Q ,
(2.10)
where A = f s,n,Q + vs,n,Q + ∇vs,n,Q + ps,n,Q . So, our task is to evaluate the last term on the right hand side of (2.10). The key point here is the duality of arguments proposed in [56] and [57]. By introducing the new notation u = ∂t w and r = ∂t q, we can derive from the equations for w and q u(·, t) − ∇r(·, t) = 0 div u(·, t) = ∂t v(·, t) · ∇ϕ(·, t) + v(·, t) · ∇∂t ϕ(·, t)
in B,
(2.11)
r(x, t)dx = 0,
u(·, t) = 0
on ∂B.
(2.12)
B
With given g ∈ Ls (B) where s = s/(s − 1), we define a function u˜ as a unique solution to the boundary value problem u˜ − ∇ r˜ = g,
div u˜ = 0
in
B,
(2.13)
r˜ (x)dx = 0,
u˜ = 0
on ∂B.
(2.14)
B
The function r˜ obeys the estimate ˜r s ,B + ∇ r˜ s ,B ≤ cgs ,B
(2.15)
Now, from (2.10)–(2.14), we find u(x, t) · g(x)dx = u(x, t) · (u(x) ˜ − ∇ r˜ (x))dx B
B
=
r˜ (x)div u(x, t)dx B
=
r˜ (x)(∂t v(x, t) · ∇ϕ(x, t) + v(x, t) · ∇∂t ϕ(x, t))dx. B
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Expressing ∂t v via the Navier–Stokes equations, we derive from the previous identity u(x, t) · g(x)dx = r˜ (x)(v(x, t) − ∇p(x, t) + f (x, t)) · ∇ϕ(x, t)dx B
B
+
r˜ (x)v(x, t) · ∇∂t ϕ(x, t)dx. B
It remains to integrate by parts and use estimate (2.15). As a result, we have u(x, t) · g(x)dx ≤ cgs ,B (v(·, t)s,B + ∇v(·, t)s,B + p(·, t)s,B ) B
and thus ∂t ws,n,Q ≤ cA.
Proposition 2.2 is proved.
In the case of boundary regularity, we proceed in a more or less similar way. The only problem is that the above integration by parts leaves non-zero boundary integrals which should be treated with the help of iterations. We refer the reader to papers [46] and [49] for details. P ROPOSITION 2.3. Assume that the condition f ∈ Ls,n (Q+ ) 2,1 1,0 holds and m ∈ ]1, s]. Let a pair of functions v ∈ Wm,n (Q+ ) and p ∈ Wm,n (Q+ ) satisfy 2,1 1,0 + + (2.2) and (2.3). Then v ∈ Ws,n (Q (1/2)) and p ∈ Ws,n (Q (1/2)) and we have the estimate
∂t vs,n,Q+ (1/2) + ∇ 2 vs,n,Q+ (1/2) + ∇ps,n,Q+ (1/2) ≤ c(m, n, s)(f s,n,Q+ + vm,n,Q+ + ∇vm,n,Q+ + pm,n,Q+ ). The difference between interior and boundary issues is that, in the boundary case, we have to start with higher regularity of v and p (probably not only for technical reasons). The following statement is a very useful tool in our approach to study boundary regularity. The reader can find its proof in [49]. P ROPOSITION 2.4. Assume that all conditions of Proposition 2.3 hold and 1 < n ≤ 2,
μ=2−
2 3 − > 0. n s
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Then 1
|v(z) − v(z )| ≤ c(m, n, s)(|x − x | + |t − t | 2 )μ (f s,n,Q+ + vm,n,Q+ + ∇vm,n,Q+ + pm,n,Q+ ) for all z = (x, t) ∈ Q+ (1/4) and for all z = (x , t ) ∈ Q+ (1/4).
3. Interior case: Ladyzhenskaya–Prodi–Serrin condition One of the first theorems on local regularity was proved by J. Serrin in [55], see also [40]. Later it was generalized by M. Struwe in [59]. The following theorem is given in Struwe’s approach. T HEOREM 3.1. Assume that a divergence free function v, defined in Q, satisfies the following three conditions v ∈ L2,∞ (Q),
∇v ∈ L2 (Q),
(3.1)
(−v · ∂t u − v ⊗ v : ∇u + ∇v : ∇u)dz = 0
(3.2)
Q ◦
∞ for all test functions u ∈ C ∞ 0 (Q) = {u ∈ C0 (Q)div u = 0}, and
v ∈ Ls,l (Q),
3 2 + = 1, s l
s > 3.
(3.3)
Then the space-time origin z = 0 is a regular point of v. R EMARK 3.2. (3.3) is called the Ladyzhenskaya–Prodi–Serrin condition. R EMARK 3.3. J. Serrin treated the case 3/s + 2/ l < 1. R EMARK 3.4. Unfortunately, there is a finite gap between functions satisfying the LPScondition and functions with finite energy, i.e., functions satisfying (3.1) only. Indeed, using multiplicative inequalities, one can show, see for instance [26], that (3.1) implies v ∈ Ls,l (Q),
3 2 3 + ≥ . s l 2
To prove Theorem 3.1, both J. Serrin and M. Struwe used the vorticity equation ω = ∇ ∧ v,
∂t ω − ω = div(v ⊗ ω − ω ⊗ v).
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G. Seregin
It is the reason why the pressure is not presented in the statement at all. The limit case s = 3 and l = +∞ was treated in [59] and [33] under the condition of smallness of the corresponding norm. In the same spirit, boundedness in the mixed Lebesgue spaces in condition (3.3) can be replaced with smallness of norms in the mixed weak spaces, see for details [61], [7], and [32]. In all papers cited above, the method is basically the same: careful analysis of the vorticity equation. However, it is not quite clear how to extend this approach up to the boundary. Our version of Theorem 3.1 is as follows. T HEOREM 3.5. Assume that we are given two functions v and p, defined in Q, and v satisfies LPS-condition (3.3). Let, in addition, 1,0 (Q), v ∈ Wm,n
(3.4)
p ∈ Lm,n (Q)
(3.5)
for some 1 < m < s,
1 < n < l,
(3.6)
and v ∈ Ld,r (Q)
(3.7)
with d=
sm , s−m
r=
ln . l−n
Suppose that v and p meet the Navier–Stokes equations in Q in the sense of distributions. Then z = 0 is a regular point of v. Before sketching the proof of Theorem 3.5, we would like to discuss conditions (3.4)– (3.7). They are natural in the following sense. If v has finite energy, i.e., (3.1) holds, then, using known multiplicative inequalities, we find two facts: v ∈ Ld ,r (Q)
(3.8)
with 3 2 3 + ≥
d r 2 and |v||∇v| ∈ Lm ,n (Q)
(3.9)
Local regularity theory of the Navier–Stokes equations
169
with 3 2 + ≥ 4,
m n see for details in [26]. In turn, condition (3.9) would imply (at least for weak solutions 1,0 to initial boundary value problems) that v is of class Wm2,1
,n and p is of class Wm ,n . Certainly, such functions satisfy conditions (3.4)–(3.6). Condition (3.7) is also valid since 3/d + 2/r ≥ 3 > 3/2. Now Theorem 3.5 can be deduced from the following proposition. P ROPOSITION 3.6. Assume that two functions v and p meet conditions (3.3)–(3.7) and satisfy the Navier–Stokes equations in the sense of distributions. There exists a positive number ε = ε(m, n, s, l) such that if vs,l,Q < ε,
(3.10)
then z = 0 is a regular point of v. P ROOF OF T HEOREM 3.5. For s > 3, we can find r > 0 such that vs,l,Q(r) < ε(m, n, s, l). Introducing new scaled functions u(y, s) = rv(ry, r 2 s),
q(y, s) = r 2 p(ry, r 2 s)
and observing that the Navier–Stokes equations are invariant with respect to this scaling and us,l,Q = vs,l,Q(r) , we deduce the statement of Theorem 3.5 from Proposition 3.6. Theorem 3.5 is proved. Regarding the proof of Proposition 3.6, we note that it is based on two facts from the linear theory. These facts generalize statements of Section 2. To formulate the first of them, consider the initial boundary value problem ∂t w + u · ∇w − w = f − ∇q, w=0
div w = 0
on ∂ Q.
in
Q,
(3.11) (3.12)
T HEOREM 3.7. Assume that we are given positive numbers n, m, l, and s, satisfying the conditions 1 < m < s,
1 < n < l,
3 2 + = 1. s l
(3.13)
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G. Seregin
Let, in addition, f ∈ Lm,n (Q).
(3.14)
There exists a positive number ε = ε(m, n, s, l) with the following property. If us,l,Q < ε,
(3.15)
2,1 1,0 then problem (3.11), (3.12) has an unique solution w ∈ Wm,n (Q) and q ∈ Wm,n (Q) satisfying the estimate
∂t wm,n,Q + ∇ 2 wm,n,Q + ∇qm,n,Q ≤ cf m,n,Q .
(3.16)
P ROOF. We start with describing of two important inequalities. The first one is Hölder’s inequality u · ∇wm,n,Q ≤ us,l,Q ∇wd,r,Q < ε∇wd,r,Q
(3.17)
with d=
sm , s −m
r=
ln . l−n
(3.18)
The second estimate comes from the embedding theorem for Sobolev’s spaces with the mixed norm, see [2] and [3], and has the form ∇wd,r,Q ≤ C(m, n, s, l)wW 2,1 (Q) ,
(3.19)
m,n
provided assumptions (3.13) and (3.18) hold. We recall to the reader the corresponding theorems. Let κ1 = 2 −
2 3 3 2 − + + , m n m1 n1
κ2 = 1 −
2 3 3 2 − + + . m n m2 n2
2,1 The space Wm,n (Q) is continuously embedded into space Lm1 ,n1 (Q) if 1 ≤ m ≤ m1 ≤ +∞, 1 ≤ n ≤ n1 ≤ +∞, and κ1 > 0, or 1 ≤ m ≤ m1 < +∞, 1 < n ≤ n1 < +∞, and κ1 = 0. It is continuously embedded into space Wm1,0 2 ,n2 (Q) if 1 ≤ m ≤ m2 ≤ +∞, 1 ≤ n ≤ n2 ≤ +∞, and κ2 > 0, or 1 ≤ m ≤ m2 < +∞, 1 < n ≤ n2 < +∞, and κ2 = 0. So, we have from (3.17)–(3.19)
u · ∇wm,n,Q ≤ εC(m, n, s, l)wW 2,1 (Q) . m,n
The latter estimate makes it possible to introduce the operator 2,1 2,1 2,1 L : Wˆ m,n (Q) → Wˆ m,n (Q) = {u ∈ Wm,n (Q)u = 0 on ∂ Q}
(3.20)
Local regularity theory of the Navier–Stokes equations
171
so that Lw = v, where v is a unique solution to the initial boundary value problem ∂t v + u · ∇w − v = f − ∇q, v=0
div v = 0
in
Q,
on ∂ Q.
2,1 By Theorem 2.1 and by (3.20), the operator L is continuous on Wˆ m,n (Q) and, for sufficiently small ε, is a contraction operator. Theorem 3.7 is proved.
Now we provide a local version of Theorem 3.7. P ROPOSITION 3.8. Let conditions (3.13) hold, τ0 ∈ ]0, 1], and numbers d and r be computed according to (3.18). Assume that we are given functions f ∈ Lm,n (Q(τ0 )), u ∈ Ls,l (Q(τ0 )), p ∈ Lm,n (Q(τ0 )), 1,0 v ∈ Wm,n (Q(τ0 )) ∩ Ld,r (Q(τ0 )) satisfying the equations ∂t v + u · ∇v − v = f − ∇p,
div v = div u = 0
in Q(τ0 ).
Suppose also that us,l,Q(τ0 ) < ε(m, n, s, l),
(3.21)
where ε(m, n, s, l) is the number of Theorem 3.7. 2,1 1,0 Then, for any τ ∈ ]0, τ0 [, v ∈ Wm,n (Q(τ )) and p ∈ Wm,n (Q(τ )) and the following estimate is valid: ∂t vm,n,Q(τ ) + ∇ 2 vm,n,Q(τ ) + ∇pm,n,Q(τ ) ≤ C(τ, τ0 )(f m,n,Q(τ0 ) + εvd,r,Q(τ0 ) + vm,n,Q(τ0 ) + pm,n,Q(τ0 ) + ∇vm,n,Q(τ0 ) ).
(3.22)
The proof of Proposition 3.8 is similar to the proof of Proposition 2.2 and can be a good exercise for the reader. P ROOF OF P ROPOSITION 3.6. Let ε be the number of Theorem 3.7. We apply Proposition 3.8 with u = v, f = 0, τ0 = 1, and τ = 1/2. According to Proposition 3.8, we have 2,1 (Q(1/2)), v ∈ Wm,n
1,0 p ∈ Wm,n (Q(1/2)).
Now we distinguish between two cases m ≥ 3 and m < 3. In the first case, without loss of generality, we may assume that 2,1 (Q(1/2)), v ∈ W3,n
Letting l1 =
2nε , 2−n
1,0 p ∈ W3,n (Q(1/2)),
1 < n < 2.
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G. Seregin
where a parameter ε ∈ ]0, 1[ satisfies the condition n>
2 , 1+ε
we have s1 =
l1 > 2,
3n 3nε > , n(1 + ε) − 2 2(n − 1)
and 3 2 + = 1. s1 l1 Now, fix m1 so that 3n < m1 < s1 , 2(n − 1) and introduce numbers d1 =
s1 m1 , s1 − m1
r1 =
l1 n . l1 − n
By the usual embedding theorem, v ∈ Wm1,0 (Q(1/2)), 1 ,n
p ∈ Lm1 ,n (Q(1/2)).
Now we use the embedding theorem for the Sobolev spaces with a mixed norm. Since 2−
3 2 2 3 2 − + + =2− >0 3 n s1 l1 n
2−
3 2 3 2 3 − + + = > 0, 3 n d1 r1 m1
and
we conclude that v ∈ Ls1 ,l1 (Q(1/2)) and v ∈ Ld1 ,r1 (Q(1/2)). Next, we can find 0 < τ0 ≤ 1/2 so that vs1 ,l1 ,Q(τ0 ) < ε(m1 , n, s1 , l1 ). Hence, all conditions of Proposition 3.8 hold if we replace m with m1 , s with s1 , and l with l1 for u = v and f = 0. This proposition says that v ∈ Wm2,1 1 ,n (Q(τ0 /2)). Since μ=2−
3 2 − > 0, m1 n
Local regularity theory of the Navier–Stokes equations
173
we can state that the function v is Hölder continuous in the completion of Q(ρ) for some 0 < ρ < 1, i.e., z = 0 is a regular point of v. So, we should consider only the case m < 3. Letting m1 = min{2m/(3 − m), 3}, we observe that p ∈ Lm1 ,n (Q(1/2)).
(Q(1/2)), v ∈ Wm1,0 1 ,n Now, for d1 = m1 s/(s − m1 ) and κ ≡2−
2 3 3 2 − + + , m n d1 r
we compute κ = 1/m1 > 0 if m1 < 3 and κ = 2 − 3/m ≥ 1/3 if m1 = 3. This means that v ∈ Ld1 ,r (Q(1/2)). Next, we determine ε1 = ε(m1 , n, s, l) and find 0 < τ∗ ≤ 1/2 such that us,l,Q(τ∗ ) < ε1 . Now, all conditions of Proposition 3.8 are fulfilled if we replace m with m1 , d with d1 , τ0 with τ∗ , and τ with τ∗ /2. If m1 = 3, the proof is finished. If m1 < 3, we should repeat with the above arguments. Obviously, after a finite number k of steps, we arrive at the case mk = 3. Proposition 3.6 is proved. 4. Boundary case: Ladyzhenskaya–Prodi–Serrin condition It is unknown if there exists a boundary version of Theorem 3.1. However, for Theorem 3.5 it is known. T HEOREM 4.1. Assume that we are given two functions v and p having the following properties 2,1 v ∈ Wm,n (Q+ ),
1,0 p ∈ Wm,n (Q+ ),
3 2 + = 1, s l
v ∈ Ls,l (Q+ ),
(4.1)
s > 3,
(4.2)
with 1 < m < s,
1 < n < l.
(4.3)
Let they obey the Navier–Stokes equations ∂t v + v · ∇v − v = −∇p,
div v = 0 in
Q+
(4.4)
and the boundary condition v|x3 =0 = 0.
(4.5)
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G. Seregin
Then z = 0 is a regular point. The experience of previous sections shows that to get a positive result in the boundary case, more assumptions on v and p should be imposed. But as it has been already pointed out, they are not very restrictive and they are valid for energy solutions to initial boundary value problems in domains with sufficiently smooth boundaries. Theorem 4.1 has been proved by Solonnikov in [58] even for curvilinear boundaries but with different assumptions. A very particular case is also presented in [46]. See also papers [62], [63], [8], and [30] in the context of weak solutions to the initial boundary value problem. As it was explained earlier, Theorem 4.1 can be deduced from the proposition below just by scaling. P ROPOSITION 4.2. Suppose that functions v and p satisfy conditions (4.1)–(4.5). There exists a positive constant ε∗ , depending on m, n, s, and l only, such that if vs,l,Q+ < ε∗ ,
(4.6)
then z = 0 is a regular point. In turn, the latter proposition easily follows from a perturbation of Proposition 2.3. P ROPOSITION 4.3. Suppose that we are given four functions v, p, f , and u having properties 2,1 (Q+ (ϑ)), v ∈ Wm,n
u ∈ Ls,l (Q+ (ϑ)),
1,0 p ∈ Wm,n (Q+ (ϑ)),
3 2 + = 1, s l
(4.7)
s > 3,
(4.8)
with 0 < ϑ ≤ 1 and 1 < m < 3 < s,
1 < n < l.
(4.9)
Let f ∈ Lm∗ ,n (Q+ (ϑ))
(4.10)
for some number m∗ satisfying two conditions m < m∗ ≤
2m , 3−m
(4.11)
m∗ < s.
It is also assumed that those functions meet the perturbed Stokes system ∂t v + u · ∇v − v = f − ∇p,
div v = div u = 0 in
Q+ (ϑ)
(4.12)
Local regularity theory of the Navier–Stokes equations
175
and the boundary condition v|x3 =0 = 0.
(4.13)
There exists a positive constant ε, depending on m, n, s, and l only, such that if us,l,Q+ (ϑ) < ε,
(4.14)
1,0 + + then v ∈ Wm2,1 ∗ ,n (Q (ϑ1 )), p ∈ Wm∗ ,n (Q (ϑ1 )) for any 0 < ϑ1 < ϑ and the following estimate is valid:
∂t vm∗ ,n,Q+ (ϑ1 ) + ∇ 2 vm∗ ,n,Q+ (ϑ1 ) + ∇pm∗ ,n,Q+ (ϑ1 ) ≤ C(m, n, s, l, ϑ, ϑ1 , vW 2,1 (Q+ (ϑ)) , pW 1,0 (Q+ (ϑ)) , f m1 ,n,Q+ (ϑ) ). m,n
m,n
We do not prove Proposition 4.3 in this survey paper. It will be published elsewhere. However, we will show how one can deduce Proposition 4.2 from it. P ROOF OF P ROPOSITION 4.2. We let ε∗ (m, n, s, l) = ε(m, n, s, l) if m < 3 and ε∗ (m, n, s, l) = 2005 if m ≥ 3. Here, ε(m, n, s, l) is the number of Proposition 4.3. If m ≥ 3, we apply the following L EMMA 4.4. Let 1 < n < 2, 1 < ϑ ≤ 1 be fixed. Assume that 2,1 v ∈ W3,n (Q+ (ϑ)),
1,0 p ∈ W3,n (Q+ (ϑ)),
f ∈ L∞,n (Q+ (ϑ)), and ∂t v + v · ∇v − v = f − ∇p,
div v = 0
in
Q+ (ϑ),
v|x3 =0 = 0. Then z = 0 is a regular point of v. The proof of Lemma 4.4 is very similar to the corresponding statement for the interior case. So, we may admit that m < 3. Setting m1 = min{2m/(3 − m), 3} and assuming that vs,l,Q+ < ε∗ (m, n, s, l), we observe that, for ϑ = 1, ϑ1 = 1/2, u = v, and f = 0, all conditions of Proposition 4.3 are satisfied and thus (Q+ (1/2)), v ∈ Wm2,1 1 ,n (1)
p ∈ Wm1,0 (Q+ (1/2)). 1 ,n (1)
Now, let ε∗ = ε(m1 , n, s, l) if m1 < 3 and ε∗ = 2005 if m1 = 3. We can find 0 < τ∗ ≤ (1) 1/2 so that vs,l,Q+ (τ∗ ) < ε∗ . If m1 = 3, the proof is over. So, assuming that m1 < 3, we
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G. Seregin
may apply Proposition 4.3, replacing m with m1 and m1 with m2 = min{2m1 /(3 − m1 ), 3}, for ϑ = τ∗ , ϑ1 = τ∗ /2, u = v, and f = 0. It is easy to see that, in a finite number k of steps, we get mk = 3. Proposition 4.2 is proved. 5. Interior case: ε-regularity theory Nonlinearity of the Navier–Stokes system might be a source of singular solutions. It is well known that so-called partial regularity of solutions is a very typical phenomenon for quasilinear parabolic systems. However, since the principal part of the Navier–Stokes system is linear, there is a hope that solutions to initial boundary value problems for the Navier– Stokes equations are smooth. We would like to emphasize that there is a difference between local regularity of solutions to the Navier–Stokes system and global regularity of solutions to the initial boundary value problem for this system. In the first case, we deal with the intrinsic properties of the Navier–Stokes equations. It is known that there is a limitation of regularity at least in time. One can observe that in Serrin’s example, see [55]: v(x, t) = c(t)∇h(x),
1 p(x, t) = −c (t)h(x) − c2 (t)|∇h(x)|2 . 2
Here, h is a harmonic function in ⊂ R3 and c is a function of t ∈ ]0, T [. From the technical point of view, such kind of limitation is caused by the impossibility to evaluate locally both ∂t v and ∇p. While treating solutions to initial boundary value problems, we may hope to derive a global estimate for ∂t v and then to find the pressure from equations. Another important point of local analysis starts from differentiability properties of solutions. A reasonable principle is to take properties from known global existence theorems for initial boundary value problems. Such properties include finiteness of energy, existence of higher derivatives and the associated pressure, the local energy inequality, etc. The fundamental role of the local energy inequality was understood by V. Scheffer in his pioneering papers [41–44]. Later on, in [4], L. Caffarelli, R.-V. Kohn, and L. Nirenberg introduced so-called suitable weak solutions. In their definition, the local energy inequality is the principal part. Before giving a precise definition, we formulate the global existence theorem, as a base. T HEOREM 5.1. Let be a bounded domain in R3 with sufficiently smooth boundary, T is a positive parameter, a ∈ L2 (),
div a = 0,
f ∈ L2 (QT ),
(5.1)
where QT = ×]0, T [. There exists at least one pair of functions v and p, which is a solution to the initial boundary value problem ∂t v + v · ∇v − v = f − ∇p, v|∂×[0,T ] = 0,
div v = 0
in QT ,
(5.2) (5.3)
Local regularity theory of the Navier–Stokes equations
v(x, 0) = a(x)
177
x∈
(5.4)
in the following sense. The velocity v has the finite energy v ∈ L2,∞ (QT ) ∩ W21,0 (QT ).
(5.5)
The function v(x, t) · w(x)dx
t→
(5.6)
is continuous on [0, T ] for any w ∈ L2 (). For any 0 < δ < T , 2,1 v ∈ Wm,n (Qδ,T ),
1,0 p ∈ Wm,n (Qδ,T )
(5.7)
where Qδ,T = ×]δ, T [, with m > 1 and n > 1 satisfying the inequality 3 2 + ≥ 4. m n The Navier–Stokes equations hold a.e. in QT . v(·, t) − a(·)L2 () → 0
(5.8)
as t → +0. The global energy inequality
1 2
t
|v(x, t)|2 dx +
|∇v|2 dxds ≤ 0
1 2
t
|a(x)|2 dx +
f · vdxds 0
(5.9) holds for all t ∈ [0, T ]. The local energy inequality
t ϕ(x, t)|v(x, t)| dx + 2 2
ϕ|∇v|2 dxds 0
t (|v|2 (∂ϕ + ϕ) + v · ∇ϕ(|v|2 + 2p) + 2f · v)dxds
≤ 0
(5.10)
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G. Seregin
holds for a.a. t ∈ [0, t] and for all non-negative cut-off functions ϕ ∈ C0∞ (R3 × R1 ) vanishing in a neighborhood of the hyperplane t = 0. Theorem 5.1 is also true for = R3 , for = R3+ , and for the compliment of smooth bounded domains. For arbitrary domains , the initial boundary value problem (5.2)–(5.4) has at least one so-called weak Leray–Hopf solution v, provided conditions (5.1) are fulfilled, see [28], [22], [23], [25], and [20]. By definition, it possesses properties (5.5), (5.6), (5.8), and (5.9) and satisfies the Navier–Stokes equations in the sense of distributions with divergence free test functions. So the definition of weak Leray–Hopf solutions does not contain the pressure p at all. However, for smooth domains, one can introduce the associated pressure transferring the convective term to the right hand side of the Navier–Stokes equations and using the coercive Lm,n -estimates and uniqueness theorems for the Stokes problem, see for details [26] and [48]. This means that in smooth domains any weak Leray–Hopf solution satisfies (5.7). However, it is unknown whether or not the local energy inequality holds for each weak Leray–Hopf solution. Now we are in a position to discuss the notion of suitable weak solutions. Contrary to weak solutions to the initial boundary value problems, suitable weak solutions (we often use abbreviation s.w.s. below) describe the local properties of the Navier–Stokes equations, although it is not completely true. In fact we have to introduce the pressure which, in a sense, is a substitution of boundary conditions. As it has been already explained, starting properties of the pressure are motivated by consideration of solutions to initial boundary value problems. Historically, V. Scheffer was the first mathematician to consider weak solutions to the initial boundary problems for the Navier–Stokes equations satisfying the local energy inequality. The next crucial step was made by L. Caffarelli, R.-V. Kohn, and L. Nirenberg [4]. They introduced so-called suitable weak solutions, which allow us to forget boundary and initial conditions (up to a certain degree, as it was noted before) and localize the regularity problem for functions satisfying the Navier–Stokes equations. Although the definition of suitable weak solutions, given by L. Caffarelli, R.-V. Kohn, and L. Nirenberg, is the most general one, we prefer to use F.-H. Lin’s definition, see [29], which seems to be more convenient for analysis. Here and below we consider the homogeneous case f = 0. The general case is treated in papers [4], [26], [48], and [49]. D EFINITION 5.2. A pair of functions v and p, defined in the space-time cylinder Q(z0 , R) = B(x0 , R)×]t0 − R 2 , t0 [, is called a suitable weak solution to the Navier–Stokes equations in Q(z0 , R) if the following conditions hold. v ∈ L2,∞ (Q(z0 , R)) ∩ W21,0 (Q(z0 , R)),
(5.11)
p ∈ L 3 (Q(z0 , R)).
(5.12)
2
Local regularity theory of the Navier–Stokes equations
179
The Navier–Stokes equations are fulfilled in Q(z0 , R) in the sense of distributions. Functions v and p satisfy the local energy inequality t
ϕ(x, t)|v(x, t)|2 dx + 2
ϕ|∇v|2 dxds
t0 −R 2 B(x0 ,R)
B(x0 ,R)
t
≤
(|v|2 (∂ϕ + ϕ) + v · ∇ϕ(|v|2 + 2p))dxds
(5.13)
t0 −R 2 B(x0 ,R)
for a.a t ∈ ]t0 − R 2 , t0 [ and for non-negative cut-off functions ϕ ∈ C0∞ (R3 × R1 ) vanishing in a neighborhood of the parabolic boundary ∂ Q(z0 , R) of the cylinder Q(z0 , R). R EMARK 5.3. Let v and p be functions of Theorem 5.1 for f = 0. Let z0 = (x0 , t0 ) be a space-time point with x0 ∈ and t0 ∈ ]0, T ]. We state that there exists R = R(z0 ) such that v and p form a suitable weak solution in the space-time cylinder Q(z0 , R). To justify that we let m = 9/8 and n = 3/2. Then 3/m + 2/n = 4 and thus p ∈ W 91,03 (QT ), which 8,2
means that p ∈ L 9 , 3 (QT ) ⊂ L 3 (QT ). 5 2
2
R EMARK 5.4. Definition 5.2 is invariant to shifts in space-time and to the scaling v λ (x, t) = λv(λx, λ2 t),
p λ (x, t) = λ2 p(λx, λ2 t).
So, without loss of generality, we may assume that our space-time cylinder is unit and centered at the space-time origin, i.e., R = 1 and z0 = 0. According to our opinion, one of the central statements in the interior ε-regularity theory can be formulated as follows. T HEOREM 5.5. Let v and p be a suitable weak solution in Q. There exit universal positive constants ε, ck , k = 0, 1, 2, . . . , with the property: if 3 3 2 |v| + |p| dz < ε,
(5.14)
Q
then, for each k = 0, 1, 2, . . . , the function z = (x, t) → ∇ k v(z) is Hölder continuous in the completion Q(1/2) of the cylinder Q(1/2) and sup
|∇ k v(z)| ≤ ck .
z∈Q(1/2)
Just by shift and scaling, we deduce from Theorem 5.5
(5.15)
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G. Seregin
C OROLLARY 5.6. Let v and p be a suitable weak solution in Q(z0 , R). If 1 R2
3 3 2 |v| + |p| dz < ε,
(5.16)
Q(z0 ,R)
then, for each k = 0, 1, 2, . . . , the function z = (x, t) → ∇ k v(z) is Hölder continuous in Q(z0 , R/2) and ck . R k+1
|∇ k v(z)| ≤
sup z∈Q(z0 ,R/2)
(5.17)
Theorem 5.5 (and an even more general statement) was demonstrated first in [4], and then in [29] and [26]. It seems that an optimal proof of Theorem 5.5 is presented in [15]. We are going to discuss such a proof of a similar theorem for the boundary case in Section 6. Now let us formulate various consequences of Theorem 5.5. To this end let us introduce the following functionals C(r) ≡
1 r2
|v|3 dz, Q(r)
A(r) ≡ ess
1 2 −r
E(r) ≡
1 r
|∇v|2 dz,
H (r) ≡
1 r3
Q(r)
|v(x, t)|2 dx, B(r)
|v|2 dz, Q(r)
sup
D(r) ≡
1 r2
3
|p| 2 dz. Q(r)
All of them are invariant to the scaling mentioned in Remark 5.4. We shall call this scaling by the natural one. P ROPOSITION 5.7. Let v and p be a suitable weak solution in Q. There is a universal positive constant ε1 such that if sup0
0, there is a positive constant ε4 (M) such that if sup0 0, there is a positive constant ε5 (M) such that if sup0
Local regularity theory of the Navier–Stokes equations
181
To the author’s knowledge, Proposition 5.7 was proved in [64], [47], and [49]. Proposition 5.8 is in fact one of the main results of the Caffarelli–Kohn–Nirenberg paper [4]. The point is that it gives the best Hausdorff dimension of the singular set of solutions to initial boundary value problems for the Navier–Stokes equations, see also [9] for the latest developments. Later on, a simpler proof was proposed by F.-H. Lin in [29], see other proofs in [26] by O. Ladyzhenksaya and G. Seregin and in [49] by G. Seregin. Proposition 5.9 was demonstrated in [64] and [53]. G. Tian and Z. Xin proved Propositions 5.10 and 5.11 in [64]. The reader can find more results in the style of Propositions 5.7–5.11 in [6]. All functionals in Propositions 5.7–5.11 do not contain the pressure p. However, reading the proof of those propositions, one can see that the maximum of |v| in a neighborhood of z = 0 is estimated by a constant proportional to D(1). For illustration, we prove Proposition 5.10. P ROOF OF P ROPOSITION 5.10. It is based on four facts. Energy inequality: 1 2 A(R/2) + E(R/2) ≤ c H (R) + C(R) + C 3 (R)D 3 (R)
(5.18)
for any 0 < R ≤ 1; Pressure decay estimate: D(r) ≤ c
2 r ρ D(ρ) + C(ρ) ρ r
(5.19)
for any 0 < r < ρ ≤ 1; Interpolation: 3 4 10 1 3 C(r) ≤ cH (r) 5 |v| dz r3 1 4
(5.20)
Q(r)
for any 0 < r ≤ 1; Multiplicative inequality: 1 r
10
2
|v| 3 dz ≤ cA 3 (r)(E(r) + H (r))
5 3
(5.21)
Q(r)
for any 0 < r ≤ 1. Inequalities (5.20) and (5.21) are elementary ones. The energy inequality (5.18) is derived from the local energy inequality for a special choice of the cut-off function ϕ there. Proof of (5.19) is also simple, see [47]. We let E(r) = E(r) + D(r),
δ = sup H (r). 0
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G. Seregin
For any 0 < ϑ < 1 and 0 < r ≤ 1, we find from (5.18) and (5.19) E(ϑ/2r) ≤ c[δ + C(ϑr) + D(ϑr)] + c[ϑD(r) + 1/ϑ 2 C(r)] ≤ c[ϑD(r) + δ + 1/ϑ 2 C(r)]. Now, (5.20) and (5.21) imply # 1 1 3$ E(ϑ/2r) ≤ c ϑD(r) + δ + 1/ϑ 2 δ 4 M 2 (E(r) + δ) 4 and thus E(ϑ/2r) ≤ cϑE(r) + δf (ϑ, M). Fix ϑ so that cϑ ≤ 1/2 and iterate the latter inequality starting with r = 1. As a result, we find E((ϑ/2)k ) ≤ (1/2)k E(1) + cδf (ϑ, M). Now, we fix k1 so that (1/2)k1 E(1) ≤ cδf (ϑ, M) and let r1 = (ϑ/2)k1 . Thus it is shown E(r1 ) ≤ cδf (ϑ, M). On the other hand, we see that C(r1 ) ≤ cδf1 (ϑ, M) and, therefore, C(r1 ) + D(r1 ) ≤ cδf2 (ϑ, M). Choosing δ sufficiently small, we deduce our statement from Corollary 5.6. Proposition 5.10 is proved. 6. Boundary case: ε-regularity theory In this section we discuss the ε-regularity theory, considering solutions in a neighborhood of the spatial boundary and assuming that the velocity is subject to the homogeneous Dirichlet boundary condition. Treating the Stokes system in Section 2, we noted a difference between interior and boundary cases. Obviously, we should expect some changes, caused by the boundary in the nonlinear theory as well. At first one needs to have a correct definition of suitable weak solutions. It was introduced in [48], see [49] and [50] for extensions and [52] for curvilinear boundaries. To formulate the corresponding definition, the additional notation is required: B + (x0 , R) ≡ {x ∈ B(x0 , R)x = (x , x3 ),
x3 > x03 },
x = (x1 , x2 ) ∈ R2 ,
Local regularity theory of the Navier–Stokes equations
B + (θ ) ≡ B + (0, θ ),
183
B + ≡ B + (1);
(x0 , R) ≡ {x ∈ B(x0 , R)x = (x , x03 )}, (θ ) ≡ (0, θ ),
≡ (1);
Q+ (z0 , R) ≡ B + (x0 , R)×]t0 − R 2 , t0 [, Q+ (θ ) ≡ Q+ (0, θ ), (v)ω ≡
1 |ω|
Q+ ≡ Q+ (1);
[p] (t) ≡
v dz,
z0 = (x0 , t0 )
ω
1 ||
p(x, t) dx.
D EFINITION 6.1. We say that a pair of functions v and p is a suitable weak solution to the Navier–Stokes equations in Q+ (z0 , R) near the boundary (z0 , R)×]t0 − R 2 , t0 ] if the following four conditions hold. The functions v and p have the differentiability properties ⎫ p ∈ L 3 (Q+ (z0 , R)), ⎪ ⎬
v ∈ L2,∞ (Q+ (z0 , R)) ∩ W21,0 (Q+ (z0 , R)), ∇ 2 v ∈ L 9 , 3 (Q+ (z0 , R)),
2
∇p ∈ L 9 , 3 (Q+ (z0 , R)).
8 2
⎪ ⎭
(6.1)
8 2
They meet the Navier–Stokes equations ∂t v + v · ∇v − v = −∇p,
div v = 0
(6.2)
a.e. in Q+ (z0 , R). The function v satisfies the boundary condition on (z0 , R) × [t0 − R 2 , t0 ].
v=0
(6.3)
The functions v and p satisfy the local energy inequality t
|v(x, t)| φ(x, t) dx + 2 2
B + (x0 ,R)
t
|∇v|2 φ dx dt
t0 −R 2 B + (x0 ,R)
≤
|v|2 (∂t φ + φ) + (|v|2 + 2p)v · ∇φ} dx dt
(6.4)
t0 −R 2 B + (x0 ,R)
for a. a. t ∈ [t0 − R 2 , t0 ] and for all non-negative functions φ ∈ C0∞ (R3 × R), vanishing in a neighborhood of ∂ Q(z0 , R).
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R EMARK 6.2. Theorem 5.1 shows that exponents m = 9/8 and n = 3/2 in (6.1) are quite reasonable, see (5.7) and also Remark 5.3. The theorem below can be regarded as the boundary analogue of Theorem 5.5 and Corollary 5.6. T HEOREM 6.3. Given τ ∈ ]0, 1[, there are positive numbers ε0 = ε0 (τ ) and c0 = c0 (τ ) with the following property. Let v and p be an arbitrary suitable weak solution to the Navier–Stokes equations in Q+ (z0 , R) near (z0 , R)×]t0 − R 2 , t0 ]. Suppose that v and p satisfy the additional condition 1 U (z0 , R; v, p) ≡ 2 R
3 3 |v| + |p| 2 dz < ε0 .
(6.5)
Q+ (z0 ,R)
Then, the function v is Hölder continuous in the closure of the set Q+ (z0 , τ R) and sup
z∈Q+ (z0 ,τ R)
|v(z)| ≤
c0 . R
(6.6)
Theorem 6.3 and various versions are proved in [48]–[50]. Here, we are going to sketch the proof of Theorem 6.3, following [50]. We start with L EMMA 6.4. Given θ ∈ ]0, 1[, there exists a positive number ε, depending on θ only, with the following property. For any suitable weak solution v and p to the Navier–Stokes equations in Q+ near ×] − 1, 0], satisfying the additional condition Y1+ (v, p) < ε,
(6.7)
the estimate Yθ+ (v, p) ≤ cθ 2 Y1+ (v, p) 1
(6.8)
is valid. Here, c is a universal positive constant and Yθ+ (v, p) = Yθ+1 (v) + Yθ+2 (p),
Yθ+2 (p) = θ
1 + |Q (θ )|
Yθ+1 (v) =
3
|p − [p]B + (θ) | 2 dz Q+ (θ)
1 + |Q (θ )|
2 3
.
1
|v| dz 3
Q+ (θ)
3
,
Local regularity theory of the Navier–Stokes equations
185
P ROOF. At first consider the case θ < 1/4. The opposite case is much easier. Assume that the statement of the lemma is false. Then a number θ ∈ ]0, 1/4[ and sequences v k and p k of suitable weak solutions in Q+ near ×] − 1, 0] exist such that Y1+ (v k , p k ) = εk → 0
(6.9)
as k → +∞, and Yθ+ (v k , p k ) > cθ 2 εk . 1
(6.10)
The constant c will be chosen later in order to get a contradiction. We introduce new functions uk = v k /εk ,
q k = (p k − [p k ]B + )/εk .
By (6.9) and (6.10), they satisfy the relations Y1+ (uk , q k ) = 1,
(6.11)
Yθ+ (uk , q k ) > cθ 2 , 1
(6.12)
the system ∂t uk + εk uk · ∇uk − uk = −∇q k ,
div uk = 0
in Q+ ,
(6.13)
and the boundary condition uk = 0
on × [−1, 0].
(6.14)
In addition, functions uk and q k have the following differentiability properties uk ∈ L2,∞ (Q+ ) ∩ W21,0 (Q+ ), ∇ 2 uk ∈ L 9 , 3 (Q+ ), 8 2
q k ∈ L 3 (Q+ ), 2
(6.15)
∇q k ∈ L 9 , 3 (Q+ ), 8 2
and the local energy inequality
|u (x, t)| φ(x, t) dx + 2 2
k
B+
≤ B + ×]−1,t[
|∇uk |2 φ dx dt
B + ×]−1,t[
k2
|u | (∂t φ + φ) + (|uk |2 εk + 2q k )uk · ∇φ dx dt
(6.16)
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G. Seregin
holds for a. a. t ∈ ] − 1, 0] and for all non-negative functions φ ∈ C0∞ (R3 × R), vanishing in a neighborhood of ∂ Q. Choosing a cut-off function φ in an appropriate way, we easily derive from (6.16) the estimate uk 2,∞,Q+ (7/8) + ∇uk 2,Q+ (7/8) ≤ c1 .
(6.17)
But then Hölder’s inequality and the known multiplicative inequality, together with (6.17), imply two bounds uk 10 ,Q+ (7/8) ≤ c2
(6.18)
uk · ∇uk 9 , 3 ,Q+ (7/8) ≤ c2 .
(6.19)
3
and 8 2
By a simple modification of Proposition 2.3 for s = m, we have ∂t uk 9 , 3 ,Q+ (5/6) + ∇ 2 uk 9 , 3 ,Q+ (5/6) + ∇q k 9 , 3 ,Q+ (5/6) 8 2 8 2 8 2 ≤ c3 uk 9 , 3 ,Q+ (7/8) + ∇uk 9 , 3 ,Q+ (7/8) 8 2
8 2
+ q k 9 , 3 ,Q+ (7/8) + uk · ∇uk 9 , 3 ,Q+ (7/8) ≤ c3 . 8 2
(6.20)
8 2
Using known compactness arguments, we select subsequences (denoting by uk and q k again) such that in L3 (Q+ ) in L 3 (Q+ ) 2 in L3 (Q+ (5/6)).
uk ! u qk ! q uk → u
(6.21)
Moreover, these limit functions are subject to the relations: Y1+ (u, q) ≤ 1,
[q]B + (t) = 0
for t ∈ ] − 1, 0[,
(6.22) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
u2,∞,Q+ (7/8) + ∇u2,Q+ (7/8) ≤ c1 u 10 ,Q+ (7/8) ≤ c2 3
u · ∇u 9 , 3 ,Q+ (7/8) ≤ c2 8 2
∂t u 9 , 3 ,Q+ (5/6) + ∇ 2 u 9 , 3 ,Q+ (5/6) + ∇q 9 , 3 ,Q+ (5/6) 8 2
8 2
8 2
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ≤ c . ⎭ 3
(6.23)
Local regularity theory of the Navier–Stokes equations
187
It remains to note that the pair u and q satisfies the following boundary value problem: ∂t u − u = −∇q,
div u = 0
in Q+
(6.24)
and u=0
on ×] − 1, 0].
(6.25)
Now we would like to use Propositions 2.3 and 2.4 for f = 0 and s = 18, replacing Q+ with Q+ (5/6). Then the function u is Hölder continuous in Q(1/4) and 1
sup |u(z)| ≤ c4 τ 2
(6.26)
z∈Q(τ )
for any 0 < τ ≤ 1/4. Now we pass to the limit in (6.12) and find with the help of (6.21) and (6.26) the following bound cθ 2 ≤ c4 θ 2 + lim sup Yθ+2 (q k ). 1
1
(6.27)
k→+∞
8 with sufficiently smooth boundary so that Let us fix a domain B 8 ⊂ B + (5/6) B + (4/5) ⊂ B and consider the initial boundary value problem ∂t u1k − u1k + ∇q 1k = −εk uk · ∇uk div u1k = 0 [q 1k ]B8(s) = 0,
%
8 = B×] 8 − (5/6)2 , 0[, in Q
s ∈ ] − (5/6)2 , 0[,
8 on ∂ Q.
u1k = 0
For the solution of this problem the coercive estimate 1k k k ∇q 1k 9 , 3 ,Q 8 + q 9 , 3 ,Q 8 ≤ c5 εk u · ∇u 9 , 3 ,Q 8 8 2
8 2
8 2
is valid. By the embedding theorem and by (6.23), we find q 1k 3 ,Q+ ( 4 ) ≤ c5 εk . 2
(6.28)
5
Introducing u2k = uk − u1k , q 2k = q k − q 1k , we have ∇ 2 u2k 9 , 3 ,Q+ ( 4 ) + ∇u2k 9 , 3 ,Q+ ( 4 ) + ∇q 2k 9 , 3 ,Q+ ( 4 ) 8 2
5
8 2
+ q 9 , 3 ,Q+ ( 4 ) ≤ c6 , 2k
8 2
5
5
8 2
5
(6.29)
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G. Seregin
∂t u2k − u2k = −∇q 2k div u2k = 0 u2k = 0
%
in Q+ (4/5),
(6.30)
on (4/5)×] − (4/5)2 , 0].
(6.31)
Now replacing Q+ with Q+ (4/5) and Q+ (1/2) with Q+ (3/4) and taking s = 18 in Proposition 2.3, we obtain the following estimate ∇q 2k 18, 3 ,Q+ ( 3 ) ≤ c6 . 2
(6.32)
4
Next, using (6.28), we can deduce from (6.27) cθ 2 ≤ c4 θ 2 + lim sup Yθ+2 (q 2k ). 1
1
(6.33)
k→+∞
In turn, Poincaré’s inequality implies
3
θ2 cθ ≤ c4 θ + c7 θ lim sup 5 k→+∞ θ 1 2
1 2
3 2
|∇q | dy 2k
ds −θ 2
2
0
3
.
B + (θ)
Now (6.32) together with Hölder inequality with respect to the spatial variables, leads to the estimate 1 2
cθ ≤ c4 θ
1 2
1
+ c7 θ
lim sup
0 −θ 2
2
1 |∇q | dy 2k 18
ds
7
θ2
k→+∞ 1
1
12
+
|B (θ )|
11 12
3
B + (θ)
1
≤ c4 θ 2 + c7
θ 2 = c8 θ 2 , where c8 is known absolute constant. In order to get a contradiction it is sufficient to let c = 2c8 . Lemma 6.4 is proved. Lemma 6.4 can be iterated in the following way. L EMMA 6.5. Let θ ∈ ]0, 1[ satisfy the condition 1
cθ 6 < 1.
(6.34)
Assume that Y1+ (v, p) < ε. Then Yθ+k−1 (v, p) < ε and Yθ+k (v, p) ≤ θ 3 Y1+ (v, p) for k = 1, 2, . . . . k
P ROOF. We argue by induction on k. For k = 1, this is the statement of Lemma 6.4. Assume now that statements of Lemma 6.5 are valid for s = 1, 2, . . . , k ≥ 2. Our goal is to prove that they are valid for s = k + 1 as well.
Local regularity theory of the Navier–Stokes equations
189
We make the natural scaling v k (y, s) = θ k v(θ k y, θ 2k s),
p k (y, s) = θ 2k p(θ k y, θ 2k s)
for (y, s) ∈ Q+ . It is easy to check that v k and p k form a suitable weak solution in Q+ . Since Y1+ (v k , p k ) = θ k Yθ+k (v, p) < ε, we can derive from Lemma 6.4 and from (6.34) Yθ+ (v k , p k ) ≤ cθ 2 Y1+ (v k , p k ) < θ 3 Y1+ (v k , p k ) 1
1
and therefore Yθ+k+1 (v, p) ≤ θ 3 Yθ+k (v, p). 1
Lemma 6.5 is proved. A direct consequence of Lemma 6.5 and the scaling v R (y, s) = Rv(x0 + Ry, t0 + R 2 s),
p R (y, s) = R 2 p(x0 + Ry, t0 + R 2 s)
is the following statement. L EMMA 6.6. Let θ and ε be as in Lemma 6.5. Let v and p be an arbitrary suitable weak solution to the Navier–Stokes equations in Q+ (z0 , R) near (x0 , R)×]t0 − R 2 , t0 ]. Assume that v and p satisfy the additional condition RY + (z0 , R; v, p) < ε. Then, for k = 1, 2, . . . , k we have Y + (z0 , θ k R; v, p) ≤ θ 3 Y + (z0 , R; v, p). The rest of the proof of Theorem 6.3 is quite routine. One needs to derive interior estimates, see [26], and, then, to “glue” them to boundary ones. The reader can find details in papers [48]–[50]. One can exclude the pressure and formulate statements similar to Propositions 5.7–5.9. On this end, we let 1 1 C + (r) ≡ 2 |v|3 dz, E + (r) ≡ |∇v|2 dz, r r Q+ (r)
A+ (r) ≡ ess
1 2 −r
Q+ (r)
|v(x, t)|2 dx.
sup
B + (r)
P ROPOSITION 6.7. Let v and p be a suitable weak solution in Q+ near ×] − 1, 0[. There is a universal positive constant ε1 such that if sup0
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G. Seregin
P ROPOSITION 6.8. Let v and p be a suitable weak solution in Q+ near ×] − 1, 0[. There is a universal positive constant ε2 such that if sup0
7. Local L3,∞ -solutions Let us go back to the Ladyzhenskaya–Prodi–Serrin condition. Just at the very beginning, we have to impose the additional condition s > 3 in (3.3). Indeed, the reduction of Theorem 3.5 to Proposition 3.6 is possible due to two important properties of Ls,l -norm provided 3/s + 2/ l = 1. This norm is invariant with respect to the natural scaling and Ls,l (Q(r)) → 0 as r → 0 for s > 3 if Ls,l (Q) is finite. If s = 3 and l = +∞, the first property is still valid while the second one is violated. However, we have T HEOREM 7.1. Consider two functions v and p with the following differentiability properties: v ∈ L2,∞ (Q) ∩ W21,0 (Q),
p ∈ L 3 (Q). 2
(7.1)
Assume that they satisfy the Navier–Stokes equations in Q in the sense of distributions. Let, in addition, v3,∞,Q < +∞.
(7.2)
Then the function v is Hölder continuous in the closure of the set Q(1/2). We explain the proof of Theorem 7.1 referring the reader to paper [15] for details. It consists of two big parts: blow-up and backward uniqueness for the heat operator. We start with a blow-up procedure. Suppose that Theorem 7.1 is not true. Without loss of generality, we can assume that z = 0 is a singular point. Just by scaling, it follows from Proposition 5.7 that sup0 ε1 /2 (7.3) Rk Q(Rk )
for all k ∈ N. Now, we may scale our functions so that uk (y, s) = Rk v(Rk y, Rk2 s),
q k (y, s) = Rk2 p(Rk y, Rk2 s).
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191
Let us see what happens if k → +∞. In a sense, see [15] for details, uk → u,
q k → q.
We call u the blow up velocity and q the blow up pressure. They are defined on Q− = R3 ×] − ∞, 0[ and satisfy the Navier–Stokes equations there in the sense of distributions. Let us list properties of u and p: u ∈ L3,∞ (Q− ),
q ∈ L 3 ,∞ (Q− );
(7.4)
2
the pair u and q forms a suitable weak solution to the Navier–Stokes equations on Q(a)
for any a > 0;
uk → u
in
L3 (Q(a))
uk → u
in
C([−a 2 , 0]; L2 (B(a)))
(7.5)
for any a > 0;
(7.6)
for any a > 0.
(7.7)
There are few important consequences of these properties. From (7.6), it follows that u is not trivial. Indeed, by scaling and by (7.3), we have |u|3 dz ≥ ε1 /2.
(7.8)
Q
Next, we state that u(·, 0) = 0.
(7.9)
To justify (7.9), we use our scaling once more
1
1 a
2
|u(x, 0)|2 dx B(a)
≤
1 a
1
2
|u (x, 0) − u(x, 0)| dx k
2
+
B(a)
√ 1 ≤ 1/ auk − uC([−a 2 ,0];L2 (B(a))) + |B| 6
1 a
1
2
|u (x, 0)| dx k
2
B(a)
1
3
|v(x, 0)|3 dx
.
B(Rk a)
It is not difficult to show that v(·, t)3,B(3/4) is bounded for t ∈ [−(3/4)2 , 0], which, together with (7.7), implies (7.9).
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G. Seregin
Now, we are going to show that (7.4) and (7.5) give us bounds of ∇ k u at infinity. Indeed, there exists a number R1 > 1000 such that 0
3 3 |u| + |q| 2 dxdt < ε,
−1002 |x|>R1
where ε is the number of Theorem 5.5. Therefore, we have 3 3 |u| + |q| 2 dxdt < ε Q(z0 ,1)
for any z0 = (x0 , t0 ) with |x0 | > R2 = 2R1 and −T2 = −1000 < t0 ≤ 0. And then Theorem 5.5 implies |∇ k u(z0 )| ≤ ck
(7.10)
for any k = 0, 1, 2, . . . and for any z0 = (x0 , t0 ) with |x0 | > R2 and −T2 < t0 ≤ 0. Now we pass to the second part of the proof: reduction to the backward uniqueness for the heat operator. Let us introduce the vorticity ω of u, i.e., ω = ∇ ∧ u. The function ω meets the equation ∂t ω + uk ω,k − ωk u,k − ω = 0
in (R3 \ B(R2 ))×] − T2 , 0].
By (7.10), there is a constant M > 0 such that the function ω satisfies two inequalities |∂t ω − ω| ≤ M(|ω| + |∇ω|)
(7.11)
|ω| ≤ M < +∞
(7.12)
and
in the set (R3 \ B(R2 ))×] − T2 , 0]. Moreover, from (7.9), it follows that ω(x, 0) = 0,
|x| > R2 .
(7.13)
As it is shown in [13], conditions (7.11)–(7.13) say that ω(x, t) = 0
if
|x| > R2
and
− T2 < t ≤ 0.
At this point the so-called unique continuation through spatial boundaries works, see [11] and [15]. As a result we have ω(x, t) = 0 for x ∈ R3 and −T2 < t ≤ 0. However, since u is a divergence free field, u(·, t) is a harmonic function in R3 with the finite L3 -norm for a.a. t ∈ ] − T2 , 0]. Therefore, u(·, t) = 0 for a.a. t ∈ ] − T2 , 0] which is in a contradiction with (7.8). This completes our proof of Theorem 7.1.
Local regularity theory of the Navier–Stokes equations
193
In the case of boundary regularity, we also can prove a result similar to Theorem 7.1. The main result of the paper [51] is as follows. T HEOREM 7.2. Let a pair of functions v and p has the following differentiability properties: v ∈ L2,∞ (Q+ ) ∩ W21,0 (Q+ ) ∩ W 92,13 (Q+ ),
p ∈ W 91,03 (Q+ ).
8,2
8,2
(7.14)
Suppose that v and p satisfy the Navier–Stokes equations ∂t v + v · ∇v − v = −∇p,
div v = 0
(7.15)
− 1 ≤ t ≤ 0.
(7.16)
in Q+ and the boundary condition x3 = 0 and
v(x, t) = 0, Assume, in addition, that v ∈ L3,∞ (Q+ ).
(7.17)
Then v is Hölder continuous in the closure of the set Q+ (1/2). Conceptually, the proof of this theorem is the same as in the interior case. We also assume that the statement of Theorem 7.2 is false and z = 0 is a singular point. Proposition 6.7 says that, in this case, there exists a sequence Rk ∈ ]0, 1[ such that Rk → 0 as k → +∞ and 1 + C (Rk ) ≡ 2 |v|3 dz > ε1 /2 (7.18) Rk Q+ (Rk )
for all k ∈ N. Next, we do blow up with the help of scaling and the sequence Rk . As a result, we arrive at the blow up velocity u and the blow up pressure q. They are defined on R3+ ×] − ∞, 0[ and have the following properties: u ∈ L3,∞ (R3+ ×] − ∞, 0[);
(7.19)
u and q form a suitable weak solution to the Navier–Stokes equations in Q+ (a)
near (a)×] − a 2 , 0[
uk → u
in L3 (Q+ (a))
uk → u
in
for any a > 0;
for any a > 0;
C([−a 2 , 0]; L2 (B + (a)))
for any a > 0.
(7.20) (7.21) (7.22)
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G. Seregin
In contrary to the interior case, we have no finite global (in space) norm of the blow up pressure q anymore (compare (7.4) and (7.19)). In the previous case, such a norm gave us bounds (7.10), which is an important condition of the backward uniqueness theorem. In paper [51] various decay estimates for the pressure q were proved, see Section 2 there. They make it possible to show that, for any h > 0, u and ∇u are bounded on sets of the form (R3+ + hi3 ) × [−50, 0], where i3 = (0, 0, 1). Similar to the interior case, conditions (7.18), (7.21), and (7.22) imply that u is not trivial |u|3 dz ≥ ε1 /2
(7.23)
Q+
and the blow up flow stops at the last moment of time, i.e.: u(·, 0) = 0
(7.24)
in R3+ . Collecting all above facts, we state that the vorticity ω = ∇ ∧ u satisfies the relations: |∂t ω − ω| ≤ M(|ω| + |∇ω|),
|ω| ≤ M
on the set (R3+ + hi3 ) × [−50, 0] for some M > 0, and ω(·, 0) = 0
in
R3+ .
In [14] the backward uniqueness result for the heat operator in a half space was proved. Hence those three conditions imply ω=0
in (R3+ + hi3 ) × [−50, 0].
Since h is taken arbitrarily, the latter means that ω=0
in R3+ × [−50, 0].
Hence, for a.a. t ∈ [−50, 0], u is a harmonic function, which satisfies the boundary condition u(x, t) = 0 if x3 = 0. But, for a.a. t ∈ [−50, 0], L3 -norm of u over R3+ is finite. This leads to the conclusion that u(·, t) = 0 in R3+ for the same t. 8. Other local regularity results In this section, we would like to mention some other issues on local regularity. In the author’s opinion, they are of great interest in the theory of the Navier–Stokes equations. We start with a local version of an interesting result proved by P. Constantin [10]. It can be formulated as follows.
Local regularity theory of the Navier–Stokes equations
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L EMMA 8.1. Assume that we are given three sufficiently smooth functions u, ω, and f . They are defined in QT = ×]0, T [, takes values into R3 , and satisfy the equation ∂t ω + uj ω,j − ω = f
in QT .
(8.1)
Then, for any δ ∈ ]0, T [, for any , and for any β > 2/3, two quantities ω1,∞,Q δ,T and Q δ,T
4
|∇ω| 3 dz β ln (1 + |ω|2 )
are bounded by a constant depending only on β, δ, T , , and by norms f 1,QT , ω2,QT , and u2,QT . In fact, equation (8.1) models the vorticity equation if we let ω = ∇ ∧ v, u = v, and f = ωj v,j . Since we do not know whether or not v is sufficiently smooth, our estimates will be of a priori type. To justify them, one can consider initial boundary value problem for the Navier–Stokes equations with a suitable regularization of the convective term, see details in [10]. The proof of Lemma 8.1 is elegant and deserves to be outlined here. The trick is the change of the unknown function G(x, t) = g(ω(x, t)), where a function g : R3 → R is sufficiently smooth. Direct calculations show that the new function G satisfies the equation ∂t G + uj G,j − G +
∂ 2g ∂g ωk,i ωl,i = fk . ∂yk ∂yl ∂yk
If the function g is non-negative, possesses bounded gradient and positively defined Hessian, then certain information about differentiability properties of ω can be extracted from that equation. We let g(y) = g0 (|y|),
y ∈ R3 ,
where r g0 (r) =
s ds
0
1 1
0
(1 + t 2 ) 2
lnγ (1 + t 2 )
dt
and γ > 1. It easy to check that the function g has the following properties: +∞ |∇g(y)| < A0 (γ ) = 0
1 1 2
(1 + t 2 ) lnγ (1 + t 2 )
dt < +∞,
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G. Seregin
c1 (γ )|y| − c2 (γ ) < g(y) < A0 (γ )|y|, ∂ 2g |ξ |2 . (y)ξi ξj ≥ 1 ∂yi ∂yj (1 + |y|2 ) 2 lnγ (2 + |y|2 ) To end up with the proof of the lemma, it is enough to multiply the equation for G by a suitable cut-off function and integrate it over the domain ×]0, t[ with arbitrary fixed t ∈ ]0, T ]. In papers [16–18], and [60], J. Frehse, M. Ružiˇcka, and M. Struwe deal with the stationary Navier–Stokes equations on R5 . In this case the equations are invariant with respect to the scaling which is very similar to the scaling of the three-dimensional non-stationary Navier–Stokes equations (this is the reason to study such a case). The authors proved many interesting local regularity results, which can be found by the reader in the papers cited above. We also would like to mention some other local regularity results on axisymmetric flows, or even in terms of one velocity component or in terms of the pressure, see for example papers [38], [39], [5], and [36]. At the end of this section, we go back to Problem 1 formulated in Introduction. We may look for singular solutions in the self-similar form
v(x, t) = λ(t)V (λ(t)x),
p(x, t) = λ2 (t)P (λ(t)x),
z = (x, t) ∈ Q, (8.2)
√ where λ(t) = 1/ −2κt and κ is a given positive parameter. The function V obeys the Leray equations κ(V + y · ∇V ) + V · ∇V − V = −∇P ,
divV = 0
(8.3)
in R3 . In order to construct singular solutions of form (8.2) to equations (1.1), we seek non-trivial solutions V to equations (8.3). Weak formulation of (8.3) is as follows: find 1 V ∈ W2,loc (R3 ) such that (κ(V · U + yi V,i · U ) − V ⊗ V : ∇U + ∇V : ∇U )dy = 0
(8.4)
R3
for any smooth compactly supported and divergence free function U . Existence or non-existence of non-trivial solutions to the Leray equations depends on the decay of V at infinity. For example, any constant C satisfies to (8.3) for a suitable choice of “pressure” P . However, such a solution generates v which has infinite energy in Q. More precisely, 1
v(·, t)2,B = |B| 2 λ(t)|C| → +∞
Local regularity theory of the Navier–Stokes equations
197
as t → 0 − 0. The first significant contribution to that problem has been made by J. Neˇcas, M. Ruziˇcka, and V. Šverák in [37]. They showed V ∈ L3 (R3 ) ⇒ V ≡ 0
R3 .
in
(8.5)
Although nowadays this statement follows from Theorem 7.1, their proof was conceptually used to extend statement (8.5). Reduction to Theorem 7.1 can be done as follows. First, we recover P in the Leray equations: 1 1 P (y) = − |V (y)|2 + 3 4π
k(y − y ) : V (y ) ⊗ V (y )dy ,
R3
where k(y) = ∇ 2 (1/|y|). Second, we extend v and p to the whole Q− = R3 ×] − ∞, 0[ with the help of (8.2) in the natural way. These functions are locally smooth and v ∈ L3,∞ (Q− ),
p ∈ L 3 ,∞ (Q− ). 2
Using the local energy inequality, we show that v has finite energy in Q and thus all conditions of Theorem 7.1 are satisfied. This means that z = 0 is a regular point of v. That is possible only if V is identically zero. In the spirit of the Neˇcas–Ruziˇcka–Šverák approach, T.-P. Tsai proved the following statements: V ∈ Lq (R3 ) ⇒ V ≡ 0
in
R3
(8.6)
if 3 ≤ q < +∞, and V ∈ L∞ (R3 ) ⇒ V ≡ constant
in
R3 .
In turn, as T.-P. Tsai shows, it follows from (8.6) that v ∈ L2,∞ (Q) ∩ W21,0 (Q) ⇒ V ≡ 0
in
R3 .
Roughly speaking, the latter says that there is no non-trivial self-similar solutions of form (8.2) with finite energy. That is a remarkable result.
Acknowledgement The work was supported by the Alexander von Humboldt Foundation, by the RFFI grant 05-01-00941-a, and by the CRDF grant RU-M1-2596-ST-04.
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References [1] H. Beirao da Veiga, On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip or non-slip boundary conditions, Comm. Pure Appl. Math. 58 (2005) 552–577. [2] O.V. Besov, V.P. Il’in and S.M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow (1975). Transl.: Vol. 1, 2, J.Wiley & Sons (1978). [3] Ya.S. Bugrov, Function spaces with mixed norm, Izv. AN SSSR 35 (1971) 1137–1158. Transl.: Math. USSRIzv. 5 (1971) 1145–1167. [4] L. Caffarelli, R.-V. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. XXXV (1982) 771–831. [5] D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier–Stokes equations, Math. Z. 239 (2002) 645–671. [6] Z.-M. Chen and W.G. Price, Morrey space techniques applied to the interior regularity problem of the Navier– Stokes equations, Nonlinearity 14 (6) (2001) 1453–1472. [7] Z.-M. Chen and W.G. Price, Blow-up rate estimates for weak solutions of the Navier–Stokes equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001) 2625–2642. [8] H.L. Choe, Boundary regularity of weak solutions of the Navier–Stokes equations, J. Differential Equations 149 (1998) 211–247. [9] H.L. Choe and J.L. Lewis, On the singular set in the Navier–Stokes equations, J. Functional Anal. 175 (2000) 348–369. [10] P. Constantin, Navier–Stokes equations and area of interfaces, Commun. Math. Phys. 129 (1990) 241–266. [11] L. Escauriaza and F.J. Fernández, Unique continuation for parabolic operators, Ark. Mat. 41 (2003) 35–60. [12] L. Escauriaza, G. Seregin and V. Šverák, On backward uniqueness for parabolic equations, Zap. Nauchn. Seminar, POMI 288 (2002) 100–103. [13] L. Escauriaza, G. Seregin and V. Šverák, On backward uniqueness for parabolic equations, Arch. Rational Mech. Anal. 169 (2) (2003) 147–157. [14] L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for the heat operator in half space, Algebra and Analyis 15 (1) (2003) 201–214. [15] L. Escauriaza, G. Seregin and V. Šverák, L3,∞ -Solutions to the Navier–Stokes equations and backward uniqueness, Uspekhi Matematicheskih Nauk 58 (2) (350) 3–44. English translation in Russian Mathematical Surveys 58 (2) (2003) 211–250. [16] J. Frehse and M. Ružiˇcka, Regularity for the stationary Navier–Stokes equations in bounded domains, Arch. Rational Mech. Anal. 128 (1994) 361–380. [17] J. Frehse and M. Ružiˇcka, On the regularity of the stationary Navier–Stokes equations, Ann. Scu. Norm. Pisa 21 (1994) 63–95. [18] J. Frehse and M. Ružiˇcka, Existence of regular solutions to the stationary Navier–Stokes equations, Math. Ann. 302 (1995) 699–717. [19] S. Friedlander and N. Pavlovich, Remarks concerning modified Navier–Stokes equations, Discrete and Contin. Dynamical Sys. 10 (2004) 269–288. [20] G.P. Galdi, An introduction to the Navier–Stokes initial-boundary value problem, in: Fundamental Directions in Mathematical Fluid Mechanics, G.P. Galdi, ed., Birkhäuser, Basel, Boston, Berlin (2000), 1–70. [21] Y. Giga and H. Sohr, Abstract Lp -estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal. 102 (1991) 72–94. [22] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachrichten 4 (1950–51) 213–231. [23] O.A. Ladyzhenskaya, Mathematical Problems of the Dynamics of Viscous Incompressible Fluids, Fizmatgiz, Moscow (1961). English translation, Gordon and Breach, New York–London (1969). [24] O.A. Ladyzhenskaya, On nonlinear problems of continuum mechanics, Proc. Internat. Congr. Math. (Moscow, 1966), Nauka, Moscow (1968), 560–573. English transl. in Amer. Math. Soc. Transl. (2), 70 (1968). [25] O.A. Ladyzhenskaya, Mathematical Problems of the Dynamics of Viscous Incompressible Fluids, 2nd edition, Nauka, Moscow (1970). [26] O.A. Ladyzhenskaya and G.A. Seregin, On partial regularity of suitable weak solutions to the threedimensional Navier–Stokes equations, J. Math. Fluid Mech. 1 (1999) 356–387.
Local regularity theory of the Navier–Stokes equations
199
[27] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uralt’seva, Linear and Quasi-Linear Equations of Parabolic Type, Moscow (1967). English translation, American Math. Soc., Providence, 1968. [28] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934) 193–248. [29] F.-H. Lin, A new proof of the Caffarelly–Kohn–Nirenberg theorem, Comm. Pure Appl. Math. 51 (3) (1998) 241–257. [30] K. Kang, On boundary regularity of the Navier–Stokes equations, Comm. Partial Differential Equations 29 (7–8) (2004) 955–987. [31] N.H. Katz and N. Pavlovich, A cheap Caffarelli–Kohn–Nirenberg inequality for the Navier–Stokes equation with hyper-dissipation, GAFA 12 (2002) 355–379. [32] H. Kim and H. Kozono, Interior regularity criteria in weak spaces for the Navier–Stokes equations, Manuscripta Math. 115 (2004) 85–100. [33] H. Kozono, Removable singularities of weak solutions to the Navier–Stokes equations, Communications in Partial Differential Equations 23 (1998) 949–966. [34] J. Malek, J. Necas, M. Rokyta and M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDE’s, Applied Mathematics and Mathematical Computation 13, Chapman and Hall/CRC (1996). [35] P. Maremonti and V.A. Solonnikov, On the estimate of solutions of evolution Stokes problem in anisotropic Sobolev spaces with a mixed norm, Zap. Nauchn. Sem. LOMI 223 (1994) 124–150. [36] J. Neˇcas and J. Neústupa, New condition for local regularity of a suitable weak solution to the Navier–Stokes equation, J. Math. Fluid Mech. [37] J. Neˇcas, M. Ruziˇcka and V. Šverák, On Leray’s self-similar solutions of the Navier–Stokes equations, Acta Math. 176 (1996) 283–294. [38] J. Neustupa and P. Penel, Regularity of a suitable weak solution to the Navier–Stokes equations as a consequence of regularity of one velocity component, in: Nonlinear Applied Analysis, H. Beirao da Veiga, A. Sequeira and J. Videman, eds, Plenum Press, New York (1999), 391–402. [39] J. Neustupa and M. Pokorny, An interior regularity criterion for an axially symmetric suitable weak solution to the Navier–Stokes equations, J. Math. Fluid Mech. 2 (2000) 381–399. [40] T. Ohyama, Interior regularity of weak solutions of the time-dependent Navier–Stokes equation, Proc. Japan Acad. 36 (1960) 273–277. [41] V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations, Pacific J. Math. 66 (1976) 535– 552. [42] V. Scheffer, Hausdorff measure and the Navier–Stokes equations, Commun. Math. Phys. 55 (1977) 97–112. [43] V. Scheffer, The Navier–Stokes equations in a bounded domain, Commun. Math. Phys. 73 (1980) 1–42. [44] V. Scheffer, Boundary regularity for the Navier–Stokes equations in a half-space, Commun. Math. Phys. 85 (1982) 275–299. [45] G.A. Seregin, Interior regularity for solutions to the modified Navier–Stokes equations, J. of Math. Fluid Mech. 1 (3) (1999) 235–281. [46] G.A. Seregin, Some estimates near the boundary for solutions to the non-stationary linearized Navier– Stokes equations, Zapiski Nauchn. Seminar, POMI 271 (2000) 204–223. [47] G.A. Seregin, On the number of singular points of weak solutions to the Navier–Stokes equations, Comm. Pure Appl. Math. 54 (8) (2001) 1019–1028. [48] G.A. Seregin, Local regularity of suitable weak solutions to the Navier–Stokes equations near the boundary, J. Math. Fluid Mech. 4 (1) (2002) 1–29. [49] G.A. Seregin, Differentiability properties of weak solutions to the Navier–Stokes equations, Algebra and Analysis 14 (1) (2002) 193–237. [50] G.A. Seregin, Remarks on regularity of weak solutions to the Navier–Stokes equations near the boundary, Zapiski Nauchn. Seminar, POMI 295 (2003) 168–179. [51] G.A. Seregin, On smoothness of L3,∞ -solutions to the Navier–Stokes equations up to boundary, Mathematische Annalen 332 (2005) 219–238. [52] G.A. Seregin, T.N. Shilkin and V.A. Solonnikov, Boundary partial for the Navier–Stokes equations, Zapiski Nauchn. Seminar, POMI 310 (2004) 158–190. [53] G.A. Seregin and V. Šverák, On solutions to the Navier–Stokes equations with lower bounds on pressure, Arch. Rational Mech. Anal. 163 (2002) 65–86.
200
G. Seregin
[54] G. Seregin and V. Šverák, The Navier–Stokes equations and backward uniqueness, Nonlinear Problems in Mathematical Physics II, In Honor of Professor O.A. Ladyzhenskaya, International Mathematical Series II (2002), 359–370. [55] J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 9 (1962) 187–195. [56] V.A. Solonnikov, Estimates of solutions to the linearized system of the Navier–Stokes equations, Trudy of the Steklov Math. Inst. LXX (1964) 213–317. [57] V.A. Solonnikov, Estimates of solutions to the non-stationary Navier–Stokes system, Zapiski Nauchn. Seminar, LOMI 28 (1973) 153–231. [58] V.A. Solonnikov, Estimates of solutions of the Stokes equations in S.L. Sobolev space with a mixed norm, Zapiski Nauchn. Seminar, POMI 288 (2002) 204–231. [59] M. Struwe, On partial regularity results for the Navier–Stokes equations, Comm. Pure Appl. Math. 41 (4) (1988) 437–458. [60] M. Struwe, Regular solutions of the stationary Navier–Stokes equations on R5 , Math. Ann. 302 (1995) 719–741. [61] S. Takahashi, On interior regularity criteria for weak solutions of the Navier–Stokes equations, Manuscripta Math. 69 (1990) 237–254. [62] S. Takahashi, On a regularity criterion up to the boundary for weak solutions of the Navier–Stokes equations, Commun. Partial Differential Equations 17 (1992) 261–285. [63] S. Takahashi, Erratum to: “On a regularity criterion up to the boundary for weak solutions of the Navier– Stokes equations”, Commun. Partial Differential Equations 19 (1994) 1015–1017. [64] G. Tian and Z. Xin, Gradient estimation on Navier–Stokes equations, Comm. Anal. Geom. 7 (2) (1999) 221–257. [65] T.-P. Tsai, On Lerays self-similar solutions of the Navier–Stokes equations satisfying local energy estimates, Arch. Rational Mech. Anal. 143 (1998) 29–51.
CHAPTER 5
On the influence of the Earth’s Rotation on Geophysical Flows Isabelle Gallagher Institut de Mathématiques de Jussieu UMR 7586, Université Paris VII, 175, rue du Chevaleret, 75013 Paris, France
and Laure Saint-Raymond Laboratoire J.-L. Lions UMR 7598, Université Paris VI, 175, rue du Chevaleret, 75013 Paris, France
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1. Modelling geophysical flows . . . . . . . . . . . . . . 1.1. Physical background . . . . . . . . . . . . . . . 1.2. Mathematical modelling . . . . . . . . . . . . . 2. A simplified model for midlatitudes . . . . . . . . . . 2.1. Introduction . . . . . . . . . . . . . . . . . . . . 2.2. Statement of the main results . . . . . . . . . . . 2.3. Uniform existence . . . . . . . . . . . . . . . . . 2.4. Weak asymptotics . . . . . . . . . . . . . . . . . 2.5. Strong asymptotics . . . . . . . . . . . . . . . . 2.6. References and remarks . . . . . . . . . . . . . . 3. Taking into account spatial variations at midlatitudes 3.1. Introduction . . . . . . . . . . . . . . . . . . . . 3.2. Statement of the main results . . . . . . . . . . . 3.3. Weak asymptotics . . . . . . . . . . . . . . . . . 3.4. Strong solutions . . . . . . . . . . . . . . . . . . 3.5. References and remarks . . . . . . . . . . . . . . 4. The tropics . . . . . . . . . . . . . . . . . . . . . . . 4.1. Introduction . . . . . . . . . . . . . . . . . . . . 4.2. Statement of the main results . . . . . . . . . . . 4.3. Weak asymptotics . . . . . . . . . . . . . . . . .
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202 4.4. Strong asymptotics 4.5. A hybrid result . . . Acknowledgement . . . . . References . . . . . . . . .
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On the influence of the Earth’s rotation on geophysical flows
203
Introduction The aim of this survey is to describe the influence of the Earth’s rotation on geophysical flows, both from a physical and a mathematical point of view. In Section 1 we gather from the physical literature the main pieces of information concerning the physical understanding of oceanic and atmospheric flows. For the scales considered, i.e., on domains extending over many thousands of kilometers, the forces with dominating influence are the gravity and the Coriolis force. The question is therefore to understand how they counterbalance each other to impose the so-called geostrophic constraint on the mean motion, and to describe the oscillations which are generated around this geostrophic equilibrium. The main equations are then introduced, along with the approximations commonly used by Physicists. The rest of the survey is devoted to the mathematical study of those equations. At mid-latitudes, on “small” geographical zones, the variations of the Coriolis force due to the curvature of the Earth are usually neglected, which leads to a problem of singular perturbation with constant coefficients. The study of that problem is the object of Section 2, which consists in the recollection of rather classical mathematical results and the methods leading to them. We are therefore interested in the wellposedness of the three dimensional Navier–Stokes system, penalized by a constant-coefficient Coriolis force, as well as in the asymptotics of the solutions as the amplitude of the force becomes large. We focus on two types of boundary conditions, which lead to two very different types of convergence results. In the case when the equations are set in R3 , we exhibit an interesting dispersive behaviour for the Coriolis operator which enables one to deduce a strong convergence result towards a vector field satisfying the two dimensional Navier–Stokes system. In the periodic case, dispersion cannot hold; it is replaced by a highly oscillatory behaviour, where the oscillations are linked to the eigenvalues of the Coriolis operator. Once those oscillations have been filtered out, a strong convergence result can also be proved. In both situations (the whole space case and the periodic case), the global existence of smooth solutions for a large enough rotation is also proved, using the special structure of the limiting system in each case. References to more general, constant coefficient situations are given at the end of Section 2. A first step in order to a get a more realistic description, is to take into account the geometry of the Earth (variations of the local vertical component of the Earth rotation). Section 3 is therefore devoted to the study of the three dimensional Navier–Stokes system with a variable Coriolis force. We assume that the direction of the force is constant (taking into account only the vertical component of the Earth’s rotation), and that its amplitude depends on the latitude only (and does not vanish). The price to pay is that the analysis can no longer be as precise as in the constant case, and in particular we have no way in general of describing precisely the waves generated by such a variable-coefficient rotation. As in Section 2, the questions of the uniform existence of weak or strong solutions are addressed, and we study their asymptotic behaviour as the amplitude of the rotation goes to infinity. In Section 4 we focus on equatorial, oceanic flows. In view of the typical horizontal and vertical length scales, it is relevant to consider in a first approximation a two dimensional model with free surface, known as the shallow-water model, supplemented with the Coriolis force. In such an approximation all the vertical oscillations are neglected; this (un-
204
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justified) simplification seems to be nevertheless consistent with experimental measures. The question here is then to understand the combination of the effects due to the free surface, and of the effects due to the variations of the Coriolis force. Contrary to Section 3, the particularity of such flows is that the Coriolis force vanishes at the equator. Note that, for the sake of simplicity, we will not discuss the effects of the interaction with the boundaries, describing neither the vertical boundary layers, known as Ekman layers, nor the lateral boundary layers, known as Munk and Stommel layers. We indeed consider a purely horizontal model, assume periodicity with respect to the longitude (omitting the stopping conditions on the continents) and infinite domain for the latitude (using the exponential decay of the equatorial waves to neglect the boundary). As in the previous sections, the questions addressed are first to solve this system, and then to understand the asymptotic behaviour of the solutions. Using the betaplane approximation of the Coriolis force, we are able to carry out computations further than in the abstract case studied in Section 3. In particular we recover rigorously the well-known trapping of the equatorial waves.
1. Modelling geophysical flows Section 1 of this survey is essentially descriptive, it aims at familiarizing the reader with the basic notions of geophysics, both from the experimental and the theoretical points of view. In the first part, we collect from the books of J. Pedlosky [50] and A.E. Gill [27] the main pieces of information concerning the physical understanding of the oceanic and atmospheric flows. This understanding is based upon a comparison between the orders of magnitude of the various measurable physical parameters. A heuristic study allows then to separate the mean flows on large time scales (which obey some strong constraint, called geostrophic equilibrium) from the deviations consisting of fast oscillations which can be classified. In the second part of Section 1 we introduce the fundamental mathematical models which should allow in the sequel to describe systematically the observed qualitative features of the geophysical flows. This formalism lies essentially on the classical fluid mechanics theory. The main points to be considered are the occurrence of the Coriolis force, and the determination of relevant boundary conditions. We will also introduce simplified models (which are expected to provide a good approximation of the fundamental ones under some conditions) to be used to analyze mathematically some precise phenomenon.
1.1. Physical background In a first approximation the atmosphere and oceans rotate with the earth with small but significant deviations which we, also rotating with the earth, identify as winds and currents. It is useful to recognize explicitly that the interesting motions are small departures from solid-body rotation by describing the motions in a rotating coordinate frame which kinematically eliminates the rigid rotation. Since such a rotating frame is an accelerating rather
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Fig. 1. Description of a thin spherical layer of fluid.
than an inertial frame, certain well-known forces will be sensed, namely the centrifugal force and the subtle and important Coriolis force. Before discussing further the effects of rotation, let us introduce some basic notation. Both in the case of the atmosphere and of oceans, the situation to be considered is that of a thin layer of fluid close to the earth’s surface. It appears therefore that the direction which is orthogonal to the earth’s surface, i.e. radial in the spherical approximation, is somewhat special. In the sequel, it is called “vertical”, and is denoted x3 . In this direction, the length scales are characterized by the parameter D. Conversely, we call “horizontal” and denote by the subscript h the vector components parallel to the earth’s surface. More precisely, we use generally the notations x1 and x2 respectively for the eastward and northward directions. The corresponding length scales are characterized by L. The coordinates considered here and depicted on Figure 1 are therefore (i) neither associated with an inertial frame because of the rotation of the earth; (ii) nor Cartesian coordinates because of the curvature of the earth. These facts have of course important repercussions on the dynamics that are naturally taken into account in the heuristic description and will be discussed in a more formal way in the second part of Section 1. 1.1.1. Geostrophic and hydrostatic approximations The gravitational force A first force with dominating influence is gravity. In the absence of relative motion, it must be balanced by the pressure p, so that the pressure is given by the hydrostatic law: ρ = ρ0 (x3 ),
p = p0 (x3 ),
with
∂p0 = −ρ0 g, ∂x3
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where ρ is the density of the fluid and g the gravitational acceleration. Note that we consider in this text atmospheric or oceanic flows, that are motions occurring in a thin layer of fluid close to the surface of the earth, so that we can assume that the gravitational acceleration is a constant g = 9.8 m s−2 . It actually comes out that the vertical distribution of density ρ0 (x3 ) in both the atmosphere and the oceans is almost always gravitationally stable, meaning that heavy fluid underlies lighter fluid. Such a stable stratification implies in particular that motion parallel to the local direction of gravity is inhibited and this constraint tends to produce large scale motions which are nearly horizontal. A measure of this stratification is given by the Burger number S=g
ρ D , ρ 42 L2
(1.1.1)
where ρ/ρ is a characteristic density-difference ratio for the fluid over its vertical scale of motion D, while L is its horizontal scale and is the angular speed of rotation of the earth. The nondimensional parameter S may be written in terms of the ratio of length scales, S=
LD L
2 ,
where the length LD is called the Rossby deformation radius. Figure 2 shows a typical height profile of density in the atmosphere: the density decrease indicates gravitational stability of vertically displaced elements even if the compressibility of air weakens this stability. Major atmospheric phenomena have a characteristic vertical scale D ∼ 10 km, while L ∼ 1000 km. For such phenomena, the Burger number is S ∼ 1. Figure 3 shows a similar depth density profile for the ocean. The depth of the ocean rarely exceeds six kilometers, and the vertical extent D of major current systems is usually much less than that. Yet the horizontal scale L is hundreds of kilometers. For such currents, the Burger number is S ∼ 0.1. R EMARK 1.1. Note that in both situations and more generally for almost all large-scale geophysical flows, there is an important disparity between horizontal and vertical scales of motion, which is measured by the aspect ratio δ=
D · L
The Coriolis force When considering winds or currents, i.e., relative motions of the oceans or atmosphere, because the reference frame is rotating, another force has to be taken into account, namely the Coriolis force. An important measure of the Coriolis force, i.e. of the significance of rotation for a particular phenomenon is the Rossby number, which
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Fig. 2. Distribution of density with height in the atmosphere (after the NASA 1962 Standard Atmosphere).
Fig. 3. Distribution of density with depth in the ocean (from Pedlosky [50]).
is defined as follows. Let L be a characteristic horizontal length scale of the motion under consideration, or in other words a length scale that characterizes the horizontal spatial variations of the dynamic fields (for instance the distance between a pressure peak and a succeeding trough). Similarly let U be a horizontal velocity scale characteristic of the motion. The time it takes a fluid element moving with speed U to cross the distance L is L/U . If that period of time is much less than the period of rotation ||−1 of the earth, the fluid can scarcely sense the earth’s rotation over the time scale of motion. For rotation to be important, then, we anticipate that L/U ≥ ||−1 , or equivalently we expect the Rossby number to be small ε=
U ≤ 1. 2||L
(1.1.2)
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Fig. 4. Distribution of wind speed along latitude circles (from Palmén and Newton [49]).
For the purpose of this text we will only consider large-scale motions, namely those which are significantly influenced by the earth’s rotation: || = 7.3 × 10−5 s−1 . Note that the smaller the characteristic velocity U is, the smaller L can be and yet still qualify for a large-scale flow. For the troposphere, the characteristic length scales are D ∼ 10 km and L ∼ 1000 km. The distribution of wind speed along latitude circles (called zonal wind ) given in Figure 4 shows that U ∼ 20 m s−1 . The Rossby number is therefore ε = 0.137 and we can expect the earth’s rotation to be important. The Gulf Stream has velocities of order U ∼ 1 m s−1 . Although its characteristic horizontal scale as shown in Figure 5 is only L ∼ 100 km, the associated Rossby number is ε = 0.07. Although the use of the local normal component of the earth’s rotation would
On the influence of the Earth’s rotation on geophysical flows
Fig. 5. Structure of the current velocity through the Gulf Stream (from Fuglister [19]).
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double this value at a latitude of 30◦ , it is still clear that such currents meet the criterion of large-scale motion. Balance between gravity and rotation • General considerations on rotating fluids allow to determine some constraints in order that motions with time scales long compared to the rotation period and with relative vorticity ω small with respect to 2|| can persist. – In the absence of friction the production of vorticity due to the pressure must indeed cancel the production of relative vorticity by the stretching and twisting terms. This constraint can be written ( · ∇)u − ∇ · u = −
(∇ρ ∧ ∇p) , 2ρ 2
where u denotes the local velocity of the fluid, ρ its density and p its pressure. – If the relative motion has a small aspect ratio δ, which is generally satisfied by currents and winds, only the local vertical component of the earth’s rotation f = || sin θ where θ denotes the latitude, is dynamically significant (the horizontal Coriolis acceleration due to the vertical motion and the vertical Coriolis acceleration due to the horizontal motion are both small terms when compared to the pressure gradients in their respective equations). The constraint states therefore (f e3 · ∇)uh = −
(∇ρ ∧ ∇p)h , 2ρ 2
f e3 ∇ · uh = 0,
(1.1.3)
where f is the Coriolis parameter or inertial frequency defined as the local component of the planetary vorticity normal to the earth’s surface. Since the density variations are commonly connected with temperature variations, the winds or currents satisfying the first equation relating the variation of the horizontal velocity to the density variation are called the thermal wind. – If in addition the fluid is barotropic, meaning that the pressure p is a function of the density ρ, then (f e3 · ∇)uh = 0, which implies that a material line once parallel to must always remain so. – If the fluid is essentially incompressible, the incompressibility constraint implies further that (f e3 · ∇)u3 = 0, so that all three components of the relative velocity are independent of the vertical coordinate. This constraint is called the Taylor–Proudman theorem. If the vertical component of the velocity is zero at some level, for example at a rigid surface, the
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motion is then completely two dimensional and can be pictured as moving in columns parallel to the rotation axis referred to as Taylor columns. The simplest situation in which such motions can occur is in the slow relative motion of a homogeneous fluid. • Specifying conservative forces leads to a more explicit constraint, expressing the balance between gravity, pressure and the Coriolis force (in the absence of friction): ρ2 ∧ u = −∇p − ρge3 .
(1.1.4)
In the absence of relative motion, such a constraint reduces to the Archimedian principle for a static fluid. – If the relative motion has a small aspect ratio δ, we have seen that only the local vertical component of the earth’s rotation f e3 is dynamically significant. Furthermore the pressure and density are small departures from their basic states, the magnitude of which is of the order of ε
2 L2 gD
with the previous notations. Then (1.1.4) can be approximated by uh =
1 e3 ∧ ∇p, fρ0 ρg = −∂3 p.
(1.1.5)
The first relation is the geostrophic approximation expressing the balance between the horizontal pressure gradient and the horizontal component of the Coriolis acceleration. It gives no direct information about the vertical velocity (without further assumption on the thermodynamic properties of the fluid). The other equation does not involve the velocity at all, it is just the hydrostatic approximation describing a balance between the vertical pressure gradient and gravity. • The geostrophic approximation is very useful to predict the motion of geophysical flows: once the pressure field is known, the horizontal velocities, their vertical shear and the vertical component of the vorticity are immediately determined. Nevertheless (i) the approximation fails in the vicinity of the equator since f cancels. A more complicated dynamical framework is then required in the equatorial regions. (ii) even at higher latitudes, the geostrophic relations do not allow to calculate the pressure field nor predict its evolution with time. Consideration of small departures from complete geostrophy is then required to complete the dynamical determination of the motion. These small departures involve either the relative acceleration terms, of the order of the Rossby number, or the frictional forces.
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1.1.2. Departures from geostrophy Waves arising in the case of shallow water In order to determine the corrections to the geostrophic motion, we first consider the case of a shallow rotating layer of homogeneous, incompressible and inviscid fluid. Such a fluid is described by its height H which is assumed to be a fluctuation η around a reference height H0 , and by its purely horizontal velocity u. The specification of incompressibility and constant density immediately decouples the dynamics from the thermodynamics, and imposes a condition of non divergence on the velocity u. The shallow-water assumption, based on the smallness of the aspect ratio δ ! 1 consists in ignoring stratification and considering only the two-dimensional motion of the fluid. Such a simple case contains some of the important dynamical features of the atmosphere and ocean. Of course it does not allow to catch physical phenomena which depend in a crucial way on stratification. In this framework, the geostrophic approximation reduces to u·∇
f H0
= 0,
where H0 is the reference depth (in absence of relative motion), meaning that streamlines are the isobaths. Of course real motions are not precisely geostrophic and we now consider what happens when the constraint of steadiness is relaxed. Perturbations (η, u) to this geostrophic approximation satisfy ∂t (∂tt2 + f 2 )η + ∇ · (C02 ∇η) − gf (∂1 H0 ∂2 η − ∂2 H0 ∂1 η) = 0, 2 (∂tt2 + f 2 )u1 = −g(∂1t η + f ∂2 η), 2 (∂tt2 + f 2 )u2 = −g(∂2t η − f ∂1 η),
√ where f denotes the Coriolis parameter and C0 = gH0 is the classical shallow-water phase speed. Wave solutions which are periodic in x and t can be sought in the form exp(i(σ t + k1 x1 + k2 x2 )) where σ is the wave frequency and k is the wave vector (which is possibly quantized if the domain under consideration is a partially bounded region). These free oscillations can actually be classified into three types, planetary waves, gravity waves and non-rotating waves, as shown in Figure 6. • Gravity waves, also known as Poincaré waves, satisfy the dispersion relation σ 2 = f 2 + C02 k 2 .
(1.1.6)
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Fig. 6. Dispersion diagram for shallow water in a channel (from Pedlosky [50]).
They depend crucially neither on the geometry of the domain, nor on the variations of f . The presence of rotation increases the wave speed. Indeed it is clear that all these waves have frequencies σ which exceed f , i.e. have periods less than half a rotation period and consequently are at frequencies considerably in excess of those of large-scale, slow atmospheric and oceanic flows. In particular these waves are far from being in geostrophic balance. For instance, in a channel of width L oriented parallel to the x1 -axis, the boundary conditions constrain k2 to take discrete values, namely nπ/L with n ∈ Z, and the corresponding modes are given by nπx2 f L k1 nπx2 η = η0 cos − sin cos(k1 x1 − σ t + φ), L nπ σ L u1 =
nπx2 fL nπx2 η0 C02 k1 cos − sin cos(k1 x1 − σ t + φ), H0 σ L nπ L
u2 = −
C 2 n2 π 2 nπx2 η0 L sin f2 + 0 2 sin(k1 x1 − σ t + φ), H0 σ nπ L L
meaning that the fluid flow is primarily in the direction of the pressure gradient. Note however that the Poincaré wave corresponding to the value k2 = 0 is not physically relevant for a rotating fluid (boundary conditions cannot be taken into account).
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• In this last case (i.e. when the domain has at least one internal boundary), the set of Poincaré modes is supplemented by the so-called Kelvin waves. They satisfy the dispersion relation σ 2 = C02 k12 ,
(1.1.7)
which is also the dispersion relation for gravity waves in a non-rotating fluid. The corresponding modes are given by f x2 η = η0 exp ± cos(k1 (x1 ± C0 t) + φ), C0 η 0 C0 f x2 cos(k1 (x1 ± C0 t) + φ), exp ± u1 = H0 C0 u2 = 0. There are several extraordinary features to note. The cross-channel velocity u2 is identically zero, whereas the flow in the x1 -direction is in precise geostrophic balance even though the frequency is not, in general, small with respect to f . More precisely, the Coriolis acceleration is balanced by a free surface slope. This cross-channel slope is exponential, with intrinsic length scale R = C0 /f which is independent of any property of the wave field. This intrinsic length scale is linked to the Rossby deformation radius. Note that R → ∞ as f → 0, so that the Kelvin waves become in that limit the missing gravest modes of the Poincaré set. These two types of waves give a complete picture of the departures from geostrophy in the simplest case, when the Coriolis parameter can be considered as a constant. Such an approximation is relevant at mid-latitudes for small geographical zone such as lakes or small portions of the oceans. • When considering more extended domains, the variations of the Coriolis parameter has to be taken into account and a third family of waves appear. The planetary waves, also called Rossby waves, whose existence requires both f and ∇f to be nonzero, have a very different dynamical structure. They are low-frequency oscillations, in the sense that their periods are greater than a rotation period, or in other words that σ/f ! 1. To lowest order in σ/f , the velocity fields, though changing with time, remain continuously in geostrophic balance with the pressure field. Thus the motion is quasigeostrophic and it is the very small cross-isobath flow, which is a nongeostrophic effect, which produces the oscillation. Another characteristic property of the Rossby waves is that, for high wave numbers, the frequency decreases as the wave number increases, in contradistinction to both the Poincaré and Kelvin waves. For instance, if the Coriolis parameter depends linearly on the northward coordinate x2 f (x2 ) = f0 + βx2
with f0 = f x20 and β = ∂2 f x20
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(which is locally a good approximation, known as beta-plane approximation), the dispersion relation for the Rossby waves states σ=
βk1 C02 f 2 + C02 k 2
·
(1.1.8)
The last feature of the Rossby waves we would like to mention here is that their phase speeds in the x1 -direction are always negative, as shown by the dispersion relation (1.1.8). In the particular case of a channel of width L oriented parallel to the x1 -axis, the explicit formulas for the Rossby modes to lowest order are nπx2 βL η = η0 sin cos(k1 x1 − σ t + φ) + O , L f g nπ nπx2 βL η0 cos cos(k1 x1 − σ t + φ) + O , u1 = − f L L f g nπx2 βL sin(k1 x1 − σ t + φ) + O . u2 = − kη0 sin f L f
Equatorial trapping As mentioned in Section 1.1, the adjustment processes are expected to be somewhat special in the vicinity of the equator when the Coriolis acceleration vanishes. A very important property of the equatorial zone is that it acts as a waveguide, i.e., disturbances are trapped in the vicinity of the equator. The waveguide effect is due entirely to the variation of the Coriolis parameter with the latitude. • The simplest wave that illustrates this property is the equatorial Kelvin wave. As for the usual Kelvin waves, the motion is unidirectional, being everywhere parallel to the equator. At each fixed latitude, the motion is exactly the same as that in a non-rotating fluid. Nevertheless, because of the variations (and the cancellation) of the Coriolis parameter f (x2 ) ∼ βe x2
with βe = ∂2 f (0) =
2 , R
rotation effects do not allow the motion at each latitude to be independent: a geostrophic balance is required between the eastward velocity and the north–south pressure gradient. The equatorial Kelvin wave shows therefore an exponential decay in a distance of order ae , where ae is given by ae =
C0 2βe
1/2 (1.1.9)
and is called the equatorial deformation radius because of its relationship with the decay scale for the usual Kelvin waves.
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Fig. 7. Dispersion diagram for shallow water in the equatorial waveguide (from Gill [27]).
• In addition to the Kelvin wave, there is an infinite set of other equatorially trapped waves, with trapping scale of the same order that for Kelvin waves, namely, the equatorial deformation radius defined by (1.1.9). Note that another important effect of the waveguide is the separation into a discrete set of modes n = 0, 1, 2, . . . as occurs in a channel. The dispersion curves for equatorial waves are given in Figure 7. – For n ≥ 1, the waves subdivide into two classes. For the upper branches, the appropriate dispersion relation has the same form as that for Poincaré waves, approximately σ 2 ∼ (2n + 1)βC0 + k12 C02 ,
(1.1.10)
and so these waves are called equatorially trapped Poincaré waves. On the lower branches of the curves, the dispersion curves are given approximately by σ∼
βC0 k1 · C0 k12 + (2n + 1)β
(1.1.11)
The corresponding waves are called equatorially trapped Rossby waves. Note that there is a large gap between the minimum gravity wave frequency and the maximum planetary wave frequency, so these waves are easily distinguished. The frequency gap for wave n involves a factor of 2(2n + 1), which is equal to 6 for the lowest value n = 1. – For n = 0, the solution is somewhat special. The dispersion curve, given by σ β + k1 − = 0, C0 σ
(1.1.12)
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is unique in that for large positive k1 it behaves like a gravity wave, whereas for large negative k1 it behaves like a planetary wave. For this reason it is called a mixed Rossby-gravity wave. The phase velocity can be to the east or west, but the group velocity is always eastward, being a maximum for short waves with eastward group velocity (gravity waves). Particles follow anticyclonic orbits everywhere. Effects of stratification The large-scale field of vertical motion in the atmosphere is of great importance because strong upward motion is associated with the development of severe weather conditions. Note that the vertical motion cannot be easily measured (due to its smallness compared to horizontal scales), but deductions can be made from properties of the pressure field. We therefore have to study the adjustment processes for continuously stratified fluids, i.e., fluids with continuously varying density. The fluids to be considered will be actually restricted to a class such that the density depends only on entropy and on composition. The motion that takes place is assumed to be isentropic and without change of phase, so that ρ is constant for a material element. Such a fluid is therefore incompressible. The equilibrium state to be perturbed is the state of rest, so the distribution of density and pressure is the hydrostatic equilibrium given by: ρ = ρ0 (x3 ),
p = p0 (x3 )
with ∂3 p0 = ρ0 g.
For such an incompressible stratified fluid, free oscillations exhibit different behaviours according to the frequency regime to be considered. A first relation between the vertical velocity u3 and the pressure perturbation p is associated with the vertical part of the motion, and thus is unaltered by rotation effects: ∂tt2 u3 + N 2 u3 = −ρ0−1 ∂t3 p , where N(x3 ) is a quantity of fundamental importance to this problem, defined by N 2 = −gρ0−1
d ρ0 . dx3
(1.1.13)
N has the dimensions of a frequency, and is known as the buoyancy frequency since it is the frequency of oscillation for purely vertical motion. The restoring force that produces the oscillation is the buoyancy force. The other equation relating u3 and p is provided by the horizontal part of the motion, and more precisely combining the equation for the vertical vorticity and the incompressibility constraint: ∂tt2 ∂3 u3 + f 2 ∂3 u3 = ρ0−1 ∂t h p , which involves the inertial frequency f but not the buoyancy frequency N .
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Fig. 8. Effects of stratification on a rotating fluid (from Gill, 1982).
• The dispersion relation for internal Poincaré waves in a rotating fluid with uniform buoyancy frequency N is therefore σ2 =
f 2 k32 + N 2 kh2 . k2
(1.1.14)
In the atmosphere and ocean, N usually exceeds f by a large factor, typically of order 100, so the contribution of the Coriolis parameter in (1.1.14) is essentially negligible, and the dispersion curves will not look any different because of rotation, except that the vertical axis would have to be labeled σ/N = 0.01 instead of zero. More precisely, when N/f is large, different regimes appear according to the value of σ/f as shown in Figure 8. – The nonhydrostatic wave regime is defined as the range of frequencies for which σ is of order N but σ ≤ N . In this range the dispersion relation is approximated by σ2 ∼
N δ2 , 1 + δ2
which is the relation obtained when rotation effects are ignored. – The hydrostatic “non-rotating” wave regime is defined as the range of frequencies for which f ! σ ! N . In this range the dispersion relation is approximated by σ 2 ∼ N 2δ2. Rotation effects do not appear to this order of approximation, which is the reason for calling this a “non-rotating” regime, although it must be remembered that rotation does have an effect at the next order of approximation, and it is sometimes important to consider this.
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– The rotating wave regime is defined as the range of frequencies for which σ is of order f but σ ≥ f . Since f/N is small, α is small and the hydrostatic approximation applies. The approximate dispersion relation reads σ 2 ∼ f 2 + N 2δ2, which is effectively the dispersion relation for Poincaré waves. • If the variations of Coriolis parameter are taken into account f (x2 ) ∼ f0 + βx2
with f0 = f x20 and β = ∂2 f x20 ,
we have to consider furthermore the vertical propagation of planetary waves. As previously, in the case of a uniformly stratified incompressible fluid, the dispersion relation for vertically propagating waves is the same as that for a single mode, but with the wave speed C0 replaced by N/k3 . In other words, for vertically propagating Rossby waves, it reads σ=
kh2
βk1 · + f02 k32 /N 2
(1.1.15)
Such upward-propagating waves have a very particular structure, with phase lines tilting toward the west with height, meaning that warm air is carried poleward and cold air equatorward, so that there is an apparent net poleward transport of heat. The corresponding buoyancy fluxes play an important role in the atmosphere, in the phenomenon known as a sudden stratospheric warming, which occurs in winter. The classification of vertically propagating waves begun previously can now be carried to larger scale k1−1 that correspond to lower encounter frequencies σ = U k1 for an observer traveling with the mean flow at speed U . If this flow is uniform, the disturbances are trapped (evanescent) at scales k1−1 larger than that given by U k1 ∼ f , i.e., for scales greater than about 100 km. This is because gravity waves (also called Poincaré waves and defined by (1.1.14)) are negligible at such frequencies (see Figure 8). If, however, the scale k1−1 is further increased, thereby reducing the encounter frequency to levels at which variations with latitude of the Coriolis parameter become important in the dynamics, the situation is changed once again because planetary waves may now be possible. It is then clear that – the f -plane quasi-geostrophic regime occupies the spectral gap defined by |U/f | ! k1−1 ! |U/β|1/2 , – the so-called β-plane quasi-geostrophic regime is a new regime to be considered for k1−1 of order |U/β|1/2 . This is about 1000 km for the atmosphere, i.e., the scale of the major topographic features of the earth’s surface, so the response to these features
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Fig. 9. Effects of stratification on an inhomogeneous rotating fluid (from Gill, 1982).
falls within this regime. The corresponding scale for the ocean is 30 to 100 km. Note that in this new regime, there is a major asymmetry between eastward and westward directions of the undisturbed flow. Westward currents are in the same direction as the phase propagation of planetary waves, so stationary waves are not possible: disturbances remain evanescent no matter how small the wavenumber. 1.1.3. Prediction of the observed motion Contribution of small scales Although a single wave of arbitrary amplitude is an exact solution of the quasigeostrophic equation, a superposition of waves will not be. The nonlinear interaction between the waves, by which the velocity field of one advects the vorticity of another, leads to a nonlinear coupling and energy transfer between the waves. When the Rossby number ε ! 1, the characteristic period of the waves describing the departures from geostrophy is much less than the advective time: the nonlinear coupling term can be therefore considered as a perturbation of the linear equation governing the waves. In particular, on can try to proceed by successive approximations and to characterize the resulting motion as a perturbation of the linear superposition of waves. The interaction of the mth and nth waves produces then a forcing term in the problem for the first correction which oscillates with the sum and difference of their two phases, i.e., a forcing term with wave vector Kmn = Km ± Kn and frequency σmn = σm ± σn .
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The problem for the first correction is a linear, forced problem, and therefore the response to each forcing term can be considered separately and the results summed. If the forcing frequency σmn is not equal to the natural frequency of oscillation of a free wave with the wave number Kmn of the forcing, such a process converges: these interactions merely produce a small-amplitude background jangle of forced waves whose amplitudes are small. Otherwise a resonance occurs, that is, two waves then combine to force a third wave with a wave number and frequency appropriate to a free, linear oscillation. A simple example is the case of the Kelvin waves. Such interactions, called resonant interactions are of great interest because of the slow growth of the first correction on the nonlinear advective time: the approximation process is then clearly invalid. This means that, filtering the high frequency waves, one obtains a motion on the advective time-scale which is nonzero. Note that to the lowest order the filtered motion conserves both the energy (defined as half the average of the square of the velocity) and the enstrophy (defined as half the average of the square of the vorticity), or in other words that the resonant interaction is an energy and enstrophy preserving mechanism. In naturally occurring situations, there is usually a whole spectrum of waves, i.e., a superposition of waves with wavenumbers varying continuously over some range of values. In such cases, wave interactions occur in the same way as they do when a small number of waves is present, and provided that the wave amplitude is not too large, the transfer of energy is dominated by those waves that are associated with the resonant triads (if such are present). The phases of the different wavenumber components in the spectrum are often assumed to be distributed randomly and this assumption can be used to calculate the evolution of the spectrum with time. This behaviour can be largely understood by considering the following three mechanisms: – Induced Diffusion occurs when two nearly identical waves interact with another wave of much lower frequency and much smaller wavenumber. The shear of the latter wave acts to diffuse wave action (wave energy divided by frequency) among vertical wavenumbers. – Elastic Scattering occurs when two waves with wavenumbers that are almost mirror images in the horizontal plane interact with a wave of much slower frequency and double the vertical wavenumber. The latter wave tends to equalize the energy between upward- and downward-propagating waves. The conditions for elastic scattering to occur are satisfied only for waves with frequency substantially greater that f , so nearinertial frequency waves are little affected. – Parametric Subharmonic Instability occurs when two waves of nearly opposite wavenumber interact with a wave of much smaller wavenumber and of twice the frequency. The process transfers energy from low-wavenumber energetic waves to high-wavenumber waves of half frequency, and so tends to produce inertial frequency waves with high vertical wavenumber. These processes have a strong influence on the internal wave spectrum, and one result is that the spectrum has a shape that varies rather little. Dissipation coming from viscosity The observed persistence over several days of largescale waves in the atmosphere, and the oceans shows that frictional forces are weak, almost
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everywhere, when compared with the Coriolis acceleration and the pressure gradient. Friction rarely upsets the geostrophic balance to lowest order. Nevertheless friction, and the dissipation of mechanical energy it implies, cannot be ignored. For the time-averaged flow, i.e., for the general circulation of both the atmosphere and the oceans, the fluid motions respond to a variety of essentially steady external forcing. The atmosphere, for example, is set in motion by the persistent but spatially nonuniform solar heating. This input of energy produces a mechanical response, namely kinetic energy of the large-scale motion, and eventually this must be dissipated if a steady state—or at least a statistically stable average state of motion—is to be maintained. Finally, even though friction may be weak compared with other forces, its dissipative nature, qualitatively distinct from the conservative nature of the inertial forces, require its consideration if questions of decay of free motions are to be studied. • To estimate the frictional force a representation of F must be specified. Considering the dissipation due to the interactions at the microscopic level, this force is proportional to the spatial derivative of the stress tensor, with a coefficient depending in principle of the thermodynamic state variables, the so-called molecular viscosity. Then νU F ∼ 2, ρ L where ρ is the local density, L the length scale characterizing the variations of the velocity field, and ν is the order of magnitude of the molecular viscosity. The ratio of the frictional force per unit mass to the Coriolis force acceleration is a nondimensional parameter, called the Ekman number, E: E=
ν νU/L2 = · 2U 2L2
(1.1.16)
If ν is the molecular kinematic viscosity of water, for example, a straightforward estimate for E for oceanic motions, would be, for L = 1000 km, ν = 10−6 m2 s−1 , E ∼ 10−14 . This is a terribly small number, and such frictional forces are clearly negligible for large scale motions. • The important issue is whether this representation of F is adequate if the state variables are to describe only the large-scale motions. Section 1.1.2 shows indeed that motions on one spatial scale interact with motions on other scales. There is therefore an a priori possibility that small scale motions, which are not the focus of our interest, may yet influence the large-scale motions. One common but not very precise notion is that small-scale motions, which appear sporadic or on longer time scales, act to smooth and mix properties on the larger scales by processes analogous to molecular, diffusive transports. For the present purposes it is only necessary to note that one way to estimate the dissipative influence of smaller-scale motions is to retain the same representation of the frictional
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force but replace ν by a turbulent viscosity, of much larger magnitude than the molecular value, supposedly because of the greater efficiency of momentum transport by macroscopic chunks of fluid. This is, of course, an empirical concept very hard to quantify. Influence of boundary conditions Rotation effects have thus far been studied in the absence of boundaries. If now a boundary is inserted that crosses the isobars, further adjustment would have to take place because no flow is possible across the boundary. This indicates that the adjustment process is strongly affected by the presence of boundaries, at least in the neighborhood of those boundaries. • Consider first the action of a stress at the horizontal surface. For instance, on the ocean surface, this stress is due to the action of the wind. It produces a direct response called the Ekman transport, which is principally confined to a thin layer near the ocean surface. In fact, the Ekman transport is thought to be usually found within the upper mixed layer of the ocean, which is mostly between 10 and 100 m deep. A sudden change of wind can cause oscillations in the Ekman transport of inertial period, or can reduce the amplitude of preexisting oscillations. If the wind stress were spatially uniform, the ocean below the mixed layer would be little affected by the wind, which would produce a time-varying Ekman transport that is confined to the near-surface region. However spatial variations in the wind (which of course occur) cause spatial variations in Ekman transport. In other words, the Ekman flow will cause mass to flow horizontally into some regions and out of others. This results in vertical motion. For instance, if the horizontal flow is converging in a particular region, vertical motion away from the boundary is required in order to conserve mass. The vertical velocity just outside the boundary layer which is so produced is called the Ekman pumping velocity. It is this velocity in the ocean that distorts the density field of the ocean and thereby causes the wind-driven currents. The stress at the underlying ocean (or land) surface, from the atmospheric point of view, is a frictional drag whose magnitude is dependent on the strength of the wind, usually called bottom friction. With the stress is associated an Ekman transport in the atmosphere whose horizontal mass flux is opposite to that in the underlying ocean. Consequently variations in Ekman transport produce Ekman pumping with a vertical mass flux that is the same in the atmosphere as in the ocean. Such a bottom friction exists also for the ocean. The boundary layer at the bottom of the ocean (the benthic boundary layer) is much thinner than is the atmospheric boundary layer, typically in the range 2 to 10 m, which affects the relative importance of topographic effects. Detailed modeling of the velocity structure of the boundary layer is therefore particularly difficult. The important feature of this process, called spin-down, is that the presence of (turbulent) friction in general tends to reduce motion and make the system tend toward a state of rest. • The second mechanism to be understood is the adjustment process in presence of side boundaries. In fact, the presence of such a boundary implies that the longshore component of the Coriolis acceleration vanishes at the boundary so that the mutual adjustment of the
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longshore velocity field and the pressure field along the boundary is more like in a nonrotating fluid than like in a rotating one. This is certainly true in the extreme case in which there are two boundaries close together, as in a narrow gulf or estuary. The rotation effects can be neglected at the first approximation because the motion is mainly along the gulf and the component of the Coriolis acceleration in this direction is negligible. At the next order of approximation, rotation modifies the flow in two ways. One is to give a cross-channel pressure gradient in order to geostrophically balance the longshore flow. The other is to produce a shear whenever the surface elevation departs from its equilibrium level, this being required in order that potential vorticity be conserved. The narrow channel approximation can be applied with success to studies of tides and seiches in gulfs, estuaries, and lakes, and even to tides in the Atlantic Ocean. When the two sides of a channel are not close together, the question arises as how far from the shore the longshore component of the Coriolis force can be neglected. The answer is a distance of the order of the Rossby deformation radius, so channels must have width small compared with this scale for the narrow-channel approximation to be valid. For wide channels, there is a special form of adjustment near the boundary by means of a Kelvin wave whose peculiarity is that it can travel along the coast in one direction only, and whose amplitude is only significant within a distance of the order of the Rossby deformation radius from the boundary. Note that the presence of boundaries also affects the Poincaré waves, but effects of the end of the channel can be quite difficult to work out. Of course details are strongly influenced by the complicated shape of the world’s oceans.
1.2. Mathematical modelling The starting point of geophysical fluid dynamics is the premise that the dynamics of meteorological and oceanographic motions are determined by the systematic application of the fluid continuum equations of motion. The dynamic variables generally required to describe the motion are the density ρ, the vector velocity u, and certain further thermodynamic variables like the temperature T or the internal energy per unit mass e. 1.2.1. Introducing a general mathematical framework In the absence of sources or sinks of mass within the fluid, the condition of mass conservation is expressed by the continuity equation ∂t ρ + ∇ · (ρu) = 0.
(1.2.1)
Newton’s law of motion written for a fluid continuum takes the form ρ(∂t + u · ∇)u = −∇p + ρ∇φ + F(u),
(1.2.2)
meaning that the mass per unit volume times the acceleration is equal to the sum of the pressure gradient force, the body force ρ∇φ where φ is the potential by which conservative force such as gravity can be represented, and the frictional force F .
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Unless the density is considered a constant, the momentum and continuity equation are insufficient to close the dynamical system. The first law of thermodynamics must be considered; it can be written as ρ(∂t + u · ∇)e = −pρ(∂t + u · ∇)ρ −1 + k∇ 2 T + χ + ρQ,
(1.2.3)
where k is the thermal conductivity, T is the temperature, Q is the rate of heat addition per unit mass by internal heat sources, and χ is the addition of heat due to viscous dissipation— which is negligible in all situations to be discussed. To complete the system, further thermodynamic state relations expressing the physical nature of the fluid are required. Mathematical features of geophysical fluids For example, in the atmosphere the state relation for dry air is well-represented by the ideal-gas law ρ=
p , RT
(1.2.4)
where R is the gas constant for dry air. The local conservation of energy then becomes % k 2 θ (∂t + u · ∇)θ = ∇ T +Q , Cp T ρ where the potential temperature θ is defined by
p0 θ =T p
R/Cp ,
for some reference pressure p0 . We have denoted by Cp the specific heat at constant pressure.Note that in the absence of conductive and internal heating θ is a conserved quantity for each fluid element. For the oceans, density differences are so slight that they have a negligible effect on the mass balance, so that the local conservation of mass can be approximated by ∇ · u = 0,
(1.2.5)
which is the incompressibility relation. Note that the incompressibility constraint does not imply that the fluid is homogeneous, meaning that (∂t + u · ∇)ρ is generally not assumed to vanish. The Navier–Stokes model with Coriolis force We noted earlier that the most natural frame for which to describe atmospheric and oceanic motions is one which rotates with the planetary angular velocity . Let r be the position vector of an arbitrary fluid element. We have dr dr = + ∧ r, dt I dt R
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where the subscript I denotes rates of change as seen by the observer in the non-rotating inertial frame. The velocity seen in the non-rotating frame uI is therefore equal to the velocity observed in the rotating frame augmented by the velocity imparted to the fluid element by the solid-body rotation ∧ r. We may write this as uI = uR + ∧ r, where uR is called the relative velocity. As Newton’s law of motion equates the applied forces per unit mass to the acceleration in inertial space, we have then to express this acceleration in terms of quantities which are directly observed in the rotating frame:
duI dt
=
I
=
duI dt duR dt
+ ∧ uI R
+ 2 ∧ uR + ∧ ( ∧ uR ) + R
d ∧ r. dt
The discrepancy between the accelerations perceived in the different frames is equal to the three additional terms on the right-hand side. They are the Coriolis acceleration 2 ∧ uR , the centripetal acceleration ∧( ∧r) and the acceleration due to variations in the rotation rate itself, which can be neglected for most oceanographic or atmospheric phenomena. Since the centrifugal force can be written as a potential ∧ ( ∧ r) = −∇
| ∧ r|2 , 2
it is included with the force potential. The Coriolis acceleration 2 ∧ r is therefore the only new term which explicitly involves the fluid velocity, and it is responsible for the structural change of the momentum equation. If we note that spatial gradients are perceived identically in rotating and non-rotating coordinate frames, the momentum equation becomes ρ((∂t + u · ∇)u + 2 ∧ u) = −∇p + ρ∇φ + F.
(1.2.6)
It is important to note that the total time rate change of any scalar such as the temperature is the same in rotating as in non-rotating frames. Thus the equation of conservation of mass and the various thermodynamic equations are unaffected by the choice of coordinate frame. Boundary conditions In most cases of interest the (turbulent) Ekman number E is sufficiently small that it might appear that friction could be neglected. However, the viscosity ν is the coefficient of the highest spatial derivatives and thus the fact that it is nonzero is quite important as regards the mathematical structure of the equations of motion, and the number of boundary conditions to be imposed.
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If the surface ∂3 O of the fluid layer is in contact with a solid surface, for instance in the case of the bottom boundary of the ocean, the natural condition to be considered is a no-slip condition: u|∂3 O ≡ 0. If the surface ∂3 O of the fluid layer is free rather than in contact with a solid surface, for instance in the case of the interface between the ocean and the atmosphere, the appropriate boundary condition is continuity of pressure and continuity of frictional stress across the fluid surface u|∂3 O · n = 0,
(ν(∇u + (∇u)T ) − p I d)|∂3 O · n = constraint.
Then, for models of atmospheric phenomena the Ekman layer or some more elaborate model of the friction layer at the lower boundary usually suffices to represent the frictional interaction of the fluid and the boundary. Models of oceanic (or lake) dynamics which explicitly recognize the fact that the water is gathered together in basins have to be supplemented by a no-slip boundary condition at the lateral boundaries: u|∂h O ≡ 0. In such a framework one has generally to introduce side-wall friction layers whose structure differ considerably from that of the Ekman layer. 1.2.2. Taking into account the geometry of the earth The situation to be described is schematically depicted in Figure 10. We consider motions on a sphere of radius r0 , meaning that we will ignore ab initio the slight departures of the figure of the earth from sphericity. The characteristic vertical scale of the motion, D, is in all cases of interest small compared to r0 so that the effective gravitational acceleration g can be considered constant over the depth of the fluid. The horizontal scale motion L is large in the sense described in the first section (i.e., L is large enough so that the Rossby number is small), but in the sequel we will focus our attention on the situation where L is considerably smaller than r0 . The equations of motion in spherical coordinates The coordinate system to be used in the spherical system is shown in Figure 11. The position of any point in the fluid is fixed by r, θ and φ, which are the distance from the earth’s centre, the latitude and the longitude respectively. The velocities in the eastward, northward, and vertical directions are uφ , uθ and ur , as shown. The equation for conservation of mass (1.2.1) in these spherical coordinates is ∂φ uφ d ∂θ (uθ cos θ ) 2ur ρ + ρ ∂r ur + + + = 0, dt r r cos θ r cos θ
(1.2.7)
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Fig. 10. Characteristic length scales for geophysical flows.
Fig. 11. Spherical coordinates for the description of geophysical flows.
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where the total time derivative is defined by uφ d uθ = ∂t + ∂φ + ∂θ + ur ∂r . dt r cos θ r The momentum equations are uφ (ur − uθ tan θ ) d uφ + − 2 sin θ uθ + 2 cos θ ur dt r Fφ 1 ∂φ p + , =− ρr cos θ ρ ur uθ + u2φ tan θ d 1 Fθ uθ + + 2 sin θ uφ = − ∂θ p + , dt r ρr ρ
(1.2.8)
u2φ + u2θ 1 Fr d ur − − 2 cos θ uφ = − ∂r p − g + , dt r ρr ρ where Fφ , Fθ , Fr are the three components of the frictional forces acting on the fluid. The equations of motion must as previously be completed with the addition of a thermodynamic equation, for example the incompressibility constraint ∂r ur +
∂φ uφ ∂θ (uθ cos θ ) 2ur + + = 0. r r cos θ r cos θ
(1.2.9)
Consider now the description of a motion, in either the ocean or the atmosphere, whose horizontal spatial scale of variation is given by the length scale L and whose horizontal velocities are characterized by the velocity scale U . Geometrical considerations imply that if the vertical scale of motion is D, the corresponding slope of a fluid element’s trajectory will not exceed D/L, so that appropriate scaling for the vertical velocity is DU/L (note that the actual scale of the vertical velocity may be less than DU/L if other dynamical constraints act to reduce the vertical motion). The scaling of the pressure and density is more subtle. For small Rossby number, the relative velocities are small and the pressure is expected to be only slightly disturbed from the value it would have in the absence of motion, whereas the horizontal pressure gradients should be of the same order as the Coriolis acceleration. Similarly we may anticipate that the buoyancy force per unit mass will be of the same order as the vertical pressure gradient, since an observed feature of large-scale motions is the excellence of the hydrostatic approximation. Such considerations allow to scale the equations so that the relative order of each term is clearly measured by the nondimensional parameter multiplying it. It is then possible to systematically exploit the smallness of the parameters ε (Rossby number), δ (aspect ratio), L/r0 and F = (2 sin θ0 )2 L2 /gD. Note that these parameters are all independent, and their relative orders will vary from phe-
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nomenon to phenomenon. The nature of the approximations will depend on these relative orders. Below we present the shallow water approximation, and refer for instance to [42], [35] or [56], for other simplified models. Some geometrical approximations • If the motion occurs in a mid-latitude region, distant from the equator, around a central latitude θ0 , it becomes convenient to introduce new longitude and latitude coordinates as follows. Define x1 and x2 by x1 = φ
r0 cos θ0 , L
x2 = (θ − θ0 )
r0 . L
(1.2.10)
They are however measures of eastward and northward distance only at the earth’s surface (r = r0 ) and at the central latitude θ0 . Although x1 and x2 are in principle simply new longitude and latitude coordinates in terms of which the equations of motion may be rewritten without approximation, they are obviously introduced in the expectation that for small L/r0 and D/r0 they will be the Cartesian coordinates of the β-plane approximation as introduced (page 215). It is also convenient to introduce x3 =
1 (r − r0 ), D
∂φ =
r0 cos θ0 ∂1 , L
so that ∂θ =
r0 ∂2 , L
∂r =
1 ∂3 . D
To this point no approximation has been made. As we focus our attention on the situation where L is considerably smaller than r0 , the trigonometric functions can be expanded about the latitude θ0 : 2 2 L x2 L sin θ0 + . . . , sin θ = sin θ0 + x2 cos θ0 − r0 r0 2 2 2 L L x2 cos θ = cos θ0 − x2 sin θ0 − cos θ0 + . . . , r0 r0 2 2 2 L L x2 −2 tan θ0 (cos θ0 )−2 + . . . . tan θ = tan θ0 + x2 (cos θ0 ) − r0 r0 2
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This allows to simplify system (1.2.7), (1.2.8) and (1.2.9) in the following way (where the Coriolis force has not yet been approximated): d ρ = 0, dt
Lx2 Lx2 1 F1 d u2 + 2 cos θ0 + u3 = u1 − 2 sin θ0 + ∂1 p + , dt r0 r0 ρL ρ d Lx2 Lx2 1 F2 u1 + 2 cos θ0 + u3 = u2 + 2 sin θ0 + ∂2 p + , dt r0 r0 ρL ρ d Lx2 Lx2 u1 + 2 cos θ0 + u2 u3 − 2 cos θ0 + dt r0 r0 1 F3 ∂3 p − g + , = ρD ρ 1 1 (∂1 u1 + ∂2 u2 ) + ∂3 u3 = 0, L D
(1.2.11)
where 1 1 d = ∂t + (u1 ∂1 + u2 ∂2 ) + u3 ∂3 . dt L D We then introduce f0 = 2 sin θ0 ,
2L β0 = cos θ0 = r0
L d f r0 dθ
θ=θ0
as the reference Coriolis acceleration and northward gradient of the Coriolis parameter at the latitude θ0 . Note that L β0 /f0 ∼ · ε r0 ε Thus while ε measures the ratio of the relative vorticity and the planetary vorticity normal to the sphere at θ0 , the magnitude of the relative-vorticity gradient and the planetary vorticity gradient is measured by the parameter εr0 /L. While ε may be small, εr0 /L may be large, order one, or small, and each of these possibilities gives rise to a quite different quasigeostrophic dynamical system. – If the geographical zone to be considered is small, meaning that εr0 /L % 1, we will neglect the variations of the Coriolis parameter and use the f -plane approximation: sin θ ∼ sin θ0 . Most of the mathematical studies on geophysical flows deal with this framework. As the Rossby operator has constant coefficients, one can make use of a powerful mathematical tool to study the asymptotic behaviour of the fluid as the rotation rate
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tends to infinity: the Fourier transform allows indeed to carry out explicit computations and to establish qualitative properties of the Poincaré waves (dispersion, resonances. . . ). Thereby, the rotating fluid equations in the f -plane approximation have been the object of a number of mathematical works in the past decade, and Section 2 of this survey aims at giving an overview of the main results as well as the methods of proof. – If the geographical zone to be considered is more extended, meaning that L/εr0 = O(1), more subtle adjustment processes due to the variations of the Coriolis parameter, and characterized by time scales large compared with f0−1 have to be taken into account, which is done using the mid-latitude β-plane approximation: sin θ ∼ sin θ0 + β0 x2 . This situation is much more complicated to study from a mathematical point of view than the previous one, since the techniques based on the Fourier transform can no longer be used. The works devoted to this study are presented in Section 3, they essentially allow to determine the mean motion of the fluid in the absence of boundaries: in particular we do not get any description of the boundary layers. Concerning the waves, we obtain some informations about the oscillating modes (which are the eigenmodes of the Rossby operator), but nothing on their shape equations. • For motions near the equator, the approximations sin θ ∼ θ,
cos θ ∼ 1
may be used, giving what is called the equatorial β-plane approximation: f ∼ β0 x2
with β0 =
2L = 2.3 × 10−11 m−1 s−1 . r0
(1.2.12)
Note that half of the earth’s surface lies at latitudes of less then 30o and the maximum percentage error in the above approximation in that range of latitudes is only 14 percent. In particular, this approximation can usefully be applied over the whole of the tropics. The shallow-water approximation – Assuming that the aspect ratio is very small δ ! 1, vertical motion can be neglected in view of the scalings imposed by the incompressibility constraint. Indeed it is natural to consider the non dimensional unknowns u˜ 1 =
τ u1 , L
u˜ 2 =
τ τ u2 and u˜ 3 = u3 , L D
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where τ is the order of the times to be considered. Rescaling time and plugging the previous formulas in (1.2.11) leads to (∂t + u˜ · ∇)ρ = 0, Lx2 Lx2 1 F1 , (∂t + u˜ · ∇)u˜ 1 − 2 sin θ0 + u˜ 2 + 2 cos θ0 + δ u˜ 3 = ∂1 p˜ + r0 r0 ρ ρ Lx2 Lx2 1 F2 u˜ 1 + 2 cos θ0 + δ u˜ 3 = ∂2 p˜ + (∂t + u˜ · ∇)u˜ 2 + 2 sin θ0 + , r0 r0 ρ ρ Lx2 Lx2 u˜ 1 + 2 cos θ0 + u˜ 2 ˜ u˜ 3 − 2 cos θ0 + (∂t + u·)δ r0 r0 =
F3 1 τ2 ∂3 p˜ − g + , ρδ L ρ ∇ · u˜ = 0.
In particular, if the vertical viscosity is strong enough (for instance independent on δ), we expect u to be asymptotically independent on the vertical variable. Thus taking formally limits as δ → 0 we obtain the horizontal momentum equations Lx2 (∂t + u˜ h · ∇h )u˜ 1 − 2 sin θ0 + u˜ 2 = r0 Lx2 (∂t + u˜ h · ∇h )u˜ 2 + 2 sin θ0 + u˜ 1 = r0
1 F˜ 1 ∂1 p˜ + , ρ ρ 1 F˜ 2 ∂2 p˜ + , ρ ρ
(1.2.13)
∂3 uh = 0. Note that this accounts for the fact that only the vertical component of the rotation of the Earth f = 2 sin θ is considered. – If we suppose moreover that the Rossby deformation radius is very small S ! 1 or in other words that the fluid is almost homogeneous ρ ∼ ρ0 , the pressure is given at leading order by the hydrostatic law p˜ = ρ0 gη, ˜ where g˜ is the non dimensional gravity constant, η is the depth variation due to the free surface, and the continuity equation is, taking into account the form for the divergence operator, ∂t η + ∇h · ((D + η)u˜ h ) = 0.
(1.2.14)
These sets of equations are the ones derived by Laplace, but with the tide-generating terms omitted. Because a shallow layer is considered, r can be taken as a constant equal to the radius of the earth.
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Note that, from a theoretical point of view, it is not clear that the use of the shallow water approximation is relevant in this context since the Coriolis force is known to generate vertical oscillations which are completely neglected in such an approach. Indeed the components of the Coriolis acceleration that are associated with the horizontal component of the rotation vector are not everywhere small compared with the terms retained. That very particular case is the matter of the mathematical works presented in Section 4. The results obtained are close to that of Section 3, but because of the equatorial trapping, the waves—in particular the Rossby waves—have decay properties which allow to get a more precise strong convergence result. That is due to the fact that explicit computations can be written in that framework.
2. A simplified model for midlatitudes 2.1. Introduction In Section 2 we intend to study a model for the movement of the ocean at midlatitudes. As explained in the introduction, at such latitudes the Coriolis acceleration can in a crude approximation be considered as a constant, which makes the analysis much simpler than in the case of the full model. Section 2 is therefore devoted to the analysis of the so-called “rotating fluid equations”, consisting in the three-dimensional Navier–Stokes system in which a constant coefficient penalization operator has been added to account for the Earth rotation. The model is the following:
(RF ε )
⎧ ⎨ ∂ u + u · ∇u − u + 1 u⊥ = − 1 ∇p t ε ε ⎩ div u = 0,
where u⊥ = (u2 , −u1 , 0). We will be interested in the wellposedness of this system for a fixed ε, as well as in the asymptotics of the solutions as ε goes to zero. We will by no means be exhaustive in the presentation, neither in the various results that can be found in the literature nor in the proofs. The aim of Section 2 is rather to give an insight to the questions usually addressed when dealing with this type of system, and to the methods commonly used to answer them. Those methods will be used in the coming sections in more realistic situations (the Coriolis force will no longer be constant), and we feel it can be useful to present them first in this easier, though unrealistic model. The question of the wellposedness of this system can be dealt with quite easily, considering the skew-symmetry of the rotation operator. This is explained in Section 2.3 below. More interesting is the question of the asymptotic behaviour of the solutions as ε goes to zero. As noted in the introduction, we expect by the Taylor–Proudman theorem a two dimensional behaviour at the limit. We show in Section 2.4 that this is indeed the case, as long as weak limits are considered, rather than strong. Section 2.5 is devoted to strong asymptotics, where we will see that it all depends on the boundary conditions imposed on the system. We will mainly focus on two types of boundary conditions, which lead to two
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very different types of convergence results. In Section 2.5.1 we consider the case when the equations are set in R3 . This is highly unrealistic, but the fact that the rotation is constant allows to write explicit calculations in Fourier space, and in particular the formulas found for the eigenvalues of the Coriolis operator enable us to exhibit an interesting dispersive behaviour for the Coriolis operator; thus we are able to deduce a strong convergence result towards a vector fields satisfying the two dimensional Navier–Stokes system. Section 2.5.2 is devoted to the periodic case: the three variables are supposed to be periodic, and in that case dispersion cannot hold; it is replaced by a highly oscillatory behaviour, where the oscillations are linked to the eigenvalues of the Coriolis operator; once again those can be explicitly computed, due to the absence of boundary conditions and to the fact that the rotation is constant. It is only once those oscillations have been filtered out that a strong convergence result can also be proved. In both situations (the whole space case and the periodic case), the global existence of smooth solutions for a large enough rotation is also proved, using the special structure of the limiting system in each case. A word on more general domains is said in Section 2.5.3, while references can be found in Section 2.6. Finally the main results of Section 2 are stated in the next section. 2.2. Statement of the main results As explained in the introduction of Section 2, we are interested in the uniform existence of solutions to (RF ε ), as well as in the asymptotic behaviour of the solutions in the limit of a fast rotation, that is, as ε goes to zero. To simplify the presentation, we will restrict our attention to the case when the equations are set in a domain with no boundary. We will call such a domain, and we will denote by h the space of horizontal coordinates xh = (x1 , x2 ). Then h will be indifferently the space R2 or T2 , and 3 , defined by = h × 3 , will be indifferently R or T, unless specified otherwise. Let us start by stating the uniform existence theorem, which will be easily proved in Section 2.3 below. T HEOREM 2.1. Let u0 be a divergence free vector field in L2 (). Then there is a solution u (in the sense of distributions) to (RF ε ) with u|t=0 = u0 , and which satisfies the following energy estimate, uniformly in ε: ∀t ≥ 0,
u(t)2L2 + 2
0
t
∇u(t )2L2 dt ≤ u0 2L2 .
In particular u is bounded in L2loc (R+ , Lq ()) for any q ∈ [2, 6]. 1
Furthermore if u0 ∈ H 2 (), then there is a time T > 0 independent of ε such that u 1 3 belongs to C([0, T ], H 2 ()) ∩ L2 ([0, T ], H 2 ()), with a norm independent of ε, and all solutions associated with u0 coincide with u on [0, T ]. R EMARK 2.1. As usual the pressure is not considered as an unknown in this system, since once u is known, p is retrieved through the formula −p = div(u · ∇u) +
1 div u⊥ . ε
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The important point to notice in that statement is the fact that all bounds are uniform in ε. It therefore makes sense to inquire on the limiting behaviour of the solution as ε goes to zero. In particular can one describe the dependence of the life span T on ε? Can one find a limit to the system as ε goes to zero? In the following we will emphasize the dependence on ε of the solutions given by Theorem 2.1 by denoting them uε . They will therefore be seen as a bounded (in L2 ) family of divergence free vector fields, whose asymptotics as ε goes to zero we want to explore. We will start by studying the weak asymptotics, and recover the Taylor–Proudman theorem, stating that as rotation increases, the mean flow becomes two dimensional. The proof of the following result can be found in Section 2.4 below. def
def
def
We have noted ∇h = (∂1 , ∂2 ), divh = ∇h ·, and h = ∂12 + ∂22 . Moreover for any vector field u = (u1 , u2 , u3 ) we define uh = (u1 , u2 ). In the next theorem we have defined || as the measure of the set if it is bounded, and ||−1 = 0 otherwise. T HEOREM 2.2. Let u0 be any divergence free vector field in L2 , and let uε be any weak solution of (RF ε ). Then uε converges weakly in L2loc (R+ × ) to a limit u which if 3 = R is zero, and if 3 = T is the solution of the two dimensional Navier–Stokes equations in h ⎧ ⎪ ⎪ ⎪ ⎨ (NS2D)
∂t u − h u + uh · ∇h u = (−∇h p, 0)
divh uh = 0 ⎪ 1 ⎪ ⎪ ⎩ u|t=0 = u0 (xh , x3 ) dx3 − (u0h (x), 0) dx. || T
R EMARK 2.2. We recall that J. Leray proved in [37] that a unique, global solution to the two dimensional Navier–Stokes equations exists, as soon as the initial data is in L2 . Once the mean flow has been described, it is natural to address the question of the strong convergence of solutions. In fact the answer to that question depends strongly on the boundary conditions. We will be mainly interested in two very different situations here: the case when the equations are posed in the whole space, and the periodic case. Let us state the theorem concerning each situation, starting by the whole space case which is studied in Section 2.5.1. T HEOREM 2.3. Let u0 and w 0 be two divergence free vector fields, respectively in L2 (R2 ) and in L2 (R3 ). Let u be the unique solution of the two dimensional Navier–Stokes equations associated with u0 , and let uε be any weak solution to (RF ε ) associated with u0 + w 0 (such a solution may be constructed as in Theorem 2.1 above). Then for any q ∈]2, 6[ and for any time T , we have lim
ε→0 0
T
uε (t) − u(t)2Lq (R3 ) dt = 0.
R EMARK 2.3. Theorem 2.3 shows that the weak convergence result stated in Theorem 2.2 is in fact strong. All x3 -dependent vector fields converge strongly to zero as ε goes to zero, and at the limit remains only the two-dimensional behaviour—note that the presence of u0
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in the initial data enables one to understand precisely that two-dimensional behaviour; if the initial data is purely three-dimensional (that is, if u0 = 0), then Theorem 2.3 states that all weak solutions uε converge strongly to zero with ε. The main ingredient in the proof of that result is a dispersive estimate, implying that the eigenvectors corresponding to the oscillatory modes created by the Coriolis operator converge strongly to zero. That fact, when applied to strong solutions, will enable us to prove the global wellposedness of (RF ε ), despite its likeness to the 3D Navier–Stokes equations for which such a result is unknown. We state the result in an unprecise way here, and refer to Theorem 2.7 for a precise statement. T HEOREM 2.4. Let u0 and w 0 be two divergence free vector fields, respectively in L2 (R2 ) 1 and H 2 (R3 ). Then a positive ε0 exists such that for all ε ≤ ε0 , there is a unique global solution uε to the system (RF ε ) associated with u0 + w 0 . We will also be interested in the periodic situation. In that case the equations are set in a def
periodic box T3 = (R/Z)3 , and we will also be able to prove the global existence of strong solutions; however the asymptotic behaviour of the solutions is less easy to describe: due to the absence of dispersion, we need to filter out the oscillatory modes before taking the strong limit. In the next theorem we have defined the operator L(t) = etL where L is the Coriolis operator L : u ∈ L2 → P(u⊥ ) ∈ L2
(2.2.1)
and P denotes the Leray projection from L2 () onto its subspace of divergence-free vector fields. In that statement, a limit system is also referred to, which will be studied in Section 2.5.2. That system is presented (page 255), and the main steps of the result are described in Section 2.5.2. 1
T HEOREM 2.5. Let u0 be a divergence free vector field in H 2 (T3 ). Then a positive ε0 exists such that for all ε ≤ ε0 , there is a unique global solution to the system (RF ε ) 1 3 in Cb (R+ ; H 2 (T3 )) ∩ L2 (R+ ; H 2 (T3 )) associated with u0 . Moreover we have 1 3 t u = 0 in L∞ R+ ; H 2 (T3 ) ∩ L2 R+ ; H 2 (T3 ) , lim sup uε − L ε ε→0 where u is the unique, global solution of the limit system (RFL) (page 255) associated with u0 . R EMARK 2.4. Let us compare this theorem with Theorem 2.4 stated above. As far as the life span of the solutions is concerned, those two theorems state essentially the same result: for any initial data, if the rotation is large enough, then the rotating fluid equations are globally wellposed, although they are very like the 3D Navier–Stokes equations, for which that is an open question. In other words, the rotation term has a stabilizing effect.
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In the case of the whole space R3 this global wellposedness for small enough ε is due to the fact that the Rossby waves go to infinity immediately; that is a dispersive effect. In the case of the torus, there is of course no such dispersive effect (at least for uniform time intervals). The global wellposedness comes in a totally different way: it is a consequence of the analysis of resonances of Poincaré waves in the non linear term v · ∇v, using again the explicit formulation of the eigenvalues of the Coriolis operator, in Fourier variables. As far as the asymptotics are concerned, the statements of Theorems 2.3 and 2.5 are very different since in the whole space case, there is no trace of the rotation at the limit whereas in the periodic case, the limit system includes spectral information on the rotation operator. The rest of Section 2 is devoted to the proof of those results. 2.3. Uniform existence In this short section we address the question of the wellposedness of system (RF ε ) and we prove Theorem 2.1. This system is very like the three dimensional Navier–Stokes system, for which it is well known that global (possibly not unique) weak solutions exist if the initial data is of finite energy (meaning it belongs to L2 ). Furthermore local in time, unique 1 solutions exist if the initial data is smooth enough (say in the Sobolev space H 2 ). The proof of both those results relies on energy estimates, the main ingredient consisting (formally) in the first case in multiplying scalarly the system by u, and by ∇u in the second—of course there is much more to the proofs than that calculation, and we refer to [36] and [20] for the original proofs, and for instance to [13] for a more recent presentation (as well as the application to (RF ε )). Since the Coriolis operator is skew-symmetric in every Sobolev space, in the sense that for any s ∈ R, (u⊥ | u)H s = 0, the previous proofs go unchanged if we add the rotation term to the Navier–Stokes equations. Theorem 2.1 follows therefore immediately, once the corresponding proofs for the three dimensional Navier–Stokes system are known. 2.4. Weak asymptotics In this section we are going to describe the weak limiting behaviour of uε , and prove Theorem 2.2. Let us start by making some general comments on the asymptotics of uε as ε goes to zero. As the family (uε )ε>0 is bounded in the energy space, up to the extraction of a subsequence it has a weak limit point u. Formally taking the limit in the equation satisfied by uε allows to expect the weak limit points u to satisfy div u = 0
and
u⊥ = −∇p
for some function p. It is easy to see that, in the absence of nonvanishing boundary conditions, this is equivalent to the fact that ∂3 u = 0 and divh uh = 0. We therefore formally
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recover the Taylor–Proudman theorem: the mean motion at the limit is governed by a twodimensional, divergence free vector field. Let us now find rigorously the nature of the weak limit points of uε . Below H˙ 1 () denotes the homogeneous Sobolev space of order one, made of the distributions f such that ∇f belongs to L2 (). P ROPOSITION 2.5. Let u0 be any divergence free vector field in L2 (). Denote by (uε )ε>0 a family of weak solutions of (RF ε ), and by u any of its limit points. Then u is a three component, divergence free vector field satisfying u ∈ L∞ (R+ ; L2 (h ) ∩ L2 ()) ∩ L2 (R+ ; H˙ 1 (h )). Moreover we have h uh (t, xh ) dxh = 0. R EMARK 2.6. If 3 = R then the only possible limit point is 0. Indeed there are no vector fields other than 0 which are in L2 (h × R) and do not depend on the vertical variable. P ROOF OF P ROPOSITION 2.5. The proof simply consists in multiplying (RF ε ) by a divergence-free test function εχ , where χ ∈ D(R+ × ). Integrating with respect to t and x gives directly, using the bounds coming from the energy estimate, that u ∈ Ker(L), where L was defined in (2.2.1). Then it is just a matter of determining the kernel of L, and an easy computation gives the following proposition. We omit the proof here (for the interested reader, it is written in Section 3 in a more general case, see Proposition 3.3). P ROPOSITION 2.7. If u is a divergence free vector field in L2 () belonging to Ker(L), then u is in L2 (h ) and satisfies the following properties: and uh dxh = 0. div uh = 0 h
h
The next question consists in finding the evolution equation satisfied by u. Due to Remark 2.6, we shall now consider only the case when 3 = T. Moreover to simplify we normalize T in the following so that T dx3 = 1. The idea to find the limit equation is to take weak limits in (RF ε ), the difficulty coming of course from the nonlinear terms. The first step of the analysis consists in proving the compactness of the vertical average of uε . The second step then consists in proving a compensated-compactness type result to show that there are no constructive interferences of x3 -dependent vector fields. 2.4.1. Compactness of vertical averages Let us start by proving the following proposition, which shows that the defect of compactness of the sequence of solutions uε can only be due to functions depending on the vertical variable. P ROPOSITION 2.8. Let u0 be any divergence free vector field in L2 . For all ε > 0, denote by uε a weak solution of (RF ε ), and define def def 1 uε (xh ) = uε (x) dx3 and uε = (uε,h (x), 0) dx. |h | h ×T T
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Then the sequence (uε − uε )ε>0 is strongly compact in L2 ([0, T ] × h ), for all times T . P ROOF OF P ROPOSITION 2.8. Let us take the vertical average of (RF ε ). Since horizontal mean free, x3 -independent vector fields are in the kernel of L due to Proposition 2.7, we have 1 ⊥ ⊥ = 0. P(uε ) dx3 − P(u⊥ ε ) dx = P (uε − uε ) |h | h ×T T It follows that ∂t (uε − uε ) − h uε + P
uε (x) · ∇uε (x) dx3 = 0.
(2.4.1)
T
Regularity with respect to space variables follows from the energy estimate, since uε is uniformly bounded in L2 ([0, T ], H 1 ()) for all times T . Regularity with respect to time is obtained classically by finding an a priori bound on ∂t (uε − uε ). It is indeed easy to see that uε · ∇uε is bounded in L2 (R+ ; H −3/2 ()), and that uε is bounded in L2 (R+ ; H˙ −1 ()) ⊂ L2 (R+ ; H −3/2 ()), so ∂t uε is uniformly bounded in L2 (R+ ; H −3/2 (h )). We can therefore infer by interpolation (using Aubin’s lemma for instance) that (uε − uε )ε>0 is strongly compact in L2loc (R+ ; L2 (h )), which proves the proposition. We infer from that result that x3 -dependent vector fields are the only obstacles to taking the limit in the non linear term. We are going to see that such vector fields do not interfere constructively in the non linear term of the equation. 2.4.2. The weak limit of the nonlinear term This section is the main step of the analysis of the weak limiting behaviour of (RF ε ), since it consists in proving that when taking the limit of the nonlinear term, there are no constructive interferences of oscillations. We will start by giving the general strategy of the proof, before going into the details. Strategy of the proof Our aim is to prove that the limit of the nonlinear term only involves the nonlinear interaction of the weak limit. More precisely we want to prove that as ε goes to zero, we have P uε · ∇uε dx3 → P(u · ∇u), T
where u is a weak limit of uε . Of course that convergence must be made more precise, in particular we need to determine in what function space it holds. In fact since we are dealing with nonlinear quantities involving weak solutions to our system, it will be convenient to start by regularizing the family uε by introducing a smooth vector field uδε which converges uniformly towards uε in L2loc (R+ ; L2 ()) as δ goes to zero. That is possible due to the additional smoothness of uε given by the Laplacian. Then we will be able to carry
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out computations on nonlinear quantities involving uδε without worrying about regularity issues (only at the very end of the argument will we let δ go to zero). Those computations consist in writing out the nonlinear term uδε · ∇uδε as the expected limit u · ∇u, to which one needs to add error terms. Those error terms naturally involve functions which are oscillatory in time, and using the algebraic properties of the wave equations associated with those oscillatory functions (see Lemma 2.9), it is possible to prove that they contribute to negligible quantities, up to gradient terms. The precise statement is given in Proposition 2.10 below. In the following we will therefore start by writing out those wave equations, applied to smoothened vector fields. It turns out that the computations are best carried out on the vorticity formulation of the equation (since the vector fields are smooth for a fixed δ, that does not create additional regularity problems). Using those equations given in Lemma 2.9, we are then able to prove the expected convergence of the quadratic term (Proposition 2.10). Convergence of the quadratic term The proof of that result requires some preparation, and we will start this study by rewriting the equations in a convenient way for future algebraic computations. Let us start by taking the rotational of the equation, by defining def
ωε = ∂1 uε,2 − ∂2 uε,1 with
8
and
8ε,h def ∂3 = (rot8 uε )h = ∇h⊥8 uε,3 − ∂38 u⊥ ε,h ,
= 0. We write, for any vector field a,
T ε,h (x) dx3
a(xh ) =
and
a(xh , x3 ) dx3
8 a = a − a,
8 a (xh , x3 ) dx3 = 0.
with
T
T
8 with In particular we have 8 a = ∂3 A, previous section implies that
8 h , x3 ) dx3 = 0. Equation (2.4.1) derived in the A(x
T
ε∂t ωε = ε(∂1 F ε,2 − ∂2 F ε,1 ) where Fε denotes the flux term def
Fε = h uε − P∇·(uε ⊗ uε ). It is easy to see that (∂1 Fε,2 − ∂2 Fε,1 ) is bounded in L2 (R+ ; H −5/2 ()) (see for instance the proof of Proposition 2.8), so we can write ε∂t ωε = εr ε ,
where r ε is uniformly bounded in L2 (R+ ; H −5/2 ()).
Similarly an easy computation yields the following equation for 8 ωε : ε∂t 8 ωε − div 8 uε,h = ε8 rε , h
where 8 rε is uniformly bounded in L2 (R+ ; H −5/2 ()).
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For the other components of the vorticity, the computations are similar: since ∇ ∧ u⊥ ε = ∂3 uε , we find after integration in the vertical variable 8ε , 8ε,h + 8 uε,h = ε R ε∂t
8ε is uniformly bounded in L2 (R+ ; H −5/2 ()). where R
∞ 3 + Now let us proceed with the regularization: let κ ∈ Cc (R ; R ) such that κ(x) = 0 if |x| ≥ 1 and κdx = 1. We define
1 . κδ : x → 3 κ δ δ
(2.4.2)
as well as def
ωεδ = ωε ∗ κδ = ωδε + 8 ωεδ ,
8δε def 8ε ∗ κδ . =
and
It is not difficult to see that the following result holds. We leave the details to the reader (see [24]). L EMMA 2.9. Let u0 be any divergence free vector field in L2 . For all ε > 0, denote by uε a weak solution of (RF ε ). Then, for all ε > 0, there is a family (uδε )δ>0 of smooth vector fields in L2 (R+ ; ∩s H s ()) such that lim uδε = uε
δ→0
in L2loc (R+ ; Lp ())
for all p ∈ [2, 6[,
uniformly in ε,
and such that the functions def
and ωεδ = ∂1 uδε,2 − ∂2 uδε,1 8δε,h (x) dx3 = 0 with
8δε,h def ∂3 = rot8 uδε h ,
T
satisfy the following equations: ε∂t ωδε = εr δε , ωεδ − divh 8 uδε,h = ε8 rεδ , ε∂t 8 δ 8ε,h 8δε,h + 8 uδε,h = ε R and ε∂t
8δ are uniformly where for all δ > 0, the functions r δε and 8 rεδ , as well as the vector field R ε,h bounded in ε in the space L2 (R+ ; L2 ()). With that lemma we are ready to study the limit of the non linear term. Let us give the main steps of the proof of the following result.
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P ROPOSITION 2.10. Let u0 be any divergence free vector field in L2 . For all ε > 0, denote by uε a weak solution of (RF ε ), and by (uδε )δ>0 the approximate family of Lemma regularisationconstant. Then for any ε > 0 and any δ > 0, we have T
δ ⊥ uε · ∇uδε h dx3 = −ωδε,h uδε,h + ∇h + ε∂t T
|uε,3 |2 |uδε |2 dx3 − ∇h 2 2 δ
T
δ ⊥ δ 8ε,h dx3 + ερε,h 8 ωεδ ,
and
δ 1 uε · ∇uδε 3 dx3 = div uδε,3 uδε,h + ε∂t h 2 T
T
δ δ ⊥ δ 8ε,h · ∂3 8ε,h dx3 + ερε,3 ,
where the vector field ρεδ satisfies ∀δ > 0,
∀T > 0,
sup ρεδ L1 ([0,T ];L6/5 ()) < +∞. ε>0
P ROOF OF P ROPOSITION 2.10. Since uδε is divergence free, we have |uδ |2 uδε · ∇uδε = ∇ · uδε ⊗ uδε = ∇ ε − uδε ∧ ∇ ∧ uδε , 2
(2.4.3)
so we shall now restrict our attention to the term uδε ∧ (∇ ∧ uδε ). We have of course T
uδε
∧ ∇ ∧ uδε dx3 = uδε ∧ ∇ ∧ uδε +
T
8 uδε ∧ ∇ ∧ 8 uδε dx3 .
(2.4.4)
Let us start by considering the first term in the right-hand side of (2.4.4). A direct computation gives 1 ⊥ uδε ∧ ∇ ∧ uδε = ∇|uδε,3 |2 + ωδε,h uδε,h − div uδε,3 uδε,h e3 . h 2 To compute the second term in the right-hand side of (2.4.4), we will use the equations derived in Lemma 2.9. To simplify the presentation we shall set to zero all remainder terms appearing in that lemma. We have uδε ) = 8 uδε ∧ (∇ ∧ 8
8δ )⊥ ) − divh 8 8δ )⊥ (8 uδε,h )⊥ 8 ωεδ − ∂3 (8 uδε,3 ( uδε,h ( ε,h ε,h 8δ −(8 uδε,h )⊥ · ∂3 ε,h
. (2.4.5)
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Let us study first the horizontal components in (2.4.5): by Lemma 2.9, neglecting all remainder terms, we have δ ⊥ δ ⊥ δ δ ⊥ δ ⊥ δ 8ε,h = −ε∂t 8ε,h 8 8ε,h . ωε − div 8 uδε,h ωε − div 8 uδε,h 8 uε,h 8 h
h
But by Lemma 2.9 again, we have − div 8 uδε,h = −ε∂t 8 ωεδ , h
so T
δ,⊥ δ δ ⊥ 8ε,h 8 uε,h 8 dx3 = −ε∂t ωε − div 8 uδε,h h
T
δ ⊥ δ 8ε,h 8 ωε dx3 .
Now we are left with the last term in (2.4.5), which is the third component: we can write, by Lemma 2.9, 8δε,h , 8 uδε,h = −ε∂t so δ ⊥ δ ⊥ 8δε,h = −ε∂t 8δε,h . 8ε,h · ∂3 8 uε,h · ∂3 Then we just need to notice that δ ⊥ 1 δ 1 δ 8δε,h = − ε∂t 8δε,h ⊥ + ∂3 8δε,h ⊥ . 8ε,h · ∂3 8ε,h · ∂3 8ε,h · ε∂t ε∂t 2 2 Putting those computations together yields finally the proposition. Note that the regularization procedure is useful here, as the (omitted) remainder terms go to zero in the expected functional space as ε goes to zero, for all δ > 0. The parameter δ will go to zero at the very end of the argument leading to the theorem. That result enables us easily to infer the following corollary. C OROLLARY 2.11. Let u0 be any divergence free vector field in L2 . For all ε > 0, denote by uε a weak solution of (RF ε ). Then for any vector field φ ∈ H 1 ∩ Ker (L), we have the following limit in W −1,1 ([0, T ]) for any T > 0: lim ∇ · (uε ⊗ uε ) · φ(xh ) dx − ∇h · (uε ⊗ uε ) · φ(xh ) dxh = 0. ε→0
h
Now we are ready to prove the convergence theorem. P ROOF OF T HEOREM 2.2. The proof of Theorem 2.2 follows quite easily from the previous results: we have seen that uε converges weakly in L2 ([0, T ] × ) towards a vector
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field u depending only on the horizontal variable and mean free, due to Proposition 2.5. Then we proved that uε − uε is strongly compact, therefore converges strongly towards u in the space L2 ([0, T ] × ) (since u = u and u = 0). Finding the equation satisfied by the limit is therefore a matter of computing the limit of Equation (2.4.1). The linear terms converge in the sense of distributions of course, and to find the limit of the nonlinear term we use Proposition 2.11 as well as the following weak-strong limit argument: we have ∇ · (uε ⊗ uε ) = ∇ · (uε ⊗ (uε − uε )) + ∇ · ((uε − uε ) ⊗ uε ))) + ∇ · ((uε − uε ) ⊗ (uε − uε )). The two first terms converge towards zero in D () since uε − uε is compact and uε converges weakly to zero, whereas the last term satisfies ∇ · ((uε − uε ) ⊗ (uε − uε )) → ∇ · (u ⊗ u)
in D ().
That gives the expected result: the limit u satisfies the two dimensional Navier–Stokes equation ∂t u − h u + P∇h · (u ⊗ u) = 0. Theorem 2.2 is proved.
We therefore recover as expected the Taylor–Proudman theorem. Now the question consists in making that convergence result more precise, by describing more finely the oscillations of uε . We have seen that they do not contribute to the limiting behaviour of the system, but it remains to understand if they are actually an obstruction to the strong convergence or not. The answer to that question depends on the boundary conditions, as shown in the following section where the case of the whole space R3 and the periodic case are studied. As pointed out in the introduction, the fact that the rotation is constant will enable us to describe very precisely the oscillations, by use of the Fourier transform.
2.5. Strong asymptotics In this section we are going to prove precised versions of Theorem 2.2, by analyzing the strong asymptotics of uε . We will mainly focus on two situations, first the case when the equations are set in the whole space R3 (proving Theorems 2.3 and 2.4), and then the periodic case (proving Theorem 2.5). Comments on more general boundary conditions can be found in Section 2.5.3. 2.5.1. The whole space case Let us suppose here that the equations are set in R3 , and let us consider again a family of weak solutions to (RF ε ). Due to the result proved in the previous section, we know that the weak limit of any such family is necessarily zero,
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and the question is to know whether it converges strongly or not. The answer is given in Theorem 2.3, where in order to give a more general statement we have considered the case when the initial data is the sum of a purely two-dimensional vector field (with possibly three components) and a three dimensional vector field. P ROOF OF T HEOREM 2.3. We shall leave as an exercise to the reader the proof of the existence of a solution to (RF ε ) with initial data u0 + w 0 in L2 (R2 ) + L2 (R3 ), which is an easy adaptation of Proposition 2.3. The main ingredient of the proof of the strong convergence result is a so-called “Strichartz estimate” on the Coriolis operator, which we will write now. Let us consider the linearized equations ⎧ v⊥ ⎪ ⎪ ∂ + ∇p = f v − v + ⎪ t ⎪ ⎨ ε (LRε )
div v = 0
⎪ ⎪ ⎪ ⎪ ⎩
v|t=0 = v 0 ,
which yields in Fourier variables ξ ∈ R3 ⎧ v∧ξ ξ3( ⎪ ⎪ v + |ξ |2( v+ = f( ⎨ ∂t ( ε|ξ |2 ⎪ ⎪ ⎩ ( v|t=0 = ( v0. We will denote by f(or Ff the Fourier transform of any function or vector field f , defined by Ff (ξ ) =
e−ix·ξ f (x) dx.
R3
def ξ3 v∧ξ |ξ |2
The matrix Mv = are
has three eigenvalues, 0 and ±i |ξξ3| · The associate eigenvectors e0 (ξ ) =t (0, 0, 1)
and
t 1 ξ1 ξ3 ∓ iξ2 |ξ |, ξ2 ξ3 ± iξ1 |ξ |, −|ξh |2 . e± (ξ ) = √ 2|ξ ||ξh |
The precise value of those vectors will not be necessary for our study; all we shall need to know is that the last two are divergence free, in the sense that ξ · e± (ξ ) = 0. Furthermore they are orthogonal, in the sense that if v belongs to L2 (R3 ) then v2L2 = v + 2L2 + v − 2L2
def v (ξ ) · e± (ξ ))e± (ξ ) . where v ± = F −1 ((
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We are now led to studying the application G ε,± (τ ): g → =
ξ3
R3ξ
( g (ξ )e±iτ |ξ | −τ ε|ξ | ξ3
R3ξ ×R3y
2 +ix·ξ
g(y)e±iτ |ξ | −τ ε|ξ |
2 +i(x−y)·ξ
dξ
dξ dy,
and we will start by considering the case when ( g is supported in Cr,R for some r < R, where Cr,R = {ξ ∈ R3 /|ξ3 | ≥ r
and
|ξ | ≤ R}.
(2.5.1)
In that situation we can multiply ( g (ξ ) in the previous formula by a function ψ in D(R3 \ {0}), such that ψ ≡ 1 in a neighborhood of Cr,R , and which is radial with respect to the horizontal variable ξh = (ξ1 , ξ2 ). For instance we suppose that ψ is supported in the set Cr/2,2R . We are now led to studying the following function: ±
def
K (t, τ, z) =
R3ξ
ψ(ξ )e±iτ a(ξ )+iz·ξ −t|ξ | dξ, 2
def
where a(ξ ) =
ξ3 · |ξ |
The following result is the main step of the proof of Strichartz estimates; it is a dispersion estimate. L EMMA 2.12. For any (r, R) such that 0 < r < R, a constant Cr,R exists such that ∀z ∈ R3 , 1
1 2
|K ± (t, τ, z)| ≤ Cr,R min{1, τ − 2 }e− 2 r t . P ROOF OF L EMMA 2.12. For the sake of simplicity we will only consider K + , the term K − being dealt with exactly in the same way. This proof is very like the proof of the more usual dispersive estimate for the wave equation. First using the rotation invariance def
in ξh , we restrict ourselves to the case when z2 = 0. Next, denoting α(ξ ) = −∂ξ2 a(ξ ) = ξ2 ξ3 /|ξ |3 , we introduce the following differential operator: def
X =
1 1 + iα(ξ )∂ξ2 , 2 1 + τ α (ξ )
which satisfies X (eiτ a ) = eiτ a . Integrating by parts, we obtain K + (t, τ, z) =
R3
2 eiτ a(ξ )+iz1 ξ1 +iz3 ξ3 t X ψ(ξ )e−t|ξ | dξ
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Easy computations yield t
2 X ψ(ξ )e−t|ξ | =
1 1 − τ α2 2 ψ(ξ )e−t|ξ | − i(∂ α) ξ2 2 2 2 1 + τα (1 + τ α )
−
iα 2 ∂ξ2 e−t|ξ | ψ(ξ ) . 2 1 + τα
As ξ belongs to the support of ψ, which is supposed to be included in Cr/2,2R as defined in (2.5.1), we can prove that " " 2 " "t " X ψ(ξ )e−t|ξ | " ≤
Cr,R − 1 tr 2 e 2 1 + τ ξ22
so we obtain, for all z ∈ R3 , 1
|K + (t, τ, z)| ≤ Cr,R e− 2 tr
2
R
dξ2 , 1 + τ ξ22
which proves Lemma 2.12.
Lemma 2.12 yields the following theorem, whose proof is quite classical in the context of Strichartz estimates (it is based on the so-called T T ∗ argument) and is omitted. T HEOREM 2.6. For any positive constants r and R such that r < R, let Cr,R be the frequency domain defined in (2.5.1). Then a constant Cr,R exists such that if v 0 ∈ L2 (R3 ) and f ∈ L1 (R+ ; L2 (R3 )) are two vector fields whose Fourier transform is supported in Cr,R , and if v is the solution of the linear equation (LRε ) with forcing term f and initial data v 0 , then for all p ∈ [1, +∞], 1 vLp (R+ ;L∞ (R3 )) ≤ Cr,R ε 4p v 0 L2 (R3 ) + f L1 (R+ ;L2 (R3 )) .
We see that the solution of the linearized equations converges strongly to zero as ε goes to zero. Now let us conclude the proof of Theorem 2.3. We define wε = uε − u, and we are going to prove that wε goes to zero as ε goes to zero, in L2loc (R+ ; Lq (R3 )) for any 2 < q < 6. One can prove, by an energy estimate on the equation satisfied by wε , that ∀t ≥ 0,
wε (t)2L2 +
t
∇wε (t )2L2 dt ≤ w 0 2L2 exp Cu0 2L2 (R2 ) .
0
Indeed we have formally wε (t)2L2
+2
t
∇wε (t 0
)2L2
" t " " "
" " dt = " 3 (wε · ∇)u · wε (t , x) dx " dt ,
0
R
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and using the two-dimensional Gagliardo–Nirenberg inequality we can write " " " " " (wε · ∇)u · wε dx " ≤ ∇uL2 wε (·, x3 )2L4 (R2 ) dx3 " 3 " R R ≤ ∇uL2 wε (·, x3 )L2 (R2 ) ∇wε (·, x3 )L2 (R2 ) dx3 R
≤
∇wε 2L2 (R3 )
+ C∇u2L2 wε 2L2 (R3 ) ,
and the above estimate follows from Gronwall’s inequality. In order to use the Strichartz estimates of Theorem 2.6, we have to get rid of high frequencies and low vertical frequencies. Let us define the following truncation operator D f, PRf = χ R def
where χ ∈ D(] − 2, 2[), χ(x) = 1 for |x| ≤ 1.
In other words we have ξ FP R f (ξ ) = χ f (ξ ). R Let us observe that, thanks to Sobolev embeddings and the energy estimate, we have, for any q ∈ [2, 6[, wε − P R wε L2 (R+ ;Lq (R3 )) ≤ Cwε − P R wε
&
1 1 ' 2−q
3 L2 R+ ;H˙
≤ CR −αq wε L2 (R+ ;H˙ 1 ) ≤ CR −αq w 0 L2 exp Cu0 2L2 (R2 ) def 3 q
with αq =
(2.5.2)
def − 12 · Now let us define χ( Dr3 )f = F −1 (χ( ξr3 )f((ξ )). We have
9 9 9 D3 9 9χ 9 P w R ε9 9 r
9 9 9 ξ3 9 9 χ F(P ≤9 w ) R ε 9 9 r L2 (R+ ;L∞ ) L2 (R+ ;L1 )
so using the fact that |ξ | ≤ R and a Cauchy–Schwartz inequality one can prove that 9 9 9 D3 9 9χ 9 P w R ε9 9 r
L2 (R+ ;L∞ )
def
1 ≤ CR r 2 w 0 L2 exp Cu0 2L2 (R2 ) .
(2.5.3)
Let us define P r,R = (Id −χ( Dr3 ))P R . The following lemma, whose proof is postponed for a moment, describes the dispersive effects due to fast rotation.
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L EMMA 2.13. For any positive real numbers r, R and T , and for any q in ]2, +∞[, 9 9 1 1− 2 ∀ε > 0, 9P r,R wε 9L2 ([0,T ];Lq (R3 )) ≤ Cε 8 q ,
the constant C above depending on r, q, R, T , u0 L2 and w 0 L2 but not on ε. Together with Inequalities (2.5.2) and (2.5.3), this lemma implies that, for any positive r, R and T , for q ∈]2, 6[, ∀ε > 0,
1
1
wε L2 ([0,T ];Lq ) ≤ CR −αq + CR r 2 + C3 ε 8
1− q2
,
the constant C3 above depending on r, R, T , u0 L2 and w 0 L2 but not on ε. We deduce that, for any positive r, R and T , for q ∈]2, 6[, 1
lim sup wε L2 ([0,T ];Lq ) ≤ CR −αq + CR r 2 . ε→0
Passing to the limit when r tends to 0 and then when R tends to ∞ gives Theorem 2.6, provided of course we prove Lemma 2.13. P ROOF OF L EMMA 2.13. Thanks to Duhamel’s formula we have, P r,R wε (t) =
3
j
P r,R wε (t)
with
j =1
t G P r,R w 0 , ε t
def 2 ε t −t P r,R wε (t) = G P r,R Q(wε (t ), wε (t )) dt and ε 0 t t − t def P 3r,R wε (t) = Gε P r,R Q(wε (t ), u(t )) + Q(u(t ), wε (t )) dt . ε 0
def P 1r,R wε (t) =
ε
We have defined Q(a, b) = P(a · ∇b). Theorem 2.6 implies that 1
P 1r,R wε L2 (R+ ;L∞ ) ≤ Cr,R ε 8 w 0 L2 . By interpolation with the energy bound, we infer that 1
P 1r,R wε L2 ([0,T ];Lq ) ≤ Cr,R,T ε 8
1− q2
w 0 L2 exp Cu0 2L2 (R2 ) .
Using again Theorem 2.6, we have 1
P 2r,R wε L2 ([0,T ];L∞ ) ≤ Cr,R ε 8 P R Q(wε , wε )L1 ([0,T ];L2 ) .
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Let us recall Bernstein’s lemma: if a function F has its Fourier transform supported in a ball of radius R, then for all k ∈ N and all 1 ≤ p ≤ q ≤ ∞, sup ∂ α F Lq (Rd ) ≤ CR
k+d
1
1 p−q
|α|=k
F Lp (Rd ) .
That lemma together with the energy estimate implies that P R Q(wε , wε )L1 ([0,T ];L2 ) ≤ CRP R (wε ⊗ wε )L1 ([0,T ];L2 ) 3
≤ CR 1+ 2 wε ⊗ wε L1 ([0,T ];L1 ) ≤ CR T w 0 2L2 (R3 ) exp Cu0 2L2 (R2 ) . By interpolation with the energy bound, we infer 1
P 2r,R wε L2 ([0,T ];Lq ) ≤ Cr,R,T ε 8
1− q2
2(1− q1 )
w 0 L2
exp Cu0 2L2 (R2 ) .
Still using Theorem 2.6, we have 1 P 3r,R wε L2 ([0,T ];L∞ ) ≤ Cr,R ε 8 P R Q(wε , u) + Q(u, wε ) L1 ([0,T ];L2 ) . Similarly, using an anisotropic-type Bernstein inequality, we can prove that 1 P 3r,R wε L2 ([0,T ];L∞ ) ≤ Cr,R,T ε 8 u0 L2 w 0 L2 exp Cu0 2L2 , and by interpolation with the energy bound, we get 1
P 3r,R wε L2 ([0,T ];Lq ) ≤ Cr,R,T ε 8 The lemma is proved.
1− q2
1− 2 u0 L2 q w 0 L2 exp Cu0 2L2 .
Once the behaviour of weak solutions has been investigated, it is natural to consider strong solutions. The question of their convergence is easily settled due to Theorem 2.3 (in 1 particular if the initial data is in H 2 (R3 ) then the strong solutions converge to zero, and if 1 it is in L2 (R2 ) + H 2 (R3 ) then they will converge towards a two-dimensional vector field satisfying the two dimensional Navier–Stokes equations). However since that limit system is known to be globally well posed, one can try to use this information to recover a better 1 bound on the life span of the solutions to the rotating fluid equations in H 2 (R3 ), depending on ε. This is achieved through the following theorem, which is a precised version of Theorem 2.4 stated in Section 2.2. T HEOREM 2.7. Let u0 and w 0 be two divergence free vector fields, respectively in L2 (R2 ) 1 and H 2 (R3 ). Then a positive ε0 exists such that for all ε ≤ ε0 , there is a unique global
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solution uε to the system (RF ε ). More precisely, denoting by u the (unique) solution of the two dimensional Navier–Stokes equations associated with u0 , by vF the solution of (LRε ) def
with initial data w 0 (with f = 0), and defining wε = uε − u, then for ε small enough, wε 1 3 is unique in L∞ (R+ , H˙ 2 (R3 )) ∩ L2 (R+ , H˙ 2 (R3 )) and we have, as ε goes to zero, 1
wε ∈ Cb (R+ ; H 2 (R3 ))
1
∇wε ∈ L2 (R+ ; H 2 (R3 )),
and
wε − v F → 0
in L∞ (R+ ; H˙ 2 (R3 ))
and ∇(wε − vF ) → 0
in L2 (R+ ; H˙ 2 (R3 )).
1
1
P ROOF OF T HEOREM 2.7. We will not give the details of the proof here but simply some indications. The first step consists in checking that there is indeed a unique solu1 tion uε = u + wε to (RF ε ) in H 2 (R3 ) for some finite time. This is achieved in a similar way to the case of the Navier–Stokes equations (up to the presence of the perturbation term 1 involving u, which is not in L∞ (R+ ; H 2 ) but only in the energy space; however it only depends on two variables so an anisotropic Gagliardo–Nirenberg inequality gives the desired estimates). To prove that wε exists globally in time one needs to use more than an energy estimate, since such an estimate would be similar to the case of the 3D Navier–Stokes system, for which the global existence in time of a unique solution is not known. The idea is to subtract from wε the solution vF of (LRε ), which we know goes to zero (at least for restricted frequencies) by Strichartz estimates. We are then led to solving the system satisfied by wε − vF , which has small data (involving the extreme frequencies of w 0 ) and small source terms (due to the Strichartz estimates of Theorem 2.6). The usual methods for the 3D Navier–Stokes equations can then be used. Of course there are a few additional difficulties, the main one being that one needs to cope with the interaction of two dimensional and three dimensional vector fields; for that an anistropic-type Strichartz estimate is needed, but we shall not pursue this question here and refer to [13] for details. 2.5.2. The periodic case Section 2.5.2 deals with the rotating fluid equations (RF ε ) in def
a purely periodic setting: we define the periodic box T3 = (R/Z )3 . All the vector fields considered in Section 2.5.2 will be supposed to be mean free. We are interested in the (strong) asymptotic behaviour of uε as ε goes to zero, proving Theorem 2.5. The first step of the analysis consists in deriving a limit system for (RF ε ), which will enable us to state and prove a convergence theorem for weak solutions. The main issue will then consist in studying the behaviour of strong solutions. The proof of Theorem 2.5 relies on the construction of families of approximate solutions. Let us state the key lemma, where we have used the following notation: u 1 = sup u(t)2
2 def 2
t≥0
1
H2
+2 0
t
∇u(t )
2 1
dt
.
H2 1
L EMMA 2.14. Let u0 be a divergence free vector field in H 2 . For any positive real number η, a family (uapp )ε,η exists such that lim supη→0 lim supε→0 uapp 1 < ∞. MoreH2
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over, the families (uapp )ε,η are approximate solutions of (RF ε ) in the sense that uapp satisfies ⎧ 1 ⎪ ⎨ ∂t uapp − uapp + P(uapp · ∇uapp ) + P(u⊥ app ) = R ε ⎪ ⎩ lim lim uapp |t=0 − u0 1 = 0,
in
T3 (2.5.4)
H2
η→0 ε→0
with lim lim RL2 (R+ ,H −1/2 ) = 0.
η→0 ε→0
R EMARK 2.15. The stability of strong solutions to the Navier–Stokes equations enables us to prove that as soon as ε and η are small enough, the solution uε to (RF ε ) remains arbitrarily close to the solution uapp of (2.5.4): indeed both equations are the same up to the initial data and forcing terms, which can be made arbitrarily close. In particular uε satisfies lim lim sup uε − uapp
η→0
ε→0
1
H2
= 0,
which implies directly the global existence result of Theorem 2.5. It follows that the construction of the families (uapp )ε,η is the main step in the analysis of (RF ε ) in a periodic box. Moreover it will also enable us to further describe the asymptotics of uε as ε goes to zero, thus achieving the proof of Theorem 2.5. Let us give the main steps of the construction of the approximate family. Fast time oscillations prevent any result of strong convergence to a fixed function. In order to bypass this difficulty, we are going to introduce a procedure of filtering of the time oscillations. This will lead us to the concept of limit system. So we start by defining the filtering operator, the limit system and to establish that the weak closure of (uε )ε>0 is included, after filtration, in the set of weak solutions of the limit system. Then we prove that the nonlinear terms in the limit system have a special structure, very close to the structure of the nonlinear term in the 2D Navier–Stokes equations, which makes it possible to prove the global wellposedness of the limit system. The families (uapp )ε,η can then be constructed. So let us start by finding a limit system. We know that there is a bounded family of solutions (uε )ε>0 associated with the initial data, so one can extract a subsequence and find a weak limit to (uε )ε>0 . Then we will need some refined analysis to understand the asymptotic behaviour of (uε )ε>0 . Let L be the evolution group associated with the Coriolis operator L defined in (2.2.1): the vector field L(t)v 0 is the solution at time t of the equation ∂t v + Lv = 0,
v|t=0 = v 0 .
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As L is skew-symmetric, the operator L(t) is unitary for all times t, in all spaces H s (T3 ). In particular the “filtered solution” associated with uε t def 8 uε = L − uε , ε is uniformly bounded in the space L∞ (R+ ; L2 (T3 )) ∩ L2 (R+ ; H˙ 1 (T3 )). It satisfies the following system: 3 uε − Qε (8 uε ,8 uε ) − 8 uε = 0 ∂t 8 :ε ) (RF 0 8 uε|t=0 = u , noticing that L(t/ε) is equal to Identity when t = 0. We have used the fact that the operator L commutes with all derivation operators, and we have noted t t t 1 def Qε (a, b) = − L − P L a · ∇L b 2 ε ε ε t t t b · ∇L a . (2.5.5) +L − P L ε ε ε The point in introducing the filtered vector field 8 uε is that one can find a limit system : ε ) (contrary to the case of (RF ε )): if u0 is in L2 (T3 ), it is not difficult to see to (RF that contrary to the original system, the family (∂t 8 uε )ε>0 is bounded, for instance in the 4 −1 3 3 space L ([0, T ]; H (T )) for all T > 0. A compactness argument enables us, up to the extraction of a subsequence, to obtain a limit to the sequence 8 uε , called u. The linear terms ∂t 8 uε and 8 uε converge towards ∂t u and u respectively in D ((0, T ) × T3 ), so the point is to find the limit of the quadratic form Qε (8 uε ,8 uε ). Let us study that term more precisely. For any vector field u = (u1 , u2 , u3 ), u is the quantity def u(xh ) = u(xh , x3 ) dx3 . T
Note that if u is divergence free, then so is uh = (u1 , u2 ). Finally we decompose u into u = u + uosc , where the notation uosc stands for the “oscillating part” of u. Now in order to derive formally the limit of Qε , let us compute more explicitly the operators L and L. As in Section 2.5.1, the eigenvalues of Ln (where n ∈ Z3 denotes the Fourier variables) are 0, in3 /|n|, and −in3 /|n|. We will call e± (n) the associate eigenvectors, as defined in Section 2.5.1. Now we are ready to find the limit of the quadratic form Qε . In the following, def
we denote σ = (σ1 , σ2 , σ3 ) ∈ {+, −}3 any triplet of pluses or minuses, and for any vector field h, its projection (in Fourier variables) along those vector fields is denoted ∀n ∈ Z3 ,
∀j ∈ {1, 2, 3},
def hσj (n) = Fh(n) · eσj (n) eσj (n).
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P ROPOSITION 2.16. Let Qε be the quadratic form defined in (2.5.5), and let a and b be two smooth vector fields on T3 . Then one can define def
Q(a, b) = lim Qε (a, b) ε→0
and we have FQ(a, b)(n) = −
in D (R+ × T3 ),
[a σ1 (k) · (n − k)] [bσ2 (n − k) · eσ3 (n)]eσ3 (n),
σ ∈{+,−}3 k∈Knσ
where Knσ is the “resonant set” defined, for any n in Z3 and any σ in {+, −}3 , as % k3 n 3 − k3 n3 σ def 3 + σ2 − σ3 =0 · (2.5.6) Kn = k ∈ Z /σ1 |k| |n − k| |n| P ROOF OF P ROPOSITION 2.16. We shall write the proof for a = b to simplify. We can write
m3 n3 t k3 −FQε (a, a)(n)(t) = e−i ε σ1 |k| +σ2 |m| −σ3 |n| [a σ1 (k) · m] (k,m)∈Z6 ,σ ∈{+,−}3 k+m=n
× [a σ2 (m) · eσ3 (n)]eσ3 (n). To find the limit of that expression in the sense of distributions as ε goes to zero, one integrates it against a smooth function ϕ(t). That can be seen as the Fourier transform k3 m3 n3 of ϕ at the point 1ε (σ1 |k| + σ2 |m| − σ3 |n| ), which clearly goes to zero as ε goes to zero,
k3 m3 n3 if σ1 |k| + σ2 |m| − σ3 |n| is not zero. That is also known as the non stationary phase theorem. In particular defining, for any (n, σ ) ∈ Z3 \ {0} × {+, −}3 , def
ωnσ (k) = σ1
k3 n 3 − k3 n3 + σ2 − σ3 , |k| |n − k| |n|
we get −FQ(a, a)(n) =
σ ∈{+,−}3
k∈Z3 ωnσ (k)=0
[a σ1 (k) · (n − k)][a σ2 (n − k) · eσ3 (n)]eσ3 (n),
and Proposition 2.16 is proved. So the limit system is the following: ∂t u − u − Q(u, u) = 0 (RFL) u|t=0 = u0 , and we have proved the following theorem.
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T HEOREM 2.8. Let u0 be a divergence free vector field in L2 (T3 ), and let (uε )ε>0 be a family of weak solutions to (RF ε ). Then as ε goes to zero, the weak closure of (L(− εt )uε )ε>0 is included in the set of weak solutions of (RFL). Now let us concentrate on the quadratic form Q: we are going to see that it has particular properties which make it very similar to the two dimensional product arising in the 2D incompressible Navier–Stokes equations. We state the following fundamental result without proof - its proof requires a careful analysis of the resonances in the nonlinear term, and is based on the fact that if the frequencies n ∈ Z3 , k1 ∈ Z and k2 ∈ Z are fixed, then there is a finite number of k3 satisfying the resonance condition (2.5.6), contrary to a classical product with no such condition, where the number of k3 is infinite. P ROPOSITION 2.17. The quadratic form Q given in Proposition 2.16 satisfies the following properties. 1) For any smooth divergence free vector field h, we have −
Q(h, h) dx3 = P(h · ∇h). T
2) If u, v and w are three divergence free vector fields, then " ∀s ≥ 0, Q(u, vosc )"(−)s vosc L2 (T3 ) = 0, and " " " "(Q(uosc , vosc )"wosc ) 1 " H2 ≤ C uosc 1 vosc H 1 + vosc 1 uosc H 1 wosc H2
H2
3
H2
.
R EMARK 2.18. 1) The first result of Proposition 2.17 is no surprise if one recalls Proposition 2.10: we saw indeed in Section 2.4.2 that the vertical average of the non linear term at the limit can only involve interactions between two-dimensional vector fields. 2) The second result of Proposition 2.17 is a typical two dimensional product rule, although the setting here is three dimensional. The estimate means indeed that one gains half a derivative when one takes into account the special structure of the quadratic form Q compared with a usual product. Notice that the limit system (RFL) can be split into two parts: indeed if u solves (RFL) then one can decompose u into u = u + uosc , where u satisfies the two dimensional Navier– Stokes equation 3 (NS2D)
∂t u − h u + Ph (uh · ∇h u) = f u|t=0 = u0 ,
where Ph denotes the two dimensional Leray projector onto two dimensional divergence free vector fields, and uosc satisfies the coupled system
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257
∂t uosc − uosc − Q(2u + uosc , uosc ) = fosc uosc|t=0 = u0osc .
Of course here u0 = u0 + u0osc where u0 the vertical mean of u0 , and similarly f = f + fosc , where f is the vertical mean of f . Using the proposition stated above on the limit Q, it is not difficult to prove the following global wellposedness result. 1
P ROPOSITION 2.19. Let u0 be a divergence free vector fields in H 2 (T3 ). Let us consider 1 also an external force f in L2 (R+ ; H − 2 (T3 )). Then there exists a unique global solution u 1 3 to the system (RFL), in Cb (R+ ; H 2 (T3 )) ∩ L2 (R+ ; H 2 (T3 )). Let us now give the strategy to describe the asymptotic behaviour of uε . It is natural to write an asymptotic expansion of uε as L( εt )u + εU 1 + . . . and to identify the powers of ε after plugging that expansion into the equation. Unfortunately a few drawbacks appear instantly. First such a method is regularity-consuming, since the equation involves derivatives and nonlinear terms, so one needs to start by smoothing out u. That is possible because of the special properties of Q pointed out above, which in particular imply the stability of the limit system. More precisely one can prove that if uN converges towards u (in our case uN will be spectrally supported in a ball of radius N ), then Q(uN , uN ) converges towards Q(u, u), in appropriate function spaces. The next difficulty, more serious than the previous one, is that the quadratic form Q is only a weak limit of the original quadratic form. So it is not clear that the next term in the expansion, εU 1 , does indeed exist (in other words it is not clear that the convergence of uε − L( εt )u to zero is strong, and is even a O(ε)). But the difference between Q and the original quadratic form is oscillatory in time, and U 1 will roughly correspond to a time integral of that difference (which again can be defined because the frequencies of u have been restricted to a fixed ball; only in the very end will we let N go to infinity). To make this sketch more precise, we are going to construct the smooth, approximate family (uapp )ε,η , and prove Lemma 2.14 using the previous results. In particular we will then only be dealing with smooth functions. As we proceed in the construction we will in fact also show that (uapp )ε,η is a (strong) approximation of the limit solution L( εt )u, by writing an Ansatz of the type sketched above, and identifying the powers of ε in the equation. In doing so we will prove the following theorem, which gives of course Lemma 2.14, and Theorem 2.5 due to Remark 2.15. 1
T HEOREM 2.9. Let u0 be a divergence free vector field in H 2 and let u be the unique, global solution of the limit system (RFL) associated with u0 constructed in Proposition 2.19 (with f = 0). For any positive real number η, a family (uapp )ε,η exists such that 9 9 9 t 9 u u9 lim lim sup9 − L = 0. app 9 η→0 ε→0 ε 9H 12 Moreover, the family (uapp )ε,η satisfies the conclusions of Lemma 2.14.
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P ROOF OF T HEOREM 2.9. Let η be an arbitrary positive number. We define, for any positive integer N , def uN = PN u = F −1 1|n|≤N ( u(n) , and obviously there is Nη > 0 such that 9 9 9 9 9L t (uN − u)9 η 9 ε 9
1
≤ ρε,η ,
H2
where ρε,η denotes from now on any non negative quantity such that lim lim sup ρε,η = 0.
η→0
ε→0
We will also denote generically by Rε,η any vector field satisfying Rε,η
L2 (R+ ;H
− 21
)
= ρε,η .
We can, from now on, concentrate on uNη , and our goal is to approximate L( εt )uNη in such a way as to satisfy system (2.5.4). So let us write t uNη + εU 1 uapp = L ε where U 1 is a smooth, divergence free vector field to be determined. To simplify we also define t uNη , U =L ε 0 def
as well as the operator 1 def Lε w = ∂t w − w + w ⊥ . ε Then we have Lε uapp + uapp · ∇uapp = Lε U 0 + εLε U 1 + uapp · ∇uapp ,
(2.5.7)
and the only point left to prove is that there is a smooth, divergence free vector field U 1 such that the right-hand side of (2.5.7) is a remainder term. We notice that by definition
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of U 0 , 1 PLε U 0 = P ∂t U 0 − U 0 + (U 0 )⊥ ε ⊥ t t t 1 1 =L (∂t uNη − uNη ) + ∂τ L uNη + P L uNη ε ε ε ε ε t PNη Q(u, u). =L ε But it is easy to prove (using the special form of Q) that 9 9 9 9 9L t PN Q(u, u) − L t Q(uN , uN )9 ≤ ρε,η . η η η 9 9 ε −1 ε L2 (R+ ;H 2 (T3 )) We infer that t Q(uNη , uNη ) + εPLε U 1 PLε uapp + P(uapp · ∇uapp ) = Rε,η + L ε + P(uapp · ∇uapp ). Now we write, by definition of Qε , t Qε (uNη , uNη ) + Fε,η , P(uapp · ∇uapp ) = −L ε where def
Fε,η = −εL
t t t t Qε uNη , L − U 1 − ε 2 Qε L − U 1 , L − U 1 . ε ε ε ε
Going back to the equation on uapp we find that t Q(uNη , uNη ) PLε uapp + P(uapp · ∇uapp ) = Rε,η + L ε t Qε (uNη , uNη ) + Fε,η + εPLε U 1 . −L ε Let us postpone for a while the proof of the following lemma. L EMMA 2.20. Let η > 0 be given. There is a family of divergence free vector fields U 1 , bounded in (L∞ ∩ L1 )(R+ ; H s (T3 )) for all s ≥ 0, such that t (Qε − Q)(uNη , uNη ) = εPLε U 1 + Rε,η . L ε
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Lemma 2.20 implies that PLε uapp + P(uapp · ∇uapp ) = Rε,η + Fε,η and the only point left to check is that Fε,η is a remainder term. But that is obvious due to the smoothness of U 1 and uNη . So the theorem is proved, up to the proof of Lemma 2.20. P ROOF OF L EMMA 2.20. We start by noticing that by definition, (Qε − Q)(uNη , uNη ) = −F −1
t
e−i ε ωn (k) 1|k|≤Nη 1|n−k|≤Nη σ
k∈ / Knσ σ ∈{+,−}3
× [uσ1 (k) · (n − k)][uσ2 (n − k) · eσ3 (n)]eσ3 (n). The frequency truncation implies that |ωnσ (k)| is bounded from below, by a constant depending on η. That enables us to define
81 def U = F −1
k∈ / Knσ σ ∈{+,−}3
t
e−i ε ωn (k) 1|k|≤Nη 1|n−k|≤Nη [uσ1 (k) · (n − k)] iωnσ (k) σ
× [uσ2 (n − k) · eσ3 (n)]eσ3 (n), t 81 def . Then and U 1 = L( )U ε 81 = (Q − Qε )(uNη , uNη ) + εRt , ε∂t U where Rt is the inverse Fourier transform of
k∈ / Knσ σ ∈{+,−}3
t
e−i ε ωn (k) 1|k|≤Nη 1|n−k|≤Nη [∂t uσ1 (k)·(n − k)][uσ2 (n − k)·eσ3 (n)]eσ3 (n). iωnσ (k) σ
81 is defined as the primitive in time of the oscillating term Qε − Q, as Notice that ε U explained in the sketch of proof above, and it is precisely the time oscillations that imply 81 is uniformly bounded in ε. We therefore have that U t 81 + ∂τ L t U 81 ∂t U ε ε ⊥ t t t 81 (Q − Qε )(uNη , uNη ) + εL Rt − P L =L , U ε ε ε
ε∂t U 1 = εL
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so finally t t ε∂t U + P(U ) = L (Q − Qε )(uNη , uNη ) + εL Rt . ε ε 1
1 ⊥
Since U 1 is arbitrarily smooth (for a fixed η) and so is Rt , Lemma 2.20 is proved, and so is Theorem 2.5. 2.5.3. More general boundary conditions We have presented above two different strong convergence results in the case of a constant rotation, depending on the boundary conditions (whole space or periodic). Those boundary conditions are of course highly unphysical, so it is natural to try to consider now more physical cases. In this short section we will only list a few cases that have been studied in the literature and give references. We will also discuss a few open questions, still in the case of a constant rotation. The first more general situation was considered by E. Grenier and N. Masmoudi in [32] where they studied the case of a fluid rotating between two horizontal plates, with vanishing Dirichlet boundary conditions. In the case of initial data independent of x3 (so-called well prepared initial data), they were able to prove the convergence of weak solutions towards a two dimensional vector field satisfying a damped, two dimensional Navier–Stokes system. The damping term is present when the initial viscosity is anisotropic (the vertical viscosity being of the order of ε, or else everything converges strongly towards zero) and is known as the Ekman pumping term (see the Introduction); it is due to the presence of boundary layers which dissipate energy. The general, ill prepared case was first investigated by N. Masmoudi in [44] in the case of periodic horizontal boundary conditions, while the study of both the periodic and the whole space horizontal boundary conditions can be found in [13]: in the case of horizontal variables in R2 , dispersion occurs which gives at the limit the same system as in the well prepared case, whereas in the periodic case, oscillating boundary conditions have to be considered, and the limit system is more complicated (though still damped). One should mention at this point the study of D. Gérard-Varet [26] who considered non smooth boundaries, meaning that the horizontal plates are replaced by rugous plates with a periodic rugosity of size ε. D. Bresch, B. Desjardins and D. GérardVaret [7] considered the case of a cylindrical domain, and under a generic assumption on the domain and a spectral assumption on the spectrum of the Coriolis operator they studied the asymptotics of the rotating fluid equations. Note that more recently, C. Bardos, F. Golse, A. Mahalov and B. Nicolaenko were able to prove in [5] that in the case of a cylindrical domain, the spectrum of the Coriolis operator is discrete. We leave out here the widely open cases concerning yet more general domains, like rotating spheres for instance, and refer to [13] or [53] for a discussion on those subjects.
2.6. References and remarks The rotating fluid equations presented in Section 2 have been the object of a number of mathematical studies in the past decade. Let us mention the pioneering works of E. Grenier [31] and of A. Babin, A. Mahalov and B. Nicolaenko [2–4], who were interested
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in the wellposedness and the limiting behaviour as ε goes to zero, in the periodic case, using S. Schochet’s method [54] presented in Section 2.5.2. The fact that the limiting system (RFL) is globally wellposed is due to [4] and putting together the works [4] and [21] gives Theorems 2.5 and 2.9. The whole space case was studied a little later, mainly in [12], where the dispersive character of the Coriolis operator was pointed out, along with the strong convergence theorems. The compensated compactness result can be found in [24]. It should be finally noted that the study of the asymptotics of rotating fluid equations in the constant case is part of a general program analyzing the asymptotics of hyperbolic, parabolic, or mixed hyperbolic-parabolic equations, penalized by a skew-symmetric operator. One has to mention here the fundamental works of J.-L. Joly, G. Métivier and J. Rauch (among other references one can refer to [33] or [34]) concerning abstract equations, as well as the study of the incompressible limit ([14,16,18,22,23,41,45]), or the gyrokinetic limit ([29]). . . Note that we have not considered other models where similar methods can be used, like for example the primitive equations (see [10,11]).
3. Taking into account spatial variations at midlatitudes 3.1. Introduction As noted in the introduction, one cannot reasonably study the movement of the atmosphere or the ocean if one neglects the spatial variations of the Coriolis force. Section 2 enabled us to go quite far in the description of the waves generated by a constant coefficient rotation; in Section 3 we shall replace that rotation by a variable one. Of course the price to pay is that the analysis can no longer be so precise, and in particular we will have no way in general of describing precisely the waves generated by a variable-coefficient rotation. We will not be considering the most general penalization operators, but with the application to geophysical flows in mind (or to magneto-hydrodynamics), we will suppose that the Coriolis operator is Lu = P(u ∧ B),
where B = b(xh )e3
and b is a smooth function, which does not vanish, and which only depends on the horizontal coordinate xh = (x1 , x2 ). We recall that P denotes the Leray projector onto divergence free vector fields. More assumptions on b will be made as we go along. We will study the system ⎧ ⎨ dt u + u · ∇u − u + 1ε u ∧ B + ∇p = 0 ∇·u = 0 on R+ × , ⎩ 0 u|t=0 = u on
on R+ × , (3.1.1)
where = h × 3 , and h denotes either the whole space R2 or any periodic domain of R2 , and similarly 3 denotes R or T. As in Section 2, we will address the questions of the uniform existence of weak or strong solutions, and we will study their asymptotic behaviour as ε goes to zero. Considering the generality of the setting, we will not be able
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to write as precise computations as in the constant case studied in Section 2, and in particular the question of the strong convergence will not be raised except for some remarks in Section 3.3.3.
3.2. Statement of the main results The first question to be addressed, and which is easily dismissed, concerns the existence of uniformly bounded weak solutions. The Coriolis operator here is no longer skewsymmetric in all Sobolev spaces, since it has variable coefficients, nevertheless it still disappears in L2 energy estimates, and it is therefore easy to prove the following theorem, which we leave as an exercise to the reader. T HEOREM 3.1. Let u0 be any divergence free vector field in L2 (). Then for all ε > 0, Equation (3.1.1) has at least one weak solution uε ∈ L∞ (R+ , L2 ) ∩ L2 (R+ , H˙ 1 ). Moreover, for all t > 0, the following energy estimate holds: uε (t)2L2
+2 0
t
∇uε (t )2L2 dt ≤ u0 2L2 .
The existence of strong solutions is a much more intricate problem, since the Coriolis operator no longer disappears, as soon as one takes derivatives of the equation. We will see that it is nevertheless possible to prove the uniform local existence and uniqueness of a solution in H s (global for small data), using the fact that B does not depend on the third variable. The precise theorem is the following; it is proved in Section 3.4. T HEOREM 3.2. Let s > 1/2 be given, and suppose that B = b(xh )e3 is a smooth, bounded function. Then there is a constant c such that the following result holds. Suppose that u0 is a divergence free vector field in H s (), such that u0 H s ≤ c. Then for all ε > 0, the system (3.1.1) has a unique, global solution uε , which is bounded in the space Cb (R+ ; H s ) ∩ L2 (R+ ; H s+1 ). Moreover if B only depends on x2 , then for any s > 1/2 and u0 in H s () divergence free, there is a time T > 0 such that for all ε > 0, the system (3.1.1) has a unique solution uε , bounded in the space C([0, T ]; H s ) ∩ L2 ([0, T ]; H s+1 ). R EMARK 3.1. A more general theorem can in fact be proved, where the Laplacian in the equation is replaced by h only. To simplify the presentation we have chosen here to state the less general result although it should be clear from the proof that the diffusion in the vertical variable plays no role in the analysis. The next question concerns the asymptotics of the solutions. We will only state results concerning the weak asymptotics of Leray-type solutions. Before stating the theorem, let us make the following additional assumption. We suppose that b has non degenerate critical
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points in the following sense: denoting by μ(X) the Lebesgue measure of any set X we suppose that lim μ {x ∈ h /|∇b(x)| ≤ δ} = 0.
(3.2.1)
δ→0
The convergence theorem is the following. It is proved in Section 3.3 below. T HEOREM 3.3. Suppose that B = be3 where b = b(xh ) is a smooth function which does not vanish, with non degenerate critical points in sense of (3.2.1). Let u0 be any divergence free vector field in L2 (), and let uε be any weak solution of (3.1.1) in the sense of Theorem 3.1. Then uε converges weakly in L2loc (R+ × ) to a limit u belonging to KerL. If 3 = R then u is identically zero, and if 3 = T it is defined as follows: the third component u3 belongs to L∞ (R+ ; L2 ) ∩ L2 (R+ ; H˙ 1 ) satisfies the transport-diffusion equation ∂t u3 − h u3 + uh · ∇h u3 = 0, u3|t=0 = u03 (xh , x3 ) dx3
∂3 u3 = 0, in R+ × ,
T
while the horizontal component uh ∈ C(R+ ; H −1 (h )) ∩ L2loc (R+ ; H 1 (h )) satisfies the following property: for any vector field ∈ H 1 (h ) ∩ Ker(L) and for any time t > 0, (uh (t)|h )L2 (h ) +
t 0
(∇h uh (t )|∇h h )L2 (h ) dt = u0h |h L2 ( ) . h
(3.2.2)
R EMARK 3.2. Formally Equation (3.2.2) can be written as a heat equation on Ker(L), as writing # the orthogonal projector in L2 onto Ker(L) the equation formally reads ∂t uh − (#h u)h = 0. That result is surprising as all non linear terms have disappeared in the limiting process. This can be understood as some sort of turbulent behaviour, where all scales are mixed due to the variation of b. Technically the result is due to the fact that the kernel of L is very small as soon as b is not a constant, which induces a lot of rigidity in the limit equation. The rest of Section 3 is devoted to the proof of those results, starting with the weak convergence result in the next section. The proof of that theorem will follow the same lines as the proof of Theorem 2.2 in Section 2, with the additional difficulty of course that the rotation vector is no longer homogeneous. The proof of Theorem 3.2 in Section 3.4 will only be sketched, as it involves techniques which do not have much to do with the fast rotation limit but consists in rather subtle anisotropic estimates, and are beyond the scope of this review article.
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3.3. Weak asymptotics In this section we are concerned with the weak asymptotics of the solutions to the rotating fluid equations with a variable Coriolis force, and we will prove Theorem 3.3. Let us start by noticing that as soon as the initial data is in L2 , it generates a bounded family uε of solutions to (3.1.1), so up to the extraction of a subsequence there exists u ∈ L∞ (R+ , L2 ) ∩ L2 (R+ , H˙ 1 ), such that uε ! u in w-L2loc (R+ × )
as ε → 0.
As in the constant case studied in Section 2, we will prove that u belongs to the kernel Ker(L) of L, so in the next section we present the operator L and study its main properties (in particular its kernel). The following section is devoted to the end of the proof of Theorem 3.3, using a compensated-compactness argument to deal with the nonlinear terms. 3.3.1. Study of the Coriolis operator The kernel Ker(L) of L is characterized in the following proposition. P ROPOSITION 3.3. If u is a divergence free vector field in L2 () belonging to Ker(L), then u is in L2 (h ) and satisfies the following properties: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
div uh = 0 h
uh · ∇h b = 0
uh ∧ Bdxh = 0. h
R EMARK 3.4. 1) In the case when 3 = R, Proposition 3.3 shows that the kernel of L is reduced to zero since L2 (h ) ∩ L2 () = {0}. 2) In the case when h = T2 , the fact that divh uh = 0 does not necessarily mean that uh can be written as uh = ∇h⊥ ϕ for some function ϕ because the horizontal mean of uh is not preserved by the equation. P ROOF OF P ROPOSITION 3.3. If u belongs to Ker(L) then we have P(u ∧ B) = 0, so in particular uh ∧ B dx = 0.
Moreover in the sense of distributions, rot(u ∧ B) = 0, which can be rewritten (∇·B)u + (B · ∇)u − (u · ∇)B − (∇·u)B = 0.
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As ∇·B = ∇·u = 0 and B = be3 , we get b∂3 u − (u · ∇)be3 = 0.
(3.3.1)
In particular, ∂3 u1 = ∂3 u2 = 0 from which we deduce that uh belongs to L2 (h ). Note that in the case where 3 = R, the invariance with respect to x3 and the fact that u belongs to L2 () imply that u1 = u2 = 0 (and therefore u3 = 0 by the divergence free condition). We suppose from now on that 3 = T. Differentiating the incompressibility constraint with respect to x3 leads then to 2 2 2 ∂33 u3 = −∂13 u1 − ∂23 u2 = 0
in the sense of distributions. The function ∂3 u3 depends only on x1 and x2 , and satisfies ∂3 u3 dx3 = 0. So ∂3 u3 = 0, u3 belongs to L2 (h ), and ∂1 u1 + ∂2 u2 = 0. Finally by (3.3.1) we get uh · ∇h b = 0 and h uh ∧ b dxh = 0. The proposition is proved. R EMARK 3.5. Before applying this result to the characterization of the weak limit u, let us just specify it in two important cases. If ∇b = 0 almost everywhere (for instance if the Coriolis operator is constant, which corresponds to the case studied in Section 2), then u ∈ L2 is a divergence free vector field in Ker(L) if and only if u = ∇h⊥ ϕ + αe3 , for some ∇h ϕ ∈ L2 (h ) and α ∈ L2 (h ). If ∇b = 0 almost everywhere, then the condition arising on u is much more restrictive: if u ∈ L2 is a divergence free vector field in Ker(L) then it can be written u=
u · ∇ ⊥b ⊥ ∇ b + αe3 |∇ ⊥ b|2
for some α ∈ L2 (h ), with the additional condition that div h
uh · ∇ ⊥ b ⊥ ∇ b =0 |∇ ⊥ b|2
and
b
uh · ∇ ⊥ b ⊥ ∇ b dx = 0. |∇ ⊥ b|2
Using this characterization of Ker(L), we deduce some constraints on the weak limit u. The proof of the following result is exactly the same as in the constant case (Proposition 2.5, page 239), so we leave it to the reader. P ROPOSITION 3.6. Let u0 be any divergence free vector field in L2 (). Denote by (uε )ε>0 a family of weak solutions of (3.1.1), and by u any of its limit points. Then u ∈ L∞ (R+ ; L2 (h )) ∩ L2 (R+ ; H˙ 1 (h ))
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and satisfies the following properties: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
div uh = 0 h
uh · ∇h b = 0
uh ∧ B dxh = 0. h
3.3.2. Proof of the weak convergence theorem In this section we shall prove Theorem 3.3. If 3 = R, then u = 0 due to Remark 3.4, so from now on we can suppose that 3 = T. The strategy of the proof is quite similar to the constant case: we have first to give a precise description of the different oscillating modes, and then to prove that these oscillations do not occur in the limiting equation. Finally we need to show that the limiting equation is in fact linear. As in the constant case, vertical modes generate fast oscillations in the system, meaning that the whole part of the velocity field corresponding to Fourier modes with k3 = 0 converges weakly to zero. The corresponding vertical oscillations depend directly on the order of magnitude of b. The main difference comes then from the fact that, in the case of a heterogeneous rotation, the kernel of the penalization is much smaller: restricting our attention to the horizontal modes (k3 = 0), we see that the Coriolis term penalizes all the fields which are parallel to ∇b, which implies in particular that the vertical average of the horizontal velocity is no longer strongly compact. The corresponding two-dimensional oscillations are then governed by ∇b, and possibly become singular if ∇b cancels. In the following we will therefore only be able to prove that the vertical average of the vertical velocity is strongly compact, and the use of that information alone, coupled with some compensated compactness argument, will enable us to establish the equation satisfied by the weak limit of the velocity field. P ROPOSITION 3.7. Let u0 be a divergence free vector field in L2 (). For all ε > 0, denote def by uε a weak solution of (3.1.1) and by uε = uε dx3 . Then, for all T > 0, (uε,3 )ε>0 is strongly compact in L2 ([0, T ] × ). P ROOF OF P ROPOSITION 3.7. The computation is similar to the constant case studied in Section 2 (Proposition 2.8, page 239), only for the fact that one must restrict one’s attention to the vertical component only. By the energy estimate, uε and consequently uε are uniformly bounded in L2 ([0, T ]; H 1 ). Integrating with respect to x3 the vertical component of the penalized Navier–Stokes equation leads to ∂t uε,3 +
∇ · (uε uε,3 ) dx3 − h uε,3 = 0,
from which we deduce that ∂t uε,3 is uniformly bounded in L2 ([0, T ], H −3/2 ()), and the result follows by Aubin’s lemma. Now let us describe the oscillations.
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L EMMA 3.8. Let u0 be a divergence free vector field in L2 (). For all ε > 0, denote by uε a weak solution of (3.1.1), by uε = uε dx3 and by 8 uε = uε − uε . Define ωε = ∂1 uε,2 − ∂2 uε,1 ,
ωε =
and
8 ωε = ωε − ω ε ,
ωε dx3 , T
8ε,h = ∇h⊥8 uε,3 − ∂38 u⊥ ∂3 ε,h ,
with
8ε,h dx3 = 0.
T
Then, regularizing by a kernel κδ as in (2.4.2), we get the following description of the oscillations ε∂t ωδε − uδε,h · ∇b = −εrεδ − sεδ ∇h · uδε,h = 0 8δε,h + b8 uδε,h = −εrεδ − sεδ ε∂t
(3.3.2)
ε∂t 8 ωεδ − ∇ · (b8 uδε,h ) = −εrεδ − sεδ denoting generically by rεδ and sεδ some quantities satisfying ∀δ > 0, and
∀T > 0, ∀T > 0,
sup rεδ L2 ([0,T ]×) < +∞ ε
sup δ −1 sεδ L2 ([0,T ]×) < +∞. ε,δ
P ROOF OF L EMMA 3.8. Denote, as in Section 2, by Fε the flux term Fε = −∇ · (uε ⊗ uε ) + uε . The energy inequality and standard bilinear estimates yield that Fε is uniformly bounded in L2 ([0, T ], H −3/2 ()). Using that notation, (3.1.1) can be simply rewritten ε∂t uε + uε ∧ B + ∇h pε = εF ε , ∇h · uε,h = 0, 8ε , ε∂t 8 uε + 8 uε ∧ B + ∇ p 8ε = ε F ∇ ·8 uε = 0, splitting the purely 2D modes (k3 = 0) and the vertical Fourier modes (k3 = 0). Using the vorticity formulation for the horizontal component of uε , we get ε∂t ωε − uε,h · ∇b = −ε∇h⊥ · F ε,h , ∇h · uε,h = 0.
(3.3.3)
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Then taking the rotational of the other part of the equation yields 8ε uε + ∇ ∧ (8 uε ∧ B) = ε∇ ∧ F ε∂t ∇ ∧ 8 and integrating the horizontal component with respect to x3 leads to 8ε )h , 8ε,h + b8 ε∂t uε,h = ε(∇ ∧ G
(3.3.4)
8ε,h ε∂t 8 ωε − ∇h · (8 uε,h b) = −ε∇h⊥ · F
8ε is just defined by ∂3 G 8ε and 8ε = F 8 where G T Gε dx3 = 0, and thus satisfies the same 8 uniform estimates as Fε . The second step of the proof consists then in regularizing the previous wave equations (3.3.3) and (3.3.4). We therefore introduce, as in Section 2, a smoothing family κδ defined by κδ (x) = δ −3 κ(δ −1 x) where κ is a function of Cc∞ (R3 , R+ ) such that κ(x) = 0 if |x| ≥ 1 and κdx = 1. By convolution, we then obtain ε∂t ωδε − uδε,h · ∇b = −ε∇h⊥ · F ε,h − uδε,h · ∇b + (uε,h · ∇b)δ , δ
∇h · uδε,h = 0, and 8δε )h + b8 8δε,h + b8 ε∂t uδε,h = ε(∇ ∧ G uδε,h − (b8 uε,h )δ , δ ωεδ + ∇h · (8 uδε,h b) = −ε∇h⊥ · F˜ε,h + ∇h · (8 uδε,h b) − ∇h · (8 uε,h b)δ . ε∂t 8
It remains only to check that the source terms satisfy the convenient a priori estimates. It is easy to see that κδ W 5/2,1 (R3 ) ≤ δ −5/2 κL1 (R3 ) , so the terms generically called rεδ satisfy a uniform bound for any fixed δ: δ 8δε )h and ∇ ⊥ · F 8δ are uniformly bounded in L2 ([0, T ] × ) ∇h⊥ · F ε,h , (∇ ∧ G h ε,h
(of order δ −5/2 ), 8ε )h and ∇ ⊥ · F 8ε,h are bounded in L2 ([0, T ]; H −5/2 ()). We then since ∇h⊥ · F ε,h , (∇ ∧ G h have to estimate quantities of the form uδε ψ − (uε ψ)δ for smooth functions ψ. We have " "
|uδε ψ(x) − (uε ψ)δ (x)| = ""
" " κδ (y)uε (x − y)(ψ(x) − ψ(x − y))dy ""
≤ δ∇ψL∞ () (κ δ ∗ |uε |)(x),
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so in particular, uδε,h · ∇b − (uε,h · ∇b)δ L2 ([0,T ]×) ≤ δD 2 bL∞ () uε L2 ([0,T ]×) b8 uδε,h − (b8 uε,h )δ L2 ([0,T ]×) ≤ δDbL∞ () uε L2 ([0,T ]×) ∇h · (8 uδε,h b) − ∇h · (8 uε,h b)δ L2 ([0,T ]×) ≤ δ(DbL∞ () + D 2 bL∞ () )uε L2 ([0,T ],H 1 ()) meaning that the terms generically called sεδ converge to 0 as δ → 0 uniformly in ε, according to the bound ∀T > 0,
sup δ −1 sεδ L2 ([0,T ]×) < +∞. ε,δ
Lemma 3.8 is proved.
Now let us compute the coupling term. As remarked in the introduction of Section 3.3.2, the fact that ∇b can get very small could lead to a defect of compactness of vertical averages. The nondegeneracy assumption (3.2.1) will enable us to deal with regions of space where ∇b is small, simply using a cut-off function. Let us state the result. P ROPOSITION 3.9. Let u0 be any divergence free vector field in L2 (). For all ε > 0, denote by uε a weak solution of (3.1.1). Define the truncation χδ by χδ (x) = χ(δ −1/4 ∇b(x)) where χ is a function of Cc∞ (R3 , R+ ) such that χ(x) = 1 if |x| ≤ 1. Then, with the same notation as in Lemma 3.8, the averaged nonlinear term in (3.1.1) can be rewritten |uδ |2 ∇ · (uδε ⊗ uδε ) − ∇ ε dx3 2 = ∇h · (uδε,h uδε,3 )e3 − ∇
|uδε,3 |2 2
+ ερεδ + σεδ
ε ∇ ⊥b ∇b δ − ∂t |ωδε |2 (1 − χδ ) − (1 − χδ )(uδε,h · ∇ ⊥ b) ω 2 2 |∇b| |∇b|2 ε ∇ ⊥b ε ε 8δε,h )⊥ dx3 + 8δε,h · ∇b)2 dx3 + ∂t 8 ∂ ωεδ ( ( t b 2b2 |∇b|2 ∇b 1 8δε,h · ∇b)dx3 8δε,h · ∇ ⊥ b)(ε∂t − 2 ( b |∇b|2 ε 8δε,h )⊥ )dx3 e3 8δε,h · (∂3 + ∂t ( 2b
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where ρεδ and σεδ are quantities satisfying the following estimates ∀δ > 0,
∀T > 0,
sup ρεδ L1 ([0,T ];L6/5 )) < +∞,
ε→0
and
∀T > 0,
lim sup σεδ L1 ([0,T ];L6/5 )) = 0.
δ→0 ε
P ROOF OF P ROPOSITION 3.9. Let us first remark that uδε ⊗ 8 uδε )dx3 ∇ · (uδε ⊗ uδε )dx3 = ∇ · (uδε ⊗ uδε ) + ∇ · (8 which allows us to consider separately purely 2D modes and vertical modes. As in the constant case, due to (2.4.3) fact further restrict our attention to the quantities δ we can in δ δ δ −uε ∧ (∇ ∧ uε ) and − 8 uε ∧ (∇ ∧ 8 uε )dx3 . We will finally simplify the computations by neglecting all the remainder terms in Lemma 3.8, and leave the precise computations to the reader, as in the constant case in Section 2. (i) We start with the study of the purely 2D modes. A simple computation leads to −uδε ∧ (∇ ∧ uδε ) = −uδε ∧ (∇h⊥ uδε,3 + ωδε e3 ) = −ωδε (uδε,h )⊥ − ∇h
|uδε,3 |2 2
(3.3.5) + ∇h · (uδε,h uδε,3 )e3 .
We can decompose uδε,h as follows uδε,h = (uδε,h · ∇b)
∇b ∇ ⊥b + (uδε,h · ∇ ⊥ b) 2 |∇b| |∇b|2
as soon as ∇b = 0, and we will actually do so, using the truncation χ , only if |∇b| ≥ δ 1/4 . Using the first identity in (3.3.2), and neglecting remainder terms, we obtain uδε,h = ε∂t ωδε
∇b ∇ ⊥b δ ⊥ + (u · ∇ b) ε,h |∇b|2 |∇b|2
and replacing in (3.3.5) provides finally
−uδε ∧ (∇ ∧ uδε ) = − (1 − χδ )ε∂t
|ωδε |2 ∇ ⊥ b ∇b δ − (1 − χδ )(uδε,h · ∇ ⊥ b) ω 2 2 |∇b| |∇b|2 ε
− χδ ωδε (uδε,h )⊥ − ∇h
|uδε,3 |2 2
+ ∇h · (uδε,h uδε,3 )e3 . (3.3.6)
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That concludes the first step of the proof since 9 9 9χδ ωδ uδ 9
ε ε,h L1 ([0,T ],L6/5 ())
≤ χδ L6 () ωδε L2 ([0,T ]×) uδε,h L2 ([0,T ],L6 ()) 1 1 ≤ C μ{x ∈ h /|∇b(x)| ≤ δ 4 } 6 ,
which goes to zero with δ according to Assumption (3.2.1), hence can be incorporated in the term σεδ . (ii) We have now to deal with the vertical modes. A simple computation leads to 8δε,h + 8 −8 uδε ∧ (∇ ∧ 8 uδε ) = −8 uδε ∧ (∂3 ωεδ e3 ) 8δε,h )⊥ )e3 8δε,h )⊥ − (8 = −8 ωεδ (8 uδε,h )⊥ + 8 uδε,3 ∂3 ( uδε,h · (∂3 so that using the divergence free condition,
8 uδε
−
∧ (∇
∧8 uδε )dx3
δ δ ⊥ 8δε,h )⊥ (∇h · 8 −8 ωε (8 uε,h ) + ( uδε,h ) dx3
=
8δε,h )⊥ )dx3 e3 . (8 uδε,h · (∂3
−
In order to determine the horizontal component, we then use the last two identities in (3.3.2) −
(8 uδε ∧ (∇ ∧ 8 uδε ))h dx3 =
1 8δε,h )⊥ dx3 8 ωεδ (ε∂t
ε = ∂t b
b
+
8δε,h )⊥ dx3 + 8 ωεδ (
1 8δε,h )⊥ (ε∂t 8 ( ωεδ − 8 uδε,h · ∇b)dx3 b 8δε,h )⊥ (
1 8δε,h · ∇bdx3 . ε∂t b2
8δ as follows We can decompose ε,h 8δε,h = ( 8δε,h · ∇b)
∇b ∇ ⊥b 8δε,h · ∇ ⊥ b) + ( 2 |∇b| |∇b|2
as soon as ∇b = 0, that is almost everywhere by assumption. Finally we get −
ε (8 uδε ∧ (∇ ∧ 8 uδε ))h dx3 = ∂t b
1 − 2 b
8δε,h )⊥ dx3 + 8 ωεδ (
ε ∂t 2b2
8δε,h · ∇b)2 dx3 (
8δε,h · ∇b)dx3 8δε,h · ∇ ⊥ b)(ε∂t (
∇b , |∇b|2
∇ ⊥b |∇b|2
(3.3.7)
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which is the expected formula. In order to determine the vertical component, we use the third identity in (3.3.2) and an integration by parts with respect to x3 , to find
8δε,h )⊥ dx3 8 uδε,h · (∂3
1 8δε,h · (∂3 8δε,h )⊥ dx3 ε∂t b 1 8δε,h ) · (∂3 8δε,h )⊥ − (ε∂t ∂3 8δε,h ) · ( 8δε,h )⊥ dx3 , =− (ε∂t 2b =−
from which we deduce 8 uδε,h
8δε,h )⊥ dx3 = − · (∂3
ε ∂t 2b
8δε,h )⊥ )dx3 . 8δε,h · (∂3 (
(3.3.8)
Combining (3.3.6), (3.3.7) and (3.3.8) gives finally the proper decomposition of the averaged nonlinear term. Proposition 3.9 is proved (up to the computation of the remainder terms). Now we are ready to take the limit. P ROPOSITION 3.10. Let u0 be any divergence free vector field in L2 (). For all ε > 0, denote by uε a weak solution of (3.1.1) and by uε = uε dx3 . Then, for all φ ∈ H 1 () ∩ Ker(L), we have the following limit in W −1,1 ([0, T ]), for all T > 0:
∇ · (uε ⊗ uε ) · φdx −
∇h · (uε,h uε,3 )φ3 dx → 0
as ε → 0.
P ROOF OF P ROPOSITION 3.10. We first introduce the same regularization as in the previous sections, and split the integral as follows
∇ · (uε ⊗ uε ) · φdx −
∇
=
∇h · (uε,h uε,3 )φ3 dx
+
· (uδε
⊗ uδε ) · φdx
−
∇h · (uδε,h uδε,3 )φ3 dx
∇ · ((uε − uδε ) ⊗ uε ) · φdx − ∇
∇h · ((uε,h − uδε,h )uε,3 )φ3 dx
+
· (uδε
⊗ (uε − uδε )) · φdx
−
∇h · (uδε,h (uε,3 − uδε,3 ))φ3 dx. (3.3.9)
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By the energy estimate, we deduce that the four last terms converge to 0 as δ → 0 uniformly in ε: indeed, 9 9 9 9 9 ∇ · ((uε − uδ ) ⊗ uε ) · φdx 9 ε 9 9 1 L ([0,T ])
≤ ∇φL2 () uε L2 ([0,T ],L6 ()) uδε − uε L2 ([0,T ],L3 ()) ≤ ω(δ)∇φL2 () uε 2L2 ([0,T ],H 1 ()) , where the function ω(δ) goes to zero as δ goes to zero. We are then interested in the difference between the first two terms. By Proposition 3.9, and using the fact that ∂3 φ = 0,
∇ ·φ =0
and
φ · ∇b = 0,
it can be rewritten
∇
· (uδε
⊗ uδε ) · φdx
−
∇h · (uδε,h uδε,3 )φ3 dx
ε ∇ ⊥b = φ · (ερε,δ + σε,δ )dx − ∂t |ωδε |2 (1 − χδ ) · φdx 2 |∇b|2 1 δ δ ⊥ 1 ∇ ⊥b 8 8δε,h · ∇b)2 8 ωε (ε,h ) · φdx + ε∂t ( · φdx + ε∂t 2 |∇b|2 b 2b 1 8δε,h · (∂3 8δε,h )⊥ )φ3 dx. + ε∂t ( 2b
(3.3.10)
We just need to check that all the terms in the right-hand side of (3.3.10) can be made arbitrarily small. For instance the second term in the right-hand side converges to 0 as δ → 0 uniformly in ε, since 9 9 9 9 9 σε,δ · φdx 9 ≤ φL6 () σε,δ L1 ([0,T ],L6/5 ()) . 9 9
L1 ([0,T ])
The other terms are dealt with as easily, and are left to the reader. Taking limits as ε → 0 and then as δ → 0 in (3.3.9)–(3.3.10) shows that, for all φ ∈ H 1 () ∩ Ker(L), ∇ · (uε ⊗ uε ) · φdx − ∇h · (uε,h uε,3 )φ3 dx → 0 as ε → 0
in W −1,1 ([0, T ]), which proves Proposition 3.10. Theorem 3.3 follows easily.
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3.3.3. Some general remarks Remarks on the strong convergence. Now that the weak asymptotic behaviour of weak solutions has been understood, one can try to study the strong asymptotics, as in the constant case. However this seems quite a difficult task, as we have no information in general on the nature of the spectrum of the variable-coefficient Coriolis operator. As Section 4 will show (in the case of a model for the tropics), it is possible to write explicit computations if one relaxes the generality of the setting, even if the Fourier transform is not available. A few general remarks are in order however. Due to the RAGE theorem [52] one can expect the continuous spectrum to have no influence on the convergence. That was clear in the constant case: in the case of the whole space, the spectrum is indeed continuous and a Strichartz theorem (which one can understand as a precise version of the RAGE theorem) enabled us to get rid of all oscillations and to find that the weak convergence was in fact strong. In the periodic case however there is no continuous spectrum and the weak convergence only becomes strong once the oscillations are filtered out. In the variable case one expects therefore to find a strong convergence once the discrete spectrum has been “filtered out” in some way, but the precise way to do so is not so clear. Furthermore even if one does manage to understand precisely the oscillations and to introduce the corresponding filtering operator, the question of the existence of solutions to the limit system is unclear: in the periodic case the terms that could cause some trouble to solve the system miraculously satisfy 2D-type energy estimates. If that were not the case then the limit system would only 1 have a short time life span (supposing the spectral projectors are continuous in H 2 , which is also far from clear). To understand the limit system better one would probably have to introduce some non resonant conditions. That program will be carried out in Section 4, in the particular case of the tropics. Remarks on the role of b and ∇b. As suggested before, the parameter b is responsible for the vertical waves, whereas ∇b rules horizontal waves. If ∇b vanishes on sets of nonzero measure, one therefore expects to recover the constant b situation, that is, a limit satisfying the 2D Navier–Stokes system inside such sets. Of course that generates transmission problems on the boundary of those sets, with a possible degeneracy of the horizontal waves. On the contrary if b vanishes on sets of nonzero measure, then the penalization itself disappears from the equation in such regions. The equation being nonlocal this will have an incidence everywhere in the system and can create coupling problems. If b vanishes at a point only, with a non degenerate singularity, then one can make the weak compactness argument work, although one must be careful with the functional setting; a special case is considered in Section 4, with a model for the tropics. The case when the direction of B is not fixed. It seems reasonable, from a physical point of view, to retain only the vertical component of the rotation vector in the Coriolis force. This should be mathematically justified by considering more general models where the direction of the rotation vector is allowed to vary. The algebraic compensated- compactness argument in that case seems to still hold (under the same type of non degeneracy condition as that required in Section 3). However serious geometrical problems appear to understand precisely the structure of the kernel of the rotation operator: the constraint established here
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on the vertical averages should be replaced by a constraint on the averages over level lines of B, which implies some geometrical understanding on those level lines (are they closed or not, have they a finite length or not. . .).
3.4. Strong solutions In this section we want to investigate the question of strong solutions to (3.1.1), and to prove Theorem 3.2. The usual methods to prove the local existence and uniqueness of solutions for the 3D Navier–Stokes equations yield the existence and uniqueness of a solution 1 to (3.1.1) as soon as the initial data is in H 2 (), but unfortunately one realizes quickly that with such methods, the life span of the solution decays to zero as ε goes to zero, while all norms (other than the energy norm) blow up. On the contrary to ensure large time existence of a unique solution one would need to require the norm of the initial data to go to zero with ε. That is due to the fact that contrary to the case of a constant rotation studied in Section 2, the Coriolis operator does not commute with derivatives, and creates large, unbounded terms in the estimates. Our aim in this section is nevertheless to prove the existence and uniqueness of a solution on a uniform time interval, or the global existence and uniqueness for small initial data, independently of ε. For technical reasons, the local in time theorem only holds if the rotation vector only depends on one variable, say x2 (which as noted in Section 1 is consistent with some models of geophysical flows, like the tropics). Let us explain the structure of the proof of Theorem 3.2. The idea is that since B does not depend on x3 , one is allowed as many vertical derivatives as one likes in the energy estimates. Only horizontal derivatives create an unbounded commutator term. So the first step of the analysis consists in proving the global existence and uniqueness of a solution for small data in an anisotropic-type Sobolev space, where derivatives are only placed on x3 . The local existence and uniqueness for arbitrary data in such an anistropic Sobolev space can also be proved, as long as B only depends on x1 —the proof is rather technical however, compared to the global existence result. Once that step is accomplished, one proves a propagation of regularity result, enabling the replacement of the anisotropic Sobolev space by H s . Those steps are explained in more detail in the next sections. To simplify the analysis we will place ourselves in the case where = R3 ; the periodic case can be proved by slight modifications of that case. Moreover we will not be giving any details of the anisotropic-type estimates involved in the proof, as they are quite technical and beyond the scope of this review article; we merely want to point out here the main ideas and estimates giving the result, and we refer to [43] for all the details. 3.4.1. Global solutions for small data Let us give the definition of the anisotropic Sobolev spaces we will be using. Calling as usual ( u the Fourier transform of u, we de fine the Hilbert space H s,s by the norm def
uH s,s =
R3
2 s
1
(1 + |ξh | ) (1 + |ξ3 | ) |( u(ξ )| dξ 2 s
2
2
.
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We will need to write an energy estimate in such spaces. The following inequality is the main ingredient to prove the next proposition. We refer to [12] and [48] for a proof. For any vector fields u and v, with u divergence free, 1 1 1 3 " " "(u · ∇v|v) 0,s " ≤ C u 2 0,s ∇h u 2 0,s v 2 0,s ∇h v 2 0,s H H H H H + ∇h uH 0,s vH 0,s ∇h vH 0,s .
Using that inequality and noticing that the Coriolis operator is skew- symmetric in H 0,s , it is not too difficult to prove the following proposition, stating the global wellposedness of (3.1.1) in H 0,s () for small enough data. P ROPOSITION 3.11. Let s > 1/2 be given. There is a constant c such that the following result holds. Suppose that u0 is a divergence free vector field in H 0,s (), such that u0 H 0,s ≤ c. Then for all ε > 0, the system (3.1.1) has a unique, global solution uε , which is bounded in the space Cb (R+ ; H 0,s ) ∩ L2 (R+ ; H 1,s ) and satisfies ∀t ≥ 0,
uε (t)2H 0,s
t
+ 0
∇h uε (t )2H 0,s dt ≤ u0 2H 0,s .
Once that result is obtained, on can infer the first part of Theorem 3.2, by writing an energy estimate in H s . Of course this time the Coriolis operator does not disappear, but since B is smooth and bounded one has C 1 |(u ∧ B|u)H s | ≤ u2H s . ε ε The main point is then that one can prove, using an anisotropic Littlewood–Paley decomposition and an anisotropic-type paraproduct algorithm (this is quite technical and omitted), that " " "(u · ∇u|u)H s " ≤ 1 ∇h u2 s + C∇h u2 0,s u2 s (1 + u2 0,s ). H H H H 2 That estimate is better than a standard H s estimate, as it involves the H 0,s norm of u and ∇u. An H s energy estimate therefore yields, using the energy estimate of Proposition 3.11 and a Gronwall lemma, Ct + Cu0 2H s + Cu0 4H s , u(t)2H s ≤ u0 2H s exp ε which allows to prove the first part of the theorem. R EMARK 3.12. It should be noted that the H s norm of the solution is unbounded with ε. The global existence result is therefore not as satisfactory as in the constant case, since one does not have a bounded family of solutions in H s , as ε goes to zero.
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3.4.2. Local solutions for large data In this section we suppose that the rotation vector B only depends on x2 . This appears like a technical assumption but it is not clear how to deal with the more general case. As in the previous section, we start by proving a result in an anisotropic space. P ROPOSITION 3.13. Suppose that B only depends on x2 , and let s > 1/2 be given. Suppose that u0 is a divergence free vector field in H 0,s (). Then there is a time T > 0 such that for all ε > 0, the system (3.1.1) has a unique solution, bounded in C([0, T ]; H 0,s ) ∩ L2 ([0, T ]; H 1,s ). P ROOF OF P ROPOSITION 3.13. The first step consists in solving the linearized equation 1 ∂t vε − vε + P(vε ∧ B) = 0 ε for smooth initial data, say vε|t=0 = χ(|D|/N )u0 , where χ is a smooth cut-off function in a ball centered at zero and N is a large enough integer. Then clearly vε is globally defined and bounded in Cb (R+ ; H 0,s ) ∩ L2 (R+ ; H 1,s ), and since B depends neither on x1 nor on x3 , its frequencies in the ξ1 and ξ3 direction are in a ball of size N . Then one needs to solve the perturbed equation satisfied by wε = uε − vε , and prove it has a solution on a uniform time interval. The equation is the following: ⎧ 1 ⎪ ⎪ ∂t wε + wε · ∇wε + wε · ∇vε + vε · ∇wε − wε + wε ∧ B + ∇p ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎨ = −vε · ∇vε ⎪ ⎪ ⎪ ∇·wε = 0 ⎪ ⎪ ⎪ ⎪ ⎩ wε|t=0 = (1 − χ(|D|/N ))u0 . This is a 3D Navier–Stokes type equation, with a non constant rotating term which is harmless since we will write an energy estimate in H 0,s . The initial data can be made arbitrarily small as soon as N is large enough. It moreover has a forcing term due to the presence of vε , and transport-reaction terms. Those latter terms classically do not cause much trouble as they contribute in an exponential in the final estimate (through a Gronwall lemma), which is independent of ε and N . More troublesome is the forcing term −vε · ∇vε , but using the fact that two frequency directions of vε are bounded, it is possible to write an estimate of the type " t " " " " (vε · ∇vε |wε ) 0,s (t ) dt " ≤ C(N, u0 2 )t 12 + 1 ∇h wε 2 0,s . H L " " H 2 0 Proving such an estimate is of course the main difficulty of the analysis and is left out. It is here that the fact that B does not depend on x2 is crucial: without that assumption, the constant C(N, u0 L2 ) above would depend on ε, which would prevent the life span from being independent of ε. Once that estimate is proved, one finds that for a small enough
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time, depending on N but not on ε, one can solve the system on wε , hence going back to the original equation, there is a solution uε on a time interval independent of ε. To infer the second part of Theorem 3.2 one uses again a propagation of regularity type result. We omit the details. 3.5. References and remarks The analysis of the weak convergence of weak solutions presented in Section 3 is probably the first attempt in understanding mathematically the behaviour of a variable coefficient Coriolis operator, and the original analysis can be found in [24]. Note that the study is not unrelated to works on the incompressible limit. As recalled in Section 2.6, the idea of using compensated compactness methods originates in the article [41] for the incompressible limit. The uniform existence of strong solutions presented in Section 3.4 is due to M. Majdoub and M. Paicu [43], and concerning the difficulty of studying strong solutions one can also refer, among other studies to the paper by G. Métivier and S. Schochet [47], concerning nonisentropic, compressible Navier–Stokes equations (see also T. Alazard [1]), or to the recent works [8] and [9]. 4. The tropics 4.1. Introduction In Section 4 we will be concerned with a shallow water system governing the movement of the ocean at the tropics, presented in the introduction (see (1.2.13). Using the Cartesian approximation (1.2.10) of the latitude and the longitude, and the shallow water approximation of the Navier–Stokes system with free surface, we obtain the following system for the depth fluctuation η and the horizontal velocity u: 1 ∂t η + ∇·((1 + εη)u) = 0, ε ∂t ((1 + εη)u) + ∇ · ((1 + εη)u ⊗ u) +
βx2 (1 + εη)u⊥ ε
1 + (1 + εη)∇η − A(1 + εη, u) = 0, ε η|t=0 = η0 ,
u|t=0 = u0 .
(4.1.1)
We will suppose that the space variable x = (x1 , x2 ) belongs to T × R. As in the previous sections, we have denoted u⊥ = (u2 , −u1 ). The operator A represents the viscous effects, and from a physical point of view, it would be relevant to model such effects by the following operator A(1 + εη, u) = ν∇ · ((1 + εη)∇u),
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meaning in particular that the viscosity cancels when 1 +εη vanishes. Then, in order for the Cauchy problem to be globally well-posed, it is necessary to get some control on the cavitation. Results by D. Bresch and B. Desjardins [6] show that capillary or friction effects can prevent the formation of singularities in the Saint-Venant system (without Coriolis force). On the other hand, in the absence of such dissipative effects, A. Mellet and A. Vasseur [46] have proved the weak stability of this same system under a suitable integrability assumption on the initial velocity field. All these results are√based on a new entropy inequality [6] which controls in particular the first derivative of 1 + εη. In particular, they cannot be easily extended to (4.1.1) since the betaplane approximation of the Coriolis force prevents from deriving such an entropy inequality. For the sake of simplicity, since we are interested in an asymptotic regime where the depth h = 1 + εη is just a fluctuation around a mean value, we will consider the following viscosity operator A(h, u) = νu, so that the usual theory of the isentropic Navier–Stokes equations can be applied. Note also that we do not consider realistic boundary conditions in the x1 variable, but that enables us to give a complete description of the asymptotics. In the presence of boundaries one would have to take into account boundary layers (namely Munk-type boundary layers; see [17] for instance). As in the previous sections, the questions we shall address are first to solve this system uniformly in ε, and then to understand the asymptotic behaviour of the solutions as ε goes to zero. The mathematical setting is not quite the one studied in Section 3, since the rotation vector vanishes for x2 = 0. However the advantage of our situation is that it is an explicit function, so it will be possible to carry out computations further than in the abstract case studied in Section 3.
4.2. Statement of the main results We obtain the following result as a consequence of the global existence of weak solutions to the isentropic Navier–Stokes equations, remarking that the penalization (which is a skewsymmetric operator) does not modify the energy inequality. T HEOREM 4.1. Let (η0 , u0 ) ∈ L2 (T × R) and consider a sequence ((ηε0 , u0ε ))ε>0 such that 1 02 0 0 2 sup |ηε | + (1 + εηε )|uε | dx ≤ E 0 ε>0 2
and (4.2.1)
(ηε0 , u0ε ) → (η0 , u0 ) in L2 (T × R). Then, for all ε > 0, System (4.1.1) has at least one weak solution (ηε , uε ) with initial data (ηε0 , u0ε ), satisfying the uniform bound
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t 1 2 sup |∇uε |2 (t , x)dxdt ≤ E 0 . ηε + (1 + εηε )|uε |2 (t, x)dx + ν 0 ε>0 2 (4.2.2) In particular, there exist η and u belonging respectively to the spaces L∞ (R+ ; L2 (T × R)) and L∞ (R+ ; L2 (T × R)) ∩ L2 (R+ ; H˙ 1 (T × R)) such that, up to extraction of a subsequence, (ηε , uε ) ! (η, u) in w-L2loc (R+ × T × R).
(4.2.3)
It therefore makes sense to inquire on the limit behaviour of the solution as ε goes to zero. We will start by studying the weak asymptotics, and establishing that as the rotation increases, the geostrophic flow is governed by a linear equation. The proof of the following result can be found in Section 4.3 below. T HEOREM 4.2. Let (η0 , u0 ) ∈ L2 (T × R) and (ηε0 , u0ε ) satisfy (4.2.1). For all ε > 0, denote by (ηε , uε ) a solution of (4.1.1) with initial data (ηε0 , u0ε ). Then up to the extraction of a subsequence, (ηε , uε ) converges weakly in L2loc (R+ × T × R) to the solution (η, u) in L∞ (R+ ; L2 (R)), with u also belonging to L2 (R+ ; H˙ 1 (R)), of the following linear equation (given in weak formulation) u2 = 0,
−βx2 u1 + ∂2 η = 0,
(4.2.4)
and for all (η∗ , u∗ ) ∈ L2 × H 1 (R) satisfying (4.2.4)
(ηη∗ + u1 u∗1 )(t, x) dx + ν
t 0
∇u1 · ∇u∗1 (t , x) dx dt =
(η0 η∗ + u01 u∗1 )(x) dx. (4.2.5)
Once the mean flow has been described, it is natural to address the question of the strong convergence of solutions. As in the case of midlatitudes (when the Coriolis penalization is assumed to be constant), for periodic boundary conditions we need to filter out the oscillatory modes before taking the strong limit. Indeed equatorial waves are known to be trapped (see Section 1), thus we cannot expect to establish any dispersion. In the next theorem we have defined the operator L(t) = e−tL where L is the Coriolis operator L : (η, u) ∈ L2 (T × R) → (∇ · u, βx2 u⊥ + ∇η).
(4.2.6)
We moreover denote by #0 the L2 projection onto the kernel of L, and by #⊥ the projection onto (KerL)⊥ . Finally for any three-component vector field , we denote by its two last components. In the following statement, a limit system is referred to, which is
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obtained as in Section 2, by a filtering method. It will be studied in Section 4.4. Special function spaces are also used, they are defined by the following norm: ∀s ≥ 0,
def
HLs = (Id − + β 2 x22 )s/2 L2 (T×R) .
(4.2.7)
The limit system is presented in Section 4.4.2, and the main steps of the result are described in Section 4.4.5. T HEOREM 4.3. Let 0 = (η0 , u0 ) ∈ L2 (T × R), and consider a family ((ηε0 , u0ε ))ε>0 such that 1 02 and |ηε | + (1 + εηε0 )|u0ε |2 dx ≤ E 0 2 (4.2.8) 1 0 0 2 0 0 0 2 |ηε − η | + (1 + εηε )|uε − u | dx → 0 as ε → 0. 2 For all ε > 0 denote by (ηε , uε ) a solution of (4.1.1) with initial data (ηε0 , u0ε ). Then • there exists a weak solution in L∞ (R+ ; L2 (T × R)) to the limit filtered system (given to simplify notation in compact formulation rather than in weak formulation as in (4.2.5) above) ∂t + QL (, ) − ν L = 0
(4.2.9)
|t=0 = 0 ,
where L and QL are defined in (4.4.11). Moreover #⊥ belongs to the space L2 (R+ ; HL1 ). If #⊥ 0 belongs to HLα for some α > 1/2, then for all but a countable number of β, the weak solution satisfies for all t ∈ R+ t ∇ · (t )L∞ (T×R) dt < +∞. (4.2.10) 0 1/2
• If we further assume that #⊥ 0 belongs to HL , then there exists a maximal time interval [0, T ∗ [, with T ∗ = +∞ under the smallness assumption #0 0 L2 (T×R) + #⊥ 0 H 1/2 ≤ Cν, L
∗ such that is the unique (strong) solution to (4.2.9), and #⊥ belongs to L∞ loc ([0, T [, 1/2 3/2 2 ∗ HL ) ∩ Lloc ([0, T [, HL ). • Finally if #⊥ 0 belongs to HLα for some α > 1/2, then for all but a countable number of β, the sequence of filtered solutions (ε ) to (4.1.1) defined by t ε = L − (ηε , uε ), ε
converges strongly towards in L2loc ([0, T ∗ [; L2 (T × R)).
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R EMARK 4.1. The limit equation (4.2.9) is obtained as usual (see Section 2) by studying resonances in the nonlinear term. It so happens that the limit quadratic form is shown to satisfy three-dimensional type estimates in HLs spaces, although the setting here is purely two-dimensional. That is due to the particular structure of the eigenvalues and eigenvectors of the penalization operator L and will be discussed in Section 4.4 below. That is the reason why we are only able to prove the local in time wellposedness of (4.2.9). The following, final result, is an intermediate statement between the two convergence theorems stated above. The proof is presented in the final section of this survey, Section 4.5 below. We have denoted by S the set of all the eigenvalues of L (which turns out to be exactly the spectrum of L). T HEOREM 4.4. Let (η0 , u0 ) ∈ L2 (T × R) and (ηε0 , u0ε ) satisfy (4.2.1). For all ε > 0, denote by (ηε , uε ) a solution of (4.1.1) with initial data (ηε0 , u0ε ), and by t ε = L − (ηε , uε ). ε s (T × Then up to the extraction of a subsequence, ε converges strongly in L2loc (R+ ; Hloc R)) (for all s < 0) to some weak solution of the following limiting filtered system: for all iλ ∈ S, there is a bounded measure υλ ∈ M(R+ × T × R) (which vanishes if λ = 0), such that for all smooth ∗λ ∈ Ker(L − iλI d),
¯ ∗λ (x) dx − ν · +
t 0
=
t 0
¯ ∗λ (t , x) dxdt L ·
¯ ∗λ (t , x) dxdt + QL (, ) ·
t 0
¯ ∗λ ) υλ (dt dx) ∇ · (
¯ ∗λ (x) dx, 0 ·
where QL and Ê L are defined by (4.4.11), and where 0 = (η0 , u0 ). R EMARK 4.2. • Note that, by interpolation with the uniform L2loc (R+ , H 1 (T × R)) bound on uε , we get the strong convergence of uε in L2loc (R+ , L2 (T × R)): up to extraction of a subsequence, 9 9 9 9 9uε − L t 9 9 9 ε
L2 (T×R)
→0
in L2loc ([0, T ]).
• The presence of the defect measure υλ at the limit is due to a possible defect of compactness in space of the sequence (ηε )ε>0 . As the proof of the theorem shows, that measure is zero if one is able to prove some equicontinuity in space on ηε , or even on εηε . Since we have been unable to prove such a result, we study in Section 4.5.4 a slightly different
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model, where capillarity effects are added in order to gain that compactness. Note that the model introduced in Section 4.5.4 is unfortunately not very physical due to the particular form of the capillarity operator (see its definition in (4.5.8) below).
4.3. Weak asymptotics In this section we intend to prove Theorem 4.2 stated above. The structure of the proof is similar to the previous sections: we study the kernel of the penalization operator and show that the limit is necessarily in that kernel, and a compensated compactness argument allows to take limits in the nonlinear terms. 4.3.1. The geostrophic constraint The first step consists in proving that the weak limit defined by (4.2.3) satisfies the geostrophic constraint (4.2.4), or in other words belongs to the kernel of L. We skip the proof of the following proposition: as in the previous sections one proves that the limit is in KerL by multiplying the system (4.1.1) by ε and taking limits in the sense of distributions thanks to the uniform bounds coming from the energy estimate. The constraint (4.3.1) is easily shown to characterize elements of KerL. P ROPOSITION 4.3. Let (η0 , u0 ) ∈ L2 (T × R) and (ηε0 , u0ε ) be initial data satisfying (4.2.1). Denote by ((ηε , uε ))ε>0 a family of solutions of (4.1.1) with respective initial data (ηε0 , u0ε ), and by (η, u) any of its limit points. Then, (η, u) ∈ L∞ (R+ ; L2 (T × R)) satisfies the constraints u2 = 0,
−βx2 u1 + ∂2 η = 0.
(4.3.1)
To go further in the description of the weak limit (η, u), we have to isolate the fast oscillations generated by the singular perturbation L, which produce “big” terms in (4.1.1), but converge weakly to 0. The idea to get the mean motion is to consider the weak form of the evolution equations, testing (4.1.1) against smooth functions of KerL. Note that contrary to the previous sections, we are missing regularity in the unknown ηε (which does not satisfy a uniform L2loc (R+ , H 1 (T × R)) bound), so we will need to use smoother functions in the kernel of L than merely H 1 functions as in Section 3. In fact a careful study of the constraint (4.3.1) indicates that the Hermite functions are naturally associated with KerL. Let us therefore introduce the Hermitian basis of L2 (R) constituted of Hermite functions (ψn )n∈N where −ψn
+ β 2 x22 ψn = β(2n + 1)ψn . We recall that βx22 Pn (x2 β) ψn (x2 ) = exp − 2
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where Pn is the n-th Hermite polynomial, as well as the identities ψn (x2 ) + βx2 ψn (x2 ) = 2βnψn−1 (x2 ), ψn (x2 ) − βx2 ψn (x2 ) = − 2β(n + 1)ψn+1 (x2 ).
285
(4.3.2)
Then decomposing any element of KerL on the Hermite basis one can show that it is a linear combination of the following ⎛ ⎞ −ψ0 (x2 ) ⎜ ⎟ (η0 , u0 ) = ⎝ ψ0 (x2 ) ⎠ and 0 ⎛ ⎞ β(n + 1) βn ψn−1 (x2 ) + ψn+1 (x2 ) ⎟ ⎜ 2 2 ⎜ ⎟ ⎜ ⎟ (ηn , un ) = ⎜ ⎟ for n ≥ 1. βn ⎜ β(n + 1) ⎟ ψn−1 (x2 ) − ψn+1 (x2 ) ⎠ ⎝ 2 2 0 We will therefore restrict our attention to those particular vector fields, which are smooth and integrable against any polynomial in x2 , and then conclude by a density argument. Using the conservations of mass and momentum (4.1.1) it is easy to see that, defining mε = (1 + εηε )uε ,
ηε ηn + mε,1 un,1 (t, x) dx + ν
∇uε,1 · ∇un,1 (t , x) dx dt
0
=
t
(ηε0 ηn + m0ε,1 un,1 )(x) dx +
t
mε · (uε · ∇un ) (t , x) dxdt .
0
The difficulty is then to take limits the nonlinear terms, which can be simply written t
mε,1 uε,2 ∂2 un,1 (t , x) dxdt .
0
This is achieved by a compensated compactness technique presented in the next section. 4.3.2. The compensated compactness argument The analysis of the nonlinear terms lies essentially on the structure of the oscillations. A rough description of those fast oscillations will be enough to prove that they do not produce any constructive interference, and therefore do not occur in the equation governing the mean (geostrophic) motion. As in the previous sections, κ denotes a regularizing kernel. L EMMA 4.4. Let us define ηεδ = κδ % ηε and mδε = κδ % ((1 + εηε )uε ) = uδε + ε(ηε uε )δ
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s (T × R)) for any which converge uniformly in ε as δ → 0 to ηε and mε in L∞ (R+ , Hloc s < 0. We also introduce the approximate vorticity
ωεδ = ∇ ⊥ · mδε . Then the following approximate wave equations hold ε∂t ηεδ + ∇·mδε = 0, ε∂t mδε + βx2 (mδε )⊥ + ∇ηεδ = εsεδ + δσεδ ,
(4.3.3)
ε∂t (ωεδ − βx2 ηεδ ) + βmδε,2 = εqεδ + δpεδ , denoting by sεδ , qεδ and σεδ , pεδ some quantities satisfying, for all T > 0, sup sup σεδ L2 ([0,T ];H 1 (T×R)) + pεδ L2 ([0,T ];L2 (T×R)) < +∞, δ>0 ε>0
sup sεδ L1 ([0,T ];H 1 (T×R)) + qεδ L1 ([0,T ];L2 (T×R)) < ∞.
∀δ > 0,
(4.3.4)
ε>0
In order to prove this lemma, we proceed in two steps as in the previous sections, first stating the wave equations for (ηε , mε ), then introducing the regularization (ηεδ , mδε ). We omit the details. Equipped with this preliminary result, we are now able to establish the compensated compactness result, which implies that the nonlinear term actually converges to zero. P ROPOSITION 4.5. With the previous notations, we have locally uniformly in t t mε,1 uε,2 ∂2 un,1 (t , x) dxdt = 0. lim ε→0 0
P ROOF. Let us define, as in Lemma 4.4, ηεδ = ηε % κδ , Then
t 0
=
uδε = uε % κδ
and
mδε = mε % κδ .
mε,1 uε,2 ∂2 un,1 (t , x) dxdt t 0
+
mδε,1 mδε,2 ∂2 un,1 (t , x) dxdt
t 0
+
t 0
+
t 0
mδε,1 (uδε,2 − mδε,2 )∂2 un,1 (t , x) dxdt mδε,1 (uε,2 − uδε,2 )∂2 un,1 (t , x) dxdt (mε,1 − mδε,1 )uε,2 ∂2 un,1 (t , x) dxdt .
(4.3.5)
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• From the energy estimates we can prove that the two last integrals converge towards zero as δ goes to zero uniformly in ε. Indeed for all α > 0 there exists some bounded subset T × α of T × R such that ∂2 un,1 W 1,∞ (R\α ) ≤ α. Then, for 0 < s < 1, " t " " " δ
" " (mε,1 − mε,1 )uε,2 ∂2 un,1 (t , x) dxdt " " 0
≤ mε,1 − mδε,1 L2 ([0,T ];H −s (T×α )) uε,2 L2 ([0,T ];H 1 (T×R)) ∂2 un,1 W 1,∞ (R) + 2αmε,1 L2 ([0,T ];H −s (T×R)) uε,2 L2 ([0,T ];H 1 (T×R)) , which goes to zero as α then δ go to zero, uniformly in ε by (4.2.2) and Lemma 4.4. Similarly, we get, for 0 < s < 1, " " t " " δ δ
" δ " (u − u )∂ u (t , x) dxdt m ε,1 ε,2 ε,2 2 n,1 " ≤ mε,1 L2 ([0,T ];H −s (T×R)) uε,2 " 0
− uδε,2 L2 ([0,T ];H s (T×α )) ∂2 un,1 W 1,∞ (T×R) + 2αmε,2 L2 ([0,T ];H −s (T×R)) uε,1 L2 ([0,T ];H 1 (T×R)) which goes to zero as α then δ go to zero, uniformly in ε by (4.2.2) and Lemma 4.4. Next we prove that for all δ > 0, the second integral in the right-hand side of (4.3.5) goes to zero as ε → 0. But ηε uε and consequently mε are uniformly bounded in L2 ([0, T ]; H s (T × R)) for s < 0. Therefore, for fixed δ > 0, (ηε uε )δ and mδε are uniformly bounded in L2 ([0, T ] × T × R). Then " " t " " δ δ δ
" " mε,1 (uε,2 − mε,2 )∂2 un,1 (t , x) dxdt " " 0
≤ εmδε,1 L2 ([0,T ]×T×R) (ηε uε,2 )δ L2 ([0,T ]×T×R) ∂2 un,1 L∞ (R) which goes to zero as ε → 0 for all fixed δ > 0. • So finally we need to consider the first term in the right-hand side of (4.3.5). We are going to prove that the limit of that term is zero using Lemma 4.4. Integrating by parts, we have t mδε,1 mδε,2 ∂2 un,1 (t , x) dxdt 0
=−
t 0
(∂2 mδε,1 )mδε,2 + mδε,1 (∂2 mδε,2 ) un,1 (t , x) dxdt
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=−
t 0
=−
t 0
δ (ωε + ∂1 mδε,2 )mδε,2 + mδε,1 (∇ · mδε − ∂1 mδε,1 ) un,1 (t , x) dxdt δ (ωε − βx2 ηεδ )mδε,2 + ηεδ (βx2 mδε,2 + ∂1 ηεδ ) + mδε,1 ∇ · mδε
× un,1 (t , x) dxdt 1 t − ∂1 (mδε,2 )2 − (mδε,1 )2 − (ηεδ )2 un,1 (t , x)dxdt 2 0 and the last term is zero because ∂1 un,1 = 0. Lemma 4.4 now implies that t 0
=
mδε,1 mδε,2 ∂2 un,1 (t , x) dxdt t 0
ε ε δ ∂t (βx2 ηεδ − ωεδ )2 + (βx2 ηεδ − ωεδ )qεδ + (βx2 ηεδ − ωεδ )pεδ 2β β β
× un,1 (t , x) dxdt t δ δ un,1 (t , x)dxdt . + − δηεδ σε,1 ε∂t (ηεδ mδε,1 ) − εηεδ sε,1 0
Now we notice that " t " " " δ δ δ
" " η − ω )p u (t , x) dxdt (βx 2 ε ε ε n,1 " " 0
≤ C(1 + x22 )1/2 un,1 L∞ (R) pεδ L2 ([0,T ];L2 (T×R)) × T 1/2 ηεδ L∞ (R+ ;L2 (T×R)) + ωεδ L2 ([0,T ];L2 (T×R)) , and similarly " " t " " δ δ
" " ηε σε,1 un,1 (t , x) dxdt " " 0
≤ CT 1/2 ηεδ L∞ (R+ ;L2 (T×R)) un,1 L∞ (R) σεδ L2 ([0,T ];L2 (T×R)) . So writing ωεδ L2 ([0,T ];L2 (T×R))) ≤ ∇ ⊥ · uδε L2 ([0,T ];L2 (T×R)) + ε∇ ⊥ · (ηεδ uδε )L2 ([0,T ];L2 (T×R)) ,
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we infer that " " " δ "" t δ δ δ
" and η − ω )p u (t , x) dxdt (βx 2 ε ε ε n,2 " " = 0, δ→0 ε→0 β 0 " " t " " δ δ
" " lim δ " ηε σε,1 un,1 (t , x) dxdt " = 0, uniformly in ε.
lim lim
δ→0
0
On the other hand, " t " " " δ δ δ
" " (βx2 ηε − ωε )qε un,2 (t , x) dxdt " " 0
≤ C ηεδ L∞ (R+ ;L2 (T×R)) + ωεδ L∞ (R+ ;L2 (T×R)) × (1 + x22 )1/2 un,2 L∞ (T×R)) qεδ L1 ([0,T ];L2 (T×R))
1 ≤ C ηεδ L∞ (R+ ;L2 (T×R)) + uε L∞ (R+ ;L2 (T×R)) δ ⊥ δ δ + ε∇ · (ηε uε )L∞ (R+ ;L2 (T×R)) × (1 + x22 )un,2 L∞ (T×R) qεδ L1 ([0,T ];L2 (T×R)) , and " " t " " δ δ
" " ηε sε,1 un,1 (t , x) dxdt " " 0
≤ Cηεδ L∞ (R+ ;L2 (T×R)) un,1 L∞ (T×R) sεδ L1 ([0,T ];L2 (T×R)) so " " " ε "" t δ δ δ
" lim (βx2 ηε − ωε )qε un,1 (t , x) dxdt " = 0, for all δ > 0, " ε→0 β 0 " " t " " δ δ
" " lim ε " ηε sε,1 un,1 (t , x) dxdt " = 0, for all δ > 0.
ε→0
0
So we simply need to let ε go to zero, then δ, and the result follows.
4.4. Strong asymptotics In this section we aim at getting a complete description of the asymptotic behaviour of the ocean in the fast rotation limit, including the various equatorial waves—thus proving the strong convergence result stated in Theorem 4.3. In Section 4.4.1 we present the various waves involved, which are eigenvectors of the singular perturbation and constitute a Hilbertian basis of L2 (T × R). That basis enables us to introduce the filtering operator and
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to formally derive the limit filtered system. Then, proving that the limit filtered system has strong solutions (see Sections 4.4.2, 4.4.3 and 4.4.4) and using the strong-weak stability of (4.1.1) (see Section 4.4.5) leads to the strong convergence result. 4.4.1. The equatorial waves In view of the structure of the rotating shallow-water equations (4.1.1), we expect the oscillations of (ηε , uε ) to be mainly governed by the singular perturbation L. The crucial point is that the description of the eigenmodes of L can be achieved using the Fourier transform with respect to x1 and the decomposition on the Hermite functions (ψn )n∈N with respect to x2 . In order to investigate the spectrum of L (which is an unbounded skew-symmetric operator), we are interested in the non trivial solutions to L(η, u) = iτ (η, u). Rewriting that equation as on equation on u2 only, one checks easily that necessarily τ 3 − (k 2 + β(2n + 1))τ + βk = 0,
(4.4.1)
for some n ∈ N. • If k = 0 and n = 0, (4.4.1) admits three solutions τ (n, k, −1) < τ (n, k, 0) < τ (n, k, 1), and one can check that these solutions are eigenvalues of L associated to the following unitary eigenvectors (the coefficient Cn,k,j ensures they are unitary) n,k,j = Cn,k,j eikx1 ⎞ ⎛ i βn β(n + 1) i ψn−1 (x2 ) + ψn+1 (x2 ) ⎟ ⎜ τ (n, k, j ) + k 2 ⎟ ⎜ k − τ (n, k, j ) 2 ⎟ ⎜ ⎟. ⎜ ×⎜ i βn β(n + 1) i ψn−1 (x2 ) − ψn+1 (x2 ) ⎟ ⎟ ⎜ τ (n, k, j ) + k 2 ⎠ ⎝ k − τ (n, k, j ) 2 ψn (x2 ) (4.4.2) The modes corresponding to τ (n, k, −1) and τ (n, k, 1) are called Poincaré modes because ! τ (n, k, ±1) ∼ ± k 2 + β(2n + 1) as |k|, n → ∞, which are the frequencies of the gravity waves. The modes corresponding to τ (n, k, 0) are called Rossby modes because τ (n, k, 0) ∼
k2
βk + β(2n + 1)
as |k|, n → ∞,
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meaning that the oscillation frequency is very small: the planetary waves n,k,0 satisfy indeed the quasigeostrophic approximation. • If k = 0 and √ n = 0, the three solutions to (4.4.1) are the two Poincaré modes τ (n, 0, ±1) = ± β(2n + 1) and the non-oscillating mode τ (n, 0, 0) = 0. The corresponding eigenvectors of L are given by (4.4.2) if j = 0 and by ⎛ ⎞ β(n + 1) βn ψn−1 (x2 ) + ψn+1 (x2 ) ⎟ ⎜ 2 2 ⎜ ⎟ ⎜ ⎟ n,0,0 = Cn,0,0 ⎜ β(n + 1) (4.4.3) ⎟. βn ⎜ ψn−1 (x2 ) − ψn+1 (x2 ) ⎟ ⎝ ⎠ 2 2 0 • If n = 0, the three solutions to (4.4.1) are the two Poincaré and mixed Poincaré–Rossby modes ! k 1 2 τ (0, k, ±1) = − ± k + 4β 2 2 with asymptotic behaviours given by τ (0, k, − sgn(k)) ∼ −k
as |k| → ∞,
β k
as |k| → ∞,
τ (0, k, sgn(k)) ∼
and the Kelvin mode τ (0, k, 0) = k. The corresponding eigenvectors of L are given by (4.4.2) if j = 0 and by ⎛ ⎞ −ψ0 (x2 ) 1 ikx1 ⎝ ψ0 (x2 ) ⎠ . 0,k,0 = √ e (4.4.4) 4π 0 One can then prove the following diagonalization result (whose technical proof is omitted here). P ROPOSITION 4.6. For all (n, k, j ) ∈ N × Z × {−1, 0, 1}, denote by τ (n, k, j ) the three roots of (4.4.1) and by n,k,j the unitary vector defined above. Then (n,k,j )(n,k,j )∈N×Z× 2 {−1,0,1} is a Hilbertian basis of L (T × R) constituted of eigenvectors of L: Ln,k,j = iτ (n, k, j )n,k,j .
(4.4.5)
Furthermore we have the following estimate: for all s > 0, there exists a nonnegative constant Cs such that, for all (n, k, j ) ∈ N × Z × {−1, 0, 1}, n,k,j L∞ (T×R) ≤ C0
and n,k,j H s ∩W s,∞ (T×R) ≤ Cs (1 + |k|2 + n)s/2 . (4.4.6)
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Moreover the following property holds : if τ (n, k, j ) = τ (n∗ , k, j ∗ ), then n = n∗ and j = j ∗ . Finally the eigenspace associated with any iλ = 0 is of finite dimension. As mentioned in the introduction, the adjustment processes are therefore somewhat special in the vicinity of the equator (when the Coriolis acceleration vanishes). A very important property of the equatorial zone is that it acts as a waveguide, i.e., disturbances are trapped in the vicinity of the equator. The waveguide effect is due entirely to the variation of Coriolis parameter with latitude. Note that another important effect of the waveguide is the separation into a discrete set of modes n = 0, 1, 2, . . . as occurs in a channel. The next definition will be useful in the following. D EFINITION 4.7. With the previous notation, let us define P = V ect{n,k,j / (n, k, j ) ∈ N∗ × Z × {−1, 1} \ {0} × {(k, −sgn(k) / k ∈ Z∗ }}, R = V ect{n,k,0 / (n, k) ∈ N∗ × Z∗ }, M = V ect{0,k,j / k ∈ Z∗ , j = −sgn(k)}, K = V ect{0,k,0 / k ∈ Z∗ }, so that L2 (T × R) = P ⊕ R ⊕ M ⊕ K ⊕ KerL. Then we denote by #P (resp. #R , #M , #K and #0 ) the L2 orthogonal projection on P (resp. on R, M, K and KerL). We are now able to define the “filtering operator” associated with the system. Let L be the semi-group generated by L: we write L(t) = exp(−tL). Then, for any three-component vector field ∈ L2 (T × R), we have
e−itλ #λ , (4.4.7) L(t) = iλ∈S
where #λ denotes the L2 orthogonal projection on the eigenspace of L corresponding to the eigenvalue iλ, and S is the set of all eigenvalues of L. Now let us consider (ηε , uε ) a weak solution to (4.1.1), and let us define t ε = L − (ηε , uε ). (4.4.8) ε Conjugating formally equation (4.1.1) by the semi-group leads to t t t t t ε , L ε − νL − L ε = Rε , ∂t ε + L − Q L ε ε ε ε ε (4.4.9)
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where and Q are the linear and symmetric bilinear operator defined by = (0, ) and Q(, ) = (∇ · (0 ), ( · ∇) )
(4.4.10)
denoting by 0 the first coordinate and by the two other coordinates of , and where εηε t 0, −ν uε . Rε = L − ε 1 + εηε We therefore expect to get a bound on the time derivative of ε in some space of distributions. A formal passage to the limit in (4.4.9) as ε goes to zero (based on formula (4.4.7) and on a nonstationary phase argument) leads then to ∂t + QL (, ) − ν L = 0, where L and QL denote the linear and symmetric bilinear operator defined by L =
iλ∈S
#λ #λ
and
QL (, ) =
#λ Q(#μ , #μ˜ ).
iλ,iμ,i μ∈S ˜ λ=μ+μ˜
(4.4.11) Note that this formulation makes a priori no sense, but should be understood in weak form. The definition of the quadratic form naturally addresses the question of the resonances induced by L, which will be studied in Section 4.4.3. 4.4.2. The quasigeostrophic motion In this section we shall investigate the wellposedness of the limit system derived formally in the previous section. The aim of this section is therefore to prove all the results of Theorem 4.3 except for the final, convergence result. Those existence results are based on a precise study of the structure of (4.2.9), and in particular of the ageostrophic part of that equation, meaning its projection onto (KerL)⊥ . One can prove in particular that the ageostrophic part of (4.2.9) is in fact fully parabolic. That should be compared to the case of the incompressible limit of the compressible Navier– Stokes equations, where again the limit system is parabolic, contrary to the original compressible system (see [15,22,44]). Note however that (4.2.9) actually satisfies the same type of trilinear estimates as the three-dimensional incompressible Navier–Stokes system, which accounts for the fact that unique solutions are only obtained for a short life span (despite the fact that the space variable x runs in the two dimensional domain T × R). We will not give all the details of the proof here but indicate the main steps, which enable us to rely on the theory of the three dimensional Navier–Stokes equations. Let us start by considering the existence of weak solutions. The main argument, as pointed out above, is that the ageostrophic part of the limit system is actually fully parabolic, in the following sense: recalling the definition of the spaces HLs given in (4.2.7) above, one can prove that for any ∈ (KerL)⊥ ∩ HLs ,
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2H s ∼
#λ 2H s (T×R) < +∞.
L
(4.4.12)
iλ∈S\{0}
Then it can be proved that for all s ≥ 0, ∀ ∈ (KerL)⊥ ,
2H s+1 (T×R) ≤ C2
HLs+1
≤ C(| − L )H s (T×R) ,
which implies in particular that once projected onto (KerL)⊥ , the system (4.2.9) is fully parabolic. The proof of those inequalities relies on three main arguments. First, the structure of the eigenmodes shows that the diffusion, acting a priori only on the velocity field, also has a smoothing effect on the depth fluctuation. Then one proves the orthogonality in H s (T × R) of the eigenmodes corresponding to the same eigenvalue iλ = 0, and finally a “quasi-orthogonality” property on the eigenmodes: one can prove that ∀ ∈ (KerL)⊥ ,
∇2H s ≤ C
∇(#λ )2H s .
iλ∈S\{0}
In the following we will also use the fact that ∀ ∈ (KerL)⊥ ,
0 2H s ≤ C
(#λ )0 2H s
(4.4.13)
iλ∈S\{0}
and 2H s ≤ C
(#λ ) 2H s .
iλ∈S\{0}
We recall that we have denoted = (0 , ). Once those results are obtained, the existence of weak solutions satisfying the usual energy estimate is obtained with a classical approximation method (the approximate sequence being a truncation to a finite number of n,k,j ’s). The proof of the uniqueness of strong solutions is more delicate. Indeed bilinear estimates have to be established on the quadratic form QL in function spaces compatible with the diffusion operator L , typically the spaces HLs . One proves the continuity of the quadratic form, and in particular an estimate of the following type: for any ∗ , and ∗ in HL1 , the following estimate is satisfied " " ∗ |QL (, ∗ ) ≤ C#⊥ ∗
L2 (T×R)
" "
3/4 1/4 3/4 3/4 #⊥ ∗ L2 (T×R) #⊥ 1 #⊥ ∗ 1 HL1 HL HL
1/4 1/4 1/4 1/4 × #⊥ ∗ 1 #⊥ L2 (T×R) + #⊥ 1 #⊥ ∗ L2 (T×R) HL
HL
+ C#⊥ ∗ L2 (T×R) #0 L2 (T×R) #⊥ ∗ H 1
L
+ #0 L2 (T×R) #⊥ H 1 . ∗
L
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This is exactly the analogue of the usual trilinear estimate for the three-dimensional Navier–Stokes equations: " " " ∗ | div( ⊗ ∗ ) 2 3 " L (R ) ≤ C∗ 3 3 3 H˙ 4 (R )
H˙ 4 (R3 )
∇∗ L2 (R3 ) + ∗
3
∇L2 (R3 )
1
∇L2 (R2 ) .
H˙ 4 (R3 )
whereas in two space dimensions one would expect " " " ∗ | div( ⊗ ∗ ) 2 2 " L (R ) ≤ C∗ 1 2 1 H˙ 2 (R )
H˙ 2 (R2 )
∇∗ L2 (R2 ) + ∗
H˙ 2 (R2 )
Similarly, three dimensional-type estimates can be derived for (∗ |QL (, ∗ ))H 1/2 , and L the usual theory of the three dimensional Navier–Stokes equations enables us to prove the expected existence and uniqueness result. We refer to [25] for all the technicalities; let us simply mention that the reason for the loss of one half derivative compared to the usual two dimensional case is√ linked to the fact that differentiation with respect to x2 corresponds to a multiplication by n instead of n. 4.4.3. Interactions between equatorial waves Unfortunately in order to prove the strong convergence result of Theorem 4.3, more regularity is required on the limit system. We postpone to Section 4.4.5 the end of the proof of the theorem, and will pursue in this section and the next the study of the limit system, in order to gather more useful information. In particular we need to study more precisely the resonance condition λ = μ + μ∗ , which can be written τ (n, k, j ) + τ (n∗ , k ∗ , j ∗ ) = τ (m, k + k ∗ , &). Recalling that the eigenvalues of the penalization operator L are defined as the roots of a countable number of polynomials whose coefficients depend (linearly) on the ratio β, we deduce that for fixed n, n∗ , m ∈ N and k, k ∗ ∈ Z, the occurrence of such a resonant triad is controlled by the cancellation of some polynomial in β. Therefore, either this polynomial has a finite number of zeros, or it is identically zero. The difficulty here is that we are not able to eliminate the second possibility using only the asymptotics β → ∞. Because of the possible resonance with j = j ∗ = & = 0 which can occur even for large β, we have to refine the previous argument introducing an auxiliary polynomial. We refer to [25] for the details, enabling one to conclude that the limit filtered system can be rewritten in the following manner for all but a countable number of β: ∂t #0 − ν#0 L #0 = 0, ∂t #R + 2Q L (#0 , #R ) + Q L (#R , #R ) − ν#R L = 0, ∂t #M + 2Q L (#0 , #M ) − ν#M L = 0,
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∂t #P + 2Q L (#0 , #P ) − ν#P L = 0, ∂t #K + 2Q L (#0 , #K ) + Q L (#K , #K ) − ν#K L = 0, (4.4.14) It can be noticed that the only nonlinear interactions are due to Kelvin or to Rossby-type waves, which will be crucial in the proof of the propagation of regularity. Let us prove estimate (4.2.10). We can decompose in the basis of eigenvectors of L, and will estimate each projection separately. Clearly we have div (#0 ) = 0, so let us consider now the projection onto Rossby modes #R . By definition of the Rossby modes we deduce the following relation ∀ ∈ R,
∇ · =
∇ · λ =
iλ∈SR
iλ(λ )0
iλ∈SR
with the notation λ = #λ , and where SR denotes the set of Roosby modes. It follows that, using (4.4.13), ∇
· 2H 2 (T×R)
92 9 9 9 9 =9 iλ(λ )0 9 9
H 2 (T×R)
iλ∈SR
≤C
λ(λ )0 2H 2 (T×R) .
iλ∈SR
But, as Rossby waves correspond to j = 0, we have (denoting by #n,k,j the orthogonal projection onto n,k,j ) λ(λ )0 2H 2 (T×R) ≤ C|λ|2
(#n,k,0 )0 2H 2 (T×R) .
τ (n,k,0)=λ
Recalling the explicit form of (n,k,j )0 , we see that (#n,k,0 )0 2H 2 (T×R) ≤ C(1 + n + k 2 )(#n,k,0 )0 2H 1 (T×R) . But for Rossby modes, the following asymptotics hold as |k| or n goes to infinity: λ = τ (n, k, 0) ∼
βk · k 2 + β(2n + 1)
So we infer that as |k| or n goes to infinity, |λ|2 (#n,k,0 )0 2H 2 (T×R) ≤ C(#n,k,0 )0 2H 1 (T×R) .
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Finally we infer that
∇ · 2H 2 ≤ C
λ(λ )0 2H 2 (T×R)
iλ∈SR
≤C
(#n,k,0 )0 2H 1 (R×T)
(n,k,0)∈SR
≤ C2H 1 . L
By the embedding of H 2 (T × R) into L∞ (T × R) we conclude that ∇ · (#R ) belongs to the space L2 ([0, T ]; L∞ (T × R)). The same result can easily be extended to the mixed Poincaré–Rossby modes (it is in fact easier since n = 0 in that case) and we obtain #M L2 ([0,T ],H 1 ) ≤ CT , L
∇ · (#M ) L2 ([0,T ],L∞ (T×R)) ≤ CT .
Finally we deduce that ∇ · ((#0 + #R + #M )) L2 ([0,T ],L∞ (T×R)) ≤ CT . Let us now consider the equation governing the Poincaré modes which can be seen as a linear parabolic equation whose coefficients depend on #0 . We can write #P =
ϕn,k,j n,k,j ,
(n,k,j )∈SP
where − SP = N∗ × Z × {−1, 1} ∪ {0} × Z+ ∗ × {1} ∪ {0} × Z∗ × {−1}.
We can use Proposition 4.6 to deduce that for each (n, k, j ) in SP the equation governing ϕn,k,j can be decoupled (recall that #0 only depends on x2 ): ∂t ϕn,k,j − νϕn,k,j (n,k,j | n,k,j )L2 (T×R) = −2ϕn,k,j (n,k,j |Q(n,k,j , #0 ))L2 (T×R) which can be rewritten ∂t ϕn,k,j exp −νt (n,k,j | n,k,j )L2 (T×R) = −2ϕn,k,j (n,k,j |Q(n,k,j , #0 ))L2 (T×R) × exp −νt (n,k,j | n,k,j )L2 (T×R) .
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From Gronwall’s lemma and the following estimates (due to (4.4.6) and to the bound of #0 in L∞ (R+ ; L2 (T × R))), " "(n,k,j |Q(n,k,j , #0 ))
L2 (T×R)
" " ≤ C1 (n + k 2 )1/2 ,
−(n,k,j | n,k,j )L2 (T×R) ≥ C2 (n + k 2 ), we then deduce that there exists a nonnegative constant Cν (depending only on ν) such that, ∀(n, k, j ) ∈ SP ,
|ϕn,k,j (t)| ≤ |ϕn,k,j (0)| exp(−Cν (n + k 2 )t).
(4.4.15)
Now we write 9 9 9∇ · (#P ) (t)9
L∞ (T×R)
≤
9 9 |ϕn,k,j (t)| 9∇ · (n,k,j ) (t)9L∞ (T×R)
(n,k,j )∈SP
≤C
|ϕn,k,j (t)|(n + k 2 )1/2
(n,k,j )∈SP
since (n,k,j ) is uniformly bounded in L∞ (T × R). Thus, by (4.4.15), we infer that 9 9 9∇ · (#P ) (t)9 ∞ L (T×R)
≤C |ϕn,k,j (0)|(n + k 2 )1/2 exp(−Cν (n + k 2 )t). (n,k,j )∈SP
Integrating with respect to time leads then to 9 9 9∇ · (#P ) 9
L1 ([0,T ];L∞ (T×R))
≤ Cν
|ϕn,k,j (0)|(n + k 2 )−1/2
(n,k,j )∈SP
≤ Cν
&
|ϕn,k,j (0)|2 (n + k 2 )α
(n,k,j )∈SP
'1/2 &
(n + k 2 )−1−α
'1/2
,
(n,k,j )∈SP
from which we deduce that for α > 1/2, 9 9 9∇ · (#P ) 9
L1 ([0,T ];L∞ (T×R))
≤ C#P 0 H˜ α (T×R) L
where C depends only on ν and α. Finally we are left with the Kelvin modes. The difficulty here is that the equation is nonlinear, and the argument of the Rossby part does not work (there is no natural smoothing of the divergence). However #K satisfies an equation which is actually one-dimensional (modulo a smooth function with respect to x2 ), and thus the energy estimate is supercritical in the sense that the H 1 norm allows to control the stability. We first note that for the
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Kelvin modes, since the decomposition of the eigenmodes of L corresponds to the Fourier decomposition, we have (#K |QL (#K , #K ))H α (T×R) = #K |Q(#K , #K ) H α (T×R) . Therefore, using the fact that H α (T) is an algebra for all α > 1/2, we get " " "(#K |Q(#K , #K ))H α (T×R) " ≤ C#K H α+1 (T×R) #K H α (R×T) #K H 1 (T×R) ≤ ν#K 2H α+1 (T×R) +
C #K 2H α (T×R) #K 2H 1 (T×R) ν
by the Cauchy–Schwarz inequality. Estimating the linear term as before, we get by Gronwall’s lemma #K (t)2H α (T×R) + ν ≤ #K 0 2H α (T×R)
t 0
#K (t )2H α+1 (T×R) dt
C 2 × exp (τ )H 1 (T×R) dτ . ν L
Then,
#K L∞ ([0,T ],H˜ α (T×R)) ≤ CT L
and
#K L2 ([0,T ],H˜ α+1 (T×R)) ≤ CT , L (4.4.16)
under the suitable initial assumption. From the orthogonality properties mentioned earlier, along with the Sobolev embeddings H α (T) ⊂ L∞ (T) we infer that ∇ · (#K ) L2 ([0,T ],L∞ (T×R)) ≤ CT , provided that α > 1/2. The estimate (4.2.10) is proved.
(4.4.17)
4.4.4. Propagation of regularity In this section we shall state without proof some useful results concerning the propagation of regularity for the limit system. Propagation of regularity for #0 Let us notice that the weak formulation of the limit equation given in the statement of Theorem 4.2 could be written in the more compact way ∂t − ν#0 #0 = 0,
(4.4.18)
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were it not for the fact that the operator #0 #0 is a priori not defined on L2 (R). The projection #0 is a pseudo-differential operator, whose symbol is ⎛
(βx2 )2 ⎜ (βx2 )2 + ξ 2 ⎜ 2 ⎜ ⎜ iβx2 ξ2 ⎜ ⎝ (βx )2 + ξ 2 2
0
2
⎞
−iβx2 ξ2 (βx2 )2 + ξ22
0⎟ ⎟ ⎟ ⎟. 0⎟ ⎠
ξ22 (βx2 )2 + ξ22 0
0
In particular extending #0 to Sobolev spaces requires some techniques of microlocal analysis like the Weyl–Hörmander calculus. The singularity at x2 = 0 unfortunately prevents one of using this theory blindfolded, but inspired by the results given by that theory, in particular its commutator estimates, one can work “by hand” (see [25] for details) to prove the following proposition. P ROPOSITION 4.8. Denote by the (unique) weak solution to the geostrophic equation ∂t − ν#0 #0 = 0,
(t = 0) = #0 0 .
Then, if the initial data satisfies the regularity assumption #0 0 HLs ≤ C0 for some s ≥ 1, the solution satisfies for all T > 0 the regularity estimate L∞ ([0,T ],HLs ) ≤ CT . Propagation of regularity for #⊥ One can prove bilinear estimates in HLs for QL for s ∈ [1/2, 1[, which allow to deduce easily the following result. P ROPOSITION 4.9. Denote by the (unique) strong solution on [0, T ∗ [ to the envelope equations ∂t + QL (, ) − ν L = 0 1/2
#⊥ (t = 0) = #⊥ 0 ∈ HL
#0 (t = 0) = #0 0 ∈ L2 (T × R). If the initial data satisfies the regularity assumption #⊥ 0 HLs ≤ C0 for some s ∈ [1/2, 1[, then the solution satisfies for all T < T ∗ the regularity estimate #⊥ L∞ ([0,T ],H s )∩L2 ([0,T ],H s+1 ) ≤ CT . L
L
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4.4.5. Stability and strong convergence In this final section we shall gather the previous results in order to prove the strong convergence of the filtered solutions. The idea is, as usual in filtering methods, to start by approximating the solution of the limit system, and then to use a weak-strong stability method to conclude. So let us consider the solution constructed in Section 4.4.4, which we truncate in the following way: N = JN #⊥ + #0 N ,
(4.4.19)
where JN is the spectral truncation defined by JN =
(4.4.20)
#λ
iλ∈SN
with ;
SN = iτ (n, k, j ) ∈ S n ≤ N, |k| ≤ N , and #⊥ denotes as previously the projection onto (KerL)⊥ . Finally #0 N solves ∂t #0 N − ν#0 #0 N = 0
#0 N |t=0 = #n,0,0 0 , 0≤n≤N
where #n,0,0 denotes the projection onto the eigenvector n,0,0 of KerL. According to Proposition 4.8, for all fixed N ∈ N we have #0 N belongs to L∞ (R+ ; HLσ ),
∀σ ≥ 0.
(4.4.21)
Recall that such a result means that #0 N is as smooth as needed, and decays as fast as needed when x2 goes to infinity. Moreover by the stability of the limit system (which is linear) we have of course lim #0 N − #0 L∞ ([0,T ];L2 (T×R)) = 0,
N →∞
∀T > 0.
Note also that for all fixed N ∈ N, using the smoothness and the decay of the eigenvectors of L, we get for any polynomial Q ∈ R[X] Q(x2 )N ∈ L∞ ([0, T ]; C ∞ (T × R)) We have moreover of course ∀T < T ∗ ,
#⊥ ( − N )L∞ ([0,T ];L2 (T×R)) → 0 as N → ∞,
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and ∀T < T ∗ ,
#⊥ ( − N )L2 ([0,T ];H α+1 ) → 0 as N → ∞. L
Finally since JN commutes with L , the vector field N satisfies the approximate limit filtered system ∂t N + JN QL (, ) − ν L N = 0, N |t=0 = JN 0 .
(4.4.22)
Conjugating this equation by the semi-group L leads then to t 1 t t t ∂t L N + L L N + JN QL L , L ε ε ε ε ε t N = 0, − ν L L ε using the definitions (4.4.11) of QL and L . We are going now to follow the same method as that used in Section 2, in the periodic case: we start by rewriting this last equation in a convenient way t 1 t t t ∂t L N + L L N + Q L N , L N ε ε ε ε ε t
− ν L N ε t t t
N , L N − ν( − L )L N = (Q − QL ) L ε ε ε t t , L + (I d − JN )QL L ε ε t t (N − ), L (N + ) . + QL L ε ε The two last terms in the right-hand side are expected to be small when N is large, uniformly in ε, using the stability of the limit system. So we are left with the first two terms, which as usual cannot be dealt with so easily since they do not converge strongly towards zero. However they are fast oscillating terms, and will be treated by introducing a small quantity εφN (which will be small when ε goes to zero, for each fixed N ), so that 1 t t t ∂t + L L εφN ∼ −(Q − QL ) L N , L N ε ε ε ε t
+ ν( − L )L N . ε
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Let us now define
φN = −
t
λ =μ+μ˜ iλ∈S,iμ,i μ∈S ˜ N
+ν
λ =μ, iλ∈S,iμ∈SN
˜ ei ε (λ−μ−μ) #λ Q(#μ N , #μ˜ N ) i(λ − μ − μ) ˜ t
ei ε (λ−μ) #λ #μ N , i(λ − μ)
(4.4.23)
and consider ε,N = N + εφN . The proof of the following result follows essentially the same lines as in the constant, periodic case of Section 2 (up to the fact that λ is not truncated here) and we refer to [25] for details. P ROPOSITION 4.10. For all but a countable number of β, the following result holds. Consider a vector field 0 = (η0 , u0 ) ∈ L2 (T × R), with #⊥ 0 in HLα for some α > 1/2. Denote by the associate solution of the limit system on [0, T ∗ [. Then there exists a family (ηε,N , uε,N ) = L( εt )ε,N such that #⊥ (ηε,N , uε,N ) is uniformly bounded in the α+1 α 2 ∗ ∗ space L∞ loc ([0, T [, HL ) ∩ Lloc ([0, T [, HL ), satisfying the following properties: • ε,N behaves asymptotically as as ε → 0 and N → ∞: ∀T < T ∗ ,
9 9 lim lim 9ε,N − 9L∞ ([0,T ];L2 (T×R)) = 0;
N →∞ ε→0
(4.4.24)
• for all N ∈ N, (ηε,N , uε,N ) is smooth: for all T ∈ [0, T ∗ [ and all Q ∈ R[X], Q(x2 )(ηε,N , uε,N ) is bounded in L∞ ([0, T ]; C ∞ (T × R)), uniformly in ε; (4.4.25) • (ηε,N , uε,N ) satisfies the uniform regularity estimate ∀T ∈ [0, T ∗ [,
sup lim ∇ · uε,N L1 ([0,T ];L∞ (T×R)) ≤ CT ;
N ∈N ε→0
(4.4.26)
• (ηε,N , uε,N ) satisfies approximatively the viscous Saint-Venant system (SWε ): 1 ∂t (ηε,N , uε,N ) + L(ηε,N , uε,N ) + Q((ηε,N , uε,N ), (ηε,N , uε,N )) ε − ν (ηε,N , uε,N ) = Rε,N
(4.4.27)
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where Rε,N goes to 0 as ε → 0 then N → ∞: lim lim (Rε,N L1 ([0,T ];L2 (T×R)) + εRε,N L∞ ([0,T ]×T×R) ) = 0.
N →∞ ε→0
(4.4.28)
Equipped with that result, we are now ready to prove the strong convergence theorem. The method relies on a weak-strong stability method which we shall now detail. We are going to prove that lim lim (ηε , uε ) − (ηε,N , uε,N )L2 ([0,T ]×T×R) = 0,
N →∞ ε→0
(4.4.29)
where (ηε,N , uε,N ) is the approximate solution to (4.1.1) defined in Proposition 4.10. Note that combining this estimate with the fact that (ηε,N , uε,N ) is close to L( εt ) provides the expected convergence, namely the fact that 9 9 9 9 t (η 9 lim 9 , u ) − L = 0. ε ε 9 2 9 ε→0 ε L ([0,T ]×T×R)
∀T ∈ [0, T ∗ [,
The key to the proof of (4.4.29) lies in the following proposition. P ROPOSITION 4.11. There is a constant C such that the following property holds. Let (η0 , u0 ) and (ηε0 , u0ε ) satisfy assumption (4.2.8), and let T > 0 be given. For all ε > 0, ¯ u) ¯ denote by (ηε , uε ) a solution of (4.1.1) with initial data (η0 , u0 ). For any vector field (η, belonging to L∞ ([0, T ]; C ∞ (T × R)) and rapidly decaying with respect to x2 , define Eε (t) =
1 2
¯ 2 + (1 + εηε )|uε − u| ¯ 2 (t, x)dx (ηε − η)
+ν
t
¯ 2 (t , x)dxdt . |∇(uε − u)|
0
Then the following stability inequality holds for all t ∈ [0, T ]: Eε (t) ≤ CEε (0) exp χ(t) + ωε (t) t 1
∂t η¯ + ∇ · u¯ + ∇ · (η¯ u) ¯ (η¯ − ηε )(t , x)dxdt +C eχ(t)−χ(t ) ε 0 t 1 χ(t)−χ(t ) ⊥ +C e ¯ + (u¯ · ∇)u¯ − νu¯ (1 + εηε ) ∂t u¯ + (βx2 u¯ + ∇ η) ε 0 × (u¯ − uε )(t , x)dxdt , where ωε (t) depends on u¯ and goes to zero with ε, uniformly in time, and where χ(t) = C 0
t
∇ · u ¯ L∞ (T×R) + ∇ u ¯ 2L2 (T×R) (t )dt .
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Let us postpone the proof of that result, and end the proof of the strong convergence. We apply that proposition to (η, ¯ u) ¯ = (ηε,N , uε,N ), where (ηε,N , uε,N ) is the approximate solution on [0, T ∗ [ given by Proposition 4.10. We will denote by χε,N and Eε,N the quantities defined in Proposition 4.11, where (η, ¯ u) ¯ has been replaced by (ηε,N , uε,N ). Because of the uniform regularity estimates on (ηε,N , uε,N ), we have ∀T ∈ [0, T ∗ [, sup lim ∇uε,N 2L2 ([0,T ],L2 (T×R)) + ∇ · uε,N L1 ([0,T ];L∞ (T×R)) ≤ CT , N ε→0
so we get a uniform bound on χε,N : sup lim χε,N L∞ ([0,T ]) ≤ CT . N ε→0
Then, from the initial convergence (4.2.8) we obtain that ∀N ∈ N, Eε,N (0) exp χε,N (t) → 0 as ε → 0 in L∞ ([0, T ]). Moreover by Proposition 4.11 we have 1 ∂t (ηε,N , uε,N ) + L(ηε,N , uε,N ) + Q((ηε,N , uε,N ), (ηε,N , uε,N )) ε − ν (ηε,N , uε,N ) = Rε,N .
(4.4.30)
Let us estimate the contribution of the remainder term. We can write t
eχε,N (t)−χε,N (t ) Rε,N · (ηε,N − ηε ), (1 + εηε )(uε,N − uε ) (t , x)dxdt 0 (1)
(2)
= Iε,N (t) + Iε,N (t), with def (1) Iε,N (t) = def (2) Iε,N (t) =
t
e
χε,N (t)−χε,N (t )
e
χε,N (t)−χε,N (t )
0
0
t
Rε,N,0 (ηε,N − ηε )(t , x)dxdt ,
and
(1 + εηε )(uε,N − uε )(t , x)dxdt . Rε,N
The first term can be estimated in the following way: (1)
|Iε,N (t)| ≤ CT Rε,N L1 ([0,T ];L2 (T×R)) ηε,N − ηε L∞ ([0,T ];L2 (T×R)) . For the second term we can write (2) |Iε,N (t)|Eˆ ≤ CT 1 + εηε (uε,N − uε )L∞ ([0,T ];L2 (T×R)) 1 + εηε × Rε,N L1 ([0,T ];L2 (T×R)) .
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Now we can write 1 + εηε Rε,N 2L2 (T×R) ≤ C Rε,N 2L2 (T×R) + εηε L2 (T×R) Rε,N 2L4 (T×R) . Since εRε,N 2L4 (T×R) ≤ εRε,N L∞ (T×R) Rε,N L2 (T×R) , 1
we infer that the quantity ε 2 Rε,N goes to zero as ε goes to zero and N goes to infinity, in the space L2 ([0, T ]; L4 (T × R)), so in particular 1
lim lim ε 2 Rε,N L1 ([0,T ];L4 (T×R)) = 0.
N →∞ ε→0
Finally by the uniform bound on ηε in L∞ ([0, T ]; L2 (T × R)) and by the smallness assumptions on Rε,N , we deduce that
t
eχε,N (t)−χε,N (t )
Rε,N · (ηε,N − ηε ), (1 + εηε )(uε,N − uε ) (t , x)dxdt
0
≤
1 ηε,N − ηε 2L∞ ([0,T ];L2 ) + 1 + εηε (uε,N − uε )2L∞ ([0,T ];L2 ) | 2 + ωε,N (t),
where lim lim ωε,N (t)L∞ ([0,T ]) = 0.
N →∞ ε→0
We now recall that by Proposition 4.11, using (4.4.30), we have Eε,N (t) ≤ CEε,N (0) exp χε,N (t) + ωε,N (t) t
+C eχε,N (t)−χε,N (t ) ×
0
Rε,N · (ηε,N − ηε ), (1 + εηε )(uε,N − uε ) (t , x)dxdt
where Eε,N (t) =
1 (ηε − ηε,N )(t)2L2 + 1 + εηε (uε − uε,N )(t)2L2 2 t +ν ∇(uε − uε,N )(t )2L2 (t )dt . 0
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Putting together the previous results we get that limN →∞ limε→0 Eε,N (t) = 0 uniformly on [0, T ], hence that lim lim ηε,N − ηε L∞ ([0,T ];L2 (T×R)) = 0,
N →∞ ε→0
lim lim 1 + εηε (uε,N − uε )L∞ ([0,T ];L2 (T×R)) = 0,
N →∞ ε→0
lim lim uε,N − uε L2 ([0,T ],H˙ 1 (T×R)) = 0.
N →∞ ε→0
By interpolation we therefore find that lim lim ηε,N − ηε L∞ ([0,T ];L2 (T×R)) + uε,N − uε L2 ([0,T ],H 1 (T×R)) = 0,
N →∞ ε→0
hence (4.4.29) is proved. To conclude the proof of Theorem 4.3 it remains to give an idea of the proof of Proposition 4.11. As the energy is a Lyapunov functional for (4.1.1), we have Eε (t) − Eε (0) t 1 2 d 1 2 ≤ η¯ − ηη ¯ − u¯ · uε (t , x)dxdt ¯ ε + (1 + εηε ) |u| 2 2 0 dt t + ¯ , x)dxdt ν(∇ u¯ − 2∇uε ) · ∇ u(t 0
≤
t
¯ η¯ − ηε ) + (1 + εηε )∂t u¯ · (u¯ − uε ) (t , x)dxdt ∂t η(
0
t ε 2 ∂t ηε η¯ + ∂t ((1 + εηε )uε ) · u¯ − ∂t ηε |u| − ¯ (t , x)dxdt 2 0 t − ν u¯ · (u¯ − uε ) − uε · u¯ (t , x)dxdt . 0
Using the conservation of mass and of momentum we get Eε (t) − Eε (0) t ≤ ¯ η¯ − ηε ) + (1 + εηε )(∂t u¯ − νu) ¯ · (u¯ − uε ) (t , x)dxdt ∂t η( 0
ε 2 1 ∇ · (1 + εηε )uε η¯ − |u| ¯ (t , x)dxdt ε 2 0 t (1 + εηε ) ⊥ + ¯ , x)dxdt (βx2 uε + ∇ηε ) + ∇ · ((1 + εηε )uε ⊗ uε ) · u(t ε 0 t + ενηε u¯ · (u¯ − uε )(t , x)dxdt . +
t
0
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Integrating by parts leads then to Eε (t) − Eε (0) t 1 ∂t η¯ + ∇ · u¯ + ∇ · (η¯ u) ¯ (η¯ − ηε )(t , x)dxdt ≤ ε 0 t 1 + ¯ + (u¯ · ∇)u¯ − νu¯ (1 + εηε ) ∂t u¯ + (βx2 u¯ ⊥ + ∇ η) ε 0 × (u¯ − uε )(t , x)dxdt t − (1 + εηε )D u¯ : (u¯ − uε )⊗2 (t , x)dxdt 0
−
t 0
1 2 ¯ + ηε u¯ · ∇ η¯ (t , x)dxdt + Rε , ηε ∇ · u¯ + (η¯ − ηε )∇ · (η¯ u) 2 (4.4.31)
where Rε (t) =
t
ενηε u¯ · (u¯ − uε )(t , x)dxdt .
0
The last term is rewritten in a convenient form by integrating by parts t
1 2 − ¯ + ηε u¯ · ∇ η¯ (t , x)dxdt η ∇ · u¯ + (η¯ − ηε )∇ · (η¯ u) 2 ε 0 t 1 2 ηε ∇ · u¯ + (η¯ − ηε )(u¯ · ∇ η¯ + η∇ =− ¯ · u) ¯ + ηε u¯ · ∇ η¯ (t , x)dxdt 2 0 t 1 2 1 ηε ∇ · u¯ + (η¯ − ηε )η∇ =− ¯ · u¯ + u¯ · ∇ η¯ 2 (t , x)dxdt 2 2 0 t 1 (ηε − η) =− ¯ 2 ∇ · u(t ¯ , x)dxdt . (4.4.32) 2 0 Plugging (4.4.32) into (4.4.31) leads to Eε (t) − Eε (0) t 1 ∂t η¯ + ∇ · u¯ + ∇ · (η¯ u) ≤ ¯ (η¯ − ηε )(t , x)dxdt ε 0 t 1 ⊥ + ¯ + (u¯ · ∇)u¯ − νu¯ (1 + εηε ) ∂t u¯ + (βx2 u¯ + ∇ η) ε 0 × (u¯ − uε )(t , x)dxdt
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−
309
(1 + εηε )D u¯ : (u¯ − uε )⊗2 (t , x)dxdt
0
t
−
1 ¯ 2 ∇ · u(t ¯ , x)dxdt + Rε (t). (ηε − η) 2
0
(4.4.33)
In order to get an inequality of Gronwall type, one has to control the right hand side in terms of Eε . We start by estimating the flux term. We have −
t 0
(1 + εηε )∇ u¯ : (u¯ − uε )⊗2 (t , x)dxdt
t
≤
∇ u ¯ L2 (T×R) + εηε L2 (T×R) ∇ u ¯ L∞ (T×R) u¯ − uε 2L4 (T×R) (t )dt
0
≤C
t
∇ u ¯ L2 (T×R) + εηε L2 (T×R) ∇ u ¯ L∞ (T×R) u¯ − uε L2 (T×R)
0
× u¯ − uε H˙ 1 (T×R) (t )dt and ¯ 2L2 (T×R) u¯ − uε 2L2 (T×R) ≤ 1 + εηε (uε − u) + εηε L2 (T×R) u¯ − uε L2 (T×R) u¯ − uε H˙ 1 (T×R) which implies ¯ 2L2 (T×R) u¯ − uε 2L2 (T×R) ≤ 2 1 + εηε (uε − u) + 16ε 2 ηε 2L2 (T×R) u¯ − uε 2H˙ 1 (T×R) . √ Therefore, using the uniform bounds on ηε , 1 + εηε uε and on uε given by the energy estimate, we gather that t − (1 + εηε )∇ u¯ : (u¯ − uε )⊗2 (t , x)dxdt 0
≤C 0
t
(∇ u ¯ L2 + εηε L2 ∇ u ¯ L∞ ) 1 + εηε
× (uε − u) ¯ L2 u¯ − uε H˙ 1 (t )dt t + Cε (∇ u ¯ L2 + εηε L2 ∇ u ¯ L∞ )u¯ − uε 2H˙ 1 (t )dt ≤
ν 4
C + ν
0
u¯ − uε 2H˙ 1 (t )dt
¯ 2L2 (t )dt + ωε (t). ∇ u ¯ 2L2 1 + εηε (uε − u)
(4.4.34)
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We also have −
t 0
1 1 (ηε − η) ¯ 2 ∇ · u(t ¯ , x)dxdt ≤ 2 2
t
∇ · u ¯ L∞ (T×R)
0
× η¯ − ηε 2L2 (T×R) (t )dt ,
(4.4.35)
so we are left with the study of the remainder Rε . We have Rε (t) ≤ ενηε L∞ (R+ ;L2 (T×R))
0
t
u ¯ L4 (T×R) u¯ − uε L4 (T×R) (t )dt .
The above estimate on u¯ − uε L2 (T×R) implies in particular that u¯ − uε L2 (T×R) is bounded in L2 ([0, T ]), hence we get that u¯ − uε L4 (T×R) is also bounded in L2 ([0, T ]). So we infer directly that Rε (t) goes to zero in L∞ ([0, T ]) as ε goes to zero. That result, joined to (4.4.34) and (4.4.35) allows to deduce from (4.4.33) the following estimate: 1 Eε (t) − Eε (0) 2 t 1 ∂t η¯ + ∇ · u¯ + ∇ · (η¯ u) ≤ ¯ (η¯ − ηε )(t , x)dxdt ε 0 t 1 ⊥ + ¯ + (u¯ · ∇)u¯ − νu¯ (1 + εηε ) ∂t u¯ + (βx2 u¯ + ∇ η) ε 0 × (u¯ − uε )(t , x)dxdt C + ¯ 2L2 (T×R) (t )dt ∇ u ¯ 2L2 1 + εηε (uε − u) ν 1 t + ∇ · u ¯ L∞ η¯ − ηε 2L2 (t )dt + ωε (t) 2 0 thus applying Gronwall’s lemma provides the expected stability inequality.
4.5. A hybrid result In this section we are going to put together some results obtained in the previous sections, to prove the strong convergence theorem presented in the introduction of Section 4, namely Theorem 4.4. Due to the unfortunate presence of a defect measure in the limit system, we propose in Section 4.5.4 an alternate model with capillarity, whose virtue is that it gives the lacking compactness on εηε . Its disadvantage is its unphysical character, along with the fact that weak solutions are only known to exist for small data.
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4.5.1. Strong compactness of #λ ε In order to characterize completely the asymptotic behaviour of (ηε , uε ) we know from the previous section that it is necessary to introduce the filtering operator t def itλ L = exp − #λ , ε ε iλ∈S
where #λ is the projection on the eigenspace of L associated with the eigenvalue iλ. L EMMA 4.12. With the notation of Theorem 4.4, the following results hold. • For all iλ ∈ S \ {0}, #λ ε is strongly compact in L2 ([0, T ], H s (T × R)) for all T > 0 and all s ∈ R; s (T × R)) for all T > 0 and all s < 0. • #0 ε is strongly compact in L2 ([0, T ], Hloc P ROOF. • For all λ = 0, we recall that by Proposition 4.6, the eigenspace of L associated with the eigenvalue iλ is a finite dimensional subspace of H ∞ (T × R). Therefore the only point to be checked is the compactness with respect to time, which is obtained as follows. Let (n, k, j ) ∈ N × Z × {−1, 0, 1} be given, such that λ = τ (n, k, j ) = 0. Multiplying (4.1.1) by n,k,j = (ηn,k,j , un,k,j ) (which is smooth and rapidly decaying as |x2 | goes to infinity) and integrating with respect to x leads to ∂t
(ηε η¯ n,k,j + mε · u¯ n,k,j )(t, x)dx iτ (n, k, j ) (ηε η¯ n,k,j + mε · u¯ n,k,j )(t, x)dx ε + ν ∇uε : ∇ u¯ n,k,j (t, x)dx − mε · (uε · ∇)u¯ n,k,j (t, x)dx
+
1 − 2
ηε2 ∇ · u¯ n,k,j (t, x)dx = 0
where u¯ denotes the complex conjugate of u, or equivalently
itτ (n, k, j ) ∂t exp (ηε η¯ n,k,j + mε · u¯ n,k,j )(t, x)dx ε itτ (n, k, j ) uε : ∇ u¯ n,k,j (t, x)dx +ν ∇ exp ε 1 2 itτ (n, k, j ) mε · (uε · ∇)u¯ n,k,j + ηε ∇ · u¯ n,k,j (t, x)dx = 0. − exp ε 2 (4.5.1)
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From the uniform estimates coming from the energy inequality we then deduce that ∂t
itτ (n, k, j ) exp (ηε η¯ n,k,j + mε · u¯ n,k,j )(t, x)dx ε
is uniformly bounded in ε. Therefore the family
itλ exp #λ (ηε , mε ) ε ε>0 is compact in L2 ([0, T ]; H s (T × R)) for any s ∈ R,
and since εηε uε converges to 0 in L2 (R+ ; H s (T × R)) for all s < 0, we deduce that itλ #λ (ηε , uε ) = #λ ε exp ε is compact in L2 ([0, T ]; H s (T × R)) for any s ∈ R. • For #0 ε = #0 (ηε , uε ) the study is a little more difficult since the compactness with respect to spatial variables has to be taken into account. From the energy estimate we have the uniform bound ε is uniformly bounded in L2loc (R+ , L2 (T × R)). Recall that we have defined
s HLs = ψ ∈ S (T × R) / (Id − + β 2 x22 ) 2 ψ ∈ L2 (T × R) . Equivalently we have
HLs = ψ ∈ S (T × R) / (1 + n + k 2 )s (n,k,j |ψ)2L2 (T×R) < +∞ , n,k,j ∈S
where S = N × Z × {−1, 0, 1}. As (n,0,0 )n∈N is a Hilbertian basis of KerL, we have for all T > 0 and all s < 0 9 9 9 9 (n,0,0 |ε )L2 (T×R) n,0,0 − #0 ε 9 9 n≤N
L2 ([0,T ],HLs )
→0
as N → ∞ uniformly in ε. Let be any relatively compact open subset of T × R. It is easy to see that, for all s > 0 H0s () ⊂ HLs ⊂ H s (T × R),
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and conversely for s < 0, H s (T × R) ⊂ HLs ⊂ H s (). Here H0s () denotes, for s ≥ 0, the closure of D() for the H s norm, and H −s () is its dual space. Thus for all s < 0 and all T > 0, we have 9 9 9 9 n,0,0 |ε L2 (T×R) n,0,0 − #0 ε 9 2 →0 9 s L ([0,T ];H ())
n≤N
as N → ∞ uniformly in ε. Moreover the same computation as previously shows that for any n ∈ N, ∂t (ηε η¯ n,0,0 + mε · u¯ n,0,0 )(t, x)dx + ν ∇uε : ∇ u¯ n,0,0 (t, x)dx −
mε · (uε · ∇)u¯ n,0,0 (t, x)dx = 0,
(4.5.2)
and, since εηε uε converges to 0 in L2 (R+ ; H s (T × R)) for any s < 0 we get
#n,0,0 (ηε , uε ) is compact in L2 ([0, T ] × T × R).
n≤N
Combining both results shows finally that s #0 ε is compact in L2 ([0, T ]; Hloc (T × R))
for all T > 0 and all s < 0.
As S is countable, we are therefore able to construct (by diagonal extraction) a subsequence of ε , and some λ ∈ Ker(L − iλI d) such that for all s < 0 and all T > 0 ∀iλ ∈ S,
s #λ ε → λ in L2 ([0, T ]; Hloc (T × R)).
Note that the λ defined as the strong limit of #λ ε can also be obtained as the weak limit of exp( itλ ε )(ηε , uε ).The following lemma is easily proved. L EMMA 4.13. With the notation of Theorem 4.4, consider a subsequence of (ε )ε>0 , and some λ ∈ Ker(L − iλI d) such that for all s < 0 and all T > 0 ∀iλ ∈ S,
s #λ ε → λ in L2 ([0, T ]; Hloc (T × R)). itλ
Then, for all iλ ∈ S, e ε (ηε , uε ) converges to λ weakly in L2 ([0, T ] × T × R). In particular, for all iλ ∈ S, λ is bounded in L2 ([0, T ]; H 1 (T × R)) uniformly in λ.
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4.5.2. Strong convergence of ε As a corollary of the previous mode by mode convergence results, we get the following convergence for ε . L EMMA 4.14. With the notation of Theorem 4.4, the following results hold. Consider a subsequence of (ε ), and some λ ∈ Ker(L − iλI d) such that for all s < 0 and all T > 0 s (T × R)). #λ ε → λ in L2 ([0, T ]; Hloc
∀iλ ∈ S, Then,
ε ! =
λ weakly in L2loc (R+ ; L2 (T × R)),
iλ∈S s (T × R)) for all s < 0. ε → strongly in L2loc (R+ ; Hloc #n,k,j , we have for any relatively compact subMoreover, defining KN = (n,k,j )∈S
and
(n+|k|2 )1/2 ≤N
set of T × R, for all T > 0 and for all s < 0, 9 9 9 9 t (I d − K ε 9 (I d − KN )ε L2 ([0,T ];H s ()) + 9 )L N 9 2 9 ε L ([0,T ];H s ()) → 0 as N → ∞,
(4.5.3)
uniformly in ε. P ROOF. The first convergence statement comes directly from the uniform bound on ε in L2loc (R+ ; L2 (T × R)) and the L2 continuity of #λ . In order to establish the strong convergence result, the crucial argument is to approximate (uniformly) ε by a finite number of modes, i.e. to prove (4.5.3). The main idea is the same as for the approximation of #0 ε in Lemma 4.12. We have for all T > 0 and all s < 0 9 9 9 9 (n,k,j |ε )L2 (T×R) n,k,j − ε 9 2 →0 9 s L ([0,T ];HL (T×R))
n≤N,|k|≤N
as N → ∞ uniformly in ε, and similarly 9 9 9 9
e
−iτ (n,k,j ) εt
(n,k,j |ε )L2 (T×R) n,k,j
n≤N,|k|≤N
→0
9 9 t ε 9 −L 9 2 ε L ([0,T ];H s (T×R)) L
as N → ∞,
uniformly in ε. Therefore for all relatively compact subsets of T × R, the embedding of HLs into H s () implies that both quantities
n≤N,|k|≤N
(n,k,j |ε )L2 (T×R) n,k,j − ε
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and
n≤N,|k|≤N
t t ε e−iτ (n,k,j ) ε n,k,j |ε L2 (T×R) n,k,j − L ε
converge strongly towards zero in L2 ([0, T ]; H s ()) as ÊN goes to infinity, uniformly in ε. So (4.5.3) is proved. The strong convergence is then directly obtained from the following decomposition: ε − = (I d − KN )ε + KN (ε − ) − (I d − KN ).
The result is proved.
4.5.3. Taking limits in the equation on #λ ε The final step is now to obtain the evolution equation for each mode λ , taking limits in (4.5.1) and (4.5.2). In the following proposition, we recall that the first result (concerning the geostrophic motion) relies on a compensated compactness argument, i.e. on both the algebraic structure of the coupling term and the particular form of the oscillating modes, which implies that there is no contribution of the equatorial waves to the geostrophic flow. That result was proved in Section 4.3. Here we will prove the second part of the statement, concerning the limit ageostrophic motion. P ROPOSITION 4.15. With the notation of Theorem 4.4, consider a subsequence of (ε ), and some family (λ )iλ∈S such that λ ∈ Ker(L − iλI d) and such that for all s < 0 and all T > 0 ∀iλ ∈ S,
s #λ ε → λ in L2 ([0, T ]; Hloc (T × R)).
Then, 0 = (η0 , u0 ) satisfies the geostrophic equation: for all (η∗ , u∗ ) belonging to KerL and satisfying u∗ ∈ H 1 (T × R),
(η0 η∗ + u0,1 u∗1 )(t, x) dx + ν =
t 0
∇u0,1 · ∇u∗1 (t , x) dxdt
(η0 η∗ + u01 u∗1 )(x) dx.
(4.5.4)
Moreover for λ = 0, λ = (0λ , λ ) satisfies the following envelope equation: there is a measure υλ in M(R+ × T × R), such that for all smooth ∗λ = (∗λ,0 , (∗λ ) ) ∈ Ker(L − iλI d),
¯ ∗λ (t, x) dx + ν λ · +
t 0
t 0
¯ ∗λ ) (t , x) dxdt ∇ λ : ∇(
¯ ∗λ ) υλ (dt , dx) ∇ · (
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+
iμ,i μ∈S ˜ λ=μ+μ˜
=
t
0
¯ ∗λ (t , x) dxdt Q(μ , μ˜ ) ·
¯ ∗λ (x) dx, 0 ·
where Q is defined by (4.4.10). P ROOF. Let us first recall that for λ = 0, Ker(L − iλI d) is constituted of smooth, rapidly decaying vector fields, so that it makes sense to apply #λ to any distribution. Starting from (4.5.1) we get that for all smooth ∗λ = (∗λ,0 , (∗λ ) ) ∈ Ker(L − iλI d) itλ ¯ ∗λ ) )(t, x)dx − (ηε0 ¯ ∗λ ) )(x)dx ¯ ∗λ,0 + mε · ( ¯ ∗λ,0 + m0ε · ( (ηε exp ε t it λ ¯ ∗λ ) (t , x)dxdt uε : ∇( +ν ∇ exp ε 0 t it λ 1 2 ∗ ∗ ¯ ¯ − exp mε · (uε · ∇)(λ ) + ηε ∇ · (λ ) (t , x)dxdt = 0. ε 2 0
(4.5.5) Taking limits as ε → 0 in the three first terms is immediate using Lemma 4.13 and the assumption on the initial data. The limit as ε → 0 in the two nonlinear terms is given in the following proposition. P ROPOSITION 4.16. With the previous notation, we have it λ ¯ ∗λ ) (t , x)dxdt mε · (uε · ∇)( exp ε t ¯ ∗λ ) (t , x)dxdt , → μ · ( μ˜ · ∇)(
t 0
0
μ+μ=λ ˜ iμ,i μ∈S ˜
and 1 2
it λ 2 ¯ ∗λ ) (t , x)dxdt exp ηε ∇ · ( ε 0 t 1 t ∗
¯ ¯ ∗λ ) υλ (dt dx). → μ,0 μ,0 ∇ · ( ˜ ∇ · (λ ) (t , x)dxdt − 2 0 0 μ+μ=λ ˜
t
iμ,i μ∈S ˜
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The fact that this result gives Proposition 4.15 is an algebraic computation left to the reader. Let us prove Proposition 4.16. The idea is to decompose ε on the eigenmodes of L, by writing
itλ t ε (t, x) = e− ε #λ ε (t, x). (ηε , uε )(t, x) = L ε iλ∈S
Note in particular that by (4.5.3), for any s < 0, t s KN ε (t) → 0 in L2loc (R+ ; Hloc (ηε , uε )(t) − L (T × R)) ε as N goes to infinity, uniformly in ε. Let us also introduce the notation t ε,N = L − (ηε,N , uε,N ) = KN ε , and ε ε,λ,N = #λ ε,N . We will start by considering the first nonlinear term in Proposition 4.16, namely t 0
it λ ¯ ∗λ ) (t , x)dxdt . mε · (uε · ∇)( exp ε
We can notice that t it λ ¯ ∗λ ) (t , x)dxdt mε · (uε · ∇)( exp ε 0 t it λ ¯ ∗λ ) (t , x)dxdt εηε uε · (uε · ∇)( = exp ε 0 t it λ ¯ ∗λ ) (t , x)dxdt . uε · (uε · ∇)( + exp ε 0 The uniform bounds coming from the energy estimate imply clearly that the first term converges to 0 as ε → 0. Then we can decompose the second contribution in the following way: it λ ¯ ∗λ ) (t , x)dxdt uε · (uε · ∇)( exp ε 0 t it λ ¯ ∗λ ) (t , x)dxdt uε · (uε · ∇)( = exp ε 0 T×(R\[−R,R]) t it λ ¯ ∗λ ) (t , x)dxdt (uε − uε,N ) · (uε · ∇)( + exp ε 0 T×[−R,R] t
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it λ ¯ ∗λ ) (t , x)dxdt uε,N · ((uε − uε,M ) · ∇)( exp ε T×[−R,R] t it λ ¯ ∗λ ) (t , x)dxdt . uε,N · (uε,M · ∇)( + exp ε 0 T×[−R,R]
t
+
0
(4.5.6)
Let us consider now all the terms in the right-hand side of (4.5.6). The uniform bound on uε and the decay of ∗λ imply that the first term on the right-hand side converges to 0 as R → ∞ uniformly in ε. By the inequality " t " " " 0
" " it λ ∗
" ¯ (uε − uε,N ) · (uε · ∇)(λ ) (t , x)dxdt " exp ε T×[−R,R]
≤ Cuε − uε,N L2 ([0,T ];H s (T×[−R,R])) uε L2 ([0,T ];H 1 (T×R)) ∗λ W 2,∞ (T×R) , with −1 < s < 0, we deduce that the third term converges to 0 as N → ∞ uniformly in ε. Now let us consider the third term on the right-hand side. Since uε,N corresponds to the projection of ε onto a finite number of eigenvectors of L, we deduce that ∀N ∈ N, ∃CN , ∀ε > 0,
uε,N L∞ (R+ ;H 1 (T×R)) ≤ CN .
Thus " t " " " 0
" " it λ ∗
" ¯ uε,N · ((uε − uε,M ) · ∇)(λ ) (t , x)dxdt " exp ε T×[−R,R]
≤ CN uε − uε,M L2 ([0,T ];H s (T×[−R,R])) ∗λ W 2,∞ (T×R) and, for all fixed N and R, the fourth term converges to 0 as M → ∞ uniformly in ε. It remains then to take limits in the last term of (4.5.6). It can be rewritten it λ ¯ ∗λ ) (t , x)dxdt uε,N · (uε,M · ∇)( exp ε T×[−R,R] t
˜ it (λ − μ − μ) (ε,μ,N ) = exp ε 0 T×[−R,R]
t 0
iμ,i μ∈S ˜
× ( ε,μ,M ˜
¯ ∗λ ) (t , x)dxdt . · ∇)(
This in turn can be written in the following way: t 0
T×[−R,R] iμ,i μ∈S ˜
˜ it (λ − μ − μ) ε,μ,N exp ε
¯ ∗λ ) (t , x)dxdt × ( ε,μ,M · ∇)( ˜
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t 0
T×[−R,R] iμ,i μ∈S ˜
319
˜ it (λ − μ − μ) ( ε,μ,N − μ,N ) exp ε
¯ ∗λ ) (t , x)dxdt · ∇)( × ( ε,μ,M ˜ t
˜ it (λ − μ − μ) μ,N | + exp ε 0 T×[−R,R] iμ,i μ∈S ˜
¯ ∗λ ) (t , x)dxdt − μ,M ) · ∇)( ˜ t
˜ it (λ − μ − μ) μ,N + exp ε 0 T×[−R,R] × (( ε,μ,M ˜
iμ,i μ∈S ˜
¯ ∗λ ) (t , x)dxdt . × ( μ,M · ∇)( ˜ We have denoted μ,N = #μ N ,
where N = KN .
The first two terms on the right-hand side go to zero as ε goes to zero, for all given N, M and R, due to the following estimates: for −1 < s < 0, t 0
T×[−R,R] iμ,i μ∈S ˜
¯ ∗λ ) (t , x)|dxdt |( ε,μ,N − μ,N ) · ( ε,μ,M · ∇)( ˜
≤ CN,M ε,N − N L2 ([0,T ];H s (T×[−R,R])) ε,M L∞ ([0,T ];H 1 (T×R)) × ∗λ W 2,∞ (T×R) , and similarly t 0
T×[−R,R] iμ,i μ∈S ˜
¯ ∗λ ) (t , x)|dxdt | μ,N · (( ε,μ,M − μ,M ) · ∇)( ˜ ˜
≤ CN,M ε,M − M L2 ([0,T ];H s (T×[−R,R])) ε,N L∞ ([0,T ];H 1 (T×R)) × ∗λ W 2,∞ (T×R) . Finally let us consider the last term, which can be decomposed in the following way: t 0
T×[−R,R] iμ,i μ∈S ˜
=
t 0
T×[−R,R]
˜ it (λ − μ − μ) ¯ ∗λ ) (t , x)dxdt μ,N · ( μ,N exp · ∇)( ˜ ε
iμ,i μ∈S ˜ λ=μ+μ˜
¯ ∗λ ) (t , x)dxdt μ,N · ( μ,M · ∇)( ˜
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+
t T×[−R,R]
0
˜ it (λ − μ − μ) μ,N · ( μ,M exp · ∇) ˜ ε iμ,i μ∈S ˜
λ =μ+μ˜
¯ ∗λ ) (t , x)dxdt . × ( For fixed N and M, the nonstationary phase theorem (which corresponds here to a simple integration by parts in the t variable) shows that the second term is a finite sum of terms converging to 0 as ε → 0. And the first term (which does not depend on ε) converges to t 0
¯ ∗λ ) (t , x)dxdt μ · ( μ˜ · ∇)(
μ+μ=λ ˜ iλ,iμ,i μ∈S ˜
as N, M, R → ∞, because N converges towards strongly in L2 ([0, T ]; L2 (T × R)) when N goes to infinity, and then by Lebesgue’s theorem when R goes to infinity. Therefore, taking limits as ε → 0, then M → ∞, then N → ∞, then R → ∞ in (4.5.6) leads to it λ ¯ ∗λ ) (t , x)dxdt mε · (uε · ∇)( exp ε t
¯ ∗λ ) (t , x)dxdt . → μ · ( μ˜ · ∇)(
t 0
0
μ+μ=λ ˜ iλ,iμ,i μ∈S ˜
Finally let us consider the second term of the proposition, namely 1 2
t 0
it λ 2 ¯ ∗λ ) (t , x) dxdt . ηε ∇ · ( exp ε
The first step of the above study remains valid, in the sense that one can write it λ 2 ¯ ∗λ ) (t , x) dxdt exp ηε ∇ · ( ε 0 it λ 2 1 t ¯ ∗λ ) (t , x) dxdt = exp ηε ∇ · ( 2 0 R\T×[−R,R] ε 1 t it λ 2 ¯ ∗λ ) (t , x) dxdt , ηε ∇ · ( + exp 2 0 T×[−R,R] ε 1 2
t
and the first term converges to zero uniformly in ε as R goes to infinity, due to the spatial decay of the eigenvectors of L. For such a result, a uniform bound of ηε in L∞ (R+ ; L2 (T× R)) is sufficient. However the next steps of the above study do not work here, as we have no smoothness on ηε other than that energy bound. In order to conclude, let us nevertheless
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decompose the remaining term as above, for any integers N and M to be chosen large enough below: 1 2
it λ 2 ¯ ∗λ ) (t , x) dxdt exp ηε ∇ · ( ε 0 T×[−R,R] 1 t it λ ¯ ∗λ ) (t , x) dxdt (ηε − ηε,N )ηε ∇ · ( = exp 2 0 T×[−R,R] ε 1 t it λ ¯ ∗λ ) (t , x) dxdt ηε,N (ηε − ηε,M )∇ · ( + exp 2 0 T×[−R,R] ε 1 t it λ ¯ ∗λ ) (t , x) dxdt . ηε,N ηε,M ∇ · ( + exp (4.5.7) 2 0 T×[−R,R] ε
t
The sequence − 12 exp( itελ )(ηε − ηε,N )ηε is uniformly bounded in N ∈ N and ε > 0 in the space L1loc (R+ × T × R), so up to the extraction of a subsequence it converges weakly, as ε goes to zero, towards a measure υλ,N , which in turn is uniformly bounded in M(R+ × T × R). Denoting by υλ its limit in M(R+ × T × R) as N goes to infinity, we find that 1 2
it λ ¯ ∗λ ) (t , x) dxdt exp (ηε − ηε,N )ηε ∇ · ( ε 0 T×[−R,R] t ¯ ∗λ ) υλ (dt dx) →− ∇ · (
t
0
T×[−R,R]
as ε goes to zero and N goes to infinity, which in turn converges to −
t 0
¯ ∗λ ) υλ (dt dx) ∇ · (
¯ ∗ ) . Note that as S is countable, one as R goes to infinity, due to the smoothness of ∇ · ( λ
can choose a subsequence such that for all iλ ∈ S, the sequence − 12 exp( itελ )(ηε − ηε,N )ηε converges towards υλ as ε goes to zero and N goes to infinity. Finally the two last terms in (4.5.7) are dealt with as in the previous case, and we leave the details to the reader. Proposition 4.16 is proved. 4.5.4. The case when capillarity is added In this final section we propose an adaptation to the Saint-Venant model which provides some additional smoothness on εηε , and which enables one to get rid of the defect measure present in the above study. The model is presented in the next part, and the convergence result stated and proved below. The model Let us define the capillarity operator K(h) = κ(−)2α h,
(4.5.8)
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where κ > 0 and α > 1/2 are given constants. The system we shall study is the following: 1 ∂t η + ∇·((1 + εη)u) = 0, ε ∂t u + u · ∇u +
βx2 ⊥ 1 ν u + ∇η − u + εκ∇(−)2α η = 0, ε ε 1 + εη η|t=0 = η0 ,
(4.5.9)
u|t=0 = u0 .
In the next part we discuss the existence of bounded energy solutions to that system of equations (under a smallness assumption), and the following part consists in the proof of the analogue of Theorem 4.4 in that setting. One should emphasize here that the additional capillarity term that is added in the system will not appear in the limit, since it comes as a O(ε) term. Moreover it is a linear term, so it should not change the other asymptotics proved in Section 4. However its unphysical character (as well as the smallness condition on the initial data) made us prefer to study the original Saint-Venant system for all the convergence results of Section 4. Existence of solutions The following theorem is an easy adaptation of the result by D. Bresch and B. Desjardins in [6] (see also [39] for the compressible Navier–Stokes system). T HEOREM 4.5. There is a constant C > 0 such that the following result holds. Let (ηε0 , u0ε ) be a family of H 2α × L2 (T × R) such that for all ε > 0, 1 0 2 (ηε ) + κε 2 |(−)α ηε0 |2 + (1 + εηε0 )|u0ε |2 (x) dx ≤ E 0 . 2 If E 0 ≤ C, then there is a family (ηε , uε ) of weak solutions to (4.5.9), satisfying the energy estimate 1 2 ηε + κε 2 |(−)α ηε |2 + (1 + εηε )|uε |2 (t, x) dx 2 t +ν |∇uε |2 (t , x) dxdt ≤ E 0 . 0
Convergence In this section our aim is to show that the capillarity term enables us to get rid of the defect measure present in the conclusion of Theorem 4.4. As the proof is very similar to that theorem, up to the compactness of ηε , we will not give the full details. The result is the following. T HEOREM 4.6. Under the assumptions of Theorem 4.5, denote by (ηε , uε ) a solution of (4.5.9) with initial data (ηε0 , u0ε ), and define t ε = L − (ηε , uε ). ε
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s (T × R)) (for Up to the extraction of a subsequence, ε converges weakly in L2loc (R+ ; Hloc all s < 0) toward some solution of the following limiting filtered system: for all iλ in S and for all smooth ∗λ in Ker(L − iλI d),
¯ ∗λ (x) dx − ν · + =
t 0
t 0
¯ ∗λ (t , x) dxdt L ·
¯ ∗λ (t , x) dxdt QL (, ) ·
¯ ∗λ (x) dx, 0 ·
where 0 = (η0 , u0 ). Let us prove that result. We will follow the lines of the proof of Theorem 4.4; the only difference consists in taking the limit as ε goes to zero, of the equation on #λ ε . Equation (4.5.5) (page 316) can be written here as follows: for all smooth ∗λ = (∗λ,0 , (∗λ ) ) belonging to Ker(L − iλI d),
itλ ¯ ∗λ ) )(t, x)dx − (ηε0 ¯ ∗λ ) )(x)dx ¯ ∗λ,0 + uε · ( ¯ ∗λ,0 + u0ε · ( exp (ηε ε t it λ α ¯ ∗λ ) (t , x)dxdt −εκ (−) exp ηε ∇ · (−)α ( ε 0 t it λ ν ¯ ∗λ ) (t , x)dxdt uε · ( − exp 1 + εη ε ε 0 t it λ ¯ ∗λ ) (t , x)dxdt (uε · ∇)uε · ( exp + ε 0 t it λ ¯ ∗λ,0 (t , x)dxdt = 0. ηε uε · ∇ exp − ε 0 (4.5.10)
R EMARK 4.17. We have chosen to keep the unknowns (ηε , uε ) and not write the analysis in terms of (ηε , mε ) as previously (recall that mε = (1 + εηε )uε ): the study of mε rather 1 than uε is indeed unnecessary here as the factor 1+εη which appears in the diffusion term ε in the equation on uε can be controlled in this situation, contrary to the previous case. The advantage of writing the equations on (ηε , uε ) is that there is no nonlinear term in ηε , contrary to the previous study, but of course the difficulty is transfered to the study of the diffusion operator; the gain of regularity in ηε will appear here. Taking limits as ε → 0 in the two first terms is immediate. For the third term, we simply recall that ηε is bounded in L∞ (R+ ; L2 (T × R)) and εηε is bounded in L∞ (R+ ; H 2α (T ×
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R)), so εηε goes strongly to zero in L∞ (R+ ; H s (T × R)) for every s < 2α. Since ∗λ is smooth, we infer that t εκ 0
it λ ¯ ∗λ ) (t , x)dxdt → 0, (−) exp ηε ∇ · (−)α ( ε α
as ε → 0.
Let us now consider the fourth term, −
t 0
it λ ν ¯ ∗λ ) (t , x)dxdt . exp uε · ( 1 + εηε ε
It is here that the presence of capillarity enables us to get a better control. Let us write it λ ν ¯ ∗λ ) (t , x)dxdt − uε · ( exp 1 + εη ε ε 0 t it λ ¯ ∗λ ) (t , x)dxdt uε : ∇( =ν ∇ exp ε 0 t εηε it λ ¯ ∗λ ) (t , x)dxdt . uε : ∇ −ν ( ∇ exp ε 1 + εηε 0 t
Clearly the first term on the right-hand side converges towards the expected limit: we have it λ ¯ ∗λ ) (t , x)dxdt ν uε : ∇( ∇ exp ε 0 t ¯ ∗λ ) (t , x)dxdt , →ν ∇ λ : ∇( t
0
as ε goes to 0. To study the second one, we can notice that εηε εηε εηε ∗ ¯ ∗λ ) + ¯ ¯ ∗λ ) , ( ∇ (λ ) = ∇ ∇( 1 + εηε 1 + εηε 1 + εηε
and since the second term on the right-hand side is obviously easier to study than the first one, let us concentrate on the first term. We have ∇
εηε ε∇ηε ε 2 ηε ∇ηε = − · 1 + εηε 1 + εηε (1 + εηε )2
Since εηε is bounded in L∞ (R+ ; H 2α (T × R)), we infer easily, by product laws in Sobolev spaces, that ε 2 ηε ∇ηε is bounded in L∞ (R+ ; H σ (T × R)), for some σ > 0.
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But on the other hand ηε is bounded in L∞ (R+ ; L2 (T × R)), so we have also ε 2 ηε ∇ηε → 0
in L∞ (R+ ; H 2α−2 (T × R)).
By interpolation we gather that ε 2 ηε ∇ηε → 0
in L∞ (R+ ; L2 (T × R)),
and the lower bound on 1 + εηε ensures that ε 2 ηε ∇ηε →0 (1 + εηε )2
in L∞ (R+ ; L2 (T × R)).
The argument is similar (and easier) for the term
ε∇ηε 1+εηε ,
so we can conclude that
it λ ν ¯ ∗λ ) (t , x)dxdt uε · ( exp − 1 + εηε ε 0 t ¯ ∗λ ) (t , x)dxdt . →ν ∇λ : ∇( t
0
Finally we are left with the nonlinear terms: let us study the limit of it λ ¯ ∗λ ) (t , x)dxdt (uε · ∇)uε · ( exp ε 0 t it λ ¯ ∗λ,0 (t , x)dxdt . ηε uε ∇ · − exp ε 0
t
The study is very similar to the case studied above (see Proposition 4.16), so we will not give all the details but merely point out the differences. First, one can truncate the integral in x2 ∈ R to x2 ∈ [−R, R], where R is a parameter to be chosen large enough in the end. As previously that is simply due to the decay of the eigenvectors of L at infinity. So we are reduced to the study of it λ ¯ ∗λ ) (t , x)dxdt (uε · ∇)uε · ( exp ε 0 T×[−R,R] t it λ ¯ ∗λ,0 (t , x)dxdt . ηε uε · ∇ − exp ε 0 T×[−R,R]
t
and
The limit of the first term is obtained in an identical way to above, since uε satisfies the same bounds, so we have it λ ¯ ∗λ ) (t , x)dxdt (uε · ∇)uε · ( exp ε T×[−R,R]
t 0
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→
t 0
μ+μ=λ ˜ iμ,i μ∈S ˜
¯ ∗λ ) (t , x)dxdt , ( μ · ∇) μ˜ · (
as ε goes to 0 and R goes to infinity. Now let us concentrate on the last nonlinear term. With the notation defined in the previous section, we can write t it λ ¯ ∗λ,0 (t , x)dxdt ηε uε · ∇ exp ε 0 T×[−R,R] t it λ ¯ ∗λ,0 (t , x)dxdt (ηε − ηε,N )uε · ∇ = exp ε 0 T×[−R,R] t it λ ¯ ∗λ,0 (t , x)dxdt ηε,N (uε − uε,M ) · ∇ + exp ε 0 T×[−R,R] t it λ ¯ ∗λ,0 (t , x)dxdt . ηε,N uε,M · ∇ + exp ε 0 T×[−R,R] The first two terms on the right-hand side converge to zero, due to the following estimates: for some −1 < s < 0 and for all t ∈ [0, T ], " t " " " it λ ∗
" " ¯ (ηε − ηε,N )uε · ∇ λ,0 (t , x)dxdt " exp " ε 0 T×[−R,R] ≤ CT ηε − ηε,N L∞ ([0,T ];H s (T×[−R,R])) uε L2 ([0,T ];H 1 (T×[−R,R])) ¯ ∗λ W 2,∞ (T×R) , × and similarly " t " " " 0
" " it λ ¯ ∗λ,0 (t , x)dxdt " ηε,N (uε − uε,M ) · ∇ exp " ε T×[−R,R]
¯ ∗λ W 2,∞ (T×R) . ≤ CT ,N uε − uε,M L2 ([0,T ];H s (T×[−R,R])) Finally the limit of the third term is obtained by the (by now) classical nonstationary phase theorem, namely we find, exactly as in the proof of Proposition 4.16, that t it λ ¯ ∗λ,0 (t , x)dxdt exp ηε,N uε,M · ∇ ε 0 T×[−R,R] t ¯ ∗λ,0 (t , x)dxdt , → μ,0 μ˜ · ∇ 0
μ+μ=λ ˜ iμ,i μ∈S ˜
as ε goes to 0 and M, N and R go to infinity. That concludes the proof of the theorem.
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Acknowledgement The authors are very grateful to D. Gérard-Varet for his careful reading of a previous version of this work, and for his useful comments. References [1] T. Alazard, Low Mach number limit of the full Navier–Stokes equations, Archive for Rational Mechanics and Analysis, to appear. [2] A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier–Stokes equations for uniformly rotating fluids, European Journal of Mechanics 15 (1996) 291–300. [3] A. Babin, A. Mahalov and B. Nicolaenko, Resonances and regularity for Boussinesq equations, Russian Journal of Mathematical Physics 4 (1996) 417–428. [4] A. Babin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier–Stokes equations for resonant domains, Indiana University Mathematics Journal 48 (1999) 1133–1176. [5] C. Bardos, F. Golse, A. Mahalov and B. Nicolaenko, Regularity of Euler equations for a class of threedimensional initial data, Trends in Partial Differential Equations of Mathematical Physics, pages 161–185, Progr. Nonlinear Differential Equations Appl. 61, Birkhäuser, Basel (2005). [6] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equation and convergence to the quasi-geostrophic model, Commun. Math. Phys. 238 (2003) 211–223. [7] D. Bresch, B. Desjardins and D. Gérard-Varet, Rotating fluids in a cylinder, Discrete and Contininous Dynamical Systems. 11 (1) (2004) 47–82. [8] D. Bresch, E. Grenier and D. Gérard-Varet, Derivation of the planetary geostrophic equations, Archive for Rational Mechanics and Analysis, to appear. [9] D. Bresch and G. Métivier, Global existence and uniqueness for the lake equations with vanishing topography: elliptic estimates for degenerate equations, Preprint (2005). [10] F. Charve, Global well-posedness and asymptotics for a geophysical fluid system, Communications in Partial Differential Equations 29 (2004) 1919–1940. [11] F. Charve, Convergence of weak solutions for the primitive system of the quasigeostrophic equations, Asymptotic Analysis 42 (2005) 173–209. [12] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity, Modélisation Mathématique et Analyse Numérique 34 (2000) 315–335. [13] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Basics of Mathematical Geophysics, Oxford University Press (2006), to appear. [14] R. Danchin, Zero Mach number limit in critical spaces for compressible Navier–Stokes equations, Annales Scientifiques de l’École Normale Supérieure 35 (1) (2002) 27–75. [15] R. Danchin, R. Zero Mach number limit for compressible flows with periodic boundary conditions, Amer. J. Math. 124 (6) (2002) 1153–1219. [16] B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, Proceedings of the Royal Society of London, Ser. A Math. Phys. Eng. Sci. 455 (1999) 2271–2279. [17] B. Desjardins and E. Grenier, On the homogeneous model of wind-driven ocean circulation, SIAM Journal on Applied Mathematics 60 (1999) 43–60. [18] B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier–Stokes equations with Dirichlet boundary conditions, Journal de Mathématiques Pures et Appliquées 78 (1999) 461–471. [19] F. C. Fuglister, Gulf Stream ‘60, Progress in Oceanography I, Pergamon Press (1963), 265–383. [20] H. Fujita and T. Kato, On the Navier–Stokes initial value problem I, Archive for Rational Mechanics and Analysis 16 (1964) 269–315. [21] I. Gallagher, Applications of Schochet’s methods to parabolic equations, Journal de Mathématiques Pures et Appliquées 77 (1998) 989–1054. [22] I. Gallagher, A Remark on smooth solutions of the weakly compressible Navier–Stokes equations, Journal of Mathematics of Kyoto University 40 (2000) 525–540.
328
B IBLIOGRAPHY
[23] I. Gallagher, Résultats récents sur la limite incompressible. (French) [Recent results on the incompressible limit] Séminaire Bourbaki, Vol. 2003/2004. Astérisque No. 299 (2005) Exp. No. 926, vii, 29–57. [24] I. Gallagher and L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations, Journal d’Analyse Mathématique, to appear. [25] I. Gallagher and L. Saint-Raymond, Study of equatorial waves for the β-plane model, Preprint (2006). [26] D. Gérard-Varet, Highly rotating fluids in rough domains, Journal de Mathématiques Pures et Appliquées 82 (2003) 1453–1498. [27] A.E. Gill, Atmosphere–Ocean Dynamics, International Geophysics Series, vol. 30 (1982). [28] A.E. Gill and M.S. Longuet-Higgins, Resonant interactions between planetary waves, Proc. Roy. Soc. London A299 (1967) 120–140. [29] F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, Journal de Mathématiques Pures et Appliquées 78 (1999) 791–817. [30] H.P. Greenspan, The Theory of Rotating Fluids, Cambridge monographs on mechanics and applied mathematics (1969). [31] E. Grenier, Oscillatory perturbations of the Navier–Stokes equations, Journal de Mathématiques Pures et Appliquées 76 (1997) 477–498. [32] E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data, Communications in Partial Differential Equations 22 (5–6) (1997) 953–975. [33] J.-L. Joly, G. Métivier and J. Rauch, Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Mathematical Journal 70 (1993) 373–404. [34] J.-L. Joly, G. Métivier and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics, Annales scientifiques de l’École Normale Supérieure 28 (1995) 51–113. [35] R. Klein and A. Majda, Systematic multi-scale models for the tropics, Journal of Atmospheric Sciences 60 (2003) 173–196. [36] J. Leray, Essai sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Matematica 63 (1933) 193–248. [37] J. Leray, Etude de diverses équations inté grales non linéaires et de quelques problèmes que pose l’hydrodynamique, Journal de Mathématiques Pures et Appliquées 12 (1933) 1–82. [38] J.-L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity 5 (1992) 237–288. [39] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. II, Compressible Models, Oxford Science Publications (1997). [40] P.-L. Lions and N. Masmoudi, Une approche locale de la limite incompressible, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 387–392. [41] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, Journal de Mathématiques Pures et Appliquées 77 (1998) 585–627. [42] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, 9, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003). [43] M. Majdoub and M. Paicu, Uniform local existence for inhomogeneous rotating fluid equations, Preprint (2005). [44] N. Masmoudi, Ekman layers of rotating fluids: the case of general initial data, Communications in Pure and Applied Mathematics 53 (2000) 432–483. [45] N. Masmoudi, Some asymptotic problems in fluid mechanics. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998) pages 395–404, Lecture Notes in Pure and Appl. Math. 215, Dekker, New York (2001). [46] A. Mellet and A. Vasseur, On the isentropic compressible Navier–Stokes equation, Preprint (2005). [47] G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal. 158 (2001) 61–90. [48] M. Paicu, Equation anisotrope de Navier–Stokes dans des espaces critiques, Revista Matematica Iberoamericana, to appear. [49] E. Palmén and C.W. Newton, Atmospheric Circulation Systems, Academic Press (1969). [50] J. Pedlosky, Geophysical Fluid Dynamics, Springer (1979). [51] S. Philander, El Nino, La Nina, and the Southern Oscillation, Academic Press (1990).
B IBLIOGRAPHY
329
[52] M. Reed and B. Simon, Methods of Modern Mathematical Physics, III, Scattering Theory, Academic Press (1979). [53] F. Rousset, Asymptotic behaviour of geophysical fluids in highly rotating balls, Preprint (2004). [54] S. Schochet, Fast singular limits of hyperbolic PDEs, Journal of Differential Equations 114 (1994) 476–512. [55] W. Thomson (Lord Kelvin), On gravitational oscillations of rotating water, Proc. Roy. Soc. Edinburgh 10 (1879) 92–100. [56] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of Mathematical Fluid Dynamics, vol. III, S. Friedlander and D. Serre, eds (2004), 535–657.
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CHAPTER 6
The Foundations of Oceanic Dynamics and Climate Modeling George R. Sell∗ School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2. Oceanic flows . . . . . . . . . . . . . . . . . . . . . . . 2.1. Boundary forces . . . . . . . . . . . . . . . . . . . 2.2. Temperature and salinity equations . . . . . . . . 2.3. Reformulation of the equations (homogenization) 2.4. The pressure . . . . . . . . . . . . . . . . . . . . . 2.5. Notation . . . . . . . . . . . . . . . . . . . . . . . 3. Solution concepts . . . . . . . . . . . . . . . . . . . . . 3.1. The B and L terms . . . . . . . . . . . . . . . . . 3.2. Bubnov–Galerkin approximations . . . . . . . . . 3.3. Weak solutions . . . . . . . . . . . . . . . . . . . 3.4. Strong solutions . . . . . . . . . . . . . . . . . . . 3.5. Mild solutions and Fréchet differentiability . . . . 4. Skew product dynamics . . . . . . . . . . . . . . . . . 4.1. Quasi periodic functions . . . . . . . . . . . . . . 4.2. Quasi Periodic Ansatz . . . . . . . . . . . . . . . 4.3. Oceanic dynamics . . . . . . . . . . . . . . . . . . 4.4. Invariant sets . . . . . . . . . . . . . . . . . . . . . 4.5. Weak solution attractor . . . . . . . . . . . . . . . 5. Thin domain dynamics . . . . . . . . . . . . . . . . . . 6. Climate modeling . . . . . . . . . . . . . . . . . . . . . 6.1. Allowable perturbations . . . . . . . . . . . . . . 6.2. The role of the planets . . . . . . . . . . . . . . . 6.3. Partial averaging . . . . . . . . . . . . . . . . . . . 7. Concluding remarks . . . . . . . . . . . . . . . . . . . 7.1. Other models . . . . . . . . . . . . . . . . . . . . 7.2. Computational issues . . . . . . . . . . . . . . . . 7.3. Stochastic disturbances . . . . . . . . . . . . . . . 7.4. Other dynamical issues . . . . . . . . . . . . . . .
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Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
Abstract The main goal of this article is to derive the mathematical foundations for oceanic dynamics, with applications to the study of the climate of the Earth. This article builds upon the work of Pliss and Sell, [55], in which the physical basis for the related mathematical model is presented. A key feature described herein is an infinitesimal Interface Model for the ocean. The role of this model is to aggregate the net effects of all the relevant atmospheric activity at the surface of the ocean. The Interface Model contains an important aspect of the boundary forces acting on the ocean. The basic model is a system of partial differential equations, consisting of the 3D Navier– Stokes equations and two transport equations for the temperature and the salinity. Since the boundary forces acting on the ocean vary with time, the oceanic equations are—of necessity— nonautonomous equations. We assume here the Quasi Periodic Ansatz (QPA), which states that all the time-dependent forces acting on the ocean are quasi periodic functions of time. The associate frequency vector is determined by the natural frequencies of the planetary motion in the solar system. The resulting inhomogeneous boundary conditions describe forces which may add energy to the ocean. This includes the transfer of radiant heat and other atmospheric effects. We calculate an upper bound on the latent energy absorbed by the ocean from the boundary forces, see (3.37). If the boundary forces are too strong, this can destabilize the dynamics of the ocean. The existence and uniqueness theory for the strong solutions is fully developed in this article. Our approach is based on the method of Leray and Hopf for the Navier–Stokes equations. The QPA enables one to obtain a skew product semiflow π(t) on T k+1 × V 1 , see Theorem 4.1. An important dynamical property of this semiflow is the QP-Herculean Theorem 4.2, see Section 4. Among other things, the latter theorem states that any bounded, invariant set in T k+1 × V 1 is contained in T k+1 × V 2 , where V 2 is the domain D(A) of the related Stokes operator for the oceanic equations. Finally we show that related research in the area of thin domain dynamics offers a hopeful sign for obtaining good information concerning the global attractor for the oceanic dynamics, and thereby for the climate of the Earth.
Keywords: planetary motion, quasi periodic forcing, skew product dynamics, thin domain dynamics
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1. Introduction Our primary objective in this article is to develop a (dynamical systems) theory of oceanic flows, as it can be applied to the study of the climate. The system of partial differential equations (PDEs) which describe the motion in the ocean appear in (1.1)–(1.3). However, before treating this aspect in more detail, we need to go to the scientific source of the theory, the climate of the Earth. As one examines the full range of the data of the climate of the Earth, one observes the multiple time-scales which are represented in the data. At the shortest time-scale, one finds the El Niño data, which occur on a 5–10 year cycle. At the other end of the spectrum, one finds the glaciation data, which describe the comings and goings of the major Ice Ages. These data occur on a 100,000–150,000 year cycle. Of course there are intermediate issues, including the “mini” Ice Ages, which seem to occur on a time-scale of 500–1500 years. It follows that such climatic phenomena need to be treated as inextricable features of the longtime dynamics of any mathematical model. In order to develop a good model for the climate, a model which incorporates the longtime dynamics of the problem, we subscribe to the thesis that the major factor underlying the changes seen in the global climate is the heat transfer occurring in the oceans of the Earth, see [27] for example. Thus, modeling of the global climate begins with the modeling of the oceanic dynamics. In other words, we adopt at the outset, the Voltaire Hypothesis, which states that “in this best-of-all-possible worlds”, our World—the Earth—is located on the global attractor of the oceanic dynamics. In the related paper [55], Pliss and Sell present an overall framework (or paradigm) for general climate models. This framework describes a broad spectrum of possible climate models, and it can be suitably adapted and simplified in order to study various climatic features, on any prescribed long term time scale. While [55] deals with the physical foundations of climate modeling, in the present paper, we focus more on the mathematical aspects behind this approach to climate modeling. The equations of motion in the ocean form the system of PDEs (1.1)–(1.3), and they include the Navier–Stokes equations for an incompressible fluid and two transport equations describing the variations of the temperature and the salinity in the ocean: Oceanic Equations of State: ∂t λ − ν'λ + (λ · ∇) λ + ∇p = F − ', ∇ · λ = 0, ∂t τ − κ2 'τ + (λ · ∇) τ = 0,
(1.1)
∂t σ − κ3 'σ + (λ · ∇) σ = 0, with F = F (t, x) = G(t, x) + a1 ∇τ + a2 ∇σ ,
(1.2)
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where t ≥ 0, x ∈ = ocean, λ = λ(t, x) = 3D velocity field p = p(t, x) = pressure τ = τ (t, x) = temperature σ = σ (t, x) = salinity
(1.3)
G(t, x) = gravity force, and ' = ' λ = Coriolis force. The 3 terms ν, κ2 , and κ3 are positive constants, and the coefficients a1 and a2 used to convert the buoyancy forces due to the temperature and salinity gradients into the units used for the velocity field λ. We use here the simplest expressions for the temperature and salinity gradient forces. The theory we develop here can be readily modified for more general nonlinear expressions, if needed. The internal forces acting on the velocity field λ = λ(t, x) consist of: (1) the force F in equation (1.2), which includes the gravity force G(t) = G(t, x) and buoyancy forces due to the temperature and the salinity gradients, and (2) the Coriolis force ' λ = a0 ω × λ, where ω is a unit-vector defining the axis of rotation of the Earth and a0 is a real scalar. See [6] and [7], for example. The theory of the solutions of these equations and the associate longtime dynamics of these solutions is a major topic, which we will treat below. However, before turning to this theory of the solutions, we make note of several important features of the equations, as well as the geometry of the oceanic domain . (The mathematical and physical justifications of these features are treated in [55], and we refer the reader to this source for more details.) • Time-dependent external forces: The assumption, that the gravity force G(t, x) is a nonautonomous force, is a key feature on our theory. We are especially interested in the dynamics of the solutions of the system (1.1)–(1.3) when the gravitational field and the various time-dependent boundary forces are quasi periodic functions of time, see Section 4. • Navier boundary conditions: We consider the oceanic flow with an infinitesimal Interface (top-surface) Model. In this model, the boundary conditions (which include slip boundary conditions for the Navier–Stokes equations) aggregate the net effect of all the relevant atmospheric activity acting on the (top) surface of the ocean, see Section 2. • Ocean geometry: While there is a global attractor for the general 3D Navier–Stokes equations (and presumably for the equations of oceanic flows) on a bounded domain, the properties of the solutions leave much to be desired, see [20] and [65]. However, in the case of thin 3D domains, Lady Fortune is beginning to smile! See Section 5 for details. The fact that the climate of the Earth might be subject to the ravages of time-dependent gravitational forces due to the planetary motion of the solar system, was explained in the 1930 seminal work of Milankovitch, wherein he described three “cycles” which affect the longtime climate of the Earth, see [46] and [33]. In the first cycle, the time-varying tilt of
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the axis of rotation of the Earth varies between 21.5 and 24.5 degrees. It is shown that this variation occurs over a time scale of 41,000 Earth-years. At the higher tilt angle, more solar radiation is absorbed in the polar regions—for 90 Earth-days—each year. At this higher tilt angle, the seasonal variations between the heating and cooling of the northern and southern hemisphere are higher. Similarly, in the second cycle the time-varying precession of the equinox/solstice features will result in a gradual switch in the roles played by the northern and southern hemisphere. For example, in about 12,000 Earth-years, the northern hemisphere will replace the role now played by the southern hemisphere. At that time the Earth will be closest to the Sun during the northern summer, as opposed to the northern winter. With the third cycle, Milankovitch showed that ε, the effective eccentricity of the orbit of the Earth about the Sun, varied over a time scale of 100,000–150,000 Earth-years. (See [78] for some figures on the time-dependent changes in the effective eccentricity.) While H , the yearly amount of the solar radiant energy received on the total surface of the Earth, does not change as the eccentricity changes, the time-varying distance between the Earth and the Sun does. As ε increases, the Earth spends more of the Earth-year farther from the Sun. Let U = U (a, ε) = 12 T + τ , where T is one Earth-year, denote the timeinterval that the Earth is farther from the Sun. One then has τ = 0, when ε = 0, and τ > 0, − 12 when ε > 0. Also one has ∂τ , for ε > 0 and small. As ε increases, we can ∂ε = const ε expect, for example, longer and colder winters followed by shorter and hotter summers, while keeping H unchanged, see [38] and [78]. The upshot of the Milankovitch study was to establish that the climate of the Earth is inherently a nonautonomous phenomenon. In the paper [55], we argue that same gravitational forces used in the analysis of Milankovitch can also lead to changes in the oceanic flows over the long time-scales used in global climate modeling. Thus the gravitational forces play two roles: they drive the Earth through the heavens, and they directly affect the heat transfer within the oceans. They form both the macrocosm and the microcosm of the climate. An interesting and extensive report on the status of what is currently known about mathematical climatology can be found in Ghil [23] and the references contained therein. The equations (1.1)–(1.3) give us only part of the story of the climate. What needs to be added is the role of the boundary forces that act on the ocean. This would include the daily heating and cooling of the surface of the ocean and the interaction between the ocean and the atmosphere, which we describe in terms of an Interface Model. In Section 2 we will rewrite the equations (1.1)–(1.3), taking into account the various boundary forces. We will also introduce the various function spaces needed to study the well-posedness of the problem. In Section 3 we examine three solution concepts for the system (1.1)–(1.3), namely, weak solutions, strong solutions, and mild solutions. The main goal here is to show that this problem (with the boundary forces) is well-posed in suitable function spaces. The longtime dynamics of the solutions is developed in Section 4. We present this theory in the context of skew product semiflows, with special emphasis on nonautonomous dynamics with quasi periodic forcing. In [55], it is shown that a Quasi Periodic Ansatz, concerning the motion of the planets and moons in the solar system, offers an appropriate context for studying a promising nonautonomous model for the longtime climate of the Earth.
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There is another feature of oceanic flows which is relevant for climate modeling. Since the depth of the ocean (about 5 km) is small in comparison to its expanse (about 17780 km in the Pacific–Arctic Ocean), the current theory of the global attractor for the Navier– Stokes equations on thin 3D domains seems of special importance for the study of the longtime dynamics of oceanic flows. See Section 5 and the papers: [13], [30], [31], [32], [35], [58], [59], [60], [61], [74], and [75], for example. (Of special interest here is the article [32], in which the Navier boundary conditions are studied.) One of the main objectives in the papers cited above is to show that the global attractor of the Navier–Stokes equations consists entirely of globally regular solutions. In Section 4, we present evidence, in lieu of complete proofs, that the weak solutions of the oceanic equations have a global attractor Aw , see [20] and [65], and we examine the issue of the global regularity of the solutions on Aw . See [67], as well as [29] and [73], for more information about global attractors in general. The reader is also invited to check the historical commentary and the more extensive referencing concerning attractors appearing in [67]. In Section 6 we present a few applications of the theory of oceanic flows—as developed herein—to the study of some of the climatic phenomena of the Earth. While our efforts there are related to the work of Milankovitch, see [46], we do not use the Milankovitch results explicitly. We use the same starting point as did Milankovitch, namely the planetary motion of the solar system. In this paper, along with the companion work [55], our approach seeks to explain a broad family of climate models and to develop mathematical tools which can be used to adapt this family to the study of specific climatic phenomena, such as the El Niño events, which occur on restricted, but relatively long, time-scales. Lastly, in Section 7 we present some additional remarks showing how the specific theory presented herein is related to other mathematical investigations into the dynamics of oceanic flows and the connections with the climate.
2. Oceanic flows In this section, we present some initial analyses of the system of equations (1.1)–(1.3) for oceanic flows. As noted above, the region represents an appropriate approximation to the global oceans of the Earth. Thus is a subset of the solid sphere in R3 . We assume that the boundary ∂ of is smooth, say of class C 4 and that ∂ = ∂top ∪ ∂bottom ∪ ∂side . Here we use ∂top and ∂bottom to refer to the top and bottom layers of the ocean. In terms of the geocentric, spherical coordinates (r, θ, φ), the surface ∂top is represented by r = r0 , where r0 is the radius of the Earth, and the surface ∂bottom is represented by r = r0 − h, where h = h(θ, φ) is a C 4 -function which satisfies 0 < c0 ≤ h(θ, φ) ≤ c1 . The coordinates (θ, φ) are in that region on the sphere that corresponds to the global oceans of the Earth. (For the applications to the climate, the parameter c1 is “small”. We will return to this feature in Section 5.)
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gravity force G(t, x), which appears in equation (1.2), is the vector sum G = The N G k=1 k , where G1 = Ge , G2 = Gs , and G3 = Gm represent the gravitational force of attraction exerted by the solar bodies: the Earth, the Sun, and the Moon, respectively, on a fluid particle located at the point x ∈ . For k ≥ 4, Gk is the gravitational force of attraction due to the other planets. The strength of these forces is given by the inversesquare-law, and each is proportional to the mass of the attracting body. The force G1 = Ge is an autonomous force of attraction towards the center of the Earth, while the other forces, G2 , · · · , GN are nonautonomous forces, which depend on the time-varying distances between the point x ∈ and the respective solar bodies. There are of course, other oceanic models studied in the mathematical literature. We will comment on some of these models in Section 7. Suffice it to say here that, as far as we are aware, none of these models make use of the full gravitational force G(t, x) acting on the oceanic flow.
2.1. Boundary forces The radiant heating from the Sun of the top surface of the ocean is one of the boundary forces that drives the oceanic flows. As a matter of fact, this heating is the major source of energy for the global climate. Since our objective here is to describe a climate model which is centered on the oceanic flows, we will use an Interface Model for ∂top . Another objective is to calculate the amount of latent energy added to the oceanic dynamics and due to the boundary forces, see (3.37). If the boundary forces are too large, this can destabilize the mathematical model. 2.1.1. Interface Model. Our goal in this article is to develop a mathematical theory of the longtime dynamics of heat transfer in the ocean, a theory based on a suitable Interface Model. We accept as a working hypothesis that this goal can be achieved without using a full coupled atmospheric–oceanic model. We do not present a detailed mathematical description of the Interface Model here. That would be the subject of a future investigation into the relevant physical and mathematical properties needed for such a model. This Interface Model is intended to describe the effective forcing of the ocean due to the atmosphere, and other external agents. One includes here the effective heating (and cooling) of the surface of the ocean, taking into account such things as the albedo, i.e., the fraction of the solar radiation that is reflected by the atmosphere and the surface of the ocean. In this Model, one also treats the effective changes in the surface salinity of the ocean due to the evaporation, the rainfall, and the melting and freezing of the polar ice. For the longtime dynamics, one needs to take into account the force induced on ∂top by the oceanic trade winds. What is needed here is a balance of the momenta of the air and the water, see [55]. Since the mass per unit volume of the air and the water differ, it follows that the velocities must differ, as well. This implies that one must use a slip boundary condition in the Interface Model. We will express this slip condition in terms of the Navier boundary conditions, which are described below. (Also see [5].)
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2.1.2. Navier boundary conditions. The Navier boundary conditions are formulated in terms of the deformation tensor D(U ), see [49]. This tensor is represented in the Euclidean coordinate system by a 3 × 3 symmetric matrix: 1 D(U )i,j = (∂i Uj + ∂j Ui ), 2
where ∂i Uj =
∂Uj . ∂xi
Let ∂ denote the boundary of and let N denote the exterior normal to ∂. The (inhomogeneous) Navier boundary conditions then read: " U · N "∂ = 0
and
" (D(U )N )tan "∂ = *,
(2.1)
where (D(U )N )tan is the tangential component of the vector field D(U )N . In equation (2.1), the function * = *(t, x) is a suitable prescribed function, which is defined for (t, x) ∈ [0, ∞) × ∂. Furthermore, * satisfies the compatibility property * · N = 0 on ∂. If * ≡ 0 on ∂, then one has " U · N "∂ = 0
and
" (D(U )N )tan "∂ = 0,
(2.2)
and one refers to the boundary conditions (2.2) as the (homogeneous) Navier boundary conditions. In (2.2), the velocity vector U is tangent to the boundary and the vector D(U )N is normal to the boundary, see [71], [79], or [32] for more details. Note that (2.2) implies that D(U )N = λ N , for some scalar λ = λ(t, x). When using the homogeneous Navier boundary conditions (2.2), it is customary to introduce the bilinear form def
E(U, V ) = 4
D(U )·D(V ) dx = 4
3
D(U )i,j D(V )i,j dx,
i,j =1
where U and V are in the space def
H01 = {U ∈ H 1 ()3 : U · N = 0 on ∂}. We also set Z0 = {U = A × x + B ∈ L2 ()3 ) ∩ H01 : A, B ∈ R3 }. For generic choices of , one has Z0 = {0}. Indeed, needs to satisfy some symmetry condition for Z0 to contain a nonzero U . We let Z⊥ 0 denote the orthogonal complement of Z0 in the space L2 ()3 . We also define H 1 = H01 ∩ Z⊥ 0. def
We use here the abbreviated notation H k ()3 = H k (; R3 ), for k ≥ 1. We also define = ClL2 ()3 V11 , where V11 is the space
V10
V11 = ClH 1 ()3 {U ∈ H 2 ()3 ∩ H 1 : U satisfies (2.2) and ∇ · U = 0 in }.
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We let P denote the Helmholtz–Leray projection of L2 = L2 ()3 , that is P is the orthogonal projection of L2 onto V10 . When Z0 = {0}, then def
V10 = H1 = {U ∈ L2 ()3 : ∇ · U = 0 in and U · N = 0 on ∂} = P L2 ()3 . (2.3) For simplicity, we assume in the sequel that Z0 = {0}. Recall that H1⊥ , the orthogonal complement (in L2 ()3 ) of H1 , is H1⊥ = Closure in L2 of {∇p : p ∈ C 1 (, R)} = {∇p : p ∈ H 1 (, R}. Thus one has L2 ()3 = H1 ⊕ H1⊥ . From the Helmholtz decomposition, one finds that, for every f ∈ L2 ()3 , there is a p ∈ H 1 (, R) and a g ∈ H1 , such that f = g + ∇p
∇p2L2 + g2L2 ≤ f ||2L2 .
and
2.1.3. Korn inequality. By means of a direct calculation, one observes that a smooth vector field U satisfies E(U, U ) = 0 if and only if U = U (x) = A × x + B, where A and B are constant vectors and x ∈ . We see then that E(U, U ) = 0, for some U ∈ H 1 , if and only if U = 0. More importantly, the Korn inequality holds, that is, there is a positive constant c2 , such that c2 U 2H 1 ≤ E(U, U ) ≤ U 2H 1 ,
for every U ∈ H 1 ,
(2.4)
where U H 1 = U H 1 ()3 denotes the H 1 -norm. It follows from the Korn inequality √ (2.4), that E(U, V ) is an inner product and that E(U ) = E(U, U ) is a norm on H 1 . Furthermore, the norm E(U ) is equivalent to U H 1 , the H 1 -norm of U , for U ∈ H 1 . See [12], [71], [79], or [32] for more details, including proofs of (2.4). It is a consequence of the Korn inequality (2.4), that E(U, V ) is a coercive, bilinear form on H 1 . It is also symmetric and satisfies |E(U, V )| ≤ U H 1 V H 1 ,
for U, V ∈ H 1 .
Furthermore, as noted in [71], the following Green Identity holds for vector fields U in H 2 ()3 and V in H 1 ()3 : 1 U · V dx = − E(U, V ) + (∇ · U )(∇ · V ) dx 2 {2(D(U )N ) · V + (∇ · U )(N · V )} dσ. (2.5) + ∂
If in addition, one has U, V ∈ V11 = H11 ∩ H 1 , then U, V L2 =
1 U · V dx = − E(U, V ), 2
(2.6)
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since (2.2) holds, with D(U )N = λ N , and ∇ · U = ∇ · V = 0 in . 2.1.4. Dual space. The dual space of V11 will be denoted by V1−1 . Recall that this dual space is the collection of all linear functionals & on V11 with the property that there is a constant c ≥ 0, such that |&(U )| ≤ c U H 1 ,
for all U ∈ V11 .
For w ∈ V1−1 , one writes the value of w at the point v ∈ V11 as def
w(v) = v, w = v, w(V 1 ,V −1 ) , 1
for all v ∈ V11 .
1
Note that V1−1 is a Banach space with the norm def
wV −1 = sup{|w(v)| : vV 1 ≤ 1}. 1
1
One has a continuous imbedding V11 → H1 , see (2.3), with V L2 ≤ V H 1 , for all V ∈ V11 . (We use the notation V L2 = V L2 ()3 for the L2 -norm.) Another continuous imbedding is H1 → V1−1 . indeed, for f ∈ H1 = P L2 ()3 , one finds that f = P f and f ∈ V1−1 , with def
f (V ) = V , f L2 ,
for all V ∈ V11 .
We see that |f (V )| ≤ V L2 f L2 ≤ f L2 V H 1 . Thus, f ∈ V1−1 with f V −1 ≤ 1
f L2 . The triple (H1 , V11 , E(·, ·)) satisfies the Lax-Milgram property, see [14], [39], [43], [67], [73], and [80]. As a result, it follows from the Lax-Milgram Lemma, see [67], page 86, that there is a unique, bounded linear transformation L ∈ L(V11 , V1−1 ) such that L is a one-to-one mapping of V11 onto V1−1 with LL(V 1 , V −1 ) ≤ 1 1
1
L−1 L(V −1 , V 1 ) ≤ c2−1 , 1
1
and one has 1 E(U, V ) = U, LV , 2
for all U, V ∈ V11 .
(2.7)
(We use the notation L(V , W ) to denote the space of bounded linear transformation from the Banach space V into the Banach space W .) 2.1.5. Stokes operator. The linear operators L and L−1 play another important role in the study of the Navier–Stokes equations. Define the space D by D = {U ∈ V11 : LU ∈ H1 } = L−1 (H1 ). def
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For V ∈ D, we let f = LV ∈ H1 . It then follows from (2.7) that 1 E(U, V ) = U, LV = U, LV L2 , 2
for all U ∈ V11 .
We now define A1 = L|D : D → H1 as the restriction of L to D, that is, def
A1 V = L V ∈ H1 ,
for all V ∈ D.
It follows from the Lax-Milgram Theorem that A1 is a selfadjoint operator with domain D(A1 ) = D. Furthermore, since the imbedding V11 +→ H1 is compact, the operator A1 has compact resolvent. (See [14] or [67] for more details.) This operator A1 is the Stokes operator for the Navier–Stokes equations with the Navier boundary conditions. It is shown in [71] that D ⊂ H 2 ()3 . Consequently, one has D = D(A1 ) ⊂ H 2 ()3 ∩ 1 V1 . It follows then that the Stokes operator A1 satisfies A1 U = P (−U ),
for U ∈ H 2 ()3 ∩ V11 .
(2.8)
The Korn inequality (2.4) implies that A1 satisfies c2 U 2L2 ≤ c2 U 2H 1 ≤ E(U, U ) = 2 A1 U, U L2 ,
(2.9)
for all U ∈ D(A1 ). Thus A1 is a positive, selfadjoint operator on the Hilbert space H1 = P L2 ()3 . Since E(U, U ) satisfies E(U, U ) ≤ ∇U 2L2 =
3
i,j =1
∂Ui /∂xj 2L2 ,
for U ∈ H 1 ,
inequality (2.9) implies that c2 U 2L2 ≤ ∇U 2L2 ,
for U ∈ H 1 ,
(2.10)
that is, the Poincaré inequality holds. 2.1.6. Analytic semigroup properties. Since A1 is a positive, selfadjoint operator with compact resolvent, it follows that A1 is a sectorial operator on the Hilbert space H1 = P L2 ()3 , and that −A1 is the infinitesimal generator of an analytic semigroup e−A1 t , see [14] or [67], pages 66–68 and 92–94. Also, this semigroup is compact, for t > 0. Consequently, the fractional powers Aα1 are defined for all α ∈ R. Let V12α denote the domain of Aα1 , for α ≥ 0. Then V12 = D ⊂ H 2 ()3 ∩ V11 , 1 V11 = D A12 ,
and
V10 = H1 = P L2 ()3 .
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As noted in [67], each space V12α is a Hilbert space, with the scalar product and the norm given by U, V V 2α = Aα1 U, Aα1 V L2
and
1
U 2V 2α = Aα1 U 2L2 . 1
1
Since A12 is selfadjoint, it follows from (2.9) that 9 1 92 1 E(U, U ) = 9A12 U 9L2 , 2
for all U ∈ V11 .
(2.11)
2β
Also, one has the compact imbeddings V1 +→ V12α , when 0 ≤ α < β < ∞. 2.1.7. Deep Ocean Model. One of the important features of the oceans of the Earth is that the temperature and the salinity are constant below a certain depth. This means that, below this depth, the equations for τ and σ are satisfied, with these constant values, and the gradient terms for τ and σ in the λ-equation vanish. In our deep ocean model, we go one step further and we temporarily drop the time-dependent gravitational terms. The resulting λ equation is ∂t λ − ν'λ + (λ · ∇) λ + ∇p = Ge (x) − ' λ, ∇ · λ = 0.
(2.12)
In equation (2.12), one may assume that the pressure p = p(t, x) satisfies the hydrostatic property (7.1), which in this case reduces to ∇p = Ge . The primary force affecting the deep ocean model is then the Coriolis force ' λ. Now assume that there is a (stable) stationary solution λ0 = λ0 (x) (i.e., ∂t λ0 ≡ 0) of the equation (2.12). This stationary vector field generates a purely circulatory motion in the deep ocean, i.e., a geostrophic flow. We will use such a stationary solution as a driving force on ∂bottom . We will return to the Deep Ocean Model in Section 4, where we reexamine this model while retaining the time-dependent gravity forces, see Section 4.4.
2.2. Temperature and salinity equations The boundary forces affecting the temperature and salinity equations in (1.1) are somewhat simpler than those forces driving the Navier–Stokes equations. For the temperature equation, one begins with inhomogeneous boundary conditions of the form: ∂τ (t, x) = 2 (t, x) dn τ (t, x) = 2 (t, x)
for (t, x) ∈ ∂top × R+ for (t, x) ∈ ∂bottom × R+ ,
(2.13)
∂τ with similar conditions given on ∂side . The derivative dn (t, x) in (2.13) is the derivative in the direction of the outward normal. For the ocean model, the functions 2 and
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2 are nonzero, and the boundary conditions in (2.13) and on ∂side are referred to as inhomogeneous boundary conditions. The Neumann condition on ∂top describes the heat flux across the interface model. This flux is driven by the radiant heating and cooling of the ocean surface during the day time and night time. The Dirichlet condition on ∂bottom arises, for example, from the Deep Ocean model, where τ = constant in the Deep Ocean. The corresponding homogeneous boundary conditions, which are used in Section 2.2, have the form: ∂τ (t, x) = 0 dn τ (t, x) = 0
for (t, x) ∈ ∂top × R+ for (t, x) ∈ ∂bottom × R+ ,
(2.14)
with similar homogeneous boundary conditions on ∂side . For the salinity equation, one begins with inhomogeneous boundary conditions of the form: ∂σ (t, x) = 3 (t, x) dn σ (t, x) = 3 (t, x)
for (t, x) ∈ ∂top × R+ for (t, x) ∈ ∂bottom × R+ ,
(2.15)
with similar conditions given on ∂side . We use the flux term ∂σ dn (t, x) in (2.15) not because the salt is “crossing the boundary” of the ocean, but rather because fresh water is coming or going due to the rainfall and the evaporation on the surface of the ocean, as well as the melting/freezing of the polar ice caps. The corresponding homogeneous boundary conditions, which are used in Section 2.2, have the form: ∂σ (t, x) = 0 dn σ (t, x) = 0
for (t, x) ∈ ∂top × R+ for (t, x) ∈ ∂bottom × R+ ,
(2.16)
with similar homogeneous boundary conditions on ∂side . The inhomogeneous boundary conditions for τ and σ on ∂top , see (2.13) and (2.15), and on ∂side are guesses. We believe that one would obtain a similar theory under different assumptions, such as using mixed inhomogeneous Dirichlet–Neumann boundary conditions, or Robin boundary conditions, instead. The inhomogeneous Dirichlet boundary conditions on ∂bottom are physically realistic. The resulting homogeneous Dirichlet boundary conditions are used in the proof of the Poincaré inequality (2.17). There is an important physical property that is relevant for selecting the appropriate inhomogeneous boundary conditions to be used in (2.13) and (2.15). In particular, these boundary conditions need to be chosen so that one obtains the needed up-welling and down-welling of the oceanic currents to get the Great Conveyor Belt—or its equivalent. See Section 7 for additional information on this matter. Next we let A2 and A3 denote linear operators on L2 ()1 that satisfy A2 = − and A3 = −, along with the boundary condition (2.14) and (2.16), respectively. By using standard methods, one can show that each A2 and A3 is a selfadjoint operator on def
H2 = H3 = L2 ()1 , with compact resolvent and being bounded below. Consequently,
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each operator −A2 and −A3 is the infinitesimal generator of an analytic semigroup e−A2 t and e−A3 t , respectively, and each semigroup is compact for t > 0. For α ∈ R, we let Aα2 and Aα3 denote the fractional powers of A2 and A3 , respectively, and we let Vj2α = D(Aαj ) denote the domain of Aαj , for j = 2, 3 and α ≥ 0. One then has V20 = H2 and V30 = H3 , 1
1
as well as, V21 = D(A22 ) and V31 = D(A32 ). As noted above, one also has the compact β imbeddings Vj +→ Vjα , for j = 2, 3, when 0 ≤ α < β. Because of the Dirichlet boundary conditions τ (t, x) = 0 and σ (t, x) = 0 on ∂bottom , it is easily verified that the Poincaré inequality ck φL2 ≤ ∇φL2 ,
1 for φ ∈ D Ak2 , k = 2, 3,
(2.17)
holds. Thus the linear operators A2 and A3 are positive. 2.3. Reformulation of the equations (homogenization) The reformulation of the equations described here is accomplished by introducing a change of variables into the system (1.1)–(1.3), a change which replaces the inhomogeneous boundary conditions with homogeneous boundary conditions, see [40]. As a result of this change, we obtain a single evolutionary equation, see (2.21). For this purpose, we set v := (u, T , S), which is a vector in R5 . Next we introduce the following change of variables λ(t, x) = u(t, x) + W1 (t, x) τ (t, x) = T (t, x) + W2 (t, x)
(2.18)
σ (t, x) = S(t, x) + W3 (t, x). We require that the vector W = (W1 , W2 , W3 ) satisfy the inhomogeneous boundary conditions (2.1), (2.13), and (2.15) on ∂. We also require that the inhomogeneous boundary data for equations (1.1)–(1.3) be sufficiently smooth so that the regularity properties given in Regularity A, see page 347, are satisfied. Among other thing, this insures that W (·, t) ∈ H 2 ()5 , and ∇ · W1 = 0 in , for all t ≥ 0. We now treat the components of W as known functions of (t, x) ∈ R+ × , where R+ = [0, ∞). Let v = (u, T , S). For U, V ∈ H 1 ()3 ∩ L∞ ()3 , we define B1 (U, V ) = P (U · ∇)V . From the change of variables (2.18), we obtain B1 (λ, λ) = B1 (u + W1 , u + W1 ) = B1 (u, u) + B1 (u, W1 ) + B1 (W1 , u) + B1 (W1 , W1 ). With this change of variables, and by applying P to the λ-equation in (1.1) and using the facts that P ∇T = P ∇S = 0, we see that u satisfies ∂t u − P νu + B1 (u, u) + L1 (t)u + L0 u = g1 (t),
(2.19)
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where L0 u = P ' u, L1 (t)u = B1 (u, W1 (t)) + B1 (W1 (t), u) # $ g1 (t) = P G(t) − ' W1 (t) − ∂t W1 (t) + νW1 (t) − B1 (W1 (t), W1 (t)) , ' W1 (t) = a0 ω × W1 (t) and G(t) denotes the gravitational forces in the ocean. Solving for the pressure p(t, x) in equation (1.1) is described below. In the case of the τ and σ equations in (1.1), we define B2 and B3 by B2 (λ, τ ) = λ · ∇τ and B3 (λ, σ ) = λ · ∇σ , respectively. One then obtains B2 (λ, τ ) = B2 (u, T ) + B2 (u, W2 ) + B2 (W1 , T ) + B2 (W1 , W2 ) B3 (λ, σ ) = B3 (u, S) + B3 (u, W3 ) + B3 (W1 , S) + B3 (W1 , W3 ). Consequently, by using the change of variables (2.18) in the τ and σ equations in (1.1), we obtain ∂t T − κ2 T + B2 (u, T ) + L2 (t)v = g2 (t) ∂t S − κ3 S + B3 (u, S) + L3 (t)v = g3 (t), where L2 (t)v = B2 (u, W2 (t)) + B2 (W1 (t), T ) L3 (t)v = B3 (u, W3 (t)) + B3 (W1 (t), S) g2 (t) = −∂t W2 (t) + κ2 W2 (t) − B2 (W1 (t), W2 (t)) g3 (t) = −∂t W3 (t) + κ3 W3 (t) − B3 (W1 (t), W3 (t)). Since L1 (t)u does not depend on T or S, we will use below the convention that L1 (t)v := L1 (t)u, for v = (u, T , S), see (2.23). Next we let A1 be given by (2.8), and let A2 and A3 be given as in Section 2.2. Let H1 , H2 , and H3 denote the respective Hilbert spaces for these operators. Consequently, for k = 1, 2 and v (k) = (u(k) , T (k) , S (k) ) ∈ H , one has v (1) , v (2) H = u(1) , u(2) H1 + T (1) , T (2) H2 + S (1) , S (2) H3 . def
On the product space H = H1 × H2 × H3 , we let ⎛ ⎞ ν A1 0 0 A = ⎝ 0 κ2 A2 0 ⎠ = diag (ν A1 , κ2 A2 , κ3 A3 ) 0 0 κ3 A3 denote the corresponding block-diagonal operator on H . Then A is a positive, selfadjoint linear operator on H , with compact resolvent, and the domain satisfies D(A) = D(A1 ) × D(A2 ) × D(A3 ) ⊂ H.
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Furthermore, −A is the infinitesimal generator of an analytic semigroup e−At on H , and e−At is compact, for t > 0. For α ∈ R, the fractional powers of A satisfy Aα = diag ν α Aα1 , κ2α Aα2 , κ3α Aα3 . For α ≥ 0, we let D(Aα ) = D Aα1 × D Aα2 × D Aα3 ⊂ H denote the domain of Aα , and let V 2α = D(Aα ). One then obtains the compact imbeddings V β +→ V α , for 0 ≤ α < β, and one has V 0 = H,
1
V 1 = D(A 2 ),
V 2 = D(A).
Since each Aα is a positive, selfadjoint operator, one has 2 α 2 λ2α 1 vL2 ≤ A vL2 ,
for all v ∈ D(Aα ),
(2.20)
where λ1 is the smallest eigenvalue of A. With this change to the homogeneous boundary conditions, the time-varying nonlinear evolutionary equation for the ocean can now be written in the form ∂t v + Av + L(t)v + B(v, v) = g(t),
where v = (u, T , S),
L(t)v = (Lˆ 1 , L2 , L3 )v,
(2.21) (2.22)
and Lˆ 1 (t)v = L1 (t)u + L0 u with L1 (t)v := L1 (t)u = B1 (u, W1 (t)) + B1 (W1 (t), u) L2 (t)v = u · ∇W2 (t) + W1 (t) · ∇T
(2.23)
L3 (t)v = u · ∇W3 (t) + W1 (t) · ∇S. Also, B(v, v) = B(v, v)| ˆ v=v and B(v, v) ˆ = (B1 , B2 , B3 ) satisfies: ˆ ˆ = P (u · ∇)uˆ B1 = B1 (u, u) B2 = B2 (u, Tˆ ) = u · ∇ Tˆ ˆ = u · ∇ S, ˆ B3 = B3 (u, S)
(2.24)
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ˆ and the term g(t) satisfies (2.26). Note that the equation (2.21) can be where vˆ = (u, ˆ Tˆ , S) rewritten as the system: ∂t u + ν A1 u + Lˆ 1 (t) u + B1 (u, u) = g1 (t) ∂t T + κ2 A2 T + L2 (t) v + B2 (u, T ) = g2 (t)
(2.25)
∂t S + κ3 A3 S + L3 (t) v + B3 (u, S) = g3 (t), where # $ g1 (t) = P G(t) − ' W1 (t) − ∂t W1 (t) + νW1 (t) − B1 (W1 (t), W1 (t)) g2 (t) = −∂t W2 (t) + κ2 W2 (t) − B2 (W1 (t), W2 (t))
(2.26)
g3 (t) = −∂t W3 (t) + κ3 W3 (t) − B3 (W1 (t), W3 (t)). Let d = diag(ν, κ2 , κ3 ) be a 3 × 3 diagonal matrix and set |d|2 = ν 2 + κ22 + κ32 . In Section 3 we describe three different classes of solutions for the oceanic equations (2.21): weak solutions, strong solutions, and mild solutions. We are interested in solutions of the Initial Value Problem. These are solutions v = v(t), for t0 ≤ t < T1 , of (2.21) that satisfy the initial condition v(t0 ) = v0 ,
where t0 ∈ R and v0 ∈ H, or v0 ∈ V 1 .
(2.27)
In the sequel, we will refer to the pair (v0 , g), where v0 is given by (2.27) and g satisfies (2.26), as the data of the problem. Regularity A: We assume (see [40]) throughout this paper that the inhomogeneous boundary conditions (2.1), (2.13), and (2.15) are smooth, in the sense that the function W = W (t) = W (t, x) can be chosen to satisfy ∂t W ∈ L∞ (R, L2 ()5 )
and
W ∈ L∞ (R, H 2 ()5 ).
We will use the following L∞ -norms, for k = 1, 2: ∂t W 0,∞ = sup ∂t W (t)L2 t∈R
and
W k,∞ = sup W (t)H k . t∈R
Because of the Quasi Periodic Ansatz, see Section 4, the boundary forces, like the function W , are well-defined for all time t ∈ R. 2.3.1. Analytic semigroup properties. The linear operator −A is the infinitesimal generator of an analytic semigroup e−At with the fractional power spaces satisfying V 2α = D(Aα ), for all α ∈ R. Furthermore, the compact imbeddings V 1 +→ H +→ V −1 give rise to a duality due to the bilinear form ·, · where 1
1
U, V = A 2 U, A− 2 V L2 , def
for U ∈ V −1 and V ∈ V 1 .
(2.28)
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In (2.28) we use the fact that the mappings 1
A− 2 : V −1 → L2
1
A 2 : V 1 → L2
and
are isomorphisms in the following sense. First, the norm · V −1 is defined by the relation 1
U V −1 = A− 2 U L2 ,
for all U ∈ V −1 .
(2.29)
This norm on V −1 is generated by the inner product 1
1
U (1) , U (2) V −1 = A− 2 U (1) , A− 2 U (2) L2 . def
(2.30) 1
Second, the isomorphism between V 1 and L2 refers to the norm A 2 U L2 on V 1 . Since 1 2
1 2
E(V , V ) = A1 V 2L2 ()3 , see (2.11), it follows from the Korn Inequality (2.4), that 1
1
A12 V 2L2 ()3 is equivalent to the norm V H 1 ()3 , for V ∈ V11 = D(A12 ). A similar re1
1
1
lation holds for A22 T and A32 S. Consequently, A 2 U L2 ()5 is equivalent to the norm 1
U H 1 ()5 , for U ∈ V 1 = D(A 2 ). Since A is an (unbounded) selfadjoint linear operator on H , with compact resolvent, there is an orthonormal basis of eigenvectors {en ; n = 1, 2, · · ·} in H , with associate eigenvalues {λn ; n = 1, 2, · · ·} that satisfy: A en = λ n e n
0 < λ1 ≤ λ2 ≤ · · ·
and
with λn → ∞, as n → ∞.
Also, V 2 = D(A) is characterized as % ∞ ∞
2 2 V = D(A) = w = cn en ∈ H : λn |cn | < ∞ . 2
n=1
n=1
The spaces V 2α = D(Aα ) satisfy % ∞ ∞
2 cn en ∈ H : λ2α |c | < ∞ , V 2α = D(Aα ) = w = n n n=1
for α ≥ 0.
n=1
The norm on V 2α is defined by v22α
=
∞
n=1
2 λ2α n |cn | ,
where v =
∞
n=1
cn en .
(2.31)
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This formalism can be extended to the dual space V −1 by using the norm and the inner 1
product given above, see (2.29)–(2.30). Let {fn ; n = 12, · · ·} satisfy fn = λn2 en , for n = 1, 2, · · ·. Then one has 1
1
1
A− 2 fn = λn2 A− 2 en = en . 1
Hence, (2.29) implies that fn V −1 = A− 2 fn L2 = en L2 = 1. Since fn , fm V −1 = en , em L2 = δn,m , it follows that {fn ; n = 12, · · ·} is a orthonormal basis for the dual space V −1 . One can then use these observations to construct an interpolation family of fractional power spaces V −1+β , where % ∞ ∞
V −1+β = w = cn fn ∈ V −1 : λβn |cn |2 < ∞ , n=1
for 0 ≤ β ≤ 1.
n=1
Thus, V −1 can be identified with the space of infinite sequences:
V
−1
% ∞
−1 2 ≡ w = (w1 , w2 , · · ·) : λn |wn | < ∞ . n=1
That is to say, V −1 is a weighted &2 space. In fact one has: % ∞
def V α = w = (w1 , w2 , · · ·) : λαn |wn |2 < ∞ ,
for α ≤ 0,
(2.32)
n=1
α 2 where the norm is w2α = ∞ n=1 λn |wn | . 2α We note that the space V , for α ∈ R, which are described here, satisfy the interpolation inequalities (2.33). Moreover, if α, β ∈ R with α > β, then one has the compact imbedding V α +→ V β . Also, V α is a dense subset of V β . Furthermore, for every γ = s α + (1 − s)β, with 0 ≤ s ≤ 1, and any v ∈ V 2α , one has 1−s , v2γ ≤ C vs2α v2β
Aγ v ≤ C Aα vs Aβ v1−s .
(2.33)
A proof of this interpolation inequality appears in [67], Example 37.1, and many other references such as [14], [43], and [73].
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G.R. Sell
2.4. The pressure Recall that equation (2.19) is derived by applying the Helmholtz–Leray projection P to the Navier–Stokes equations in (1.1). Since P ∇p = 0, the pressure term p = p(t, x) does not appear (2.19), nor in (2.21) or (2.25). However, p can be recovered by applying the complementary projection (I − P) to the Navier–Stokes equations. In this case, one has (I − P)∂t λ = 0 and (I − P)∇p = ∇p. As a result, one finds that ∇p = (I − P)h + a1 ∇τ + a2 ∇σ,
(2.34)
where def
h = h(t) = (I − P)(νλ − (λ · ∇)λ + G(t) − ' u − ∂t W1 (t) − νW1 (t) − (W1 (t) · ∇)W1 (t)). By setting q = p − a1 τ − a2 σ , it follows from the Helmholtz decomposition that, for “smooth” terms, there is a scalar field q such that (2.34) can be written in the form ∇ q = h.
(2.35)
We take the divergence of equation (2.35) and obtain q = ∇ · h.
(2.36)
With appropriate boundary conditions, the elliptic problem (2.36) is solvable. It follows from standard sources, see [14], [22], [28], [36], [68], [72], or [80], for example, that, under reasonable assumptions, there is a good solution q(t) = q(t, x). In order to solve (2.36), with the prescribed boundary conditions, we make the following assumptions, which will guarantee that the known function h = h(t) is in L2loc [0, T0 ; L2 ()3 ), for some T0 ∈ (0, ∞). 1. G = G(t) = G(t, x) is a smooth function, say of class C 2 on × R, and G0,∞ < ∞; 2. Regularity A holds, which implies that ∂t W1 0,∞ < ∞, W1 0,∞ < ∞, as well as (W1 · ∇)W1 0,∞ < ∞; and 3. The solution λ = λ(t) of the oceanic equation (2.21) is a strong solution on an interval 0 ≤ t < T0 , where T0 ∈ (0, ∞], see Section 3.4. It follows that λ ∈ L2loc [0, T0 ; V 2 ) ∩ 2 1 L∞ loc [0, T0 ; V ). This implies that λ ∈ Lloc [0, T0 ; H ), and with (3.60), one has 8
7 P(λ · ∇)λ = B(λ, λ) ∈ L4loc [0, T0 ; H ) → L2loc [0, T0 ; H ) → Lloc [0, T0 ; V 1 ).
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2.5. Notation In this section, we will collect some of the notation for the various function spaces that arise in the analysis of the solutions of the system of equations (1.1)–(1.3). The Hilbert spaces L2 ()3 = L2 (, R3 )
and
H 1 ()3 = H 1 (, R3 )
have been introduced above. More generally one has L2 ()k = L2 (, Rk )
and
H m ()k = H m (, Rk ),
where m and k are positive integers. We will use H 0 ()k = L2 ()k on occasion, and we let · L2 ,
· H m ,
·, ·L2 ,
and
·, ·H m
denote the respective norms and inner products on these spaces. In reference to the equations (2.21) and (2.25), one typically has u ∈ H m ()3 ,
T ∈ H m ()1 ,
S ∈ H m ()1 ,
v = (u, T , S) ∈ H m ()5 ,
and (2.37)
for various choices of m. Since the domain will be fixed throughout, and since the constant k is understood by the context—see (2.37), for example,—we will use below the abbreviated notation L2 = L2 ()k
and
H m = H m ()k
below, with similar abbreviations for the norms. (See (2.4), for example.) The fractional power Sobolev spaces H α ()k are defined in the standard manner, see [1] or [43]. We use the notation U α = U H α
and
U, V α = U, V H α ,
for the norms and scalar products on the fractional power spaces H α ()k . Due to the smoothness properties described above, one has D(Aα ) ⊂ H 2α ()5 , for 0 ≤ α ≤ 1, and there exist constants d1 and d2 , which depend on α, such that d1 U 2α ≤ Aα U L2 ≤ d2 U 2α ,
(2.38)
for all U ∈ V 2α = D(Aα ). In other words, the norms Aα u = Aα uL2 and uH 2α are equivalent on D(Aα ).
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G.R. Sell
Because the inner products and norms on the L2 spaces L2 ()k occur so frequently, we will use the abbreviated notation: ·, ·0 = ·, ·,
and
· 0 = ·
for these expressions. The context should make it clear which value the integer k assumes in each usage. Typically, we use k = 1, 3, 5 in this article. We will also use an interpolation inequality. Let γ = sα +(1−s)β, where 0 ≤ β ≤ α ≤ 2 and 0 ≤ s ≤ 1. Then there is a constant C > 0 such that U γ ≤ C U sα U β1−s ,
for all U ∈ H α ()k .
(2.39)
Also one has U β ≤ U α ,
when β ≤ α.
(2.40)
If V ∈ H α+1 ()k , where α ≥ 0, we define ∇V α by d def
∇V 2α =
Di V 2α .
i=1
Note that one has ∇V 2α ≤ V 2α+1 , for all V ∈ H α+1 . We will use the following Fréchet spaces below: q
Lloc (t0 , T0 ; W ),
or
q
Lloc [t0 , T0 ; W ),
where 1 ≤ q ≤ ∞
and t0 ∈ R with T0 ∈ (t0 , ∞]. To explain this notation, we let W be a Banach space and let I denote, respectively, either the open interval (t0 , T0 ) in R, or the half-open interval [t0 , T0 ). q We say that a sequence {fn } in Lloc (I ; W ) converges to a function f —in the same space— provided that, for every compact interval J = [a, b] in I , the sequence {fn } converges to f in the Banach space Lq (J, W ). In the case where I = (0, ∞), the interval J = [a, b] must satisfy 0 < a < b < ∞, while for I = [0, ∞), J must satisfy 0 ≤ a < b < ∞. See [67], Appendices A and B, for more details. We also consider the spaces C(t0 , T0 ; W ),
C[t0 , T0 ); W ),
0,z Cloc (t0 , T0 ; W ),
0,z Cloc [t0 , T0 ; W )
of continuous and Hölder continuous functions, where z ∈ (0, 1] is the Hölder exponent. The topologies used here is uniform convergence on compact subsets of I , where I has the same meaning as used in the previous paragraph. For the Hilbert space H , we let Hw denote the Hilbert space H , with the weak topology. Thus, the space C[0, T ; Hw ) will denote the collection of all functions u : [0, T ) → H , that are continuous in the weak topology on H , i.e., for each v ∈ H , the mapping t → u(t), vH is a continuous mapping from [0, T ) into the scalar field, R or C. We will use
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Table 1 Function spaces Symbol
Key features
H1 H1⊥ H2 = H3 H H1
{U ∈ L2 ()3 : ∇ · U = 0 in and U · N = 0 on ∂} {∇p : p ∈ H 1 ()1 } L2 ()1 H1 × H2 × H3 ⊂ L2 ()5 {U ∈ H 1 ()3 : U satisfies U · N = 0 in ∂}
V11
ClH 1 ()3 {U ∈ H 2 ()3 : (2.2) holds and ∇ · U = 0 in }
V1−1 Vk2α Vk1 Vk2 Vα
dual space to V11 D(Aαk ), for α ≥ 0 and k = 1, 2, 3 1
D(Ak2 ), for k = 1, 2, 3 D(Ak ), for k = 1, 2, 3 V1α × V2α × V3α , for k = 1, 2, 3 and α ∈ R
V0
H
V1
D(A 2 ) D(A) dual space to V 1 fractional power Sobolev space subset of H 2α ()5 ∩ H
1
V2 V −1 H α ()k D(Aα )
s
w
the symbols → and → to denote, respectively, strong and weak convergence in either a Hilbert space, or a Fréchet space. Let V and W be two Banach spaces, or Fréchet spaces. We will use the notation V → W
or
V +→ W
to denote that there is a continuous imbedding, or a compact imbedding, respectively, of V into W . For the convenience of the reader, we present in Table 1 a listing of some of the function spaces which are used in this article, when Z0 = {0}. 2.5.1. The ubiquitous C. We will use the symbol C to denote a local variable in the sense that it may change from line to line, sometimes more often. We interpret C as a positive “constant”. Depending on the usage, it may depend on time. However, it will always be independent of n when referring to the Bubnov–Galerkin approximations.
3. Solution concepts In this section we present the various solution concepts which arise in the study of oceanic dynamics. Rather than using the oceanic equations (1.1)–(1.3), with the inhomogeneous boundary conditions (2.1), (2.13) and (2.15), we will use the formulation (2.21), where
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G.R. Sell
u, T , and S, now satisfy the homogeneous boundary conditions (2.2), (2.14), and (2.16), respectively. As noted above, some of the time-dependent terms in (2.21) and (2.25), arise as a result of the homogenization of the physical boundary conditions used in the original equations (1.1)–(1.3). The dynamical issues, such as the Quasi Periodic Ansatz, which are related to the time-dependent terms L(t) and g(t) in equation (2.21), will be treated in Section 4. As we will see, the Navier–Stokes equations in the oceanic model (2.21) and (2.25) will play a dominant role in the theory of the solutions for the oceanic problem. In fact, it should not be surprising that theory of solutions of the oceanic problem is based on suitable modifications of the Navier–Stokes theory. The main modifications we require are due to (1) the two transport equations in (2.25) and to (2) the influence of the linear term L(t)v. See [50] for a related application. (Recall that L(t) arises in the homogenization process used above.) The related Navier–Stokes theory can be found in: [2], [14], [21], [36], [39], [67], [68], [72], [73], and [80], for example. Throughout this section, we will assume, without further reference, that the following Standing Hypotheses are satisfied. Standing Hypotheses. The oceanic region is a bounded, smooth region, as described in Sections 1 and 2; the gravity force G(t, x) is C 2 -valued vector field on × R with G0,∞ < ∞; and the properties in Regularity A hold. 3.1. The B and L terms In this subsection we examine, in more detail, the quadratic term B(v, v) and the linear term L(t)v that appear in the equations (2.21) and (2.25). 3.1.1. The quadratic term. Since the vector v = (u, T , S) includes both vector fields and scalar fields on , it is convenient to treat the components of B = (B1 , B2 , B3 ) separately. We begin with the inertial term B1 (U, U ) which appears in the Navier–Stokes equations (2.19). We define the trilinear form b1 by def b1 (U, V , Vˆ ) =
3
Ui (∂i Vj )Vˆj dx = (U · ∇)V , Vˆ ,
i,j =1
where U , V , and Vˆ are vector fields which lie in appropriate subspaces of L2 ()3 , and ∂i = ∂x∂ i . Note that in the case where Vˆ ∈ H1 , one has P Vˆ = Vˆ and B1 (U, V ), Vˆ = P (U · ∇)V , Vˆ = (U · ∇)V , P Vˆ = (U · ∇)V , Vˆ = b1 (U, V , Vˆ ). By integrating by parts, we find that b1 (U, V , Vˆ ) = −b1 (U, Vˆ , V ),
(3.1)
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1 for U ∈ H1 ∩ H 2 ()3 and V , Vˆ ∈ H 1 ()3 . Hence,
b1 (U, V , V ) = 0,
1
for U ∈ H1 ∩ H 2 ()3 and V ∈ H 1 ()3 .
(3.2)
The following lemma is basic to deriving all of the auxiliary inequalities for the trilinear form b1 . L EMMA 3.1. Let be an open, bounded set in R3 of class C & , where & is an integer with & ≥ 2. Let αi , i = 1, 2, 3, satisfy 0 ≤ α1 ≤ &, 0 ≤ α2 ≤ & − 1, 0 ≤ α3 ≤ &, 3 and 2 ' & 3 ' & & 3 3 ' (α1 , α2 , α3 ) ∈ , 0, 0 , 0, , 0 , 0, 0, . / 2 2 2 α1 + α2 + α3 ≥
Then there is a positive constant C = C(α1 , α2 , α3 , ) such that |b1 (U, V , Vˆ )| ≤ CU α1 V α2 +1 Vˆ α3 ,
(3.3)
for all U ∈ H α1 ()3 , V ∈ H α2 +1 ()3 , and Vˆ ∈ H α3 ()3 . The proof of (3.3) is based on estimating each integral Ui (∂i Vj )Vˆj dx separately and then taking the sums. The details can be found in [14]. (Also see [67].) As a result, the estimate (3.3) remains valid when some of the vector fields U, V , Vˆ are replaced by scalar fields. We will use this observation below. We note that there is a constant C3 > 0 such that ⎧ 1 1 ⎪ 2 2 ⎪ C u v v w0 , 3 1 ⎪ 1 2 ⎪ ⎪ ⎪ for u ∈ H 1 , v ∈ H 2 , and w ∈ L2 , ⎪ ⎪ ⎨ |b1 (u, v, w)| ≤ C3 u1/2 v2 w0 , ⎪ ⎪ for u ∈ H 1/2 , w ∈ L2 , and v ∈ H 2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ C3 u2 v1 w0 , 2 for u ∈ H , v ∈ H 1 , and w ∈ L2 .
(3.4)
The first inequality in (3.4) comes about by using the interpolation inequality (2.39) and (3.3) with α1 = 1, α2 = 12 and α3 = 0, while the second follows with α1 = 12 , α2 = 1, and α3 = 0. For the third inequality one uses α1 = 2 and α2 = α3 = 0. For the components B2 and B3 , we will use the trilinear forms b2 and b3 , where b2 (U, ϕ, φ) = b3 (U, ϕ, φ) =
3
i=1
Ui (∂i ϕ)φ dx = U · ∇ϕ, φL2 ()1 ,
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where U and ϕ, φ are smooth functions in L2 ()3 and L2 ()1 , respectively. As noted in the paragraph following (3.3), the trilinear forms bk satisfy the counterpart of (3.3) and one has |bk (U, ϕ, φ)| ≤ C U α1 ϕα2 +1 φα3 ,
for k = 2, 3,
(3.5)
and all U ∈ H α1 ()3 , ϕ ∈ H α2 +1 ()1 and φ ∈ H α3 ()1 , where the α’s satisfy the hypotheses of Lemma 3.1. It then follows from (3.5), that if U ∈ H 1 ()3 and ϕ ∈ H 1 ()1 , then |bk (U, ϕ, ϕ)| ≤ C U 1 ϕ1 ϕ1/2 . If in addition, one has ϕ ∈ W 1,4 ()1 ⊂ W 1,2 ()1 = H 1 ()1 , then bk (U, ϕ, ϕ) = 0,
for k = 2, 3,
(3.6)
since ∇ϕ 2 ∈ L2 ()3 , ∇ϕ 2 ∈ H1⊥ , with U ∈ H1 , and
1 bk (U, ϕ, ϕ) = ϕ(U · ∇ϕ) dx = 2
U · ∇ϕ 2 dx = 0,
for k = 2, 3.
Now the space W 1,4 ()1 is a dense set in H 1 ()1 , in the H 1 ()1 -topology. Since the mapping ϕ → bk (U, ϕ, ϕ) ≡ 0 is uniformly continuous on W 1,4 ()1 , in the same topology, it follows that (3.6) is valid for all U ∈ H 1 ()3 and ϕ ∈ H 1 ()1 . If U ∈ H1 ∩ H 1 ()3 and ϕ, φ ∈ H 1 ()1 , then one has bk (U, ϕ, φ) = −bk (U, φ, ϕ),
for k = 2, 3.
(3.7)
For U ∈ H 1 ()5 , we will use the notation U = (U1 , U2 , U3 ), where U1 ∈ H 1 ()3 and U2 , U3 ∈ H 1 ()1 . For U, V , Vˆ ∈ H 1 ()5 , we define def b(U, V , Vˆ ) = B1 (U1 , V1 ), Vˆ1 H1
+ B2 (U1 , V2 ), Vˆ2 H2 + B3 (U1 , V3 ), Vˆ3 H3 .
(3.8)
Note that (3.8) implies that b(U, V , Vˆ ) = B(U, V ), Vˆ H ,
for U, V , Vˆ ∈ V 1 ,
where B(U, V ) is given by (2.24). It then follows from (3.2) and (3.6) that b(U, V , V ) = 0,
for all U, V ∈ V 1 .
(3.9)
Likewise, (3.1) and (3.7) imply that b(U, V , Vˆ ) = −b(U, Vˆ , V ),
for U, V , Vˆ ∈ V 1 .
(3.10)
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By using (3.8), with (3.3) and (3.5), one can readily verify that there is a constant C > 0 such that |b(U, V , Vˆ )| ≤ C U α1 V α2 +1 Vˆ α3 ,
(3.11)
for all U ∈ H α1 ()5 , V ∈ H α2 +1 ()5 , and Vˆ ∈ H α3 ()5 , where the α’s satisfy the hypotheses of Lemma 3.1. With the help of (2.39), this implies that 1
1
|b(U, V , Vˆ )| ≤ C3 U 1 V 12 V 22 Vˆ 0 ,
(3.12)
for all U ∈ H 1 ()5 , V ∈ H 2 ()5 , and Vˆ ∈ H 0 ()5 . 1 Next we derive an estimate of A− 2 B(V (1) , V (2) ), where V (k) = (u(k) , T (k) , S (k) ) ∈ V 1 ,
for k = 1, 2, 3.
From equation (2.24) we obtain −1
1
A− 2 B(V (1) , V (2) )2H = A1 2 B1 (u(1) , u(2) )2H1 −1
−1
+ A2 2 B2 (u(1) , T (2) )2H2 + A3 2 B3 (u(1) , S (2) )2H3 . − 12
Since A1
is selfadjoint, it follows from (3.1) that −1
−1
A1 2 B1 (u(1) , u(2) ), u(3) H1 = b1 (u(1) , u(2) , A1 2 u(3) ) −1
= − b1 (u(1) , A1 2 u(3) , u(2) ). Hence, (3.3) implies that there is a constant C > 0 such that −1
A1 2 B1 (u(1) , u(2) )H1 ≤ C u(1) 3/4 u(2) 3/4 . Similarly, one shows that bk (u, φ, ϕ) = −bk (u, ϕ, φ), for k = 2, 3, where u ∈ V11 and φ, ϕ ∈ H 1 ()1 . It then follows from (3.5) that there is a constant C > 0 such that −1
Ak 2 Bk (u, φ)Hk ≤ C u3/4 φ3/4 ,
for k = 2, 3.
Using (2.38) and (2.39), we find that 1
1
1
3
3
for V ∈ V 1 .
1
1
3
A− 2 B(V (1) , V (2) )2 ≤ C 2 V (1) 2 A 2 V (1) 2 V (2) 2 A 2 V (2) 2 ,
(3.13)
which implies that 1
1
1
A− 2 B(V , V ) ≤ C V 2 A 2 V 2 ,
(3.14)
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There are additional similar properties which arise in the study of the solutions of the oceanic equations. We present the following inequalities without detailed proofs. In the setting of the Navier–Stokes equations, one can find complete proofs in [67], pages 365– 369. By using the methodology developed above, see for example the proof of (3.11), one can readily extend these arguments in [67] to the setting of the oceanic equations. 5
5
1
3
1
1
3
1
B(U, V ) ≤ C A 8 U A 8 V ≤ C A 2 U 4 AU 4 A 2 V 4 AV 4 (3.15) and 1
1
1
7
A 2 B(U, U ) ≤ C A 2 U 4 AU 4 ,
(3.16)
for U, V ∈ V 2 . Similarly, one has 1
1
1
1
A− 4 B(U, V ), A− 4 B(V , U ) ≤ C A 2 U A 2 V
(3.17)
and 1
1
A 2 B(U, V ), A 2 B(V , U ) ≤ C AU AV ,
(3.18)
for U, V ∈ V 2 . 3.1.2. The linear term. In this section we will derive an estimate of the linear term L(t)V that appears in equation (2.21). First, we consider L0 u, where V = (u, T , S). Since ' u = a0 ω × u satisfies ' u ⊥ u, for any u ∈ R3 , one has ' u, uL2 = a0 ω × u, uL2 = 0,
for any u ∈ L2 ()3 .
(3.19)
From the Schwarz inequality one finds that there is constant C > 0 such that ˆ ≤ C u1/2 u ˆ 0, |' u, u| ˆ = |a0 ω × u, u|
(3.20)
for (u, u) ˆ ∈ V 1 ()3 × L2 ()3 . For the sequel we will use W = (W1 , W2 , W3 ) as defined in Section 2.3. L EMMA 3.2. Under the formulation given above, there is a constant C > 0 such that one has L(t) V 0 ≤ C W 2,∞ V 1 ,
for all V ∈ V 1 and t ∈ R,
(3.21)
where L is given by (2.22). In addition, there is a constant C > 0 such that 1
|A− 2 L(t) V (1) , V (2) 0 | ≤ C W 2,∞ V (1) 3/4 V (2) 0 , for all V (1) , V (2) ∈ V 1 .
(3.22)
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P ROOF. To prove (3.21), it suffices to show that |L(t)V , Vˆ 0 | ≤ C W 2,∞ V 1 Vˆ 0 ,
for all t ∈ R,
(3.23)
ˆ ∈ H . Next note that equation (2.23) implies where V = (u, T , S) ∈ V 1 and Vˆ = (u, ˆ Tˆ , S) that ˆ H3 , L(t)V , Vˆ H = Lˆ 1 (t)V , u ˆ H1 + L2 (t)V , Tˆ H2 + L3 (t)V , S and that the expression Lˆ 1 (t)V , u ˆ H1 contains 3 terms. Since one has ˆ H1 = (u · ∇)W1 (t), u ˆ H1 = b1 (u, W1 (t), u) ˆ B1 (u, W1 (t)), u B1 (W1 (t), u), u ˆ H1 = (W1 (t) · ∇)u, u ˆ H1 = b1 (W1 (t), u, u), ˆ it follows from (2.40) and (3.3), that there is a constant C > 0 such that ˆ H1 + (W1 (t) · ∇)u, u ˆ H1 | ≤ C W1 (t)2 V 1 Vˆ 0 . |(u · ∇)W1 (t), u Therefore, it follows from Regularity A that sup |L1 (t)V , Vˆ H1 | ≤ C W1 2,∞ V 1 Vˆ 0 . t∈R
It then follows from (3.19) that sup |Lˆ 1 (t)V , Vˆ H1 | ≤ C W1 2,∞ V 1 Vˆ 0 . t∈R
In order to estimate L2 (t)V , Tˆ H2 , we use again the equations (2.23). It follows from (3.5) that there is a constant C > 0 such that |u · ∇W2 (t), Tˆ H2 | ≤ C W2 (t)2 u1 Tˆ 0 and |W1 (t) · ∇T , Tˆ H2 | ≤ C W1 (t)2 T 1 Tˆ 0 . As a result, one obtains |L2 (t)V , Tˆ H2 | ≤ C W2 2,∞ u1 + W1 2,∞ T 1 Tˆ 0 . Similarly, one finds that ˆ 0. ˆ H2 | ≤ C W3 2,∞ u1 + W1 2,∞ S1 S |L3 (t)V , S
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By combining these estimates with the Schwartz inequality, we obtain a constant C > 0 such that (3.23) holds. The proof of (3.22) involves a straightforward, but rather lengthy calculation using the definition of L(t), see (2.23), and a routine modification of the argument leading up to (3.13). We omit the details. The inequality (3.21) implies that the linear operator L = L(t) satisfies L ∈ L∞ (R, L(V 1 , H )),
(3.24)
where L(V, W ) denotes the space of bounded linear operators from the Banach space V into the Banach space W . Because of the continuous imbeddings 5
1
L(V 4 , H ) → L(V 1 , H )) → L(V 1 , V − 2 ), it follows from (3.24) that 5
1
L ∈ L∞ (R, L(V 4 , H )) ∩ L∞ (R, L(V 1 , V − 2 )).
(3.25)
3.2. Bubnov–Galerkin approximations The basic strategy we use in describing the various solution concepts for the oceanic equations (2.21), or (2.25), is based on the Leray–Hopf approach in the study of the Navier– Stokes equations. In particular we will follow closely the methodology of [14] or [67] and use the theory of the Bubnov–Galerkin approximations, as this theory applies in the study of the oceanic equations (2.21). Since A is a positive, selfadjoint operator with compact resolvent, it follows that the spectrum σ (A) of A consists of real, positive eigenvalues, say 0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · , where each eigenvalue is repeated according to its multiplicities and λk → ∞, as k → ∞. We let B = {e(1) , e(2) , e(3) , · · ·} denote the respective eigenvectors of A. Without loss of generality, we assume that B is an orthonormal basis for H . For each integer n ≥ 1 we let P = Pn denote the orthogonal projection of H onto Span{e(1) , · · · , e(n) }, and we set Q = Qn = I − P . The projections P and Q are called the spectral projections determined by the operator A. For each v ∈ H we define p = pn and q = qn by p = P v = Pn v and q = Qv = Qn v. These spectral projections commute with A, i.e., P A = AP and QA = AQ, where the last equation is restricted to D(A), the domain of A. Next we apply P and Q to the oceanic evolutionary equation (2.21) to obtain the equivalent system
∂t p + Ap + P B(p + q, p + q) + P L(t)(p + q) = P g(t), ∂t q + Aq + QB(p + q, p + q) + QL(t)(p + q) = Qg(t),
(3.26)
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where the p-equation in (3.26) is an n-dimensional equation, while the q-equation is infinite dimensional. The nth order Bubnov–Galerkin approximation of (2.21) is given by the solutions of the th n order ordinary differential equation ∂t p + Ap + P B(p, p) + P L(t)p = P g(t),
(3.27)
which is obtained from (3.26) by setting q = 0 in the p-equation and ignoring the qequation. More precisely, if v = v(t) is a solution of (2.21) with initial condition v(0) = v0 ∈ H , then the solution p(t) of (3.27) that satisfies p(0) = p0 = P v0 ,
(3.28)
where v0 is a given element in H , is said to be the nth order Bubnov–Galerkin approximation of v(t). We will sometimes write vn (t) = p(t) to emphasize the dependence on n. The strategy then is to let n go to ∞ and to show that the nth order approximate solutions vn (t) = p(t) have suitable limits. These limits, in turn, describe the various solution concepts for the oceanic equations. We will use the notation Dt vn := ∂t∂ vn below. As the first step in this process is to show that the nth order Bubnov–Galerkin approximation p(t) satisfies various estimates which are independent of the order n. We begin by taking the scalar product in H of the Bubnov–Galerkin equation (3.27) with p = (λ, τ, σ ), where λ ∈ V11 ⊂ H1 , τ ∈ H 1 ()1 ⊂ H2 , and σ ∈ H 1 ()1 ⊂ H3 . We use (3.19) and then obtain ∂t p, p + Ap, p + P B(p, p), p + P L(t)p, p = P g(t), p. By using the selfadjointness of A and P , one then obtains 1 1 ∂t p2 + A 2 p2 + B(p, p), p + L(t)p, p = g(t), p, 2
(3.29)
where the scalar products and norms are taken in H = H1 × H2 × H3 ⊂ L2 ()5 . The nonlinear term B(p, p), p becomes B(p, p), pH = B1 (λ, λ), λH1 + B2 (λ, τ ), τ H2 + B3 (λ, σ ), σ H3 . Since v ∈ H11 , it follows from (3.2) that B1 (λ, λ), λH1 = 0. Moreover, since τ, σ are in H 1 ()1 , it follows from (3.6) that B2 (λ, τ ), τ H2 = 0
and
B3 (λ, σ ), σ H3 = 0.
Thus one has B(p, p), pH = 0, since p ∈ H 1 ()5 . Next we turn to the time-varying linear term P L(t)p, p. Since P is an orthogonal projection on L2 ()5 , it follows that P L(t)p ≤ L(t)p. Therefore, (2.40) and (3.23) imply that there is a constant C1 > 0, such that |P L(t)p, p| ≤ C1 W 2,∞ p1 p0 .
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G.R. Sell
For the term g(t), pH , we use (2.26) and (2.40). Since (3.3) and (3.5) imply that there is a constant C > 0 such that |Bk (W1 (t), Wk (t), φ| ≤ C W1 (t)1 Wk (t)2 φ0 ,
for k = 1, 2, 3,
it follows that there is a constant C0 > 0 such that B1 (W1 , W1 )H1 + B2 (W1 , W2 )H2 + B3 (W1 , W3 )H3 ≤ C0 W1 1,∞ W 2,∞ . By combining these estimates with the definition of g(t) and using the Schwartz inequality, we obtain |g(t), p| ≤ g0,∞ p0 , where g0,∞ ≤ G0,∞ + |a0 | W1 0,∞ + ∂t W 0,∞ + |d| W 2,∞ + C0 W1 1,∞ W 2,∞ .
(3.30)
Let us return to the equation (3.29). By using the relations derived above and (2.38), with α = 12 , we find that 1 1 1 ∂t p20 + d12 p21 ≤ ∂t p20 + A 2 p20 2 2
(3.31)
≤ C1 W 2,∞ p1 p0 + g0,∞ p0 . Next it follows from the Young inequality and (2.40) that C1 W 2,∞ p1 p0 ≤ g0,∞ p0 ≤ g0,∞ p1 ≤
d12 p21 + d1−2 C12 W 22,∞ p20 , 4 d12 p21 + d1−2 g20.∞ . 4
Therefore (3.31) becomes ∂t p20 + d12 p21 ≤ C72 W 22,∞ p20 + 2 d1−2 g20,∞ ,
(3.32)
where C72 = 2 d1−2 C12 . Next we define β as def
β = d12 − C72 W 22,∞ .
(3.33)
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It follows from (2.40) and (3.32) that ∂t p20 + β p20 ≤ 2 d1−2 g20.∞ .
(3.34)
By replacing β with a small negative number, if necessary, there is no loss in generality in assuming in (3.34) that β = 0. Furthermore, when β > 0, then (2.40) and (3.32) imply that ∂t p20 + β p20 ≤ ∂t p20 + β p21 ≤ 2 d1−2 g20.∞ ,
when β > 0.
(3.35)
It then follows from the Gronwall inequality, as applied to equations (3.35) and (3.34), and the fact that p(0) ≤ v0 , that one has 3 p(t) ≤ 2
e−βt v0 2 + 2 d1−2 |β|−1 g20,∞ (1 − e−βt ), v0 2 + 2 d1−2 |β|−1 g20,∞ (e|β|t + 1),
when β > 0 when β < 0, (3.36)
for all t ≥ 0. Note that the right side of (3.36) does not depend on n, the order of the Bubnov–Galerkin approximation. Also note that β >0
⇐⇒
W 22,∞ < d12 C7−2 .
(3.37)
R EMARK . The term W 22,∞ is measure of the latent energy put into the model by the boundary forces. We will say that the boundary forces are L2 -stable when β > 0. In the next lemma we present various estimates of the Bubnov–Galerkin approximations. These estimates will be used to show the existence of weak solutions of the oceanic equations (2.21). It is important to note that, in this lemma, the right sides of all the inequalities do not depend on n, the order of the Bubnov–Galerkin approximation. L EMMA 3.3. Let vn = p = p(t) be a solution of the Bubnov–Galerkin equation (3.27) that satisfies (3.28), for some v0 ∈ H . Let β be given by (3.33) and let g satisfy (3.30). Then p(t) is well-defined for all t ≥ 0 and satisfies the following properties: 1. The inequalities (3.36) are satisfied, for all t ≥ 0. In addition, when β > 0, then it follows that, for all t ≥ t0 ≥ 0, one has t 1 (3.38) A 2 p2 ds ≤ d22 e−βt0 v0 2 + 2 d1−2 [β −1 + (t − t0 )] g20,∞ . β t0
For β < 0, there is a constants cˆ1 = cˆ1 (β, W ) and c1 = c1 (β, W ), which do not depend on n nor on time, such that, for all t ≥ t0 ≥ 0, one has t 1 A 2 p2 ds ≤ cˆ1 e|β|t p(t0 )2 + g20,∞ t0 (3.39) |β|t 2 2 v0 + g0,∞ . ≤ c1 e
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G.R. Sell
2. For every T1 , with 0 < T1 < ∞, the sequence vn is in a bounded set in L2 (0, T1 ; V 1 )∩ L∞ (0, T1 ; H ), the sequence Avn lies in a bounded set in L2 (0, T1 ; V −1 ). Also, there is a constant b = b(T1 ) = b(T1 ; v0 , g), which does not depend on n, such that
T1
1
4
A− 2 Dt vn (s) 3 ds ≤ b(T1 ),
for all n ≥ 1,
0 4
and the sequence Dt vn lies in a bounded set in L 3 (0, T1 ; V −1 ). Furthermore, none of these bounded sets depend on n. 3. For t ≥ t0 ≥ 0, the following energy equality holds: p(t) + 2 2
t
1
A 2 p(s)2 ds
t0
= p(t0 ) + 2 2
t
g(s), p(s) ds − 2
t0
t
(3.40) L(s) p(s), p(s) ds.
t0
4. The Variation of Constants Formula is valid, that is, p(t) = e−A(t−t0 ) p(t0 ) +
t
# $ e−A(t−s) P g(s) − B(p(s), p(s)) − L(s)p(s) ds
t0
(3.41) for all t ≥ t0 ≥ 0. P ROOF. The validity of the inequality (3.36) was established above. When β > 0, then we use (2.38), with α = 12 , and integrate (3.35) to obtain
t
β t0
1 A 2 p(s)2 ds ≤ d22 p(t0 )2 + 2 d1−2 (t − t0 ) g2o,∞ .
By using (3.36) to estimate p(t0 )2 , we then obtain (3.38). A similar argument yields (3.39) when β < 0. It then follows from the inequalities (3.36), (3.38), and (3.39), that the sequence of Bubnov–Galerkin approximations {vn = p} that satisfy (3.28) lie in bounded sets in both L∞ (0, T2 ; H ) and L2 (0, T2 ; V 1 ), for each T2 ∈ (0, ∞). (Note that the sets are bounded in n, but not necessarily in time.) Now (2.29) implies that 1
AU V −1 = A 2 U L2 ,
for U ∈ V −1 .
It follows then from (3.38) and (3.39) that {Avn = Ap} lies in a bounded set in L2 (0, T2 ; V −1 ), for each T2 ∈ (0, ∞). Let us now turn to the estimate for {Dt vn = Dt p}. By rewriting equation (3.27) as Dt p = −Ap − P B(p, p) − P L(t)p + P g(t),
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we will now show that each of the 4 terms on the right side of the last equation lie in a 4 bounded set in L 3 (0, T2 ; V −1 ), for each T2 ∈ (0, ∞). First we use the continuous imbed4 ding L2 (0, T2 ; V −1 ) → L 3 (0, T2 ; V −1 ) with Item 1 to conclude that {Avn = Ap} lies in a 4 bounded set in L 3 (0, T2 ; V −1 ), for each T2 ∈ (0, ∞). The term g = g(t) does not depend on n, and one has 4
4
g ∈ L∞ (0, T2 ; H ) → L 3 (0, T2 ; H ) +→ L 3 (0, T2 ; V −1 ), for each T2 ∈ (0, ∞). Since P commutes with Aα , for any α, one has 1
1
P B(p, p)V −1 = A− 2 P B(p, p)0 ≤ A− 2 B(p, p)0 . It then follows from (3.14) that there is a constant C > 0 such that 4 2 1 3 4 1 1 P B(p, p)V3 −1 ≤ C p02 A 2 p02 3 = C p03 A 2 p20 .
It then follows from Item (1) that {P B(vn , vn ) = P (p, p)} lies in a bounded set in 4 L 3 (0, T2 ; V −1 ), for each T2 ∈ (0, ∞). It remains to check P L(t)p. Since P is an orthogonal projection on H , it follows from Lemma 3.2 that P L(t) p0 ≤ L(t) p0 ≤ C W 2,∞ p1 ,
for all t ∈ R.
Inequalities (3.38) and (3.39) then imply that {P L(t) vn = P L(t) p} lies in a bounded set in 4
L2 (0, T2 ; H ) → L 3 (0, T2 ; V −1 ),
for each T2 ∈ (0, ∞).
The proof of (3.40) follows by integrating equation (3.29) and using the observations that Pp = p and P B(p, p), pH = b(p, p, p) = 0, see (3.9). Finally the Variation of Constants Formula (3.41) is a standard property from the theory of ordinary differential equations.
3.3. Weak solutions As the reader may suspect at this point, the theory of weak solutions for the oceanic equations (2.21) proceeds down a track running parallel to the Leray–Hopf theory for the Navier–Stokes equations. We will pass many familiar landmarks on this journey, see [14] or [67]. The main objective we face here is to show that there is a subsequence of the Bubnov–Galerkin approximations {vn = p} that converges to a weak solution of equation (2.21). We begin with the definition of a weak solution, as it is used in this article.
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G.R. Sell
DEFINITION . We will say that a function v = v(t) on [0, ∞) is a weak solution of the oceanic equations (2.21), of Class LH, provided that v(0) = v0 , where v0 ∈ H , there exist positive constants c0 and c2 , which do not depend on t, and the following four properties hold: 1. One has 2 1 v ∈ L∞ loc [0, ∞; H ) ∩ Lloc [0, ∞; V ).
(3.42)
Consequently, there is a subset E in (0, ∞), with Lebesgue measure zero, such that v(t) ∈ V 1 , for all t ∈ E c = {t ∈ [0, ∞) : t ∈ / E}. 2. The function v has a time-derivative h in the space V −1 , so that (3.47) is valid with q h = Dt v, where h = Dt v ∈ Lloc [0, ∞; V −1 ), for some q with 1 ≤ q < ∞. 3. For β > 0, for all t0 ∈ E c , and all t with 0 < t0 < t, one has v(t)2 ≤ e−β(t−t0 ) v(t0 )2 + c02 g20,∞ ,
(3.43)
where β is given by (3.33). For β < 0, for all t0 ∈ E c , and all t ≥ t0 , one has v(t)2 ≤ c2 e|β|(t−t0 ) v(t0 )2 + g20,∞ .
(3.44)
4. For all t0 ∈ E c and all t ≥ t0 , the function v satisfies v(t) + 2 2
t
1 2
A v(s) ds + 2
t0
≤ v(t0 ) + 2 2
2
t
L(s) v(s), v(s) ds
t0 t
g(s), v(s) ds.
(3.45)
t0
5. For t0 ∈ E c , the function v satisfies v(t) − v(t0 ), w + +
t
1
t0 t
L(s) v(s), w ds =
t0
1
t
A 2 v, A 2 w ds +
b(v(s), v(s), w) ds t0
t
g(s), w ds,
(3.46)
t0
for all w ∈ V 1 , and for all t with t ≥ t0 ≥ 0. In this definition, we make reference to a specialized concept of a time-derivative Dt v. In particular, a function v : (0, T1 ) → H is said to have a time derivative h in the space q V −1 , provided that for some q with 1 ≤ q < ∞, there is a function h ∈ Lloc [0, T1 ; V −1 ) such that one has v(t) − v(t0 ) =
t
h(s) ds, t0
for 0 ≤ t0 ≤ t < T1 ,
(3.47)
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where now the integral exists in the space V −1 . An equivalent formulation of (3.47) is t 1 1 that A− 2 (v(t) − v(t0 )) = t0 A− 2 h ds, where the integral exists in the space H . Note that (3.47) implies that for all w ∈ V 1 one has v(t) − v(t0 ), w =
t
h(s), w ds =
t0
t
1
1
A− 2 h(s), A 2 w ds.
(3.48)
t0
By differentiating (3.48) with respect to t, one obtains a.e.
Dt v(t), w = h(t), w,
for all w ∈ V 1 .
(3.49)
We will use the integrated form (3.46), or (3.48), instead of (3.49), where Av, w has been 1 1 replaced by A 2 v, A 2 w, in the sequel. If (3.47) or (3.48) holds, then we will write h in the form h = Dt v. Notice that Dt v refers to a weak derivative. Once one has a weak solution in the sense described above, then many good things happen! L EMMA 3.4. Let v = v(t) be any weak solution of Class LH on [0, ∞) and let c0 and c2 be given as in the definition. Then the following properties are valid. 1. One has v ∈ L2loc [0, ∞; H ) and inequalities (3.43)–(3.44) hold. 2. The solution v satisfies v ∈ C[0, ∞; Hw ). In particular, v = v(t) is uniquely determined, for all t and t0 in [0, ∞), with t ≥ t0 , by equation (3.46). 3. For t0 ∈ E c and all t ≥ t0 , the following inequalities hold:
t
β t0
1 A 2 v2 ds ≤ d22 e−βt0 v0 2 + 2 d1−2 [β −1 + (t − t0 )] g20,∞ ,
for β > 0, and for β < 0, one has
t
t0
1 A 2 v2 ds ≤ cˆ1 e|β|t v(t0 )2 + g20,∞ ≤ c1 e|β|t v0 2 + g20,∞ ,
where the constants c1 and cˆ1 do not depend on time. 4. For β > 0, the solution v satisfies the inequality v(t)2 ≤ e−β(t−t0 ) v2∞ + c02 g20,∞ ,
for all t ≥ t0 ≥ 0,
where v∞ = vL∞ (0,∞;H ) . 5. The function v satisfies (3.46) in the space L1loc [0, ∞; V −1 ), for all t > 0. Consequently, one has ˆ = g(t), Dt v + Av + B(v, v) + L(t)v a.e.
in the space V −1 .
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G.R. Sell 4
0,θ 3 6. Also, Dt v is in the space Lloc [0, ∞; V −1 ), and v ∈ Cloc [0, ∞; V −1 ), for some θ > 0. 7. The function v is a mild solution in the space V −1 , i.e., the Variation of Constants Formula t ˆ v(s)] ds, e−A(t−s) [g(s) − B(v(s), v(s)) − L(s) v(t) = e−A(t−t0 ) v(t0 ) + t0
is valid in the space V −1 , for 0 ≤ t < ∞. Since this lemma is a standard result, we do not include a proof here. A detailed argument is given in [67], pages 375–378. We will use the following elementary fact in the sequel: Let {vn } be any sequence in L2loc [0, ∞; V α ), for some α ∈ R, that satisfies the following two properties: • there is a function a ∈ C[0, ∞; R+ ), such that, for each n, one has vn (t)α ≤ a(t),
for almost all t ≥ 0,
s
• and one has vn → u in L2loc [0, ∞; V α ), as n → ∞. Then by (2.38), one has d2−1 Aα/2 v(t)0 ≤ v(t)α ≤ a(t),
for almost all t ≥ 0.
(3.50)
We will need a compactness lemma to study sequences of Bubnov–Galerkin approxiq mations. Recall that a set B in a Fréchet space X = Lloc (I, V α ) is said to be a bounded set if for every neighborhood U of the origin in X, there is an r > 0 such that B ⊂ rU , where rU = {ru ∈ X : u ∈ U }, see [67], pages 601–602. We will make use of the compact imbeddings V α+1 +→ V α +→ V α−1 ,
where α ∈ R.
As usual we let ·, ·α = ·, ·, V α and · α = · V α denote the inner product and the norm on V α . L EMMA 3.5. Let wn be a sequence in L2loc (0, ∞; V α+1 ), for some α ∈ R, with the properties that: • wn is in a bounded set in L2loc (0, ∞; V α+1 ), • each wn has a time derivative in the space V α−1 , and q • the sequence Dt wn is in a bounded set in Lloc (0, ∞; V α−1 ), for some q satisfying 1 < q < ∞. Then there exists a subsequence of wn , which we relabel as wn , and functions w ∈ q L2loc (0, ∞; V α+1 ) and h ∈ Lloc (0, ∞; V α−1 ) such that the following properties hold: w
1. One has wn → w in L2loc (0, ∞; V α+1 ). w
q
2. One has Dt wn → h in Lloc (0, ∞; V α−1 ). s
3. One has wn → w in L2loc (0, ∞; V α ) and in L2loc (0, ∞; V α−1 ).
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369
s
4. For almost every t ∈ (0, ∞), one has wn (t) → w(t) in V α . q In addition, the functions w and h are in L2loc [0, ∞; V α ) and Lloc [0, ∞; V α−1 ), respectively. Furthermore, the limits in Items (1), (2), and (3) are valid, respectively, in the spaces q
L2loc [0, ∞; V α+1 ), Lloc [0, ∞; V α−1 ), L2loc [0, ∞; V α ), and L2loc [0, ∞; V α−1 ). A proof of this lemma, in the setting of the Navier–Stokes equations, appears in [67], pages 378–387. By using the modification described above concerning the linear term L(t)v, the same argument applies in the setting of the oceanic equations. We omit the details. In the next lemma, we apply Lemma 3.5, with α = 0, to a sequence of Bubnov–Galerkin approximations. Among other things, we will show that the limit functions v and h, that occur in Lemma 3.5, satisfy Dt v = h. L EMMA 3.6. For n = 1, 2, . . ., let vn = p be solutions of (2.21) with the property that the initial conditions vn (0) = p(0) lie in a bounded set in H , i.e., there is a positive constant C0 , which does not depend on n, such that vn (0)0 ≤ C0 , for all n ≥ 1. Then the sequence vn = vn (t) lies in a bounded set in L2loc [0, ∞; V 1 ), and the sequence of derivatives Dt vn 4
3 lies in a bounded set in Lloc [0, ∞; V −1 ). (Thus, the hypotheses of Lemma 3.5 are in play.)
w
Choose any subsequence of vn so that vn (0) → v0 ∈ H , and choose any further subsequence, which we relabel as vn , such that the conclusions in Lemma 3.5 are valid, with 4
3 limit functions v ∈ L2loc [0, ∞; V 1 ) and h ∈ Lloc [0, ∞; V −1 ). Then, after a possible change of v(t) on a set of measure zero, the following hold: 2 1 1. one has v ∈ L∞ loc [0, ∞; H ) ∩ Lloc [0, ∞; V ); 2. equation (3.46) is valid, for all t and t0 , with t ≥ t0 ≥ 0; 3. one has Dt v = h in V −1 , where
def
h(t) = −Av(t) − B(v(t), v(t)) − L(t)v(t) + g(t);
and
(3.51)
4. the three inequalities (3.43)–(3.45) are valid. P ROOF. For the Navier–Stokes equations alone, a proof of this lemma can be found in the several references cited above. Since this lemma is the most crucial aspect of the Leray– Hopf approach, we present the proof here for the convenience of the reader. The main new feature arising here is in the handling of the linear term L(t)v. Item (1): Since vn (0) ≤ C0 , for all n ≥ 1, it follows from (3.36) that vn is a bounded set in the Fréchet space L∞ loc [0, ∞; H ). Furthermore, from Lemma 3.3, Items (1) and (2), it follows that vn is a bounded set in L2loc [0, ∞; V 1 ) and Dt vn is in a bounded set in 4
3 Lloc [0, ∞; V −1 ). Therefore the hypotheses of Lemma 3.5 are satisfied. Next let vn now denote the related subsequence and let v and h denote the two limit functions satisfying the conclusions of Lemma 3.5.
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G.R. Sell
Item (2): Let w ∈ V 1 be given. By taking the scalar product of (2.21) in H with w and integrating in time, for t ≥ t0 ≥ 0, we obtain
t
vn (t) − vn (t0 ), w + +
1
1
A 2 vn (s), A 2 w ds
t0 t
t
b(vn (s), vn (s), Pn w) ds =
t0
g(s) − L(s) vn (s), Pn w ds.
(3.52)
t0
Now the last equation can be written in the form 1
1
A− 2 (vn (t) − vn (t0 )), A 2 w =
<
t
= 1 1 A− 2 M(vn ) ds, A 2 w ,
t0
where M(vn ) = M(vn )(s) = −Avn + Pn [−B(vn , vn ) − L(s) vn + g(s)]. 4
3 We will now show that M(vn ) has an appropriate weak limit in the space Lloc [0, ∞; V −1 ).
s
Since Pn w → w in H , one has
t
g(s), Pn w ds →
t0
t
g(s), w ds,
as n → ∞.
t0
w
Since vn → v in L2loc [0, ∞; V 1 ), one has
t
lim
n→∞ t 0
1
1
A 2 vn (s), A 2 w ds =
t
1
1
A 2 v(s), A 2 w ds,
for 0 ≤ t0 ≤ t < ∞.
t0
Next we will show that t t lim Pn L(s)vn (s), w ds = L(s)v(s), w ds. n→∞ t 0
(3.53)
t0
Note that Lvn , Pn w − Lv, w = L(vn − v), Pn w + Lv, Pn w − w. Since 1
1
L(vn − v), Pn w = A− 2 L(vn − v), A 2 Pn w, Lemma 3.2 and the interpolation inequality (2.33) imply that 1
1
1
3
|L(vn − v), Pn w| ≤ C W 2,∞ A 2 Pn w0 vn − v04 A 2 (vn − v)04 .
Oceanic dynamics and climate modeling
The Hölder inequality implies that
1 2
C A Pn w
2
t
t t0
371
|L(vn − v), Pn w|2 ds is bounded by
vn − v ds 2
14
t0
t
1 2
A (vn − v) ds 2
38 .
t0 s
Notice that the first integral above converges to 0, as n → ∞, since vn → v in L2loc [0, ∞; H ). The second integral is uniformly bounded in n by (3.39). Since 1
1
1
A 2 Pn w = Pn A 2 w ≤ A 2 w,
for w ∈ V 1 ,
(3.54)
we see that
t
lim
n→∞ t 0
L(vn − v), Pn w ds = 0.
The equality (3.53) now follows from the observation that (1) Pn w − w → 0, as n → ∞ and (2) that Lemma 3.2 implies that |Lv, Pn w − w| ≤ C W 2,∞ v1 Pn w − w0 . For the next step, we first assume that the trilinear term b(vn , vn , Pn w) satisfies
t
lim
n→∞ t 0
b(vn (s), vn (s), Pn w) ds =
t
b(v(s), v(s), w) ds,
(3.55)
t0
for 0 ≤ t0 ≤ t < ∞. Then by using Lemma 3.5, Item (5), one can take the limit as n → ∞ in (3.52) to obtain v(t) − v(t0 ), w + +
t
1
t0 t
1
A 2 v(s), A 2 w ds + a.e.
L(s) v(s), w ds =
t0
t
b(v(s), v(s), w) ds t0
t
g(s), w ds
(3.56)
t0
for all t0 ∈ E c and all t ≥ t0 . Now each of the terms with integrals in (3.56) is continuous in t and t0 . Therefore, by changing v on a set of measure zero, if necessary, we can assume, and we do assume, that (3.56) is valid for all t and all t0 with 0 ≤ t0 ≤ t < ∞, i.e., v satisfies (3.46). It remains to verify (3.55). Note that the trilinear form b satisfies b(vn , vn , Pn w) − b(v, v, w) = b(vn − v, vn , Pn w) + b(v, vn − v, Pn w) + b(v, v, Pn w − w).
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G.R. Sell
Now (3.10) implies that b(vn − v, vn , Pn w) = −b(vn − v, Pn w, vn ), it follows from (3.3) and (3.5) that 1
|b(vn − v, vn , Pn w)| ≤ C vn − v3/4 vn 3/4 A 2 Pn w. Next (3.13) implies that 1
1
3
1
1
3
1
|b(vn − v, vn , Pn w)| ≤ C vn − v 4 A 2 (vn − v) 4 vn 4 A 2 vn 4 A 2 Pn w. Similarly, one finds that 1
1
3
1
1
3
1
|b(v, vn − v, Pn w)| ≤ C v 4 A 2 v 4 vn − v 4 A 2 (vn − v) 4 A 2 Pn w. Now the Hölder inequality implies that product
1
C A 2 Pn w
t
t t0
vn − v2 ds
|b(vn − v, vn , Pn w)| ds is bounded above by the
18
t0
×
t
1 2
t
vn 2 ds
18
t0
A (vn − v) ds 2
38
t0
t
1 2
A vn ds 2
38 .
t0 s
Notice that the first integral above converges to 0, as n → ∞, since vn → v in L2loc [0, ∞; H ). On the other hand, the other three integrals are uniformly bounded in n by inequalities (3.36), (3.38), and (3.39). From (3.54), we see that
t
lim
n→∞ t 0
|b(vn − v, vn , Pn w)| ds = 0.
A similar argument shows that lim
t
n→∞ t 0
|b(v, vn − v, Pn w)| ds = 0.
We now examine the term b(v, v, Pn w). Let s ∈ E c be chosen so that t0 < s < t, v(s) ∈ and B(v(s), v(s)) ∈ V −1 , by (3.14). Since w ∈ V 1 , one has (I − Pn )w ∈ V 1 , and therefore
V1,
|b(v(s), v(s), Pn w − w)| = |B(v(s), v(s)), (I − Pn )wH | 1
1
= |A− 2 B(v(s), v(s)), A 2 (I − Pn )wH | 1
1
≤ A− 2 B(v(s), v(s)) (I − Pn )A 2 w → 0,
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1
as n → ∞, since (I − Pn )A 2 w → 0, as n → ∞. It follows then that a.e.
lim b(v, v, Pn w) = b(v, v, w).
n→∞
Due to the duality property (2.28), one has 1
1
b(v, v, Pn w) = A− 2 B(v, v), Pn A 2 wH . Therefore, (3.14) implies that there is a C > 0 such that 1
1
1
3
|b(v, v, Pn w)| ≤ C A 2 w v 2 A 2 v 2 . 1
1
3
From the Hölder inequality we infer that v 2 A 2 v 2 ∈ L1loc [0, ∞; R). Therefore one has
t
|b(v, v, Pn w − w)| ds → 0,
as n → ∞,
t0
by the Lebesgue Dominated Convergence Theorem. This in turn implies that (3.55) is valid. Item (3): We now use (3.56) to show that Dt v = h, where h satisfies (3.51). The ar4
w
3 gument leading up to (3.56) implies that M(un ) → M(u) in the space Lloc [0, ∞; H ).
w
4
3 However, from Lemma 3.5 one has Dt vn = M(un ) → h in Lloc [0, ∞; V −1 ). From the uniqueness of the limits one then obtains M(u) = h. Since (3.56) implies that Dt v = M(v) 4
3 [0, ∞; V −1 ), we see that Dt v = h. in Lloc Item (4): The inequality (3.36) shows that {vn : n ≥ 1} is in a bounded set in the Fréchet space L∞ loc [0, ∞; H ). It follows from Lemma 3.5, Item (5) that the limit function v is in the space L∞ loc [0, ∞; H ), as well. Moreover, inequality (3.36) holds in the limit, as n → ∞, by (3.50), with α = 0. Hence v satisfies (3.43)–(3.44). We claim that
lim
t
n→∞ t 0
L(s)vn (s), vn (s) ds =
t
L(s)v(s), v(s) ds.
(3.57)
t0
Indeed, one has Lvn , vn − Lv, v = L(vn − v), v + Lvn , vn − v. By using Lemma 3.2 with the Hölder inequality, and arguing as in the proof of (3.53), one readily obtains (3.57). We will omit the details.
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G.R. Sell
In order to prove the energy inequality (3.45), we return to equation (3.40). For t0 ∈ E c and t ≥ t0 , it follows from Lemma 3.5, Item (5), and (3.57) that
&
lim vn (t0 ) + 2 2
n→∞
= v(t0 )2 + 2
t
g(s), vn (s) ds − 2
t0 t
g(s), v(s) ds − 2
t0
t
L(s)vn (s), vn (s) ds
'
t0 t
L(s)v(s), v(s) ds.
t0 w
In order to take limits on the left side of (3.40), we use the fact that, since vn → v in L2loc [0, ∞; V 1 ), the norms satisfy the semicontinuity relationship
t
1
A 2 v2 ds ≤ lim inf
t
n→∞ t 0
t0
1
A 2 vn 2 ds,
for 0 ≤ t0 ≤ t.
By using: lim sup(an + bn ) ≥ lim sup an + lim inf bn , we see that v(t)2 + 2
t t0
1
A 2 v(s)2 ds is bounded above by
lim sup vn (t)2 + lim inf 2 n→∞
n→∞
t
1
A 2 vn (s)2 ds
t0
t 1 A 2 vn (s)2 ds ≤ lim sup vn (t)2 + 2 n→∞
≤ v(t0 )2 + 2
t0 t
g(s), v(s) ds − 2
t0
which gives (3.45).
t
L(s)v(s), v(s) ds,
t0
The main result on weak solutions of Class LH is the following theorem. This is an extension of the Leray–Hopf theory to oceanic flows. The proof is a direct consequence of Lemma 3.6. T HEOREM 3.7 (Weak solutions). For every v0 ∈ H there is a weak solution (of Class LH) v = v(t) of (2.21) satisfying v(0) = v0 , and one has 4
3 Dt v ∈ Lloc [0, ∞; V −1 ).
(3.58)
3.3.1. The uniqueness problem. As in the case of the weak solutions of the 3D Navier– Stokes equations, the weak solutions (of Class LH) of the 3D oceanic equations (2.21) are not known to be uniquely determined by the data (v0 , g). As is seen in the proof of
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the Theorem 3.7, for every choice of data, there is a sequence of Bubnov–Galerkin approximations which converge to a corresponding weak solution of the oceanic equations. However, if the weak solution is not uniquely determined by the data, then it can happen, for example, that there exists weak solutions which are not the limit of a sequence of Bubnov–Galerkin approximations. In such a case, one cannot use the Bubnov–Galerkin method to derive properties (other than existence) of the class of weak solutions for the oceanic equations.
3.4. Strong solutions In this section we turn to the theory of strong solutions of the oceanic equations (2.21). The major feature, that distinguishes a strong solution v = v(t) from a weak solution, is that v satisfies (3.59), in addition to (3.42). There are several equivalent definitions of a strong solution. We choose the following. DEFINITION . Let I = [t0 , T1 ), where 0 ≤ t0 < T1 ≤ ∞. A function v = v(t) is said to be a strong solution of the oceanic equations (2.21) on the interval I , provided that the following hold: 1. The function v is the restriction of a weak solution of Class LH to the interval [t0 , T1 ). 2. One has v(t0 ) ∈ V 1 . 3. The solution v satisfies 1 2 2 v ∈ L∞ loc [t0 , T1 ; V ) ∩ Lloc [t0 , T1 ; V ).
(3.59)
(Recall that D(A) = V 2 .) A strong solution v(t) of (2.21), on an interval [t0 , T0 ), is said to be maximally defined if either T0 = ∞, or v(t) has no proper extension, as a strong solution, to an interval [t0 , T1 ), where T1 > T0 . The relation (3.59) seems like a modest beginning for the theory of strong solutions, and it is. However, many good features follow from it. In particular, there are several properties of strong solutions which follow directly from the definition. First of all, all the properties given in Lemma 3.4 are valid on the interval [t0 , ∞), since a strong solution is also a weak solution of Class LH. Also, it follows from (3.15), (3.16), and (3.59) that if v is a strong solution on [t0 , T1 ), then one has 8
7 B(v, v) ∈ L4loc [t0 , T1 ; H ) ∩ Lloc [t0 , T1 ; V 1 ).
(3.60)
Additional properties are given in the following lemma. L EMMA 3.8. Let v = v(t) be any strong solution of (2.21) on the interval [t0 , T1 ). Then the following properties are valid. 1. The properties given in Lemma 3.4 are valid on [t0 , T1 ). 2. One has Dt v = h in H , where h satisfies (3.51) and h ∈ L2loc [t0 , T1 ; H ).
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G.R. Sell
3. The function v is a mild solution in the spaces H and V 1 , i.e., the Variation of Constants Formula v(t) = e
−A(t−t0 )
v(t0 ) +
t
e−A(t−s) [g(s) − B(v(s), v(s)) − L(s) v(s)] ds,
t0
is valid in these spaces, for t0 ≤ t < T1 . 4. The strong solution v on the interval [t0 , T1 ) satisfies 0,θ
0,θ1 v ∈ Cloc 0 [t0 , T1 ; H ) ∩ Cloc (t0 , T1 ; V 1 ).
P ROOF. Item (1) follows from the definition of a strong solution. For Item (2), we note that Av ∈ L2loc [t0 , T1 ; H ) since v ∈ L2loc [t0 , T1 ; D(A)). By Lemma 3.2, one has Lv ∈ L2loc [t0 , T1 ; H ) and, as noted above g ∈ L2loc [t0 , T1 ; H ). Since (3.60) implies that B(v, v) ∈ L4loc [t0 , T1 ; H ) → L2loc [t0 , T1 ; H ), we see that h ∈ L2loc [t0 , T1 ; H ) and Dt v = h in H . Items (3) and (4): The argument for the Navier–Stokes equations is found in [67], pages 365–369, as well as in [14]. The same argument is readily adapted to the oceanic flows. It is shown in Theorem 3.11 that the strong solutions of the oceanic equations (2.21) are uniquely determined by the data (v0 , g). We will let v = v(t) = S(g, t)v0 denote the maximally defined strong solution v of (2.21) with v(0) = v0 , where v0 ∈ V 1 , and we let [0, T0 ) denote the time interval of definition of this solution, where T0 = T0 (v0 , g) satisfies 0 < T0 ≤ ∞. As in the case of the weak solutions, the construction of strong solutions for the oceanic equations is based on the Bubnov–Galerkin method. In this case, we will use Lemma 3.5 with α = 1, which differs from the theory of weak solutions presented above. We begin the analysis for strong solutions with the sequence of Bubnov–Galerkin approximations s vn used in the proof of Theorem 3.7. One then has vn → v in L2loc [0, ∞; H ), where v is a weak solution (of Class LH) satisfying v(0) = v0 . We will then argue that vn satisfies the hypotheses of Lemma 3.5 for α = 1. It then follows from this lemma that there is a s w subsequence, which we will relabel as vn , such that vn → v in L2loc [0, T0 ; V 1 ) and vn → v in L2loc [0, T0 ; V 2 ), for a suitable T0 with 0 < T0 ≤ ∞. Now for the details. L EMMA 3.9. Let vn = p = p(t) be a solution of the Bubnov–Galerkin equation (3.27) that 1 satisfies A 2 p(t0 )2 ≤ d2−2 v0 21 , where v0 ∈ V 1 . Then the following properties hold: 1. There exist positive constants Ck , for k = 4, 5, 6, which do not depend on time nor on n, such that for t > 0, one has 1
1
1
∂t A 2 p2 ≤ C4 A 2 p6 + C5 A 2 p2 + C6 g20,∞ .
(3.61)
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2. There exists a time T1 ∈ (t0 , ∞) such that 1
A 2 p(t)2 ≤ ρ(t)2 ,
for t0 ≤ t < T1 ,
(3.62)
where ρ(t)2 = r(t) and r(t) is the maximally defined solution of the ordinary differential equation (ODE) ∂t r = C4 r 3 + C5 r 2 + c6 g20,∞ ,
(3.63)
that satisfies r(t0 ) = A 2 p(t0 )2 ≤ d2−2 A 2 v0 2 . 3. In addition, there is a constant C > 0, such that 1
t
1
1
Ap(s)2 ds ≤ A 2 p(t1 )2 + C
t1
t#
t1
$ 1 1 A 2 p6 + A 2 p2 + g20,∞ ds (3.64)
for almost all t1 ∈ [t0 , T1 ) and all t ∈ [t1 , T1 ). 4. There is a continuous function b(t), which does not depend on n and which is defined on the interval [t0 , T1 ), such that, for almost all t1 ∈ [t0 , T1 ), one has
t
Dt p2 ds ≤ b(t),
for all n and for all t ∈ [t1 , T1 ).
(3.65)
t1
Thus, Dt p ∈ L2loc [t1 , T1 ; H ). P ROOF. We return to the Bubnov–Galerkin equation (3.27), and take the scalar product with Ap to obtain 1 1 ∂t A 2 p2 + Ap2 = −b(p, p, Ap) − L(t)p, Ap + g(t), Ap. 2
(3.66)
(Note that A and P = Pn commute on the space R(P ).) By using (3.12), the Young inequality, (2.38) and (2.39), we find that 1 1 |b(p, p, Ap)| ≤ Ap2 + C A 2 p6 . 6
By using instead (3.23), with the other inequalities, we obtain 1 1 |L(t)p, Ap| ≤ Ap2 + C A 2 p2 , 6
and 1 |g(t), Ap| ≤ Ap2 + C g20,∞ . 6
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By combining these three inequalities with (3.66), we obtain >5 A 2 p2 + C6 g20,∞ . ∂t A 2 p2 + Ap2 ≤ C4 A 2 p6 + C 1
1
1
(3.67)
1
Using (2.20) where λ1 A 2 p2 ≤ Ap2 , with (3.67), we thus obtain (3.61). Item (2) now follows by using a standard nonlinear Gronwall inequality on the differential inequality (3.61). (It should be noted that the solution r(t) of the ODE (3.63) does not depend on n. However, since C4 > 0, r(t) always blows up in finite time whenever |r(t0 )| + g20,∞ > 0.) Inequality (3.64) follows by integrating (3.67). Item (4) follows from Lemma 3.8, Item (2). L EMMA 3.10. For every v0 ∈ V 1 and t0 ≥ 0, there is a T1 > 0 and there is a strong solution v = v(t) of (2.21) on [t0 , t0 + T1 ) that satisfies v(t0 ) = v0 . Moreover, the inequalities (3.62), (3.64), and (3.65) hold, where p = p(t) is replaced by v = v(t). P ROOF. When A 2 p(t0 ) ≤ d2−2 v0 1 , it follows from Lemma 3.9 that the hypotheses of Lemma 3.5 are satisfied, for v0 ∈ V 1 and α = 1. Therefore, there exists a subs w sequence of vn , which we relabel as vn , such that vn → v in L2loc [t0 , T1 ; V 1 ), vn → v 1
w
in L2loc [t0 , T1 ; D(A)), and ∂t vn → ∂t v in L2loc [t0 , T1 ; H ). Let v and h denote the limit functions given by Lemma 3.5. Since the Bubnov–Galerkin approximants satisfy inequalities (3.62), (3.64), and (3.65) it follows from (3.50), with α = 1, that the strong solution v = v(t) satisfies these inequalities where p = p(t) is replaced by v = v(t). The main result for the existence of strong solutions for the oceanic equations is the following. T HEOREM 3.11 (Strong solutions). For every v0 ∈ V 1 and every time t0 ≥ 0, there exists a maximally defined strong solution v = v(t) of (2.21) on the interval [t0 , T0 ), where T0 = T0 (v0 , t0 ) ∈ (t0 , ∞]. Furthermore, the following hold: 1. The strong solution v = v(t) is uniquely determined on every subinterval of [t0 , T0 ). Moreover, the inequalities (3.62), (3.64), and (3.65) are satisfied on any subinterval [t1 , T1 ) on which the solution r(t), of the ODE (3.63) satisfying 0 ≤ r(t1 ) ≤ 1 A 2 v(t1 )2 , is defined. 2. The conclusions of Lemmas 3.8 and 3.10 are satisfied. 3. Every strong solution is the limit of a subsequence of Bubnov–Galerkin approximations. 4. If T0 < ∞, then one has 1
lim A 2 v(t)2 = ∞.
t→T0−
(3.68)
The proof of this important result is a straightforward adaptation of the argument used for the Navier–Stokes equations, see [14] or [67], for example. We will not include the details here.
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R EMARK . By using the Variation of Constants Formula in (3.70)–(3.71), Fujita and Kato [21] have shown that the Initial Value Problem for the weak solution of the Navier–Stokes 1
equations is well-posed when the initial datum u0 is in the fractional power space V12 ⊂ 1
H 2 ()3 . Thus one obtains (local) existence and uniqueness for the weak solution in this case. We leave it to the reader to determine whether the same is valid for the oceanic 1 1 problem when v0 = (u0 , T0 , S0 ) ∈ V 2 ⊂ H 2 ()5 . 3.4.1. The global regularity problem. It is noteworthy that when studying the strong solu1 tions of the oceanic equations, one derives the bound A 2 p(t)2 ≤ ρ(t)2 , for the Bubnov– Galerkin approximations p = p(t). This upper bound is independent of n only on a finite time interval [t0 , T0 ), since ρ(t) → ∞ in finite time. What this suggests is that a given weak solution v(t) may be a strong solution on a finite interval [t0 , T1 ) and then loose its regularity in the sense that the blowup (3.68) occurs with T1 = T0 < ∞. When blowup does not occur, i.e., when T0 = ∞, we say that a maximally defined, strong solution v(t) of (2.21) is globally regular, i.e., one has 1 2 2 v ∈ L∞ loc [t0 , ∞; V ) ∩ Lloc [t0 , ∞; V ).
(3.69)
The theory of the oceanic equations shares the same problem seen in the theory of the Navier–Stokes equations. The Global Regularity Problem: Is every maximally defined, strong solution of the 3D Navier–Stokes equations (or the oceanic equations) globally regular ? What is known about the Global Regularity Problem as it relates to the oceanic equations 1 (2.21)? First of all, let us note what inequality A 2 v(t)2 ≤ ρ(t)2 does not say. While this inequality holds on the finite interval [t0 , T1 ), and while the term ρ(t)2 does blow up in 1 finite time, this does not say that A 2 v(t)2 blows up in finite time. To the best of our knowledge, there does not exist any example of such equations, (either the oceanic or the Navier–Stokes equations) on a smooth bounded domain in R3 , which have a strong solution that blows up in finite time.
3.5. Mild solutions and Fréchet differentiability An alternate theory of solutions of the Navier–Stokes equations is the theory of mild solutions. This theory is presented in the seminal work of Fujita and Kato [21]. Subsequently, many researchers have used the theory of mild solutions to study the Navier–Stokes equations, see for example: [2], [29], [53], and [59]. A mild solution on the interval [t0 , T0 ) of the oceanic equations (2.21) is a continuous solution of the Variation of Constants Formula t v(t) = e−A(t−t0 ) v(t0 ) + e−A(t−s) N (s) ds, for t ∈ [t0 , T0 ), (3.70) t0
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where N (s) = N (s, v(s)) = −B(v(s), v(s)) − L(s) v(s) + g(s).
(3.71)
An in depth study of the connections between the three solution classes (weak, strong, and mild) in the general context of evolutionary equations appears in [67], Sections 4.2 and 4.7. The approach used in this cited work is to rewrite the equation (2.21) as a linear inhomogeneous equation ∂t w + Aw = N (t),
(3.72)
and to treat N = N (t) as a known function with a prescribed degree of regularity, for −At example, N ∈ L∞ loc [0, ∞; H ). One then uses the properties of the analytic semigroup e to derive temporal and spatial regularity properties for the mild solution w = w(t) of w(t) = e−A(t−t0 ) w(t0 ) +
t
e−A(t−s) N (s) ds,
t0
with appropriate assumptions on w(t0 ). In some cases, one can repeat this process, as a part of the bootstrap method, to derive higher regularity of the solution v = v(t) of (3.70). We do this, in fact, in the proof of Theorem 4.2. (Also see [43] for related results.) It is known, for example, that the weak solutions of Class LH for the oceanic equations, as well as the Navier–Stokes equations, are also mild solutions in the space V −1 . By this statement, one means that the (Bochner) integral, in equation (3.70), exists in the space V −1 . (This is in contrast to the fact that the term v(t) is in V 1 , almost everywhere.) Similarly, the strong solutions for the oceanic equations, like the Navier–Stokes equations, are mild solutions in the spaces H and V 1 . The proofs of these assertions are based entirely on certain properties of the nonlinear evolutionary equations, see [67] for the details. In particular, the arguments use the analyticity of the semigroup e−At and appropriate Lipschitz continuity of the linear and nonlinear terms in (2.21) (for the oceanic equations) and in (2.19) (for the Navier–Stokes equations). Fortunately, the oceanic equations (2.21), like the Navier–Stokes equations, have Fréchet differential linear and nonlinear terms, and consequently, they have the desired Lipschitz continuity property. In particular, it follows from inequalities (3.15), (3.17), and (3.18) that the bilinear term B = B(U, V ) satisfies B : V 2 × V 2 → V 1, and
5
5
B : V 4 × V 4 → H = V 0, 1
B : V1 × V1 → V− 2 .
This implies that the B = B(U, U ) satisfies 5
1
2 2 2 (V 2 , V 1 ) ∩ CLip (V 4 , H ) ∩ CLip (V 1 , V − 2 ). B ∈ CLip
(3.73)
We adopt here the notation used in [67]. In particular, for the Banach spaces V and W , k (V, W ) = C (V, W ) ∩ C k (V, W ). The space C k (V, W ) denotes the space we let CLip Lip F F
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of nonlinear/linear functions that are mappings of V into W , that are k-times Fréchet differentiable. The space CLip (V, W ) denotes the space of Lipschitz-continuous mappings of V into W , where the Lipschitz constant is bounded on bounded sets in V. It follows from (3.25) that for each t ∈ R, one has 5
1
L(t) ∈ L(V 4 , H ) ∩ L(V 1 , V − 2 ). Hence N(v, t)—see (3.71)—is Fréchet differentiable with 5 1 N (·, v) ∈ L∞ R, CF2 (V 4 , H ) ∩ CF2 (V 1 , V − 2 ) .
(3.74)
The importance of the Fréchet differentiability in the study of the longtime dynamics of oceanic flows cannot be overstated. The primary tool used in the study of the nonlinear dynamics is: the difference between two nearby solutions of the nonlinear problem is well-approximated by a suitable solution of the linearized problem. In order to get a good approximation, one needs a good estimate of the error. This means that the solutions of the nonlinear problem need to be Fréchet differentiable functions of the data. (In short, one needs a good estimate of the “closeness” of the linear and the nonlinear dynamics.) However, the solution of the nonlinear problem is Fréchet differentiable precisely when the nonlinear terms in the system (4.10), or (4.11), are Fréchet differentiable. For the oceanic equations, this property is implied by (3.73) and (3.74). It is important to note that the Fréchet differentiability of the nonlinear terms is valid only for the strong solutions. Weak solutions do not have this property. At this time, there is no known theory of linearization of the weak solutions that can be used in the study of the longtime dynamics of the oceanic equations.
4. Skew product dynamics The key assumption regarding the time-varying forces acting on the ocean is that they satisfy the Quasi Periodic Ansatz (QPA), see below. In this section we examine the basics of quasi periodic functions and the related connection with dynamical systems, in general, and climate modeling, in particular. The rationale behind the QPA and the central role it plays in the climate modeling of the Earth is presented in [55]. The main point to note here is the apparent quasi periodic behavior of the solution of the N -body problem of the solar system. In particular, the quasi periodic behavior can be seen in: • the historical record of the positions of the celestial bodies of the solar system beginning with the observations of Tycho Brahe; • the extensive calculations of the longtime behavior of the solution of the N -body problem as seen [78], for example; and • the theoretical implications of the KAM theory, see [47] and [48]. Based on these facts, the QPA is a very reasonable assumption for studying the ocean dynamics. Furthermore, by using the QPA and the associate property of Partial Averaging, see Section 6, one is able to make a rigorous separation of the time-scales to study different aspects of the climate, see [55].
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4.1. Quasi periodic functions Let g = g(t) be a function g : R → X from R into a Banach space X. This function g is said to be a quasi periodic function if: 1. there is a K-dimensional torus T K = S 1 × S 1 × · · · × S 1 ; 2. there is a frequency vector ω = (ω1 , · · · , ωK ) in RK ; and 3. there is a continuous function G : T K → X from T K into X, such that g(t) = G(θ 0 + ωt),
for all t ∈ R,
(4.1)
where θ 0 = (θ10 , · · · , θk0 ) is a point in T K and θ 0 + ωt is calculated using toroidal arithmetic, i.e., θ 0 + ωt = θ10 + ω1 t, · · · , θK0 + ωK t (mod 1). An equivalent way of describing the function G(θ ) = G(θ1 , · · · , θK ) is to say that G, viewed as a mapping of RK into X, is periodic of period 1 in each of the K-variables. There is no loss in generality in assuming the coordinates ω1 , · · · , ωK of the frequency vector ω to be independent. That is to say, ω satisfies the independence condition: n · ω = 0 ⇐⇒ n = 0,
for all n ∈ ZK ,
(4.2)
where n · ω = n1 ω1 + · · · + nK ωK and n = (n1 , · · · , nK ) is a vector with integer entries. Indeed, if (4.2) is not satisfied, then the torus T K can be replaced by a lower dimensional torus T L , where (4.1)–(4.2) hold, with a different function G, where θ ∈ T L , ω ∈ RL and n ∈ ZL . 4.1.1. Ergodic theorem. When g and G satisfy (4.1), then the time-averages 1 T →∞ 2T lim
T −T
g(s) ds =
1 (T −S)→∞ T − S
T
lim
g(s) ds S
exist. Moreover, when (4.2) holds, then 1 lim (T −S)→∞ T − S
T
g(s) ds =
S
TK
=
0
1
G(θ ) dθ ···
1
(4.3) G(θ1 , · · · , θK ) dθ1 · · · dθK .
0
That is to say, the Ergodic Theorem holds, and the time-average of g equals the spaceaverage of G, see [8], [18], and [69].
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4.1.2. Fourier series. Both of the functions g and G have related Fourier series representations. We begin with G. Since G is continuous, it is an element of the Lebesgue space L2 (T K , X), and it satisfies G(θ ) =
where θ ∈ T K ,
Gn e2π i n·θ ,
(4.4)
n∈Zk
where n = (n1 , · · · , nK ) ∈ ZK and n · θ = n1 θ2 + · · · + nK θK . The coefficients Gn , which are elements of the Banach space X, satisfy G(θ )e−2π i n·θ dθ, for n ∈ ZK . (4.5) Gn = TK
It follows from (4.1) that the function g = g(t) satisfies g(t) = G(θ 0 + ωt) =
0 Gn e2π i n·θ e2π i n·ωt ,
for t ∈ R.
(4.6)
n∈ZK
On the other hand, one can begin with the quasi periodic function g = g(t). By using the theory of almost periodic functions, we note that the function g has the Fourier series expansion g(t) =
where t ∈ R,
gn e2π i n·ωt ,
n∈ZK
and the coefficients gn , which are also elements of the Banach space X, satisfy gn =
1 (T −S)→∞ T − S
T
lim
g(s) e−2π i n·ωs ds,
for n ∈ ZK .
S
When (4.2) holds and (thus) the Ergodic Theory is applicable, the associate function G must satisfy (4.4). Consequently, g satisfies (4.6). Since the Fourier series expansions are unique, one finds that 0
gn = Gn e2π i n·θ ,
for n ∈ ZK .
(4.7)
This identity (4.7) then determines the coefficients of G, as well as the toroidal phase, the point θ 0 in T K . Also, we note that since G(θ ) is continuous in θ , the function g(t)—that satisfies (4.1)—also satisfies g ∈ C(R, X) ∩ L∞ (R, X).
(4.8)
Let us turn to some examples of quasi periodicity that have been introduced above. The linear term L = L(t) in (2.21) is assumed to satisfy the Quasi Periodic Ansatz, see below. It then follows that L satisfies (4.8) with X = L(V 1 , H ), see (3.24). Also, one
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G.R. Sell
has L(t) = L(θ 0 + ωt), where L is a mapping of the torus T K into the Banach space L(V 1 , H ). In terms of the Fourier series, the coefficients Ln are elements of the Banach space L(V 1 , H ), for n ∈ ZK . Next consider the gravity function G(t, x). Here we view G = G(t, ·) as a time dependent mapping of R into H α ()3 , for some α ≥ 0. When G is quasi periodic in t, this means that there is a mapping G : T K × → R3 with the property that θ → G(θ, ·) is a continuous mapping of T K into H α ()3 . In terms of the Fourier series, the coefficients Gn are elements of the Banach space H α ()3 , for n ∈ ZK . Because of the smoothness assumptions on the geometry of , one can also view the mapping θ → G(θ, ·) as a continuous mapping of T K into C 4 (, R3 ) ⊂ H 4 ()3 . 4.2. Quasi Periodic Ansatz In this article, we assume that all of the time-dependent forces acting on the ocean are quasi periodic functions of t. This includes, for example, the internal force of gravity G(t, x), the time-dependent heating and cooling of the surface of the ocean due to radiation, as well as the time-dependent changes in the salinity on surface of the ocean due to rainfall and evaporation. The time-dependent behavior of the gravity force is directly related to the timedependent motion of the Sun, the Earth, the Moon, and the other planets in the solar system. This behavior, in turn, is driven by the vector solution X = X(t) of the N -body problem that describes the motion of the solar system. The assumption of the quasi periodicity of X(t) leads directly to the quasi periodicity of G(t, x) as a result of the Newtonian inverse-square law for the gravity force. Quasi Periodic Ansatz (QPA): Let H (t) = H : R → W be given, where W is a Banach space. We say that H satisfies the QPA provided the following hold: • H (t) is a quasi periodic function of time, that is, H (t) = H(θ 0 + ωt),
for t ∈ R,
where H : T k+1 → W is a continuous function from the torus T k+1 into the space W and θ 0 ∈ T k+1 is the toroidal phase at t = 0; • For each θ ∈ T k+1 , the function H(θ + ωt) is Hölder continuous in t, that is, there are constants K > 0 and z, with 0 < z ≤ 1, such that, for all θ ∈ T k+1 , one has H(θ + ωt1 ) − H(θ + ωt2 ) ≤ K |t1 − t2 |z ,
for |t1 − t2 | < 1;
• The frequency vector ω ∈ Rk+1 satisfies ω = (ω0 , ω1 , ω2 , · · · , ωk ) ∈ Rk+1 |ω0 | > |ω1 | > |ω2 | ≥ |ωj | > 0,
for 3 ≤ j ≤ k,
where the frequencies ω0 , ω1 , · · ·, ωk are well-approximated by the observed frequencies for the daily rotation of the Earth, the rotation of the Moon about the Earth, the
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rotation of the Earth about the Sun, and the rotations of the other planets about the Sun, see (6.1) and Table 2. • The frequency vector ω satisfies the independence condition (4.2).
4.3. Oceanic dynamics The Standing Hypotheses appearing in Section 3 have served their purpose in giving us a basis for the existence and uniqueness theory of strong solutions of the oceanic equations (2.21). For the study of the longtime dynamics of such solutions, we now replace the Standing Hypotheses with the stronger QP-Hypotheses. Therefore, in the sequel, we will assume that the following hold: QP-Hypotheses: In addition to the Standing Hypotheses stated in Section 3, we assume that the following hold: All the time-dependent forces and other terms appearing in the oceanic equations, including G(t, x), W (t, x), L(t), and g(t), satisfy the Quasi Periodic Ansatz (QPA) with the same Hölder exponent z and the same frequency vector ω. For the time-scales occurring in the modeling of the climate of the Earth, the QPA is a good approximation to the time-dependent forces acting on the ocean. Furthermore, as noted in [55], one can use the method of partial averaging, with the QPA, to give a rigorous foundation to using a reduced planetary model, such as the SEM (Sun–Earth–Moon) system for studying some climatical features, such as the El Niño events, see Section 6. We now return to the evolutionary equation (2.21), where we now impose the QPA. In particular, the quasi periodic functions of time L(t) and g(t) will now be written in the form L(t) = L(θ 0 + ωt)
and
g(t) = G(θ 0 + ωt),
for t ∈ R.
(4.9)
Since L and g are mappings, L : R → L and g : R → X, the functions L and G are mappings, L : T k+1 → L and G : T k+1 → X. As a result of the QPA, the evolutionary equation (2.21) can be written in the form ∂t v + Av + L(θ )v + B(v, v) = G(θ )
(4.10)
∂t θ = ω, or equivalently, that ∂t v + Av + L(θ + ωt) v + B(v, v) = G(θ + ωt),
where θ ∈ T k+1 .
(4.11)
Note that the single equation (2.21) has been replaced by a family of evolutionary equations (4.10), or (4.11), where the toroidal phase θ varies over the torus T k+1 . Note also that the Coriolis term ' u is included as a part of the linear term L(θ )v. The Coriolis term does not depend on θ ∈ T k+1 .
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The QPA has the effect of turning the nonautonomous problem (2.21) into an autonomous problem (4.10). Since the θ -coordinate in (4.10) does not depend on the vcoordinate, this equation is one of the standard examples that gives rise to a skew product flow, see [34], [54], [61], [63], [67], and [69]. By setting θ equal to θ 0 , as is done in (4.9), we refer to the toroidal phase of the climate model of our Earth, at the time t = 0. One may use the Ergodic Theorem and choose to replace the functions L and G in (4.10) by their spatial averages L0 and G0 , see (4.3), thereby obtaining ∂t v + Av + L0 v + B(v, v) = G0
(4.12)
∂t θ = ω.
(The symbol L0 used here is a local variable, which should not be confused with the same symbol appearing in (2.19).) Notice that the v and the θ variables are fully decoupled in (4.12). It is equation (4.12), sometimes with periodic forcing (as opposed to quasi periodic forcing) that is frequently used to study various features of the climate of the Earth. As is explained in [55], we do not opt for using equation (4.12), with, or without, periodic forcing. Instead of using the full spatial average, as is used in (4.5) and (4.12), we describe below a method of partial averaging. This partial average seems more appropriate for building various models of the climate, see [55]. 4.3.1. Skew product semiflow. The basic existence and uniqueness theorem for strong solutions, Theorem 3.11, applies to the equation (4.11), for each θ ∈ T k+1 . For each v0 ∈ V 1 and each θ ∈ T k+1 , we let S(θ, t)v0 represent the maximally defined strong solution of (4.11) that satisfies S(θ, 0)v0 = v0 . Set T0 = T0 (θ, v0 ), where T0 ∈ (0, ∞] and [0, T0 ) denotes the interval of definition of S(θ, t)v0 , see Theorem 3.11. The semiflow generated by the strong solutions of the oceanic equations (4.10), or (4.11), is denoted by π(t), where def
π(t)(θ, v0 ) = (θ · t, S(θ, t)v0 ),
for t ∈ [0, T0 (θ, v0 )),
(4.13)
and θ · t = θ + ωt. The mapping (θ, t) → θ + ωt is referred to as the twist flow on the torus T k+1 . Next define + and M by: + = {(θ, v0 , t) ∈ T k+1 × V 1 × [0, ∞) : t ∈ [0, T0 (θ, v0 ))} M = {(θ, v0 ) ∈ T k+1 × V 1 : T0 (θ, v0 ) = ∞}. The proof of the following result is given on [67], Theorem 47.5. T HEOREM 4.1. Under the QP-Hypotheses the following hold: 1. The semiflow mapping π : (θ, v0 , t) −→ π(t)(θ, v0 ) = (θ · t, S(θ, t)v0 ) is a continuous mapping of + into T k+1 × V 1 with π(0)(θ, v0 ) = (θ, v0 ).
(4.14)
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2. The set + is an open set in T k+1 × V 1 × [0, ∞). 3. Whenever one has τ ∈ [0, T0 (θ, v0 )) and t ∈ [0, T0 (θ · τ, S(θ, τ )v0 )) then one has τ + t ∈ [0, T0 (θ, v0 )) and S(θ, τ + t)v0 = S(θ · τ, t)S(θ, τ )v0 .
(4.15)
In other words, whenever the right-side of equation (4.15) is defined, then the left side of (4.15) is defined and equality holds. 4. One has π(t) M ⊂ M, for all t ≥ 0, i.e. M is a positively invariant set in T k+1 × V 1 . The strong solutions v = v(t) = S(θ, t)v0 have additional properties, see [67]. In particular, they are Lipschitz continuous functions of the data (v0 , L, G), uniformly on bounded intervals. The solution v also inherits whatever smoothness properties in θ that the quasi periodic terms L = L(θ ) and G = G(θ ) have. The dependence of the v on the frequency vector ω is another matter. However, for our application, this frequency vector is fixed. It does not change. One uses the same frequency vector for all climate models. Note that θ · τ is well-defined for all (θ, τ ) ∈ T k+1 × R, which brings us to two important concepts. A continuous mapping φ : (−∞, T1 ) → V 1 is said to be a negative continuation of the solution S(θ, t)v0 provided that φ satisfies: (1) φ(0) = v0 , (2) T1 = T0 (θ, v0 ), and (3) for all τ ∈ (−∞, T1 ), φ satisfies DEFINITION .
S(θ · τ, t)φ(τ ) = φ(τ + t),
for all t ∈ [0, T1 − τ ).
(4.16)
A global solution through the point (θ, v0 ) ∈ M is a continuous mapping φ : R → V 1 such that: (1) φ(0) = v0 and (2) φ satisfies S(θ · τ, t)φ(τ ) = φ(τ + t),
for all τ ∈ R and all t ∈ [0, ∞).
It is important to note that, when a negative continuation of a solution S(θ, t)v0 exists, it need not be unique. This lack of uniqueness is a major complication that arises in infinite dimensional dynamical systems, such as the dynamics generated by solutions of partial differential equations. Nevertheless, it is convenient to adopt a notational convention here. For τ ≤ 0, we set S(θ, τ )v0 := φ(τ ), where φ is some negative continuation of S(θ, t)v0 . In this way, (4.16) reads S(θ · τ, t)S(θ, τ )v0 = S(θ, τ + t)v0 ,
for all t ∈ [0, T1 − τ ).
Since a global solution, S(θ, τ )φ(0) = S(θ, τ )v0 = φ(τ ) is defined for all τ ∈ R, it satisfies: S(θ · τ, t)S(θ, τ )v0 = S(θ, τ + t)v0 ,
for all τ ∈ R and all t ≥ 0.
(4.17)
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4.3.2. Longtime dynamics. Assume for the time being that, there is a single datum (θˆ , v(0 ) ∈ T k × V 1 , with the property that the strong solution S(θˆ , t)( v0 satisfies v0 2 = ρ 2 < ∞. sup A 2 S(θˆ , t)( 1
t≥0
It follows then that for every r ≥ ρ, the set
1 K = Kr = (θ, v0 ) ∈ T k × V 1 : sup A 2 S(θ, t)v0 2 ≤ r 2 t≥1
is a nonempty, compact, positively invariant set for the oceanic semiflow π = π(t) and Kr ⊂ M. Furthermore, the omega limit set K = ω(Kr ) is a nonempty, compact, invariant set, that is to say, π(t) K = K,
for all t ≥ 0.
(4.18)
The identity (4.18) implies that for every (θ, v0 ) ∈ K, there is a global solution v(t) = S(θ, t)v0 ; v is defined for all t ∈ R; and (θ + ωt, v(t)) ∈ K, for all t ∈ R. Since K ⊂ M, it follows that for all (θ, v0 ) ∈ K, S(θ, t)v0 satisfies (4.17), see [29] and [67]. 4.4. Invariant sets Let K be an invariant set in M, that is, (4.18) holds. Assume that K is nonempty and compact. Then for each θ in T k , the fiber K(θ ) = {v0 ∈ V 1 : (θ, v0 ) ∈ K} is a nonempty, compact set in V 1 . The fact that each fiber K(θ ) is nonempty is a conseˆ of the twist flow on the torus T k+1 is dense in quence of the fact that every trajectory γ (θ) k+1 , which in turn follows from the independence condition (4.2). The invariance condiT tion (4.18) implies that the fibers satisfy: π(t)(θ, K(θ )) = (θ · t, K(θ · t)),
for t ≥ 0.
That is to say, for t fixed in R+ , the semiflow π maps the fiber K(θ ) onto the fiber K(θ · t). One possible scenario that might arise is: each fiber K(θ ) contains precisely one point, i.e., K(θ ) = {(θ )}, for all θ ∈ T k . In this case, one has π(t)(θ, (θ )) = (θ · t, (θ · t)) = (θ · t, S(θ, t)(θ )),
for t ≥ 0.
This implies that the solution v = v(t) = S(θ, t)(θ ) satisfies v(0) = (θ ). It follows from [62] and [63] that is a continuous mapping of T k+1 into V 1 . Since θ · t = θ + ωt, we see that v(t) = S(θ, t)(θ ) = (θ + ωt),
for all (θ, t) ∈ T k × R.
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Consequently, v is a quasi periodic function, see (4.1). The Hölder continuity of (θ + ωt) with respect to t, follows from (4.19). As a possible application, let us return to the deep ocean flow λ0 described in Section 2.1.7. Consider the perturbation of equation (2.12) caused by replacing the gravity term Ge (x) by the quasi periodic gravity force G(t, x). It follows from Table 3 that Ge (x) − G(t, x) can be viewed as a small perturbation. If λ0 has a “good” stability property, then the perturbed equation has a quasi periodic solution that is close to λ0 , for all t ∈ R. For a proof and a formula, see [67], Theorem 45.7 and equation (45.40). 4.4.1. Herculean theorem. Because of the importance of strong solutions in the study of the longtime dynamics of oceanic models, we need the following result. This theorem is a variation of the (autonomous) Herculean Theorem appearing in [67], Theorem 47.6. What we need is a nonautonomous version which is based on the quasi periodic forcing of the oceanic equations. T HEOREM 4.2 (QP-Herculean Theorem). Let the QP-Hypotheses be satisfied. Let K be a bounded set in T k+1 × V 1 , and assume that K is an invariant set for the semiflow π(t). Let K¯ denote the closure of K in T k+1 × V 1 . Then K¯ is a bounded, invariant set, and one ¯ the global mild solution S(θ, t)v0 has K¯ ⊂ T k+1 × V 2 . Moreover, for every (θ, v0 ) ∈ K, is both a strong solution and a classical solution of equation (4.11) in V 2r , for every r with 0 ≤ r < 1, for all t ∈ R. In addition, one has 0,1−r S(θ, ·)v0 ∈ Cloc (R; V 2r ) ∩ C(R; V 2 ).
(4.19)
Furthermore, K¯ is a compact, invariant set in T k+1 × V 2r , for each r with 0 ≤ r < 1. P ROOF. By using the fact that K is a bounded, invariant for π(t), we will first show that: 1. the conclusions in the theorem, including (4.19), are valid for all (θ, v0 ) ∈ K; 2. one has K ⊂ T k+1 × V 2s , for 0 ≤ s ≤ 1; 3. the set K is a bounded set in T k+1 × V 2r , for 0 ≤ r < 1; and 4. the closure K¯ is a compact invariant set in T k+1 × V 2r , for 0 ≤ r < 1. Since K is an invariant set for π(t), it follows that K¯ is an invariant set as well. Of course, ¯ which K¯ is bounded too. Consequently, the argument in Items (1)-(4) now applies to K, will complete the proof. Items (1)–(2). First note that Item (2) follows from (4.19). In order to verify Item (1), we will use Lemma 4.3, see below, which is merely a reformulation of [67], Lemma 47.2. In Lemma 4.3, reference is made to the mild solution v(t) = e
−A(t−t1 )
v1 +
t
e−A(t−s) F (s, v(s)) ds,
(4.20)
t1
where t1 ∈ R and F = F (t, v) ∈ CLip;z (R × V s+2β ; V s ),
where 0 ≤ β < 1,
(4.21)
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and s ∈ R. The space CLip;z refers to continuous mappings F = F (t, v) from R × V s+2β into V s , that are Lipschitz continuous in v, with Lipschitz coefficients being bounded on bounded sets in V s+2β and that are (locally) Hölder continuous in t with Hölder exponent being z, with 0 < z ≤ 1. The initial condition v1 is assumed to be in V s+2β . The solution v = v(t) in (4.20) is a mild solution of ∂t v + Av = F (t, v),
with v(t1 ) = v1 .
(4.22)
The existence and uniqueness of a mild solution of (4.22) are well-known, see [67], Lemma 47.1 for example. For our application, we set F (t, v) = F(θ + ωt, v), where F(θ, v) = −B(v, v) − L(θ )v + G(θ ), see (3.71) and (4.9). We let Sm (θ · t1 , t)v1 denote the maximally defined mild solution of (4.22) in V s+2β , with interval of definition [0, T1 ), where T1 ∈ (0, ∞]. L EMMA 4.3. Let the QP-Hypotheses hold and let F ∈ CLip;z (R+ × V s+2β ; V s ), where 0 ≤ β < 1 and 0 < z ≤ 1. Then for every v1 ∈ V s+2β , there is a time T1 ∈ (0, ∞] such that the mild solution Sm (θ · t1 , t)v1 of (4.20) in V s+2β is a strong solution of (4.22) in V s+2β on the interval 0 ≤ t < T1 . Furthermore, this solution satisfies 0,1−r (0, T1 ; V s+2r ) ∩ C(0, T1 ; V s+2 ), Sm (θ · t1 , ·)v1 ∈ C[0, T1 ; V s+2α ) ∩ Cloc
for all α and r with 0 ≤ α ≤ β and 0 ≤ r < 1. We continue now with the proof of Item (1). Let (θ, v0 ) ∈ K and let S(θ, t)v0 be a global (strong) solution in K, see (4.17). Set t1 = −1 and fix v1 = S(θ, t1 )v0 , so that S(θ · t1 , 1)v1 = v0 . Next we use a bootstrap argument and apply Lemma 4.3 twice. First we use (3.73) and set s = − 12 and 2β = 32 . From (3.73) and (3.74) and the QP-Hypotheses, it follows that for every (θ, v0 ) ∈ K, F satisfies (4.21). Hence Lemma 4.3 implies that the mild solution Sm (θ · t1 , t)v1 of (4.20), and (4.22), is a strong solution on [0, T1 ). Because of the uniqueness of strong solutions, for t ≥ 0, one has Sm (θ · t1 , t)v1 = S(θ · t1 , t)v1 = S(θ · t1 , t)S(θ, t1 )v0 = S(θ, t1 + t)v0 , see (4.17). Since S(θ, t1 + t)v0 is a bounded global solution, one has T1 = ∞. Consequently, S(θ, t1 + ·)v0 = Sm (θ · t1 , ·)v1 satisfies 0,1−r Sm (θ · t1 , ·)v1 ∈ Cloc (0, ∞; V s+2r ) ∩ C(0, ∞; V s+2 ),
for every r with 0 ≤ r < 1. In particular, (4.23) holds for r = 5 8
v0 = S(θ, t1 + 1)v0 , it follows from (4.23) that v0 ∈ V .
9 16 ,
(4.23)
where s + 2r = 58 . Since
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We now apply Lemma 4.3 a second time, with s = 0 and 2β = 58 . Then Lemma 4.3 implies that Sm (θ, t)v0 satisfies (4.23) with s = 0. Hence one has v0 ∈ V 2 . Since the argument above applies to any point (θ, v0 ) ∈ K, we can replace (θ, v0 ) by (θ · t1 , S(θ, t1 )v0 ), for any t1 < 0. This implies that (4.19) holds. Items (3)–(4). Next we claim that K is a bounded set in V 2r , for each r with 0 ≤ r < 1. By a routine calculation, as is used in the proof of Lemma 47.6 in [67], one finds that there is a positive constant k(r), which does not depend on v0 , such that Ar v0 ≤ k(r), for 0 ≤ r < 1 and all (θ, v0 ) ∈ K. Finally, we see that the closure K¯ is a compact set in V 2r , for 0 ≤ r < 1, by using the compact imbedding V 2s +→ V 2r , for 0 ≤ r < s < 1. 4.4.2. Attractors. We return to the semiflow π = π(t) on the set M in T k+1 × V 1 , see (4.13), (4.14), and Theorem 4.1. DEFINITION .
A set A in M is said to be an attractor for π provided that • A is a compact, invariant set in M, and • there is a bounded neighborhood U of A in M, such that A attracts U . To say that A attracts U, we mean that d(π(t) U, A) → 0,
as t → ∞,
where d(B, A) is defined, for every bounded set B in M, as d(B, A) = inf{ > 0 : B ⊂ N (A)}, and N (A) is the -neighborhood of A in the (T k+1 × V 1 )-topology. (See [67], Chapter 2, for additional information.) The word “attractor” has had many incarnations in the theory of dynamical systems. (The “attractor” concept is treated in the following references: [19], [20], [29], [32], [37], [50], [59], [60], [61], [65], and [73]. For a historical discussion, with additional references, see [67], pages 53–59.) The concept we use here is widely accepted because, when A is an attractor—in the sense used above—then A has two very important stability properties: • A is Lyapunov stable, and • A is asymptotically stable, see [67], Theorem 23.10. Because of the stability properties, an attractor is “robust” under small changes in the parameters of the model, see [67], Section 2.3.6. While the attractor A is robust, this does not mean that the corresponding solution S(θ 0 , t)v0 on A is either boring, or non-chaotic, or non-turbulent. One of the tasks facing a mathematical climatologist is to describe the attractor and to explain the future behavior of our favorite solution, S(θ 0 , t)v0 . Let the chaos be damned!! 4.4.3. Global attractors. The theory of global attractors for the 3D oceanic equations is similar to the theory for the 3D Navier–Stokes equations, provided that the oceanic problem is L2 -stable, that is, β > 0, see (3.37). As noted above, if the linear term L(t) in (2.21) is
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too large, that is if β < 0, this can lead to a non-dissipative weak solution problem. For this reason, we make the assumption here that β > 0. How does one find the global attractor for the weak solutions? Since the weak solutions (θ · t, S(t)v0 ) are not uniquely determined by the data (θ, v0 ), when v0 ∈ H , one is led to the technique of treating each weak solution ϕ(t) = S(t)v0 as a point in a suitable function space, see [64] and [65]. In the case of the oceanic equations, the appropriate function space is the Fréchet space L2loc [0, ∞; H ). This induces a semiflow σ (τ ) on T k+1 × L2loc [0, ∞; H ), where σ (τ )(θ, ϕ) = (θ · τ, ϕτ ),
for τ ≥ 0,
and ϕτ (t) = ϕ(τ + t). The global attractor A is a nonempty, compact, invariant subset of T k+1 × L2loc [0, ∞; H ). The arguments used for the autonomous 3D Navier–Stokes equations can be readily adapted for the quasi periodically forced oceanic problem, see [65], [19], and [20]. For related work, see [10], [13], [15], [16], [50], and [77]. The argument used in the study of the Navier–Stokes equations extends readily to the oceanic problem (4.10), when β > 0. While this theory of the global attractor A for the weak solutions of the oceanic equations has many interesting properties (see [20] for the Navier–Stokes equations setting), there remains one important open issue: It is not known whether every solution S(θ, t)v0 , with (θ, v0 ) ∈ A, is globally regular, see (3.69). Nevertheless, there is hope that, by using the theory of thin domain dynamics, one may be able to show that the Global Regularity Problem—when restricted to A—has a positive answer. That being the case, we turn to the study of thin-domain dynamics for the oceanic equations. 4.5. Weak solution attractor Next we focus on the 3D Navier–Stokes equations by decoupling the u-equation from the T -equation and the S-equation in (2.25). One then has ∂t u + ν A1 u + Lˆ 1 (t) u + B1 (u, u) = g1 (t), which we will rewrite in the skew product notation (see (4.11)) as L1 (θ + ω t) u + B1 (u, u) = G1 (θ + ω t), ∂t u + ν A1 u + (
where θ ∈ T K . (4.24)
The existence of the global attractor for the autonomous 3D Navier–Stokes equations; where (4.24) is modified by requiring that: (1) g1 does not depend on t, (2) Lˆ 1 = L1 + L0 satisfies L1 ≡ 0 (see (2.19)), and (3) the homogeneous Dirichlet boundary conditions hold; is developed and proved in [65], also see [67]. It is a straightforward task to extend this argument to our case where: (1) the homogeneous Navier boundary conditions hold and (2) ( L1 and G1 satisfy the QP-Hypothesis with ( L1 = 0; provided that β (see (3.33) on page 362) is positive. As a consequence, we obtain the following result:
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T HEOREM . When β > 0, there is a global attractor Aw for the weak Leray–Hopf solutions of equation (4.24). We will denote an individual point in Aw as (θ, φ), where θ ∈ T K and φ is a global weak solution of (4.24) that satisfies φ ∈ L∞ (R, V10 ) ∩ L2loc (R, V11 ).
(4.25)
The semiflow πw on Aw is defined by πw (τ )(θ, φ) = (θ · τ, φτ ),
for all τ ∈ R,
where φτ (t) = φ(τ + t). It is important to note that if (θ, φ) ∈ Aw , then φτ is a global solution of the shifted equations, wherein θ is replaced by θ · τ in (4.24). By using the methodology of Foias and Temam [20], one can readily show that 0 ), φ ∈ C(R, V1,w 0 denotes the Hilbert space V 0 with the weak topology. where V1,w 1 From Lemma 3.4 and the Leray–Hopf Theorem 3.7, it follows that if u = u(t) is a weak solution of (4.24), then def
(θ, u) ∈ T K × L2loc = T K × L2loc [0, ∞; V10 ). It follows then that Aw ⊂ T K × L2loc . We invite the reader to show that, when the QPHypothesis holds and β > 0, then for every θ ∈ T K , there is a global weak solution φ of (4.24) such that φ ∈ L∞ (R, V10 ) and (θ, φ) ∈ Aw . Because of (4.25), we see that if (θ, φ) ∈ Aw , then φ(t) ∈ V11 , for almost all t ∈ R. This brings us to the LRP, which is formulated here for the 3D Navier–Stokes equations. LRP (Little Regularity Problem): Is it the case that whenever, β > 0, then for every (θ, φ) ∈ Aw , one has 1 φ ∈ L∞ loc (R, V1 ) ?
(4.26)
If the answer is positive, then one can show that whenever φ(τ ) ∈ V11 , then one has φ ∈ L2loc [τ, ∞; V12 ). (This is accomplished by reverting to the Bubnov–Galerkin approximations, once again. See [14]or [67], for more information.) Hence φ(t) is a strong solution for t ≥ τ . Furthermore, the blowup relation (3.68) is never satisfied. Also, one finds that φτ ∈ C[0, ∞; V11 ). Since φτ is also a mild solution, one finds that (θ · τ, φτ ) ∈ M, see (4.14). By letting τ → −∞, we see that Aw ⊂ M. The LRP is the small brother of the Million Dollar Navier–Stokes Problem. However, the LRP does give some information about the bigger problem. When the answer to the LRP is positive, then the blowup phenomenon (3.68) is a transient feature only. The point
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being, there is a growing body of studies in which it is shown, among other things, that the LRP does have positive answers. Thin 3D domains is one place where one finds such results. 5. Thin domain dynamics As noted in Section 1, the depth of the ocean (about 5 km) is small when compared to the expanse (about 17780 km). The depth given here is just a guess. What is needed is that the depth is fixed so that one can model, for example, the counter-currents that return the northward flow of the Gulf Stream in the Atlantic ocean back to the equatorial region. Such a model may be possible with a depth of even less than 6 km. The thinness of the oceanic domain is connected with the aspect ratio: depth divided by expanse. Let us now explore the potential significance of treating the oceanic domain as a thin domain. The theory of the Navier–Stokes equations on thin 3D domains was initiated in the three articles by Raugel and Sell, [59], [60], and [61]. Extensions of this thin domain theory appear in several sources, including: [13], [30], [31], [32], [35], [58], [74], [75], and [76]. This study of the Navier–Stokes equations on thin 3D domains is a part of a larger program in the general area of partial differential equations, a program concerned with the dynamics on thin domains. An excellent survey of the existing theory of thin domain dynamics for partial differential equations, appears in the recent paper by Raugel, [58]. Instead of repeating this historical account of the theory here, we will focus on the major mathematical themes that arise, especially in the context of Navier–Stokes equations, with the related implications for the oceanic equations. There are several basic mathematical issues which motivate much of the existing and current research into thin domain dynamics, as this theory applies to our problem. In the thin domain setting, the 3D domain = 3 is geometrically “close to” some 2D domain 2 . Therefore, one expects that the theory of the Navier–Stokes equations, or the oceanic equations, on 2 may play an important role in the 3D problem on 3 . However, when one rewrites the 3D equations as a perturbation of the 2D problem, one obtains a singular perturbation, see [30] and [59], for example. In order to remove the singularity, one needs to regularize the perturbation with an appropriate change of variables. The success of this regularization process—as it is now understood—depends heavily on some geometric properties of the domains (such as the shape of the domains and the nature of the thinness), as well as the specific boundary conditions used for the domain 3 . In terms of the notation used in (4.11), the change of variables used for the thin domain study has the form v = V + W . In this decomposition, V denotes an appropriate spatial average, such as the one-dimensional average in the thin direction. In this way, V depends only on two spatial variables, and the time-evolution of V should be determined by a 2D problem on 2 . In addition, one wants the term W = v − V to satisfy some strong stability property which forces W = W (t) to decrease very rapidly in short time. Also, one wants this all to work well, when the initial datum (v0 , g) is large. As it stands now, the theory of thin domain dynamics has been successfully applied in [61] to the (nonautonomous) Navier–Stokes equations ∂t u + νA1 u + B1 (u, u) = G1 (θ + ωt),
where u(0) = u0 .
(5.1)
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This equation agrees with (2.19) (or the u-coordinate of (4.11)) with L1 ≡ 0. Nevertheless, we hope to see appropriate extensions of this theory to the oceanic equations (4.10), at least in the case where β > 0, see (3.37). We let S1 (θ, t)u0 denote the maximally defined strong solution of (5.1), where u0 ∈ H11 . Let > 0 denote the measure of thinness of the domain 3 . (For the oceanic problem one would use the aspect ratio: depth/expanse for .) The typical theorems one finds in the Navier–Stokes literature are the following: • For small, there is a very large set B1 in H11 such that for all (θ, u0 ) ∈ T k+1 × B1 the strong solution S1 (θ, t)u0 is globally regular. • There is an attractor A1 for the strong solutions with A1 ⊂ T k+1 × B1 , and (θ · t, S1 (θ, t)u0 ) → A1 ,
as t → ∞,
for all (θ, u0 ) ∈ T k+1 × B1 . In particular, A1 ⊂ M, see (4.14). • A1 is the global attractor for the weak solutions of (5.1). • A1 is the global attractor for the globally regular strong solutions of (5.1). The methodology used in [61] is useful in establishing the results given here dealing with the attractor A, when the QPA is in play. While rigorous proofs are still wanting, it is our expectation that a similar theory will be developed for the oceanic equations.
6. Climate modeling The connections between the mathematical theory presented above and the global climate of the Earth are described in some detail in [55]. However, there are some features which are very important and which we address here. In particular, we examine the concept of an allowable perturbation, the role of the Planets in the climate model, and the concept of Partial Averaging.
6.1. Allowable perturbations The mathematical model described here is an omnibus model, which is intended to cover climate issues on many time-scales. However, this model is much too large, as it stands, and one wants to be able to modify (and simplify) this model in such a way as to have only “minimal impact” on the longtime dynamics of the problem. This brings us to the concept of an “allowable perturbation”. The mathematical issues underlying allowable perturbations form a rather long story, which begins with the earlier works of Pliss and Sell, see [51], [52], [53], and [54]. We omit the details here and simply note that the allowable perturbations form a wardrobe with several cloaks. For the first cloak, a perturbation of the equations of motion with a small (local) C 1 -norm is allowable. (The point being that a small C 1 -perturbation preserves the underlying hyperbolic structures in the model, even in the infinite dimensional setting.) Sometimes one needs another cloak, such as an integrated C 1 -norm. The latter cloak has been used successfully to show that the spatial discretization that occurs in the Bubnov–Galerkin approximations is allowable. (For an application
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G.R. Sell Table 2 Planetary periods and frequencies. The approximate periods and frequencies of the various planets are given here. The unit of time used in Table 2 is one Earth-year Name
Period
Frequency
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto
0.2408 0.6152 1.0000 1.8808 11.862 29.457 84.020 164.77 248.40
4.1528 1.6255 1.0000 5.3169 × 10−1 8.4303 × 10−2 3.3948 × 10−2 1.1902 × 10−2 6.0691 × 10−3 4.0258 × 10−3
of this type of norm to the study of the Couette–Taylor flow, see [53].) Sometimes one needs to use a nonlinear change of variables, as is used in [55], to show that the method of Partial Averaging can be used to make an allowable perturbation that removes the timedependent daily rotation of the Earth from any climate model. (We note that the Coriolis force ' u, which is caused by the daily rotation of the Earth, is an autonomous term. It is not time-dependent.)
6.2. The role of the planets In the QPA we introduced a frequency vector ω = (ω0 , ω1 , ω2 , · · · , ωk ) for the quasi periodic dependence on time t. Each of these frequencies ωi is approximated by 1/Pi , where Pi is the nominal period arising in the solar system, see Table 2. For example, one has: t = 1 ⇔ time unit: day, year, century, etc. ω0 ⇔ daily axial rotation of Earth ω1 ⇔ monthly rotation of Moon
(6.1)
ω2 ⇔ yearly rotation of the Earth about the Sun ωm ⇔ the other planets, for 3 ≤ m ≤ 11. One of the interesting features of the Solar System is the near resonance caused by the two largest planets, Jupiter and Saturn. From Table 2, we see that the periods of these planets satisfy the relation 5 × PeriodJupiter ) 2 × PeriodSaturn . This implies that once every 59 Earth years, or so, these big fellows align themselves on the same side of the Earth and “pull hard” in that direction. Can one find evidence of this near resonance in the historical climate data? It would be nice to know.
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Table 3 Relative gravity forces for the ocean. The relative sizes of the gravity forces acting on the ocean appear in column 5 Name
Mass (in kg)
Axis (in km)
Eccen
Rel. gravity
Earth Sun Moon Jupiter Saturn Venus Mercury Uranus Mars Neptune Pluto
5.97 × 1024 2.00 × 1030 7.35 × 1022 1.90 × 1027 5.68 × 1026 4.87 × 1024 3.33 × 1023 8.72 × 1025 6.42 × 1023 1.02 × 1026 6.60 × 1023
1.50 × 108
0.0168
3.84 × 105 7.78 × 108 1.43 × 109 1.08 × 108 5.79 × 107 2.87 × 109 2.28 × 108 4.50 × 109 5.90 × 109
0.0549 0.0483 0.0559 0.0068 0.2056 0.0471 0.0933 0.0085 0.2494
1.00 6.04 × 10−4 3.31 × 10−6 2.13 × 10−8 1.90 × 10−9 1.16 × 10−9 9.84 × 10−11 7.23 × 10−11 7.14 × 10−11 3.43 × 10−11 1.30 × 10−13
In Table 3, which is taken from [55], we present the relative strengths of the gravity forces (acting on the waters of the oceans) for the Sun, the Earth, the Moon, and the other 8 planets. While these forces may seem small, one should keep in mind that, it is not only the norm of the gravity vector, but also the direction of this vector and the time-scales for the various climatic events, that are important, see [78]. For example, since the Sun and Moon affect the daily tidal activity of the ocean, the role played by these bodies in the El Niño events may be even more significant, see [66], for example.
6.3. Partial averaging The method of Partial Averaging is well-suited for the construction of allowable perturbations in the context of a climate model satisfying the QPA, see [55]. Our interest here is to give an example and to illustrate how this method might be used for constructing a climate model for the El Niño events. For this purpose, we now make a small change in notation. In particular, we write 8 θ = (θ0 ; θ ), where 8 θ = (θ0 , θ1 , θ2 , · · · , θk ) = (θ0 ; θ ) ∈ T k+1 θ = (θ1 , θ2 , · · · , θk ) ∈ T k . Similarly, we write the frequency vector as 8 ω = (ω0 ; ω), where 8 ω = (ω0 , ω1 , ω2 , · · · , ωk ) = (ω0 ; ω) ∈ Rk+1 ω = (ω1 , ω2 , · · · , ωk ) ∈ Rk .
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The equation (4.10) is now written in the form ∂t y + Ay = H(θ0 , θ, y)
(6.2)
∂t (θ0 , θ ) = (ω0 , ω), where H(θ0 , θ, y) = −B(y, y) − L(θ0 , θ )y + G(θ0 , θ ), and y replaces v. The El Niño model assume the form ∂t v + Av = F(θ1 , θ2 , v)
(6.3)
∂t (θ1 , θ2 ) = (ω1 , ω2 ), where F(θ1 , θ2 , v) = 0
1 1 0
···
1
H(θ0 , θ1 , θ2 , θ3 , · · · , θk , v) dθ0 dθ3 · · · dθk .
(6.4)
0
The integral with respect to dθ0 removes the daily rotation of the Earth from the model (6.3), see [55]. The multiple integral in (6.4) should be treated as an iterated integral. Each 1 1 1 of the single integrals: 0 · dθ0 , 0 · dθ3 , · · ·, 0 · dθ10 generates an allowable perturbation. The composition of these allowable perturbations reduces the full problem (6.2) to the El Niño model (6.3). We claim that the difference def
K(θ0 , θ, v) = H(θ0 , θ, v) − F(θ1 , θ2 , v) is an allowable perturbation for the El Niño events, see Table 3. By using F and K, the equation (6.2) can be rewritten as ∂t y + Ay = F(θ1 , θ2 , y) + K(θ0 , θ, y) ∂t (θ0 , θ ) = (ω0 , ω). A related, but simpler, El Niño model is studied in [66]. 7. Concluding remarks We have presented here a theory of partial differential equations, which we propose as the Foundations of Oceanic Dynamics. There are several other issues of climate modeling which are related. These issues include: other mathematical models, computational matters, stochastic disturbances, and other dynamical matters. 7.1. Other models The system of equations (1.1)–(1.3) is a commonly accepted starting point for other models of oceanic dynamics. (We will refer to these equations as the Navier–Stokes model.) For
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example, in the study of the interaction between the weather and the ocean, one may use the Navier–Stokes model as part of a jointly coupled atmospheric-oceanic model, see [17], [24], [42], [44], and [27]. Also see [41]. As noted above, we believe that a good Interface Model is a better way to handle the role of the atmosphere. In short, the trade winds on the ocean surface seem to be far more important for the climate than the tornados of Kansas, see [55]. Another class of oceanic models comes under the title of the “primitive equations”. An extensive analysis of the primitive equations (as they occur in geophysical fluids studies of the ocean and the coupled atmospheric-oceanic problem), along with an informative history of these equations, appears in Temam and Ziane [76]. Also see the recent study of Cao and Titi [11] wherein the global regularity of the strong solutions of the primitive equations is established. The primitive equations for the ocean arise as a simplification of the Navier–Stokes model. The main simplification leading to the primitive equations is the hydrostatic assumption, (7.1). In particular, one assumes that the pressure term p = p(ρ, θ, φ), which is now expressed in the spherical coordinates of the oceanic region , must satisfy ∂p = g0 = g0 (θ, φ), ∂ρ
in ,
(7.1)
see [76]. Another common simplification is that the gravity force G = Ge does not depend on either the Sun, the Moon, or the other planets. Thus G is an autonomous forcing term and the constant g0 appearing in (7.1) does not depend explicitly on time. Now the raison d’etre for the Navier–Stokes model is the role to be played in the study of the global climate of the Earth. Consequently, one seeks to study the longtime dynamics of heat transfer in the ocean. What can be said about heat transfer in the primitive equation model? In particular, is the perturbation of the Navier–Stokes model, that leads to the primitive equation model, an allowable perturbation, in the sense used in Section 6.1 and in [55]? That is to say, is this perturbation “small” in a relevant C 1 -norm? To the best of our knowledge, these questions have not been addressed in the existing literature on oceanic dynamics, which brings us to an open research area that needs to be studied. If it can be shown • that the primitive equations can be modified to handle the quasi periodic gravity field G(t, x) described above, and • that the resulting perturbation of the Navier–Stokes model is allowable; this certainly would be a very important contribution to oceanic dynamics. (In this connection, we believe, for example, that the paper of Smolarkiewicz, Margolin, and Wyszogrodzki [70] on non-hydrostatic models is a significant first step in this direction.) However, having said that, we must express our reservations about successfully showing that the unmodified hydrostatic assumption (7.1) results in an allowable perturbation of the Navier–Stokes model, even in the context of the autonomous oceanic problem (4.12). In particular, the hydrostatic assumption appears to rule out some essential physical properties that must be present in any meaningful mathematical model of heat transfer in the ocean. To be more precise, consider for example, the role of the Gulf Stream in the northern Atlantic Ocean. Because of the interactions between the related trade winds and the
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oceanic surface flow, one finds that the warm waters from the western equatorial zone of the Atlantic are carried to the northeast to warm the beaches and the climate of Europe. However, these waters which travel on the surface of the ocean must return—presumably at a lower level in the ocean—to the equatorial zone. The oceanic physicists have proposed a theory of a Great Conveyor Belt (GCB, for short) to explain these different currents, see http://science.nasa.gov/headlines/y2004/05mar_arctic.htm. The GCB theory is a thought experiment which describes the physical mechanisms of the up-welling of waters in the equatorial zone and the resulting down-welling in the polar zone. Both the heavy fresh-water rainfall in the equatorial zone and the resulting cooling of the water as it approaches the polar zone play a role in the GCB theory. It is then the resulting changes in the temperature and salinity gradients over the global ocean that form the driving force for the GCB. These changes in the temperature and salinity gradients directly affect the pressure gradient, see (2.34). At first glance, it appears that such changes in these gradients may be inconsistent with the hydrostatic assumption (7.1). If so, this would preclude the use of the hydrostatic assumption for the study of the longtime dynamics of heat transfer in the oceans of the Earth. R EMARK . There are, of course, many other aspects of oceanic dynamics which we do not address in this article. To be more specific, the theory we present here is valid for any value of the real scalar a0 used in defining the Coriolis force, see (1.1)–(1.3). However, it has been noted that when |a0 | is large, the geostrophic Coriolis force tends to introduce some helical patterns in the Navier–Stokes equations. For example, it is shown in [6] and [7] that the Navier–Stokes equations in some geometries have a 2D-like behavior. It is interesting to speculate on whether one can find similar behavior in the Navier–Stokes model of the ocean.
7.2. Computational issues There is of course, a multitude of computational issues that need to be addressed. With apologies to the many who have worked in this area, we cite only those references which we have used in preparing this article: [5], [9], [23], [24], [25], [26], [27], [33], [38], [46], [56], [57], [66], [70], and [78]. Because the model we propose involves the planetary motion coupled with the partial differential equations, one may need to do both these calculations in parallel. The references [56], [57], and [78] are especially helpful for the planetary computations. While it is shown in [53] that a spatial discretization of the partial differential equations in (4.10) by means of a high-order Bubnov–Galerkin approximation results in an allowable perturbation, it is not known to what extent other spatial discretizations—such as the finite element methods, the finite difference methods, or other spectral methods—have the same property. A possible extension of the theory of allowable perturbations, as formulated in [55], to other spatial discretization methods, is worthy of further study.
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7.3. Stochastic disturbances We take the following quote from [55]. There are several sources of the stochastic events, some being frequent events, and others rare. For example, volcanic activity on the ocean floor adds a source of heat to equations of the oceanic flow. (Too much heat.) On the other hand, a major volcanic eruption on the land, such as the great Tambora eruption in 1815 or the Krakatau eruption of 1883, can create a global blanket of volcanic ash which in turn disrupts the heating of the Earth’s surface by the Sun for many months. (Too little heat.) Similarly, earthquakes in the ocean can spawn tsunamis, as we have seen recently. These are examples of rare stochastic events. Frequent stochastic events would include the interaction between the oceans and the atmosphere which gives rise to the great storms: hurricanes, typhoons, and monsoons.
From the point of view of mathematical modeling, the theory of stochastic disturbances can be modeled in the general context of random dynamical systems, see [3]. Applications to mathematical climatology appear in many sources, see [5], [15], [16], [19], [24], and [38], for example. An interesting connection between longtime dynamics in stochastic and deterministic climate models appears in the definitive work of L. Arnold, [4]. In our approach to mathematical climatology, as developed herein, we maintain that the influence of the atmosphere, acting on the surface of the ocean, can be best modeled as a stochastic disturbance on the Interface Model. Thus the role of stochastic disturbances in the equations described above is via a stochastic boundary force. How such forces affect the global attractor of the oceanic dynamics generated by (1.1)–(1.3) is a topic deserving further study.
7.4. Other dynamical issues There are several questions which have arisen when the author was lecturing on this topic. Here are two examples. Q1. What about adding other periodic-like behavior to the model, such as the coming and going of sun-spots?
If the periodic-like behavior is regular enough to satisfy the QPA, perhaps with a higher dimensional frequency vector, there is no problem in adding such activity to the model. Q2. Why limit the time-dependent forcing to quasi periodic behavior only? Why not use almost periodic forcing, or forcing terms with more general dynamical properties?
1. There is no problem in making the step from quasi periodic to Bohr almost periodic forcing. This would mean only that one would replace the torus T k+1 with a finite dimensional solenoid, see [45], or perhaps an infinite dimensional almost periodic minimal set. The main issue is how such a forcing function may be related to climate modeling. 2. A related and rather interesting variation of the dynamical theory presented here is formulated in terms of almost automorphic functions, see [69]. Such functions are more general than, but closely related to, almost periodic functions. Furthermore, they possess a type of Fourier series theory, and they represent the most “general
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type” functions with such a theory. In addition, the hull of some almost automorphic functions, unlike the almost periodic functions, can have some very complicated dynamics, including chaotic behavior. This is certainly worthy of further study because it may help to explain some of the chaotic features seen in various climate models, see [81]. See [81] for more details on the underlying dynamics.
Acknowledgements We are sincerely grateful to the Institute for Mathematics and its Applications (IMA) at the University of Minnesota for its role in creating a stimulating scientific environment that led to a number of investigations and publications that have now culminated in this present work. In particular, we benefitted greatly during the 1990–1991 IMA program on Dynamical Systems and Their Applications and the 2001–2002 IMA program on Mathematics in the Geosciences. We also express our sincere appreciation to Ciprian Foias, Michael Ghil, Luan Hoang, Marta Lewicka, and Victor A. Pliss for their kind comments and their very helpful suggestions. We are especially grateful to the referee for an in-depth reading of an earlier version of the manuscript, for many helpful suggestions, which led to improvements in this paper, and for identifying some mis-statements.
References [1] [2] [3] [4]
R.A. Adams, Sobolev Spaces, Academic Press, New York (1975). H. Amann, On the strong solvability of the Navier–Stokes equations, J. Math. Fluid Mech. 2 (2000) 16–98. L. Arnold, Random Dynamical Systems, Monographs in Mathematics, Springer Verlag, Berlin (1998). L. Arnold, Hasselmann’s program revisited: the analysis of stochasticity in deterministic climate models, in: Stochastic Climate Models, Progress in Probability, 49, Birkhäuser, Basel (1999), 141–157. [5] L. Arnold, P. Imkeller and Y. Wu, Reduction of deterministic coupled atmosphere—ocean models to stochastic ocean models: a numerical case study of the Lorenz–Maas system, Dynamical Systems 18 (2003) 295–350. [6] A. Babin, A. Mahalov and B. Nicolaenko, 3D Navier–Stokes equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J. 50 (2001) 1–35. [7] A. Babin, A. Mahalov and B. Nicolaenko, Fast singular oscillating limits of stably-stratified 3D Euler and Navier–Stokes equations and ageostrophic wave fronts, in: Large-Scale Atmosphere–Ocean Dynamics, vol. 1, Cambridge Univ. Press, Cambridge (2002), 126–201. [8] A.S. Besicovitch, Almost Periodic Functions, Dover, New York (1932/1955). [9] D. Brower and M. Clemence, Methods of Celestial Mechanics, Academic Press, New York (1961). [10] A.V. Busuioc and D. Iftimie, A non-Newtonian fluid with Navier boundary conditions, J. Dynamics Differential Equations 18 (2006) 357–379. [11] C. Cao and E.S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmospheric dynamics, Ann. Math. (2007), to appear. [12] P.G. Ciarlet, Mathematical Elasticity, vol III, Part A, North Holland Publishing, Amsterdam (2001). [13] I. Chueshov, G. Raugel and A. Rekalo, Interface boundary value problem for the Navier–Stokes equations in thin two-layer domains, J. Differential Equations 208 (2004) 449–493. [14] P. Constantin and C. Foias, Navier–Stokes Equations, University of Chicago Press, Chicago (1988). [15] N.J. Cutland, Global attractors for small samples and germs of 3D Navier–Stokes equations, Nonlinear Anal. 62 (2005) 265–281. [16] N.J. Cutland and H.J. Keisler, Global attractors for 3-dimensional stochastic Navier–Stokes equations, J. Dynamics Differential Equations 16 (2004) 205–266.
Oceanic dynamics and climate modeling
403
[17] V.P. Dymnikov and A.N. Filatov, Mathematics of Climate Modeling, Birkhäuser, Boston (1997). [18] A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math. 377, Springer, New York/Heidelberg (1974). [19] F. Flandoli and B. Schmalfuss, Weak solutions and attractors for 3-dimensional Navier–Stokes equations with nonregular force, J. Dynamics Differential Equations 11 (1999) 355–398. [20] C. Foia¸s and R. Temam, The connection between the Navier–Stokes equations, dynamical systems, and turbulence theory, in: Directions in Partial Differential Equations, Academic Press, New York (1987), 55–73. [21] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964) 269–315. [22] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I and II, Springer, New York (1994). [23] M. Ghil, Cryothermodynamics: The chaotic dynamics of paleoclimate, Physica D 77 (1994) 130–159. [24] M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics, Springer-Verlag, New York (1987). [25] H. Gildor and E. Tziperman, Sea ice as the glacial cycles climate switch: Role of seasonal and orbital forcing, Paleoceanography 15 (2000) 605–615. [26] H. Gildor and E. Tziperman, A sea ice climate switch mechanism for the 100 kyr cycles, J. Geophys. Res. (2001). [27] S.M. Griffies, Fundamentals of Ocean Climate Models, Princeton Univ. Press, Princeton, NJ (2005). [28] P. Grisvard, Elliptic Problems in Non-Smooth Domains, Pitman, London (1985). [29] J.K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI (1988). [30] D. Iftimie, The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier–Stokes equations, Bull. Soc. Math. France 127 (1999) 473–517. [31] D. Iftimie and G. Raugel, Some results on the Navier–Stokes equations in thin 3D domains, J. Differential Equations 169 (2001) 281–331. [32] D. Iftimie, G. Raugel and G.R. Sell, The Navier–Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J., Bloomington (2007), to appear. [33] J. Imbrie and K.P. Imbrie, Ice Ages: Solving the Mystery, Harvard Univ. Press, Cambridge (1986). [34] R.A. Johnson, K.J. Palmer and G.R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal. 18 (1987) 1–33. [35] I. Kukavica and M. Ziane, Regularity of the Navier–Stokes equations in a thin periodic domain with large data, Discrete Continuous Dynamical Systems 16 (2006) 67–86. [36] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York (1963). [37] O.A. Ladyzhenskaya, On the dynamical system generated by the Navier–Stokes equations, J. Soviet Math. 3 (1975) 458–479. [38] H. Le Treut and M. Ghil, Orbital forcing, climates interactions, and glaciations cycles, J. Geophys. Res. (1983) 5157–5190. [39] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Gauthier Villars, Paris (1969). [40] J.L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer Verlag, New York (1972). [41] J.L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity 5 (1992) 1007– 1053. [42] J.L. Lions, R. Temam and S. Wang, Mathematical theory of the coupled models of atmosphere and ocean, J. Math. Pure App. 74 (1995) 105–163. [43] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel (1995). [44] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics 9, American Mathematical Society, Providence, RI (2003). [45] L. Markus and K.R. Meyer, Periodic orbits and solenoids in generic Hamiltonian systems, Amer. J. Math. 102 (1980) 25–92.
404
G.R. Sell
[46] M. Milankovitch, Mathematische Klimalehre und Astromoische Theorie der Klimaschwankungen, I, Handbuch der Klimataologie, Part A, Verlag Borntrager, Berlin (1930). [47] J. Moser, On the theory of quasi periodic motions, SIAM Review 8 (1966) 145–172. [48] J. Moser, Convergent series expansions for quasi periodic motions, Math. Ann. 169 (1967) 136–176. [49] C.L.M.H. Navier, Sur les lois de l’équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France 6 (1827) 369. [50] D. Norman, Chemically reacting fluid flows: strong solutions and global attractors, J. Dynamics Differential Equations 12 (2000) 273–307. [51] V.A. Pliss and G.R. Sell, Perturbation of attractors of differential equations, J. Differential Equations 92 (1991) 100–124. [52] V.A. Pliss and G.R. Sell, Approximation dynamics and the stability of invariant sets, J. Differential Equations 149 (1998) 1–51. [53] V.A. Pliss and G.R. Sell, Perturbations of normally hyperbolic invariant manifolds with applications to the Navier–Stokes equations, J. Differential Equations 169 (2001) 396–492. [54] V.A. Pliss and G.R. Sell, Perturbations of foliated bundles and evolutionary equations, Ann. Mat. Pura. Appl. 185 (2005/2006), S325–S388. DOI: 10.1007/s10231-004-0148-y. [55] V.A. Pliss and G.R. Sell, Planetary motion and climate modeling, preprint (2006). [56] G.D. Quinlan, Celestial Mech. Dyn., J. Astron. 58 (1994) 339. [57] T.R. Quinn, S. Tremaine and M. Duncan, Astrophysical J. 101 (1991) 2287. [58] G. Raugel, Dynamics of partial differential equations on thin domains, in: CIME Course, Montecatini Terme, Lecture Notes in Mathematics 1609, Springer Verlag (1995), 208–315. [59] G. Raugel and G.R. Sell, Navier–Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503–568. [60] G. Raugel and G.R. Sell, Navier–Stokes equations on thin 3D domains. II: Global regularity of spatially periodic solutions, in: Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. XI, Longman Sci. Tech., Harlow (1994), 205–247. [61] G. Raugel and G.R. Sell, Navier–Stokes equations on thin 3D domains III: Global and local attractors, in: Turbulence in Fluid Flows: A Dynamical Systems Approach, IMA Volumes in Mathematics and its Applications 55, Springer (1993), 137–163. [62] R.J. Sacker and G.R. Sell, Finite extensions of minimal transformation groups, Trans. Amer. Math. Soc. 190 (1974) 325–334. [63] R.J. Sacker and G.R. Sell, Lifting properties in skew product flows with applications to differential equations, Memoir Amer. Math. Soc. 190 (1977). [64] G.R. Sell, Differential equations without uniqueness and classical topological dynamics, J. Differential Equations 14 (1973) 42–56. [65] G.R. Sell, Global attractors for the three-dimensional Navier–Stokes equations, J. Dynamics Differential Equations 8 (1996) 1–33. [66] G.R. Sell and B. Cockburn, On the return of El Niño, under preparation. [67] G.R. Sell and Y. You, Dynamics of Evolutionary Equations. Applied Mathematical Sciences 143, Springer, New York (2002). [68] J. Serrin, Mathematical principles of classical fluid mechanics, in: Handbuch der Physics, vol. 8/1, Springer, Berlin (1959), 125–263. [69] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew product semiflows, Mem. Am. Math. Soc. 136 (647) (1998). [70] P.K. Smolarkiewicz, L.G. Margolin and A.A. Wyszogrodzki, A class of nonhydrostatic global models, J. Atmospheric Sciences 58 (2001) 349–364. [71] V.A. Solonnikov and V.E. Šˇcadilov, On a boundary value problem for a stationary system of Navier–Stokes equations, Trudy Mat. Inst. Steklov 125 (1973) 186–199. [72] R. Temam, Navier–Stokes Equations, 3rd edn, North-Holland, Amsterdam (1984). [73] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, New York (1988). [74] R. Temam and M. Ziane, Navier–Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996) 499–546.
Oceanic dynamics and climate modeling
405
[75] R. Temam and M. Ziane, Navier–Stokes equations in thin spherical domains, Contemporary Math. 209 (1997) 281–314. [76] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, in: Handbook of Mathematical Fluid Dynamics, vol. III, North-Holland, Amsterdam (2004), 535–657. [77] J. Valero, Attractors of parabolic equations without uniqueness, J. Dynamics Differential Equations 13 (2001) 711–744. [78] F. Varadi, B. Runnegar and M. Ghil, Successive refinements in long-term integration of planetary orbits, Astrophysical J. 592 (2003) 620–630. [79] R. Verfürth, Finite element approximation of incompressible Navier–Stokes equations with slip boundary conditions, Numer. Math. 50 (1987) 697–721. [80] W. von Wahl, The Equations of Navier–Stokes and Abstract Parabolic Equations, Friedr. Vieweg & Sohn, Braunschweig (1985). [81] Y. Yi, On almost automorphic oscillations, in: Differences and Differential Equations, Fields Inst. Commun. 42 (2004) 75–99.
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CHAPTER 7
Mathematical Properties of the Solutions to the Equations Governing the Flow of Fluids with Pressure and Shear Rate Dependent Viscosities∗ J. Málek Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83, 186 75 Prague 8, Czech Republic
K.R. Rajagopal Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA
Contents 1. Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. On the Navier–Stokes and non-Newtonian fluid . . . . . . . . . . . . . . . . 1.2. Literature supporting the dependence of viscosity on pressure . . . . . . . . . 1.3. Viscosity-pressure relationship within the explicit/implicit constitutive theory 1.4. A brief survey of mathematical results . . . . . . . . . . . . . . . . . . . . . . 2. Models and their physical attributes . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Models within the framework of implicit constitutive theories . . . . . . . . . 2.3. A thermodynamic framework for fluids with pressure dependent viscosities . 3. Mathematical analysis of the models . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Systems of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. On a datum for the pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. On analysis of models when ν = ν(p) . . . . . . . . . . . . . . . . . . . . . . 3.5. On analysis of models when ν = ν(p, |D(v)|2 ) . . . . . . . . . . . . . . . . . 4. Special solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The case (b1 , b2 ) = (0, 0) and ν(p, |D|2 ) = ν(p) . . . . . . . . . . . . . . . 4.2. The case (b1 , b2 ) = (0, g), g ∈ R and ν(p, |D|2 ) = exp p . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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∗ The contribution of J. Málek to this work is a part of the research projects MSM 0021620839 and LC06052
financed by MSMT. J. Málek thanks also the Czech Science Foundation, the project GACR 201/06/0321, for its support. K.R. Rajagopal thanks the National Science Foundation for its support. HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME IV Edited by S.J. Friedlander and D. Serre © 2007 Elsevier B.V. All rights reserved. 407
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Keywords: incompressible fluid, pressure dependent viscosity, shear rate dependent viscosity, implicit constitutive theory
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1. Introductory remarks What do we mean by a fluid? A meaningful discussion of the mathematical properties of the equations governing the flow of fluids first requires a proper understanding of what we mean by a fluid as well as a clear understanding of the nature of the specific fluid that we have in mind. Unfortunately, it is impossible to provide a definition for what one means by a “fluid” within a sentence or two, or for that matter even with the aid of an extended discussion, without the definition’s inadequacy being laid bare with an easy counterexample. Many of the definitions, including those in renowned dictionaries are circular; a fluid being defined as a material that flows and flow being defined as an innate property of the fluid. The Oxford English Dictionary defines a fluid as “A substance whose particles move freely among themselves, so as to give way before the slightest pressure”. This definition is far from satisfactory and does not reflect what is the common understanding of the term, for it is the inability to support a shear stress that is considered as the quintessential feature of a fluid. Though an idealization, we do model a fluid such as water as incompressible with infinite ability to sustain the pressure rather than to succumb to its slightest application. The whole field of hydrostatics is built on such a basis. No fluid is perfectly incompressible and a sufficiently large pressure will lead to a change of volume, but in most liquids such changes in volume are of little consequence (see Málek and Rajagopal [31] for a discussion of relevant issues). Also, things are not as simple as defining a fluid to be a body that is incapable of resisting a shear stress; we have to contend with what we mean by resistance and how that resistance is observed? What are the time, length and force scales that are being considered? At the molecular level we always have “flow” in that the molecules are moving about, even in the absence of applied shear stresses1 . Another common definition for a fluid is that it takes the shape of the container. Such a definition does not differentiate between a gas that expands to fill the container and a liquid that partially fills a container, nor does it address the issue of how much time it takes to attain the shape of a container. For our purposes here we will regard a body as a fluid if in the time scale of observation of interest, it undergoes a flow that is discernible to the naked eye due to the application of a shear stress (that can be measured with the aid of reasonably unsophisticated instruments; i.e., the forces in question are robust, not mere picoNewtons). Based on how they are constituted, different fluids respond differently to the application of external stimuli. Many mathematical models have been developed to describe the diverse response exhibited by fluids and it would be fair to say that the Navier–Stokes fluid model (also referred to as the Newtonian fluid model2 or the linearly viscous fluid model) enjoys a central place amongst them. 1 Even from the macroscopic point of view, if the time scale is sufficiently large, then all materials could be viewed as flowing due to the effects of gravity. Deborah remarks (see Old Testament, JUDGES V, 4) that “Even mountains quaked in the presence of the Lord”. The Hebrew sentence in question “Harim nazloo mipnay yahway” has been translated as: “Even mountains flow in the presence of the Lord”. While one cannot fail to recognize that the notion of quaking and flowing are quite different, the statement of Deborah is used to imply that everything flows. 2 The attribution of the nomenclature Newtonian Fluid to the Navier–Stokes fluid is not merely a far stretch, it is inappropriate. Truesdell [55] rightly remarks “Newton’s theories of fluids are largely false” and nowhere is there a discussion of the various subtleties involved in the development of the Navier–Stokes fluid even alluded to in his immortal book.
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1.1. On the Navier–Stokes and non-Newtonian fluid No fluid model has been scrutinized and studied by mathematicians, physicists and engineers as intensively as the Navier–Stokes fluid model. While this model describes adequately a large class of flows (primarily laminar flows) of the most ubiquitous fluids, air and water, it is inadequate in describing the laminar response of a variety of polymeric liquids, geological fluids, food products, and biological fluids, or for that matter the response of air and water undergoing turbulent flow. The departure from the behavior exhibited by the Navier–Stokes fluid is referred to as non-Newtonian behavior. Amongst the many points of departure, one that is encountered commonly is the dependence of viscosity on the shear rate (or to be more precise on the euclidean norm of the symmetric part of the velocity gradient), stress relaxation, non-linear creep, the development of normal stress differences in a simple shear flow and yield-like behavior (see Rajagopal [38] or Málek and Rajagopal [31]). When one confines attention to incompressible Navier–Stokes fluids, the Cauchy stress is given to within an arbitrary spherical part −pI, where p is referred to as the pressure and is the mean normal stress, and the viscosity that appears in the constitutively determined part is assumed to be constant. The compressible Navier–Stokes fluid is characterized by two material moduli, the bulk viscosity λ and the shear viscosity μ, and a spherical part −pI, both the moduli as well as p (referred to as the thermodynamic pressure) being functions of the density -. In general, the material moduli and the pressure can also depend on the temperature θ . To be precise, the Cauchy stress has the representation T = −p(-, θ )I + λ(-, θ )(tr D)I + 2μ(-, θ )D,
(1.1)
where D = D(v) =
1 2
T ∂v ∂v , + ∂x ∂x
(1.2)
v being the velocity, ( ∂v ∂x ) the velocity gradient. The relationship between thermodynamic pressure, temperature and the density is referred to as an equation of state. The ideal gas law is one such equation of state. If the functional relationship between p and - is invertible, then the moduli λ and μ can be expressed as functions of thermodynamic pressure. In the incompressible case, the Cauchy stress takes the form T = −pI ˆ + 2μ(θ )D,
(1.3)
where μ is a constant and pˆ is the mean normal stress. It is important to note that the viscosity in the case of an incompressible Navier–Stokes fluid does not depend on the pressure. In this article, we shall be interested in incompressible fluids whose viscosity depends on the mean normal stress (pressure) and the symmetric part of the velocity gradient (shear rate) and thus the model that we are interested in studying is a non-Newtonian fluid model. While in the case of the incompressible Navier–Stokes fluid the pressure is the mean normal stress, such is not the case in general. It would be more appropriate to consider the
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viscosity as depending on the mean normal stress3 than just the thermodynamic pressure, in the case of compressible fluids4 . In general compressible non-Newtonian fluids the mean normal stress would also involve kinematical variables such as the symmetric part of the velocity gradient. We shall not get into a detailed discussion of these issues here as we are only interested in discussing certain mathematical results concerning incompressible fluids in this article. A detailed discussion of the same can be found in the recent paper by Rajagopal [39].
1.2. Literature supporting the dependence of viscosity on pressure Stokes, in his celebrated paper on the motions of fluids and the equilibrium of solids [53], clearly recognized that the viscosity of fluids such as water could depend on the mean normal stress, as evidenced by his comments: “. . . If we suppose μ to be independent of the pressure . . . ” and “Let us now consider in what cases it is allowable to suppose μ to be independent of the pressure”. That is, according to Stokes, one cannot suppose μ to be independent of pressure in all processes that the fluid undergoes, and in this he was exactly on the mark over a century and half ago. Elastohydrodynamics is an example wherein one can sensibly approximate the fluid as being incompressible with the viscosity depending on the pressure. It is however worth emphasizing that there is a large class of flows, not those merely restricted to flows in pipes and channels wherein the viscosity can be assumed to be constant, so much so that this is the assumption that is usually made in fluid mechanics. The classical incompressible Navier–Stokes model bears testimony to the same. A considerable amount of experimental work has been carried out concerning the pressure dependence of the material moduli, and the pertinent literature prior to 1930 can be found in the authoritative book by Bridgman [10]. Andrade [3] proposed the following relationship between the viscosity μ, the pressure p, the density - and the temperature θ : B 2 (p + D- ) , μ(p, -, θ ) = A- exp θ 1 2
(1.4)
where A, B and D are constants. We note that Andrade [3] did not consider the possibility that the viscosity of the fluids which he experimented could depend upon the shear rate. We cannot be sure that the fluids that he tested did not posses a viscosity depending on the shear rate; the experiments that he carried out are inadequate to speak to this matter. Also, Vogel [59] had recognized a dependence of the viscosity on the temperature, namely a , μ(θ ) = μ0 exp b+θ
(1.5)
3 An even more general possibility is that the viscosity depend on the stress and not just on the mean normal
stress. In fact, Saal and Koens [48] assumed that the viscosity of asphaltic bitumen depends on both the shear and normal stresses. 4 Note that in (1.1) p is the mean normal stress if 3λ + 2μ = 0.
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where μ0 , a and b are positive constants. Interestingly while in Andrade’s model the viscosity becomes infinite as the temperature tends to zero, in Vogel’s model it attains a constant finite value. As early as 1893 Barus [6] suggested the following relationship for the viscosity for liquids: μ(p) = μ0 exp(αp),
α > 0.
(1.6)
Such an expression has been used for several decades in elastohydrodynamics where the fluid is subject to a wide range of pressures and consequently a significant change in the viscosity occurs (see Szeri [54]). There are several recent experimental studies that indicate that the pressure gets to be so large that the fluid is very close to undergoing glass transition and at such high pressures the Barus equation (1.6) becomes inappropriate (see Bair and Kottke [4]). In fact the viscosity varies even more drastically than exponential dependence (see Figure in Bair and Kottke [4]). Other formulae for the variation of the viscosity with pressure can be found in the literature but they invariably involve an exponential relationship of sorts (see Cutler et al. [13], Griest et al. [19], Johnson and Cameron [24], Johnson and Greenwood [25], Johnson and Tevaarwerk [26], Bair and Winer [5], Roelands [47], Paluch et al. [35], Irving and Barlow [23], Bendler et al. [7]). The precise relationship between the viscosity and the pressure is not of consequence, what is important is the fact that the viscosity depends on the pressure. The density changes in liquids such as water and the pressure correlates well with the empirical expression (see Dowson and Higginson [14])
0.6p - = -0 1 + , 1 + 1.4p
(1.7)
where -0 is density in the liquid as the pressure tends to zero. When liquids such as water and many organic fluids5 are subject to a wide range of pressures, say from 2 GPa to 3 GPa, it is found that while the density of the fluid varies (say 3 to 10%) slightly, the viscosity of the fluid can change by as much as a factor of 108 ! (see recent experiments of Bair and Kottke [4]). This suggests that it would be reasonable to model such fluids as incompressible fluids with the viscosity depending upon the pressure. We would be remiss if we did not emphasize that liquids are compressible and that the scatter amongst the compressibility of liquids can be quite large. As Bridgman points out, due to a certain pressure difference, while glycerin can have a change of volume of approximately 13.5%, mercury changes by only 4%. The marginal compressibility, that is the change of density due to a change in pressure decreases as the pressures increase, as can be inferred from (1.7). Despite recognizing the importance of the dependence of viscosity on pressure, the elastohydrodynamicists have failed to systematically incorporate the pressure dependence until recently. The classical Reynold’s approximation for lubrication which forms one of the 5 The variation of the viscosity of water is somewhat different from those of many organic liquids in that in certain range of pressure it exhibits anomalous response.
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corner stones of fluid mechanics is derived under the assumption that the viscosity is constant. This approximation has been subsequently generalized to the field of elastohydrodynamics. While the elastohydrodynamicist recognizes that the viscosity depends on the pressure, he merely substitutes this dependence after the approximation has been derived rather than incorporating the dependence of the viscosity, à priori, and subsequently deriving the approximations. Rajagopal and Szeri [43] have recently derived a consistent set of approximate equations that are the appropriate generalization of the celebrated Reynold’s lubrication approximation.
1.3. Viscosity-pressure relationship within the explicit/implicit constitutive theory If one starts with the assumption that the Cauchy stress T in a fluid depends on the density, ∂v ), then standard arguments in continuum mechantemperature and the velocity gradient ( ∂x ics lead to the following representation for the stress (see Truesdell [56], Serrin [50]): T = α1 (-, θ, ID , II D , III D )I + α2 (-, θ, ID , II D , III D )D + α3 (-, θ, ID , II D , III D )D2 , where ID = tr D,
1 II D = [(tr D)2 − tr D2 ], 2
III D = det D.
If one requires that the stress depend linearly on D, then one obtains equation (1.1), the classical compressible Navier–Stokes fluid, for which α1 (-, θ, ID , II D , III D ) = −p(-, θ ) + λ(-, θ )(tr D), α2 (-, θ, ID , II D , III D ) = μ(-, θ ),
α3 ≡ 0.
If we require that the fluid be incompressible, as well as linear in D, we then obtain the representation for the Cauchy stress in (1.3). If we ignore temperature effects, we obtain T = −pI + 2μD,
(1.8)
where μ is a constant. The notion of incompressibility is an idealization6 , all bodies being compressible. However, when the effects of compressibility are sufficiently small so as to be ignorable, we can use idealized models such as (1.8). Another interesting approximation that is made concerning liquids is that in isothermal processes they can only undergo isochoric motions but the motion need not be volume 6 Another idealization which is even more common, so much so that the assumption is hardly acknowledged, is that of the fluid being homogeneous. The motion of inhomogeneous fluids has not been clearly grasped in most of studies that purport to address the flows of such bodies (see Anand and Rajagopal [2] and Málek and Rajagopal [33]).
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preserving if the processes are not isothermal. In such bodies, the density and the determinant of the deformation gradient would be a function of temperature. The balance of mass, momentum and energy, within the context of such an approximation was discussed first by Oberbeck [34] and then by Boussinesq [9]. Several attempts have been made to provide a rigorous basis for such an approximation, the details of which can be found in Rajagopal et al. [40]. It is worth noting that if det F = f (θ ), where F is the deformation gradient, then it immediately follows that div v =
f (θ ) θ˙ , f (θ )
where the prime denotes differentiation with respect to the argument. At this juncture, it is worth observing that the constraint of incompressibility requires that all motions meet det F = 1.
(1.9)
It is customary in fluid mechanics to interpret the constraint to mean div v = 0.
(1.10)
This is in virtue of the relation d (det F) = (det F)(div v). dt Thus, in order for (1.9) and (1.10) to be equivalent, we need to assume a certain smoothness concerning F. It is also worth remarking that incompressibility does not mean that the density is the same constant everywhere in the body, it merely means that the density of a particular material particle remains fixed with respect to time. Thus, we can have inhomogeneous fluids that are incompressible wherein the density of different particles are different (see Truesdell [56], Málek and Rajagopal [33]). The fact that in the current configuration, the density, viscosity or some other property of the fluid particle occupying different points x in a three dimensional Euclidean space is not constant does not mean that the fluid is inhomogeneous. For a fluid to be inhomogeneous different particles P belonging to the abstract body need to have some property that is not the same7 . This fact becomes obvious when we consider a homogeneous fluid whose viscosity depends on the shear rate. Thus, 7 The current definition of what is meant by inhomogeneity needs to be re-examined in the light of biological matter that can grow and remodel. We shall not get into these issues here but use the notion of inhomogeneity as it is currently used (see Truesdell [56]).
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while the viscosity might be different from point to point, at a some time t when it is being sheared it might have a constant viscosity in the limit of zero shear rate, i.e., if μ = μ0 + μ1 (tr D2 )n , then in the limit of zero shear rate the fluid would have constant viscosity μ0 . In this article we are interested in discussing mathematical issues concerning the response of incompressible fluids whose Cauchy stress has the following representation T = −pI + 2μ(p, θ, |D|2 )D.
(1.11)
The model (1.11) is markedly different from the models (1.1) or (1.3) in that while relations (1.1) and (1.3) are explicit relationships between the stress and D, equation (1.11) is an implicit relationship between T and D of the form f(T, D, θ ) = 0.
(1.12)
Since we shall be interested in the flows of incompressible fluids, we have to ensure that the flows under consideration are isochoric. It follows from (1.11) and (1.10) that 1 ˆ T, θ, |D|2 )D, T = (tr T)I + μ(tr 3
(1.13)
which can be expressed in the form (1.12). We note that, in general, neither can T be expressed explicitly in terms of D nor vice-versa. A more general class of implicit relationships concerns rate type models given by the constitutive specification: (n)
(m)
f(-, θ, T, . . . , T , D, . . . , D ) = 0,
(1.14)
where the superscript (n) , (m) denote n, m material time derivatives (or some appropriate frame indifferent time derivatives). We note that the implicit rate type constitutive relation defined by Truesdell and Noll [57]: (n)
(n−1)
·
(n)
T = f(-, θ, T, . . . , T , F, F, . . . , F ),
(1.15)
does not include the model (1.12) as a special subcase, while it is a special case of (1.14). Saal and Koens [48] while developing models to describe the response of asphaltic bitumens assumed that the viscosity depends on the shear stress as well as the normal stresses thereby introducing a truly implicit constitutive theory. If f in (1.11) is sufficiently smooth, we can then obtain ∂f ˙ ∂f ˙ ∂f T+ D+ θ˙ = 0. ∂T ∂D ∂θ The above model belongs to the class of implicit models that have the form ˙ + BD ˙ + Cθ˙ = 0, AT
(1.16)
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where A(T, D, θ ) and B(T, D, θ ) are fourth order tensors and C(T, D, θ ) is a second order tensor (see Rajagopal [36] and [37] for further details). The difference between (1.12) and (1.16) lies in the fact that (1.16) may not be integrable. Appropriate constitutive choices for A and B lead to the class of fluids with shear rate and pressure dependent viscosity whose mathematical properties will be discussed in the second part of this chapter. One has to make constitutive choices for the fourth order tensors A(T, D, θ ) and B(T, D, θ ), and the second order tensor C(T, D, θ ). Let us for the moment ignore temperature effects. Suppose that we have a function μ = μ(tr T, |D|2 ). Consider % 1 2∂μ(. . .) A(T, D) := J − I ⊗ I − (D ⊗ I) 3 ∂(tr T) and % ∂μ(. . .) D⊗D , B(T, D) := − 2μ(. . .)J + 4 ∂(|D|2 ) where J is the fourth order identity tensor and ⊗ denotes the usual tensor product. On substituting the expressions for A and B given above into (1.16), and subsequently integrating, we will obtain 1 T = (tr T)I + 2[μ(tr T, |D|2 )]D + T0 , 3
(1.17)
where T0 is a constant tensor. If we require that the fluid should be in a spherical state of stress when at rest, then it follows that 1 T = (tr T)I + 2[μ(tr T, |D|2 )]D. 3
(1.18)
If we define p to be mean normal stress, i.e., 1 p = − tr T, 3 then we have T = −pI + 2[μ(p, ˜ |D|2 )]D.
(1.19)
It is important to note that implicit theories expressed in the form (1.12) do not give precedence to one amongst the many variables that appear in the implicit relation. The same status is accorded to the stress, kinematical and thermal quantities. Interestingly such is the case in the classical ideal gas law wherein one may choose to express the pressure,
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specific volume or temperature in terms of the other quantities and in the classical linear theory of elasticity wherein either the stress or the strain can be expressed in terms of the other variables. Implicit constitutive theories seem to be ideally suited for describing a fluid phenomenon that has defied the most creative and resolute of modelers, turbulence. Classical theoretical models that have been put into place approach turbulence by assuming that there is a “mean” field about which “fluctuations” take place, a closure being provided for the “fluctuations” by prescribing a constitutive theory. The mean flow is assumed to satisfy an explicit constitutive relation, the Navier–Stokes equations. On the other hand, implicit models accord the possibility of totally distinct constitutive relations in different regimes of flow, precisely what is needed in turbulence. This for instance would be ideal to describe phenomena such as “intermittancy”. Also, the fact that the invariants of the stress also appear in the constitutive modeling allows one far greater flexibility with regard to modeling. We shall not get into a discussion of issues concerning turbulence modeling here.
1.4. A brief survey of mathematical results To date there have been few mathematically rigorous studies concerning fluids with pressure dependent viscosity. To our knowledge, there is no global existence theory that is in place for both steady and unsteady flows of fluids whose viscosity depends purely on the pressure. Previous studies by Renardy [45], Gazzola [17] and Gazzola and Secchi [18] either addressed existence of solutions that are short-in-time and for small data or assumed structures for the viscosity that are clearly contradicted by experiments, namely ν(p)/p → 0 as p → ∞. Experiments suggest that the kinematic viscosity ν(p) is such that ν(p)/p → +∞ as p → ∞, a condition met by the formula (1.6) due to Barus [6]. The paper by Renardy [45] also presumes à priori that the solution ought to have certain properties while trying to establish the existence of the solutions. We discuss this in more detail later (see Section 3). Recently, there has been some resurgence of interest in studying the flows of fluids with pressure dependent viscosities. Hron et al. [21] in addition to obtaining some explicit solutions have obtained numerical solutions for the flow of such fluids in special geometries. Vasudevaiah and Rajagopal [58] considered the fully developed flow of a fluid that has a viscosity that depends on the pressure and shear rate and were able to obtain explicit exact solutions for the problem. The recent study by Kannan and Rajagopal [27] points to a very interesting feature concerning the flows of fluids with pressure dependent viscosities, namely the capability of such fluids to develop boundary layers in that the vorticity is concentrated by virtue of an increase in pressure due to gravitational effects and hence an increase in the viscosity. Málek, Neˇcas and Rajagopal [30], Hron, Málek, Neˇcas and Rajagopal [20] and Franta, Málek and Rajagopal [16] have established existence results concerning the flows of fluids whose viscosity depends on both the pressure and symmetric part of the velocity gradient in an suitable manner8 . Franta et 8 These studies suffer the same drawback of earlier studies in that the dependence of the viscosity on the pressure is of the form that implies that ν(p)/p → 0 as p → ∞, contrary to experimental findings. The exact solutions established by Hron et al. [21], Vasudevaiah and Rajagopal [58] and Kannan and Rajagopal [27] do not suffer from such a drawback.
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al. [16] established the existence of weak solutions for the steady flows of fluids whose viscosity depends on both pressure and the symmetric part of the velocity gradient, that satisfy Dirichlet boundary conditions. Earlier, Málek et al. [30] and Hron et al. [20] established global-in-time existence for unsteady flows of such fluids under spatially periodic boundary conditions. The extension of these results to flows in bounded domains subject to the Navier’s slip are due to Bulíˇcek, Málek and Rajagopal [12]. The organization of the chapter is as follows. In the next section we introduce the models that will be the object of our study, taking two points of view into account. First, we identify the models that will be considered within the framework of implicit constitutive theories. Then we derive the constitutive equation for the relevant form of the Cauchy stress using a suitable thermodynamic framework. In Section 3 we shall provide a brief survey of the mathematical results concerning the analysis of the selected models; both the steady flows of fluids satisfying Dirichlet boundary conditions and the unsteady flows of these fluids under spatially periodic or Navier’s slip boundary conditions will be considered. Finally, in Section 4 we shall list explicit solutions for the flows of fluids with pressure and shear rate dependent viscosities in special geometries. 2. Models and their physical attributes 2.1. Kinematics Let B denote the abstract body and let P ∈ B denote a material point. Let κr and κt be one-to-one mappings, referred to as placers, that place the body into the reference configuration κr (B) and the configuration at time t, κt (B), in a three-dimensional euclidean space. In elasticity theory, it is common to use the stress-free configuration as the reference configuration, while in classical fluids the current configuration is used as the reference configuration. Let κp(t) (B) denote the configuration that the body would attain when all the external stimuli that act on the body in κt (B) are removed. We shall refer to this configuration as the preferred natural configuration. The preferred configuration that the body attains depends on the process class that is permissible for the body under consideration. Thus, the body may attain a particular natural configuration if it is only allowed to undergo isothermal processes and another natural configuration if it is only subject to adiabatic processes, that is, the natural configuration attained depends on how the external stimuli are removed (for example instantaneously or slowly, etc.). We shall be interested in modeling the response of fluids whose current configuration is the natural configuration, i.e., removal of the external stimuli leaves the fluid in the configuration that it is in. The Navier–Stokes fluid is one such fluid. Since the placers are one-to-one mappings, one can introduce a mapping χ := χκR : κR (B) × R+ 0 → κt (B) such that x = χ(X, t)
for X ∈ κR (B), x ∈ κt (B).
If the placers are differentiable, the deformation gradient F is defined through F=
∂χ , ∂X
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and the velocity field v(x, t) is defined through v := vˆ (X, t) :=
∂χ . ∂t
Since the motion χ is invertible, it immediately follows that v = vˆ (X, t) = vˆ (χ −1 (x, t), t) = v˜ (x, t). Any (scalar) property ϕ associated with material point P ∈ B can be expressed as ϕ = ϕ(P , t) = ϕ(X, ˆ t) = ϕ(x, ˜ t). We can analogously define any vector or tensorial quantity. We introduce the following derivatives: ϕ˙ :=
∂ ϕˆ , ∂t
ϕ,t :=
∂ ϕ˜ , ∂t
∇X ϕ =
∂ ϕˆ , ∂X
∇x ϕ =
∂ ϕ˜ . ∂x
The symmetric part of the velocity gradient is denoted by 1 D = [∇x v + (∇x v)T ]. 2 2.2. Models within the framework of implicit constitutive theories Let us start by considering fluids described by implicit constitutive relations of the form (1.12). If we require the function f to be isotropic, then f has to satisfy the restriction f(QTQT , QDQT , θ ) = Qf(T, D, θ )QT
∀Q ∈ Q,
where Q denotes the orthogonal group. It immediately implies (see Spencer [52]) that α0 I + α1 T + α2 D + α3 T2 + α4 D2 + α5 (DT + TD) + α6 (T2 D + DT2 ) + α7 (TD2 + D2 T) + α8 (T2 D2 + D2 T2 ) = 0, where αi , i = 0, . . . 8 depend on the temperature and the invariants tr T, tr D, tr T2 , tr D2 , tr T3 , tr D3 , tr(TD), tr(T2 D), tr(D2 T), tr(T2 D2 ). We note that if 1 α0 = − tr T, 3 α1 = 1,
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and 1 α2 = −2μ(− tr T, |D|2 ) 3
(μ > 0),
we obtain the model 1 1 2 tr T I + 2μ − tr T, |D| D. T= 3 3
(2.1)
On defining 1 p := − tr T, 3 equation (2.1) reduces to T = −pI + 2μ(p, |D|2 )D,
(2.2)
the model whose mathematical properties we shall discuss in this chapter. We note that the model (2.2) automatically meets the constraint (1.10). That is, the model is only capable of undergoing isochoric motions; it describes an incompressible fluid. We obtained the model as a consequence of our constitutive choice for α0 . . . α8 , and not by imposing (1.10) as a constraint and thereby obtaining p as a Langrange multiplier that enforces the constraint. It is important to note that the classical approach that is employed in most continuum mechanics textbooks to enforce internal constraints is to require that they do no work, and splitting the stress into a constraint response stress TC and an extra constitutively determinate stress TE (see Truesdell [56]), i. e., T = TE + TC will not lead to the material moduli that appear in TE to depend on the Lagrange multiplier, in our case that mean normal stress p (see Rajagopal and Srinivasa [42]). However, models such as (2.2) appear naturally within the context of the implicit relation of the form (1.12).
2.3. A thermodynamic framework for fluids with pressure dependent viscosities A thermodynamic framework has been developed by Rajagopal and co-workers for describing the response of bodies that produce entropy as they undergo a thermodynamic process. We shall not discuss the framework in detail here, it suffices to say that the structure is sufficiently robust to describe the response of viscoelastic fluids and solids, inelastic solids, anisotropic fluids, shape memory alloys, as well as describe phenomena such as twinning of crystals and crystallization of polymers. The framework has been used to describe a variety of fluid models and amongst them are rate type fluids with shear rate and
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pressure dependent material moduli (see Rajagopal and Srinivasa [41] for the development of rate type fluids). Here, we shall follow the work by Málek and Rajagopal [32] for rate type fluids whose material moduli are pressure, shear rate and density dependent, and extend it to those fluids where the material moduli depends also on temperature. For simplicity, we deal with homogeneous fluids where the density is uniformly constant. For the sake of completeness, we record the balance laws for mass, linear and angular momentum and energy. The balance of mass in its local version takes the form ∂+ div(-v) = 0. ∂t For a homogeneous, incompressible fluid this equation is automatically fulfilled. The balance of linear and angular momentum (in the absence of body couples) are given by div TT + -b = -
dv , dt
(2.3)
and T = TT ,
(2.4)
respectively, where b is the specific body force and the superscript T denotes the transpose. The balance of energy is expressed as -
∂v d =T· − div q + -r, dt ∂x
(2.5)
where is the specific internal energy, q is the heat flux vector and r is the specific radiant heating. The term T · ( ∂v ∂x ) is usually referred to as stress-power and indicates the rate at which work is done due to the stresses. Note that forming the scalar product of (2.3) with v and adding the result to (2.5) one obtains |v|2 d + = div(Tv) + -b · v − div q + -r, dt 2
(2.6)
a form that is more suitable for the mathematical analysis of the corresponding partial differential equations. Equation (2.6) is in fact the local form of the energy equation which in its global form reads d dt
|v|2 dv = - + t · vda + -b · vdv − q · nda 2 Pt ∂Pt Pt ∂Pt + -rdv ∀Pt ⊂ κt (B), Pt
(2.7)
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J. Málek and K.R. Rajagopal
which on using the divergence theorem and (2.3), continuity arguments and arbitrariness of Pt , leads to (2.6). This equivalence is however not true anymore if we deal with weak solutions of problems where v is not the suitable test function of the weak formulation of the balance of linear momentum. See Feireisl and Málek [15] for details. The specific Helmholtz potential ψ is defined through ψ = − θ η,
(2.8)
where η denotes the entropy. We shall find it convenient to work with the specific Helmholtz potential in what follows. The second law of thermodynamics is usually expressed as the following inequality9 (the Clausius–Duhen inequality): -r q − ≥ 0. -η˙ + div θ θ We shall choose to express it in the form q -r -η˙ + div − := -ξ ≥ 0, θ θ
(2.9)
where ξ is the rate of entropy production per unit mass. It follows from (2.5) and (2.9) that T·
∂v q · ∇x θ − -˙ + -θ η˙ − = -θ ξ ≥ 0. ∂x θ
(2.10)
Let us define ζ := -θ ξ. If one restricts oneself to a body at uniform temperature θˆ undergoing an isothermal process, then (2.10) would reduce to T·
∂v − -ψ˙ = -θˆ ξ = ζ ≥ 0. ∂x
The quantity ζ which in this case is the product of the entropy production, the density and the constant temperature, is referred to as the rate of dissipation. That is, the rate of dissipation is the entropy production due to mechanical working multiplied by the density and temperature. The rate of dissipation denotes the amount of mechanical working that 9 There is considerable debate as to the proper form of the entropy production inequality, whether it should be required to hold locally or globally for the body in question or whether it should be formulated for the system as a whole. There is also the point of view that the inequality does not hold in the “very small”. We shall not discuss such issues here.
Fluids with pressure and shear rate dependent viscosities
423
is converted into energy in its thermal form (heat). In general, the product -θ ξ is not the rate of dissipation though it is usual to refer to it as such even when thermal effects are involved. In general, by virtue of (2.8) and (2.10) we have q · ∇x θ ∂v − -ψ˙ − -ηθ˙ − = ζ ≥ 0. T· ∂x θ
(2.11)
The above inequality is referred to as the reduced energy-dissipation inequality. A few words of caution concerning the reduced energy-dissipation inequality are warranted, especially in view of its blind and widespread use in continuum mechanics. In arriving at the inequality we have completely eliminated the radiant heating r, and have reduced two equations (or to be more precise one equation and one inequality) into a single inequality. This in fact creates serious difficulties. First, one cannot take into account the different responses which materials have with respect to radiant heating. We know from every day experience that a bowl of soup and a block of wood respond to radiant heating (say in a microwave oven) in very distinctive ways and this cannot be taken into consideration in the modeling. This might not be viewed as a deficiency with respect to what we are considering as radiant heating is altogether ignored in the classical Navier–Stokes theory. However, such an approach does not have universal applicability. We cannot forget that there would be no life on our planet, as we know it, if such radiant heating is not accounted for. Second, using (2.11) to obtain restrictions would lead to the undesirable result that the radiant heating ought to be exactly that which leads to the balance of energy, as we have no flexibility with any of the other terms that appear in the balance of energy. This is similar to requiring body forces to be what we would like them to be so that the stresses can be balanced. Nonetheless, we often neglect the effect of gravity to study a large class of problems, and this is appropriate provided we are sure that the effects of gravity are small with respect to other effects and thus they can indeed be ignored. An excellent example is the development of homogeneous deformations within the context of homogeneous compressible isotropic elastic bodies. No such deformations are possible in the presence of gravity, but such homogeneous solutions serve as excellent approximations in numerous technologically important problems. Neither of the objections recorded above seriously hamper our further discussion as the results obtained by ignoring radiant heating seem to be valid for a very large class of problems. However, we should bear the above objections in mind when we tackle problems wherein radiant energy is an important component (see Rajagopal and Tao [44] for a discussion of problems involving microwave radiation). Unlike the usual approach in thermodynamics of obtaining restrictions on constitutive relations for the stress and other relevant quantities by requiring that the reduced energydissipation inequality be met in arbitrary processes, we shall use the equality defined in (2.11) as the second law and assume a constitutive form for the rate of entropy production that automatically ensures that the second law is met. While we can indeed vary the mathematical functions arbitrarily we ought not to lose sight of the fact that these functions describe some property of a particular body of interest, and that the particular body in question might not be describable by the variables in question if one were to subject it to arbitrary processes. For instance, while a block of rubber may behave as an elastic
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J. Málek and K.R. Rajagopal
body in that there is no dissipation when the deformations are sufficiently slow, the same block of rubber would not behave as an elastic body if the deformations are sufficiently large and rapid. The rubber would crystallize and there would be entropy production due to the crystallization and the effect of the rate of strain would have to be taken into account to model such a block of rubber. Thus, the Helmholtz potential for rubber would depend only on the deformation gradient F, provided F˙ is sufficiently small. The same is true for all bodies, their constitutive relations hold only for an appropriate class of processes. This issue is of particular relevance in fluid mechanics. Arbitrary processes may make the flow turbulent and the constitutive equations which we assume in the first place may cease to be valid. We shall assume general forms for the specific Helmholtz potential, the heat flux vector and the rate of entropy production. Instead of turning a mathematical crank and considering arbitrary processes we choose to make appropriate choices for the constitutive relations based on some physical insight. Simply put, there is no substitute to good physical understanding. From amongst the class of admissible constitutive forms, admissible in that they meet all the balance laws and the reduced energy-dissipation equation in the form (2.11), we shall pick that which maximizes the rate of dissipation, subject to (2.11) as a constraint in addition to other constraints such as incompressibility. For the fluid under consideration, entropy is produced due to mechanical working as well as conduction. We shall assume that -θ ξ := ζ = ζW + ζC ,
ζW , ζC ≥ 0,
(2.12)
where ζW and ζC represent the rate of entropy production (times -θ ) with respect to the mechanical working and conduction, respectively. Furthermore, we shall assume that ζC = −
q · ∇x θ . θ
(2.13)
An appropriate choice has to be made for the constitutive expression for heat flux so that ζC is non-negative. We notice that q := −k(θ, T, D)∇x θ,
k(θ, T, D) ≥ 0,
automatically ensures that ζC ≥ 0. While in our discussions we have been primarily concerned with the effect of pressure on the viscosity of liquids, pressure is an extremely important factor in determining the response of liquids and solids (its importance with regard to gases being universally recognized). For instance one can find an extended discussion concerning the variation in the thermal conductivity of various organic liquids in pp. 307–329 in Bridgman’s treatise [10]. It is appropriate to point out that the coefficient of thermal expansion of liquids also changes with the pressure and details for the same can be found in Table III on page 133 in Bridgman’s book. We mention these in particular as a fully thermodynamic theory of liquids has to take these aspects of the liquid’s response into account. Pressure also affects a variety of other properties of fluids as well as solids that are not germane to
Fluids with pressure and shear rate dependent viscosities
425
this paper but we mention it none the less. It affects the solubility, electrical conductivity, dielectric constant, refractive index, plane of polarization, magnetic permeability, electromotive force, and even radioactive disintegration. While a great deal has been done since the authoritative treatise of Bridgman in 1931, it is yet a great source of information for the important role that pressure has to play in the response of materials. There seems to be no data, at least to our knowledge, on if, and if so how, the thermal conductivity depends on the shear rate. We shall thus assume that k = k(θ, p).
(2.14)
On substituting (2.13) into (2.11) we are left with T · L − -ψ˙ − -ηθ˙ = ζW .
(2.15)
We shall assume, as is common, that ψ = ψ(θ )
(2.16)
ζW = ν(θ, p, |D|2 )D · D.
(2.17)
and
Inserting (2.16) into (2.15) leads to ∂ψ + η θ˙ = ζW . T·L−∂θ This suggests that we set η=−
∂ψ ∂θ
(2.18)
which reduces (2.15) to T · L = ζW .
(2.19)
We shall now appeal to the assumption that constitutive choices are arrived at by maximizing the rate of dissipation subject to whatever constraint is appropriate. We thus need to maximize ζW with respect to all possible D with equations (2.19) and (1.10) appended as constraints. It is fairly straightforward to show that (see for example [31] for details) T = −pI + ν(θ, p, |D|2 )D.
(2.20)
It follows from (2.18) that ∂ ∂η ∂ 2ψ =θ = −θ 2 . ∂θ ∂θ ∂θ
(2.21)
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J. Málek and K.R. Rajagopal
The left hand side of (2.21) is usually referred to as the specific heat that can be mea∂ = cV , the specific heat at constant volume as we are enforcing the constraint sured. Here ∂θ of incompressibility. The specific heat cV is positive. If we assume that it is a constant then = cV θ
η = cV log θ
and
ψ = cV θ (log θ − 1).
(2.22)
The above model, i.e. (2.20) is capable of describing shear thinning or shear thickening as well as pressure thinning or pressure thickening. While there is considerable amount of evidence for pressure thickening, there is no evidence to date with regard to pressure thinning. The model (2.20), (2.22) is however incapable of describing stress relaxation, non-linear creep, the development of normal stress differences in simple shear, and yieldlike phenomena. The rest of the paper is devoted to the discussion of mathematical results concerning the flow of fluids modeled by (2.20).
3. Mathematical analysis of the models There is a significant need to understand the mathematical properties of the solutions to the equations governing the flows of incompressible fluids with the viscosity depending on pressure, temperature and shear rate, both due to their use in various areas of engineering applications, and due to difficulties that occur during numerical simulations of the relevant systems of partial differential equations (PDEs). This system of PDEs is of independent interest on its own in virtue of its structural simplicity on the one hand and the very complicated relation between the velocity and the pressure on the other hand that does not permit one to eliminate the pressure from the analysis of the problem by projecting the equations to the set of divergenceless functions, as it is frequently done in the analysis of NSEs and similar systems.
3.1. Systems of PDEs We consider the case when flows take place inside a fixed container. It means that κt (B) occupies for all t ≥ 0 the same open bounded set ⊆ Rd , i.e. κt (B) = for all t ≥ 0. We consider for simplicity the case when the boundary ∂ is smooth. On substituting (2.1) into the balance of linear momentum (2.3), and (2.14), (2.22) and r = 0 into the balance of energy (2.6), we obtain the system of governing equations10 −∇x p + div[μ(p, θ, |D(v)|2 )D(v)] + -b ¯ = -¯
dv dt
− div(pv) + div[μ(p, θ, |D(v)|2 )D(v)v] − div(k(θ, p)∇x θ ) |v|2 d cV θ + = -¯ dt 2 10 We deal with a homogeneous fluid having constant density -. ¯
Fluids with pressure and shear rate dependent viscosities
427
that holds in (0, T ) × . To the above set of equations we add the conservation of mass that reduces to the constraint (1.10) that implies that the flow is isochoric. It is convenient to divide these forms of the balance of linear momentum and energy by the constant positive value of the density -. ¯ Then, relabeling -ν¯ , p-¯ and k(θ,p) by ν, p and k, θ respectively, we can rewrite the above system as div v = 0,
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
S = ν(p, θ, |D(v)|2 )D(v)
v,t + div(v ⊗ v) − div S = −∇x p + b in (0, T ) × . |v|2 |v|2 ⎪ θ+ v − div(k(θ, p)∇x θ ) ⎪ + div p + θ + ⎪ ⎪ 2 ,t 2 ⎪ ⎪ ⎪ ⎭ − div(Sv) = 0 (3.1) If the flows are steady the system reduces to the form div v = 0,
⎫ ⎪ ⎪ ⎪ ⎪ ⎬
S = ν(p, θ, |D(v)|2 )D(v)
div(v ⊗ v) − div S = −∇x p + b in . ⎪ ⎪ |v|2 ⎪ ⎭ − div(k(θ, p)∇x θ ) − div(Sv) = 0 ⎪ v · ∇x p + θ + 2
(3.2)
For flows that take place at uniform temperature the relevant systems simplify to div v = 0,
S = ν(p, |D(v)|2 )D(v)
4
div(v ⊗ v) − div S = −∇x p + b
in ,
(3.3)
and div v = 0,
S = ν(p, |D(v)|2 )D(v)
v,t + div(v ⊗ v) − div S = −∇x p + b
4 in (0, T ) × ,
(3.4)
respectively. The system (3.1) should be rendered complete by prescribing proper initial conditions 2 for v and θ (or v and θ + |v|2 ). Since there are no results concerning either (3.1) or (3.2) in place11 , we will not delve any further in this direction and we will restrict further discussions to (3.3) and (3.4). Naturally, if v0 is a given initial velocity field, the relevant initial condition for the system (3.4) takes the form v(0, x) = v0 (x)
at almost all x ∈ and div v0 = 0 weakly.
11 Bulíˇcek [11] considers this case in his thesis written after the submission of this article.
(3.5)
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J. Málek and K.R. Rajagopal
3.2. Boundary conditions Here we shall restrict ourselves to a discussion of internal flows which meet12 v · n = 0.
(3.6)
This condition is considered on ∂ if we deal with (3.3), and on [0, T ] × ∂ if the evolutionary model (3.4) is studied. Regarding the tangential components of the velocity, we shall consider the no-slip boundary condition where vτ = v − (v · n)n = 0,
(3.7)
or we take slip into account, according to what is usually referred to as Navier’s slip, i.e., (Sn)τ + αvτ = 0.
(3.8)
We shall also present results corresponding to solutions of (3.4) that are spatially periodic and thus we shall assume that (v, p) : (0, T ) × Rd → Rd × R are L-periodic for each spatial variable and that v(x, t) dx = 0, := (0, L)d .
(3.9)
3.3. On a datum for the pressure Unlike the classical Navier–Stokes equations wherein one only encounters the gradient of the pressure, in the problem under consideration the actual value of the pressure appears as the viscosity depends on the pressure. Within the context of the mathematical framework that we are using, it does not make sense to merely prescribe p at a specific point. Thus, in order to fix the pressure, for stationary problems we use the condition 1 ||
p dx = p0 ,
(3.10)
where p0 ∈ R is given and || denotes the volume of , while for the evolutionary problems we incorporate the condition 1 ||
p(t, x) dx = Q(t)
for all t ∈ (0, T ),
where Q : (0, T ) → R is given. 12 Here, n = n(x) denotes the outer normal to the boundary at point x.
(3.11)
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429
When the viscosity does not depend on the pressure, the constant that fixes the pressure is irrelevant, such is not the case in the current situation as the value of the viscosity depends on the value of the pressure. Thus, the flow characteristics corresponding to a flow in pipe due to a pressure of 100 psi at the inlet and 99 psi at the outlet would be significantly different than the flow due to an inlet pressure of 105 + 1 psi and on outlet pressure 105 psi, although the pressure difference in the two cases is exactly the same. The effect of the pressure on the viscosity in the latter case is however significantly different than in the former. While, from physical considerations it might be best to fix the value of the pressure by knowing its value at one point, as we are interested in dealing with integrable functions fixing it on a set of measure zero is not meaningful. In view of this we fix the pressure by requiring it satisfies a certain mean value as defined through equation (3.10). One can also specify an average value for p over a set of non-zero measure. An alternate possibility of fixing the pressure is to prescribe the normal traction on a relevant portion of the boundary with non-zero area measure, or to require (besides of (3.7) or (3.8)) that p + (v · n)2 − Sn · n = g
on [0, T ] × 1 ,
(3.12)
1 being a part of ∂, g being given. If the boundary is fixed this is possible provided that the boundary is porous. In such a case however one cannot ask for v · n = 0 on [0, T ] × 1 , since there may be inflow and outflow through 1 . The mathematical investigation of the systems of PDEs completed by (3.12) is a subject of current research and we will not discuss it further in this chapter.
3.4. On analysis of models when ν = ν(p) Up until now there have been few studies dealing with the mathematical analysis of incompressible fluid models wherein the viscosity is pressure-dependent. We first provide a survey of observations related to (3.3) or (3.4) where however ν is a function only of the pressure, i.e., ν = ν(p).
(3.13)
Renardy [45] seems to be the first who dealt with theoretical analysis of (3.3) and (3.4). First, he asked the question whether a given velocity field uniquely determines the pressure and he showed that this happens if ∗ ν∞ := lim ν (p) < ∞,
(3.14)
p→+∞
i.e., the viscosity grows at most linearly with the pressure, and if eigenvalues of D(v) are strictly less then
1 . ∗ ν∞
(3.15)
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J. Málek and K.R. Rajagopal
The assumption (3.15) implies that the velocity gradients associated with the flow are sufficiently small. Let us remark that experimental results unequivocally indicate that ν (p) > 0 for all p ∈ R.
(3.16)
In fact, if the pressure is sufficiently high glass transition takes place. The condition (3.14) is evidently not fulfilled if the viscosity depends on the pressure exponentially, as is the case in (1.4) or (1.6). One can however argue that the viscosity can be truncated at some higher value of p0 % 1. Thus, for example, ν can be approximated by ∗
ν (p) =
ν(p)
if p ∈ (−∞, p0 ],
ν(p0 )
if p > p0 ,
(3.17)
or it can be extended smoothly at p0 , but with sublinear growth so that (3.14) holds. A much more severe restriction seems to be the condition (3.15) since not all velocity fields are admissible. Working with the higher order Sobolev spaces, and assuming the restriction (3.14) and (3.15), Renardy proves the existence and uniqueness of solution to (3.4). Since the initial data also has to fulfill (3.15), this is a result that is restricted to small-data. Also, since the energy estimate (b ≡ 0 for simplicity) 1 v(t)22 + 2
t 0
1 ν(p)|D(v)|2 dx dτ = v0 22 2
(3.18)
is available, it seems it would be more natural to construct a weak solution in the spaces determined by (3.18). Such a result is however not in place to our knowledge. Gazzola [17] while treating the evolutionary problem (3.4), and Gazzola and Secchi [18] when dealing with the stationary problem (3.3), established results without assuming (3.15). Their results are in some sense straightforward even if the treatment of the pressure requires an approach totally different from that used usually in Navier–Stokes theory where one frequently eliminates the pressure and works with the spaces of functions that are divergenceless. The results established in [17] and [18] are an immediate consequence of the fact that only small initial conditions and almost potential external body forces (b ∼ ∇x g) 0 are considered; one can immediately observe that b = ∇x g and v ≡ 0 leads to the energy identity ( bv = g div v = 0) 1 v(t)22 + 2
t 0
ν(p)|D(v)|2 dx dτ = 0,
which implies that v ≡ 0, p = g is the unique trivial solution of the problem. One obtains a similar conclusion for the stationary problem. Thus, if b is close to ∇x g and (v0 is small for the time-dependent case) it is not “surprising” that in the case of (3.4), Gazzola obtains in virtue of all these restriction, the existence of solution only on a certain interval [0, T0 ], T0 determined by a suitable (small) norm of v0 . Málek, Neˇcas and Rajagopal [30] observed that the feature “to a given velocity field there is a uniquely defined pressure” can be achieved if one “allows” that the viscosity is also dependent on the shear rate in a suitable
Fluids with pressure and shear rate dependent viscosities
431
manner. The structure of the viscosities and the results established for such models will be discussed next. 3.5. On analysis of models when ν = ν(p, |D(v)|2 ) Málek, Neˇcas and Rajagopal [30] recognized that the dependence of ν on |D(v)|2 may help, particularly if such a dependence is sublinear. To see this, let us assume that there are p 1 , p 2 for a given v that fulfill (3.4). Then, by taking the divergence of (3.4) for (v, p 1 ) and (v, p 2 ) we come to the relation for the difference of the form p 2 − p 1 = (−')−1 div div([ν(p 1 , |D(v)|2 ) − ν(p 2 , |D(v)|2 )]D(v)) ∂ν(p 1 + δ(p 2 − p 1 ), |D(v)|2 ) −1 1 2 D(v)(p − p ) , = (−') div div ∂p which on simplifying leads to " " " ∂ν " p 1 − p 2 q ≤ max "" (Q, |D(v)|2 )D(v)""p 1 − p 2 q . Q,D(v) ∂p Thus, if " " " ∂ν " 2 " max" (Q, |D| )D"" < 1, Q,D ∂p we obtain p 1 = p 2 . This observation motivates the following assumptions. We assume that the viscosity ν is a C 1 -mapping of R × R+ 0 into R+ satisfying for some fixed (but arbitrary) r ∈ [1, 2] and d×d d×d all D ∈ Rsym , B ∈ Rsym and p ∈ R the following inequalities C1 (1 + |D|2 )
r−2 2
|B|2 ≤
r−2 ∂ν(p, |D|2 )Dij Bij Bkl ≤ C2 (1 + |D|2 ) 2 |B|2 , ∂Dkl
" " " ∂ν(p, |D|2 ) " " "|D| ≤ γ0 (1 + |D|2 ) r−2 4 , " " ∂p
(3.19) (3.20)
where γ0 is a positive constant whose value will be restricted in the formulations of the particular results. Since r = 2 is included in the range of parameters, we see that (3.19) (and naturally also (3.20)) includes the Navier–Stokes model. Also, if ν is independent of p then (3.20) is irrelevant and (3.19) is fulfilled by generalized power-law-like fluids. We refer to Málek and Rajagopal [31] for a survey of mathematical results related to these special cases. Note that our assumptions do not permit us to consider any model where the viscosity depends only on the pressure. The following forms of the viscosities fulfill the assumptions (3.19) and (3.20).
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J. Málek and K.R. Rajagopal
E XAMPLE 1. Consider for r ∈ (1, 2] νi (p, |D|2 ) = (1 + γi (p) + |D|2 )
r−2 2
,
i = 1, 2,
(3.21)
where γi (p) have the form (s ≥ 0) γ1 (p) = (1 + α 2 p 2 )−s/2 , (1 + exp(αp))−s γ2 (p) = 1 Then (3.19) holds with C1 = 2 (see [30] for details).
r−2 2
⎫ ⎪ ⎬
if p > 0 ⎪ ⇒ 0 ≤ γi (p) ≤ 1 (i = 1, 2). (3.22) ⎭ if p ≤ 0
(r − 1) and C2 = A
r−2 2
and (3.20) holds with γ0 = αs 2−r 2
E XAMPLE 2. Within the class of the viscosities of the type ν(p, |D|2 ) = γp (p)νD (|D|2 ) + ν∞ , with ν∞ positive, consider ν(p, |D|2 ) :=
γ3 (p) |D|2 + ε
+ ν∞
(3.23)
such that for some γ∞ , γ0 > 0 0 ≤ γ3 (p) ≤ γ∞
and
|γ3 (p)| ≤ γ0 .
(3.24)
Then (3.23) satisfies the assumptions (3.19)–(3.20) with parameters γ0 and r = 2. Setting ε = ν∞ = 0 and γ3 (p) = α(p) in (3.23), one obtains a model introduced by Schaeffer [49] in order to describe certain flows of granular materials. In order to clearly formulate the results that have been recently established for fluids with pressure and shear rate dependent viscosity, we first introduce the notation for particular problems and for suitable function spaces. We set ⎫ (Psteady )dir ⎪ ⎪ ⎪ ⎪ (Psteady )nav⎪ ⎪ ⎬ (Pevol )per ⎪ ⎪ ⎪ (Pevol )dir ⎪ ⎪ ⎪ ⎭ (Pevol )nav
for the problem consisting of
⎧ (3.3), (3.6), (3.10) and (3.7), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨(3.3), (3.6), (3.10) and (3.8), (3.4), (3.11) and (3.5), ⎪ ⎪ ⎪ ⎪ (3.4), (3.6), (3.11), (3.5) and 3.7), ⎪ ⎪ ⎩ (3.4), (3.6), (3.11), (3.5) and (3.8).
We write that ∈ C 0,1 if ⊆ Rd , d ≥ 2 is a bounded open connected set with Lipschitz boundary ∂. If in addition the boundary ∂ is a locally C 1,1 mapping then we write ∈ C 1,1 . Let r ∈ [1, ∞]. The Lebesgue spaces Lr () equipped with the norm · r and the Sobolev spaces W 1,r () with the norm · 1,r are defined in the standard way. If X is a Banach space then X d = X ? ×X× @A. . . × XB. The trace of a Sobolev function u is denoted d-times
Fluids with pressure and shear rate dependent viscosities
433
through tr u, if v ∈ (W 1,r ())d then tr v := (tr v1 , . . . , tr vd ). We introduce the subspaces of vector-valued Sobolev functions which have zero normal part on the boundary. Let 1 ≤ q ≤ ∞. We define 1,q
W0
·1,q
:= {v ∈ (C ∞ ())d ; supp v ⊂ }
1,q
,
1,q
W0,div := {v ∈ W0 ; div v = 0}, 1,q
Wn
:= {v ∈ (C ∞ ())d ∩ (C())d ; tr v · n = 0 on ∂}
1,q
·1,q
,
1,q
Wn,div := {v ∈ Wn ; div v = 0}, 1,q
q
Ln := {v ∈ Wn,div }
·q
.
We also introduce the notation for the dual spaces:
W −1,q := (W0 )∗ , −1,q
Wn
1,q
:= (Wn )∗ 1,q
−1,q
Wdiv
:= (W0,div )∗ , 1,q
−1,q
Wn,div := (Wn,div )∗ .
and
1,q
All the spaces introduced above are Banach spaces. Moreover, if 1 < q < ∞ then they are also reflexive and separable. The first theorem discusses the results dealing with stationary problems (Psteady )dir and (Psteady )nav .
T HEOREM 1. Let ∈ C 0,1 and p0 ∈ R be given. Let b ∈ Wn−1,r for the problem
(Psteady )nav or b ∈ W0−1,r for the problem (Psteady )dir . Assume that ν satisfies (3.19)–(3.20) with the parameters r and γ0 such that 2d
and
0 ≤ γ0 <
C1 , 2C∗ (, 2)(C1 + C2 )
(3.25)
where C∗ (, 2) is specified below. Then there exists a weak solution to the problem (Psteady )dir and (Psteady )nav such that 3 v∈
1,r W0,div 1,r Wn,div
for (Psteady )dir
for (Psteady )nav ⎧ 3d
r ⎪ ⎪ ,2 , for r ∈ ⎨ L () d +2 p∈ dr ⎪ 3d 2d ⎪ ⎩ L 2(d−r) () for r ∈ , , d +1 d +2
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and the weak formulation −(v ⊗ v, ∇x ϕ) + (ν(p, |D(v)|2 )D(v), D(ϕ)) + α
v · ϕ dS = (p, div ϕ) + b, ϕ ∂
(3.26)
is valid for all ϕ having the property13 ϕ∈
⎧ ⎨ W 1,q
0,div ⎩ W 1,q n,div
for (Psteady )dir for (Psteady )nav
where q = max r,
dr (d + 2)r − 2d
% =
⎧ ⎪ ⎪ ⎨r ⎪ ⎪ ⎩
dr (d + 2)r − 2d
3d ,2 , if r ∈ d +2 3d 2d . if r ∈ , d +1 d +2
The constant C∗ (, q) has relevance to the following problem: for a given g ∈ Lq () with zero mean value, find z by solving div z = g
in ,
z = 0 on ∂.
(3.27)
It is known (see Bogovskii [8] or Amrouche and Girault [1]) that there is a bounded linear 1,q operator B that maps Lq () into W0 (), for every q ∈ (1, ∞), such that z := B(g) solves (3.27). Particularly, we have z1,q = B(g)1,q ≤ C∗ (, q)gq . 3d , 2] the result for (Psteady )dir is established Comments concerning the proof: For r ∈ ( d+2 2d 3d ] and for (Psteady )nav can be deduced in Franta et al. [16]. The results for r ∈ ( d+1 , d+2 in a straightforward way from Bulíˇcek et al. [12] where the authors treat a more complicated evolutionary model (Pevol )nav . While it is not at all clear how to extend the result to (Pevol )dir , for steady flows the extension from (Psteady )nav to (Psteady )dir requires only the modifications in the definitions of relevant function spaces (see also Comments concerning the proof of Theorem 3). Partial regularity of weak solution to (Psteady )dir for d = 2, 3 is established in [29], see the announcement in [28]. This study initiated the analysis of a generalized Stokes system whereas the generalization consists in replacing the Laplace operator by a general elliptic operator of second order and by replacing the gradient of the pressure by a general first order operator. Such generalized Stokes systems are studied by Huy and Stará in [22].
1,q 13 Note that the boundary integral α ∂ v · ϕ dS = 0 for ϕ ∈ W0 .
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The second theorem deals with the problem (Pevol )per . For this purpose we denote Lrper 1,r and Wper the standard Lebesgue and Sobolev spaces of L-periodic functions. The Sobolev spaces contain only functions with zero mean value over the periodic cell := (0, L)d .
−1,r T HEOREM 2. Let d = 2, 3. Let ∈ C 0,1 , b ∈ Lr (0, T ; Wper ), v0 ∈ L2per,div and Q ∈ L2 (0, T ) be given. Assume that ν satisfies (3.19)–(3.20) with the parameters r and γ0 such that
⎧4 ⎪ if d = 2 ⎪ ⎨ 3,2 r∈ ⎪ 9 ⎪ ⎩ ,2 if d = 3 5
% 1 C1 . and γ0 = min , 2 4C2
Then there exists a weak solution to the problem (Pevol )per such that 1,r ), v ∈ C(0, T ; L2weak ) ∩ Lr (0, T ; Wper,div ⎧ −1,r ⎨ Lr (0, T ; Wper ) if d = 2, v,t ∈ 5r 5r −1, ⎩ 6 L (0, T ; Wper 6 ) if d = 3, 3 r if d = 2, L (0, T ; Lr ()) p∈ 5r 5r L 6 (0, T ; L 6 ()) if d = 3,
and the weak formulation
v,t , ϕ − (v ⊗ v, ∇x ϕ) + (ν(p, |D(v)|2 )D(v), D(ϕ)) = (p, div ϕ) + b, ϕ (3.28) is valid for almost all t ∈ (0, T ) and all ϕ having the property
ϕ∈
⎧ 1,r ⎨ Lr (0, T ; Wper ) ⎩
L
5r 5r−6
1, 5r (0, T ; Wper5r−6 )
if d = 2, if d = 3.
Comments concerning the proof: The result in three dimensions is established in [30], and the two-dimensional case is treated in [20]. The final theorem deals with the problem (Pevol )nav .
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T HEOREM 3. Let d = 2, 3. Let ∈ C 1,1 , b ∈ Lr (0, T ; Wn−1,r ), v0 ∈ L2n,div and Q ∈ L2 (0, T ) be given. Assume that ν satisfies (3.19)–(3.20) with the parameters r and γ0 such that ⎧ 3 ⎪ ,2 if d = 2 ⎪ ⎨ 2 C1 1 r∈ , and γ0 < ⎪ C (, 2) C + C2 8 # 1 ⎪ ⎩ ,2 if d = 3 5 where C# (, 2) appears in (3.30) below. Then there exists a weak solution to the problem (Pevol )nav such that 1,r v ∈ C(0, T ; L2weak ) ∩ Lr (0, T ; Wn,div ), r(d+2)
1,
r(d+2)
v,t ∈ (L r(d+2)−2d (0, T ; Wn r(d+2)−2d ))∗ , 3 r L (0, T ; Lr ()) if d = 2, p∈ 5r 5r L 6 (0, T ; L 6 ()) if d = 3, and the weak formulation v,t , ϕ − (v ⊗ v, ∇x ϕ) + (ν(p, |D(v)|2 )D(v), D(ϕ)) v · ϕ dS = (p, div ϕ) + b, ϕ +α
(3.29)
∂
is valid for almost all t ∈ (0, T ) and all ϕ having the property ⎧ ⎨ Lr (0, T ; Wn1,r ) if d = 2, ϕ∈ 5r 5r 1, ⎩ 5r−6 (0, T ; Wn 5r−6 ) if d = 3. L Moreover, if d = 2 and r = 2 then the weak solution is unique. The constant C# (, q) occurs in the solvability of the following problem: for q ∈ (1, ∞) and ∈ C 1,1 and an arbitrary ϕ ∈ Lq () with zero mean value to find g ∈ W 2,q () solving g(x) dx = 0 'g = ϕ in , ∇x g · n = 0 on ∂,
satisfying g2,q ≤ C# (, q)ϕq .
(3.30)
Comments: Even for steady flows, there is a remarkable difference in how the pressure is introduced. While for the problems where the viscosity is independent of pressure, the
Fluids with pressure and shear rate dependent viscosities
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pressure can be easily identified using for example de Rham’s theorem, the same method cannot be used for the problems with pressure dependent viscosity since one needs to have knowledge of the nature of the pressure à priori. There is also another crucial difference in introducing the pressure for the evolutionary NSEs and time-dependent models with non-constant viscosity. For the NSEs, we can identify the model with an evolutionary Stokes system, where the convective term is included in the right-hand side, and apply results concerning the Lp -estimates available for such systems (see [51]). For the models where ν is not constant (and may depend on p or |D(v)|2 ) an analogous theory for generalized Stokes system is not available thus far. In the article [12] (see Theorem 3), Bulíˇcek et al. show that the problem with Navier’s slip boundary conditions, contrary to that with no-slip boundary conditions, does not suffer such a deficiency and it is possible to introduce the pressure globally. Theorem 3 establishes the first result concerning long-time and large-data existence of weak solutions to any incompressible fluid model where the viscosity depends on the pressure and where flows take place in a bounded container. Theorem 3 covers several interesting results even for fluids whose viscosity is independent of pressure (as are for example Navier–Stokes or power-law fluids). We refer the reader to [12] for details. 4. Special solutions In this section we shall show that unidirectional plane flows corresponding to Couette or Poiseuille flow are possible only for special forms of viscosity. The aim is to show that these special solutions have markedly different characteristic than the corresponding solutions to the classical Navier–Stokes fluid. We illustrate this feature dealing with steady flows between two parallel plates. Similar features can be also shown in other geometries: for the flows in a pipe we refer to Vasudevaiah and Rajagopal [58] or Renardy [46], for example. It is not our aim here to discuss in detail exact solutions established for the flows fluids with pressure dependent viscosity in various geometries. Our aim is to merely emphasize the distinctly different nature of the solutions from that for the classical Navier– Stokes fluid. Consider plane steady flows between two parallel lines located at y = ±1 in a Cartesian coordinate system (x, y) of the form v(x, y) := (u(y), 0)
and
p(x, y).
(4.1)
Inserting (4.1) into (3.3) the governing equations reduce to px = (ν(p, |u |2 )u )y + b1 (x, y),
(4.2)
py = (ν(p, |u |2 )u )x + b2 (x, y).
(4.3)
At this juncture we distinguish two cases, b = 0 and b = (0, g), in order to point out that the presence of body forces (gravity) may have a significant impact on the solvability of the problem. More precisely, we shall show, among other results, that if b = 0 and ν(p) = α exp(p) then there is no solution to (4.1), while for the same ν and b = (0, g) the problem has smooth nontrivial solution.
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4.1. The case (b1 , b2 ) = (0, 0) and ν(p, |D|2 ) = ν(p) Considering (b1 , b2 ) = (0, 0) and ν(p, |D|2 ) = ν(p) we shall show that either p is independent of x or ν (p) is constant. The latter implies that ν(p) = αp + β. We study the consequences of this result. We note that the first part of this result was established by Renardy in [46]. Here, we reprove Renardy’s result using different arguments. With appropriate assumptions on b and ν the governing equations (4.3) take the form px = (ν(p)u )y = ν (p)py u + ν(p)u
,
(4.4)
py = (ν(p)u )x = ν (p)px u ,
(4.5)
which is equivalent to (just by substituting (4.5) to (4.4) and (4.4) into (4.5)) (1 − [ν (p)]2 [u ]2 )py = ν (p)ν(p)u u
,
(4.6)
(1 − [ν (p)]2 [u ]2 )px = ν(p)u
.
(4.7)
Then (4.7) implies px [ν (p)]2 [u ]2 − px = u
. ν(p) ν(p)
(4.8)
Since the right-hand side of (4.8) depends only on y, then necessarily px = C1 (y) + h(x, y), ν(p) px 2 = C2 (y) + h(x, y). [ν (p)] u ν(p)
(4.9) (4.10)
Substituting (4.9) into (4.10) we obtain [C1 (y) + h(x, y)](u )2 [ν (p)]2 = C2 (y) + h(x, y)
(4.11)
which leads to C1 (y)(u )2 = h(x, y)
1
− u (y) . (ν (p))2
(4.12)
Then (4.11) implies ν (p) = g(y). But this happens only if ν (p) ≡ α(> 0) or p is independent of x. In the latter case (4.5) then simplifies to py = 0 and (4.4) implies that u(y) = Ay + B where A, B are constants fixed by boundary conditions, it means we obtain the solution to the classical Couette
Fluids with pressure and shear rate dependent viscosities
439
problem for the Navier–Stokes fluid. The nontrivial result of our computation is given in the former case implying ν(p) = αp + β. Particularly, we have shown that in this geometry and if b = 0 then the relationship between viscosity and pressure cannot be exponential (or polynomial). Next we consider the subcase when ν (p) = α, namely ν(p) := αp. Then prescribing u(±1) = 0 (Poiseuille flow) one obtains the derivative of u (see [21] for details) as u (y) =
1 sinh(C0 y) , α cosh(C0 y)
C0 < 0,
(4.13)
1 cosh(C0 y) u(y) = , ln αC0 cosh(C0 )
(4.14)
which implies
p(x, y) = L exp(C0 x) cosh(C0 y),
L > 0.
For some set of parameters C0 the corresponding solutions are drawn in Figure 1. It is worth mentioning that for a viscosity of the form ν(p, |D|2 ) := αp|D|r−2 , that reduces to previous case by setting r = 2, we can show (see [21]) that unidirectional flows between plates where the upper plate moves with constant speed V subjected thus to u(−1) = 0 and u(1) = V allow for multiplicity of solutions. This type of non-uniqueness does not occur in either Navier–Stokes fluids or power-law fluids (with viscosity independent of pressure). For example, if r = 32 then there are three solutions (ui , p i ), i = 1, 2, 3, where (u3 , p 3 ) corresponds to the classical linear profile, and the analytic forms for i = 1, 2 are 2 1 2 ui (y) = 2 , + C (y + 1) − 0 α C0 Mi e2C0 y + 1 Mi e−2C0 + 1 p i (x, y) = L(Mi e2C0 y + 1)eC0 (x−y) ,
L > 0,
with Mi =
−C0 (αV − 2)(e4C0 + 1) − 2(e4C0 − 1) + (−1)i 2C0 (α 2 V − 2)e2C0 1
[(C0 (αV − 2)(e4C0 + 1) + 2(e4C0 − 1))2 − 4α 2 C02 (α 2 V − 2)e4C0 ] 2 × . 2C0 (α 2 V − 2)e2C0
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Fig. 1. Velocity profiles for different values of C0 = −0.1, −1, −10 and −100.
These solutions are depicted for C0 = −1.9 in Figure 2. It is worth mentioning that the pressure that can be computed explicitly and consequently also the viscosity are positive although this was not required à priori.
4.2. The case (b1 , b2 ) = (0, g), g ∈ R and ν(p, |D|2 ) = exp p For simplicity we also assume that p = p(y). Under all these assumptions (4.4) and (4.5) simplify to (exp(p(y))u (y))y = 0,
(4.15)
py = g.
(4.16)
Since g is a (positive) constant (4.16) implies that p(y) = gy + C, where C can be fixed, for example, by requiring that p(−1) = p0
or
1
−1
p(y) dy = 0.
(4.17)
In the first case we obtain p0 = −g + C, and consequently p(y) = g(y + 1) + p0 , while the second case implies C = 0, which leads to p(y) = gy. Then from (4.15) we have K exp(gy)u (y) = C0 or equivalently u (y) =
C0 exp(−gy), K
(4.18)
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Fig. 2. Couette flow. Velocity profiles (a, c) and pressure across the channel (b, d) for C0 = −1.9 and r = 32 ; there are two (of three) possible solutions.
where K = exp(g + p0 ) if we consider the first case in (4.17), and K = 1 if the second case in (4.17) is treated. The integration of the last equation leads to u(y) − u(−1) =
C0 1 (exp g − exp(−gy)). K g
Assuming u(−1) = 0 and V = u(1), we obtain from (4.19) that C0 =
KgV 2sinhg
and consequently u(y) =
g V e − e−gy . 2sinhg 2
(4.19)
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References [1] Ch. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J. 44 (119) (1) (1994) 109–140. [2] M. Anand and K.R. Rajagopal, A note on the flows of inhomogeneous fluids with shear-dependent viscosities, Archives of Mechanics 57 (2005) 417–428. [3] C. Andrade, Viscosity of liquids, Nature 125 (1930) 309–310. [4] S. Bair and P. Kottke, Pressure-viscosity relationships for elastohydrodynamics, Tribology transactions 46 (2003) 289–295. [5] S. Bair and W.O. Winer, The high pressure high shear stress rheology of liquid lubricants, J. Tribology 114 (1992) 1–13. [6] C. Barus, Isothermals, isopiestics and isometrics relative to viscosity, American Jour. Sci. 45 (1893) 87–96. [7] J.T. Bendler, J.J. Fontanella and M.F. Schlessinger, A new Vogel-like law: ionic conductivity, dialectric relaxation and viscosity near the glass transition, Phys. Rev. Lett. 87 (19) (2001) 1955. [8] M.E. Bogovski˘ı, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (5) (1979) 1037–1040. [9] J. Boussinesq, Théorie Analytique de la Chaleur, Gauthier-Villars (1903). [10] P.W. Bridgman, The Physics of High Pressure, MacMillan, New York (1931). [11] M. Bulíˇcek, Navier’s slip and evolutionary Navier–Stokes systems with pressure, shear-rate and temperature dependent viscosity, PhD thesis, Faculty of Mathematics and Physics, Charles University, Czech Republic (2006). [12] M. Bulíˇcek, J. Málek and K.R. Rajagopal, Navier’s slip and evolutionary Navier–Stokes systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J. (2006) (to appear). [13] W.G. Cutler, R.H. McMickle, W. Webb and R.W. Schiessler, Study of the compressions of several high molecular weight hydrocarbons, J. Chem. Phys. 29 (1958) 727–740. [14] D. Dowson and G.R. Higginson, Elastohydrodynamic Lubrication, The Fundamentals of Roller and Gear Lubrication, Pergamon, Oxford (1966). [15] E. Feireisl and J. Málek, On the Navier–Stokes equations with temperature-dependent transport coefficients, Differential Equations and Nonlinear Mechanics (2006), Article ID 90616, 14 pages (electronic). [16] M. Franta, J. Málek and K.R. Rajagopal, On steady flows of fluids with pressure- and shear-dependent viscosities, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2055) (2005) 651–670. [17] F. Gazzola, A note on the evolution of Navier–Stokes equations with a pressure-dependent viscosity, Z. Angew. Math. Phys. 48 (5) (1997) 760–773. [18] F. Gazzola and P. Secchi, Some results about stationary Navier–Stokes equations with a pressure-dependent viscosity, in: Navier–Stokes Equations: Theory and Numerical Methods, R. Salvi, editor, volume 388 of Pitman Res. Notes Math. Ser., Longman, Harlow (1998), 31–37. [19] E.M. Griest, W. Webb and R.W. Schiessler, Effect of pressure on viscosity of higher hydrocarbons and their mixtures, J. Chem. Phys. 29 (1958) 711–720. [20] J. Hron, J. Málek, J. Neˇcas and K.R. Rajagopal, Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities, Mathematics and Computers in Simulation 61 (2003) 297–315. [21] J. Hron, J. Málek and K.R. Rajagopal, Simple flows of fluids with pressure dependent viscosities, Proc. R. Soc. A 457 (2001) 1603–1622. [22] N.D. Huy and J. Stará, On existence and regularity of solutions to a class of generalized stationary Stokes problem, Comment. Math. Univ. Carolin. 47 (2006) 241–264. [23] J.B. Irwing and A.J. Barlow, An automatic high pressure viscometer, J. Phys. E 4 (1971) 232–236. [24] K.L. Johnson and R. Cameron, Shear behaviour of elastohydrodynamic oil films at high rolling contact pressures, Proc. Instn. Mech. Engrs. 182 (14) (1967) 307–319. [25] K.L. Johnson and J.A. Greenwood, Thermal analysis of an Eyring fluid in elastohydrodynamic traction, Wear 61 (1980) 355–374. [26] K.L. Johnson and J.L. Tevaarwerk, Shear behaviour of elastohydrodynamic oil films, Proc. R. Soc. Lond. A 356 (1977) 215–236. [27] K. Kannan and K.R. Rajagopal, A thermomechanical framework for the transition of a viscoelastic liquid to a viscoelastic solid, Mathematics and Mechanics of Solids 9 (2004) 37–59.
Fluids with pressure and shear rate dependent viscosities
443
[28] J. Málek, G. Mingione and J. Stará, Fluids with pressure dependent viscosity: partial regularity of steady flows, in: EQUADIFF 2003 Proceedings of the International Conference on Differential Equations, F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede and S. Verduyn Lunel, eds, World Scientific, Singapore (2005), 380–386. [29] J. Málek, G. Mingione and J. Stará, Fluids with pressure dependent viscosity: partial regularity of steady flows (2007), in preparation. [30] J. Málek, J. Neˇcas and K.R. Rajagopal, Global analysis of the flows of fluids with pressure-dependent viscosities, Arch. Ration. Mech. Anal. 165 (3) (2002) 243–269. [31] J. Málek and K.R. Rajagopal, Mathematical issues concerning the Navier–Stokes equations and some of their generalizations, in: Evolutionary Equations, vol. II, C. Dafermos and E. Feireisl, eds, Handbook of Differential Equations, Elsevier/North Holland, Amsterdam (2005), 371–459. [32] J. Málek and K.R. Rajagopal, Incompressible rate type fluids with pressure and shear-rate dependent material moduli, Nonlinear Analysis: Real World Applications 8 (2007) 156–157. [33] J. Málek and K.R. Rajagopal, On the modeling of inhomogeneous incompressible fluid-like bodies, Mechanics of Materials 38 (2006) 233–242. [34] A. Oberbeck, Ueber die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen, Ann. Phys. Chem. VII (1879) 271–292. [35] M. Paluch, Z. Dendzik and S.J. Rzoska, Scaling of high-pressure viscosity data in low molecular-weight glass-forming liquids, Phys. Rev. B 60 (1999) 2979–2982. [36] K.R. Rajagopal, On implicit constitutive theories, Appl. Math. 48 (2003) 279–319. [37] K.R. Rajagopal, On implicit constitutive theories for fluids, Journal of Fluid Mechanics 550 (2006) 243– 249. [38] K.R. Rajagopal, Mechanics of non-newtonian fluids, in: Recent Developments in Theoretical Fluid Mechanics, G.P. Galdi and J. Neˇcas, eds, volume 291 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, Essex (1993), 129–162. [39] K.R. Rajagopal, Remarks on the notion of pressure and implicit constitutive theories, Appl. Math. (2007), to appear. [40] K.R. Rajagopal, M. R˚užiˇcka and A.R. Srinivasa, On the Oberbeck–Boussinesq approximation, Math. Models Methods Appl. Sci. 6 (1996) 1157–1167. [41] K.R. Rajagopal and A.R. Srinivasa, A thermodynamic framework for rate type fluid models, J. NonNewtonian Fluid Mech. 88 (2000) 207–227. [42] K.R. Rajagopal and A.R. Srinivasa, On the nature of constraints for continua undergoing dissipative processes, Proc. R. Soc. A 461 (2005) 2785–2795. [43] K.R. Rajagopal and A.Z. Szeri, On an incosistency in the derivation of the equations of elastohydrodynamic lubrication, Proc. R. Soc. A 459 (2003) 2771–2786. [44] K.R. Rajagopal and L. Tao, Modeling of microwave drying process of aqueous dielectrics, Z. Angew. Math. Phys. 9 (2004) 305–312. [45] M. Renardy, Some remarks on the Navier–Stokes equations with a pressure-dependent viscosity, Comm. Partial Differential Equations 11 (7) (1986) 779–793. [46] M. Renardy, Parallel shear flows of fluids with a pressure dependent viscosity, J. Non-Newtonian Fluid Mech. 114 (2003) 229–236. [47] C.J.A. Roelands, Correlation aspects of the viscosity–temeperature–pressure relationship of lubricating oils, PhD thesis, Technische Hogeschool Delft, The Netherlands (1966). [48] R.N.J. Saal and G. Koens, Investigation into the plastic properties of asphaltic bitumen, J. Inst. Pet. 19 (1933) 176–212. [49] D.G. Schaeffer, Instability in the evolution equations describing incompressible granular flow, J. Differential Equations 66 (1) (1987) 19–50. [50] J. Serrin, Mathematical principles of classical fluid mechanics, in: Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin (1959), 125–263. [51] V.A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier–Stokes equations, Trudy Mat. Inst. Steklov. 70 (1964) 213–317. [52] A.J.M. Spencer, Theory of invariants, in: Continuum Physics I, A.C. Eringen, editor, Academic Press, New York (1971), 292–352.
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J. Málek and K.R. Rajagopal
[53] G.G. Stokes, On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids, Trans. Cambridge Phil. Soc. 8 (1845) 287–305. [54] A.Z. Szeri, Fluid Film Lubrication: Theory and Design, Cambridge University Press (1998). [55] C.A. Truesdell, A program towards Rediscovering the Rational Mechanics of the Age of Reason, Archive for the history of the Exact Sciences 1 (1960) 1–30. [56] C.A. Truesdell, A First Course in Rational Continuum Mechanics, volume 1, 2nd edition, Academic Press, New York (1991). [57] C.A. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, volume III, Handbuch der Physik, Springer-Verlag, Berlin–Heidelberg–New York (1965). [58] M. Vasudevaiah and K.R. Rajagopal, On fully developed flows of fluids with a pressure dependent viscosity in a pipe, Appl. Math. 50 (2005) 341–353. [59] H. Vogel, Die Bedeutung der Temperatur Abhangigkeit der Viscosität f.d. Beurteilung von Oelen, Zeit. Angew. Chem. 35 (1922) 561–563.
CHAPTER 8
The Navier–Stokes System in Domains with Cylindrical Outlets to Infinity. Leray’s Problem K. Pileckasa,b,∗ a Vilnius University, Faculty of Mathematics and Informatics, Naugarduko Str., 24, Vilnius, Lithuania b Institute of Mathematics and Informatics, Akademijos 4, Vilnius, Lithuania
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1. Chapter I. Function spaces and auxiliary results . . . 1.1. Function spaces and related inequalities . . . . . 1.2. Solvability of the divergence equation . . . . . . 2. Chapter II. Poiseuille type flows . . . . . . . . . . . . 2.1. Steady Poiseuille flows . . . . . . . . . . . . . . 2.2. Unidirectional time-dependent Poiseuille flow . 2.3. Generalized time-dependent Poiseuille flow . . . 2.4. Behavior of the Poiseuille flow as t → ∞ . . . . 2.5. Time-periodic Poiseuille flow . . . . . . . . . . 2.6. Final remarks . . . . . . . . . . . . . . . . . . . 3. Chapter III. Steady problems . . . . . . . . . . . . . . 3.1. Formulation of the problem . . . . . . . . . . . . 3.2. Construction of the flux carrier . . . . . . . . . . 3.3. The Stokes problem . . . . . . . . . . . . . . . . 3.4. The Steady Navier–Stokes problem . . . . . . . 4. Chapter IV. Time-dependent problems . . . . . . . . 4.1. Formulation of the problem . . . . . . . . . . . . 4.2. Construction of the flux carrier . . . . . . . . . . 4.3. The Time-dependent Stokes problem . . . . . . 4.4. The two-dimensional Navier–Stokes problem . . 4.5. The Three-dimensional Navier–Stokes problem 4.6. Uniqueness of the solution to problem (4.2) . . . 4.7. Remarks on weak Hopf’s solution . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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* The work is supported in part by Lithuanian State Science and Studies Foundation, T-05176.
HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME IV Edited by S.J. Friedlander and D. Serre © 2007 Elsevier B.V. All rights reserved. 445
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The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
447
Introduction The solvability of boundary and initial boundary value problems for Navier–Stokes equations is of high importance in the mathematical hydrodynamics. It has been studied in many papers and monographs (e.g. [28], [98], [20]). The existence theory developed there deals mainly with flows of viscous fluids in domains with compact boundaries (i.e., in bounded and exterior domains). Although some of these results do not depend on the shape of the boundary, many problems of scientific and practical interest related to flows of viscous incompressible fluids in domains with noncompact boundaries are still unsolved. Therefore, it is not surprising that during the last 30 years special attention was given to problems of such domains. The attention to the correct formulation of boundary value problems for Navier–Stokes equations in domains with noncompact boundaries has been drawn by J. Heywood. In 1976 he has shown [23] that in domains with noncompact boundaries the motion of a viscous fluid is not always uniquely determined by the applied external forces and usual initial and boundary conditions. Moreover, certain physically important variables (such as fluxes of the velocity field or limiting values of the pressure at infinity) have to be prescribed additionally. J. Heywood [23] has considered the aperture domain = {x ∈ R3 : x3 = 0 or
x3 = 0, x = (x1 , x2 ) ∈ S},
(0.1)
where S is a bounded domain in R2 . For such a domain it is shown in [23] that the Navier– Stokes problem (and even linear the Stokes problem) admits infinitely many solutions with finite Dirichlet integrals and the unique solution may be specified by prescribing either the total flux F (t) of the velocity field u(x, t) through the aperture S: u3 (x, t) dx = F (t), S
or the “pressure drop”: p ∗ = p + − p− =
lim
|x|→∞, x3 >0
p(x, t) −
lim
|x|→∞, x3 <0
p(x, t).
The existence under such conditions of a weak solution to the nonlinear Navier–Stokes problem was proved in [23] for small data. After the paper of J. Heywood the theory of Navier–Stokes equations in domains with noncompact boundaries has made great progress in a number of papers. Such problems were investigated in a wide class of domains having “outlets” to infinity2 . First, it was proved (see [30], [31], [94], [88], [25]) that looking for solutions with a finite Dirichlet integral it is necessary to prescribe additional conditions (fluxes over the sections of outlets to infinity or pressure drops) in all “wide” outlets j which grow at infinity “sufficiently rapidly”. The weak solvability of the steady Navier–Stokes problem and twodimensional time-dependent Navier–Stokes problem was proved for arbitrary data in papers mentioned above. For the three-dimensional time-dependent Navier–Stokes problem 2 I.e., in domains ⊂ Rn , n = 2, 3, splitting outside the ball {x : |x| < R } into J > 1 unbounded disjoint 0 components 1 , . . . , J which are called outlets to infinity.
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K. Pileckas
the unique solvability is obtained, of course, only for small data. Note that the aperture domain has two “wide” outlets to infinity. The results of J. Heywood [23] concerning nonlinear problems have been extended in [30], [31], [94], [88], [25] for the case of arbitrary large data. The next step in the mathematical theory of viscous incompressible fluids in domains with outlets to infinity was to consider the physical problems with prescribed fluxes in “narrow” outlets to infinity (for example, in pipes). However, since any divergence-free vector-field v(x) with finite Dirichlet integral has obligatory zero fluxes over the sections of “narrow” outlets to infinity, the usual energy estimates method becomes insufficient in this case. The Navier–Stokes problem with additionally prescribed fluxes in “narrow” outlets to infinity has to be studied in a class of functions having infinite Dirichlet integrals. The basic results concerning such problems were obtained by O.A. Ladyzhenskaya and V.A. Solonnikov [32], where a special technique of integral estimates (so-called “techniques of Saint-Venant’s principle”) has been developed and the existence of solutions having infinite Dirichlet integral was proved. The steady Navier–Stokes problem was considered in domains with the “parabolic-like” structure of outlets to infinity j , i.e., assuming that in certain coordinate systems the outlets j have the form [32] j = {x ∈ R3 : |x | < gj (x3 ), x3 > 1}.
(0.2)
The existence of a solution having prescribed fluxes Fj of the velocity field over sections of all outlets to infinity has been proved [32]. These solutions have either finite or infinite Dirichlet integral over j dependent of properties of the function gj . This result was obtained without any restrictions on the data, assuming only the necessary compatibility condition that the total flux is equal to zero, i.e. Jj=1 Fj = 0. In particular, the solvability of the steady Navier–Stokes problem was proved for arbitrary data in domains with cylindrical outlets to infinity and it was shown that for sufficiently small fluxes this solution is unique and exponentially tends (as |x| → ∞) in each cylinder j to the corresponding Poiseuille flow. Note that for domains with two cylindrical outlets to infinity the steady Navier–Stokes problem with prescribed flux F has been also studied by C.J. Amick [4], where a solution approaching the Poiseuille flow as |x| → ∞ was constructed for small |F |. The similar results were obtained in domains with “layer-like” outlets to infinity, i.e., with outlets j of the form [58]: j = {x ∈ R3 : |x3 | < hj (x ), |x | > 1}.
(0.3)
The time-dependent Navier–Stokes system in domains with outlets to infinity was studied in [33], [34], [90], [92], [93], where the existence of solutions with prescribed fluxes Fj (t) was proved. These solutions have finite or infinite energy integrals, dependent on the geometry of the outlets to infinity. In particular, if outlets are cylindrical, the energy integral is infinite. Note that the solvability of both two- and three-dimensional nonlinear time dependent Navier–Stokes problems was proved in [33], [34], [90], [92], [93] either for small data or for small time intervals. The global solvability of the two-dimensional
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
449
time dependent Navier–Stokes problem in domains with cylindrical outlets to infinity was mentioned in [93] as unsolved problem. V.A. Solonnikov has studied both steady and time-dependent Navier–Stokes problems in a very general class of domains with outlets to infinity ([89,90,92]). He developed an axiomatic approach for such problems without making assumptions on the shape of the outlets and imposing only certain general restrictions. The function spaces used in [89], [90], [92] are also very general. The method used by V.A. Solonnikov is similar to that used in [32]. However, in [89], [90], [92] the differential inequalities for energy integrals (due to the technique of Saint-Venant’s principle) are replaced by more convenient (in our opinion) difference inequalities. Many papers were devoted to the investigation of related questions, such as regularity, asymptotic behavior and uniqueness of solutions to the steady Stokes and Navier–Stokes problems in noncompact domains. It is evident that the behavior of solutions to Navier– Stokes problem as |x| → ∞ strongly depends on the geometry of outlets to infinity. Therefore studying the properties of solutions to Navier–Stokes problem in such domains it is necessary to specify geometrical properties of the outlets to infinity. It is convenient to study the problem in weighted function spaces which reflect the decay properties of the solutions as |x| → ∞. In this context we refer to papers [62], [64], [46] where steady Stokes and Navier–Stokes problems were studied in weighted function spaces assuming the “parabolic-like” structure of the outlets to infinity. Weights were replaced by the powers of functions gj (xn ) describing the shape of outlets to infinity (see (0.2)). Moreover, in [46] the precise asymptotic representation of the solution is constructed and justified 1−γ assuming, in addition, gj (xn ) = g0 xn j , 0 < γj < 1, g0 = const. We mention also the papers [47], [48], [50], [67], [80] where the asymptotic properties of solutions to steady Stokes and Navier–Stokes problems were studied in domains with outlets to infinity coinciding for large |x| with an infinite layer L = {x ∈ R3 : 0 < x3 < 1, x ∈ R2 }
(0.4)
and papers [12–15], [21], [42], [52], [84], [16–19], where the analogous questions were studied for the aperture domain (0.1) and in the domain having sector-like outlets to infinity. Finally, we mention the papers [59–61], [77], [78], [91], [45], [1], [75], [76], [24], [43], where certain existence theorems for regular solutions were proved in domains with striplike outlets to infinity and in domains with outlets having periodically varying sections. For more general two-dimensional domains with outlets to infinity certain results concerning the pointwise decay and asymptotic properties of the solutions were obtained in [5], [6]. Some of these papers deal with noncompact free boundary value problems and there arise additional difficulties due to the presence of an unknown boundary and more complicated boundary conditions. More detailed description of results mentioned above can be found also in my review papers [65], [66]. I would like to apologize in advance if I have missed some references, because the activity in different topics of mathematical fluid mechanics became very high during the last years.
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In this paper we study both steady and time-dependent Navier–Stokes equations in domains with cylindrical outlets to infinity, describing the motion of viscous incompressible fluid in a system of pipes.
Such problems are of a great importance both from applied and theoretical points of view. An example of an application can be flow of oil in a system of pipelines. Time periodic flows have also obvious applications in hemodynamics: blood flow simulation in large arteries and veins. In this paper we always assume that the walls of pipes are rigid. However, in reality arteries and veins have elastic walls and such problems have to be further investigated (concerning blood flow problems see, for instance, [55]). The special attention in the paper is given to so-called Professor J. Leray problem. Let be a “disturbed pipe” with two semi-infinite cylindrical ends: = 0 ∪ 1 ∪ 2 , where 0 is a bounded subset of , while j , j = 1, 2, coincide for large |x| with straight pipes #j , j = 1, 2, i.e., is a domain with two cylindrical outlets to infinity. The problem of determining a solution u(x) of steady Navier–Stokes system having a given flux F over cross-sections of and tending in each outlet to infinity j to the Poiseuille flow corresponding to the flux F and the pipe #j is known as Leray’s problem. This problem of Leray is one of the most challenging problems in the mathematical hydrodynamics. However, until now this problem is open (the problem seems to have been proposed by J. Leray himself to O.A. Ladyzhenskaya during his visit to St. Petersburg in 1958 (see [27])). The steady Poiseuille flow is an exact solution of steady Navier–Stokes system in an infinite straight pipe # = {x = (x , x3 ) ∈ R3 : x ∈ σ ⊂ R2 , −∞ < x3 < ∞} of constant cross-section σ and prescribed flux F over the cross-section σ . The Poiseuille flow is characterized by the fact that the associated velocity field has only one nonzero component U (x ) directed along the axis of # which depends only of variables x of cross-section σ .
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
451
In an obvious way, Leray’s problem could be generalized for domains with J > 2 outlets to infinity. Moreover, it is natural to study time-dependent analogues of Leray’s problem. However, in this case there already appears a problem of the definition of time-dependent Poiseuille flows. There are two ways of describing a Poiseuille flow, namely, either prescribing the axial pressure gradient (pressure drop) q(t) or the flux F (t) = σ U (x , t) dx . In the first case the problem reduces to an initial-boundary value problem for heat equation for U = U (x , t) with time-dependent forcing q(t). The solvability of such problems and properties of corresponding solutions are well known. If, conversely, we prescribe the flux F (t), we have to solve a more complicated nonstandard inverse parabolic problem for U (x , t) and q(t), where the relation between q(t) and F (t) in principle depends on the solution of the inverse problem. This paper consists of four chapters. Chapter I contains auxiliary matterials. We introduce function spaces and formulate necessary embedding theorems and multiplicative inequalities. We also present results concerning the problem of representing a given scalar function g(x) as the divergence of a vector-field u(x) which are important for further considerations. Chapters II–IV have “local” introductions, where we describe the results in detail and give additional references related to the material of these chapters. Moreover, in the end of each chapter we indicate some important open problems. In Chapter II we study the existence of steady and time-dependent Poiseuille flows in an infinite straight cylinder #. The results concerning the steady Poiseuille solutions in principle are well known, while the results concerning the evolutionary case are new and belong mainly to the author [74], [68], [70]. The exceptions are the results of Section 2.5 concerning the existence of time-periodic Poiseuille flow obtained by H. Beirão da Veiga [9] and by J.P. Galdi and A.M. Robertson [22]. Finally, in Section 2.6 we briefly discuss analogous questions for parabolic-like outlets to infinity having the form (0.2) and for an infinite layer (0.4). In Chapter III we study steady Stokes and Navier–Stokes problems in domains with cylindrical outlets to infinity. More detailed description of the results and the corresponding references are presented in the introduction to this chapter. Note that only for the steady case Leray’s problem is solved only for small fluxes. Finally, in Chapter IV time-dependent Stokes and Navier–Stokes problems are considered in domains with cylindrical outlets to infinity. For the time-dependent Stokes problem we prove the existence and uniqueness a solution with prescribed fluxes Fj (t) which tends in all outlets to the corresponding time-dependent Poiseuille flows both for two- and three-dimensional cases. The decay rate of the solution is dependent only on the decay rate of an external force and initial data. For the nonlinear Navier–Stokes problem the global unique solvability, i.e., for arbitrary data (in particular, for arbitrary fluxes) and the infinite time interval, is proved only in the two-dimensional case assuming that fluxes Fj (t) vanish as t → ∞. If limt→∞ Fj (t) = Fj , where Fj are constants, the global unique solvability is proved only under condition that Fj are “small”. For large Fj the existence of a unique solution is proved in two-dimensional domains for arbitrary data and arbitrary finite time interval (0, T ), T < ∞ (we will call such result long time solvability). In the case of three-dimensional domains only local solvability could be proved, i.e., either for “small” data or for “small” time intervals. The results presented in Chapter IV are obtained in author’s papers [69], [71–73], [79]. Note that the essential point in these proofs is the existence of time dependent Poiseuille flows (see Chapter II) that are exact solutions of Navier–Stokes system in a straight cylinder.
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This gives the possibility to construct a “flux carrier” V(x, t) which coincides for large |x|, x ∈ j , j = 1, . . . , J , with corresponding Poiseuille flows. Therefore the remainder left by V(x, t) in Navier–Stokes equations has compact support and for the resulting perturbed problem it is possible to obtain solvability by the usual methods of energy estimates.
1. Chapter I. Function spaces and auxiliary results In this chapter we introduce function spaces which are used in the paper and formulate certain embedding theorems and multiplicative inequalities which are important for further considerations. In particular, we introduce weighted function spaces in domains with cylindrical outlets to infinity, where the weight-function regulates the behavior of elements of these spaces as |x| → ∞. The problem of representing a given scalar function g(x) as the divergence of a vectorfield u(x) is of a great importance in mathematical hydrodynamics. Here we present results concerning this problem. More information about related results can be found in references presented in [20].
1.1. Function spaces and related inequalities 1.1.1. Definitions of basic function spaces By c, C, cj , j = 1, 2, . . . , etc., we denote different constants whose possible dependence of parameters a1 , . . . , am will be specified whenever it is necessary. In such a case, we shall write c = c(a1 , . . . , am ). We shall use the symbols c, C, etc., to denote constants whose numerical values or whose dependence on parameters is unessential to our considerations. In such case c may have several different values in a single computation. Let V be a Banach space. The norm of an element u in the function space V is denoted by u; V . Vector-valued functions are denoted by bold letters and the spaces of scalar and vector-valued functions are not distinguished in notations. The vector-valued function u = (u1 , . . . , un ) belongs to the space V , if ui ∈ V , i = 1, . . . , n, and u; V = ni=1 ui ; V . Let G be an arbitrary domain in Rn , n ≥ 1, with the boundary ∂G. As usual, denote by ∞ C (G) the set of all infinitely many times differentiable in G functions and by C0∞ (G) the subset of functions from C ∞ (G) with compact supports in G. For given integer nonnegative l and q > 1, Wql (G) indicates the Sobolev space of functions with the finite norm u; Wql (G) =
where Dxα =
∂ |α| α ∂x1 1 ...∂xnαn
l
1/q |D u(x)| dx α
q
(1.1)
,
|α|=0 G ◦
, |α| = α1 + · · · + αn , Wq0 (G) = Lq (G) and W lq (G) is the closure
l of C0∞ (G) in the norm (1.1). We shall write u ∈ Wq,loc (G), if u ∈ Wql (G ) for any bounded
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
453
subdomain G with G ⊂ G. L∞ (G) is a linear space of all real Lebesgue measurable functions defined in G with the norm u; L∞ (G) = ess sup |u(x)| < ∞. x∈G
¯ (l is an integer) is a Banach space of functions u(x) for which D α u(x) is bounded C l (G) ¯ is defined by the ¯ for all 0 ≤ |α| ≤ l. The norm in C l (G) and uniformly continuous in G formula ¯ = u; C l (G)
l
sup |D α u(x)|.
¯ |α|=0 x∈G
¯ is the subspace of C l (G) ¯ consisting of all functions whose derivLet δ ∈ (0, 1). C l+δ (G) ¯ C l+δ (G) is a Banach space with the atives up to the order l are Hölder continuous in G. norm ¯ = u; C l (G) ¯ + u; C l+δ (G)
|α|=l
|D α u(x) − D α u(y)| . |x − y|δ ¯ x,y∈G sup
Consider now functions dependent on x ∈ G and t ∈ (0, T ). Let GT = G × (0, T ), T ∈ (0, ∞]. W22l,l (GT ), l ≥ 0—an integer, is a Hilbert space of functions that have generalized derivatives Dtr Dxα with every r and α such that 2r + |α| ≤ 2l. The norm in W22l,l (GT ) is defined by the formula u; W22l,l (GT ) =
2l
j =0 2r+|α|=j 0
T
1/2
G
|Dtr Dxα u(x, t)|2 dxdt
.
W21,1 (GT ) and W21,0 (GT ) are spaces of functions with finite norms u; W21,1 (GT ) =
T 0
|ut (x, t)|2 + |u(x, t)|2 + |∇u(x, t)|2 dxdt
1/2
G
and u; W21,0 (GT ) = ◦
◦
T 0
|u(x, t)|2 + |∇u(x, t)|2 dxdt
1/2 .
G
1,0 1,1 1,0 1,1 W 2 (GT ) and W 2 (GT ) are subspaces of W2 (GT ) and W2 (GT ) consisting of function satisfying the condition u(x, t)|∂G = 0. C 2l+2δ, l+δ (G¯T ), l ≥ 0—an integer, δ ∈ (0, 1/2), is the Hölder space of continuous functions having in GT continuous derivatives Dxα with respect to x up to the order 2l and
454
K. Pileckas
continuous derivatives Dtr with respect to t up to the order l and the finite norm u; C
2l+2δ, l+δ
(GT )
=
2l
|Dxα u(x, t)| +
sup
|α|=0 (x,t)∈GT
+
l
|Dxα u(x, t) − Dyα u(y, t)|
sup
|α|=2l (x,t),(y,t)∈GT
+
sup |Dtr u(x, t)|
|r=0 (x,t)∈GT
sup
r=l (x,t),(x,τ )∈GT
|x − y|2δ
|Dtr u(x, t) − Dτr u(x, τ )| . |t − τ |δ
1.1.2. Embedding and multiplicative inequalities Inequalities and facts listed below are well known. The proofs of these results can be found in many books on Sobolev spaces (see, e.g., [3], [29]). L EMMA 1.1 (Poincaré inequality). Let G be bounded domain in Rn . Then for any u ∈ W21 (G) such that u(x)|S = 0, where S ⊂ ∂G, |S| = mes(S) > 0, holds the inequality 2 |u(x)| dx ≤ c |∇u(x)|2 dx. (1.2) G
G
The constant c in (1.2) depends only on G. ◦ If u ∈ W 12 (G), then |u(x)|2 dx ≤ G
1 μ1
|∇u(x)|2 dx,
(1.3)
G
where μ1 is the smallest eigenvalue of the problem −u(x) = μu(x),
u(x)|∂G = 0.
L EMMA 1.2 (Sobolev embedding theorem). Let G ⊂ Rn be a bounded domain and let u ∈ Wql (G), l ≥ 1, q > 1. If n ≥ ql,
r≤
qn , n − ql
then u ∈ Lr (G) and u; Lr (G) ≤ cu; Wql (G).
(1.4)
¯ with h ≤ (ql − n)/q and If n < ql, then u ∈ C h (G) ¯ ≤ cu; Wql (G). u; C h (G) The constants in inequalities (1.4) and (1.5) depend only on G.
(1.5)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
455
qn If n ≥ ql and r < n−ql , then the embedding operator I : Wql (G) +→ Lr (G) is compact. ¯ is compact. If n < ql, h < (ql − n)/q, then the embedding operator I : Wql (G) +→ C h (G)
L EMMA 1.3. If u ∈ W22l,l (GT ), then Dtr Dxα u(x, t) with 2r + |α| < 2l − 1 belongs to the 2l−2r−|α|−1 (G) and there holds the inequality space W2 2l−2r−|α|−1
Dtr Dxα u(·, t); W2
(G) ≤ cu; W22l,l (GT )
(1.6)
with the constant c independent of t ∈ [0, T ]. ◦
L EMMA 1.4 (Ladyzhenskaya inequalities). Let u ∈ W 12 (G). If G ⊂ R2 , then u; L4 (G)4 ≤ 2u; L2 (G)2 ∇u; L2 (G)2 ,
(1.7)
and if G ⊂ R3 , then u; L4 (G)4 ≤ (4/3)3/2 u; L2 (G)∇u; L2 (G)3 .
(1.8)
L EMMA 1.5. Let G ⊂ R3 be a bounded domain. For any u ∈ W21 (G) holds the multiplicative inequality u; L3 (G) ≤ cu; W21 (G)1/2 u; L2 (G)1/2 .
(1.9)
If u ∈ W22 (G), then the following multiplicative inequality u; L∞ (G) ≤ c∇u; L6 (G)1/2 u; L6 (G)1/2 ≤ c∇u; W21 (G)1/2 u; W21 (G)1/2
(1.10)
is valid. The constants in (1.9), (1.10) depend only on G. L EMMA 1.6. Let G ⊂ R2 be a bounded domain. For any u ∈ W21 (G) holds the multiplicative inequality u; L4 (G) ≤ cu; W21 (G)1/2 u; L2 (G)1/2
(1.11)
where the constant c dependents only on G. L EMMA 1.7. Let G ⊂ R2 be a bounded domain, T < ∞. Then the embedding operator I : W22,1 (GT ) +→ L4 (GT ) is compact. The proof of this lemma can be found in [28], Ch.1, Lemma 10.
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K. Pileckas
L EMMA 1.8. Let G ⊂ R2 be a bounded domain, T < ∞ and let {un (x, t)} be a weakly convergent in the space W22,1 (GT ) sequence of functions. Then
T
∇un (·, t) − ∇um (·, t); L4 (G)2 dt → 0 as n, m → ∞.
0
P ROOF. Since {un (x, t)} is weakly convergent in W22,1 (GT ), there exists a number A such that un ; W22,1 (GT ) ≤ A, ∀n ≥ 1. Moreover, the embedding W21,1 (GT ) +→ L2 (GT ) is compact, and thus, un − um ; L2 (GT ) → 0 as n, m → ∞. Applying (1.11), Young and interpolation inequalities, we get
T
∇un (·, t) − ∇um (·, t); L4 (G)2 dt
0
T
≤c 0
∇un (·, t) − ∇um (·, t); L2 (G)∇un (·, t) − ∇um (·, t); W21 (G)dt
T
≤ c1 δ 0
c2 δ ≤ c1 δ
+
c2 δ
T
∇un (·, t) − ∇um (·, t); L2 (G)2 dt
0 T 0
+
un (·, t) − um (·, t); W22 (G)2 dt
0
un (·, t) − um (·, t); W22 (G)2 dt T#
δ 2 un (·, t) − um (·, t); W22 (G)2
$ + c3 δ −2 un (·, t) − um (·, t); L2 (G)2 dt T ≤ c4 δ un (·, t) − um (·, t); W22 (G)2 dt 0
c5 T + 3 un (·, t) − um (·, t); L2 (G)2 dt δ 0 ≤ c4 δ un ; W22,1 (GT )2 + um ; W22,1 (GT )2 c5 T + 3 un (·, t) − um (·, t); L2 (G)2 dt δ 0 c5 T 2 ≤ 2c4 A δ + 3 un (·, t) − um (·, t); L2 (G)2 dt. δ 0
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
457
Therefore, for any given ε > 0 we can take δ = 4c εA2 and then chose n, m so large that 4 c5 T 2 dt < ε/2. Thus, for any ε we can find sufficiently large u (·, t) − u (·, t); L (G) n m 2 3 0 δ n, m such that T ∇un (·, t) − ∇um (·, t); L4 (G)2 dt ≤ ε. 0
1.1.3. Domains with cylindrical outlets to infinity and weighted function spaces Let ⊂ Rn , n = 2, 3, be a domain with J outlets to infinity, i.e. outside the sphere |x| = r0 the domain splits into J disjoint components j (outlets to infinity) which in some coordinate systems x (j ) are given by the relations j = {x (j ) ∈ Rn , x (j ) ∈ σj , xn > 0}, (j )
j = 1, . . . , J,
(1.12)
where x (j ) = (x1 , x2 ) for n = 3, x (j ) = x1 for n = 2 and σj ⊂ Rn−1 is a bounded (j ) domain, i.e., for xn > 0 outlets to infinity coincide with infinite pipes #j = {x (j ) ∈ Rn , (j ) x (j ) ∈ σj , −∞ < xn < ∞} (if n = 2, the outlets j coincide with infinite strips while cross-sections σj = (0, hj ) are bounded intervals). Such a domain we call domain with cylindrical outlets to infinity. We introduce the following notations: (j )
(j )
(j )
(j )
j k = {x ∈ j : xn < k}, ωj k = j k+1 \ j k , j = 1, . . . , J, ( ωj k = ωj k−1 ∪ ωj k ∪ ωj k+1 , j = 1, . . . , J, ' ' &C C&CJ J (0) = \ (k) = (0) j =1 j k , j =1 j ,
(1.13)
where k ≥ 0 is an integer. (j ) Denote β = (β1 , . . . , βJ ) and let Eβj (x) = Eβj (xn ) be smooth monotone weightfunctions in j such that Eβj (x) > 0,
a1 ≤ E−βj (x)Eβj (x) ≤ a2
∀x ∈ j ,
Eβj (0) = 1, (1.141 )
b1 Eβj (k) ≤ Eβj (x) ≤ b2 Eβj (k) |∇Eβj (x)| ≤ b3 γ∗ Eβj (x) lim
(j )
xn →∞
Eβj (x) = ∞,
∀x ∈ ωj k ,
∀x ∈ j ,
(1.142 ) (1.143 )
if βj > 0,
(1.144 )
where the constants a1 , a2 , b1 , b2 are independent of k and b3 is independent of βj . Simple examples of such weight-functions are (j ) β Eβj (x) = 1 + δ|xn |2 j
and
(j ) Eβj (x) = exp 2βj xn .
458
K. Pileckas
The conditions (1.141 ), (1.142 ) and (1.144 ) for these functions are obvious. The inequality (1.143 ) holds for the first weight-function with γ∗ = |βj |δ, and for the second one with γ∗ = |βj |. Below, in proofs of solvability for Navier–Stokes system, we need γ∗ to be “sufficiently small”. For the exponential weight-function this is the case, if we assume that |βj | is sufficiently small. Thus, in the case of exponential weight-functions we obtain a restriction on the on weight-exponents βj . In the case of the power weight-function, i.e., if (j ) Eβj (x) = (1 + δ|xn |2 )βj , this assumption can be satisfied taking sufficiently small δ and there are no restrictions on βj . Set 3 Eβ (x) =
1,
x ∈ (0) ,
(j ) Eβj (xn ),
x ∈ j , j = 1, . . . , J ,
(1.15)
and define in weighted function spaces. Let C0∞ () be the set of all functions from l (), l ≥ 0, the space of funcC ∞ () that are equal to zero for large |x|. Denote by W2,β tions obtained as a closure of C0∞ () in the norm l u; W2,β () =
l
1/2 Eβ (x)|D α u(x)|2 dx
|α|=0
0 (). and put L2,β () = W2,β l () and their If βj > 0, weight-indices βj show the decay rate of elements u ∈ W2,β (j )
derivatives as |x| → ∞, x ∈ j . For example, if Eβj (x) = exp(2βj xn ), βj > 0, then elements from L2,β (j ) vanish exponentially as |x| → ∞, x ∈ j . If Eβj (x) = (1 + (j )
(j )
δ|xn |2 )βj , then elements u ∈ L2,β (j ) vanish as a power of (1 + |xn |2 ). Obviously, l l W2,β () ⊂ W2l () ⊂ W2,−β ()
for βj ≥ 0, j = 1, . . . , J.
l () and W l () Note that for functions with compact supports the norms of spaces W2,β 2
are equivalent, i.e., if supp u(x) ⊂ , where is a bounded subdomain of , then l l c1 u; W2,β () ≤ u; W2l () ≤ c2 u; W2,β (),
where constants c1 and c2 depend on | | = mes . The norms in weighted function spaces on subdomains of are defined by the same formulas with the only difference that the integrals are taken over instead of , e.g. (j )
u; L2,βj (ωj k ) = Eβj (xn )u; L2 (ωj k ).
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
459
2l,l 1,1 1,0 Analogously, W2,β (T ) (l ≥ 0 is an integer), W2,β (T ) and W2,β (T ) are the spaces of functions obtained as closures of the set of all infinitely many times differentiable with respect to x and t functions equal to zero for large |x| in the norms
2l,l (T ) = u; W2,β
2l
T
j =0 2r+|α|=j 0
1,1 u; W2,β (T )
T
=
0
1/2
Eβ (x)|Dtr Dxα u(x, t)|2 dxdt
,
Eβ (x) |ut (x, t)|2 + |u(x, t)|2
1/2
+ |∇u(x, t)| dxdt 2
and 1,0 u; W2,β (T ) =
T
0
Eβ (x) |u(x, t)|2 + |∇u(x, t)|2 dxdt
1/2 ,
respectively. Finally, L2,β (T ) is the space of functions with the finite norm u; L2,β ( ) = T
T
1/2
2
Eβ (x)|u(x, t)| dxdt 0
.
We will need also a “step” weight-function ⎧ 1, ⎪ ⎨ (k) Eβ (x) = Eβj (xn(j ) ), ⎪ ⎩ Eβj (k),
x ∈ (0) , x ∈ j k , j = 1, . . . , J,
(1.16)
x ∈ j \ j k , j = 1, . . . , J.
(k)
It is easy to see that Eβ (x) = Eβ (x) for x ∈ (k) , " " "∇E (k) (x)" ≤ b3 γ∗ E (k) (x), β
β
and, if βj ≥ 0, then (k)
Eβj (x) ≤ Eβj (x)
∀ x ∈ .
(1.17)
460
K. Pileckas
Moreover, by the definition
∂ (j )
∂xl
Eβ(k) (x) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
l = 1, . . . , n, x ∈ (0) , l = 1, . . . , n, x ∈ \ (k) , l = 1, . . . , n − 1, x ∈ j k , j = 1, . . . , J,
0, 0, 0,
l = n,
∂ (j ) Eβ (x), ∂xl
x ∈ j k , j = 1, . . . , J.
Thus, (k)
supp ∇Eβ ⊂
J D
(1.18)
j k .
j =1 1 () which is equal to zero on ∂ holds the folL EMMA 1.9. For any function u ∈ W2,β lowing weighted Poincaré inequality
Eβ (x)|u(x)|2 dx ≤ c
Eβ (x)|∇u(x)|2 dx.
(1.19)
(j )
(k)
P ROOF. Since Eβ (x) depends only on xn in j and is equal to 1 in (0) , we get (1.19) applying Poincaré inequality (1.2) in the domain (0) and on the cross-sections σj : Eβ (x)|u(x)|2 dx
=
|u(x)| dx + 2
(0)
|u(x)| dx + 2
J
j =1 0
(0)
≤c
|∇u(x)|2 dx + c (0)
≤c
Eβ (x)|u(x)|2 dx
j =1 j
=
J
∞
(j ) Eβj (xn )
J
j =1 0
∞
(j )
|u(x)|2 dx (j ) dxn σj (j )
Eβj (xn )
|∇ u(x)|2 dx (j ) dxn
(j )
σj
Eβ (x)|∇u(x)|2 dx. (k)
R EMARK 1.1. Obviously, in (1.19) one may take (l) and Eβ (x) instead of and Eβ (x). The constant in the obtained weighted Poincaré inequality is the same as in (1.19) and does not depend on k and l.
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
461
1.2. Solvability of the divergence equation 1.2.1. Divergence equation with the right-hand side depending only on x bounded domain in Rn , g ∈ L2 (G) and g(x)dx = 0.
Let G be a
(1.20)
G
Consider in G the following problem: For given g ∈ L2 (G) satisfying condition (1.20) to find a vector-field w(x) such that div w(x) = g(x), (1.21) w(x)|∂G = 0. Note that (1.20) represents necessary compatibility condition and follows from (2.2) and the Gauss theorem. L EMMA 1.10. Let G ⊂ Rn be a bounded domain with the Lipschitz boundary ∂G, g ∈ L2 (G) and let there holds (1.20). Then problem (1.21) admits at least one solution w ∈ ◦ W 12 (G) satisfying the estimate ∇w; L2 (G) ≤ cg; L2 (G).
(1.22)
The constant c in (1.22) depends on G, but it is independent of g. Lemma 1.10 is proved in [30] (see also [56], [57]). L EMMA 1.11. Let G ⊂ Rn be a bounded domain with the Lipschitz boundary ∂G, g ∈ ◦ W 12 (G) and let there holds (1.20). Then problem (1.21) admits at least one solution w ∈ ◦
W 22 (G) satisfying the estimate w; W22 (G) ≤ cg; W21 (G)
(1.23)
with a constant c independent of g. This result is obtained in [11] using an explicit representation formula for the solution. It adapts for the divergence operator a well-known Sobolev’s representation formula. If the boundary ∂G and g(x) are more regular it is possible to find a solution w(x) which also has more regularity. L EMMA 1.12. Let G ⊂ Rn be a bounded domain, ∂G ∈ C l+2 , l ≥ 0. Then, for every given ◦ g ∈ W2l (G) satisfying (1.20), there exists at least one solution w ∈ W2l+1 (G) ∩ W 12 (G) of problem (1.21) obeying the estimate w; W2l+1 (G) ≤ cg; W2l (G).
(1.24)
462
K. Pileckas
¯ 1 , then Moreover, if G1 ⊂ G and supp g(x) ⊂ G ¯ 2, supp w(x) ⊂ G
(1.25)
where G1 ⊂ G2 ⊂ G. Lemma 1.12 is proved in [26]. Note that here we do not require g(x) to be equal to zero in the neighborhood of the whole boundary ∂G, i.e., boundaries ∂G1 , ∂G2 and ∂G can have nonzero intersections. (k) Let be a domain with J cylindrical outlets to infinity and Eβ (x) be a step weightfunction defined by (1.16). Investigating decay properties of the solutions to Stokes and Navier–Stokes problems we will need the following ◦
L EMMA 1.13. Let u ∈ W 12 (), div u(x) = 0 and u(x) · n(x) ds = 0,
j = 1, . . . , J.
(1.26)
σj ◦
Then there exists a vector-function W(k) ∈ W 12 () such that supp W(k) (x) ⊂
J D
(1.27)
j k
j =1
and (k)
div W(k) (x) = − div(Eβ (x)u(x)),
x ∈ .
(1.28)
Moreover, there holds the estimate (k) (k) E−β (x)|∇W(k) (x)|2 dx ≤ cγ∗2 Eβ (x)|u(x)|2 dx
≤ cγ∗2
(k)
Eβ (x)|∇u(x)|2 dx,
(1.29)
In (1.29) γ∗ is a constant from the inequality (1.143 ) for Eβ(k) (x). The constant c in (1.29) is independent of k and u(x). (k)
(k)
P ROOF. The support of the function div(Eβ (x)u(x)) = ∇Eβ (x) · u(x) is in the bounded C C domain Jj=1 j k (see (1.18)). Each j k can be represented as a union j k = k−1 s=0 ωj s . Moreover, in virtue of condition (1.26) we have ωj s
(k)
∇Eβ (x) · u(x) dx =
s
s+1
∂
(k)
(j )
E (xn ) (j ) βj
∂xn
un (x (j ) ) dx (j ) = 0. σj
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
463
◦
Therefore, by Lemma 1.10 there exist vector-fields wj s ∈ W 12 (ωj s ) such that div wj s (x) = −
∂
(k)
(j )
∂xn
(j )
Eβj (xn )un (x),
x ∈ ωj s ,
and because of (1.22) and (1.143 ) we obtain the estimates (k)
∇wj s ; L2 (ωj s )2 ≤ cγ∗2 Eβj u; L2 (ωj s )2 .
(1.30)
The domains ωj s can be transformed into the fixed domain ωj 0 by the translation. There(k) fore, the constant in (1.30) may be chosen independent of s. Multiplying (1.30) by E−βj (s) (k)
and taking into account the conditions (1.141 )–(1.142 ) for Eβj (x) we derive ωj s
(k)
E−βj (x)|∇wj s (x)|2 dx ≤ cγ∗2
(k)
ωj s
Eβj (x)|u(x)|2 dx.
(1.31)
Extend the functions wj s (x) by zero into \ ωj s and define W(k) (x) =
J k−1
wj s (x).
j =1 s=0
Obviously, W(k) (x) satisfies the equation (1.28) and supp W(k) (x) ⊂
CJ
j =1 j k .
mate (1.29) follows from (1.31), the definition (1.16) of the step weight-function and weighted Poincaré inequality (1.19).
Esti-
(k) Eβ (x)
1.2.2. Divergence equation with time-dependent right-hand side Let G be a bounded domain in Rn . Consider in G the problem (1.21) with the right-hand side g(x, t) which depends also on t, i.e., consider the problem 3 div w(x, t) = g(x, t), (1.32) w(x, t)|∂G = 0 assuming that g(x, t)dx = 0 ∀t ∈ [0, T ].
(1.33)
G
L EMMA 1.14. Let G ⊂ Rn be a bounded domain with the Lipschitz boundary ∂G. If g(·, t), gt (·, t) ∈ L2 (G) and the condition (1.33) holds ∀t ∈ [0, T ], then equation (1.32) and estimate (1.22) may be “differentiated” with respect to t, i.e., the solution w(x, t) from Lemma 1.10 satisfies the relation div wt (x, t) = gt (x, t),
(1.34)
464
K. Pileckas
and, in addition to (1.22), there holds the estimate ∇wt (·, t); L2 (G) ≤ cgt (·, t); L2 (G).
(1.35)
with a constant c independent of g(x, t) and t ∈ [0, T ]. L EMMA 1.15. Let G ⊂ Rn be a bounded domain with the Lipschitz boundary ∂G, ◦ g(·, t) ∈ W 12 (G), gt (·, t) ∈ L2 (G) and (1.33) holds ∀t ∈ [0, T ]. Then problem (1.32) ◦
◦
admits at least one solution w(·, t) ∈ W 22 (G) with wt (·, t) ∈ W 12 (G) satisfying equation (1.34). There holds the estimate (1.35) and the estimate w(·, t); W22 (G) ≤ cg(·, t); W21 (G)
(1.36)
with a constant c independent of g(x, t) and t ∈ [0, T ]. For the proofs of Lemmata 1.14 and 1.15 an explicit representation formula proposed in [11] may be used. This formula admits the differentiation with respect to t, i.e., for wt (x, t) holds the same representation formula as for w(x, t) with g(x, t) changed to gt (x, t) and, therefore, the statements of Lemmata 1.14 and 1.15 follow from this representation formula. L EMMA 1.16. Let G ⊂ Rn be a bounded domain, ∂G ∈ C 2l+3 , l ≥ 0. If g ∈ W22l+2,l+1 (GT ) and g(x, t) satisfies (1.33) for all t ∈ [0, T ], then there exists at least one solution ◦ l+1 w ∈ W22l+2,l+1 (GT ) ∩ W 12 (G) of problem (1.32) such that ∂t∂ l+1 ∇w ∈ L2 (GT ). There holds the estimate 9 l+1 92 9∂ 9 T 9 w; W22l+2,l+1 (GT )2 + 9 ∇w; L (G ) 2 9 ∂t l+1 9 ≤ cg; W22l+2,l+1 (GT )2 . ¯1 Moreover, if G1 is a subdomain of G and suppx g(x, t) ⊂ G ¯2 suppx w(x, t) ⊂ G
(1.37) ∀t ∈ [0, T ], then
∀t ∈ [0, T ],
(1.38)
where G1 ⊂ G2 ⊂ G. P ROOF. From the inclusion g ∈ W22l+2,l+1 (GT ) it follows that s = 0, . . . , l, and
2l+1−2s ∂s (G), ∂t s g(·, t) ∈ W2
9 l 9 s
9∂ 9 2l+2,l+1 2l+1−2s 9 (G)9 (GT ) 9 ∂t s g(·, t); W2 9 ≤ cg; W2 s=0
(1.39)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
465 ◦
(see Lemma 1.3). By Lemma 1.12, there exist a vector-field w(·, t) ∈ W22l+2 (G) ∩ W 12 (G) such that ¯2 suppx w(·, t) ⊂ G and w(·, t); W22l+2 (G)2 ≤ cg(·, t); W22l+1 (G)2 .
(1.40)
The constructions of Lemma 1.12 use only linear operations and admit differentiation with respect to t (see [26]). Therefore, div
∂s ∂s w(x, t) = g(x, t), ∂t s ∂t s
∂s w(x, t)|∂G = 0, ∂t s
x ∈ G,
s = 1, . . . , l,
and 9 s 92 9 s 92 9∂ 9 9 9 2l+2−2s 9 9 ≤ c9 ∂ g(·, t); W 2l+1−2s (G)9 , w(·, t); W (G) 2 2 9 ∂t s 9 9 ∂t s 9 s = 1, . . . , l. On the other hand,
∂ l+1 g ∂t l+1
all t ∈ [0, T ]. The condition (1.33) for fore,
∈ L2 (GT ). Hence, ∂ l+1 g(x, t) ∂t l+1
∂ l+1 g(·, t) ∈ L2 (G) ∂t l+1
(1.41) for almost
is valid for almost all t ∈ [0, T ]. There-
∂ l+1 ∂ l+1 ∂ l+1 w(x, t) = g(x, t), w(x, t)|∂G = 0, ∂t l+1 ∂t l+1 ∂t l+1 9 l+1 92 9 l+1 92 9∂ 9 9∂ 9 1 9 9 9 9 9 ∂t l+1 w(·, t); W2 (G)9 ≤ c9 ∂t l+1 g(·, t); L2 (G)9 .
div
(1.42)
Summing inequalities (1.40)–(1.42) and integrating them over t we derive the estimate l
9 s 92 9 9 ∂ w(·, t); W 2l+2−2s (G)9 dt + 2 9 ∂t s 9
T9
s=0 0
0
+ 0
≤c
9 92 T 9 ∂ l+1 9 9 9 9 ∂t l+1 ∇w(·, t); L2 (G)9 dt
l s=0 0
9 l+1 92 9 9∂ 9 dt w(·, t); L (G) 2 9 ∂t l+1 9
T9
9 s 92 9 9 ∂ g(·, t); W 2l+1−2s (G)9 dt 2 9 ∂t s 9
T9
9 l+1 92 9 2l+2,l+1 9∂ 9 (GT )2 9 ∂t l+1 g(·, t); L2 (G)9 dt ≤ g; W2
T9
+ 0
which is equivalent to (1.37).
466
K. Pileckas
From Lemma 1.14, just in the same way as in Lemma 1.13, follows ◦
L EMMA 1.17. Let u(·, t) ∈ W 12 (), ut (·, t) ∈ L2 (), div u(x, t) = 0 ∀t ∈ [0, T ], u(x, t) · n(x) ds = 0,
j = 1, . . . , J.
(1.43)
σj ◦
(k)
◦
Then there exists a vector-field W(k) (·, t) ∈ W 12 () with Wt (·, t) ∈ W 12 () such that C suppx W(k) (x, t) ⊂ Jj=1 j k and div W(k) (x, t) = − div(Eβ(k) (x)u(x, t)).
(1.44)
Moreover, there hold the estimates (k) (k) E−β (x)|∇W(k) (x, t)|2 dx ≤ cγ∗2 Eβ (x)|u(x, t)|2 dx
≤ cγ∗2
(k)
(k)
E−β (x)|∇Wt (x, t)|2 dx ≤ cγ∗2
(k)
(1.45)
(k)
(1.46)
Eβ (x)|∇u(x, t)|2 dx, Eβ (x)|ut (x, t)|2 dx.
The constants in (1.45) and (1.46) are independent of k and t ∈ [0, T ]. P ROOF. As in Lemma 1.13, the solution W(k) (x, t) to (1.44) is defined as a sum W(k) (x, t) =
J k−1
wj s (x, t),
j =1 s=0 ◦
where wj s (·, t) ∈ W 12 (ωj s ) are solutions of the equation div wj s (x, t) = −
∂ (j ) ∂xn
(k)
(j )
Eβj (xn )un (x, t),
x ∈ ωj s .
By Lemma 1.14, in addition to (1.30), wj s (x, t) satisfy the estimates (k)
∇wj st (·, t); L2 (ωj s )2 ≤ cγ∗2 Eβj ut (·, t); L2 (ωj s )2 . Therefore, we get (1.45) and
(k) (k) E−β (x)|∇Wt (x, t)|2 dx
≤ cγ∗2
(k)
Eβ (x)|ut (x, t)|2 dx.
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
467
2. Chapter II. Poiseuille type flows The motion of a viscous incompressible fluid in an infinite straight pipe # = {x = (x , xn ) ∈ Rn : x ∈ σ ⊂ Rn−1 , −∞ < xn < ∞} of constant cross-section σ is among the most important problems in fluid mechanics, see, e.g., [36], [8]. The flows characterized by the fact that the associated velocity field has only one nonzero component U (x , t) (t is time) directed along the axis of # which depends only of variables x on cross-section σ are often called unidirectional Poiseuille flows. There are two ways of determining a Poiseuille flow, namely, either prescribing the axial pressure gradient (pressure drop) q or the flow rate (flux) F = σ U dx . In the first case the problem reduces to solving of a boundary value problem for Poisson equation for U = U (x ) with the constant forcing q, if the motion is steady, or initial-boundary value problem for heat equation for U = U (x , t) with time-dependent forcing q(t) in the evolutionary case. The solvability of such problems and properties of corresponding solutions are well known. If, conversely, we prescribe the flux F (t), then we have to solve for U (x , t) and q(t) more complicated nonstandard inverse parabolic problem, at least in time-dependent case. Note that in the steady case the problem still remains very simple, since the flux F and the pressure gradient q are proportional. In the time-dependent case relation between q(t) and F (t) in principle depends on the solution of inverse problem mentioned above. Poiseuille flows are also important in the study of motion in “bent” pipes or in pipes of varying cross-sections which end up in the shape of semi-infinite straight pipes #+ . In such problems, Poiseuille flows are attained at very large distances in #+ . In this chapter we focus on existence and properties of unidirectional Poiseuille flows in steady and time-dependent cases. We shall study also so-called generalized Poiseuille flows for which associated velocity field has all components (U1 (x , t), . . . , Un (x , t)) that depend only of x ∈ σ . Such a flow could occur under special forcing which is independent of xn . We prove the existence of Poiseuille type flows in a straight pipe #. Results concerning steady Poiseuille flows (see Sections 2.1.1 and 2.1.2) are classical and well known. In Section 2.1.3 we prove the existence of a steady flow with prescribed pressure drop in a periodic pipe. This subsection is based on the paper [24] of L.V. Kapitanskii. Results concerning the existence of time-dependent Poiseuille flows (Sections 2.2 and 2.3) and results on the behavior of such flows as t → ∞ (Section 2.4) belong to the author (see [68], [70], [74]). In Section 2.5 we study time-periodic Poiseuille flows. The existence of such flows was first proved by H. Beirão da Veiga [9]. However, in Section 2.5 we present the proofs belonging to G.P. Galdi and A.M. Robertson [22], since they are more simple and give also the elementary relationship between the pressure drop q(t) and the flux F (t). Finally, in Section 2.6 we briefly discuss analogous questions for “pipes” with paraboliclike structure and for the infinite layer.
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K. Pileckas
2.1. Steady Poiseuille flows 2.1.1. Unidirectional steady Poiseuille flow Consider in an infinite pipe # = {x = (x , xn ) ∈ Rn : x ∈ σ ⊂ Rn−1 , −∞ < xn < ∞}, where n ≥ 2 and σ is a bounded domain3 , the steady Navier–Stokes system ⎧ ⎪ ⎨−νu(x) + (u(x) · ∇)u(x) + ∇p(x) = 0, div u(x) = 0, ⎪ ⎩ u(x)|∂# = 0
(2.1)
with additionally prescribed total flux
un (x) dx = F.
(2.2)
σ
Classical unidirectional steady Poiseuille flow is an exact solution of problem (2.1), (2.2) having a simple form (e.g., [36], [8]) UF (x) = (0, . . . , 0, qF U0 (x )),
PF (x) = −qF xn + p0 ,
where p0 is an arbitrary constant and U0 (x ) is the solution of the Poisson equation on the cross-section σ : 3
−ν U0 (x ) = 1, U0 (x )|∂σ = 0.
(2.3)
In (2.3) is the Laplace operator with respect to variables x . It is well known that problem (2.3) admits a unique solution U0 (x ) and that the regularity of U0 (x ) depends only on the regularity of the boundary ∂σ . There holds the following T HEOREM 2.1. Let ∂σ ∈ C l+2+δ . Problem (2.4) admits a unique solution U0 ∈ C l+2+δ (σ ) satisfying the estimate U0 ; C l+2+δ (σ ) ≤ c0 ,
(2.4)
where the constant c0 depends only on σ . Since
U0 (x ) dx = ν σ
|∇ U0 (x )|2 dx := νκ0 > 0,
σ
3 If n = 2, # is an infinite strip in R2 , σ = (0, h ) is a bounded interval. 0
(2.5)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
the constant qF could be chosen so that Poiseuille flow has the given flux F : U0 (x ) dx = F, qF
469
(2.6)
σ
i.e. qF = F κ0−1 ν −1 .
(2.7)
Note that κ0 represents geometrical characteristics of the domain σ . In the two-dimensional case there hold the following formulas U0 (x1 ) =
1 x1 (h0 − x1 ), 2ν
qF =
12F ν h30
.
(2.8)
2.1.2. Generalized steady Poiseuille flow Consider the nonhomogeneous steady Navier– Stokes problem in the cylinder # ⊂ R3 : ⎧ −νu(x) + (u(x) · ∇)u(x) + ∇p(x) = f(x), ⎪ ⎪ ⎪ ⎨ div u(x) = 0, ⎪ u(x)|∂# = 0 ⎪ ⎪ ⎩
σ un (x) dx = F,
(2.9)
assuming that the external force f(x) does not depend on x3 , i.e., f(x) = (f1 (x ), f2 (x ), f3 (x )). We look for the solution (U(x), P (x)) having the following form U(x) = (U1 (x ), U2 (x ), U3 (x )),
P (x) = p 8(x ) − qF x3 + p0 .
(2.10)
Substituting (2.10) into (2.9) we see that (2.9) decomposes into two problems. For 8(x )) = ((U1 (x ), U2 (x )), p 8(x )) we obtain the two-dimensional Navier–Stokes (U (x ), p problem on the cross section σ : ⎧
8(x ) = f (x ), ⎪ ⎨−ν U (x ) + (U (x ) · ∇ )U (x ) + ∇ p div U (x ) = 0, (2.11) ⎪ ⎩
U (x )|∂σ = 0, and for U (x ) = U3 (x ) we get the boundary value problem: ⎧
⎪ ⎨−ν U (x ) + (U (x ) · ∇ )U (x ) = qF + f (x ),
U (x )|∂σ = 0, ⎪ ⎩
σ U (x ) dx = F,
(2.12)
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K. Pileckas
where f (x ) = (f1 (x ), f2 (x )), f (x ) = f3 (x ), ∇ = ∂x∂ 1 , ∂x∂ 2 , div U (x ) = ∇ · U (x ). Results concerning the existence and regularity of solutions to problem (2.11) are well known [28]: T HEOREM 2.2. If f ∈ L2 (σ ), then problem (2.11) admits at least one weak solution U ∈ ◦ W 12 (σ )4 and there holds the estimate U ; W21 (σ ) ≤ cf ; L2 (σ ).
(2.13)
If ∂σ ∈ C l+2 , f ∈ W2l (σ ), l ≥ 0, then U ∈ W2l+2 (σ ), ∇ p˜ ∈ W2l (σ ) and U ; W2l+2 (σ ) + ∇ p; ˜ W2l (σ ) ≤ C(f ; W2l (σ )).
(2.14)
If ∂σ ∈ C l+2+δ , f ∈ C l+δ (σ ), then U ∈ C l+2+δ (σ ), ∇ p˜ ∈ C l+δ (σ ) and ˜ C l+δ (σ ) ≤ C(f ; C l+δ (σ )). U ; C l+2+δ (σ ) + ∇ p;
(2.15)
Constants C(f ; W2l (σ )) and C(f ; C l+δ (σ )) in (2.14), (2.15) depend nonlinearly on f ; W2l (σ ) (respectively on f ; C l+δ (σ )) and vanish as f ; W2l (σ ) → 0 (respectively f ; C l+δ (σ ) → 0). For sufficiently small f ; L2 (σ ) the solution U (x ) is unique. ((x ) + U 8(x ) We look for the solution U (x ) of problem (2.12) in the form U (x ) = U
8(x ) satisfies the problem where U 3
8(x ) = f (x ), 8(x ) + (U (x ) · ∇ )U −ν U 8(x )|∂σ = 0, U
(2.16)
(0 (x ), ((x ) = qF U U 3
(0 (x ) = 1, (0 (x ) + (U (x ) · ∇ )U −ν U (0 (x )|∂σ = 0, U
(2.17)
and qF is chosen to satisfy the condition
(= F − (0 (x ) dx = F U
qF σ
8(x ) dx . U
σ ◦
4 Weak solution to problem (2.11) is a divergence-free vector-field U ∈ W 12 (σ ) satisfying for any divergence◦ 1 free η ∈ W 2 (σ ) the integral identity
ν σ
∇ U (x ) · ∇ η(x ) dx −
σ
(U (x ) · ∇ )η(x ) · U (x ) dx =
σ
f (x ) · η(x ) dx .
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
471
Obviously, for the fixed U (x ) both problems (2.16) and (2.17) are uniquely solvable and (0 (x ) and U 8(x ) depend on the regularity of ∂σ , the regularity properties of solutions U U (x ) and f (x ). Moreover, since
(0 (x ) · U (0 (x ) dx = 0, (U (x ) · ∇ )U
σ
we get
(0 (x ) dx = ν U
σ
(0 (x )|2 dx := νκ1 > 0. |∇ U
σ
(κ −1 ν −1 , we satisfy the flux condition Therefore, taking qF = F 1 qF
(. (0 (x ) dx = F U
σ
There holds the following T HEOREM 2.3. For arbitrary f ∈ L2 (σ ) problem (2.12) admits a unique weak solution ◦ U ∈ W 12 (σ ) and U ; W21 (σ ) ≤ c(f ; L2 (σ ) + |F |).
(2.18)
If ∂σ ∈ C l+2 , f ∈ W2l (σ ), U ∈ W2l+2 (σ ), then U ∈ W2l+2 (σ ) and U ; W2l+2 (σ ) ≤ c(f ; W2l (σ ) + |F |).
(2.19)
If ∂σ ∈ C l+2+δ , f ∈ C l+δ (σ ), U ∈ C l+2+δ (σ ), then U ∈ C l+2+δ (σ ) and U ; C l+2+δ (σ ) ≤ c(f ; C l+δ (σ ) + |F |).
(2.20)
The constants in (2.19) and (2.20) depend on U (x ), however, they are independent of |F |. Solutions of problem (2.9) having the form (2.10) we call generalized Poiseuille flows. R EMARK 2.1. The uniqueness of the solution to problem (2.9) having the form (2.10) follows from Theorems 2.2 and 2.3 only for small norms of f (x ), since for large f (x ) the solution of problem (2.11) may be non unique.
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K. Pileckas
R EMARK 2.2. Because of the divergence-free condition (div u(x) = 0), in the twodimensional case the generalized Poiseuille flow is unidirectional, i.e., U(x1 ) = (0, U2 (x1 )). x Moreover, p 8(x1 ) = 0 1 f1 (y1 ) dy1 , and problem (2.12) takes the form ⎧ 2 ⎪ −ν d U (x ) = qF + f2 (x1 ), ⎪ ⎨ dx12 2 1 U2 (0) = U2 (h0 ) = 0, ⎪ ⎪ h0 ⎩ 0 U2 (x1 ) dx1 = F. Note that in the two-dimensional case the generalized Poiseuille flow is unique also for large norms of f1 (x1 ). This follows from the fact that U (x1 ) = 0. 2.1.3. Steady flow with prescribed pressure drop in a periodic pipe Results of this section are based on the paper [24]. Let = {x ∈ Rn : |x | < R(xn )}, where R is a periodic function with period L > 0. By L denote the part of which is situated between the planes xn = 0 and xn = L, by σs denote the cross-section of by the plane xn = s and by SL – the “lateral surface” of the domain L , i.e., SL = ∂ ∩ ∂L . Consider in the domain the steady Navier–Stokes problem ⎧ ⎪ ⎨−νu(x) + (u(x) · ∇)u(x) + ∇p(x) = f(x), div u(x) = 0, (2.21) ⎪ ⎩ u(x)|∂ = 0. An external force f(x) is assumed now to be periodic with respect to xn with period L. If (u(x), p(x)) is a solution of (2.21) with L-periodic in xn velocity field u(x), then, as it can be easily seen, the pressure p(x) has a constant drop over the period, i.e., 1 [p(x , xn + L) − p(x , xn )] ≡ −q∗ = const L
∀x ∈ .
(2.22)
Consider the following problem: For given q∗ and f(x) to find a solution (u(x), p(x)) of problem (2.21) such that the velocity field u(x) is L-periodic with respect to xn and the pressure p(x) has the prescribed pressure drop q∗ . Let H (L ) be a closure in the Dirichlet norm |∇u(x)|2 dx u; H (L )2 = L
of the set of all smooth divergence-free L-periodic vector-fields equal to zero in the neighborhood of ∂. By a weak solution of problem (2.21), (2.22) we understand a vector-field u ∈ H (L ) satisfying the following integral identity ν ∇u(x) · ∇η(x) dx − (u(x) · ∇)η(x) · u(x) dx L
L
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
= q∗ L
ηn (x) dx +
σ0
473
f(x) · η(x) dx
∀η ∈ H (L ).
(2.23)
L
Obviously, each classical solution to (2.21), (2.22) satisfies the integral identity (2.23). T HEOREM 2.4. For any L-periodic f(x) such that f ∈ L2 (L ) and any q∗ ∈ R problem (2.21), (2.22) has at least one weak solution u ∈ H (L ). Moreover, p(x) = −q∗ xn + p 8(x), where ∇ p 8 ∈ G(L ) (G(L ) is the closure in L2 (L )-norm of gradients of smooth L-periodic functions defined in ). There holds the estimate u; H (L ) ≤ c(f; L2 (L ) + |q∗ |).
(2.24)
For sufficiently small f; L2 (L ) and |q∗ | the weak solution is unique. P ROOF. By standard arguments (see [28], Ch. V) integral identity (2.23) reduces to an operator equation in the space H (L ): u − ν −1 Au = ν −1 (F + Q).
(2.25)
The nonlinear operator A and the elements F, Q ∈ H (L ) are defined by: (u(x) · ∇)η(x) · u(x) dx,
[Au, η] = L
f(x) · η(x) dx,
[F, η] =
[Q, η] = q∗ L
L
ηn (x) dx ,
σ0
where [u, η] = L ∇u(x) · ∇η(x) dx is the scalar product in H (L ). The existence of Q ∈ H (L ) follows from the fact that the elements of H (L ) are divergence free and, therefore, σ0 ηn (x) dx = L−1 L ηn (x) dx while the last integral is estimated by Poincaré inequality, since η(x)|∂ = 0. The operator A is compact (see [28]) and, in order to prove the existence of a solution to (2.25), it is necessary to show that all possible solutions of the operator equation u(λ) − λν −1 Au(λ) = ν −1 (F + Q),
λ ∈ [0, ν −1 ],
(2.26)
are bounded by the same constant. Equation (2.26) is equivalent to the integral identity
∇u
ν
(λ)
L
= q∗ L
(x) · ∇η(x) dx − λ
σ0
ηn (x) dx +
(u(λ) (x) · ∇)η(x) · u(λ) (x) dx L
f(x) · η(x) dx L
∀η ∈ H (L ).
(2.27)
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K. Pileckas
Taking in (2.27) η(x) = u(λ) (x) and observing that 0, we get u(λ) ; H (L )2 = ν −1 q∗ L
σ0
L (u
(λ) (x) · ∇)u(λ) (x) · u(λ) (x) dx
−1 u(λ) n (x) dx + ν
=
f(x) · u(λ) (x) dx. L
Therefore, using Cauchy–Schwarz and Poincaré inequalities we derive u(λ) ; H (L ) ≤ cν −1 (f; L2 (L ) + |q∗ |) with constant c independent of u(λ) . Now, the existence of a solution to (2.25) follows from the Leray–Schauder theorem. The uniqueness of the solution for sufficiently small f; L2 (L ) and |q∗ | is evident. Now, consider the relation between the pressure drop q∗ and the flux F =
∀s. σs un (x) dx
σ0
un (x) dx =
T HEOREM 2.5. There exists a constant q0 ≥ 0 determined by the geometry of the domain and by the external force f(x) such that for any weak solution u(x) of problem (2.21), (2.22) with q∗ > q0 (or q∗ < −q0 ) the flux F = σ0 un (x) dx is positive (respectively, negative). P ROOF. First, assume that the flux is equal to zero:
un (x) dx = 0.
σ0
Put into integral identity (2.23) η(x) = u(x). Then the nonlinear term and the term with q∗ disappear (i.e., [Au, u] = 0 and [Q, u] = 0), and from (2.23) follows the inequality u; H (L ) ≤ c1 ν −1 f; L2 (L ).
(2.28)
Let us estimate |q∗ |. Denote by η0 (x) the vector-field from H (L ) with the unite flux over σ0 and having a minimal H (L )-norm among all such fields. Let η0 ; H (L ) = a0 . Inserting into (2.23) η(x) = η0 (x), we get in virtue of (2.28), (I. 1.8) (or (I. 1.7), if n = 2) and Poincaré inequality (I. 1.2): " " " " " " " " −1 " " " ∇u(x) · ∇η0 (x) dx " + L " (u(x) · ∇)η0 (x) · u(x) dx "" |q∗ | = L ν " L L " " " " + L−1 "" f(x) · η0 (x) dx "" −1
L
≤L
−1
a0 (νu; H (L ) + u; L4 (L )2
+ c1 f; L2 (L ))
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
475
≤ L−1 a0 (νu; H (L ) + c2 u; H (L )2 + c1 f; L2 (L )) ≤ L−1 a0 c1 f; L2 (L )(2 + c1 c2 ν −2 f; L2 (L )) ≡ q0 . Thus, it is proved that for |q∗ | > q0 the flux of u(x) is different from zero. Moreover, we have also proved that the condition |q∗ | > q0 implies the inequality u; H (L ) > c1 ν −1 f; L2 (L ). Let |q∗ | > q0 and F = q∗ F = L
−1
σ0
un (x) dx . Taking in (2.23) η(x) = u(x) we obtain
|∇u(x)| dx − L 2
ν
(2.29)
−1
L
f(x) · u(x) dx L
≥ L−1 u; H (L )(νu; H (L ) − c1 f; L2 (L )). Therefore, by (2.29) the product q∗ F is positive.
R EMARK 2.3. If the force f(x) is missing, then q0 = 0. Indeed, in this case we have L−1 νu; H (L )2 = q∗ F, from where it follows that for q∗ = 0 we have u(x) = 0 and for q∗ = 0: F = L−1 νu; H (L )2 q∗−1 . R EMARK 2.4. Note that the exact relation between the pressure drop q∗ and the flux F is unknown in the periodic case. The flux depends in principle on the weak solution u(x) of problem (2.21), (2.22). The existence in a periodic pipe L of solutions to the Navier– Stokes equations with the given flux F is proved in [43] for sufficiently small |F |. R EMARK 2.5. In the case of a straight pipe # and f(x) independent of xn from Theorem 2.4 follows the existence of periodic in xn solution u(x) with arbitrary period L > 0. On the other hand, there exists the Poiseuille flow corresponding to the same pressure drop. This solution is independent of xn and, thus, periodic with any period L. For sufficiently small data both solutions, obviously, coincide. However, for large data this problem is open, i.e., it is not known whether periodic in xn solution is different from Poiseuille one or they coincide.
2.2. Unidirectional time-dependent Poiseuille flow
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K. Pileckas
2.2.1. Definition of time-dependent Poiseuille flow Let us consider in a pipe # initialboundary value problem for the Navier–Stokes equations ⎧ ⎪ ⎨ut (x, t) − νu(x, t) + (u(x, t) · ∇)u(x, t) + ∇p(x, t) = f(x, t), div u(x, t) = 0, ⎪ ⎩ u(x, t)|∂# = 0, u(x, 0) = u0 (x)
(2.30)
with additionally prescribed time-dependent flux F (t):
un (x, t) dx = F (t).
(2.31)
σ
Assume that the initial data u0 (x) and the external force f(x, t) have the form u0 (x) = (0, . . . , 0, u0n (x )),
f(x, t) = (0, . . . , 0, fn (x , t))
and that there holds the necessary compatibility condition
u0n (x ) dx = F (0).
(2.32)
σ
We look for the solution (u(x, t), p(x, t)) of problem (2.30), (2.31) in the form U(x, t) = (0, . . . , 0, Un (x , t)),
P (x, t) = −q(t)xn + p0 (t),
(2.33)
where p0 (t) is an arbitrary function. Substituting (2.33) into (2.30) we derive for U (x , t) = Un (x , t) and q(t) the following initial-boundary value problem on the cross-section σ : ⎧ Ut (x , t) − ν U (x , t) = q(t) + fn (x , t), ⎪ ⎪ ⎪ ⎨ U (x , t)|∂σ = 0, ⎪ U (x , 0) = u0n (x ), ⎪ ⎪ ⎩
σ U (x , t) dx = F (t).
(2.34)
The solution (u(x, t), p(x, t)) of problem (2.30), (2.31) having the form (2.33), where U (x , t) and q(t) satisfy 2.34), we call unidirectional time-dependent Poiseuille flow. The function q(t) in (2.341 ) is unknown and has to be found, in order to satisfy the flux condition (2.343 ). Thus, (2.34) is an inverse problem: For given u0n (x ) and F (t) to find a pair of functions (u(x , t), q(t)) solving the initialboundary value problem (2.341 )–(2.342 ) and satisfying the flux condition (2.343 ). Unlike the steady case where the constant qF defining the pressure gradient is proportional to the flux F (see (2.7)), in the time-dependent case the function q(t) has to be found as a solution of the inverse problem (2.34). Therefore, in the time-dependent case the formulation of the problem with the prescribed pressure gradient q(t) is not equivalent
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
477
to the formulation with the prescribed flux F (t). By prescribing the pressure gradient q(t) one has to solve the direct parabolic problem (2.34 1 )–(2.342 ) where q(t) is known while the solution u(x , t) has a nonzero flux Q(t) = σ u(x , t) dx . Prescribing the flux condition (2.343 ) it is necessary to solve the inverse problem (2.341 )–(2.343 ) and the relation between q(t) and F (t) depends on the solution U (x , t) of (2.34). Similar to (2.34) inverse problems for parabolic equations were studied by many authors (see the book [81] and the literature cited there). However, in [81] only the case of weighted “flux” condition
w0 (x )U (x , t)dx = F (t)
(2.35)
σ ◦
with the weight function w0 ∈ W 12 (σ ) is considered. The restriction w0 (x )|∂σ = 0 is essential for methods used in [81]. In next sections we prove the solvability of unidirectional and generalized timedependent Poiseuille flows in Sobolev and Hölder spaces. 2.2.2. Construction of approximate solutions to problem (2.34) Consider the inverse ◦ problem (2.34) in the case where fn (x , t) = 0, u0n (x) = 0, F (0) = 0. Let uk (x ) ∈ W 12 (σ ) and λk be eigenfunctions and eigenvalues of the Laplace operator: 3
−νuk (x ) = λk uk (x ), uk (x )|∂σ = 0.
(2.36)
Note that λk > 0 and {λk } → ∞ as k → ∞. The eigenfunction uk (x ) are orthogonal in L2 (σ ) and we assume that uk (x ) are normalized in L2 (σ ). Then ν
|∇ uk (x )|2 dx = λk ,
σ
∇ uk (x ) · ∇ ul (x ) dx = 0,
k = l.
σ
Moreover, {uk (x )} form a basis in L2 (σ ). Since the constant function 1 belongs to L2 (σ ), it could be decomposed into Fourier series: 1=
∞
βk uk (x ),
k=1
2 where βk = σ uk (x )dx , k = 1, 2, . . . , and ∞ k=1 βk = |σ |. We look for an approximate solution (U (N ) (x , t), q (N ) (t)) of problem (2.34) in the form
U
(N )
(x , t) =
N
k=1
wk (t)uk (x ), (N )
(2.37)
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K. Pileckas
and chose the function q (N ) (t) in order to satisfy the flux condition U (N ) (x , t) dx = F (t) ∀t ∈ [0, T ].
(2.38)
σ
Substituting the sum (2.37) into (2.341 ), (2.342 ) we find (N ) wk (t) = βk
t
exp(−λk (t − τ ))q (N ) (τ ) dτ,
0
i.e., U
(N )
(x , t) =
N
t
βk
exp(−λk (t − τ ))q
(N )
(τ ) dτ uk (x )
(2.39)
0
k=1
and the condition (2.38) yields the relation
U (N ) (x , t) dx =
σ
N
βk
k=1
t
exp(−λk (t − τ ))q (N ) (τ ) dτ
0
uk (x ) dx
σ
= F (t), which is equivalent to the Volterra integral equation of the first type for the function q (N ) (t): N
k=1
βk2
t
exp(−λk (t − τ ))q (N ) (τ ) dτ = F (t).
(2.40)
0
Let us assume that there exists a derivative F (t). Differentiating (2.40), we derive q (N ) (t) −
t N 1 2 βk λk exp(−λk (t − τ ))q (N ) (τ ) dτ = ϕ (N ) (t), 1N 0
(2.41)
k=1
2 (N ) (t) = F (t)/1 . Equation (2.41) is the Volterra integral equation were 1N = N N k=1 βk , ϕ of the second type with the kernel −1 K (N ) (t, τ ) = 1N
N
βk2 λk exp(−λk (t − τ )).
k=1
The solvability of such equation is well known (see, e.g., [39], [99]). If ϕ (N ) ∈ L2 (0, T ), T ∈ (0, ∞], the equation (2.41) has a unique solution q (N ) ∈ L2 (0, T ) and q (N ) ; L2 (0, T ) ≤ c(N)ϕ (N ) ; L2 (0, T ).
(2.42)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
479
Moreover, if ϕ (N ) (0) = 0, then q (N ) (0) = 0. In the case where ϕ (N ) ∈ C δ ([0, T ]), T ∈ (0, ∞), and ϕ (N ) (0) = 0 the solution q (N ) ∈ δ C ([0, T ]), q (N ) (0) = 0 and q (N ) ; C δ ([0, T ]) ≤ c(N)ϕ (N ) ; C δ ([0, T ]).
(2.43)
However, it is not possible to pass to a limit as N → ∞ directly in the integral equa 2 (∞) (t, t) at t = τ is tion (2.41), since the series |σ1 | ∞ k=1 βk λk defining the limit kernel K divergent. Differentiating equation (2.41) we find that
q (N ) (t) −
N 1 2 βk λk q (N ) (t) 1N k=1
−
t N 1 2 d
(exp(−λk (t − τ )))q (N ) (τ ) dτ = ϕ (N ) (t). βk λk 1N 0 dt k=1
Integrating by parts in the third term and using the condition q (N ) (0) = 0, we rewrite this equation as
q (N ) (t) −
t N 1 2
βk λk exp(−λk (t − τ ))q (N ) (τ ) dτ = ϕ (N ) (t). 1N 0
(2.44)
k=1
Thus, the derivative q (N ) (t) is the solution of the same equation (2.41) with the right-hand
side ϕ (N ) (t). Analogously, if d s (N ) ϕ (0) = 0, dt s the derivatives d s (N ) (t): dt s ϕ
d s (N ) (t), dt s q
s = 0, . . . , l, s = 1, . . . , l, satisfy (2.41) with the right-hand sides equal to
t N d s (N ) 1 2 d s (N ) q (t) − β λ exp(−λ (t − τ )) q (τ ) dτ k k k dt s 1N dt s 0 k=1
ds = s ϕ (N ) (t), dt
d s (N ) q (0) = 0, dt s
s = 0, . . . , l.
(2.45)
480
K. Pileckas
The approximations (U (N ) (x , t), q (N ) (t)) satisfy the following initial-boundary value problems ⎧ (N ) ⎪ Ut (x , t) − ν U (N ) (x , t) ⎪ ⎪ ⎪ ⎨ = q (N ) (t) − q (N ) (t) ∞
k=N +1 βk uk (x ), " (N )
(N )
⎪ U (x , 0) = 0, U (x , t)"∂σ = 0, ⎪ ⎪ ⎪ (N ) ⎩ (x , t) dx = F (t). σu
(2.46)
In order to prove the convergence of the approximate solutions (U (N ) (x , t), q (N ) (t)) to the solution (U (x , t), q(t)) of problem (2.34) we shall use a priori estimates of the solution (U (N ) (x , t), q (N ) (t)) to (2.46). Note that the limit function q(t) = limN →∞ q (N ) (t) is not a solution of the limiting integral equation with the kernel K ∞ (t, τ )5 . 2.2.3. A priori estimates in Sobolev spaces Let (U (x , t), q(t)) be a solution (sufficiently regular) of the following problem ⎧
⎪ ⎨Ut (x , t) − ν U (x , t) = q(t) − q(t)h(x ), U (x , t)|∂σ = 0, U (x , 0) = 0, ⎪ ⎩
F (0) = 0. σ U (x , t) dx = F (t),
(2.47)
L EMMA 2.1. Let F ∈ W21 (0, T ), T ∈ (0, ∞], h ∈ L2 (σ ) and h; L2 (σ )2 ≤ δ, where δ is sufficiently small. Then for the solution (U (x , t), q(t)) of problem (2.47) holds the estimate U (·, t); W21 (σ )2 + U ; W21,1 ( t )2 dτ + q; L2 (0, t)2 ≤ cF ; W21 (0, t)2 ,
(2.48)
where T = σ × (0, T ). If, in addition ∂σ ∈ C 2 , then U ; W22,1 ( t )2 + q; L2 (0, t)2 ≤ c 1 + |1 − h(x )|2 dx F ; W21 (0, t)2 .
(2.49)
σ
The constants in (2.48), (2.49) do not depend on t. P ROOF. Multiply equation (2.47) by U (x , t), integrate by parts in σ and then integrate with respect to t: 1 2
σ
|U (x , t)|2 dx + ν
t 0
|∇ U (x , τ )|2 dx dτ
σ
◦
1 5 In the case of the weighted flux condition
σ w0 (x )U (x , t) dx = F (t) with w0 (x ) ∈ W 2 (σ ) it is possible
to pass to a limit as N → ∞ in the integral equation that replaces (2.41) in this case.
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
t
=
0
σ t
=
U (x , τ ) dx dτ −
q(τ )
q(τ )F (τ ) dτ −
0
q(τ )
t
1 2ε
|q(τ )|2 dτ +
0
t
≤ε
1 2ε
|q(τ )|2 dτ +
0
h(x )U (x , τ ) dx dτ
σ
h(x )U (x , τ ) dx dτ
q(τ ) 0
≤ε
0
t
t
σ
t
|F (τ )|2 dτ +
0
t
481
1 2ε
t 0
h(x )U (x , τ ) dx
2 dτ
σ
|F (τ )|2 dτ
0
t 1
2
|h(x )| dx |U (x , τ )|2 dx dτ + 2ε σ 0 σ t ≤ε |q(τ )|2 dτ 0
+
1 2ε
t
|F (τ )|2 dτ +
0
δν 2λ1 ε
t 0
|∇ U (x , τ )|2 dx dτ,
σ
where λ1 is the first eigenvalue of problem (2.36). Here we have applied Young, Cauchy– Schwarz and Poincaré (see (I.1.3)) inequalities. Let δ ≤ ελ1 . Then the last relation yields 1 2
ν t |U (x , t)| dx + |∇ U (x , τ )|2 dx dτ 2 0 σ σ t 1 t |q(τ )|2 dτ + |F (τ )|2 dτ. ≤ε 2ε 0 0
2
(2.50)
Now, multiply (2.47) by Ut (x , t) and integrate by parts in σ :
ν d |Ut (x , t)| dx + |∇ U (x , t)|2 dx 2 dt σ σ = q(t)F (t) − q(t) h(x )Ut (x , t) dx
2
σ
≤ ε|q(t)|2 + ≤ ε|q(t)|2 +
1 2 1 |F (t)| + 2ε 2ε 1 2 δ |F (t)| + 2ε 2ε
h(x )Ut (x , t) dx σ
|Ut (x , t)|2 dx .
σ
If δ ≤ ε, from (2.51) it follows that 1 2
t 0
ν |Ut (x , τ )| dx dτ + 2 σ
2
σ
2
|∇ U (x , t)|2 dx
(2.51)
482
K. Pileckas
t
≤ε
|q(τ )|2 dτ +
0
1 2ε
t
|F (τ )|2 dτ.
(2.52)
0
Let U0 (x ) be the solution of problem (2.4), i.e., 3
−ν U0 (x ) = 1, U0 (x )|∂σ = 0.
(2.53)
Multiplying equation (2.47) by U0 (x ) and integrating twice by parts in the second term, we derive
Ut (x , t)U0 (x ) dx − ν U (x , t) U0 (x ) dx σ
σ
U0 (x ) dx − q(t)
= q(t)
σ
h(x )U0 (x ) dx .
(2.54)
σ
Denote κ∗ =
U0 (x ) dx −
σ
≥ν
h(x )U0 (x ) dx
σ
|∇ U0 (x )|2 dx −
σ
≥ν
σ
|∇ U0 (x )|2 dx −
σ
1/2
= κ0
|h(x )|2 dx
1/2
κ0
−
δ
1/2 1/2
δ 1/2 ν
1/2
1/2
λ1
1/2
|U0 (x )|2 dx σ
|∇ U0 (x )|2 dx
1/2
1/2
σ
,
λ1
√ where κ0 = ν σ |∇ U0 (x )|2 dx = σ U0 (x ) dx . If δ ≤ λ1 κ0 / 2, then κ∗ ≥ κ0 /2 > 0. From (2.54) and (2.53) follows the relation (2.55) κ∗ q(t) = Ut (x , t)U0 (x ) dx + F (t). σ
Hence, applying again Cauchy–Schwarz and Poincaré inequalities, we get κ∗2
t
|q(τ )|2 dτ ≤ 2
0
t 0
≤2
t 0
+2 0
Ut (x , τ )U0 (x ) dx σ
|Ut (x , τ )| dx σ
t
|F (τ )|2 dτ
2
2
dτ + 2
t
|F (τ )|2 dτ
0
2
|U0 (x )| dx σ
dτ
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
2ν λ1
≤
t 0
+2
|Ut (x , τ )|2 dx
σ t
|∇ U0 (x )|2 dx dτ
σ
|F (τ )|2 dτ
0
2κ0 ≤ λ1
483
t 0
|Ut (x , τ )|2 dx dτ + 2
σ
t
|F (τ )|2 dτ.
(2.56)
0
Inequalities (2.56), (2.50) and (2.52) yield
|U (x , t)|2 + ν|∇ U (x , t)|2 dx
σ
+
t 0
(ν|∇ U (x , τ )|2 + |Ut (x , τ )|2 ) dx dτ
σ
1 t |q(τ )| dτ + (|F (τ )|2 + |F (τ )|2 ) dτ ≤ 4ε ε 0 0 t t 8κ0 |Ut (x , τ )|2 dx dτ + c(ε) (|F (τ )|2 + |F (τ )|2 ) dτ ≤ 2 ε κ∗ λ1 0 σ 0 t t 16 ε |Ut (x , τ )|2 dx dτ + c(ε) (|F (τ )|2 + |F (τ )|2 ) dτ. (2.57) ≤ κ0 λ1 0 σ 0 t
2
Fixing in (2.57) ε = κ0 λ1 /32, gives
(|U (x , t)|2 + |∇ U (x , t)|2 ) dx
σ
+
t
≤c
0 t
(|∇ U (x , τ )|2 + |Ut (x , τ )|2 ) dx dτ
σ
(|F (τ )|2 + |F (τ )|2 ) dτ
(2.58)
0
and from (2.56) we derive the estimate
t
0
|q(τ )|2 dτ ≤ c
t
(|F (τ )|2 + |F (τ )|2 ) dτ.
(2.59)
0
Let ∂σ ∈ C 2 . Consider U (x , ·) as a solution of the Poisson equation 3
−ν U (x , t) = q(t)(1 − h(x )) − Ut (x , t), U (x , t)|∂σ = 0.
(2.60)
484
K. Pileckas
The right-hand side of (2.60) belongs to L2 (σ ) for almost all t. Therefore, the solution U (·, t) ∈ W22 (σ ) and there holds the estimate σ
(|U (x , t)|2 + |∇ U (x , t)|2 + |∇ 2 U (x , t)|2 ) dx ≤ c |q(t)|2 |1 − h(x )|2 dx + |Ut (x , t)|2 dx , σ
σ
where |∇ 2 u(x , t)|2 = |α|=2 |Dxα u(x , t)|2 . Integrating the last inequality with respect to t, we obtain in virtue of (2.58) and (2.59) t 0
(|U (x , τ )|2 + |∇ U (x , τ )|2 + |∇ 2 U (x , τ )|2 ) dx dτ σ
t
2
(|F (τ )|2 + |F (τ )|2 ) dτ. ≤ c 1 + |1 − h(x )| dx 0
σ
Thus, if δ≤
κ0 λ1 min(1, λ1 ), 32
then there hold estimates (2.48) and (2.49). L EMMA 2.2. Let ∂σ ∈ C 2l+2 , h ∈ W22l (σ ), l ≥ 0, and let F ∈ W2l+1 (0, T ), F (0) = 0, . . . ,
dl F (0) = 0. dt l
(2.61)
Assume that (U, q) ∈ W22l+2,l+1 ( T ) × W2l (0, T ) be a solution to (2.47) with q(t) satisfying the conditions q(0) = 0, . . . ,
dl g(0) = 0. dt l
(2.62)
If h; W22l (σ ) ≤ δ with δ being sufficiently small, then there holds the estimate U ; W22l+2,l+1 ( t )2 + q; W2l (0, t)2 ≤ c(1 + 1 − h; W22l (σ )2 )F ; W2l+1 (0, t)2 .
(2.63)
P ROOF. For l = 0 estimate (2.63) coincides with (2.49). Let l ≥ 1. Assume that (2.63) is true for l ≤ r − 1 and prove it for l = r. Denote U {s} (x , t) =
∂s U (x , t), ∂t s
q {s} (t) =
ds q(t), dt s
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
F {s} (t) =
ds F (t), dt s
485
s = 1, . . . , r.
Differentiating (2.47) s times with respect to t we derive6 ⎧ {s}
{s} {s}
⎪ ⎨Ut (x , t) − ν U (x , t) = q (t)(1 − h(x )), ⎪ ⎩
U {s} (x , t)|∂σ = 0, U {s} (x , 0) = 0, {s}
{s} σ U (x , t) dx = F (t).
(2.64)
Obviously, F {s} ∈ W2r+1−s (0, T ). Therefore, from (2.63) at l = r − s, s = 1, . . . , r, follow the inequalities 92 9 s 92 9 s 9 9d 9 9∂ 2r+2−2s,r+1−s r−s t 9 9 9 9 U ; W ( ) + q; W (0, t) 2 2 9 9 dt s 9 9 ∂t s 9 s 92 9 9 2r−2s r+1−s 2 9 d ≤ c(1 + 1 − h; W2 (σ ) )9 s F ; W2 (0, t)9 9 dt ≤ c(1 + 1 − h; W22r (σ )2 )F ; W2r+1 (0, t)2 ,
s = 1, . . . , r.
(2.65)
Now, considering (U {s} (x , t), q {s} (t)) as a solution of the Poisson equation 3 {s} −ν U {s} (x , t) = q {s} (t)(1 − h(x )) − Ut (x , t), U {s} (x , t)|∂σ = 0, we obtain the estimates 9 s 92 9∂ 9 2r+2−2s 9 9 U (·, t); W (σ ) 2 9 ∂t s 9 "2 " s "d " ≤ c "" s q(t)"" 1 − h; W22r−2s (σ )2 dt 9 s+1 92 9∂ 9 2r−2s 9 + 9 s+1 U (·, t); W2 (σ )9 9 , ∂t
s = 1, . . . , r.
Integrating these inequalities from 0 to t and using (2.65) yields 92 t9 s 9∂ 9 2r+2−2s 9 (σ )9 9 ∂τ s U (·, τ ); W2 9 dτ 0
≤ c 1 − h; W22r−2s (σ )2
0
" " " " " dτ s " dτ
t " d s q(τ ) "2
6 Note that deriving the initial condition U {s} (x , 0) = 0 we have used (2.62). For the approximate solution q (N ) (t) relation (2.62) follows from conditions (2.61) (see (2.45)).
486
K. Pileckas
92 t 9 s+1 9 ∂ U (·, τ ) 9 2r−2s 9 + (σ )9 9 ∂τ s+1 ; W2 9 dτ 0
≤ c(1 + 1 − h; W22r (σ )2 )F ; W2r+1 (0, t)2 ,
s = 1, . . . , r.
(2.66)
Finally, considering (U (x , t), q(t))) as a solution of Poisson problem (2.60) we conclude that U (·, t) ∈ W22r+2 (σ ) and
t
0
U (·, τ ); W22r+2 (σ )2 dτ ≤c
t
t
|q(τ )|2 dt +
0
0
Uτ (·, τ ); W22r (σ )2 dτ
≤ c(1 + 1 − h; W22r (σ )2 )F ; W2r+1 (0, t)2 .
(2.67)
Estimate (2.63) at l = r follows from inequalities (2.65)–(2.66), s = 1, . . . , r, (2.67) and the definition of the norm in the space W22r+2,r+1 ( T ). 2.2.4. Existence of a solution in Sobolev spaces By weak solution of problem (2.34) ◦ we understand a pair of functions (U, q) ∈ W 21,1 ( T ) × L2 (0, T ) satisfying the initial condition U (x , 0) = u0n (x ), the flux condition
U (x , t)dx = F (t)
σ
and the integral identity t 0
Uτ (x , τ )η(x , τ )dx dτ + ν σ
=
0
t
+
t 0
∇ U (x , τ ) · ∇ η(x , τ )dx dτ
σ
η(x , τ )dx dτ
q(τ ) 0
t
σ
fn (x , τ )η(x , τ )dx dτ ∀t ∈ (0, T ],
(2.68)
σ
◦
for all η ∈ W 21,0 ( T ). First, consider problem (2.34) assuming that u0n (x ) = 0, fn (x , t) = 0. T HEOREM 2.6. Let F ∈ W21 (0, T ), u0n (x ) = 0, fn (x , t) = 0 and let there holds the compatibility condition F (0) = 0.
(2.69)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
487
Then for arbitrary T ∈ (0, ∞] problem (2.34) admits a unique weak solution (U (x , t), q(t)). There holds the estimate sup U (·, t); W21 (σ )2 + U ; W21,1 ( T )2 + q; L2 (0, T )2
t∈[0,T ]
≤ cF ; W21 (0, T )2 .
(2.70)
Moreover, if ∂σ ∈ C 2l+2 , F ∈ W2l+1 (0, T ), l ≥ 0, and F (t) satisfies compatibility conditions (2.61), then (U, q) ∈ W22l+2,l+1 ( T ) × W2l (0, T ) and there holds the estimate U ; W22l+2,l+1 ( T )2 + q; W2l (0, T )2 ≤ cF ; W2l+1 (0, T )2 .
(2.71)
P ROOF. The approximate solution (U (N ) (x , t), q (N ) (t)) constructed in Section 2.2.2 is a solutions to problem (2.47) with ∞
h(x ) = h(N ) (x ) =
βk uk (x )
k=N +1
(N ) ; L (σ ) = 0 and, thus, for (see (2.46)). Since ∞ 2 k=1 βk uk (x ) = 1, we have limN →∞ h sufficiently large N h(N ) ; L2 (σ ) ≤ δ with δ satisfying conditions of Lemma 2.1. The approximate solution is sufficiently regular, i.e. (U (N ) (x , t), q (N ) (t)) is such that all considerations of Section 2.2.3 are legitimate. By Lemma 2.1, (U (N ) (x , t), q (N ) (t)) obey estimate (2.48) with constant independent of N . Therefore, there exists a subsequence {(U (Nl ) (x , t), q (Nl ) (t))} such that (N ) {U (Nl ) (x , t)}, {∇ U (Nl ) (x , t)}, {Ut l (x , t)}, converge weakly in L2 ( T ) and {q (Nl ) (t)} converges weakly in L2 (0, T ). For (U (Nl ) (x , t), q (Nl ) (t)) holds the integral identity
T
0
σ
(Nl )
Ut
T
=
(x , t)η(x , t)dx dt + ν
0
T
∇ U (Nl ) (x , t) · ∇ η(x , t)dx dt
σ
(1 − h(Nl ) (x ))η(x , t)dx dt.
q (Nl ) (t)
0
(2.72)
σ
Since h(Nl ) ; L2 (σ ) → 0, we can pass in (2.72) to a limit as Nl → ∞ and we receive for the limit pair of functions (U (x , t), q(t)) integral identity (2.68). Moreover, each U (N ) (x , t) satisfies the flux condition and, therefore, the same condition is true for U (x , t). Estimate (2.48) remains valid for (U (x , t), q(t)). Let us prove the uniqueness of the weak solution (U (x , t), q(t)). Assume that F (t) = 0. Taking in (2.68) η(x , t) = U (x , t) we obtain 1 2
|U (x , t)| dx + ν σ
2
t 0
σ
|∇ U (x , τ )|2 dx dτ
488
K. Pileckas
=
t
U (x , τ ) dx dτ.
q(τ ) 0
σ
Since
U (x , τ ) dx = F (t) = 0,
σ
we have, 1 2
|U (x , t)|2 dx + ν
t 0
σ
|∇ U (x , τ )|2 dx dτ = 0
σ
and, therefore, U (x , t) = 0. From equation (2.34) it follows that q(t) = 0. Finally, if ∂σ ∈ C l+2 , F ∈ W2l+1 (0, T ) and F satisfies the compatibility conditions (2.61), then q (Nl ) ∈ W2l (0, T ) and for {q (Nl ) (t)} hold conditions (2.62) (see 2.45)). By Lemma 2.2, approximate solutions satisfy, in addition, estimate (2.63): U (Nl ) ; W22l+2,l+1 ( T )2 + q (Nl ) ; W2l (0, T )2 ≤ c(1 + 1 − h(Nl ) ; W22l (σ )2 )F ; W2l+1 (0, T )2 and, therefore, for the limit functions U (x , t) and q(t) holds (2.71).
Now, consider the general case of nonzero u0n (x ) and fn (x , t). ◦
T HEOREM 2.7. Let u0n ∈ W 12 (σ ), fn ∈ L2 ( T ) and let there holds the compatibility condition F (0) = u0n (x ) dx. (2.73) σ
Then for arbitrary T ∈ (0, ∞] there exists a unique solution (U (x , t), q(t)) of problem (2.34) such that sup U (·, t); W21 (σ )2 + U ; W21,1 ( T )2 dt + q; L2 (0, T )2
t∈[0,T ]
≤ c(F ; W21 (0, T )2 + u0n ; W21 (σ ) + fn ; L2 ( T )2 ).
(2.74)
Moreover, if ∂σ ∈ C 2l+2 , F ∈ W2l+1 (0, T ), u0n ∈ W22l+1 (σ ), fn ∈ W22l,l ( T ), l ≥ 0, and if there hold the compatibility conditions of order l: ds F (0) = dt s
u0 (x ) dx , (s)
σ
u0 (x )|∂σ = 0, (s)
s = 0, . . . , l,
(2.75)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
489
where u0 (x ) = u0n (x ),
u0 (x ) = ν u0
(0)
(s)
(s−1)
(x ) + f0
(s) f0 (x ) =
(x ),
s = 1, . . . , l,
(x , 0),
s = 0, . . . , l,
(s−1)
∂s ∂t s
fn
then (U, q) ∈ W22l+2,l+1 ( T ) × W2l (0, T ) and the following estimate U ; W22l+2,l+1 ( T )2 + q; W2l (0, T )2 ≤ c(F ; W2l+1 (0, T )2 + u0n ; W22l+1 (σ )2 + fn ; W22l,l ( T )2 )
(2.76)
is valid. P ROOF. We look for the solution (U (x ), q(t)) of problem (2.34) in the form (U (x , t), q(t)) = (U (1) (x , t), 0) + (U (2) (x , t), q(t)), where U (1) (x , t) is the solution of the heat equation 3 Ut(1) (x , t) − ν U (1) (x , t) = fn (x , t), U (1) (x , t)|∂σ = 0,
U (1) (x , 0) = u0n (x )
(2.77)
and (U (2) (x , t), q(t)) is the solution of the inverse problem ⎧ (2)
(2) ⎪ ⎨ Ut (x , t) − ν U (x , t) = q(t), U (2) (x , t)|∂σ = 0, U (2) (x , 0) = 0, ⎪ ⎩ (2)
8 σ U (x , t) dx = F (t)
(2.78)
(1)
8(t) = F (t) − with F σ U (x , t) dx . The solvability of (2.77) is well know (see, e.g. [29]). There exists a unique weak solution U (1) (x , t) obeying the estimate sup U (1) (·, t); W21 (σ )2 + U (1) ; W21,1 ( T )2
t∈[0,T ]
≤ c(u0n ; W21 (σ )2 + fn ; L2 ( T )2 ).
(2.79)
Moreover, if ∂σ ∈ C 2l+2 and u0 (x )|∂σ = 0, s = 0, . . . , l, then U (1) ∈ W22l+2,l+1 ( T ), (s)
U (1) ; W22l+2,l+1 ( T )2 ≤ c(u0n ; W22l+1 (σ )2 + +fn ; W22l,l ( T )2 ) 8(t) in (2.78) admits the estimate (see [87], [29]). The flux function F 8; W l+1 (0, T )2 F 2
(2.80)
490
K. Pileckas
l+1
≤ c F ; W2l+1 (0, T )2 +
j =0 0
9 j 92 9 9 ∂ U (1) (·, t); L2 (σ )9 dt 9 ∂t j 9
T9
≤ c(F ; W2l+1 (0, T )2 + u0n ; W22l+1 (σ )2 + +fn ; W22l,l ( T )2 ). (2.81) Moreover, we conclude from (2.73) and (2.75) that 8(0) = F (0) − F
u0 (x ) dx = 0, σ
ds 8 F (0) = 0, dt s
s = 0, . . . , l.
Therefore, by Theorem 2.6, problem (2.78) admits a unique weak solution (U (2) (x , t), q(t)) satisfying estimate (2.70). If ∂σ ∈ C 2l+2 and F ∈ W2l+1 (0, T ), then (U (2) , q) ∈ W22l+2,l+1 ( T ) × W2l (0, T ) and there holds (2.71). Estimate (2.74) follows from inequalities (2.79), (2.81), (2.70) and estimate (2.76) follows from (2.80), (2.81), (2.71). 2.2.5. Uniform estimates in Hölder spaces for the solution of integral equation (2.41) In this section we prove uniform (with respect to N ) estimates in Hölder spaces for the solution q (N ) (t) of the integral equation (2.41). Consider first the auxiliary integral equation f (N ) (t) −
t N 1 2 βk λk exp(−λk (t − τ ))f (N ) (τ ) dτ = g(t). 1N 0
(2.82)
k=2
This equation differs from (2.41) by the second term where the sum starts from k = 2. L EMMA 2.3. Suppose that g ∈ C δ ([0, T ]) and g(0) = 0. Then equation (2.82) admits a unique solution f (N ) ∈ C δ ([0, T ]). Moreover, f (N ) (0) = 0 and |f (N ) (t)| ≤
1N sup |g(τ )|, β12 τ ∈[0,t]
f (N ) ; C δ ([0, T ]) ≤ 2
∀t ∈ [0, T ],
(2.83)
1N g; C δ ([0, T ]). β12
(2.84)
P ROOF. Define (N )
f0
(t) = g(t), . . . , fn(N ) (t) t N 1 2 (N ) = βk λk exp(−λk (t − τ ))fn−1 (τ ) dτ, . . . , 1N 0 k=2
f
(N )
(t) =
∞
n=1
fn(N ) (t).
(2.85)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
491
Then (N )
|f0
(t)| ≤ sup |g(τ )|, τ ∈[0,t]
1 sup |g(τ )| βk2 λk 1N τ ∈[0,t] N
|f1(N ) (t)| ≤
k=2
t
exp(−λk (t − τ )) dτ
0
γN 1 sup |g(τ )| βk2 (1 − exp(−λk t)) ≤ sup |g(τ )|, . . . , 1N τ ∈[0,t] 1N τ ∈[0,t] N
≤
k=2
|fn(N ) (t)| ≤
γN 1N
n sup |g(τ )|, . . . ,
τ ∈[0, t]
γN 2 2 where γN = N k=2 βk = 1N − β1 . Since 1N < 1, the series (2.85), determining the solution to (2.82), converges absolutely and uniformly on the interval [0, T ]. Moreover, |f
(N )
(t)| ≤ sup |g(τ )| τ ∈[0,t]
=
∞
γN n n=0
1N
1N 1N sup |g(τ )| = 2 sup |g(τ )|, 1N − γN τ ∈[0,t] β1 τ ∈[0,t]
and (2.82) implies that f (N ) (0) = 0. (N ) Now, estimate the Hölder norm of f (N ) (t). Since fn−1 (0) = 0, we have " t+h N " 1 (N ) |fn(N ) (t + h) − fn(N ) (t)| = "" βk2 λk exp(−λk (t + h − τ ))fn−1 (τ ) dτ 1N 0 k=2
" t N " 1 2 (N ) − βk λk exp(−λk (t − τ ))fn−1 (τ ) dτ "" 1N 0 k=2
t N 1 2 (N ) (N ) ≤ βk λk exp(−λk (t − τ ))|fn−1 (τ + h) − fn−1 (τ )| dτ 1N 0 k=2
h N 1 2 (N ) (N ) + βk λk exp(−λk (t + h − τ ))|fn−1 (τ ) − fn−1 (0)| dτ, 1N 0 k=2
n = 1, 2, . . . . Estimating the right-hand sides here, we find for all t, t + h ∈ [0, T ] the inequalities (N )
|f0
(N )
(t + h) − f0
(t)| = |g(t + h) − g(t)| ≤ hδ g; C δ ([0, T ]),
492
K. Pileckas
γN ,..., 1N n γN (N ) (N ) δ δ |fn (t + h) − fn (t)| ≤ 2h g; C ([0, T ]) ,.... 1N (N )
|f1
(N )
(t + h) − f1
(t)| ≤ 2hδ g; C δ ([0, T ])
Hence, fn(N ) ; C δ ([0, T ]) ≤ 2g; C δ ([0, T ]) and consequently, for f (N ) (t) =
∞
(N ) n=1 fn (t)
γN 1N
n
holds estimate (2.84).
(δ ([0, T ]) = { ∈ C δ ([0, T ]) : (0) = 0} an operator BN by the Define on the space C formula t N 1 2 βk λk exp(−λk (t − τ ))(τ ) dτ. BN (t) = (t) − 1N 0 k=2
−1 (δ : C ([0, T ]) → Lemma 2.3 implies that there exists a bounded inverse operator BN δ ( C ([0, T ]) and −1 (δ (δ ([0, T ]) ≤ 2 1N . BN ; C ([0, T ]) → C β12
(2.86)
In addition, −1 |BN g(t)| ≤
1N sup |g(τ )|. β12 τ ∈[0,t]
(2.87)
Rewrite the equation (2.41) in the form BN q
(N )
1 2 (t) − β λ1 1N 1
t
exp(−λ1 (t − τ ))q (N ) (τ ) dτ = ϕ (N ) (t).
(2.88)
0
−1 (N ) g (t), we obtain Introducing the notation g (N ) (t) = BN q (N ) (t) and q (N ) (t) = BN from (2.88) the integral equation
g
(N )
1 2 (t) − β λ1 1N 1
t 0
−1 (N ) exp(−λ1 (t − τ ))(BN g )(τ ) dτ = ϕ (N ) (t).
(2.89)
(δ ([0, T ]). Then equation (2.89) admits a unique solution g (N ) ∈ L EMMA 2.4. Let ϕ (N ) ∈ C δ ( ([0, T ]). There holds the estimates C |g (N ) (t)| ≤ exp(λ1 t) sup |ϕ (N ) (τ )|, τ ∈[0,t]
(2.90)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
g (N ) ; C δ ([0, T ]) ≤ c exp(2λ1 T )ϕ (N ) ; C δ ([0, T ]).
493
(2.91)
P ROOF. Set g (N ) (t) =
∞
gn(N ) (t),
(2.92)
n=0
where (N )
g0 (t) = ϕ (N ) (t), (N ) g1 (t) =
1 2 β λ1 1N 1
gn+1 (t) =
1 2 β λ1 1N 1
t
−1 exp(−λ1 (t − τ ))(BN g0 )(τ ) dτ, . . . , (N )
0
t 0
−1 (N ) exp(−λ1 (t − τ ))(BN gn )(τ ) dτ, . . . .
Using (2.87), we obtain by induction that (N )
|g1 (t)| ≤
1N 1 sup |ϕ (N ) (τ )| β12 λ1 2 1N β1 τ ∈[0,t]
t
exp(−λ1 (t − τ )) dτ
0
≤ λ1 t sup |ϕ (N ) (τ )|, . . . , τ ∈[0,t]
(N )
|gn+1 (t)| ≤
1 2 β λ1 1N 1
t
exp(−λ1 (t − τ ))
0
λn+1 sup |ϕ (N ) (τ )| ≤ 1 n! τ ∈[0,t]
t
1N sup |gn(N ) (s)| dτ β12 s∈[0,τ ]
τ n dτ =
0
(λ1 t)n+1 sup |ϕ (N ) (τ )|, . . . . (n + 1)! τ ∈[0,t]
Thus, ∞
n=0
|gn(N ) (t)| ≤ sup |ϕ (N ) (τ )| τ ∈[0,t]
∞
(λ1 t)n n=0
n!
= exp(λ1 t) sup |ϕ (N ) (τ )|,
(2.93)
τ ∈[0,t]
and consequently, the series (2.92), determining the solution to (2.89), converges absolutely and uniformly over each finite interval [0, T ]. Thus, |g (N ) (t)| ≤ exp(λ1 t) sup |ϕ (N ) (τ )| ≤ exp(λ1 T ) sup |ϕ (N ) (t)|. τ ∈[0,T ]
t∈[0,T ]
(2.94)
494
K. Pileckas
Using (2.86), it is easy to show that for the difference |g (N ) (t + h) − g (N ) (t)| there holds the inequality |g (N ) (t + h) − g (N ) (t)| ≤ chδ exp(2λ1 T )g (N ) ; C δ ([0, T ])
∀t, t + h ∈ [0, T ].
(δ ([0, T ]), and the estimate (2.91) is valid. Therefore, g (N ) ∈ C
(δ ([0, T ]). Then integral equation (2.12) admits a unique soluL EMMA 2.5. Let ϕ (N ) ∈ C (N ) δ ( ([0, T ]) and ∈C tion q 1N exp(λ1 t) sup |ϕ (N ) (τ )|, β12 τ ∈[0,t]
(2.95)
q (N ) ; C δ ([0, T ]) ≤ c exp(2λ1 T )ϕ (N ) ; C δ ([0, T ]).
(2.96)
|q (N ) (t)| ≤
−1 (N ) P ROOF. To prove this lemma, it suffices to take q (N ) (t) = BN g (t) and to apply estimates (2.86), (2.87), (2.90) and (2.91).
(l+δ ([0, T ]) the subspace of functions from C l+δ ([0, T ]) satisfying the conDenote by C ditions h(0) = 0,
dl d h(0) = 0, . . . , l h(0) = 0. dt dt
(2.97)
(l+δ ([0, T ]), l ≥ 0. Then there exists a unique solution q (N ) ∈ L EMMA 2.6. Let ϕ (N ) ∈ C l+δ ( ([0, T ]) to (2.41) and C " s " s " " " d (N ) " 1N " d (N ) " " " "≤ ", q (t) exp(λ t) sup ϕ (τ ) 1 " dt s " s " β2 " τ ∈[0,t] dτ 1
s = 0, . . . , l,
(2.98)
q (N ) ; C l+δ ([0, T ]) ≤ c exp(2λ1 T )ϕ (N ) ; C l+δ ([0, T ]).
(2.99) s
d (N ) (t), P ROOF. The results of the lemma follow from the fact that the derivatives dt sq s = 1, . . . , l, satisfy the same integral equation (2.41) with the right-hand sides equal to d s (N ) (t) (see (2.45)). dt s ϕ
2.2.6. Existence of a solution in Hölder spaces Let us prove first that the sequence {q (N ) (t)} converges in the norm of C l+δ ([0, T ]). (l+1+δ ([0, T ]) with l ≥ 0 and δ ∈ (0, 1), then the sequence {q (N ) (t)} L EMMA 2.7. If F ∈ C converges in the norm of C l+δ ([0, T ]) and the limit function q(t) satisfies the following inequalities " s " s+1 " " "d " " " " " ≤ 1 exp(λ1 t) sup " d ", q(t) F (τ ) " dt s " dτ s+1 " β2 " τ ∈[0,t] 1
s = 0, . . . , l,
(2.100)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
q; C l+δ ([0, T ]) ≤ c exp(2λ1 T )F ; C l+1+δ ([0, T ]).
495
(2.101)
P ROOF. It is easy to see that the difference Q(N,M) (t) = q (N +M) (t) − q (N ) (t) satisfies the relation Q
(N,M)
t N 1 2 (t) − βk λk exp(−λk (t − τ ))Q(N,M) (τ ) dτ 1N 0 k=1
=
1 1N +M
+
−
1 1N
N
N +M
1 1N +M
βk2 λk
k=1
βk2 λk
k=N +1
t
t
exp(−λk (t − τ ))q (N +M) (τ ) dτ
0
exp(−λk (t − τ ))q (N +M) (τ ) dτ
0
+ (ϕ (N +M) (t) − ϕ (N ) (t)) ≡ I1
(N,M)
(N,M)
(t) + I2
(N,M)
(t) + I3
(t).
Since lim 1N =
N →∞
∞
βk2 = |σ |,
ϕ (N ) (t) = F (t)/1N ,
k=1
we find using (2.99) that (N,M)
; C l+δ ([0, T ]) " N " " 1 1 "" 2 (N +M) l+δ − βk q ; C ([0, T ]) ≤ c"" 1N +M 1N "
I1
k=1
" " ≤ c""
1
1N +M " " 1 ≤ c"" 1 N +M
" N 1 "" 2 βk exp(2λ1 T )ϕ (N +M) ; C l+δ ([0, T ]) 1N " k=1 " 1 "" exp(2λ1 T )F ; C l+δ ([0, T ]) → 0, − 1N " −
(N,M) ; C l+δ ([0, T ]) I2
≤
c 1N +M
exp(2λ1 T )F ; C l+δ ([0, T ])
N +M
βk2 → 0,
k=N +1
" " 1 (N,M) l+δ I3 ; C ([0, T ]) = "" 1
N +M
" 1 "" l+δ F ; C ([0, T ]) → 0 − 1N "
as N, M → ∞. Since the function Q(N,M) (t) is a solution to (2.41) with the right-hand (N,M) (N,M) (N,M) side equal to I1 (t) + I2 (t) + I3 (t), by Lemma 2.6 (see estimate (2.99)) we
496
K. Pileckas
get Q(N,M) ; C l+δ ([0, T ]) (N,M) 1N (N,M) (N,M) ; C l+δ ([0, T ]) → 0 + I2 + I3 ≤ 2 2 exp(2λ1 T ) I1 β1 (l+δ ([0, T ]), and there exists as N, M → ∞. Thus, {q N (t)} is a Cauchy sequence in C l+δ ( ([0, T ]). Estimates (2.100) and (2.101) for q(t) follow from the a limit function q ∈ C corresponding estimates for q N (t). T HEOREM 2.8. Let ∂σ ∈ C 2l+2+2δ , F ∈ C l+1+δ ([0, T ]), u0n ∈ C 2l+2+2δ (σ ), fn ∈ C 2l+2δ,l+δ ( T ) with l ≥ 0, δ ∈ (0, 1/2), T ∈ (0, ∞). Suppose that there hold the compatibility conditions of order l + 1: dsF (0) = dt s
σ
u0 (x ) dx , (s)
u0 (x )|∂σ = 0, (s)
s = 0, . . . , l + 1,
(2.102)
(see the notations after the formula (2.75)). Then problem (2.34) admits a unique solution (l+δ ([0, T ]) and the following estimate (U, q) ∈ C 2l+2+2δ,l+1+δ ( T ) × C U ; C 2l+2+2δ,l+1+δ ( T ) + q; C l+δ ([0, T ]) ≤ c(T ) F ; C l+1+δ ([0, T ]) (2.103) + u0n ; C 2l+2+2δ (σ ) + fn ; C 2l+2δ,l+δ ( T ) is valid. P ROOF. First, consider the case u0n (x ) ≡ 0, fn (x , t) ≡ 0. Suppose that F ∈ (l+1+δ ([0, T ]). By construction C U (N ) (x , t) =
N
k=1
t
βk
exp(−λk (t − τ ))q (N ) (τ ) dτ uk (x )
0
solves the initial-boundary value problem for the heat equation 3
(N )
Ut
(x , t) − ν U (N ) (x , t) = q (N ) (t)(1 − h(N ) (x )), U (N ) (x , t)|∂σ = 0
where h(N ) (x ) =
∞
k=N +1 βk uk (x
).
U (N ) (x , 0) = 0,
(2.104)
Hence,
U (N ) ; W22,1 ( T ) ≤ cq (N ) (t)(1 − h(N ) (x )); L2 ( T ),
(2.105)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
497
and the constant c is independent of N . Using Lemma 2.7, it is easy to show that the sequence % N
(N )
(N ) (N )
q (t)(1 − h (x )) = q (t) βk uk (x ) k=1
converges in the norm of L2 ( T ) to q(t). Since problem (2.104) is linear, the difference V (N,M) (x , t) = U (N +M) (x , t) − U (N ) (x , t) is a solution to the analogous problem with the right-hand side equal to q (N +M) (t)
N +M
βk uk (x ) − q (N ) (t)
k=1
N
βk uk (x ).
k=1
From estimate (2.105) we conclude that {U (N ) (x , t)} is a Cauchy sequence in the space W22,1 ( T ), and the limit function U ∈ W22,1 ( T ) is a solution of the problem 3
Ut (x , t) − ν U (x , t) = q(t), U (x , t) = 0, U (x , 0) = 0.
(2.106)
The right-hand side q(t) of (2.106) belongs to the space C l+δ ([0, T ]) and depends only on t. Thus, q ∈ C 2l+2δ,l+δ ( T ), and q; C 2l+2δ,l+δ ( T ) ≤ cq; C l+δ ([0, T ]). Furthermore, q(t) satisfies the compatibility conditions q(0) = · · · =
dl q(0) = 0. dt l
Consequently, see [87], [29], U ∈ C 2l+2+2δ,l+1+δ ( T ), and U ; C 2l+2+2δ,l+1+δ ( T ) ≤ cq; C 2l+2δ,l+δ ( T ) ≤ cq; C l+δ ([0, T ]) ≤ c exp(2λ1 T )F ; C l+1+δ ([0, T ]). By construction, for all N the functions U (N ) (x , t) satisfy the condition σ
U (N ) (x , t) dx = F (t)
∀t ∈ [0, T ],
498
K. Pileckas
which holds for the limit function U (x , t) as well. Therefore, (U (x , t), q(t)) is a solution of inverse problem (2.34) for u0n (x ) ≡ 0, fn (x , t) ≡ 0 and estimate (2.103) is valid. Consider the case of nonzero initial data fn ∈ C 2l+2δ,l+δ ( T ), u0n ∈ C l+2+δ (σ ). We look for a solution to (2.34) in the form of a sum (U (x , t), q(t)) = (U (1) (x , t), 0) + (U (2) (x , t), q(t)), where U (1) (x , t) is a solution of the initial-boundary value problem 3
Ut (x , t) − ν U (1) (x , t) = fn (x , t), (1)
U (1) (x , t)|∂σ = 0,
U (1) (x , 0) = u0n (x ).
(2.107)
Since u0n (x ) and fn (x , t) satisfy the compatibility conditions of order l + 1 (see (2.102)), there exists a unique solution U (1) (x , t) to (2.107) for which holds the estimate (e.g. [87], [29]) U (1) ; C 2l+2+2δ,l+1+δ ( T ) ≤ c u0n ; C 2l+2+2δ (σ ) + fn ; C 2l+2δ,l+δ ( T ) .
(2.108)
For (U (2) (x , t), q(t)) we obtain the inverse problem ⎧ ⎪ ⎨
Ut(2) (x , t) − ν U (2) (x , t) = q(t),
⎪ ⎩
U (2) (x , 0) = 0, U (2) (x , t) = 0, (2)
(1) 8 σ U (x , t)dx = F (t) − σ U (x , t)dx = F (t).
(2.109)
Because of the second equalities in compatibility conditions (2.102) we see that ds 8 F (0) = 0, dt s
s = 0, . . . , l + 1.
8∈ C (l+1+δ ([0, T ]) and Moreover, F 8; C (l+1+δ ([0, T ]) ≤ c F ; C (l+1+δ ([0, T ]) + U (1) ; C 2l+2+2δ,l+1+δ ( T ) F (l+1+δ ([0, T ]) + u0n ; C 2l+2+2δ (σ ) ≤ c F ; C + fn ; C 2l+2δ,l+δ ( T ) . (l+δ ([0, T ]) to Therefore, there exists a solution (U (2) , q) ∈ C 2l+2+2δ,l+1+δ ( T ) × C (2.109) satisfying the estimate U (2) ; C 2l+2+2δ,l+1+δ ( T ) + q; C l+δ ([0, T ]) 8; C l+1+δ ([0, T ]) ≤ c(T )F
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
(l+1+δ ([0, T ]) + u0n ; C 2l+2+2δ (σ ) ≤ c(T ) F ; C + fn ; C 2l+2δ,l+δ ( T ) .
499
(2.110)
Thus, (U (x , t), q(t)), where U (x , t) = U (1) (x , t) + U (2) (x , t), is a solution to (2.34), and estimate (2.103) for it follows from (2.108), (2.110). The uniqueness of the solution follows from Theorem 2.6.
2.3. Generalized time-dependent Poiseuille flow 2.3.1. Definition of generalized time-dependent Poiseuille flow Consider now in the pipe # ⊂ R3 the following problem ⎧ ut (x, t) − νu(x, t) + (u(x, t) · ∇)u(x, t) + ∇p(x, t) = f(x, t), ⎪ ⎪ ⎪ ⎨ div u(x, t) = 0, ⎪ u(x, t)|∂ = 0, u(x, 0) = u0 (x), ⎪ ⎪ ⎩
σ u3 (x, t) dx = F (t),
(2.111)
where the data do not depend on x3 , i.e. f(x, t) = (f1 (x , t), f2 (x , t), f3 (x , t)),
u0 (x) = (u01 (x ), u02 (x ), u03 (x )).
As in the steady case, we look for the solution in the form U(x, t) = (U1 (x , t), U2 (x , t), U3 (x , t)), P (x, t) = p 8(x , t) − q(t)x3 + p0 (t).
(2.112)
Then (2.111) decomposes into two problems. For (U (x , t), p 8(x , t)) = ((U1 (x , t),
8(x , t)) we obtain the two-dimensional Navier–Stokes problem on the crossU2 (x , t)), p section σ : ⎧
8(x , t) = f (x , t), ⎪ ⎨Ut (x , t) − ν U (x , t) + (U (x , t) · ∇ )U (x , t) + ∇ p div U (x , t) = 0, ⎪ ⎩ U (x , t)|∂σ = 0, U (x , 0) = U 0 (x ), (2.113) and for U (x , t) = Un (x , t) we get the inverse problem: ⎧
⎪ ⎨Ut (x , t) − ν U (x , t) + (U (x , t) · ∇ )U (x , t) = q(t) + f (x , t),
U (x , t)|∂σ = 0, U (x , 0) = u0 (x ), ⎪ ⎩
σ U (x , t) dx = F (t),
500
K. Pileckas
(2.114) where f (x , t) = (f1 (x , t), f2 (x , t)), U 0 (x ) = (u01 (x ), u02 (x )), u0 (x ) = u03 (x ), f (x , t) = f3 (x , t). Solution of problem (2.111) having the form (2.112) we call generalized time-dependent Poiseuille flow. Generalized time-dependent Poiseuille flow could be defined also in a two-dimensional infinite strip. As in the steady case, such flow is unidirectional (see Remark 2.2). 2.3.2. Solvability of the Navier–Stokes problem (2.113) Let us consider in the bounded two-dimensional domain σ the Navier–Stokes problem (2.113). The following result is well known (see [28], Ch.VI). ◦
T HEOREM 2.9. Let ∂σ ∈ C 2 , U 0 ∈ W 12 (σ ), div U 0 (x ) = 0, f ∈ L2 ( T ). Then for arbitrary T ∈ (0, ∞] problem (2.113) admits a unique solution (U (x , t), p 8(x , t)) such that 2,1
T
T U ∈ W2 ( ), ∇ p 8 ∈ L2 ( ) and there holds the estimate 8; L2 ( T )2 ≤ A0 , U ; W22,1 ( T )2 + ∇ p
(2.115)
The constant A0 in (2.115) depends on the norms U 0 ; W21 (σ ) and f ; L2 ( T ). If T = ∞, then U (·, t); L2 (σ )2 + ∇ U (·, t); L2 (σ )2 → 0
as t → ∞.
(2.116)
2.3.3. Construction of Galerkin approximations for the solution of problem (2.114) Consider the case u0 (x ) = 0, f (x , t) = 0, i.e., consider the following problem ⎧
⎪ ⎨Ut (x , t) − ν U (x , t) + (U (x , t) · ∇ )U (x , t) = q(t), U (x , t)|∂σ = 0, U (x , 0) = 0, (2.117) ⎪ ⎩
F (0) = 0. σ U (x , t) dx = F (t), We construct the solution U (x , t) of (2.117) using Galerkin approximations. Let {uk (x )} ◦ be the same basis in L2 (σ ) ∩ W 12 (σ ) as in Section 2.2.2, i.e., uk (x ) are the eigenfunctions of problem (2.36). We look for the approximate solutions (U (N ) (x , t), q (N ) (t)) in the form U (N ) (x , t) =
N
yk (t)uk (x ), (N )
(2.118)
k=1 (N )
find the coefficients yk (t) from the following integral relations # (N ) Ut (x , t)uk (x ) + ν∇ U (N ) (x , t) · ∇ uk (x ) σ
+ (U (x , t) · ∇ )U
(N )
$ (x , t)uk (x ) dx = q (N ) (t)
uk (x ) dx , σ
(2.119)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
k = 1, . . . , N, and chose q (N ) (t), in order to satisfy the flux condition U (N ) (x , t) dx = F (t) ∀t ∈ [0, T ].
501
(2.120)
σ
Using the properties of the eigenfunctions uk (x ) and remembering that σ uk (x ) dx = βk , (N ) we derive for yk (t) the Cauchy problem for the system of ordinary differential equations 3
Y(N ) (t) + (J(N ) + A(N ) (t))Y(N ) (t) = β (N ) q (N ) (t), Y(N ) (0) = 0, (N )
(2.121)
(N )
where Y(N ) (t) = colomn(y1 (t), . . . , yN (t)), β (N ) = colomn(β1 , . . . , βN ), J(N ) = diag(λ1 , .. . , λN ) is a diagonal matrix and A(N ) (t) is a (N × N )-matrix with elements αlk (t) = σ (U (x , t)·∇ )ul (x )uk (x ) dx . Obviously, the linear system (2.121) is uniquely solvable. The fundamental matrix Z(N ) (t) of system (2.121) satisfies the equation
Z(N ) (t) + (J(N ) + A(N ) (t))Z(N ) (t) = O,
Z(N ) (0) = E(N ) ,
(2.122)
where E(N ) is the unit matrix. System (2.122) is equivalent to the integral equation Z(N ) (t) = exp(−J(N ) t)E(N ) −
t
exp(−J(N ) (t − τ ))A(N ) (τ )Z(N ) (τ ) dτ.
0
Therefore, there holds the estimate t (N ) (N ) A (τ ) dτ . Z (t) ≤ c exp −λ1 t + c1
(2.123)
0
The elements αlk (t) of the matrix A(N ) (t) satisfy the estimates " " " "
" " |αlk (t)| = " (U (x , t) · ∇ )ul (x )uk (x ) dx " σ
≤ U (·, t); L4 (σ )∇ ul ; L2 (σ )uk ; L4 (σ ) ≤ cU (·, t); L2 (σ )1/2 ∇ U (·, t); L2 (σ )1/2 ∇ ul ; L2 (σ ) × ∇ uk ; L2 (σ ) 1/2 1/2
≤ cλk λl
sup (∇ U (·, t); L2 (σ )1/2 )U (·, t); L2 (σ )1/2 .
t∈[0,T ]
Here we have used the relations ν∇ uk ; L2 (σ )2 = λk and estimate (I.1.7) together with Poincaré inequality (I.1.3). From Lemma I.1.3 we have ◦
U (·, t); W 12 (σ ) ≤ cU ; W22,1 ( T ).
502
K. Pileckas
Therefore, from Theorem 2.9 it follows that the coefficients αlk (t) are bounded and (see (2.115), (2.116)) lim αlk (t) = 0 ∀l, k.
t→∞
Hence, for each ε > 0 there exists T (ε) such that
t
|αlk (τ )| dτ ≤ εt
∀t ≥ T (ε).
(2.124)
0
Let us choose ε∗ > 0 so that c1
t
A(N ) (τ ) dτ ≤
0
λ1 t 2
∀t ≥ T∗ .
Then from (2.123) follows the estimate Z(N ) (t) ≤ c exp(−λ1 t/2).
(2.125)
The solution Y(N ) (t) of system (2.121) has the form
t
Y(N ) (t) =
Z(N ) (t)(Z(N ) (τ ))−1 β (N ) q (N ) (τ ) dτ.
0
Substituting (2.118) into (2.120) and using (2.126) we obtain
U (N ) (x , t) dx =
F (t) = σ
= β (N ) ·
N
k=1
t
(N )
yk (t)
uk (x ) dx = Y (N ) (t) · β (N )
σ
Z(N ) (t)(Z(N ) (τ ))−1 β (N ) q (N ) (τ ) dτ.
0
Thus, we have derive for q (N ) (t) the Volterra integral equation of the first type:
t
β (N ) · Z(N ) (t)(Z(N ) (τ ))−1 β (N ) q (N ) (τ ) dτ = F (t).
0
Differentiating this equation and using (2.122) we get |β (N ) |2 q (N ) (t) −
t
β (N ) · (J(N )
0
+A
(N )
(N )
(t))Z
(t)(Z(N ) (τ ))−1 β (N ) q (N ) (τ ) dτ = F (t),
(2.126)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
503
equivalently, q (N ) (t) −
(N ) 2 | 0 |β
1
=
β (N )
t
|β (N ) |2
· (J(N ) + A(N ) (t))Z(N ) (t)(Z(N ) (τ ))−1 β (N ) q (N ) (τ ) dτ
F (t).
(2.127)
(2.127) is the Volterra integral equation of the second type with the kernel K (N ) (t, τ ) =
β (N ) |β (N ) |2
· (J(N ) + A(N ) (t))Z(N ) (t)(Z(N ) (τ ))−1 β (N ) .
In virtue of (2.125) K (N ) (t, τ ) is bounded for all 0 ≤ τ ≤ t and, therefore (e.g. [39], [99]), for every F (t) ∈ L2 (0, T ) equation (2.127) has a unique solution q (N ) ∈ L2 (0, T ) and there holds the estimate q (N ) ; L2 (0, T ) ≤ CN F ; L2 (0, T ).
(2.128)
Note that Galerkin approximations U (N ) (x , t) for every N satisfy the flux condition:
U (N ) (x , t) dx = F (t).
(2.129)
σ
Multiplying relations (2.119) by yk(N ) (t) and summing them from k = 1 to k = N we find in virtue of (2.129) the following equality 1 d 2 dt
|U
(N )
σ
(N )
|Ut
|∇ U (N ) (x , t)|2 dx = q (N ) (t)F (t).
(2.130)
σ
Analogously, multiplying (2.119) by
σ
(x , t)| dx + ν 2
(x , t)|2 dx +
d (N ) dt yk (t)
ν d 2 dt
and summing over k, we get the relation
|∇ U (N ) (x , t)|2 dx
σ
(U (x , t) · ∇ )U (N ) (x , t)Ut
(N )
+ σ
(x , t) dx = q (N ) (t)F (t).
◦
(2.131)
Let U0 ∈ W 12 (σ ) be a solution to problem (1.3). If ∂σ ∈ C 2 , we have U0 ∈ W22 (σ ). Therefore, U0 (x ) can be represented as a sum
U0 (x ) =
∞
k=1
γk uk (x ),
γk = σ
U0 (x )uk (x ) dx ,
504
K. Pileckas
and this series converges in W22 (σ ). Multiplying (2.119) by γk and summing the obtained relations over k from 1 to N , we obtain (N ) (N ) (N ) Ut (x , t)U0 (x ) dx + ν ∇ U (N ) (x , t) · ∇ U0 (x ) dx σ
σ
(U (x , t) · ∇ )U (N ) (x , t)U0 (x ) dx (N )
+ σ
=q
(N )
U0 (x ) dx , (N )
(t) σ
N
where U0 (x ) = (N )
k=1 γk uk (x
).
(2.132)
Since
−ν U0 (x ) = 1, integrating by parts in the second term at the left-hand side of (2.132), we derive (N ) ν ∇ U (N ) (x , t) · ∇ U0 (x ) dx σ
U (N ) (x , t) U0 (x ) dx (N )
= −ν σ
U (N ) (x , t) U0 (x ) dx
= −ν σ
=
U (N ) (x , t) (U0 (x ) − U0 (x )) dx (N )
−ν σ
U (N ) (x , t) dx − ν
σ
= F (t) − ν σ
σ
U (N ) (x , t) (U0 (x ) − U0 (x )) dx (N )
U (N ) (x , t) (U0 (x ) − U0 (x )) dx . (N )
We have −ν (U0 (x ) − U0 (x )) = −ν (N )
∞
γk uk (x ) =
k=N +1
∞
γk λk uk (x )
k=N +1
and using the orthogonality in L2 (σ ) of eigenfunctions uk (x ), we conclude that (N ) ν U (N ) (x , t) (U0 (x ) − U0 (x )) dx = 0. σ
Therefore, relation (2.132) can be rewritten as follows: (N ) (N ) (N )
Ut (x , t)U0 (x ) dx + F (t) + (U (x , t) · ∇ )U (N ) (x , t)U0 (x ) dx σ
σ
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
U0 (x ) dx . (N )
= q (N ) (t) σ
505
(2.133)
2.3.4. A priori estimates L EMMA 2.8. Let ∂σ ∈ C 2 , F ∈ W21 (0, T ). Then for sufficiently large N there holds the estimate U (N ) ; W21,1 ( t ) + q (N ) ; L2 (0, t) ≤ cF ; W21 (0, t),
(2.134)
where the constant c does not depend on N and t. P ROOF. Arguing as in Lemma 2.1 from relation (2.130) we receive the estimate (compare with (2.50)) t 1 |U (N ) (x , t)|2 dx + ν |∇ U (N ) (x , τ )|2 dx dτ 2 σ 0 σ 1 t ε t (N ) 2 |q (τ )| dτ + |F (τ )|2 dτ. (2.135) ≤ 2 0 2ε 0 From (2.131) we get ν d (N ) 2
|Ut (x , t)| dx + |∇ U (N ) (x , t)|2 dx 2 dt σ σ (N ) = − (U (x , t) · ∇ )U (N ) (x , t)Ut (x , t) dx + q (N ) (t)F (t) σ
≤ sup(|U (x , t)|2 ) x∈σ
1 + 4
σ
|∇ U (N ) (x , t)|2 dx σ
(N )
|Ut
ε 1 (x , t)|2 dx + |q (N ) (t)|2 + |F (t)|2 . 2 2ε
Inequality (2.136) yields the estimate 2 d |∇ U (N ) (x , t)|2 dx ≤ sup(|U (x , t)|2 ) |∇ U (N ) (x , t)|2 dx dt σ ν x∈σ σ ε 1 + |q (N ) (t)|2 + |F (t)|2 ν εν and, therefore, |∇ U (N ) (x , t)|2 dx σ
≤e
2 t ν 0
m(τ ) dτ
t 0
2 τ ε (N ) 1 2 2 |q (τ )| + |F (τ )| e− ν 0 m(s) ds dτ ν εν
(2.136)
506
K. Pileckas
ε 2 t ≤ e ν 0 m(τ ) dτ ν
t
|q (N ) (τ )|2 dτ +
0
1 2 t m(τ ) dτ eν 0 εν
t
|F (τ )|2 dτ,
(2.137)
0
where m(t) = supx ∈σ (|U (x , t)|2 ). In virtue of (2.115) and Sobolev embedding theorem (see Lemma I.1.2)
T
m(t) dt =
0
T
sup (|U (x , t)| ) dt ≤ c 2
x ∈σ
0
T
0
U (·, t); W22 (σ )2 dt ≤ c A0 .
Hence, (2.137), (2.136) yield
|∇ U (N ) (x , t)|2 dx
σ
ε ≤ ecA0 ν
1 2
t 0
σ
t
|q (N ) (τ )|2 dτ +
0
(N ) |Ut (x , τ )|2 dx dτ
1 cA0 e εν
ν + 2
t
|F (τ )|2 dτ,
|∇ U (N ) (x , t)|2 dx
σ
t 1 t m(τ ) dτ ε |q (N ) (τ )|2 dτ + |F (τ )|2 dτ ε 0 0 0 t t ε 1 + |q (N ) (τ )|2 dτ + |F (τ )|2 dτ 2 0 2ε 0 t c3 (A0 ) t |q (N ) (τ )|2 dτ + |F (τ )|2 dτ. ≤ εc2 (A0 ) ε 0 0
t
≤ c1 (A0 )
Now, consider the relation (2.133): (N ) (N ) κ0 q (N ) (t) = Ut (x , t)U0 (x ) dx + F (t) σ
(U (x , t) · ∇ )U (N ) (x , t)U0 (x ) dx (N )
+ σ
+ q (N ) (t) σ
where κ0 =
κ02
σ
U0 (x ) dx . Thus,
t
(U0 (x ) − U0(N ) (x )) dx ,
|q (N ) (τ )|2 dτ
0
≤c
t 0
(2.138)
0
σ
(N ) (N ) Ut (x , τ )U0 (x ) dx
2
dτ + 0
t
|F (τ )|2 dτ
(2.139)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
507
& '2 t (N ) + sup |U0 (x )| |U (x , τ )|2 dx x ∈σ
0
|∇ U (N ) (x , τ )|2 dx dτ
× σ
t
+
|q
(N )
2
(τ )| dτ
(U0 (x
0
≤c
σ
σ
t 0
σ
(N ) |Ut (x , τ )|2 dx
(N ) ) − U0 (x )) dx
σ
2
(N ) |U0 (x )|2 dx
'2 & (N ) sup |U (x , τ )|2 dx + sup |U0 (x )| x ∈σ
×
t 0
τ ∈[0,t]
|∇ U
(N )
2
σ
σ
dτ +
t
|F (τ )|2 dτ
0
(x , τ )| dx dτ
σ
+ |σ |
|U0 (x ) − U0(N ) (x )|2 dx
t
|q (N ) (τ )|2 dτ.
0
By Lemma I.1.6 and by (2.115) sup U (·, τ ); W21 (σ ) ≤ cU ; W22,1 ( T ) ≤ cA0 .
t∈[0,T ]
Since U0 ∈ W22 (σ ), we have U0(N ) ; L2 (σ )2 + sup |U0(N ) (x )|2 x ∈σ
(N )
≤ cU0 ; W22 (σ )2 ≤ cU0 ; W22 (σ )2 := cB0 . (N )
Moreover, since U0 − U0 ; L2 (σ ) → 0 as N → ∞, for sufficiently large N holds
|U0 − U0 |2 dx ≤ (N )
σ
κ02 . 2|σ |
Therefore, 0
t
|q
(N )
t t (N ) 2
(τ )| dτ ≤ c4 (A0 , B0 ) |Ut (x , τ )| dx dτ + |F (τ )|2 dτ 2
+
0
0
σ
(N ) 2
|∇ U (x , τ )| dx dτ ,
t σ
0
(2.140)
508
K. Pileckas
where the constant c4 (A0 , B0 ) depends on A0 , B0 but is independent of t and N . Estimates (2.135), (2.139), (2.140) yield t 0
σ
(|∇ U (N ) (x , τ )|2 + |Ut(N ) (x , τ )|2 ) dx dτ
≤ εc5 (A0 )
t
0
≤ εc6 (A0 , B0 )
|q
(N )
t 0
c7 (A0 , B0 ) + ε
c6 (A0 ) (τ )| dτ + ε
t
(|F (τ )|2 + |F (τ )|2 ) dτ
0
(|∇ U (N ) (x , τ )|2 + |Ut
(N )
σ t
2
(x , τ )|2 ) dx dτ
(|F (τ )|2 + |F (τ )|2 ) dτ.
(2.141)
0
Taking in (2.141) ε = 1/2c6 (A0 , B0 ), we conclude t 0
(|∇ U (N ) (x , τ )|2 + |Ut
(N )
σ
t
≤ c8 (A0 , B0 )
(x , τ )|2 ) dx dτ
(|F (τ )|2 + |F (τ )|2 ) dτ
(2.142)
0
and from (2.140), (2.142) obtain the estimate
t
t
|q (N ) (τ )|2 dτ ≤ c9 (A0 , B0 )
0
(|F (τ )|2 + |F (τ )|2 ) dτ.
(2.143)
0
Estimate (2.134) follows now from (2.142), (2.143) and Poincaré inequality.
2.3.5. Solvability of problem (2.114) First, consider problem (2.117). T HEOREM 2.10. Let ∂σ ∈ C 2 , F ∈ W21 (0, T ), F (0) = 0. Then for arbitrary T ∈ (0, ∞] problem (2.117) admits a unique solution (U, q) ∈ W22,1 ( T ) × L2 (0, T ) and there holds the estimate U ; W22,1 ( t )2 + q; L2 (0, t)2 ≤ cF ; W21 (0, t)2
∀t ∈ (0, T ]
(2.144)
with constant c independent of t and T . P ROOF. The Galerkin approximations (U (N ) (x , t), q (N ) (t)) constructed in Section 2.3.3 satisfy integral relations (2.119). Let
η
(M)
(x , t) =
M
k=1
dk (t)uk (x ),
(2.145)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
509
where dk (t) are arbitrary continuous on [0, T ] functions. Multiplying relations (2.119) by dk (t), summing them by k = 1, . . . M, and integrating over t, we obtain the integral identity t # (N ) Ut (x , τ )η(M) (x , τ ) + ν∇ U (N ) (x , τ ) · ∇ η(M) (x , τ ) 0
σ
$ + (U (x , τ ) · ∇ )U (N ) (x , τ )η(M) (x , τ ) dx dτ t (N ) = q (τ ) η(M) (x , τ ) dx dτ ∀t ∈ (0, T ]. 0
(2.146)
σ
By Lemma 2.8, Galerkin approximations (U (N ) (x , t), q (N ) (t)) satisfy estimate (2.134). Hence, there exists a subsequence {(U (Nl ) (x , t), q (Nl ) (t))} such that {U (Nl ) (x , t)} converges weakly in W21,1 ( T ) and {q (Nl ) (t)} converges weakly in L2 (0, T ). Fixing M and passing in (2.146) Nl → ∞, we obtain for the limit functions U (x , t) and q(t) the integral identity t # Ut (x , τ )η(M) (x , τ ) + ν∇ U (x , τ ) · ∇ η(M) (x , τ ) 0
σ
$ + (U (x , τ ) · ∇ )U (x , τ )η(M) (x , τ ) dx dτ t = q(τ ) η(M) (x , τ ) dx dτ ∀t ∈ (0, T ]. 0
(2.147)
σ ◦
The set of functions η(M) (x , t) given by (2.145) is dense in the space W 21,0 ( T ) which consists of functions η ∈ W21,0 ( T ) with η(x , t)|∂σ = 0. Therefore, identity (2.147) is ◦
true not only for η(M) (x , t) having the form (2.145) but also for all η ∈ W 21,0 ( T ). Thus, (U, (x , t)q(t)) is a weak solution of problem (2.117) and, obviously, estimate (2.134) for (U (x , t), q(t)) remains valid. From the integral identity (2.147) it follows that for almost all t ∈ (0, T ] holds the identity
ν ∇ U (x , t) · ∇ ξ(x ) dx = [−Ut (x , t) − (U (x , t) · ∇ )U (x , t) σ
σ
+ q(t)]ξ(x ) dx
◦
∀ξ ∈ W 12 (σ ).
Therefore, U (x , t) may be considered as a weak solution of the Poisson equation 3 −ν U (x , t) = q(t) − (U (x , t) · ∇ )U (x , t) − Ut (x , t), U (x , t)|∂σ = 0. Since ∂σ ∈ C 2 , we have U (·, t) ∈ W22 (σ ) and (|U (x , t)|2 + |∇ U (x , t)|2 + |∇ 2 U (x , t)|2 ) dx σ
510
K. Pileckas
≤c
(|q(t)|2 + |Ut (x , t)|2 ) dx +
σ
|(U (x , t) · ∇ )U (x , t)|2 dx
σ
& '2 ≤ c |q(t)|2 + |Ut (x , t)|2 dx + sup |U (x , t)| |∇ U (x , t)|2 dx , x ∈σ
σ
σ
where |∇ 2 U (x , τ )|2 = |α|=2 |∂ α U/∂x1α1 ∂x2α2 |2 . Integrating the last inequality with respect to t and using (2.134) we obtain t t 2 2 U (·, τ ); W2 (σ ) dτ ≤ c |q(τ )|2 dτ + c(A0 )U ; W21,1 ( t )2 0
0
≤c
t
(|F (τ )|2 + |F (τ )|2 ) dτ.
0
This inequality together with (2.134) gives (2.144). Taking into account that σ (U (x , t) · ∇ )U (x , t)U (x , t) dx = 0, the uniqueness of the solution (U (x , t), q(t)) is proved just in the same way as in Theorem 2.6. Consider now the general case of problem (2.114). ◦
T HEOREM 2.11. Let ∂σ ∈ C 2 , u0 ∈ W 12 (σ ), f ∈ L2 ( T ), F ∈ W21 (0, T ) and let there hold the compatibility conditions F (0) = u0 (x ) dx . (2.148) σ
Then for arbitrary T ∈ (0, ∞] problem (2.114) admits a unique solution (U, q) ∈ W22,1 ( T ) × L2 (0, T ). There holds the estimate U ; W22,1 ( t )2 + q; L2 (0, t)2 ≤ c(f ; L2 ( t )2 + u0 ; W21 (σ )2 + F ; W21 (0, t)2 )
∀t ∈ (0, T ] (2.149)
with constant c independent of t and T . P ROOF. We look for the solution of problem (2.114) in the form (U (x , t), q(t)) = (U (1) (x , t), 0) + (U (2) (x , t), q(t)), where U (1) (x , t) is the solution of the initial-boundary value problem for the parabolic equation ⎧ (1)
(1)
(1) ⎪ ⎨Ut (x , t) − ν U (x , t) + (U (x , t) · ∇ )U (x , t) (2.150) = f (x , t), ⎪ ⎩ (1)
(1)
U (x , 0) = u0 (x ) U (x , t)|∂σ = 0,
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
511
and (U (2) (x , t), q(t)) is the solution of the inverse problem ⎧ (2)
(2)
(2) ⎪ ⎨Ut (x , t) − ν U (x , t) + (U (x , t) · ∇ )U (x , t) = q(t), ⎪ ⎩
σ
U (2) (x , 0) = 0, U (2) (x , t)|∂σ = 0, 8(t) := F (t) − U (1) (x , t). U (2) (x , t) = F
(2.151)
σ
The unique solvability of the parabolic problem (2.150) can be proved by standard methods, (e.g., [29]). For the solution U (1) (x , t) holds the estimate U (1) ; W22,1 ( t )2 ≤ c u0 ; W21 (σ )2 + f ; L2 ( T )2 .
(2.152)
8(t) in (2.151) obviously belongs to the space W 1 (0, T ) and The flux F 2 t 8; W21 (0, t)2 ≤ c F ; W21 (0, t)2 + F U (1) (·, τ ); L2 (σ )2 dτ + 0
0
t
(1)
Ut (·, τ ); L2 (σ )2 dτ
≤ c(F ; W21 (0, T )2 + u0 ; W21 (σ )2 + f ; L2 ( T )2 ).
(2.153)
Moreover, in virtue of the compatibility condition (2.148): 8(0) = F (0) − F
u0 (x ) dx = 0.
σ
Therefore, by Theorem 2.10 there exists a unique solution (U (2) , q) ∈ W22,1 ( T ) × L2 ( T ) of problem (2.151). Estimate (2.149) follows from inequalities (2.144), (2.152) and (2.153). R EMARK 2.6. All obtained results remain valid also for two-dimensional infinite strip # = {(x1 , x2 ) ∈ R2 ; x1 ∈ (0, h0 ), x2 ∈ R}. However, in the two-dimensional case the generalized Poiseuille flow is always unidirectional (see Remark 2.2). R EMARK 2.7. Note that because of the structure of the nonlinear term in Navier–Stokes equations, unidirectional Poiseuille solutions (both steady and time-dependent) solve also the linear Stokes (or time-dependent Stokes) problem. For the generalized Poiseuille solutions having all three components of the velocity field U(x , t) this is not the case. These solutions satisfy only nonlinear steady and non-steady Navier–Stokes problems (1.9) and (2.111). However, it is obvious, that all result remain true, if we consider the Stokes or time-dependent Stokes problems (the corresponding proofs only simplify) instead of (1.9) and (2.111). We refer to such solutions as to generalized Poiseuille flows corresponding to the linear Stokes problem.
512
K. Pileckas
2.4. Behavior of the Poiseuille flow as t → ∞ In this section we study a behavior of the Poiseuille flows as t → ∞ assuming some additional decay of the data. First, we consider the case of unidirectional flows, i.e., we study the behavior as t → ∞ of the solution U (x , t) to problem (2.34). For simplicity we restrict ourselves by the case fn (x , t) = 0. l (0, ∞) be the space of functions with the finite norm Let W2,μ l (0, ∞) = exp(μt)F (t); W2l (0, ∞). F ; W2,μ ◦
1 (0, ∞) with T HEOREM 2.12. Let ∂σ ∈ C 2 , fn (x , t) = 0, u0n ∈ W 12 (σ ) and F ∈ W2,μ μ > 0. Assume that there holds the compatibility condition (2.73). Then the solution (U (x , t), q(t)) of problem (2.34) satisfies the estimate
exp(γ∗ t) |U (x , t)|2 dx + ν |∇ U (x , t)|2 dx +
σ
0
σ
t
exp(γ∗ τ )
|Uτ (x , τ )|2 dx dτ +
t 0
σ
exp(γ∗ τ )|q(τ )|2 dτ
1 (0, ∞)2 + u0n ; W21 (σ )2 ), ≤ c(F (t); W2,μ
(2.154)
where γ∗ = min{λ1 , 1, 2μ}. P ROOF. Arguing as in Lemma 2.1 we derive for the approximate solutions (U (x , t), q(t)) of problem (2.34) the estimates 1 d |U (x , t)|2 dx + ν |∇ U (x , t)|2 dx 2 dt σ σ ≤ ε|q(t)|2 + cε |F (t)|2 , ν d 2 dt
|∇ U (x , t)|2 dx +
σ
(2.155)
|Ut (x , t)|2 dx
σ
≤ ε|q(t)| + cε |F (t)| , 2
2
(2.156)
and |q(t)|2 ≤ c
|Ut (x , t)|2 dx + |F (t)|2
(2.157)
σ
where ε is an arbitrary positive number. For sufficiently small ε (2.155)–(2.157) yield 1 d 2 dt
(|U (x , t)|2 + ν|∇ U (x , t)|2 ) dx σ
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
|∇ U (x , t)|2 dx +
+ν σ
1 2
513
|Ut (x , t)|2 dx
σ
≤ c(|F (t)|2 + |F (t)|2 ).
(2.158)
In the domain σ holds the Poincaré inequality (see I.1.3)):
ν |U (x , t)| dx ≤ λ 1 ω
2
|∇ U (x , t)|2 dx , ω
where λ1 is the first eigenvalue of problem (2.36). Therefore, from (2.158) it follows that
d dt
(|U (x , t)|2 + ν|∇ U (x , t)|2 ) dx
σ
+ γ∗
|U (x , t)| dx + ν 2
σ
|∇ U (x , t)| dx 2
+
σ
|Ut (x , t)|2 dx
σ
≤ c(|F (t)|2 + |F (t)|2 )
(2.159)
with γ∗ = min{λ1 , 1, 2μ}. Multiplying (2.159) by exp(γ∗ t) and integrating over t we receive |U (x , t)|2 dx + ν |∇ U (x , t)|2 dx exp(γ∗ t) σ
t
+
exp(γ∗ τ )
0
t
≤c 0
σ
|Uτ (x , τ )|2 dx dτ
σ
exp(γ∗ τ )(|F (τ )| + |F (τ )| ) dτ 2
2
+ u0n ; W21 (σ )2
◦ ≤ c F ; W 12,μ (0, ∞)2 + u0n ; W21 (σ )2 .
(2.160)
Relations (2.160) and (2.157) yield 0
t
◦
exp(γ∗ τ )|q(τ )|2 dτ ≤ cF ; W 12,μ (0, ∞)2 .
(2.161)
Estimate (2.154) follows from (2.160) and (2.161). ◦
2 (0, ∞), u ∈ W 3 (σ ) ∩ T HEOREM 2.13. Let ∂σ ∈ C 4 , F ∈ W2,μ W 12 (σ ), fn (x , t) = 0 0n 2 and let there holds the compatibility conditions (2.75) of the first order. Then the solution (U (x , t), q(t)) of problem (2.34) obeys the estimate
Ut (·, t); L2 (σ )2 + U (·, t); W22 (σ )2 + |q(t)|2 2 ≤ c exp(−2γ∗ t) F ; W2,μ (0, ∞)2 + u0n ; W23 (σ )2 ,
(2.162)
514
K. Pileckas
where ⎧ ⎪ ⎨μ for 0 < μ < λ1 , γ∗ = λ1 for μ > λ1 , ⎪ ⎩ λ1 − ε for μ = λ1 . Here ε is an arbitrary positive number. The constant c in (2.162) tends to infinity as μ → λ1 in first two cases and c tends to infinity as ε → 0 in the third case. P ROOF. According to Theorem 2.6, problem (2.34) admits a unique solution (U, q) ∈ W24,2 (σ × (0, ∞)) × W21 (0, ∞). Differentiating equation (2.34) with respect to t we get Utt (x , t) − ν Ut (x , t) = q (t).
(2.163)
Moreover, using conditions (2.75) we find from (2.341 ) that q(0) = 0 and Ut (x , 0) = ν u0n (x ). Multiplying (2.163) by Ut (x , t) and integrating by parts over σ gives the relation
d dt
|Ut (x , t)|2 dx + 2ν
σ
|∇ Ut (x , t)|2 dx = 2q (t)F (t).
σ
Applying Poincaré inequality to the second summand on the left-hand side of the last relation, we get
d dt
|Ut (x , t)|2 dx ≤ 2q (t)F (t).
|Ut (x , t)| dx + 2λ1 2
σ
σ
Therefore,
|Ut (x , t)|2 dx
exp(2λ1 t) σ
t
≤2
exp(2λ1 τ )q (τ )F (τ ) dτ +
0
= −2
|Ut (x , 0)|2 dx
σ t
exp(2λ1 τ )q(τ )F (τ ) dτ − 4λ1
0
+ 2 exp(2λ1 t)q(t)F (t) + ν 2
t
exp(2λ1 τ )q(τ )F (τ ) dτ
0
| u0n (x )|2 dx .
σ
Since 2 (0, ∞)2 , sup(exp(μt)(|F (t)|2 + |F (t)|2 )) ≤ cF ; W2,μ t≥0
(2.164)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
515
from (2.164) follows the estimate 2 |Ut (x , t)|2 dx ≤ c∗ F ; W2,μ (0, ∞) exp(−μt)|q(t)| σ
t 1/2 + exp(−2λ1 t) exp((4λ1 − 2μ)τ )|q(τ )|2 dτ 0
+ c∗ exp(−2λ1 t)u0n ; W22 (σ )2 .
(2.165)
By estimate (2.76), sup |q(t)| ≤ cq; W21 (0, ∞) ≤ c(F ; W22 (0, ∞) + u0n ; W23 (σ )) t≥0
2 (0, ∞) + u0n ; W23 (σ )). ≤ c1 (F ; W2,μ
(2.166)
If μ ≥ 2λ1 , then (2.165), (2.166) yield |Ut (x , t)|2 dx σ
t 1/2 2 ≤ cF ; W2,μ (0, ∞) exp(−2λ1 t)|q(t)| + exp(−2λ1 t) |q(τ )|2 dτ 0
+ c∗ exp(−2λ1 t)u0n ; W23 (σ )2 2 ≤ c exp(−2λ1 t)(F ; W2,μ (0, ∞)2 + u0n ; W23 (σ )2 )
(2.167)
and, since q(t) satisfies the inequality
|Ut (x , t)|2 dx
|q(t)| ≤ c
1/2
σ
|Ut (x , t)| dx
≤ c2 σ
2
+ |F (t)|
1/2
2 + exp (−μt)F ; W2,μ (0, ∞)
,
(2.168)
we obtain from (2.166) 2 (0, ∞)2 + u0n ; W23 (σ )2 ). |q(t)|2 ≤ c exp(−2λ1 t)(F ; W2,μ
Let λ1 < μ < 2λ1 . Then we conclude from (2.165), (2.166) that 2 |Ut (x , t)|2 dx ≤ c exp(−μt)(F ; W2,μ (0, ∞)2 + u0n ; W23 (σ )2 ). σ
Therefore, μ 2 |q(t)| ≤ c exp − t (F ; W2,μ (0, ∞) + u0n ; W23 (σ )), 2
(2.169)
516
K. Pileckas
and (2.165) gives σ
|Ut (x , t)|2 dx 3 2 ≤ c exp(−2λ1 t)u0n ; W23 (σ )2 + cF ; W2,μ (0, ∞)2 exp − μt 2 t 1/2 . exp((4λ1 − 3μ)τ )| dτ + exp(−2λ1 t) 0
If μ ≥ 4λ1 /3, then from the last estimate and from (2.168) follow inequalities (2.167) and (2.169). If λ1 < μ < 4λ1 /3, then 3 2 |Ut (x , t)|2 dx ≤ c exp − μt (F ; W2,μ (0, ∞)2 + u0n ; W23 (σ )2 ), 2 σ
and we obtain from (2.167) that 3 2 |q(t)| ≤ c exp − μt (F ; W2,μ (0, ∞) + u0n ; W23 (σ )). 4 Thus, if μ ≥ 8λ1 /7, then from (2.165), (2.168) follow (2.167) (2.169) and, if λ1 < μ < 8λ1 /7, then 7 2 |Ut (x , t)|2 dx ≤ c exp − μt (F ; W2,μ (0, ∞)2 + u0n ; W23 (σ )2 ), 4 σ
7 2 |q(t)| ≤ c exp − μt (F ; W2,μ (0, ∞) + u0n ; W23 (σ )). 8 Repeating these arguments it is easy to show that inequalities (2.167), (2.169) hold, if k k μ ≥ 2k2−1 λ1 , and if λ1 < μ < 2k2−1 λ1 , then k 2 −1 2 |Ut (x , t)| dx ≤ c(k) exp − k−1 μt (F ; W2,μ (0, ∞)2 2 σ
2
+ u0n ; W23 (σ )2 ), k 2 −1 2 |q(t)| ≤ c(k) exp − k μt (F ; W2,μ (0, ∞) + u0n ; W23 (σ )). 2 k
For every μ > λ1 , there is k such that μ ≥ 2k2−1 λ1 and consequently by finite number of steps we derive inequalities (2.167) and (2.169) for every μ > λ1 . Note that the constant c(k) tends to infinity as k → ∞, i.e., as μ → λ1 .
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
517
Consider the case 0 < μ < λ1 . It follows from (2.168) and (2.166) that |Ut (x , t)|2 dx σ 2 2 ≤ c∗ c1 F ; W2,μ (0, ∞)(F ; W2,μ (0, ∞) + u0n ; W23 (σ )) t 1/2 exp((4λ1 − 2μ)τ ) dτ × exp(−μt) + exp(−2λ1 t) 0
+ c∗ exp(−2λ1 t)u0n ; W23 (σ )2
≤ c1 c∗ 1 +
1 2 2 exp(−μt)F ; W2,μ (0, ∞)(F ; W2,μ (0, ∞) (4λ1 − 2μ)1/2
+ u0n ; W23 (σ )) + c∗ exp(−μt)u0n ; W23 (σ )2 2 ≤ c∗ c∗∗ γ a0 exp(−μt)(F ; W2,μ (0, ∞) + u0n ; W23 (σ ))2 ,
(2.170)
where a0 = 1,
c∗∗ = max{c1 , c2 , 1},
γ =2+
1 (4λ1 − 4μ)1/2
and c2 is the constant from (2.168). Now, applying (2.170) and (2.168) we obtain μ 2 |q(t)| ≤ c2 [(c∗ c∗∗ γ a0 )1/2 + 1] exp − t (F ; W2,μ (0, ∞) + u0n ; W23 (σ )) 2 μ 2 (0, ∞) + u0n ; W23 (σ )), = c2 a1 exp − t (F ; W2,μ 2 where a1 = [(c∗ c∗∗ γ a0 )1/2 + 1]. By induction, using (2.165) and (2.168), we derive the relation k 2 −1 2 (0, ∞) + u0n ; W23 (σ )), (2.171) |q(t)| ≤ c2 ak exp − k μt (F ; W2,μ 2 k = 1, 2, . . . , with ak = (c∗ c∗∗ γ ak−1 )1/2 + 1. The sequence {ak } is bounded. Indeed, if the inequality ak+1 ≥ ak holds for some k, then ak2 − (2 + c∗ c∗∗ γ )ak + 1 ≤ 0 2 γ 2 /4 + c c γ ). Since a = 1, we and consequently ak ≤ y0 = (1 + c∗ c1 γ /2 + c∗2 c∗∗ ∗ ∗∗ 0 have ak ≤ y0 for every k. Thus, we can pass to the limit in (2.171) as k → ∞. As a result we obtain the estimate 2 (0, ∞) + u0n ; W23 (σ )). |q(t)| ≤ c exp(−μt)(F ; W2,μ
(2.172)
518
K. Pileckas
Finally, it follows from (2.165) that σ
2 |Ut (x , t)|2 dx ≤ c exp(−2μt)(F ; W2,μ (0, ∞) + u0n ; W23 (σ ))2 .
(2.173)
Consider the case μ = λ1 . Arguing as above, by induction, we derive the inequality (2.171) with ak = (c∗ c1 ak−1 γk−1 )1/2 + 1, γk = 2k λ1 + 2 and a0 = 1. In this case we cannot pass k to the limit in inequality (2.171), since γk → ∞ as k → ∞. However, for every ε > 0, there exists k such that λ∗ = λ1 − ε ≤ λ1 (1 − 2−k ) and instead of (2.172), (2.173) we get the following estimates 2 |q(t)| ≤ c exp(−(λ1 − ε)t)(F ; W2,μ (0, ∞) + u0n ; W23 (σ )),
σ
(2.174)
2 |Ut (x , t)|2 dx ≤ c exp(−2(λ1 − ε)t)(F ; W2,μ (0, ∞)
+ u0n ; W23 (σ ))2 ,
∀ε > 0.
(2.175)
The solution U (x , t) may be considered as a solution to the first boundary value problem for the Poisson equation 3
−ν U (x , t) = q(t) − Ut (x , t), U (x , t)|∂σ = 0.
Then for U (x , t) holds the estimate U (·, t); W22 (σ ) ≤ cq(t) + Ut (·, t); L2 (σ ) ≤ c(|q(t)||σ |1/2 + Ut (·, t); L2 (σ )).
(2.176)
Thus, inequality (2.162) follows from estimates (2.176), (2.165), (2.167) in the case μ > λ1 , from estimates (2.176), (2.172), (2.173) in the case μ < λ1 and from estimates (2.176), (2.174), (2.175) in the case μ = λ1 . From Theorems 2.12, 2.13 it follows that, if the flux F (t) is represented as a sum F (t) = ((t), where F∗ ∈ R and F ((t) vanishes as t → ∞, then the time-dependent Poiseuille F∗ + F flow tends as t → ∞ to the steady one. In particular, from Theorem 2.13 follows ((t), F∗ ∈ R, F ( ∈ W 2 (0, ∞), T HEOREM 2.14. Let ∂σ ∈ C 4 . Suppose that F (t) = F∗ + F 2,μ ◦
((t) and ( u0n ∈ W23 (σ ) ∩ W 12 (σ ), fn (x , t) = 0 and let the functions F u0n (x ) = u0n (x ) −
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
519
UF∗ (x ), where UF∗ (x ) is the steady Poiseuille flow corresponding to the flux F∗ (see Section 2.1.1), satisfy the compatibility conditions (2.75) of the first order. Then problem (2.34) has a unique solution (U (x , t), q(t)) admitting the representation ((x , t), U (x , t) = UF∗ (x ) + U
q(t) = νqF∗ + ( q (t).
(2.177)
There holds the estimate ((·, t); W22 (σ )2 + |( (t (·, t); L2 (σ )2 + U q (t)|2 U 2 (; W2,μ ≤ c exp(−2γ∗ t)(|F∗ |2 + F (0, ∞)2 + u0n ; W23 (σ )2 ),
(2.178)
where γ∗ is the same as Theorem 2.13. Now, we consider the case of generalized Poiseuille flow. ◦
1 (0, ∞) with T HEOREM 2.15. Let ∂σ ∈ C 2 , fn (x , t) = 0, u0n ∈ W 12 (σ ) and F ∈ W2,μ μ > 0. Assume that there holds the compatibility condition (2.148). Then the solution (U (x , t), q(t)) of problem (2.114) satisfies estimate (2.154) with the same γ∗ as in Theorem 2.12.
P ROOF. The solution (U (x , t), q(t)) of problem (2.114) admits the estimate (see (2.135), (2.136)) 1 d
2
|U (x , t)| dx + ν |∇ U (x , t)|2 dx 2 dt σ σ ≤ ε|q(t)|2 + cε |F (t)|2 , ν d 2 dt
3 |∇ U (x , t)| dx + 4 σ
2
(2.179)
|Ut (x , t)|2 dx
σ
≤ ε|q(t)|2 + cε |F (t)|2 + cm(t)
|∇ U (x , t)|2 dx ,
(2.180)
σ
|q(t)| ≤ c 2
|Ut (x , t)|2 dx + |F (t)|2
σ
|∇ U (x , t)|2 dx
+ m(t)
(2.181)
σ
where m(t) = U (·, t); W22 (σ )2 and U (x , t) is the solution of the Navier–Stokes problem (2.113). Taking in (2.179)–(2.181) ε sufficiently small, we obtain d (|U (x , t)|2 + ν|∇ U (x , t)|2 ) dx dt σ
520
K. Pileckas
+ (γ∗ − cm(t))
(|U (x , t)|2 + ν|∇ U (x , t)|2 ) dx + σ
|Ut (x , t)|2 dx
σ
≤ c(|F (t)|2 + |F (t)|2 ). Therefore, t exp γ∗ t − c m(τ ) dτ (|U (x , t)|2 + ν|∇ U (x , t)|2 ) dx 0
t
+ 0
σ
exp γ∗ τ − c
τ
exp γ∗ τ − c
τ
m(s) ds
0
t
≤c 0
+
|Uτ (x , τ )|2 dx dτ
σ
m(s) ds (|F (τ )|2 + |F (τ )|2 ) dτ
0
(|U (x , 0)|2 + ν|∇ U (x , 0)|2 ) dx .
σ
Since,
∞ 0
m(t) dt ≤ A0 (see (2.115)), from the last inequality follows the estimate exp(γ∗ t) (|U (x , t)|2 + ν|∇ U (x , t)|2 ) dx + 0
σ t
exp(γ∗ τ )
|Uτ (x , τ )|2 dx dτ σ
1 (0, ∞)2 ≤ c(F ; W2,μ
+ u0n ; W21 (σ )2 )
and, therefore, there holds the inequality (2.154).
R EMARK 2.8. We do not prove the analog of Theorem 2.13 for the solution of problem (2.114). This result requires the existence of more regular solutions than it was obtained in Section 2.3. The existence of such solutions can be proved under additional compatibility conditions both for data of Navier–Stokes problem (2.113) and of problem (2.114). ((t), R EMARK 2.9. In the case when the flux F (t) is represented as a sum F (t) = F∗ + F ((t) vanishes as t → ∞, it is possible to prove that the solution of where F∗ ∈ R and F problem (2.114) tends to the steady Poiseuille flow as t → ∞. However, the decay rate of the difference between the time-dependent and steady Poiseuille solutions is conditioned ((t) but also by the decay of the solution U (x , t) of Navier– not only by the decay of F Stokes problem (2.113). This requires additional considerations and we omit this result. 2.5. Time-periodic Poiseuille flow In this section we study a time-periodic Poiseuille flow. Results presented below have been obtained in [22]. We restrict ourselves to the case where the external force is identically
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
521
zero and the Poiseuille flow is unidirectional, i.e., consider in a pipe # the following problem for the Navier–Stokes equations ⎧ ut (x, t) − νu(x, t) + (u(x, t) · ∇)u(x, t) + ∇p(x, t) = 0, ⎪ ⎪ ⎪ ⎨ div u(x, t) = 0, (2.182) ⎪ u(x, t)|∂# = 0, u(x, −π) = u(x, π), ⎪ ⎪ ⎩
σ un (x, t) dx = F (t) with the flux F (t) satisfying the periodicity condition F (−π) = F (π).
(2.183)
Here, without loss of generality, we take the period T = 2π . If we look for the solution (u(x, t), p(x, t)) of problem (2.182) in the form U(x, t) = (0, . . . , 0, Un (x , t)),
P (x, t) = −q(t)xn + p0 (t),
(2.184)
then for U (x , t) = Un (x , t) and q(t) we receive the inverse problem on the cross-section σ: ⎧ Ut (x , t) − ν U (x , t) = q(t), ⎪ ⎨
U (x , t)|∂σ = 0, U (x , −π) = U (x , π), (2.185) ⎪ ⎩
σ U (x , t) dx = F (t). We shall prove the existence of a unique time periodic solution to problem (2.185). First, let us consider the heat equation with a given right-hand side q(t): 3 Ut (x , t) − ν U (x , t) = q(t), (2.186) U (x , t)|∂σ = 0, U (x , −π) = U (x , π). The following result is well known (see, e.g. [100]). T HEOREM 2.16. Let ∂σ ∈ C 2 and let q(t) be a 2π -periodic function with q ∈ L2 (−π, π). Then problem (2.186) has a unique 2π -periodic solution such that π sup U (·, t); W21 (σ )2 + (Ut (·, t); L2 (σ )2 + U (·, t); W22 (σ )2 ) dt −π
t∈[−π,π]
≤c
π
−π
|q(t)|2 dt.
(2.187)
to investigate the relation between the pressure gradient q(t) and the flux F (t) = In order
, t) dx , we consider the following sequence of problems U (x σ 3
Unt (x , t) − ν Un (x , t) = cos(nt), (c)
Un (x , t)|∂σ = 0, (c)
(c)
Un (x , −π) = Un (x , π). (c)
(c)
(2.1871 )
522
K. Pileckas
3
Unt (x , t) − ν Un (x , t) = sin(nt), (s)
Un(s) (x , t)|∂σ = 0,
(s)
Un(s) (x , −π) = Un(s) (x , π).
(2.1872 )
It is easy to compute that the solutions Un(c) (x , t), Un(s) (x , t) to (2.1871 ) and (2.1872 ) have the form Un (x , t) = ϕn (x ) cos(nt) − ψn (x ) sin(nt), (c)
(2.188)
Un (x , t) = ψn (x ) cos(nt) + ϕn (x ) sin(nt), (s)
where the pair (ϕn (x ), ψn (x )) is a solution to the following elliptic problem ⎧ −nψn (x ) = ν ϕn (x ) + 1, ⎪ ⎨ nϕn (x ) = ν ψn (x ), ⎪ ⎩ ϕn (x )|∂σ = 0, ψn (x )|∂σ = 0.
(2.189)
It can be proved by standard methods that for each non-negative integer n problem (5.8) ◦ possesses a unique solution (ϕn , ψn ) ∈ W22 (σ ) ∩ W 12 (σ ). Set an = ϕn (x ) dx , bn = − ψn (x ) dx , n = 0, 1, 2, . . . . (2.190) σ
σ ◦
L EMMA 2.9. Let (ϕn , ψn ) ∈ W22 (σ ) ∩ W 12 (σ ) be a solution to (2.189). Then, the following inequality holds 1 ν ϕ; L2 (σ )2 + ν ψ; L2 (σ )2 ≤ |σ | ∀n = 0, 1, 2, . . . . ν
(2.191)
Moreover, the numbers an and bn satisfy the following properties (a)
an > 0 ∀n = 0, 1, 2, . . . ;
(b) (c)
b0 = 0, bn > 0 ∀n = 1, 2, . . . ; |σ | |σ | , bn ≤ ∀n = 1, 2, . . . ; an ≤ n n lim (nbn ) = |σ |. n→∞
P ROOF. If we multiply (2.1891 ) by ϕ( x ), then (2.1892 ) by ψn (x ), add both resulting equalities and integrate by parts in σ , we find the relation
2
2 ν ϕn ; L2 (σ ) + ν ψn ; L2 (σ ) = − ϕn (x ) dx σ
≤ from which immediately follows (2.191).
ν 1 ϕn ; L2 (σ )2 + |σ | 2 2ν
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
523
To show (a), we multiply (2.1892 ) by ϕn (x ) and (2.1891 ) by ψn (x ) and integrate by parts in σ . This delivers −nψn ; L2 (σ )2 = −ν ∇ ϕn (x ) · ∇ ψn (x ) dx − bn σ
and −nϕn ; L2 (σ ) = ν 2
∇ ϕn (x ) · ∇ ψn (x ) dx .
σ
Summing these two last displayed equations furnishes bn = n(ϕn ; L2 (σ )2 + ψn ; L2 (σ )2 ).
(2.192)
We now multiply (2.1891 ) by ϕn (x ) and (2.1892 ) by ψn (x ), integrate by parts and add two resulting relations. We thus deduce an = ν(∇ ϕn ; L2 (σ )2 + ∇ ψn ; L2 (σ )2 ).
(2.193)
Since (2.189) does not allow an identically zero solution, the proof of (a) follows from (2.192), (2.193). Let us show (b). By Cauchy–Schwarz inequality from (2.190) we get an ≤ |σ |1/2 ϕn ; L2 (σ ),
bn ≤ |σ |1/2 ψn ; L2 (σ ),
n = 0, 1, 2, . . . . (2.194)
Using the relation (2.192) from (2.194) easily follows that |σ | , n
an ≤
bn ≤
|σ | , n
n = 1, 2, . . . .
(2.195)
It remains to show property (c). From the second relation in (2.195) and from (2.192) it follows that nψn ; L2 (σ ) ≤ |σ |1/2 ,
n = 0, 1, 2 . . . .
(2.196)
Using the first relation in (2.195) together with (2.193) we obtain that lim ∇ ϕn ; L2 (σ ) = 0.
(2.197)
n→∞
Let us multiply both sides of (2.1891 ) by arbitrary χ ∈ C0∞ (σ ) and integrate by parts over σ :
−n ψn (x )χ(x ) dx = −ν ∇ ϕn (x ) · ∇ χ(x ) dx + χ(x ) dx . σ
σ
σ
524
K. Pileckas
By passing to the limit n → ∞ in this relation and with the help of (2.197) we have lim
n→∞
−n
ψn (x )χ(x ) dx = χ(x ) dx .
σ
σ
The set C0∞ (σ ) is dense L2 (σ ). Therefore, the last relation remains valid for any χ ∈ L2 (σ ). Taking in it χ(x ) ≡ 1 we get property (c). We are now in a position to determine the relation between q(t) and F (t). To this end, let us write Fourier series of both quantities:
∞ (c) q0 (s) n=1 qn cos(nt) + qn sin(nt) , 2 + (c)
(c) F (s) F (t) = 02 + ∞ n=1 Fn cos(nt) + Fn sin(nt) . q(t) =
(c)
(c)
(2.198)
(s)
(c)
(s)
L EMMA 2.10. The Fourier coefficients (qn , qn ) of q(t) and those (Fn , Fn ) of F (t) are related to each other by the following formulas
Fn(c) = an qn(c) − bn qn(s) ,
Fn(s) = bn qn(c) + an qn(s) ,
n = 0, 1, 2, . . . , (2.199)
or, equivalently, by their inverse qn(c) =
an Fn(c) + bn Fn(s) , an2 + bn2
qn(s) =
an Fn(s) − bn Fn(c) . an2 + bn2
(2.200)
P ROOF. We begin to observe that it is enough to show the validity of (2.199), since (2.200) follows directly from this latter (note that an2 + bn2 > 0 ∀n). Set (n(c) (x , t) ≡ Un(c) (x , −t), U
(n(s) (x , t) ≡ Un(s) (x , −t), U
t ∈ [−π, π]. (2.201)
Then 3
3
(c) (nt (n(c) (x , t) = − cos(nt), U (x , t) + ν U (n(c) (x , −π) = U (n(c) (x , π). (n(c) (x , t)|∂σ = 0, U U
(2.2021 )
(s) (nt (n(s) (x , t) = − sin(nt), U (x , t) + ν U (n(s) (x , −π) = U (n(s) (x , π). (n(s) (x , t)|∂σ = 0, U U
(2.2022 )
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
525
(n (x , t) and integrate by part over σ : Multiply the both sides of (2.185) by U (c)
d dt
U (x σ
= σ
, t)Un(c) (x , t) dx
∂ ((c)
((c) − U (x , t) U (x , t) + ν Un (x , t) dx ∂t n σ
(n(c) (x , t) dx . q(t)U
Furthermore, integrate this relation over t ∈ [−π, π]. By taking into account (2.1881 ), (2.190), (2.2011 ), (2.2021 ) and the fact that U (x , −π) = U (x , π) , we get
π
−π
F (t) cos(nt) dt =
π
−π
q(t)(an cos(nt) − bn sin(nt)) dt
which proves the first relation in (2.199). The second relation in (2.199) is obtained exactly (n(c) (x , t) with U (n(s) (x , t). by the same procedure, we replace the above argument U Our next task is to investigate the convergence of the series (2.198) and the relation between the norms of q(t) and F (t). Specifically, we have the following (c)
(s)
(c)
(s)
L EMMA 2.11. Assume that the numbers (Fn , Fn ) and (qn , qn ), n = 0, 1, 2, . . . , satisfy (2.199) (or, equivalently, (2.200)). Then, if the Fourier series (2.1981 ) converges to some q ∈ L2 (−π, π), also the Fourier series (2.1982 ) converges to some F ∈ L2 (−π, π) and F ; L2 (−π, π) ≤ c1 q; L2 (−π, π).
(2.203)
Conversely, if the Fourier series (2.1982 ) converges to some F ∈ W21 (−π, π), then also the Fourier series (2.1981 ) converges to some q ∈ L2 (−π, π) and q; L2 (−π, π) ≤ c2 F ; W21 (−π, π),
(2.204)
where the constants c1 and c2 depend only on |σ |. P ROOF. From (2.199) and Lemma 2.9 (a) it follows that (c)
(c)
F0 = a 0 q 0 ,
|Fn(c) |2 + |Fn(s) |2 = (an2 + bn2 )(|qn(c) |2 + |qn(s) |2 ),
n ≥ 1. (2.205)
Therefore, from Lemma 2.9 (b) we find ∞
∞
|F0 |2 (c) 2 |q | + |Fn | + |Fn(s) |2 ≤ a02 0 + 2|σ |2 |qn(c) |2 + |qn(s) |2 2 2 (c)
n=1
(c)
n=1
526
K. Pileckas
which proves that the series (2.1982 ) converges in L2 (−π, π), if q ∈ L2 (−π, π). Moreover, (2.203) follows from Parseval’s equality. Conversely, we note that (2.205) implies |qn(c) |2 + |qn(s) |2 ≤
1 (c) 2 |Fn | + |Fn(s) |2 , bn2
n ≥ 1.
(2.206)
From Lemma 2.9 (c) we have that there exists a positive integer N0 such that bn ≥
|σ | 2n
∀n ≥ N0 .
Setting b∗ = min{b1 , . . . , bN0 }, in view of Lemma 2.9 (a), it follows that b∗ > 0. Thus, from the latter displayed equations, from Lemma 2.9 (b) and from (2.206) we find ∞ ∞ (c) (c)
|q0 |2 (c) 2 1 (c) 2 1 |F0 |2 + + 2 |qn | + |qn(s) |2 ≤ |Fn | + |Fn(s) |2 2 a0 2 b∗ n=1
n=1
+
4 |σ |2
∞
n2 |Fn(c) |2 + |Fn(s) |2
n=1
from which, by the assumptions on F (t) follows that series (2.1981 ) converges in L2 (−π, π). Finally (2.204) is a consequence of Parseval’s equality. An immediate consequence of the previous results is the existence and uniqueness of a periodic Poiseuille flow under a given periodic flux F (t). T HEOREM 2.17. Let ∂σ ∈ C 2 , and let F (t) be a 2π -periodic function with F ∈ W21 (−π, π). Then problem (2.185) admits one and only one 2π -periodic solution (U (x , t), q(t)) such that sup t∈[−π,π]
U (·, t); W21 (σ )2
≤ cF ; W21 (−π, π)2 . (c)
(s)
+
π −π
(Ut (·, t); L2 (σ )2 + U (·, t); W22 (σ )2 )dt (2.207)
P ROOF. Let Fn , Fn be the Fourier coefficients of F (t), and consider the series (2.1981 ) (c) with coefficients qn , qn (s) given in (2.200). From Lemma 2.10 we know that the series (2.1981 ) is convergent to some q ∈ L2 (−π, π) and that inequality (2.204) holds. We then solve problem (2.186) with the given q(t). The existence part along with the validity of the estimate (2.207) is a consequence of Theorem 2.16, Lemma 2.10 and of inequality (2.204). To show the uniqueness, we assume that F (t) = 0 and multiply both sides of (2.1851 ) by U (x , t), integrate by parts over σ and then integrate by t over the inπ terval [−π, π]. Using that F (t) = 0 then furnishes −π σ |∇ U (x , t|2 dx dt = 0, that is
U (x , t) = 0. Going back to the equations (2.185), we get q(t) = 0.
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
527
R EMARK 2.10. Solution of problem (2.185) has the following simple representation U (x , t) =
∞
q0 (qn(c) ϕn (x ) + qn(s) ψn (x )) cos(nt) ϕ0 (x ) + 2 n=1
+ (qn(s) ϕn (x ) − qn(c) ψn (x )) sin(nt) , (c)
(2.208)
where, for n = 0, 1, 2, . . ., the functions ϕn (x ), ψn (x ) satisfy (2.189) while the num(c) (s) bers qn , qn are given in (2.200). This easily follows from (2.187)–(2.189) along with inequalities (2.191) and (2.193). R EMARK 2.11. In the special case when the flux F is a constant, U does not depend on time and we find U (x ) = ϕ0 (x )/2a0 . 2.6. Final remarks The question of construction of solutions (or approximate solutions) of Navier–Stokes problem having a given flux may be posed also for “pipes” with growing cross-sections, for the infinite layer, etc. Let us briefly review known results in this direction. In the domain which is parabolic-like7 , i.e., = {x ∈ Rn : |x | < g0 xn
1−γ
, 0 < γ < 1, xn > 1}
(2.209)
approximate solutions (for large xn ) to steady Stokes and Navier–Stokes problems (also with right-hand sides having a special representation) are constructed by S.A. Nazarov and K. Pileckas in [46]. The procedure which is used in [46] is based on the known algorithm of constructing the asymptotic representation of solutions to elliptic equations in slender domains developed by S.A. Nazarov [40], [41]. Note that for the time-dependent problem analogous results are not known. In the three-dimensional infinite layer L = {x ∈ R3 : x ∈ R2 , 0 < x3 < 1}
(2.210)
the solution of steady Stokes problem with different from zero flux F over the cylindrical surface Sη = {x ∈ # : |x | = η} has the form (see [58], [48]) 3F x1 x2 x3 (1 − x3 ) , ,0 , UF (x) = π |x |2 |x |2
PF (x) = −
6F ν ln |x |. π (2.211)
This solution satisfies Stokes system everywhere in L excepting the axis |x | = 0. 7 If γ = 1, this domain coincides with the straight semi-infinite pipe # with circular cross-section, while for + γ ∈ (0, 1) the cross-section σ (s) = {x ∈ Rn : |x | < g0 s 1−γ , xn = s} grows as s → ∞, i.e. |σ (s)| = πg02 s 2(1−γ ) .
528
K. Pileckas
The solution with prescribed flux F (t) and the initial data u0 (x), which admits the representation x1 x2 , , 0 , u0 (x) = v0 (x3 ) |x |2 |x |2 of time-dependent Stokes problem in the layer L has the form x1 x2 U(x, t) = v(x3 , t) , ,0 , |x |2 |x |2
P (x, t) = −q(t) ln |x |,
where (v(x3 , t), q(t)) is the solution of the following inverse problem on the interval (0, 1): ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
vt (x3 , t) − ν
∂2 v(x3 , t) = q(t), ∂x32
v(0, t) = v(1, t) = 0∀t ∈ [0, T ], ⎪ ⎪ v(x3 , 0) = v0 (x3 ), ⎪ ⎪ ⎪ ⎩ 1 2π 0 v(x3 , t) dx3 = F (t) ∀t ∈ [0, T ].
(2.212)
The flux F (0) and the function v0 (x3 ) have to satisfy the compatibility condition
1
2π
v0 (x3 ) dx3 = F (0).
0
The existence of a unique solution to inverse problem (2.212) follows from results of Section 2.2.
3. Chapter III. Steady problems In this chapter we study the steady Stokes and Navier–Stokes problems in domains with cylindrical outlets to infinity—a model for a system of channels and pipes. Such flows of viscous incompressible fluid are of a great importance both from the point of view of applications and from the theoretical point of view. The main point of our considerations is the Leray’s problem, i.e., we are looking for the solutions of the steady Navier–Stokes system which have given fluxes Fj , j = 1, . . . , J , over sections of all outlets to infinity j and which tend in each j to the corresponding steady Poiseuille flow having the same flux Fj . As it is already mentioned in Introduction, first results concerning the existence of a solution to Leray’s problem was obtained by C.J. Amick [4], [5] who proved that in the domain with two cylindrical outlets this problem has a unique solution, if the flux |F | is sufficiently small (in comparison with the viscosity ν). The most general results concerning stationary Leray’s problem were obtained by O.A. Ladyzhenskaya and V.A. Solonnikov [32]. They considered domains with J cylindrical outlets to infinity j = #j , j = 1, . . . , J, and proved that the steady Navier–Stokes problem admits at least one solution with an infinite Dirichlet integral which has given fluxes Fj , j = 1, . . . , J , over cross-sections σj of all
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
529
outlets to infinity. This result is obtained without any restrictions on values of fluxes Fj , assuming only the necessary condition that the total flux is equal to zero, i.e. Jj=1 Fj = 0. ! N 2 Moreover, it is proved in [32] that for sufficiently small |F| = j =1 |Fj | the solution is unique and tends exponentially in each pipe #j to the Poiseuille flow corresponding to Fj . However, the main question still remains open: we do not know, whether Leray’s problem is solvable for any value of F . To this end O.A. Ladyzhenskaya and V.A. Solonnikov [32] reformulated Leray’s problem and proposed to describe a set of all solutions with nonzero fluxes to steady Navier–Stokes problem in straight pipe # and to show that the solution in a “disturbed pipe” tends to one of the elements of this set. However, in such formulation the problem also has no answer until now. It is even not known whether for large |F | the Poiseuille flow is the only solution with given flux of Navier–Stokes problem in a straight pipe or there exist other different from Poiseuille flow solutions. One approach to solve the formulated problem could be to compare the Poiseuille solution with the periodic solution of Kapitanskii (see Section 1.1.3). Since in the straight pipe the cross-section is constant, we may consider it as domain with periodically changing cross-section and the period L may be taken arbitrary. Hence, for any period L there exists a periodic solution uL (x) having the same pressure drop as the Poiseuille solution. Thus, it is necessary either to find a periodic solution different from the Poiseuille flow or to show that for arbitrary flux |F | and arbitrary period L the periodic solution does not depend on xn and, therefore, coincides with the Poiseuille one. However, pro tempore this question has no answer. The chapter consists of five sections. In Section 3.1 we present the exact formulation of the linear Stokes and the nonlinear steady Navier–Stokes problems. Section 3.2 is devoted to the construction of divergence-free vector-fields equal to zero on ∂ and having prescribed fluxes over sections of outlets j , j = 1, . . . , J . Such vector-fields we call flux carriers. In Section 3.2.1 we construct the flux carrier that coincide in each outlet to infinity with the Poiseuille flow (or generalized Poiseuille flow) corresponding to this outlet, and in Section 3.2.2 we present the construction of a general flux carrier A(x) having certain “good” properties (in particular, for A(x) holds estimate (2.47)) which allow to prove the solvability of the nonlinear steady Navier–Stokes problem for arbitrary fluxes Fj . Such flux carrier was first constructed in [31] (see also [32], [94]). Here we present the results from [94]. In Section 3.3 we study the linear Stokes system. We assume that in each outlet to
f(x), infinity j the external force f(x) admits the representation f(x) = f(j ) (x (j ) ) + ( f(x) belongs to a certain weighted space of vanishing as |x| → ∞ funcx ∈ j , where ( l () with β ≥ 0). In this case we prove the existence of a unique tions (e.g., ( f ∈ W2,β j solution (u(x), p(x)) to the Stokes problem that tends in each outlet to infinity j to the
(j ) (j ) (j ) steady Poiseuille type flow (UF (x (j ) ), PF (xn )) corresponding to the pipe #j , the flux
Fj and the external force f(j ) (x (j ) ). Moreover, in each outlet j the decay rate of the dif (j ) ference between the solution u(x) and the Poiseuille type flow UF (x (j ) ) is conditioned only by the decay rate of an external force ( f(x). These results are proved for arbitrary large data. Note that the linear Stokes problem in domains with cylindrical outlets was treated in weighted function spaces in many papers. For example, it was studied in [32] on basis of Sobolev spaces generated by L2 -norms. On the other hand, the Stokes system could be
530
K. Pileckas
analyzed in weighted function spaces with exponential weights applying the theory of general elliptic systems as presented in the book [51]. In [53] the Stokes system was treated in weighted spaces with polynomial weights. We have chosen the approach presented below by two reasons. First, the weighted spaces defined in Chapter I allow both exponential and polynomial weight functions and, second, this approach is also convenient for the study of time-dependent problems. We present here only results concerning the solvability of Stokes problem in weighted spaces generated by L2 -norms. However, all results could be generalized for weighted Lq , q > 1, and weighted Hölder spaces as it was done in [62], [63] for spaces with exponential weighted. In Section 3.4 we study the steady Navier–Stokes problem. First, using Banach conl (), we prove that for sufficiently small data traction principle in weighted spaces W2,β there exists a unique solution u(x) which tends as |x| → ∞, x ∈ j , to the correspond ing Poiseuille flow U(j ) (x (j ) ) (see Section 3.4.1). In Section 3.4.2 we prove the existence of solutions with infinite Dirichlet integral having prescribed nonzero fluxes (without any smallness assumption on the data). For small data this solution is unique, and, therefore, if the external force admits the mentioned above representation, this solution coincides with the strong solution found in Section 3.4.1 and, thus, tends in each outlet to infinity j to the
corresponding Poiseuille flow U(j ) (x (j ) ). Furthermore, we also study the regularity properties of this solution. Mainly these results are based on the paper of V.A. Solonnikov [90]. Finally, we prove the result which shows the decay rate of the solution to Poiseuille flows
U(j ) (x (j ) ) as |x| → ∞ along the outlets to infinity. This result does not require the norm ( f; L2,β () to be small, i.e., only the fluxes Fj and norms f(j ) ; L2 (σj ) of functions
f(j ) (x (j ) ), participating in determination of Poiseuille flows U(j ) (x (j ) ), have to be sufficiently small. In relation to results presented in Section 3.4 it is necessary to mention also the paper [49] dealing with systematic investigation of setting of adequate asymptotic conditions at infinity (not necessary the fluxes) for steady Stokes and Navier–Stokes equations in domains with cylindrical outlets. The obtained there results use function spaces with exponential weights. The paper [49] is based on the general theory for elliptic problems in cylindrical and conical domains as presented in [51]. In [54] investigations of [49] are extended to a larger class of data. While in [49] right-hand sides exponentially vanish, in [54] is considered the case of forces which do not vary along the axes of the cylindrical outlets additive components. A convincing example for such setting is a domain situated in a gravitational field. The solutions having nonzero fluxes and bounded pressure [54] are also constructed. The problem of approximation of the solutions of steady Stokes and Navier– Stokes systems in unbounded domains with cylindrical outlets to infinity by solutions on bounded subdomains is considered in [10] and [96]. The various types of artificial boundary conditions on truncated domains are studied and the precise estimates of truncation error are obtained. The asymptotic representation for solutions of steady Stokes and Navier–Stokes systems in domains with outlets to infinity that are parabolic-like (i.e., in some coordinate systems have the form (II.2.209)) is justified in [46]. Note that for “wide” parabolic-like outlets (i.e., if γ > 1/4) the asymptotic representation of the solution to nonlinear steady Navier– Stokes problem is justified without any restriction on the value of data. For the domain which has layer-like (of form (II.2.210)) outlet to infinity the full asymptotic representation
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
531
for steady Stokes problem is constructed and justified in [48], [50]. It is proved in [67] that any solution with the prescribed flux of steady nonlinear Navier–Stokes problem in the domain with layer-like outlet to infinity tends to the solution of Stokes problem and the main asymptotic term is given by formula (II.2.211). This result is also obtained without any restriction on the value of data. The detailed presentation of mentioned above results would increase enormously the volume of the paper, therefore we do not include these results.
3.1. Formulation of the problem Let us consider in the domain the linear Stokes ⎧ −νu(x) + ∇p(x) = f(x), ⎪ ⎪ ⎪ ⎪ ⎨ div u(x) = 0, u(x)|∂ = 0, ⎪ ⎪ ⎪ ⎪ ⎩ u(x) · n(x) dx (j ) = Fj , j = 1, . . . , J,
(3.1)
σj
and the steady nonlinear Navier–Stokes ⎧ −νu(x) + (u(x) · ∇)u(x) + ∇p(x) = f(x), ⎪ ⎪ ⎪ ⎪ ⎨ div u(x) = 0, u(x)|∂ = 0, ⎪ ⎪ ⎪ ⎪ ⎩ u(x) · n(x) dx (j ) = Fj , j = 1, . . . , J,
(3.2)
σj
problems with additionally prescribed fluxes Fj over cross-sections σj of outlets to infinity (conditions (3.14 ) and (3.24 )). Because of the incompressibility of the fluid (div u(x) = 0) the fluxes Fj must satisfy the compatibility condition J
Fj = 0
(3.3)
j =1
(the total flux must be zero). Concerning the external force, we assume that f(x) admits the representation f(x) =
J
(j ) ζ (xn )f(j ) (x (j ) ) +( f(x),
(3.4)
j =0
where
(j )
(j )
f(j ) (x (j ) ) = (f1 (x (j ) ), . . . , fn (x (j ) )),
(3.5)
532
K. Pileckas
ζ (τ ) is a smooth cut-off function with ζ (τ ) = 0 for τ ≤ 1 and ζ (τ ) = 1 for τ ≥ 2, f(j ) ∈ W2l (σj ), l ≥ 0, and ( f(x) belongs to a certain weighted space of vanishing as |x| → ∞ l () with β ≥ 0). ( functions (e.g., f ∈ W2,β j We look for the solutions (u(x), p(x)) of problems (3.1) and (3.2) that tend in each outlet
(j ) (j ) (j ) to infinity j to the steady Poiseuille type flow (UF (x (j ) ), PF (xn )) corresponding to
(j ) (j ) the pipe #j , the flux Fj and the external force f (x ). In the case of the linear Stokes problem (3.1) we prove the existence of such a solution for arbitrary data and in the case of nonlinear problem (3.2) for sufficiently small data.
3.2. Construction of the flux carrier 3.2.1. The flux carrier coinciding with Poiseuille flows in j , j = 1, . . . , J If the righthand side f(x) in equations (3.1), (3.2) has the representation (3.4)–(3.5), then in each
cylinder #j there exists at least one generalized Poiseuille flow (U(j ) (x (j ) ), P (j ) (x (j ) ))
corresponding to the external force f(j ) (x (j ) ) and the flux Fj and having the form (II.2.10) ◦
(see Section 2.1.2). Moreover, if f(j ) ∈ L2 (σj ), then U(j ) ∈ W 12 (σj ) and there hold estimates (II.2.13), (II.2.18), and if ∂σj ∈ C l+2 , f(j ) ∈ W2l (σj ), then U(j ) ∈ W2l+2 (σj ) ∩ ◦
8(j ) ∈ W2l (σj ) and there hold estimates (II.2.14), (II.2.19). Set W 12 (σj ), ∇ p U(x) =
J
(j ) ζ (xn )U(j ) (x (j ) ), P (x, t) =
j =1
J
(j )
ζ (xn )P (j ) (x (j ) ),
(3.6)
j =1
where ζ is the same cut-off function as in the representation (3.4). Then g(x) = div U(x) =
J
(j ) (j ) (j ) ∂ ), j =1 ∂x (j ) ζ (xn )Un (x n
suppx g(x) ⊂ (2) \ (1) . Moreover, from the condition
J
j =1 Fj
(3.7)
= 0 it follows that
g(x) dx = 0.
(3.8)
(2)
From (II.2.18) we get the estimate g; W21 ((3) ) ≤ c
J
(j )
Un ; W21 (σj )
j =1
≤c
J
(j ) (fn ; L2 (σj ) + |Fj |) j =1
(3.9)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
533
(j )
and, if ∂σj ∈ C l+2 , fn ∈ W2l (σj ), then (II.2.19) gives the estimate g; W2l+2 ((3) )
≤c
J
(j )
Un ; W2l+2 (σj )
j =1
≤c
J
(j ) (fn ; W2l (σj ) + |Fj |).
(3.10)
j =1 ◦
◦
Since g ∈ W 12 ((3) ), by Lemma I.1.11, there exits a vector-field W ∈ W 22 ((3) ) such that W(x)|∂ = 0
div W(x) = g(x),
(3.11)
and W; W22 ((3) ) ≤ cg; W21 ((3) ) ≤ c
J
(j )
(fn ; L2 (σj ) + |Fj |).
(3.12)
j =1
Assuming, in addition, that ∂ ∈ C l+4 , by Lemma I.1.12 we find in the case (3.10) a ◦ vector-field W ∈ W2l+3 ((3) ) ∩ W 12 ((3) ) satisfying (3.11) and such that (j )
supp W(x) ⊂ {x ∈ : xn ≤ 5/2, j = 1, . . . , J },
W; W2l+3 ((3) ) ≤ cG; W2l+2 ((3) ) ≤c
J
(j ) (fn ; W2l (σj ) + |Fj |).
(3.13)
j =1
Extend the functions U(x) and W(x) by zero into the whole and set V(x) = U(x) + W(x).
(3.14)
Then, div V(x) = 0,
V(x)|∂ = 0,
V(x) · n(x) ds = Fj ,
j = 1, . . . , J,
σj
and for x ∈ j \ j 3 the vector-field V(x) coincides with the velocity part U(j ) (x (j ) ) of the corresponding Poiseuille flow.
534
K. Pileckas
R EMARK 3.1. In the case of the linear Stokes problem (3.1) in representation formula (3.6) for (U(x), P (x)) we take either the unidirectional Poiseuille flows, if the right hand sides f(j ) (x (j ) ) allow the existence of a such solution, or the generalized Poiseuille flow corresponding to the linear steady Stokes problem (see Remark (II.2.7)). 3.2.2. General flux carrier Let γik be a contour consisting of the symmetry axes γi and γik which joins them being γj of pipes i and k (i, k = 1, . . . , J ) and of a smooth curve 8 situated in so that the distance from γik to the surface ∂ is not less than a number d0 > 0. First we consider the case n = 3. We introduce a divergence-free vector-field Aik (x) = curl(ζ ik (x) · Bik (x)) = ∇ζ ik (x) × Bik (x),
(3.15)
where E
x −y × dl |x − y|3
B (x) = ik
γik
and
ρ(δ ik (x)) . ζ (x) = ψ ε ln (x) ik
Here ψ(t) and ρ(t) are smooth monotone functions with ψ(t) = 0 for t ≤ 0, ψ(t) = 1 for t ≥ 1, ρ(t) = a1 d0 /2 for t ≤ a2 d0 /2, ρ(t) = t for t ≥ a2 d0 , a1 and a2 are positive constants, δ ik (x) and (x) are regularized distances from the point x to γik and ∂ respectively. The regularized distance G (x) from x to the closed set G ⊂ Rn is infinitely many times differentiable in Rn \ G function, satisfying the inequalities a1 dG (x) ≤ G (x) ≤ a2 dG (x),
1−|α|
|D α G (x)| ≤ a3 dG
(x),
where dG (x) is the real distance from x to G (see [97]). L EMMA 3.1. The vector-field Bik (x) is divergence-free in R3 \ γik , curl Bik (x) = 0 and the circulation of Bik (x) along any closed contour, enveloping γik , is equal to −4π , if the direction of integration along this contour and along γik are connected by the right-hand rule. If this contour does not envelop γik , then the circulation of Bik (x) along it is equal to zero. At points x whose distance from γik is not less than ρ0 one has |Dxα Bik (x)| ≤
c(α, ρ0 ) 1+|α|
dik
(x)
,
|α| ≥ 0,
(3.16)
where dik (x) is the distance from the point x to γik . P ROOF. The first three assertions of the lemma are well known and can be easily verified8 . To prove (3.16) we represent Dxα Bik (x) in the form E Dxα Bik (x)
=
Dxα
γi
x−y × dl + Dxα |x − y|3
E γk
x−y × dl |x − y|3
8 Bik (x) is a magnetic field, which generates, upon passage through γ , an electric flow of unit intensity, ik e.g. [35].
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
E + Dxα
γ˜ik
535
x −y × dl. |x − y|3
Let us estimate the first two terms at the right-hand side. Without loss of generality we can assume that the half-line γi lies on the x3 -axis for x3 > R. Hence, " E " ∞ " α " x −y dy3 c "D " × dl" ≤ c ≤ 1+|α| , |α| " x 3 1+ 2 2 |x − y| R (x + x + (x3 − y3 )2 ) γi 2 di (x) 1 2 where di (x) = infy3 >R " E " α "D " x
γik 8
!
x12 + x22 + (x3 − y3 )2 is the distance from x to γi . Moreover,
" " γik | x−y c " ≤ c|8 × dl ≤ 1+|α| , " 81+|α| 3 |x − y| 8 dik (x) dik (x)
γik , d8ik (x) is the distance from x to 8 γik . Now (3.16) follows where |8 γik | is the length of 8 from the last two inequalities. L EMMA 3.2. The function ζ ik (x) is equal to zero at those points of where ρ(δ ik (x)) ≤ (x), while the d0 /2-neighborhood of the curve γik is contained in this set, and ζ ik (x) = 1 at those points where (x) ≤ e−1/ε ρ(δ ik (x)). For the derivatives of the functions ζ ik (x) there hold the inequalities " " " ∂ ik " cε " " " ∂x ζ (x)" ≤ (x) , k
" " " ∂2 " cε ik " " " ∂x ∂x ζ (x)" ≤ 2 (x) . k l
(3.17)
P ROOF. Obviously ζ ik (x) = 0, if ρ(δ ik (x)) ≤ (x) and ζ ik (x) = 1, if (x) ≤ e−1/ε ρ(δ ik (x)). Let d(x) and dik (x) be the distances from x to ∂ and γik , respectively. If dik (x) ≤ d0 /2, then d(x) ≥ d0 − dik (x) ≥ d0 /2, δ ik (x) ≤ a2 d0 /2. Hence, ρ(δ ik (x)) = a1 d0 /2 ≤ a1 d(x) ≤ (x) and ζ ik (x) = 0. It remains to prove (3.17). We have ∂ζ ik (x) ρ(δ ik (x)) ik Sk (x), = εψ ε ln ∂xk (x) ρ(δ ik (x)) ik ∂ 2 ζ ik (x) 2
Sk (x)Slik (x) = ε ψ ε ln ∂xk ∂xl (x) ρ(δ ik (x)) ∂S ik (x)
+ εψ ε ln , (x) ∂xl where Skik (x) =
1 ∂(x) 1 ∂δ ik (x) , − ρ (δ (x)) ∂xk ρ(δ ik (x)) (x) ∂xk
ik
536
K. Pileckas
so that ∂Skik (x) = ∂xl
+
ρ
(δ ik (x)) ρ 2 (δ ik (x)) ∂δ ik (x) ∂δ ik (x) − 2 ik ρ(δ ik (x)) ∂xk ∂xl ρ (δ (x)) ρ (δ ik (x)) ∂ 2 δ ik (x) ρ(δ ik (x)) ∂xk ∂xl
% 1 ∂(x) ∂(x) 1 ∂ 2 (x) . + 2 − ∂xl (x) ∂xk ∂xl (x) ∂xk Using the basic properties of the regularized distance and of the function ρ(t), it is easy to derive the inequalities 1 , ρ(δ ik (x)) (x) " ik " " ∂Sk (x) " 1 1 " "≤c + . " ∂x " ρ 2 (δ ik (x)) 2 (x) l
|Skik (x)| ≤ c
1
+
(3.18)
One can choose the function ρ(t) so that |ρ (t)| ≤ c, |ρ
(t)| ≤ ca2−1 d0−1 , where c is an absolute constant. Then the constants in (3.18) depend only on the constants ai . Since supp ψ is contained in the set {x : (x) ≤ ρ(δ ik (x))}, inequality (3.17) follows from (3.18). L EMMA 3.3. The vectors-fields Aik (x) given by (3.15) are infinitely many times differentiable in . Vectors-fields Aik (x) vanish near the boundary ∂ and near the contour γik . The supports of Aik (x) are contained in the set of points x ∈ satisfying the inequalities ρ(δ ik (x))e−1/ε ≤ (x) ≤ ρ(δ ik (x)).
(3.19)
Moreover, Aik (x) · n(x) dS = ±4π(δij + δkj ),
j = 1, . . . , J,
(3.20)
σj
(here at the right-hand side of (3.20) one must have the sign “−”, if j = i or j = k and if the integration over the part γj of the contour γik is in the direction of increase of the (j ) coordinate x3 , and the vector n(x) points in the same direction). Finally, there hold the inequalities |Aik (x)| ≤
c1 ε , d(x)
x ∈ ,
|Dxα Aik (x)| ≤ const,
x ∈ ,
(3.21) |α| ≥ 0,
where d(x) = dist(x, ∂) and c1 does not depend on ε.
(3.22)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
537
P ROOF. It follows from (3.15) and Lemma 3.2 that the support of the vector-field Aik (x) is contained in the set (3.19). Relation (3.20) is a consequence of Stokes formula Aik (x) · n(x)ds = Bik (x) · dl σj
∂σj
and Lemma 3.1. Finally, from (3.16) and (3.17) we have cε , (x)δik (x) " ik " " ∂A (x) " 1 1 " " ≤ cε + . " ∂x " 2 (x) 2 (x)δik (x) (x)δik k |Aik (x)| ≤
From these inequalities, (3.19), definition of the function ρ(t) and properties of the generalized distance follow (3.21) and (3.22). ◦
L EMMA 3.4. For every function u ∈ W 12 ((l) ) there hods the inequality |Aik (x)|2 |u(x)|2 dx ≤ εc |∇u(x)|2 dx (l)
(3.23)
(l)
where the constant c is independent of l, ε and u(x). P ROOF. By (3.21)
|Aik (x)|2 |u(x)|2 dx ≤ cε
|u(x)|2 dx d 2 (x)
|u(x)|2 dx + 2 d (x) J
≤ cε (0)
j =1 j l
|u(x)|2 dx . d 2 (x)
Let S0 = ∂ ∩ ∂(0) and δ(0) = {x ∈ (0) : dist(x, S0 ) ≥ δ > 0}. If δ is sufficiently small, the domain (0) \ δ(0) can be covered by a finite number of domains Ts defined in some coordinate systems {y (s) } by relations Ts = {(y1(s) , y2(s) ) ∈ B s ⊂ R2 , 0 < (s) (s) (s) y3 − Gs (y1 , y2 ) < δ1 }, where Bs is a ball in R2 . For simplicity we omit the index s in notations for y (s) . For x ∈ Ts we have d(x) ≥ c(y3 − Gs (y1 , y2 )) and, therefore, Ts
|u(x)|2 dx ≤ c d 2 (x)
dy1 dy2
Bs
Gs (y )+δ1 Gs (y )
|u(y)|2 dy3 . (y3 − Gs (y )2
By Hardy’s inequality, the last integral is bounded by
dy1 dy2
c Bs
" " " " dy3 ≤ c |∇u(x)|2 dx. " ∂y " 3 Ts
Gs (y )+δ1 " ∂u(y) "2 Gs (y )
538
K. Pileckas
Further, by Poincaré inequality (0) \δ(0)
|u(x)|2 c dx ≤ 2 d 2 (x) δ
|u(x)| dx ≤ c(δ)
|∇u(x)|2 dx.
2
(0)
(0)
Now, consider the integrals
|u(x)|2 dx = d 2 (x) l
j l
k=0 ωj k
|u(x)|2 dx. d 2 (x)
( (j ) ) from x to the lateral In each ωj k the distance d(x) is equivalent to the distance d(x surface of ωik . Therefore, ωj k
|u(x)|2 dx ≤ c d 2 (x)
k+1
k
(j ) |u(x (j ) , x3 )|2 (j ) (j ) dx dx3 .
d(2 (x (j ) )
σj
Arguing as above, we obtain
σj
(j )
|u(x (j ) , x3 )|2 (j ) ≤c dx
d(2 (x (j ) )
σj
|∇ u(x (j ) , x3 )|2 dx (j ) (j )
and, thus, ωj k
|u(x)|2 dx ≤ c d 2 (x)
|∇u(x)|2 dx. ωj k
Collecting all obtained estimates, we get (3.23). Define the vector-field A(x) =
J −1
αj Aj,j +1 (x),
(3.24)
j =1
where αj = 4π
j
Fl ,
j = 1, . . . , J − 1.
(3.25)
l=1
Obviously, div A(x) = 0,
A(x)|∂ = 0,
A(x) · n(x) ds = Fj , σj
j = 1, . . . , J.
(3.26)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
539
Moreover, there holds the estimates |Dxα A(x)| ≤ c
J
1/2 |Fj |
2
,
x ∈ ,
|α| ≥ 0,
(3.27)
j =1
|A(x)|2 |u(x)|2 dx ≤ εc
J
(l)
|Fj |2
j =1
|∇u(x)|2 dx.
(3.28)
(l) ◦
The constant in (3.28) is independent of l, ε and u ∈ W 12 (). The analogous results are valid also for the two-dimensional case. The role of vectorsfields (3.15) is played now by A (x) = ik
∂ζ ik (x) ∂ζ ik (x) , ,− ∂x1 ∂x2
(3.29)
where the function ζ ij (x) is defined by the same formula as for n = 3. Inequalities (3.27), (3.28) are valid for the flux carrier A(x) in the two-dimensional case as well.
3.3. The Stokes problem 3.3.1. Stokes problem in the case of zero fluxes Consider in the Stokes problem (3.1) assuming that all fluxes Fj , j = 1, . . . , J , are equal to zero, i.e., consider the problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎨
−νu(x) + ∇p(x) = f(x), div u(x) = 0, u(x)|∂ = 0, ⎪ ⎪ ⎪ ⎪ ⎩ u(x) · n(x) ds = 0, j = 1, . . . , J. σj
(3.30)
◦
A weak solution of problem (3.30) is a divergence-free vector-field u ∈ W 12 () satisfy◦
ing for every divergence-free η ∈ W 12 () the integral identity
∇u(x) · ∇η(x) dxd =
ν
f(x) · η(x) dx.
(3.31)
The following theorem is well known (see, e.g., [28]). T HEOREM 3.1. Let f ∈ L2 (). Then problem (3.30) admits a unique weak solution u(x) and there holds the following estimate u; W21 () ≤ cf; L2 ().
(3.32)
540
K. Pileckas
Let us prove the decay properties as |x| → ∞, x ∈ j , of the weak solution u(x) to l () with (3.30) assuming that the right-hand side f(x) belongs to the weighted space W2,β βj > 0, j = 1, . . . , J . l (), β ≥ 0, j = 1, . . . , J . Suppose that T HEOREM 3.2. Let ∂ ∈ C l+2 and let f ∈ W2,β j the number γ∗ in the inequality (I.1.143 ) for the weight-function Eβ (x) is sufficiently small. l+2 () and there exists a function p(x) Then the weak solution u(x) belongs to the space W2,β l with ∇p ∈ W2,β () such that the pair (u(x), p(x)) satisfies equations (3.30) almost everywhere in . There holds the estimate l+2 l l u; W2,β () + ∇p; W2,β () ≤ cf; W2,β (.
(3.33)
(k)
P ROOF. Let us take in the integral identity (3.31) η(x) = Eβ (x)u(x) + W(k) (x), where (k)
Eβ (x) is the “step” weight-function (I.1.16) and W(k) (x) is a vector-field constructed in Lemma I.1.13. Then div η(x) = 0, η(x)|∂ = 0. We get
(k)
ν
∇u(x) · ∇(Eβ (x)u(x) + W(k) (x)) dx
=
f(x) · (Eβ(k) (x)u(x) + W(k) (x)) dx.
(3.34)
(k)
Note that all integrals in (3.34) are finite since Eβ (x) is equal to a constant for large |x| and W(k) (x) has a compact support. Using Hölder and Young inequalities, weighted Poincaré inequality (I.1.19), properties (k) (I.1.14) of the weight-function Eβ (x) and inequality (I.1.29) for W(k) (x), we get the estimates " " " " (k) " f(x) · (E (k) (x)u(x) + W(k) (x)) dx " ≤ 1 Eβ (x)|f(x)|2 dx β " 4ε " (k) (k) 2 (k) 2 Eβ (x)|u(x)| dx + c1 E−β (x)|W (x)| dx +ε ≤
1 4ε
(k)
Eβ (x)|f(x)|2 dx
+ c2 ε ≤
1 4ε
(k) Eβ (x)|∇u(x)|2 dx
(k)
Eβ (x)|f(x)|2 dx + c3 ε
+
(k) E−β (x)|∇W(k) (x)|2 dx
(k)
Eβ (x)|∇u(x)|2 dx,
" " " " " " " " (k) (k) " " " ν " ∇u(x) · ∇Eβ (x) · u(x) dx " + ν " ∇u(x) · ∇W (x) dx ""
(3.35)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
≤ν
(k) Eβ (x)|∇u(x)|2 dx
541
1/2
1/2 1/2 (k) × c 4 γ∗ Eβ(k) (x)|u(x)|2 dx + c5 E−β (x)|∇W(k) (x)|2 dx
≤ c 6 γ∗
(k)
Eβ (x)|∇u(x)|2 dx.
(3.36)
Relations (3.34)–(3.36) yield the inequality 1 (k) (k) ν Eβ (x)|∇u(x)|2 dx ≤ Eβ (x)|f(x)|2 dx 4ε (k) + (c3 ε + c6 γ∗ ) Eβ (x)|∇u(x)|2 dx.
Let γ∗ ≤
Choosing ε =
ν 4c6 .
ν 2
ν 4c3 ,
from the last inequality we obtain
Eβ (x)|∇u(x)|2 dx ≤ (k)
≤c
(k)
ν 2
Eβ (x)|f(x)|2 dx ≤ c
(k)
Eβ (x)|∇u(x)|2 dx
Eβ (x)|f(x)|2 dx.
(3.37)
Since the right-hand side of (3.37) is independent of k, we pass k → ∞ in (3.37) and we get ν 2 Eβ (x)|∇u(x)| dx ≤ c Eβ (x)|f(x)|2 dx. (3.38) 2 In order to prove weighted estimates for the higher derivatives of u(x) and for the pressure p(x), we use local estimates for ADN-elliptic problems [2], [85]. It is well known that l+2 l+1 Stokes problem is ADN-elliptic [85]. Therefore, u ∈ W2,loc (), p ∈ W2,loc (), Stokes system is valid almost everywhere in and the following local estimates are true u; W2l+2 (ωj k )2 + ∇p; W2l (ωj k )2 ≤ c f; W2l (( ωj k )2 + ∇u; L2 (( ωj k )2 + p − p¯ j k ; L2 (( ωj k )2 , j = 1, . . . , J ;
k = 0, 1, 2, . . . ,
(3.39)
ωj k |−1 ( not depend on k. Let us where p¯ j k = |( ωj k p(x) dx. The constant c in (3.39) does estimate the last norm at the right-hand side of (3.39). Since ( ωj k (p(x) − p¯ j k ) dx = 0, ◦
by Lemma I.1.10 there exists a vector-field w(j k) ∈ W 12 (( ωj k ) such that div w(j k) (x) = p(x) − p¯ j k in ( ωj k . There holds the estimate ∇w(j k) ; L2 (( ωj k ) ≤ cp − p¯ j k ; L2 (( ωj k ),
(3.40)
542
K. Pileckas
where c is independent of k. Then, using the Stokes system (3.30), integrating by parts over ( ωj k and applying (3.40), we derive
|p(x) − p¯ j k | dx =
(p(x) − p¯ j k ) div w(j k) (x) dx
2
( ωj k
( ωj k
=− =ν
∇p(x) · w(j k) (x) dx
( ωj k
( ωj k
∇u(x) · ∇w(j k) (x) dx
−
( ωj k
f(x) · w(j k) (x) dx
ωj k )∇w(j k) ; L2 (( ωj k ) ≤ c ∇u; L2 (( + f; L2 (( ωj k )w(j k) ; L2 (( ωj k ) ≤ c ∇u; L2 (( ωj k ) + f; L2 (( ωj k ) p − p¯ j k ; L2 (( ωj k ). Thus, p − p¯ j k ; L2 (( ωj k )2 ≤ c ∇u; L2 (( ωj k )2 + f; L2 (( ωj k )2 .
(3.41)
Substituting (3.41) into (3.39) furnishes u; W2l+2 (ωj k )2 + ∇p; W2l (ωj k )2 ≤ c f; W2l (( ωj k )2 + ∇u; L2 (( ωj k )2 , j = 1, . . . , J ;
k = 0, 1, 2, . . . .
(3.42)
Analogously can be proved the estimate u; W2l+2 ((1) )2 + ∇p; W2l ((1) )2 ≤ c f; W2l ((2) )2 + ∇u; L2 ((2) )2 .
(3.43)
Multiplying the both sides of inequalities (3.42) by Eβj (k), using the property (I.1.142 ) of the weight-functions Eβj (x) and summing the obtained relations, we derive the estimate l+2 l u; W2,β (j )2 + ∇p; W2,β (j )2 l ≤ c f; W2,β (j )2 + ∇u; L2,β (j )2 ,
Estimate (3.33) follows now from (3.38), (3.43), (3.44).
j = 1, . . . , J.
(3.44)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
543
3.3.2. Stokes problem in the case of nonzero fluxes Now, consider problem (3.1). T HEOREM 3.3. Let ∂ ∈ C l+2 and let f(x) admits the representation (3.4), (3.5) with l (), β ≥ 0, j = 1, . . . , J . Suppose that the number γ in the f ∈ W2,β f(j ) ∈ W2l (σj ), ( j ∗ inequality (I.1.143 ) for the weight-function Eβ (x) is sufficiently small. Then there exists a unique solution (u(x), p(x)) of problem (3.1) admitting the asymptotic representation u(x) = V(x) + v(x),
p(x) = P (x) + p 8(x).
(3.45)
In (3.45) V(x) is the flux carrier constructed in Section 3.2.1, i.e., V(x) is defined
by (3.14) and (U(x), P (x)) by (3.6), where U(j ) (x (j ) ) are taken either unidirectional l+2 (), (if possible), or as solutions to linear Stokes problem (see Remark II.2.7), v ∈ W2,β l ∇p 8 ∈ W2,β (). There holds the estimate l+2 l v; W2,β () + ∇ p 8; W2,β ()
J J
l f; W2,β ≤ c ( ( + f(j ) ; W2l (σj ) + |Fj | . j =1
(3.46)
j =1
P ROOF. We look for the solution (u(x), p(x)) of problem (3.1) in the form (3.45). Then for (v(x), p 8(x)) we obtain the problem (3.30) with the right-hand side8 f(x) =( f(x)+νV(x)− ∇P (x). Since (V(x), P (x)) coincides for large |x|, x ∈ j with the Poiseuille solution
(U(j ) (x (j ) ), P (j ) (x)), we have supp(νV(x) − ∇P (x)) ⊂ (3) (see Section 3.2.1). Moreover, from the construction of (U(x), P (x)) (see (3.6)) and W(x, t) (see (3.11)) and from estimates (II.2.19), (3.13) it follows that l νV − ∇P ; W2,β () ≤ cνV − ∇P ; W2l ((3) )
≤c
J
f(j ) ; W2l (σj ) +
j =1
J
|Fj | .
j =1
l (), Therefore, 8 f ∈ W2,β
J J
l l (j ) l ( 8 f ; W2 (σj ) + |Fj | f; W2,β () ≤ f; W2,β () j =1
(3.47)
j =1
and, in virtue of Theorem 3.2, there exists a unique solution (v(x), p 8(x)) of problem (3.30) with the right-hand side 8 f(x). Estimate (3.33) holds for this solution and, thus (3.46) follows from (3.33), (3.47).
544
K. Pileckas
3.4. The Steady Navier–Stokes problem 3.4.1. Solvability of Navier–Stokes problem in weighted function spaces Assume that l (), the external force f(x) admits representation (3.4), (3.5) with f(j ) ∈ W2l (σj ), ( f ∈ W2,β l ≥ 0, βj ≥ 0, j = 1, . . . , J . Consider the Navier–Stokes problem (3.2). We look for the solution (u(x), p(x)) in the form u(x) = V(x) + v(x),
p(x) = P (x) + p 8(x),
(3.48)
where V(x) is defined by (3.14) and P (x) by (3.6). Then for (v(x), p 8(x)) we get the following problem ⎧ −νv(x) + (v(x) · ∇)v(x) + (V(x) · ∇)v(x) ⎪ ⎪ ⎪ ⎪ ⎪ +(v(x) · ∇)V(x) + ∇ p 8(x) =8 f(x), ⎪ ⎨ div v(x) = 0, ⎪ ⎪ ⎪ v(x)|∂ = 0, ⎪ ⎪ ⎪
⎩ (j ) = 0, j = 1, . . . , J, σj v(x) · n(x) dx
(3.49)
where 8 f(x) =( f(x) + f(1) (x) + f(2) (x), f(1) (x) = (f(1) (x), fn(1) (x)),
f(1) (x) = (1)
fn (x) =
J
(j ) (j ) (x (j ) ) − ζ (x (j ) )ζ (x (j ) )U (j ) (x (j ) )U(j ) (x (j ) ) n n n j =1 (νζ (xn )U
(j ) (j ) (j ) (j )
(j ) − ζ (xn )(ζ (xn ) − 1)(U (x ) · ∇ )U (x (j ) )), J
(j ) (j ) (j ) (j ) (j )
(j ) ) − ζ (xn )ζ (xn )|Un (x (j ) )|2 j =1 (νζ (xn )Un (x
(j ) (j ) (j ) − ζ (xn )(ζ (xn ) − 1)(U(j ) (x (j ) ) · ∇ )Un (x (j ) ) (j ) − ζ (xn )P (j ) (x)),
f(2) (x) = νW(x) − (W(x) · ∇)W(x) − (U(x) · ∇)W(x) − (W(x) · ∇)U(x). (3.50) From the construction of (U(x), P (x)) (see 3.6) and W(x, t) (see 3.11) and estimates (II.2.19), (3.13) it follows that supp(f(1) (x) + f(2) (x)) ⊂ (3) and l f(1) + f(2) ; W2,β ()2 ≤ cf(1) + f(2) ; W2l ((3) )2 ≤ cA1 (1 + A1 ),
(3.51)
where A1 =
J j =1
f(j ) ; W2l (σj )2 +
J
j =1
|Fj |2 .
(3.52)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
545
T HEOREM 3.4. Let ∂ ∈ C l+2 and let f(x) admits the representation (3.4), (3.5) with l (), l ≥ 0, β ≥ 0, j = 1, . . . , J . Suppose that the number γ f(j ) ∈ W2l (σj ), ( f ∈ W2,β j ∗ in inequality (I.1.143 ) for the weight-function Eβ (x) is sufficiently small. Denote A2 = l ()2 . There exists a number ε > 0 such that, if ( f; W2,β 0 A1 (1 + A1 ) + A2 ≤ ε0 ,
(3.53)
then problem (3.49) admits a unique solution (v(x), p 8(x)) satisfying the estimate l+2 l v; W2,β ()2 + ∇ p 8; W2,β ()2 ≤ c(A1 (1 + A1 ) + A2 ).
(3.55)
◦
l+2 P ROOF. Let v ∈ W2,β () ∩ W 12 () with l ≥ 0, βj ≥ 0. Denote Mv(x) = (v(x) · ∇)v(x). Since by the Sobolev imbedding theorem (see Lemma I.1.2) v ∈ L∞ (), it is straightforl () and ward to show that Mv ∈ W2,β l+2 l () ≤ cv; W2,β ()2 . Mv; W2,β
(3.56)
l+2 () there holds the inequality Moreover, for v, w ∈ W2,β l l () ≤ (v · ∇)(v − w); W2,β () Mv − Mw; W2,β l+2 l () ≤ c(v; W2,β () + ((v − w) · ∇)v; W2,β l+2 l+2 + w; W2,β ())(v − w); W2,β ().
(3.57)
Further, using estimates (II.2.19), (3.13) for U(j ) (x) and W(x), one easily gets that l l (V · ∇)v; W2,β ()2 + (v · ∇)V; W2,β ()2 l+2 ≤ cA1 v; W2,β ()2 .
(3.58)
Problem (3.49) is equivalent to an operator equation v = L−1 (Mv + N v +8 f) ≡ Av
(3.59)
l+2 in the space W2,β (). In (3.59) L is the Stokes operator (3.30) and N v(x) = (V(x) · l+2 () and there exits a ∇)v(x) + (v(x) · ∇)V(x). By Theorem 3.2, L is defined on W2,β
l () → W l+2 (). From (3.51), (3.53)–(3.59) it folbounded inverse operator L−1 : W2,β 2,β lows that
l+2 l l Av; W2,β ()2 ≤ c Mv; W2,β ()2 + N v; W2,β ()2 l + 8 f; W2,β ()2
546
K. Pileckas
l+2 l+2 ≤ c1 v; W2,β ()4 + A1 v; W2,β ()2 + (A2 + A1 (1 + A1 )) , l+2 l+2 ()2 ≤ c2 v; W2,β ()2 Av − Aw; W2,β l+2 l+2 + w; W2,β ()2 + A1 v − w; W2,β ()2 . Let % r0 1 . ε0 < min , 4c1 2c2
% 1 1 , ,1 , r0 = min 2c1 4c2
Then, from the latter displayed inequalities we conclude that A is a contraction in the l+2 l+2 ball Br0 = {u ∈ W2,β () : u; W2,β ()2 ≤ r0 }. Thus, the solvability of operator equation (3.59) and estimate (3.55) for the solution follow from the Banach contraction principle. 3.4.2. Existence of a weak solutions to Navier–Stokes problem for large fluxes Now, consider problem (3.2) in the case of “large” data. Assume that f ∈ L2,loc (). Results of this section are based on the paper of V.A. Solonnikov [90]. By a weak solution of problem (3.2) we mean a divergence free vector field v(x) ∈ 1 () vanishing on ∂, satisfying for any divergence-free η ∈ C ∞ () the integral W2,loc 0 identity
∇u(x) · ∇η(x) dx −
ν
(u(x) · ∇)η(x) · u(x) dx =
and the flux conditions
u(x) · n(x) dx (j ) = Fj ,
f(x) · η(x) dx
(3.60)
j = 1, . . . , J.
σj
We look for the solution u(x) in the form u(x) = A(x) + v(x), where A(x) is a flux carrier constructed in Section 3.2.2. Then for (v(x), p(x)) we get the following problem ⎧ −νv(x) + (v(x) · ∇)v(x) + (A(x) · ∇)v(x) + (v(x) · ∇)A(x) ⎪ ⎪ ⎪ ⎪ ⎪ + ∇p(x) = f(x) + νA(x) − (A(x) · ∇)A(x), ⎪ ⎨ div v(x) = 0, ⎪ ⎪ ⎪ v(x)|∂ = 0, ⎪ ⎪ ⎪
⎩ (j ) = 0, j = 1, . . . , J, σj v(x) · n(x) dx
(3.61)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
547
Correspondingly, the weak solution to (3.61) is a divergence-free vector field v(x) ∈ 1 (), vanishing on ∂, satisfying for any divergence-free η ∈ C ∞ () the integral W2,loc 0 identity
∇v(x) · ∇η(x) dx −
ν
((A(x) + v(x)) · ∇)η(x) · v(x) dx
(v(x) · ∇)η(x) · A(x) dx =
−
∇A(x) · ∇η(x) dx +
−ν
f(x) · η(x) dx
(A(x) · ∇)η(x) · A(x) dx
(3.62)
and having zero fluxes:
v(x) · n(x) dx (j ) = 0,
j = 1, . . . , J.
σj
In order to prove the existence of at least one weak solution v(x), we will need the following lemma. L EMMA 3.5. Assume that numbers yk satisfy the inequalities yk+1 ≥ yk > 0,
∀k ≥ 1,
yk ≤ c∗ (yk+1 − yk ) + c∗∗ (yk+1 − yk )3/2 + kc0 ,
k = 1, 2, . . . , l,
where c∗ , c∗∗ and c0 do not depend on k. If yl ≤ lc0 , then for any k = 1, 2, . . . , l, there holds the estimate yk ≤ kc0 . If limk→∞ k −3 yk = 0 and yk ≤ c∗ (yk+1 − yk ) + c∗∗ (yk+1 − yk )3/2 , then lim yk = 0.
k→∞
The proof of Lemma 3.5 can be found in [90].
∀k = 1, 2, . . . ,
548
K. Pileckas
T HEOREM 3.5. Assume that the right-hand side f(x) satisfies the condition sup(l −1 f; L2 ((l) )2 ) ≡ B1 (f) < ∞. l≥0
Then problem (3.61) admits at least one weak solution v(x) and there holds the estimate 2 J J
2 2 |∇v(x)| dx ≤ cl B1 (f) + |Fj | + |Fj | ,
2
(l)
j =1
(3.63)
j =1
where the constant c is independent of l. P ROOF. First, let us prove the existence of a weak solution v(l) (x) for any bounded do◦ main (l) , i.e., we prove the existence of divergence-free vector-field v(l) ∈ W 12 ((l) ) satisfying the integral identity ∇v(l) (x) · ∇η(x) dx − ((A(x) + v(l) (x)) · ∇)η(x) · v(l) (x) dx ν (l)
(l)
−
(v(l) (x) · ∇)η(x) · A(x) dx (l)
=
f(x) · η(x) dx − ν (l)
+
∇A(x) · ∇η(x) dx (l)
(A(x) · ∇)η · A(x) dx
(3.64)
(l) ◦
for any divergence-free η ∈ W 12 ((l) ). It is well known that for the existence of such v(l) (x) it is enough to prove an a priori estimate. Then, the existence follows from the Leray– Schauder theorem (see [28]). Taking in (3.64) η(x) = v(l) (x), we obtain |∇v(l) (x)|2 dx = f(x) · v(l) (x) dx − ν ∇A(x) · ∇v(l) (x) dx ν (l)
(l)
+
(l)
(A(x) · ∇)v(l) (x) · A(x) dx + (l)
(v(l) (x) · ∇)v(l) (x) · A(x) dx (l)
≤ c f; L2 ((l) )2 + ∇A; L2 ((l) )2 + A; L4 ((l) )4 ν +c |A(x)|2 |v(l) (x)|2 dx + |∇v(l) (x)|2 dx. 4 (l) (l)
(3.65)
In virtue of (3.27) we have ∇A; L2 ((l) )2 + A; L4 ((l) )4 ≤ cl
J j =1
|Fj |2 +
J j =1
2 |Fj |2
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
549
and, by (3.28), |A(x)| |v (x)| dx ≤ cε 2
2
(l)
(l)
J
|Fj |
|∇v(l) (x)|2 dx.
2 (l)
j =1
Therefore, (3.65) furnishes 2 J J
2 2 |∇v (x)| dx ≤ cl B1 (f) + |Fj | + |Fj |
(l)
ν
2
(l)
j =1
+
ν + c1 ε 4
j =1
J
|Fj |2
|∇v(l) (x)|2 dx.
(3.66)
(l)
j =1
Fixing in the construction of the flux carrier A(x) the number ε = ν 4c1
J
|Fj |
2
−1
j =1
(see Section 3.2.2), we obtain from (3.66) the estimate 2 J J
2 2 , |∇v (x)| dx ≤ cl B1 (f) + |Fj | + |Fj |
(l)
ν
2
(l)
j =1
(3.67)
j =1
which guaranties the existence of at least one v(l) (x) satisfying integral identity (3.64) in a bounded domain (l) . Next, we estimate the norm ∇v(l) ; L2 ((k) ) with any k < l. Let
(l)
Vk (x) =
⎧ ⎪ ⎨ ⎪ ⎩
v(l) (x), x ∈ (k) , (l) χk (x)v(l) (x) + Jj=1 wj k (x), x ∈ ∪Jj=1 ωj k , 0,
(3.68)
x ∈ \ (k+1) ,
where χk (x) is a smooth cut-off function with χk (x) = 1 in (k) , χk (x) = 0 in \ (k+1) (l) and |∇χk (x)| ≤ const, wj k (x) are the solutions of the problems ⎧ ⎨div w(l) (x) = −∇χk (x) · v(l) (x), jk ⎩
jk
(l) ∇wj k ; L2 (ωj k ) ≤ c∇v(l) ; L2 (ωj k ),
where the constant c does not depend of k. Since (l)
(l) " wj k "∂ω = 0,
x ∈ ωj k ;
(k)
j = 1, . . . , J,
(3.69)
∇χk (x) · v(l) (x)dx = 0, the exis(l)
tence of wj k (x) follows from Lemma I.1.10. It easy to see that the vector-field Vk (x) is
550
K. Pileckas ◦
(l)
(l)
divergence-free and Vk ∈ W 12 ((l) ). Taking η(x) = Vk (x) in the integral identity (3.64), we obtain |∇v (x)| dx = −ν 2
(l)
ν (k)
J
j =1 ωj k
∇v(l) (x) · ∇V(l) k (x) dx (l)
+ (k+1)
((v(l) (x) + A(x)) · ∇)Vk (x) · v(l) (x) dx
(l)
+ (k+1)
(v(l) (x) · ∇)Vk (x) · A(x) dx +
−ν
(k+1)
(l)
(k+1)
f(x) · Vk (x) dx
∇A(x) · ∇V(l) k (x) dx (l)
+ (k+1)
(A(x) · ∇)Vk (x) · A(x) dx.
(3.70)
We estimate the integrals at the right-hand side of (3.70) by using the definition of V(l) k (x), continuous embedding of W21 (ωj k ) into L4 (ωj k ) (see Lemma I.1.2), Hölder, Young and Poincaré inequalities, inequality (3.69) and inequalities (3.27), (3.28) for the flux carrier A(x): J "
" " ν " j =1
"
∇v
ωj k
(l)
" (l) (x) · ∇Vk (x) dx ""
≤ν
J
(l)
∇v(l) ; L2 (ωj k )∇Vk ; L2 (ωj k )
j =1
≤c
J
∇v(l) ; L2 (ωj k )2 ;
j =1
" " " "
(k+1)
" " (l) " ((v(l) (x) + A(x)) · ∇)V(l) (x) · v (x) dx k "
J "
" " ≤ " j =1
≤
"
((v
ωj k
(l)
" (l) (l) (x) + A(x)) · ∇)Vk (x) · (v(l) (x) − Vk (x)) dx ""
J
(l) (v(l) ; L4 (ωj k )(Vk − v(l) ); L4 (ωj k ) j =1 (l)
(l)
+ (v(l) − Vk )A; L2 (ωj k ))∇Vk ; L2 (ωj k ) ≤ c1
J
j =1
∇v(l) ; L2 (ωj k )3 + c2 ε 1/2
J j =1
|Fj |2
1/2 J j =1
∇v(l) ; L2 (ωj k )2 ;
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
" " " "
(k+1)
551
" " (l) (v(l) (x) · ∇)Vk (x) · A(x) dx ""
1/2
(l)
≤ ∇Vk ; L2 ((k+1) ) ≤ cε 1/2
J
|v(l) (x)|2 |A(x)|2 dx (k+1)
1/2 |Fj |2
∇v(l) ; L2 ((k+1) )2
j =1
≤ cε
1/2
J
1/2 |Fj |
∇v ; L2 ((k) ) +
2
2
(l)
j =1
" " " "
"
" " (l) f(x) · Vk (x) dx "" + ν ""
" " + ""
(k+1)
∇v ; L2 (ωj k ) (l)
;
2
j =1
"
(k+1)
J
"
(k+1)
" (l) ∇A(x) · ∇Vk (x) dx "" "
" (l) (A(x) · ∇)Vk (x) · A(x) dx ""
≤ c f; L2 ((k+1) ) + ∇A; L2 (k+1) (l) + A; L4 (k+1) 2 ∇Vk ; L2 ((k+1) ) ≤ ck
1/2
1/2 J J (l) 1/2 2 2 B1 (f) + |Fj | + |Fj | ∇Vk ; L2 ((k+1) ) j =1
j =1
2 J J
ν + ∇v(l) ; L2 ((k) )2 |Fj |2 + |Fj |2 ≤ ck B1 (f) + 2 j =1
+c
J
j =1
∇v(l) ; L2 (ωj k )2 .
j =1
For sufficiently small ε it follows from these inequalities and from (3.70) that |∇v (x)| dx ≤ c∗ (l)
(k)
2
J
|∇v(l) (x)|2 dx
j =1 ωj k
+ c∗∗
J
3/2 |∇v(l) (x)|2 dx
j =1 ωj k
2 J J
. + ck B1 (f) + |Fj |2 + |Fj |2 j =1
j =1
(3.71)
552
K. Pileckas
Denote (k) |∇v(l) (x)|2 dx = yk . Then Jj=1 ωj k |∇v(l) (x)|2 dx = yk+1 − yk , and we rewrite (3.71) in the form yk ≤ c∗ (yk+1 − yk ) + c∗∗ (yk+1 − yk )3/2 2 J J
. + kc B1 (f) + |Fj |2 + |Fj |2 j =1
(3.72)
j =1
By Lemma 3.5, inequality (3.72) together with (3.66) implies for all k < l 2 J J
2 2 . |∇v (x)| dx ≤ kc B1 (f) + |Fj | + |Fj |
yk =
2
(l)
(k)
j =1
(3.73)
j =1
Since, due to Lemma I.1.2, for every bounded domain (k) the embedding W21 ((k) ) +→ L4 ((k) ) is compact, estimate (3.73) guarantees the existence of a subsequence {v(lm ) (x)} ◦
which converges weakly in W 12 ((k) ) and strongly in L4 ((k) ) (such subsequence could be constructed by the diagonalization process). Therefore, taking in integral identity (3.64) ◦ a test function η ∈ W 12 ((l) ), extending it by zero into \ (l) , and considering all integrals in (3.64) as integrals over , we can pass in (3.64) to a limit as lm → ∞. As a result we get integral identity (3.62) for the limit vector-field v(x). Obviously, estimate (3.63) remains valid for v(x). T HEOREM 3.6. Assume that the following norm f; L ((0) ) + sup 2
2
J
f; L2 (ωj k )2 ≡ B0 (f)
k≥0 j =1
is finite. Then problem (3.61) admits at least one weak solution v(x) and there holds the estimate J
|∇v(x)| dx ≤ c B0 (f) + 2
j =1 ωj k
with the constant c independent of k ≥ 0. P ROOF. First, note that |f(x)|2 dx ≤ clB0 (f). (l)
Thus, we have B1 (f) ≤ cB0 (f).
J
j =1
|Fj | + 2
J j =1
2 |Fj |
2
(3.74)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
553
Moreover, by virtue of (3.27), J
∇A; L2 (ωj k )2 +
j =1
J
A; L4 (ωj k )4
j =1
≤c
J
|Fj | + 2
j =1
J
2 |Fj |
2
(3.75)
j =1
with the constant c independent of k. Therefore, due to Theorem 3.5, there exists at least one weak solution v(x) of problem (3.61) and there holds estimate (3.63). Let us fix k and take in integral identity (3.62) η(x) = Vk+s (x) − Vk−s (x), where 1 ≤ s ≤ k − 1 and Vk±s (x) are defined by formula (3.68) with v(l) (x) changed to v(x). Note that Vk+s (x) − Vk−s (x) = v(x) in (k+s) \ (k−s+1) and Vk+s (x) − Vk−s (x) = 0 in \ (k+s+1) and in (k−s) . Identity (3.62) takes the form ν
(k+s) \(k−s+1)
|∇v(x)|2 dx = −ν
− Vk−s (x)) dx + ν
J
j =1 ωj k+s ∪ωj k−s
J
j =1 ωj k+s ∪ωj k−s
∇v(x) · ∇(Vk+s (x)
((v(x) + A(x)) · ∇)(Vk+s (x)
− Vk−s (x)) · (v(x) − Vk+s (x) + Vk−s (x)) dx + (v(x) · ∇)(Vk+s (x) − Vk−s (x)) · A(x) dx +
(k+s+1) \(k−s)
(k+s+1) \(k−s)
f(x) · (Vk+s (x) − Vk−s (x)) dx
−ν +
(k+s+1) \(k−s)
(k+s+1) \(k−s)
∇A(x) · ∇(Vk+s (x) − Vk−s (x)) dx
(A(x) · ∇)(Vk+s (x) − Vk−s (x)) · A(x) dx.
(3.76)
We estimate the right-hand side of (3.76) with the help of the same arguments as that of Theorem 3.5 (for more details see the similar proofs in [90]). For sufficiently small ε we obtain then, in view of (3.75), the inequality (k)
(k)
c∗ (zs+1 − zs(k) ) + ( c∗∗ (zs+1 − zs(k) )3/2 zs(k) ≤ ( 2 J J
, + sc B0 (f) + |Fj |2 + |Fj |2 j =1
j =1
(3.77)
554
K. Pileckas
where zs(k) =
(k) zs+1
(k+s) \(k−s+1)
− zs(k)
=
|∇v(x)|2 dx,
J
j =1 ωj k+s ∪ωj k−s
|∇v(x)|2 dx.
Note that in (3.77) k is fixed and (3.77) is a difference inequality with respect to s. By virtue of (3.63) (k) zk−1
=
|∇v(x)| dx ≤
|∇v(x)|2 dx
2
(2k−1) \(2)
(2k−1)
≤ kcB1 (f) ≤ kcB0 (f). Therefore, from Lemma 3.5 it follows that zs(k) ≤ scB0 (f)
(3.78)
with c being independent of s and k. Taking in (3.78) s = 1, we get (k)
z1 =
(k+1) \(k)
|∇v(x)|2 dx =
J
|∇v(x)|2 dx ≤ cB0 (f).
j =1 ωj k
R EMARK 3.2. Note that the weak solution u(x) = A(x) + v(x) of problem (3.2) also obeys estimates (3.63), (3.74). This fact follows from estimate (3.27) for the flux carrier A(x) and from estimates (3.63), (3.74) for v(x). Let us study the regularity of the weak solution u(x). T HEOREM 3.7. Let ∂ ∈ C 2 and let f(x) satisfies the conditions of Theorem 3.6. Then 2 () ∩ C 1/2 (), p ∈ W 1 () and the following estimate is valid u ∈ W2,loc 2,loc u; W22 ((2) ) + sup
J
k≥1 j =1
+ sup
J
u; W22 (ωj k ) + ∇p; L2 ((2) )
∇p; L2 (ωj k ) + u; C 1/2 () ≤ Cd ,
k≥1 j =1
where Cd depends only on B0 (f) and
J
2 j =1 |Fj | .
(3.79)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
555
P ROOF. Consider the weak solution u(x) in a bounded domain ωj k . According to local regularity results for solutions to the Navier–Stokes problem, u ∈ W22 (ωj k ) and there holds the estimate u; W22 (ωj k ) ≤ c(u; W21 (( ωj k )4 + f; L2 (( ωj k )),
(3.80)
where the constant c is independent of k (see [28], estimate (36) in Chapter V). Recall that ( ωj k = ωj k−1 ∪ ωj k ∪ ωj k+1 . By the Sobolev embedding theorem (see Lemma I.1.2) from (3.80) follows the inequality u; C 1/2 (ωj k )| ≤ cu; W22 (ωj k ) ≤ c(u; W21 (( ωj k )4 + f; L2 (( ωj k )).
(3.81)
Finally, estimating ∇p from the Navier–Stokes equations and using (3.80), (3.81), we get ∇p; L2 (ωj k ) ≤ c(u; L2 (ωj k ) + (u · ∇)u; L2 (ωj k ) + f; L2 (ωj k )) ≤ c(u; W22 (ωj k ) + u; C 1/2 (ωj k )u; W21 (ωj k ) + f; L2 (ωj k )) # ≤ c u; W21 (( ωj k )4 + (u; W21 (( ωj k )4 + f; L2 (( ωj k ))u; W21 (( ωj k ) $ ωj k ) . + f; L2 ((
(3.82)
Taking in (3.80)–(3.82) the supremum over all k ≥ 1 and applying estimate (3.74), we derive sup u; W22 (ωj k ) + sup ∇p; L2 (ωj k ) + u; C 1/2 (j ) ≤ Cd , k≥1
(3.83)
k≥1
j = 1, . . . , J . Analogously, can be proved the inequalities u; W22 ((2) ) + u; C 1/2 ((2) ) ≤ c(u; W21 ((3) )4 + f; L2 ((3) )) ≤ Cd ,
(3.84)
# ∇p; L2 ((2) ) ≤ c u; W21 ((3) )4 + (u; W21 ((3) )4 + f; L2 ((3) ))u; W21 ((3) ) $ + f; L2 ((3) ) ≤ Cd . Now, estimate (3.79) follows from (3.83)–(3.85) and (3.63).
(3.85)
Now, we are going to prove the uniqueness for small data of the solution to problem (3.2) in a class of function that may grow at infinity.
556
K. Pileckas
T HEOREM 3.8. There exists a number δ0 > 0 such that if B0 (f) +
J
|Fj |2 +
J
j =1
2 |Fj |2
≤ δ0 ,
j =1
then a weak solution of problem (3.2) is unique in a class of functions satisfying the condition −3 lim k
k→∞
|∇w(x)| dx = 0. 2
(3.86)
(k)
P ROOF. Let u(x) be the solution of problem (3.2) found Theorem 3.6 and u(1) (x) be another weak solution of the same problem, satisfying condition (3.86). The difference w(x) = u(1) (x) − u(x) satisfies the integral identity
∇w(x) · ∇η(x) dx =
ν
(w(x) · ∇)η(x) · w(x) dx
(u(x) · ∇)η(x) · w(x) dx +
+
(w(x) · ∇)η(x) · u(x) dx
(3.87)
with any divergence-free η ∈ C0∞ () and
w(x) · n(x) dx (j ) = 0,
j = 1, . . . , J.
(3.88)
σj 1 () which compact Obviously, (3.87) remains valid for any divergence-free η(x) ∈ W2,loc support. Let us take in (3.87)
η(x) = Wk (x), (l)
where Wk (x) is related to w(x) in the same way as Vk (x) is related to v(l) (x), i.e., Wk (x) is defined by formula (3.68) with v(l) (x) changed to w(x). Then (3.87) takes the form |∇w(x)|2 = −ν
ν (k)
J
∇w(x) · ∇Wk (x) dx
j =1 ωj k
+
J
((w(x) + u(x)) · ∇)Wk (x) · (w(x) − Wk (x)) dx
j =1 ωj k
+
(w(x) · ∇)Wk (x) · u(x) dx. (k+1)
(3.89)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
557
Arguing as before, for the first two integrals at the right-hand side of (3.89) we obtain the estimate J "
" " " j =1
ωj k
" " ((w(x) + u(x)) · ∇)Wk (x) · (w(x) − Wk (x)) dx ""
+ν
J "
" " "
ωj k
j =1
+ c2 1 +
" J
" ∇w(x) · ∇Wk (x) dx "" ≤ c1 ∇w; L2 (ωj k )3 j =1
J
∇u; L2 (ωj k )
J
j =1
∇w; L2 (ωj k )2 .
(3.90)
j =1
For the last integral there holds the inequality " " " "
" " (w(x) · ∇)Wk (x) · u(x) dx ""
(k+1)
" " ≤ ""
(k)
" " (w(x) · ∇)Wk (x) · u(x) dx ""
J "
" " + "
" " ≤ ""
j =1
(k)
+ c3
ωj k
" " (w(x) · ∇)Wk (x) · u(x) dx ""
" " (w(x) · ∇)Wk (x) · u(x) dx ""
J
J ∇u; L2 (ωj k ) ∇u; L2 (ωj k )2 .
j =1
j =1
Applying for the first term estimates (3.63), (3.71), we have " " " "
" " (w(x) · ∇)Wk (x) · u(x) dx "" (k) " " " " " ≤" (w(x) · ∇)Wk (x) · u(x) dx "" (2)
+
J k−1 "
" " " j =1 s=1
ωj s
" " (w(x) · ∇)Wk (x) · u(x) dx ""
≤ c4 ∇u; L2 ((2) )
|∇w(x)|2 dx (2)
(3.91)
558
K. Pileckas
+ c5
J k−1
∇u; L2 (ωj s )
|∇w(x)|2 dx ωj s
j =1 s=1
≤ c4 ∇u; L2 ((2) )
|∇w(x)|2 dx (2)
+ c5
J
sup ∇u; L2 (ωj k )
k−1
≤ c6 B0 (f)
1/2
+
J
1/2 |Fj |
2
+
j =1
+
J
j =1 j k
|∇w(x)|2 dx
s=1 ωj s
j =1 k≥1
J
|Fj |
|∇w(x)|2 dx
2 (2)
j =1
1/2 |∇w(x)|2 dx ≤ c6 δ0
|∇w(x)|2 dx.
(3.92)
(k)
Inequalities (3.89)–(3.92) furnish
1/2
ν (k)
|∇w(x)|2 dx ≤ c6 δ0
|∇w(x)|2 dx + c1 (k)
J
∇w; L2 (ωj k )3
j =1 1/2
+ c7 (1 + δ0 )
J
∇w; L2 (ωj k )2 .
j =1 1/2
If c6 δ0
≤ ν/2, this relation yields |∇w(x)|2 dx ≤ 8 c∗
(k)
Denote Zk =
J
∇w; L2 (ωj k )2 + 8 c∗∗
j =1
(k)
J
∇w; L2 (ωj k )3 .
j =1
|∇w(x)|2 dx and rewrite the last inequality as
Zk ≤ 8 c∗ (Zk+1 − Zk ) + 8 c∗∗ (Zk+1 − Zk )3/2 .
(3.93)
By condition (3.86), limk→∞ k −3 Zk = 0. Therefore, applying the second assertion of Lemma 3.5, we conclude from (3.93) that lim Zk = |∇w(x)|2 dx = 0 k→∞
and, hence, w(x) = 0.
R EMARK 3.3. From Theorem 3.8 it follows that, if the right-hand side f(x) of (3.2) is represented in the form (3.4), (3.5) and the norms of data are sufficiently small, then the
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
559
weak solution u(x) of problem (3.2) coincides with a strong solution found in Theorem 3.4 and, thus, u(x) is represented in the form (3.48) and tends in each outlet to infinity j to
the corresponding Poiseuille flow U(j ) (x (j ) ). We finish this section by proving the following assertion. T HEOREM 3.9. Assume that f(x) admits the representation (3.4) with f(j ) ∈ L2 (σj ), j = 1, . . . , J , ( f ∈ L2,β (), βj ≥ 0. If A1 =
J
f(j ) ; L2 (σj )2 +
j =1
J
|Fj |2
j =1
and the number γ∗ in inequality (I.1.143 ) for the weight-function Eβ (x) are sufficiently small, then problem (3.2) admits at least one weak solution u(x) which is represented in the form u(x) =
J
(j )
ζ (xn )U(j ) (x (j ) ) + W(x) + v(x),
(3.94)
j =1 ◦
where U(j ) ∈ W 12 (σj ) are Poiseuille flows in j corresponding to the external forces
f(j ) (x (j ) ) and the fluxes Fj , j = 1, . . . , J , ζ (t) is a smooth cut-off function, W(x) is the solution of the problem div W(x) = − Jj=1 " W(x)"∂ = 0,
(j ) (j ) (j ) ∂ ), (j ) ζ (xn )Un (x ∂xn
supp W(x) ⊂ (3) ,
and ∇v ∈ L2,β (). There holds the estimate ∇v; L2,β ()2 ≤ c(A1 (1 + A1 ) + ( f; L2 ()2 ).
(3.95)
2 (), ∇p ∈ L If ∂ ∈ C 2 , then v ∈ W2,β 2,β () and 2 v; W2,β ()2 + ∇p; L2,β ()2 ≤ Cd ,
(3.96)
f; L2 ()2 . where Cd depends on A1 and ( P ROOF. We look for the week solution of problem (3.2) in the form (3.94) and introduce a (j ) 8(x), where P (j ) (x (j ) ) are presnew pressure function p(x) = Jj=1 ζ (xn )P (j ) (x (j ) ) + p (j )
sure functions associated to Poiseuille flows Un (x (j ) ), j = 1, . . . , J . Then we get for (v(x), p 8(x)) problem (3.49) with the right-hand side 8 f(x) defined by (3.50). The weak
560
K. Pileckas
solution v(x) of problem (3.49) satisfies the integral identity
∇v(x) · ∇η(x) dx −
ν
((V(x) + v(x)) · ∇)η(x) · v(x) dx
(v(x) · ∇)η(x) · V(x) dx =
−
8 f(x) · η(x) dx
(3.97)
with any divergence-free η ∈ C0∞ (). In order to prove the existence of at least one v(x) satisfying (3.97), we first prove the existence of a sequence of divergence-free vector-fields ◦ v(l) ∈ W 12 ((l) ), satisfying the integral identities
∇v(l) (x) · ∇η(x) dx −
ν (l)
((V(x) + v(l) (x)) · ∇)η(x) · v(l) (x) dx (l)
−
8 f(x) · η(x) dx
(v(l) (x) · ∇)η(x) · V(x) dx = (l)
(3.98)
(l) ◦
for any divergence-free η ∈ W 12 ((l) ). As in Theorem 3.5, for the existence of such v(l) (x) it is enough to prove an a priori estimate. Taking in (3.98) η(x) = v(l) (x), we get
8 f(x) · v(l) (x) dx
|∇v(l) (x)|2 dx =
ν (l)
(l)
+
(v(l) (x) · ∇)v(l) (x) · V(x) dx (l)
ν ≤ c8 f; L2 ((l) )2 + |∇v(l) (x)|2 dx 4 (l) + |V(x)|2 |v(l) (x)|2 dx. (l)
Using estimate (II.2.18) for U(j ) (x (j ) ) and estimate (3.13) for W(x), we obtain |V(x)|2 |v(l) (x)|2 dx (l)
≤
|W(x)|2 |v(l) (x)|2 dx + (3)
J l−1
|U(j ) (x (j ) )|2 |v(l) (x)|2 dx
j =1 s=0 ωj s
≤ W; L4 ((3) )2 v(l) ; L4 ((3) )2 +
J l−1
j =1 s=0
U(j ) ; L4 (ωj s )2 v(l) ; L4 (ωj s )2
(3.99)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
561
≤ cW; W21 ((3) )2 +c
J l−1
|∇v(l) (x)|2 dx (3)
U(j ) ; W21 (ωj s )2
j =1 s=0
|∇v (x)| dx +
≤ cA1
(l)
2
(3)
|∇v(l) (x)|2 dx ωj s
J
|∇v (x)| dx (l)
2
j =1 j l
|∇v(l) (x)|2 dx.
≤ cA1 (l)
Therefore, (3.99) and estimate (3.51) for 8 f(x) yield |∇v(l) (x)|2 dx ≤ cA1 (1 + A1 )
ν (l)
+
ν + c1 A1 4
|∇v(l) (x)|2 dx + c( f; L2 ()2 .
(l)
If c1 A1 ≤ ν/4, the last relation furnishes
|∇v(l) (x)|2 dx ≤ cA1 (1 + A1 ) + c( f; L2 ()2
(3.100)
(l)
which guaranties the existence of at least one v(l) (x) satisfying integral identity (3.98) in a bounded domain (l) . Since the right-hand side of (3.100) does not depend on l, ◦
there exists a subsequence {v(lm ) (x)} which converges weakly in W 12 ((k) ) and strongly in L4 ((k) ) for every domain (k) . Therefore, taking in integral identity (3.98) a test function η ∈ C0∞ () and passing to a limit as lm → ∞, we get integral identity (3.97) for the limit vector-field v(x). Moreover, the following estimate
|∇v(x)|2 dx ≤ c A1 (1 + A1 ) + c( f; L2 ()2
(3.101)
is valid. Now we argue as in Theorem 3.2. We take in integral identity (3.97) η(x) = (k) (k) Eβ (x)v(x) + W(k) (x), where Eβ (x) is the “step” weight-function (I.1.16) and W(k) (x) is a vector-field constructed in Lemma I.1.13. As a result we get
(k)
ν
∇v(x) · ∇(Eβ (x)v(x) + W(k) (x))dx
=
(k) 8 f(x) · (Eβ (x)v(x) + W(k) (x))dx
562
K. Pileckas
(k)
+
((V(x) + v(x)) · ∇)(Eβ (x)v(x) + W(k) (x)) · v(x) dx
(k)
+
(v(x) · ∇)(Eβ (x)v(x) + W(k) (x)) · V(x) dx.
(3.102)
There holds the estimate (see the proof of Theorem 3.2) " " " " " " " " (k) (k) (k) " " " " 8 " f(x) · (Eβ (x)v(x) + W (x)) dx " + ν " ∇v(x) · ∇Eβ (x) · v(x) dx " " " " " 1 (k) (k) " " + ν " ∇v(x) · ∇W (x) dx " ≤ E (x)|8 f(x)|2 dx 4ε β (k) (k) (3.103) + cε Eβ (x)|∇v(x)|2 dx + cγ∗ Eβ (x)|∇v(x)|2 dx.
Let us estimate the last two summands at the right-hand side of (3.102). Using Hölder (k) inequality and properties (I.1.14) of the weight-function Eβ (x) we obtain " " " " " ((V(x) + v(x)) · ∇) E (k) (x)v(x) + W(k) (x) · v(x) dx " β " " " " " (k) 1" ≤ "" ((V(x) + v(x)) · ∇) Eβ (x) |v(x)|2 dx "" 2 " " " " + "" ((V(x) + v(x)) · ∇)W(k) (x) · v(x) dx "" (k) (k) 2 ≤ cγ∗ Eβ (x)|V(x)||v(x)| dx + cγ∗ Eβ (x)|v(x)|3 dx
+c
(k) Eβ (x)|v(x)|4 dx
1/2
(k)
(k) E−β (x)|∇W(k) (x)|2 dx
1/2
+c
Eβ (x)|V(x)|2 |v(x)|2 dx
1/2 1/2
(k)
E−β (x)|∇W(k) (x)|2 dx
= I1 + I2 + I3 + I4 . Further, applying Lemma I.1.4 together with Poincaré inequality and using again (I.1.14), we derive
(k)
Eβ (x)|v(x)|4 dx ≤c
|v(x)|4 dx + (3)
J ∞
j =1 s=0
(k)
Eβj (s)
|v(x)|4 dx ωj s
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
2
≤c
|∇v(x)|2 dx
+
J ∞
(3)
j =1 s=0
≤c
|∇v(x)| dx
2
(3)
and
|∇v(x)| dx 2
ωj s
2
(k) Eβj (s)
Eβ(k) (x)|∇v(x)|2 dx
|∇v(x)| dx
≤c
J ∞
j =1 s=0
|∇v(x)|2 dx
ωj s
|∇v(x)| dx +
2
2
(k) Eβj (s)
563
(k)
Eβ (x)|V(x)|2 |v(x)|2 dx |W(x)|2 |v(x)|2 dx
≤c (3)
+
J ∞
j =1 s=0
(k) Eβj (s)
|U
(j )
(x
)| |v(x)| dx
1/2
1/2
|W(x)|4 dx
|v(x)|4 dx
(3)
(3)
J ∞
(k) + Eβj (s)
|U
|∇W(x)| dx + 2
(3)
(x
1/2
(j ) 4
1/2 |v(x)| dx 4
)| dx ωj s
×
(j )
ωj s
j =1 s=0
2
ωj s
≤c
≤c
(j ) 2
J
|∇ U
(j )
(x
(j ) 2
)| dx
(j )
j =1 σj (k)
Eβ (x)|∇v(x)|2 dx.
Therefore, I1 + I2 ≤ cγ∗
(k) Eβ (x)|v(x)|2 dx
1/2
(k) Eβ (x)|V(x)|2 |v(x)|2 dx (k)
≤ cγ∗
1/2
+ |∇W(x)|2 dx +
(3)
1/2 |∇v(x)| dx 2
Eβ (x)|v(x)|4 dx
|∇ U(j ) (x (j ) )|2 dx (j )
j =1 σj
+
J
(k)
1/2
Eβ (x)|∇v(x)|2 dx,
564
K. Pileckas
I3 + I4 ≤ c
|∇W(x)|2 dx + (3)
J
|∇ U(j ) (x (j ) )|2 dx (j )
j =1 σj
1/2
+
|∇v(x)|2 dx
1/2
(k)
×
Eβ (x)|∇v(x)|2 dx
≤ cγ∗
|∇W(x)|2 dx + (3)
|∇ U(j ) (x (j ) )|2 dx (j )
j =1 σj
1/2
+
J
1/2
(k)
E−β (x)|∇W(k) (x)|2 dx
|∇v(x)| dx 2
(k)
×
Eβ (x)|∇v(x)|2 dx.
Note that deriving the last inequality we have used also estimate (I.1.29) for the function W(k) (x). From the two last displayed inequalities, from estimate (II.2.18) for
U(j ) (x (j ) ), from estimate (3.13) for W(x) and from (3.101) it follows that " " " " " ((V(x) + v(x)) · ∇)(E (k) (x)v(x) + W(k) (x)) · v(x) dx " β " " (k) ≤ cγ∗ [A1 (1 + A1 ) + c( f; L2 ()2 ]1/2 Eβ (x)|∇v(x)|2 dx.
Analogously, it can be proved that " " " " " (v(x) · ∇)(E (k) (x)v(x) + W(k) (x)) · V(x) dx " β " "
1/2
(k)
≤c
Eβ (x)|∇v(x)|2 dx
1/2
(k)
×
|∇W(x)| dx + 2
(3)
1/2
(k)
E−β (x)|∇W(k) (x)|2 dx
Eβ (x)|V(x)|2 |v(x)|2 dx
≤c
+
J
|∇ U
(j )
(x
(j ) 2
)| dx
(j )
1/2
j =1 σj (k)
×
Eβ (x)|∇v(x)|2 dx
1/2 ≤ cA1
(k)
Eβ (x)|∇v(x)|2 dx.
(3.104)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
565
Relations (3.102)–(3.104) yield the inequality # 1/2 $ (k) (k) ν Eβ (x)|∇v(x)|2 dx ≤ c2 ε + γ∗ + A1 Eβ (x)|∇v(x)|2 dx
# $1/2 f; L2 ()2 + c3 γ∗ A1 (1 + A1 ) + c( 1 + 4ε
Eβ(k) (x)|∇v(x)|2 dx
(k) Eβ (x)|8 f(x)|2 dx.
(3.105)
Let us fix ε = ν/8c2 . If ν2 , A1 ≤ 64c22
% ν ν , γ∗ ≤ min , 8c2 2c3 (A1 (1 + A1 ) + c( f; L2 ()2 )1/2
then (3.105) furnishes ν (k) (k) Eβ (x)|∇v(x)|2 dx ≤ c Eβ (x)|8 f(x)|2 dx ≤ c Eβ (x)|8 f(x)|2 dx. 8 Since the right-hand side of this inequality is independent of k, we can pass k → ∞ and we get ν Eβ (x)|∇v(x)|2 dx ≤ c Eβ (x)|8 f(x)|2 dx. (3.106) 8 Now, we estimate the norm of second derivatives of the solution v(x). Consider the weak solution v(x) in a bounded domain ωj s . If ∂ ∈ C 2 , then by local regularity results (see [28], Chapter V) the solution to the Navier–Stokes problem v belongs to W22 (ωj s ) and there holds the estimate ωj s )8 + 8 f + (v · ∇)V + (V · ∇)v; L2 (( ωj s )2 v; W22 (ωj s )2 ≤ c v; W21 (( ≤ c v; W21 (( ωj s )8 + 8 f; L2 (( ωj s )2 + ∇V; L4 (( ωj s )2 v; L4 (( ωj s )2 + sup (|V(x)|2 )∇v; L2 (( ωj s )2 x∈
ωj s )8 + 8 f; L2 (( ωj s )2 ≤ c ∇v; L2 ((
+ ∇V; W22 (( ωj s )2 ∇v; L2 (( ωj s )2 ,
(3.107)
where ( ωj s = ωj s−1 ∪ ωj s ∪ ωj s+1 . Multiplying (3.107) by Eβj (s) and summing the obtained relations over s from 1 to ∞, we obtain 2 v; W2,β (j )2 ≤ c ∇v; L2 ()6 ∇v; L2,β (j )2 + 8 f; L2,β ()2
566
K. Pileckas
+ A1 ∇v; L2,β (j )2 ≤ c 8 f; L2,β ()8 + (1 + A1 )8 f; L2,β ()2 ,
(3.108)
j = 1, . . . , J . Proving (3.108) we have used properties (I.1.14) of the weight-function Eβ (x), inequality (3.106) and inequalities (II.2.18), (3.13) in order to estimate the norm ωj s ). Analogously, we get that V; W22 (( v; W22 ((3) )2 ≤ c 8 f; L2,β ()8 + (1 + A1 )8 f; L2,β ()2 .
(3.109)
It follows from (3.108), (3.109) and from estimate (3.51) for the norm of 8 f(x) that 2 v; W2,β ()2 ≤ Cd ,
where Cd depends on A1 and on the norm ( f; L2,β ()2 . Finally, ∇p can be estimated from Navier–Stokes equations and we get 8d . ∇p; L2,β ()2 ≤ C
R EMARK 3.4. Note that in Theorem 3.9 we do not require the norm ( f; L2 () to be small. For the existence of a weak solution u(x), which tends in each outlet to infinity j to the corresponding Poiseuille flow, only the fluxes Fj and norms f(j ) ; L2 (σj ) of functions f(j ) (x (j ) ), participating in determination of Poiseuille flows, have to be sufficiently small. In this regard we should mention the papers [82], [83], where the opposite results are obtained. In [82], [83] the invertibility of the linearization of steady Navier–Stokes equations near the Poiseuille solution with the flux F in infinite and semi-infinite straight two dimensional channels is investigated under certain symmetry conditions. It is proved that in the case of an infinite channel such “Poiseuille linearization” is invertible for all fluxes while for semi-infinite case this result holds only off a discrete set of singular points (possibly empty). Using these results in [82], [83] the existence of a solution to nonlinear Navier–Stokes system approaching at infinity the Poiseuille flow is proved for fluxes of arbitrary large magnitude if the perturbation data is small enough in a suitable norm.
4. Chapter IV. Time-dependent problems As it is already mentioned in Introduction, properties of solutions to time-dependent Navier–Stokes system in domains with noncompact boundaries are studied not perfectly. It is known (see [33], [34], [90], [92], [93]) that in domains with outlets to infinity there exist solutions with prescribed fluxes Fj (t) and, dependently on the geometry of the outlets, solutions have finite or infinite energy integral. In particular, if outlets are cylindrical, the energy integral is infinite. Note that the solvability of both two and three-dimensional nonlinear time-dependent Navier–Stokes problems is proved in [33], [34], [90], [92], [93] either for small data or for small time intervals. The global solvability of the two-dimensional
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
567
Navier–Stokes problem in domains with cylindrical outlets to infinity is mentioned in [93] as unsolved problem. The Saint-Venant type estimates giving spatial decay bounds for solutions of timedependent Stokes problem in an infinite cylinder were obtained also in [37], [38], [95]. Finally, we have to mention the paper of A.V. Babin [7] where the time-dependent perturbation of the steady Poiseuille flow is studied in weighted spaces with polynomial weights. In [7] it is assumed that the flux F is independent of t and it is proved that for sufficiently small |F | the solution of time-dependent Navier–Stokes problem tends to the steady Poiseuille solution as |x| → ∞. In this chapter we study time-dependent Stokes and Navier–Stokes problems in the domain with cylindrical outlets to infinity. We assume that in each outlet to infinity j the initial data u0 (x) and the external force f(x, t) admit the representations (j )
u0 (x), u0 (x, t) = u0 (x (j ) ) + (
f(x, t) = f(j ) (x (j ) , t) +( f(x, t),
where ( f(x, t) and ( u0 (x, t) belong to weighted spaces of vanishing as |x| → ∞ functions. We look for the solutions to time-dependent Stokes and Navier–Stokes systems which have prescribed fluxes Fj (t) over sections of all outlets to infinity and which tend as |x| → ∞, x ∈ j , to time-dependent Poiseuille solution corresponding to the outlet j . The considered problem may be treated as a time-dependent generalization of the steady Leray’s problem. For time-dependent Stokes problem we prove for arbitrary data and the infinite time interval (both in two and three-dimensional cases) the existence and uniqueness of regular solutions which tend in all outlets to infinity to the corresponding time dependent Poiseuille flows, assuming only the necessary for regularity compatibility conditions (Section 4.3). In Section 4.4 for nonlinear Navier–Stokes problem the global unique solvability, i.e., for arbitrary data (in particular, for arbitrary fluxes) and the infinite time interval, is proved in the two-dimensional case assuming that fluxes Fj (t) vanish as t → ∞. If limt→∞ Fj (t) = Fj , where Fj are constants, the global unique solvability is proved only under the condition that |Fj | are “small”. For large |Fj | the existence of a unique solution is proved in the twodimensional domain for arbitrary data and arbitrary finite time interval [0, T ], T < ∞ (long time solvability). In the case of three-dimensional domains (Section 4.5) only local solvability could be proved, i.e., either for “small” data or for “small” time intervals. Note that the essential point in the proofs is the existence of time-dependent Poiseuille flows (see Chapter II) which are exact solutions of Navier–Stokes system in a straight cylinder. This gives the possibility to construct a “flux carrier” V(x, t) which coincides for large |x|, x ∈ j , j = 1, . . . , J , with corresponding Poiseuille flows (Section 4.2). Therefore, the remainder left by V(x, t) in Navier–Stokes equations has a compact support and it is possible to prove the solvability for the obtained perturbed problem by the method of weighted energy estimates. In the case where coincides with the straight three-dimensional cylinder # we prove the global unique existence for the perturbation of the time-dependent generalized Poiseuille flow assuming only that the norms of ( f(x, t) and ( u0 (x) are “sufficiently small”, i.e., in this case the fluxes and norms of the parts of an initial data and an external force participating in the definition of Poiseuille solution may be arbitrary (see Section 4.5.4). In Section 4.6 we prove the uniqueness of the solution to time-dependent
568
K. Pileckas
Navier–Stokes problem in a class of functions that in certain sense are bounded at infinity. Note that in Sections 4.4 and 4.5 the uniqueness result is proved only for solutions which have the special asymptotic representation u(x, t) = v(x, t) + V(x, t),
p(x, t) = p 8(x, t) + P (x, t),
where (V(x, t), P (x, t)) coincides in each outlet to infinity j with the Poiseuille flow corresponding to this outlet. In particular, from results of Section 4.6 follows the uniqueness of the time-dependent Poiseuille flow in a straight cylinder. Note that for the steady case the uniqueness of Poiseuille flow is not known and perhaps is not true. In all cases the decay rate of the solutions is conditioned only by the decay rate of an external force and initial data. In particular, if ( f(x, t) = 0, ( u0 (x) = 0, then in each outlet to infinity j the solution exponentially tends as |x| → ∞ to the corresponding Poiseuille flow. From the obtained results it follows also that in a two-dimensional domain with cylindrical outlets to infinity the solution of time-dependent Navier–Stokes system having constant fluxes Fj ∈ R exponentially tends in each outlet to infinity to the corresponding steady Poiseuille flow. This result is true for arbitrary fluxes and arbitrary finite time interval [0, T ]. All results of this chapter belong to the author (see [69], [71–73], [79]).
4.1. Formulation of the problem Let us consider in the domain T the time-dependent Stokes ⎧ ⎪ ⎪ ⎪ ⎪ ⎨
ut (x, t) − νu(x, t) + ∇p(x, t) = f(x, t), div u(x, t) = 0, " " u(x, t) = 0, u(x, 0) = u0 (x),
⎪ ∂ ⎪ ⎪ ⎪ ⎩ u(x, t) · n(x) dx (j ) = F (t), j σj
(4.1)
j = 1, . . . , J,
and the nonlinear Navier–Stokes ⎧ u (x, t) − νu(x, t) + (u(x, t) · ∇)u(x, t) + ∇p(x, t) = f(x, t), ⎪ ⎪ ⎪ t ⎪ ⎨ div u(x, t) = 0, " " u(x, t) ∂ = 0, u(x, 0) = u0 (x), ⎪ ⎪ ⎪ ⎪ ⎩ u(x, t) · n(x) dx = Fj (t), j = 1, . . . , J,
(4.2)
σj
problems with additionally prescribed fluxes Fj (t) over cross-sections σj of outlets to infinity. The fluxes Fj (t) must satisfy the compatibility condition Fj (0) =
σj
u0 (x) · n(x) dx , j = 1, . . . , J, J j =1 Fj (t) = 0 ∀t ∈ [0, T ].
(4.3)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
569
Assume that the initial data u0 (x) and the external force f(x, t) admit the representations
(j ) (j ) u0 (x, t) = Jj=0 ζ (xn )u0 (x (j ) ) + ( u0 (x), J (j ) (j ) (j ) f(x, t), f(x, t) = j =0 ζ (xn )f (x , t) +(
(4.4)
where
(j )
(j )
(j )
u0 (x (j ) ) = (u01 (x (j ) ), . . . , u0n (x (j ) )),
(j )
(4.5)
(j )
f(j ) (x (j ) , t) = (f1 (x (j ) , t), . . . , fn (x (j ) , t)),
ζ (τ ) is a smooth cut-off function with ζ (τ ) = 0 for τ ≤ 1 and ζ (τ ) = 1 for τ ≥ 2, (j ) 2l,l f ∈ W2,β (T ) and ( u0 ∈ u0 ∈ W22l+1 (σj ), f(j ) ∈ W22l,l (jT ), where jT = σj × (0, T ); ( 2l+1 f(x, t) and ( u0 (x) vanish as |x| → ∞. W2,β () with βj ≥ 0, j = 1, . . . , J , i.e., ( We look for the solutions (u(x, t), p(x, t)) of problems (4.1) and (4.2) that tend in each
outlet to infinity j to time-dependent Poiseuille flow (U(j ) (x (j ) , t), P (j ) (x (j ) , t)) corre sponding to the pipe #j , the flux Fj (t), the external force f(j ) (x (j ) , t) and the initial data
(j ) u0 (x (j ) ) (see Chapter II).
4.2. Construction of the flux carrier 4.2.1. Construction of the flux carrier Let ∂ ∈ C 2 , Fj ∈ W21 (0, T ). Assume that the initial data u0 (x) and the external force f(x, t) admit the representation (4.4), (4.5) with (j )
u0
(j )
◦
(j )
= (u10 , . . . , u0n−1 ) ∈ W 12 (σj ),
2 ( u0 ∈ W2,β () ⊂ W22 (),
◦
(j )
u0n ∈ W 12 (σj ),
f(j ) ∈ L2 (jT ), (4.6)
( f ∈ L2,β (T ) ⊂ L2 (T ),
(4.7)
i.e., βj ≥ 0, j = 1, . . . , J . Moreover, assume that there hold the compatibility conditions (j )
div u0 (x (j ) ) = 0,
Fj (0) = σj
(j )
u0n (x (j ) ) dx (j ) ,
j = 1, . . . , J.
(4.8)
Then in each cylinder #j exists the generalized time dependent Poiseuille flow (U(j ) (x (j ) , t), P (j ) (x (j ) , t)) having the form (II.2.112) and satisfying estimates (II.2.115) and (II.2.149). Set U(x, t) =
J
j =1
(j )
ζ (xn )U(j ) (x (j ) , t), P (x, t) =
J
j =1
(j )
ζ (xn )P (j ) (x (j ) , t).
(4.9)
570
K. Pileckas
Let g(x, t) = − div U(x, t) = −
J
(j ) (j ) (j ) ∂ , t). j =1 ∂x (j ) ζ (xn )Un (x n
Then
suppx g(x, t) ⊂ (2) \ (1) . J
j =1 Fj (t) = 0
From the condition
(4.10)
we get
g(x, t) dx = 0 ∀t ∈ [0, T ].
(4.11)
(2)
Moreover, in virtue of (II.2.149)
T
0
g( ·, t); W22 ((3) )2 dt +
≤c
J
0
j =1
≤c
T
T
gt (·, t); L2 ((3) )2 dt
0
(j ) Un (·, t); W22 (σj )2 dt
+ 0
T
(j ) Unt (·, t); L2 (σj )2 dt
J
(j ) (j ) fn ; L2 (jT )2 + u0n ; W21 (σj )2 + Fj ; W21 (0, T )2 .
(4.12)
j =1 ◦
◦
Since g ∈ W 12 ((3) ), by Lemma I.1.15, there exits a vector-field W(·, t) ∈ W 22 ((3) ) with ◦
Wt (·, t) ∈ W 12 ((3) ) such that div W(x, t) = g(x, t),
div Wt (x, t) = gt (x, t),
and there holds the estimate 0
T
W(·, t); W22 ((3) )2 dt
≤c 0
≤c
T
+
T
0
g(·, t); W21 ((3) )2 dt
Wt (·, t); W21 ((3) )2 dt
T
+
gt (·, t); L2 ((3) ) dt 2
0
J
(j ) (j ) fn ; L2 (jT )2 + u0n ; W21 (σj )2 + Fj ; W21 (0, T )2 .
(4.13)
j =1
Define V(x, t) = U(x, t) + W(x, t). Then, div V(x, t) = 0,
" V(x, t)"∂ = 0,
(4.14)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
571
V(x, t) · n(x) ds = Fj (t),
j = 1, . . . , J,
σj
and for x ∈ j \ j 3 , j = 1, . . . , J , the vector-field V(x, t) coincides with the velocity part
U(j ) (x (j ) , t) of the corresponding generalized time-dependent Poiseuille flow. Consider now the case of unidirectional time-dependent Poiseuille flows, i.e., assume
(j ) that u0 (x (j ) ) = 0, f(j ) (x (j ) , t) = 0. Moreover, assume that the data are more regular: ∂ ∈ C 2l+3 ,
(j )
u0n ∈ W22l+1 (σj ),
Fj ∈ W2l+1 (0, T ),
l ≥ 0,
(j )
fn ∈ W22l,l (jT ), j = 1, . . . , J,
(4.15)
l ≥ 0,
and let there hold the compatibility conditions of order l (see (II.2.75)). According to Theorem II.2.7, in each #j there exists a unique unidirectional time-dependent Poiseuille flow (U (j ) , q (j ) ) ∈ W22l+2,l+1 (jT ) × W2l (0, T ) satisfying the estimate (II.2.76). Then the function g ∈ W22l+2,l+1 (T(3) ) and g; W22l+2,l+1 (T(3) )2 ≤ c
J
(j )
Un ; W22l+2,l+1 (σj )2
j =1
≤c
J
(j ) (j ) fn ; W22l,l (jT )2 + u0n ; W22l+1 (σj )2 j =1
+ Fj ; W2l+1 (0, T )2 .
(4.16) ◦
By Lemma I.1.16, there exist a vector-field W ∈ W22l+2,l+1 (T(3) ) ∩ W 12 ((3) ) such that ∂ l+1 ∇W ∈ L2 (T(3) ), ∂t l+1
suppx W(·, t) ⊂ (3) and
9 l+1 92 9∂ 9 T 9 ∇W; L ( ) W; W22l+2,l+1 (T(3) )2 + 9 2 (3) 9 9 ∂t l+1 ≤ cg; W22l+2,l+1 (T(3) )2 ≤c
J
(j ) (j ) fn ; W22l,l (jT )2 + u0n ; W22l+1 (σj )2 j =1
+ Fj ; W2l+1 (0, T )2 .
(4.17)
We define the flux carrier V(x, t) again by formula (4.14), however, now U(x, t) and W(x, t) are more regular. 4.3. The Time-dependent Stokes problem
572
K. Pileckas
4.3.1. Time dependent Stokes problem in the case of zero fluxes Consider in T the timedependent Stokes problem (4.1) assuming that all fluxes Fj (t), j = 1, . . . , J , are equal to zero, i.e., consider the problem ⎧ ut (x, t) − νu(x, t) + ∇p(x, t) = f(x, t), ⎪ ⎪ ⎪ ⎪ ⎨ div u(x, t) = 0, " (4.18) u(x, t)"∂ = 0, u(x, 0) = u0 (x), ⎪ ⎪ ⎪ ⎪ ⎩ u(x, t) · n(x) ds = 0, j = 1, . . . , J. σj
◦
By a weak solution of problem (4.18) we understand the function u ∈ W 21,1 (T ) satisfying the conditions u(x, 0) = u0 (x),
div u(x, t) = 0
(4.19)
and the integral identity t 0
uτ (x, τ ) · η(x, τ ) dxdτ + ν
0
=
t 0
t
f(x, τ ) · η(x, τ ) dxdτ
∇u(x, τ ) · ∇η(x, τ ) dxdτ
∀t ∈ (0, T ]
(4.20)
◦
for every divergence-free vector-field η ∈ W 21,0 (T ). Note that each divergence-free vector field u(x, t) equal to zero on ∂ and such that u(·, t), ∇u(·, t) ∈ L2 () has zero fluxes over all cross-sections σj , i.e. u(x, t) · n(x) dS = 0, j = 1, . . . , J. σj ◦
T HEOREM 4.1. Let f ∈ L2 (T ), u0 ∈ W 12 () and let there hold the compatibility conditions div u0 = 0, u0 (x) · n(x) dS = 0, j = 1, . . . , J. (4.21) σj
Then problem (4.18) admits a unique weak solution u(x, t) and the following estimate is valid u; W21,1 (T ) ≤ c f; L2 (T ) + u0 ; W21 () , (4.22) where the constant c is independent of f(x, t), u0 (x) and T . The proof of the theorem is standard (e.g. [28]). First, using the Galerkin method one could prove the existence of a weak solution u(k) (x, t) in the bounded domain (k) and then to pass k → ∞ in integral identity (4.19).
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
573
Let us prove the solvability of problem (4.18) in weighted spaces. 1 (), β ≥ 0, j = 1, . . . , J , and let there T HEOREM 4.2. Let f ∈ L2,β (T ), u0 ∈ W2,β j hold the compatibility conditions (4.21). If the number γ∗ in the inequality (I.1.143 ) for the weight-function Eβ (x) is sufficiently small, then the weak solution u(x, t) belongs to the 1,1 space W2,β (T ) and there holds the estimate 1,1 1 sup u(·, t); W2,β () + u; W2,β (T )
t∈[0,T ]
1 ≤ c u0 ; W2,β () + f; L2,β (T ) .
(4.23)
2,1 If, in addition, ∂ ∈ C 2 , then u ∈ W2,β (T ) and there exists a function p(x, t) with ∇p ∈ T L2,β ( ) such that the pair (u(x, t), p(x, t)) satisfies equations (4.18) almost everywhere in T . There holds the estimate 2,1 u; W2,β (T ) + ∇p; L2,β (T ) 1 ≤ c u0 ; W2,β () + f; L2,β (T ) .
(4.24) (k)
P ROOF. Take in the integral identity (4.20) η(x, t) = Eβ (x)u(x, t) + W(k) (x, t), where (k)
Eβ (x) is the “step” weight-function (I.1.16) and W(k) (x, t) is a vector-field constructed in Lemma I.1.17. Note that div η(x, t) = 0, η(x, t)|∂ = 0. We get t 0
(k) Eβ (x)uτ (x, τ ) · u dxdτ
+ν =
t
t 0
0
+
t 0
uτ (x, τ ) · W(k) (x, τ ) dxdτ
(k) ∇u(x, τ ) · ∇ Eβ (x)u(x, τ ) + W(k) (x, τ ) dxdτ
f(x, τ ) · Eβ(k) (x)u(x, τ ) + W(k) (x, τ ) dxdτ.
Since t 0
(k)
Eβ (x)uτ (x, τ ) · u(x, τ ) dxdτ =
1 2
1 − 2
(k)
Eβ (x)|u(x, t)|2 dx
(k)
Eβ (x)|u0 (x)|2 dx,
relation (4.25) takes the form 1 2
(k) Eβ |u(x, t)|2 dx
+ν
t 0
(k)
Eβ (x)|∇u(x, τ )|2 dxdτ
(4.25)
574
K. Pileckas
=
1 2
(k)
−ν
Eβ (x)|u0 (x)|2 dx −
t 0
+
0
0
uτ (x, τ ) · W(k) (x, τ )dxdτ
(k) ∇u(x, τ ) · ∇Eβ (x)u(x, τ ) + ∇W(k) (x, τ ) dxdτ
t
t
(k) f(x, τ ) · Eβ (x)u(x, τ ) + W(k) (x, τ ) dxdτ
= I1 (t) + I2 (t) + I3 (t) + I4 (t).
(4.26)
Arguing as in Theorem III.3.2, we derive 1 t (k) |I3 (t)| + |I4 (t)| ≤ E (x)|f(x, τ )|2 dxdτ 4ε 0 β t (k) Eβ (x)|∇u(x, τ )|2 dxdτ. + c1 (ε + γ∗ ) 0
We estimate I2 (t) using the Hölder and Young inequalities, weighted Poincaré inequal(k) ity (I.1.19), properties (I.1.14) of the weight-function Eβ (x) and inequality (I.1.45) for W(k) (x, t): |I2 (t)| ≤c
t 0
×
t 0
≤c
t 0
×
(k) E−β (x)|W(k) (x, τ )|2 dxdτ
≤ c 2 γ∗
1/2
(k)
Eβ (x)|uτ (x, τ )|2 dxdτ
1/2
(k)
E−β (x)|∇W(k) (x, τ )|2 dxdτ
t 0
1/2
(k) Eβ (x)|uτ (x, τ )|2 dxdτ
t 0
1/2
(k)
Eβ (x)|∇u(x, τ )|2 dxdτ +
t 0
(k) Eβ (x)|uτ (x, τ )|2 dxdτ .
From these estimates and from (4.26) it follows that t 1 (k) (k) Eβ (x)|u(x, t)|2 dx + ν Eβ (x)|∇u(x, τ )|2 dxdτ 2 0 t (k) Eβ (x)|∇u(x, τ )|2 dxdτ ≤ c3 (ε + γ∗ ) 0
+ c 2 γ∗
t 0
(k) Eβ (x)|uτ (x, τ )|2 dxdτ
1 + 2
(k)
Eβ (x)|u0 (x)|2 dx
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
+
t
1 4ε
0
(k)
Eβ (x)|f(x, τ )|2 dxdτ.
575
(4.27)
Now, take in (4.20) (k)
(k)
η(x, t) = Eβ (x)ut (x, t) + Wt (x, t). This gives t
ν (k) Eβ (x)|∇u(x, t)|2 dxdt 2 0 t ν (k) E (x)|∇u0 (x)|2 dx − uτ (x, τ ) · W(k) = τ (x, τ ) dxdτ 2 β 0 t (k) ∇u(x, τ ) · ∇Eβ (x)uτ (x, τ ) + ∇W(k) −ν τ (x, τ ) dxdτ (k)
Eβ (x)|uτ (x, τ )|2 dxdτ +
0
+
t 0
(k) f(x, τ ) · Eβ (x)uτ (x, τ ) + W(k) τ (x, τ ) dxdτ.
(4.28)
Estimating the right-hand side of (4.28) as above and using (I.1.46) we derive t ν (k) (k) Eβ (x)|uτ (x, τ )|2 dxdτ + Eβ (x)|∇u(x, t)|2 dx 2 0 t (k) Eβ (x)|uτ (x, τ )|2 dxdτ ≤ c4 (γ∗ + ε) 0
+ c 5 γ∗
t 0
ν + 2
Eβ(k) (x)|∇u(x, τ )|2 dxdτ
(k) Eβ (x)|∇u0 (x)|2 dx
1 + 4ε
t 0
(k)
Eβ (x)|f(x, τ )|2 dxdτ.
(4.29)
Summing inequalities (4.27) and (4.29) yields 1 (k) E (x)(|u(x, t)|2 + ν|∇u(x, t)|2 ) dx 2 β t (k) + Eβ (x)(|uτ (x, τ )|2 + ν|∇u(x, τ )|2 ) dxdτ 0
t
≤ c6 (γ∗ + ε) 1 + 2
1 + 2ε
0
(k)
Eβ (x)(|uτ (x, τ )|2 + |∇u(x, τ )|2 ) dxdτ
(k)
Eβ (x)(|u0 (x)|2 + ν|∇u0 (x)|2 ) dx
t 0
(k)
Eβ (x)|f(x, τ )|2 dxdτ.
(4.30)
576
K. Pileckas
Let us fix ε = min{ 4cν6 , 4c16 }. If the number γ∗ is sufficiently small, i.e., if γ∗ ≤
1 min{ν, 1}, 4c6
then from (4.30) follows the inequality
(k)
Eβ (x)(|u(x, t)|2 + ν|∇u(x, t)|2 ) dx
+
t 0
(k)
Eβ (x)(|uτ (x, τ )|2 + ν|∇u(x, τ )|2 ) dxdτ
(k)
≤
Eβ (x)(|u0 (x)|2 + ν|∇u0 (x)|2 ) dx + c
1 ()2 + f; L2,β (T )2 . ≤ c u0 ; W2,β
t 0
(k)
Eβ (x)|f(x, τ )|2 dxdτ (4.31)
The right-hand side of (4.31) does not depend on k. Passing in (4.31) to a limit as k → ∞ we get 1,1 1 u(·, t); W2,β ()2 + u; W2,β (T )2 1 ≤ c u0 ; W2,β ()2 + f; L2,β (T )2 .
(4.32)
Let now ∂ ∈ C 2 . From integral identity (4.20) it follows that for almost all t ∈ [0, T ] there holds the identity
∇u(x, t) · ∇η(x) dx =
ν
(f(x, t) − ut (x, t)) · η(x) dx ◦
where η(x) is arbitrary divergence-free function belonging to the space W 12 (). Hence, ◦
u(·, t) ∈ W 12 () could be considered as a weak solution of the stationary Stokes problem with right-hand side equal to f(x, t) − ut (x, t): ⎧ −νu(x, t) + ∇p(x, t) = f(x, t) − ut (x, t), ⎪ ⎪ ⎪ ⎪ ⎨ div u(x, t) = 0, " u(x, t)"∈∂ = 0, ⎪ ⎪ ⎪ ⎪ ⎩ u(x, t) · n(x) ds = 0, j = 1, . . . , J.
(4.33)
σj
We have f(·, t) − ut (·, t) ∈ L2,β () for almost all t ∈ [0, T ]. If γ∗ is sufficiently small, 2 () (see Theothen the weak solution u(x, t) of (4.33) belongs to the space W2,β rem III.3.2). Moreover, there exists a function p(x, t) with ∇p(·, t) ∈ L2,β () such that
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
577
(u(x, t), p(x, t)) satisfy equations (4.33) almost everywhere in and there holds the estimate 2 ()2 + ∇p(·, t); L2,β ()2 u(·, t); W2,β
≤ c(f(·, t) − ut (·, t)); L2,β ()2 ≤ c f(·, t); L2,β ()2 + ut (·, t); L2,β ()2 .
(4.34)
Integrating inequality (4.34) with respect to t and using (4.32) we derive t 0
Eβ (x)(|u(x, τ )|2 + |∇u(x, τ )|2 +
+
t 0
|Dxα u(x, τ )|2 ) dxdτ
|α|=2
Eβ (x)|∇p(x, τ )|2 dxdτ
1 ()2 + f; L2,β (T )2 . ≤ c u0 ; W2,β
The last estimate together with (4.32) is equivalent to (4.24).
R EMARK 4.1. In the proof of Theorem 4.2 we have allowed an “inaccuracy”. For the function u ∈ W21,1 (T ) the derivative ut (x, t) does not belong to the space W21,0 (T ) (k) (k) and, formally, we are not allowed to take η(x, t) = Eβ (x)ut (x, t) + Wt (x, t) in integral identity (4.20). However, since the norm of ∇ut (x, t) is not involved into final estimates, ◦ we can first approximate the data f ∈ L2 (T ), u0 ∈ W 12 () by functions fn with fn , fnt ∈ ◦
L2 (T ) and u0n ∈ W22 () ∩ W 12 (). To each pair (fn (x, t), u0n (x)) there exists a unique solution un ∈ W21,1 (T ) with ∇unt ∈ L2 (T ) (see, e.g., considerations of Ch. 6 in [28]). For such un (x, t) all considerations of Theorem 4.2 are legitimate. Thus, for un (x, t) we get estimates (4.23), (4.24) and then pass n → ∞. ◦
R EMARK 4.2. Theorem 4.2 remains valid also if f ∈ L2,β (T ), u0 ∈ W 12,β (), with βj < ◦
◦
0, j = 1, . . . , J . However, in this case L2,β (T ) L2 (T ), W 12,β () W 12 (), and in order to prove the theorem, we, first, approximate f(x, t) and u0 (x) by smooth functions fn (x, t) and u0n (x) having compact supports. To each pair (fn (x, t), u0n (x)) corresponds ◦ a unique solution un ∈ W 21,1 (T ) to problem (4.18). Arguing as above we get for un (x, t) estimate (4.23) with constant independent of n and pass n → ∞. Note that in the case 1,1 (T ) and satisfies integral identity (4.20) βj < 0 the weak solution u(x, t) belongs to W2,loc ◦
for all divergence-free η ∈ W 21,0 (T ) with compact supports. 4.3.2. Higher order compatibility conditions for Stokes problem It is well known that smooth solutions of problem (3.1) could exist only if the data f(x, t), u0 (x) satisfy the
578
K. Pileckas
compatibility conditions. Let us describe these conditions following the arguments of papers [86], [87]. Denote g (k) (x) =
" ∂k g(x, t)"t=0 . k ∂t
If the solution u(x, t) of (4.18) is a smooth function, it should satisfy the compatibility conditions at the edge ∂ × {t = 0} of the domain × (0, T ): " u(k) (x)"∂ = 0.
(4.35)
Let us assume that " u(k) (x) · n(x)"∂ = 0.
(4.36)
Then u(k) (x) could be defined from the system (4.18). Differentiating (4.18) k − 1 times with respect to t and passing t → 0 we get u(k) (x) = νu(k−1) (x) − ∇p (k−1) (x) + f(k−1) (x).
(4.37)
Obviously, div u(k) (x) = 0 and, therefore, from (4.18) it follows that p (k−1) (x) = div f(k−1) (x)
(4.38)
" ∂ (k−1) "" (x) ∂ = (νu(k−1) (x) · n(x) + f(k−1) (x) · n(x))"∂ . p ∂n
(4.39)
and
Thus, p (k−1) (x) is defined over u(k−1) (x) as a solution of the Neumann problem (4.38), (4.39). Note that the necessary compatibility condition for the existence of a solution to problem (4.38), (4.39) with the finite Dirichlet integral (νu(k−1) (x) · n + f(k−1) (x) · n(x))dS ∂
=−
div(u(k−1) (x) + f(k−1) (x))dx
=−
div f(k−1) (x) dx
(4.40)
is valid. Hence, assuming that u(k) (x) · n(x)|∂ = 0, we could define all u(k) (x) by recurrent formulas (4.37)–(4.39), where u(0) (x) = u0 (x). As usually, we assume that data f(x), u0 (x) of problem (4.18) satisfy the compatibility conditions of the order s, if the functions u(k) (x), k = 0, 1, . . . , s, defined from recurrent formulas (4.37)–(4.39) satisfy condition (4.35).
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
579
2l,l 2l+1 2l+3−2k Let f ∈ W2,β (T ), u0 ∈ W2,β (), k ≤ l, and let us assume that u(k−1) ∈ W2,β (). Then 2l−2k div f(k−1) ∈ W2,β (),
" 2l−2k+1/2 (νu(k−1) · n + f(k−1) · n)"∂ ∈ W2,β (∂),
and for sufficiently small γ∗ for the solution p (k−1) (x) of the problem (4.38), (4.39) holds 2l+1−2k the inclusion ∇p (k−1) ∈ W2,β () and the estimate 2l+1−2k () ∇p (k−1) ; W2,β 2l−2k 2l+3−2k ≤ c div f(k−1) ; W2,β () + u(k−1) ; W2,β () 2l,l 2l+3−2k (T ) + u(k−1) ; W2,β () . ≤ c f; W2,β
This estimate may be proved using the same method as in the proof of Theorem III.3.2 (see [51] for results in weighted spaces with exponential weights). From (4.37) we get now 2l+1−2k u(k) ∈ W2,β () and 2l,l 2l+1−2k 2l+3−2k u(k) ; W2,β () ≤ c f; W2,β (T ) + u(k−1) ; W2,β () . 2l+1 (), from the last two inequalities it follows that Since u(0) = u0 ∈ W2,β
2l,l 2l+1−2k 2l+1 u(k) ; W2,β () ≤ c f; W2,β (T ) + u0 ; W2,β () .
(4.41)
Finally, we note that u(k) (x) · n(x) ds = 0 ∀j = 0, . . . , J, σj
because divergence-free vector-fields with finite Dirichlet integrals have zero fluxes in domains with cylindrical outlets to infinity. 2l+2,l+1 (T ) 4.3.3. Solvability of problem (4.18) in spaces W2,β 2l,l 2l+1 T HEOREM 4.3. Let ∂ ∈ C 2l+2 . Assume that f ∈ W2,β (T ), u0 ∈ W2,β (), βj ≥ 0, j = 1, . . . , J , l ≥ 1, and let there hold the compatibility conditions of the order l. If γ∗ is sufficiently small, then problem (4.18) admits a unique solution (u(x, t), p(x, t)) with 2l+2,l+1 2l,l u ∈ W2,β (T ) and ∇p ∈ W2,β (T ). There holds the estimate 2l+2,l+1 2l,l u; W2,β (T ) + ∇p; W2,β (T ) 2l,l 2l+1 () + f; W2,β (T ) . ≤ c u0 ; W2,β
(4.42)
580
K. Pileckas
P ROOF. First of all, note, if the compatibility conditions of the order l are valid, then we have, by classical regularity results for linear time-dependent Stokes problem (4.18), u ∈ W22l+2,l+1 (T ), ∇p ∈ W22l,l (T ) (see [86], [28]) and according to Theorem 4.1 the solution is unique. Thus, we need only to prove estimate (4.42). For l = 0 estimate (4.42) coincides with (4.24) (see Theorem 4.2). Let l ≥ 1. We assume 2r,r 2r+1 that (4.42) is true for l ≤ r − 1 and prove it for l = r. Let f ∈ W2,β (T ), u0 ∈ W2,β () 2r,r 2r−2,r−1 and let u ∈ W2,β (T ) ∩ W22r+2,r+1 (T ), ∇p ∈ W2,β (T ) ∩ W22r,r (T ) be a solution of problem (4.18) satisfying estimate (4.42) at l = r − 1. Denote
u{s} (x, t) =
∂s u(x, t), ∂t s
f{s} (x, t) =
∂s f(x, t), ∂t s
p {s} (x, t) =
∂s p(x, t), ∂t s
s = 1, . . . , r.
Differentiating equation (4.18) s times with respect to t we derive the problem ⎧ {s} ut (x, t) − νu{s} (x, t) + ∇p {s} (x, t) = f{s} (x, t), ⎪ ⎪ ⎪ ⎪ ⎨ div u{s} (x, t) = 0, " ⎪ u{s} (x, 0) = u(s) (x), u{s} (x, t)"∂ = 0, ⎪ ⎪ ⎪ ⎩ u{s} (x, t) · n(x) ds = 0, j = 1, . . . , J.
(4.43)
σj
2r−2s,r−s (T ). Moreover, in virtue of the compatibility conditions of Obviously, f{s} ∈ W2,β 2r+1−2s () and the order r, we have u(s) ∈ W2,β
" div u(s) (x) = 0, u(s) (x)"∂ = 0, u(s) (x) · n(x) ds = 0, j = 1, . . . , J, σj
s = 1, . . . , r. Therefore, from estimate (4.42) at l = r − s, s = 1, . . . , r, follow the inequalities 2r+2−2s,r+1−s 2r−2s,r−s u{s} ; W2,β (T ) + ∇p {s} ; W2,β (T ) 2r−2s,r−s 2r+1−2s (T ) + u(s) ; W2,β () ∀t ∈ [0, T ]. ≤ c f{s} ; W2,β
(4.44)
Estimating the right-hand side of (4.44) with the help of (4.41) at k = s, we derive 9 9 s 9 9 s 9 9∂ 9 9∂ 2r+2−2s,r+1−s 2r−2s,r−s T 9 T 9 9 9 + u; W ( ) ∇p; W ( ) 2,β 2,β 9 9 ∂t s 9 9 ∂t s 2r,r 2r+1 ≤ c f; W2,β (T ) + u0 ; W2,β () , s = 1, . . . , r.
(4.45)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
581
Consider now (u{s} (x, t), p {s} (x, t)) as a solution of the steady Stokes problem ⎧ {s} ⎪ −νu{s} (x, t) + ∇p {s} (x, t) = f{s} (x, t) − ut (x, t), ⎪ ⎪ ⎪ ⎨ div u{s} (x, t) = 0, " ⎪ u{s} (x, t)"∂ = 0, ⎪ ⎪ ⎪ ⎩ {s} σj u (x, t) · n(x) ds = 0, j = 1, . . . , J.
(4.46)
By Theorem III.3.2, there hold the estimates 2r+2−2s 2r−2s ()2 + ∇p {s} (·, t); W2,β ()2 u{s} (·, t); W2,β {s} 2r−2s 2r−2s ≤ f{s} (·, t); W2,β ()2 + ut (·, t); W2,β ()2 ,
s = 1, . . . , r.
Integrating the last inequality from 0 to t and using (4.45), we obtain 92 92 t9 s t9 s 9 ∂ u(·, τ ) 9 ∂ p(·, τ ) 9 9 2r+2−2s 2r−2s 9 9∇ 9 dτ + 9 dτ ; W () ; W () 2,β 2,β 9 ∂τ s 9 9 9 s ∂τ 0 0 2r,r 2r+1 (4.47) (T )2 + u0 ; W2,β ()2 ∀t ∈ [0, T ], ≤ c f; W2,β s = 1, . . . , r. Finally, considering (u(x, t), p(x, t)) as a solution of the Stokes prob2r+2 () and using (4.47) we obtain the lem (4.46) at s = 0, we conclude that u(·, t) ∈ W2,β estimate
t
2r+2 u(·, τ ); W2,β ()2 dτ +
0
t
≤c 0
t
0
2r f(x, τ ); W2,β ()2 dt
2r ∇p(·, τ ); W2,β ()2 dτ
+ 0
t
2r uτ (x, τ ); W2,β ()2 dτ
2r,r 2r+1 (T ) + u0 ; W2,β ()2 . ≤ c f; W2,β
(4.48)
Estimate (4.42) at l = r follows from inequalities (4.45), (4.47), (4.48) and from the defin2r,r (T ). ition of the norm in the space W2,β 4.3.4. Time dependent Stokes problem with nonzero fluxes Let us prove the main results of Section 4.3. T HEOREM 4.4. Let ∂ ∈ C 2 , Fj ∈ W21 (0, T ) and let the initial data u0 (x) and the external force f(x, t) are represented in the form (4.4), (4.5) with (j )
u0 (x (j ) ) = 0,
f(j ) (x (j ) , t) = 0,
(j )
◦
u0n ∈ W 12 (σj ),
(j )
fn ∈ L2 (jT ), (4.49)
582
K. Pileckas 1 ( u0 ∈ W2,β (),
( f ∈ L2,β (T ),
βj ≥ 0,
(4.50)
j = 1, . . . , J . Moreover, assume that there holds the condition (1.32 ) and the compatibility conditions
(j ) u0n (x (j ) ) dx (j ) , j = 1, . . . , J, (4.51) Fj (0) = σj
div( u0 (x) +
J (j )
∂ζ (xn ) j =1
(j ) ∂xn
(j )
u0n (x (j ) ) = 0,
" ( u0 (x)"∂ = 0.
(4.52)
If the number γ∗ is sufficiently small, then problem (1.1) has a unique solution (u(x, t), p(x, t)) admitting the asymptotic representation u(x, t) = V(x, t) + v(x, t),
p(x, t) = P (x, t) + p 8(x, t),
(4.53)
where V(x, t) is the flux carrier defined by (4.14) and P (x, t) is the corresponding pressure 2,1 defined by (4.9), v ∈ W2,β (T ), ∇ p ((x, t) ∈ L2,β (T ). There holds the estimate 2,1 1 v; W2,β (T ) + ∇ p 8; L2,β (T ) ≤ c ( u0 ; W2,β () + ( f; L2,β (T ) +
J
(j ) (j ) (u0n ; W21 (σj ) + fn ; L2 (jT ) + Fj ; W21 (0, T )) .
(4.54)
j =1
P ROOF. We look for the solution (u(x, t), p(x, t)) of (1.1) in the form (3.36). Then for (v(x, t), p ((x, t)) we get problem (4.18) with the new external force 8 f(x, t) = ( f(x, t) + f(1) (x, t) + f(2) (x, t), where f(1) (x, t) = −Ut (x, t) + νU(x, t) − ∇P (x, t), f(2) (x, t) = −Wt (x, t) + νW(x, t) u0 (x) − W(x, 0). Since for x ∈ j \ j 3 the and the new initial data 8 u0 (x, t) = (
pair (V(x, t), P (x, t)) coincides with the unidirectional Poiseuille flow (U(j ) (x (j ) , t), (j ) P (x, t)), j = 1, . . . , J , which is the exact solutions of time-dependent Stokes system
in a pipe #j corresponding to the right-hand side f(j ) (x (j ) , t), the initial data u(j ) (x (j ) ) and the flux Fj (t), we conclude that suppx (f(1) (x, t) + f(2) (x, t)) ⊂ (3) (recall that suppx W(x, t) ⊂ (3) ). From the construction of (U(x, t), P (x, t)) and W(x, t) and from estimates (II.2.149), (2.8), it follows that f(1) + f(2) ; L2,β (T ) ≤ cf(1) + f(2) ; L2 (T )
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
≤c
583
J
(j ) (j ) (u0n ; W21 (σj ) + fn ; L2 (jT ) j =1
+ Fj ; W21 (0, T )). u0 (x) · n(x) ds = 0, j = 1, . . . , J , 8 u0 (x)|∂ = 0 (see (4.51), Moreover, div8 u0 (x) = 0, σj 8 (4.52)). Therefore, by Theorem 4.2, there exists a unique solution (v(x, t), p 8(x, t)) of problem (4.18) with the right-hand side 8 f(x, t) and the initial data 8 u0 (x). There holds the estimate 2,1 (T ) + ∇ p 8; L2,β (T ) v; W2,β 1 ≤ c 8 u0 ; W2,β () + 8 f; L2,β (T )
≤c
J
(j )
(j )
(u0n ; W21 (σj ) + fn ; L2 (jT ) + Fj ; W21 (0, T ))
j =1 1 () + ( f; L2,β (T ) + ( u0 ; W2,β
.
T HEOREM 4.5. Let ∂ ∈ C 2l+3 , l ≥ 0, Fj ∈ W2l+1 (0, T ), j = 1, . . . , J, and let the initial data u0 (x) and the external force f(x, t) admit the representation (4.4), (4.5) with (j )
u0 (x (j ) ) = 0, ◦
(j )
u0n ∈ W22l+1 (σj ) ∩ W 12 (σj ), ◦
2l+1 ( u0 ∈ W2,β () ∩ W 12 (),
f(j ) (x (j ) , t) = 0,
(4.55)
(j )
fn ∈ W22l,l (jT ), 2l,l ( f ∈ W2,β (T ),
βj ≥ 0. (j )
(4.56)
(j )
Suppose that there holds condition (1.32 ) and that Fj (t), u0n (x (j ) ) and fn (x (j ) , t)), j = 1, . . . , J , satisfy compatibility conditions (II.2.75) of the order l and ( f(x, t),( u0 (x) satisfy the compatibility conditions of order l for the Stokes problem (see (4.35), (4.37)– (4.39)). If the number γ∗ is sufficiently small, then problem (4.1) has a unique solution 2l+2,l+1 (T ), (u(x, t), p(x, t)) admitting asymptotic representation (4.53) where v ∈ W2,β 2l,l ∇p 8(∈ W2,β (T ) and there holds the estimate 2l+2,l+1 2l,l v; W2,β (T ) + ∇ p 8; W2,β (T )
J
(j ) 2l,l 2l+1 ≤ c ( u0 ; W2,β () + ( f; W2,β (T ) + (u0n ; W22l+1 (σj ) j =1
584
K. Pileckas
(j ) + fn ; W22l,l (jT ) + Fj ; W2l+1 (0, T )) .
(4.57)
P ROOF. As in Theorem 4.4, we look for the solution (u(x, t), p(x, t)) of problem (4.1) in the form (4.53) and obtain for (v(x, t), p 8(x, t)) problem (4.18) with the right-hand sides 8 f(x, t) and 8 u0 (x). Moreover, from the construction of (U(x, t), P (x, t)) and W(x, t) and from estimates (II.2.76) and (2.11), it follows that 2l,l (T ) ≤ cf(1) + f(2) ; W22l,l (T ) f(1) + f(2) ; W2,β
≤c
J
(j )
(u0n ; W22l+1 (σj ) + f(j ) ; W22l,l (jT )
j =1
+ Fj ; W2l+1 (0, T )). Let us show that 8 f(x, t) and 8 u0 (x) = ( u0 (x) − W(x, 0) satisfy the compatibility conditions ds of order l for problem (4.18). First of all, we mention that dt s q(t)|t=0 = 0, s = 1, . . . , l s (see the proof of Theorem II.2.7). Hence, ∂t∂ s ∇P (x, t)|t=0 = 0, s = 1, . . . , l, and, in virtue of the second compatibility conditions in (II.2.75), "
(j ) ( )s Un (x (j ) , 0)"∂σ = 0, j
∂ s (j ) (j ) "" Un (x , 0) ∂σ = 0, j ∂t s
s = 1, . . . , l + 1.
Therefore, the function f(1) (x, t) does not participate in compatibility conditions (4.36)– s (4.39). Further, by the construction ∂t∂ s W(x, t)|∂ = 0, s = 1, . . . , l + 1, and, since W(x, t) is involved in f(2) (x, t) as νW(x, t) and in the initial condition as −W(x, 0), the corresponding terms on the right-hand sides of (4.37) and (4.38) cancel. Finally,
∂ ∂s
(j ) ∂ (j ) W(x, 0) = − ζ (xn ) s Un (x (j ) , 0), s (j ) ∂t ∂t ∂xn J
div
s = 1, . . . , l + 1,
j =1
(s)
(s)
(s)
(s)
and, hence, div(f(1) (x) + f(2) (x)) = 0, s = 1, . . . , l. Thus, f(1) (x), f(2) (x) does not appear in the right-hand side of (4.38). Therefore, using the assumptions of the theorem on ( f(x, t) and ( u0 (x), we conclude that 8 f(x, t) and 8 u0 (x) satisfy the compatibility conditions of the order l for problem (4.18). Due to Theorem 4.3, there exists a unique solution (v(x, t), p 8(x, t)) of problem (4.18) with the right-hand side 8 f(x, t) and the initial data 8 u0 (x). There holds the estimate 2l+2,l+1 2l,l (T ) + ∇ p 8; W2,β (T ) v; W2,β 2l,l 2l+1 () + 8 f; W2,β (T )) ≤ c(8 u0 − W(·, 0); W2,β 2l,l 2l+1 () + ( f; W2,β (T ) ≤ c ( u0 ; W2,β
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
+
J
585
(j ) (j ) (u0n ; W22l+1 (σj ) + fn ; W22l,l (jT ) + Fj ; W2l+1 (0, T )) .
j =1
4.4. The two-dimensional Navier–Stokes problem 4.4.1. Reduction of problem (4.2) to a problem with zero fluxes Consider problem (4.2) in a two-dimensional domain , i.e., the outlets to infinity j coincide for large |x| with (j ) (j ) infinite strips #j = {x (j ) ∈ R2 : x1 ∈ σj , −∞ < x2 < ∞} while cross-sections σj = 2 (0, hj ) are finite intervals. Let ∂ ∈ C . Assume that the initial velocity u0 (x, t) and the external force f(x, t) admit the representations (j ) (j ) (j ) u0 (x) = Jj=0 ζ (x2 )(0, u02 (x1 )) + ( u0 (x), J (j ) (j ) (j ) f(x, t), f(x, t) = j =0 ζ (x2 )(0, f2 (x1 , t)) +(
(4.58)
where ζ (τ ) is a smooth cut-off function with ζ (τ ) = 0 for τ ≤ 1 and ζ (τ ) = 1 for τ ≥ 2, ◦ ◦ (j ) 1 () ∩ u02 ∈ W 12 (σj ), f2 ∈ L2 (jT ) and ( f ∈ L2,β (T ), ( u0 ∈ W2,β W 12 (). Moreover, we suppose that there hold the compatibility conditions J
div u0 (x) = 0,
Fj (0) =
hj 0
j =1 Fj (t) = 0 (j ) (j ) (j ) u02 (x1 ) dx1 ,
∀t ∈ [0, T ],
(4.59)
j = 1, . . . , J.
By Theorem II.2.7, in each infinite strip #j there exists a time-dependent unidirectional Poiseuille flow: (j )
(j )
U(j ) (x, t) = (0, U2 (x1 , t)),
(j )
(j )
P (j ) (x, t) = −q (j ) (t)x2 + p0 (t).
There hold the estimates (j )
U2 ; W22,1 (jT )2 + q (j ) ; L2 (0, T )2 (j ) (j ) ≤ c Fj ; W21 (0, T )2 + u02 ; W21 (σj )2 + f2 ; L2 (jT )2 , j = 1, . . . , J.
(4.60)
Using these Poiseuille flows for the construction of a flux carrier (see Section 4.2), we define (U(x, t), P (x, t)) by formula (4.9) and V(x, t) by formula (4.14). We look for the solution (u(x, t), p(x, t)) of problem (4.2) in the form u(x, t) = v(x, t) + V(x, t),
p(x, t) = p 8(x, t) + P (x, t).
(4.61)
586
K. Pileckas
Then for (v(x, t), p 8(x, t)) we derive the following problem ⎧ vt (x, t) − νv(x, t) + (v(x, t) · ∇)v(x, t) + (V(x, t) · ∇)v(x, t) ⎪ ⎪ ⎪ ⎪ ⎪ +(v(x, t) · ∇)V(x, t) + ∇ p 8(x, t) =8 f(x, t), ⎪ ⎨ div v(x, t) = 0, " ⎪ ⎪ ⎪ v(x, t)"∂ = 0, v(x, 0) = 8 u0 (x), ⎪ ⎪ ⎪ ⎩ v(x, t) · n(x) ds = 0, j = 1, . . . , J,
(4.62)
σj
where 8 u0 (x) = ( u0 (x) − W(x, 0), 8 f(x, t) =( f(x, t) + f(1) (x, t) + f(2) (x, t), J
ν
(j ) (j ) (j ) d2 (j ) ζ (x2 )(0, U2 (x1 , t)) dx2 2 (j ) (j ) (j ) (j ) (j ) (j ) −ζ (x2 ) d(j ) ζ (x2 )(0, |U2 (x1 , t)|2 ) − d(j ) ζ (x2 )x2 (0, q (j ) (t)) , dx dx
f(1) (x, t) =
j =1
2
2
f(2) (x, t) = −Wt (x, t) + νW(x, t) − (W(x, t) · ∇)W(x, t) −(U(x, t) · ∇)W(x, t) − (W(x, t) · ∇)U(x, t). (4.63) Note that for deriving (4.63) we have used the fact that (U(j ) (x, t), P (j ) (x, t)), j = (j ) (j ) 1, . . . , J , is an exact solution of system (4.2) in #j with the right-hand side (0, f2 (x1 , t)), (j ) (j ) the initial data (0, u02 (x1 )) and the flux Fj (t). By construction, suppx [f(1) (x, t) + f(2) (x, t)] ⊂ (3) . Moreover, using Sobolev embedding theorem (see Lemma I.1.2) on the one-dimensional interval σj , we obtain from (4.60) the estimate t 0
|f(1) (x, τ )|2 dx dτ
≤c
J t
j =1 0
j 3
(j ) (j ) (j ) (j ) |U2 (x1 , τ )|2 + |U2 (x1 , τ )|4 + |q (j ) (τ )|2 dxdτ
J t
(j ) (j ) U2 (·τ ); L2 (σj )2 + U2 (·, τ ); W21 (σj )4 dτ ≤c j =1 0
+c
t
σj
|q (j ) (τ )|2 dτ ≤ cA1 (1 + A1 ),
(4.64)
0
where A1 =
J
(j ) (j ) f2 ; L2 (jT )2 + u02 ; W21 (σj )2 + Fj ; W21 (0, T )2 . j =1
(4.65)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
587
Analogously, using again Sobolev embedding theorem in the domain (3) , estimate (4.13) for W(x, t) and (4.60), we derive t 0
|f(2) (x, τ )|2 dx dτ
≤c
t 0
|Wτ (x, τ )|2 + |W(x, τ )|2
(3)
+ (|W(x, τ )|2 + U(x, τ )|2 )|∇W(x, τ )|2 + |W(x, τ )|2 |∇U(x, τ )|2 dxdτ % t sup (|W(x, τ )|2 + |U(x, τ )|2 ) ≤ cA1 + c (|∇W|2 + |∇U|2 ) dx dτ 0
(3)
x∈(3)
≤ cA1 + c sup
J
(j ) U2 (·, t); W21 (σj )2 W(·, t); W21 ((3) )2 +
t∈[0,T ]
j =1
t J
(j ) 2 2 1 2 W(·, τ ); W2 ((3) ) + × U2 (·, τ ); W2 (σj ) dτ 0
j =1
2 J
(j ) 2,1 2,1 T 2 T 2 ≤ cA1 + c W; W2 ((3) ) + U2 ; W2 (j ) j =1
≤ cA1 (1 + A1 ).
(4.66)
4.4.2. A priori estimates of the solution L EMMA 4.1. Let ( u0 ∈ L2 (), ( f ∈ L2 (T ), T ∈ (0, ∞], and let v ∈ W22,1 (T ) be a solution to problem (4.62). Then there holds the estimate |v(x, t)|2 dx + ν
t 0
|∇v(x, τ )|2 dx dτ
≤ c(1 + A1 ecA1 )[A2 + A1 (1 + A1 )] := cB1 ,
∀t ∈ [0, T ],
(4.67)
where A1 is defined by (4.65) and
A2 =
|( u0 (x)| dx + 2
0
T
|( f(x, τ )|2 dx dτ.
The constant c in (4.67) does not depend on t ∈ (0, T ] and T . P ROOF. Multiply (4.62) by v(x, t) and integrate by parts over : 1 d 2 |v(x, t)| dx + ν |∇v(x, t)|2 dx 2 dt
(4.68)
588
K. Pileckas
=−
(v(x, t) · ∇)V(x, t) · v(x, t) dx +
8 f(x, t) · v(x, t) dx.
(4.69)
Let us estimate the right-hand side of (4.69). We have " " " " " (v(x, t) · ∇)V(x, t) · v(x, t) dx " " " " " " " " ≤" (v(x, t) · ∇)W(x, t) · v(x, t) dx "" (3)
J "
" " + "
" " (v(x, t) · ∇)U(x, t) · v(x, t) dx "" = J1 (t) + J2 (t).
j
j =1
(4.70)
Using (I.1.11), Poincaré, Hölder and Young inequalities, the first integral at the right-hand side of (4.70) is estimated as follows 1/2
|J1 (t)| ≤
1/2
|v(x, t)| dx
|∇W(x, t)| dx
4
(3)
2
(3)
1/2
≤c
1/2
|v(x, t)| dx
|∇v(x, t)| dx
2
(3)
2
(3)
1/2
×
|∇W(x, t)|2 dx (3)
≤ cm1 (t)
|v(x, t)|2 dx +
ν 6
|∇v(x, t)|2 dx,
where m1 (t) =
|∇W(x, t)|2 dx. (3)
Analogously, the second term admits the estimate J "
" " |J2 (t)| = " j =1
j
" " (v(x, t) · ∇)U(x, t) · v(x, t) dx ""
J ∞ "
" " ≤ " j =1 s=0
≤c
"
ωj s
" (j ) (j ) (v(x, t) · ∇)(ζ (x2 )U(j ) (x1 , t)) · v(x, t) dx ""
J ∞
j =1 s=0
ωj s
1/2 |v(x, t)|2 dx
1/2 |∇v(x, t)|2 dx
ωj k
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
×
(|U
(j )
ωj s
≤c
J ∞
j =1 s=0
(j ) (x1 , t)|2
+ |∇ U
(j )
|∇ U(j ) (x1 , t)|2 dx1 (j )
σj
(j ) (x1 , t)|2 ) dx
589
1/2
(j )
|v(x, t)|2 dx ωj s
J ∞ ν
|∇v(x, t)|2 dx ≤ cm2 (t) |v(x, t)|2 dx 6 j =1 s=0 ωj s ν + |∇v(x, t)|2 dx, 6
+
where m2 (t) =
J
|∇ U(j ) (x1 , t)|2 dx1 . (j )
j =1 σj
(j )
Finally, using Young and Poincaré inequalities, we get " " " " ν 2 " " 8 8 |∇v(x, t)|2 dx. " f(x, t) · v(x, t) dx " ≤ c |f(x, t)| dx + 6 Therefore, from (4.69) it follows that d dt
|v(x, t)| dx + ν
|∇v(x, t)|2 dx
2
|v(x, t)|2 dx + c
≤ cM(t)
|8 f(x, t)|2 dx
(4.71)
with M(t) = m1 (t) + m2 (t). Thus, d dt
|v(x, t)|2 dx ≤ cM(t)
|v(x, t)|2 dx + c
|8 f(x, t)|2 dx
(4.72)
and by Gronwall inequality |v(x, t)|2 dx dt
≤e ≤e
c
c
t 0
t 0
M(τ ) dτ
M(τ ) dτ
|v(x, 0)| dx + c 2
e
−c
0
|8 u0 (x)| dx + c 2
t
0
M(s) ds
|8 f(x, τ )|2 dx dτ
2 8 |f(x, τ )| dx dτ .
t 0
τ
(4.73)
590
K. Pileckas
Substituting (4.73) into (4.71) and integrating with respect to t, we derive |v(x, t)| dx + ν 2
t 0
≤
|∇v(x, τ )|2 dx dτ
|8 u0 (x)|2 dx + c
t 0
+
t
M(τ )ec
τ 0
M(s) ds
|8 f(x, τ )|2 dx dτ
τ
|8 u0 (x)|2 dx + c
0
0
|8 f(x, s)|2 dx ds dτ.
(4.74) It follows from (4.60) and (4.13) that
T
T
M(t) dt =
0
0
+
|∇W(x, t)|2 dxdt (3)
J
T
j =1 0
|∇ U(j ) (x1 , t)|2 dx1 dt ≤ cA1 . (j )
σj
(j )
Further, in virtue of Lemma I.1.3, (4.13), (4.64) and (4.66) |8 u0 (x)|2 dx ≤ 2 |( u0 (x)|2 dx + 2 |W(x, 0)|2 dx
≤
t 0
|8 f(x, τ )|2 dxdτ ≤ c
t 0
(3)
2A2 + cW; W22,1 (T(3) )2
≤ 2A2 + cA1 ,
(|( f(x, τ )|2 + |f(1) (x, τ )|2 + |f(2) (x, τ )|2 ) dxdτ
≤ c(A2 + A1 + A21 ). Collecting the obtained estimates, from (4.74) we conclude the inequality (4.67). Consider problem (4.62) as the linear time-dependent Stokes problem (4.18): ⎧ vt (x, t) − νv(x, t) + ∇ p(x, ˜ t) = g(x, t, v(x, t)), ⎪ ⎪ ⎪ ⎪ ⎨ div v(x, t) = 0, " " = 0, v(x, t) v(x, 0) = 8 u0 (x), ⎪ ⎪ ∂ ⎪ ⎪ ⎩ v(x, t) · n(x) ds = 0, j = 1, . . . , J, σj
with the right-hand side g(x, t; v(x, t)) = 8 f(x, t) − (v(x, t) · ∇)v(x, t) − (V(x, t) · ∇)v(x, t) − (v(x, t) · ∇)V(x, t)
(4.75)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
591
and the initial data 8 u0 (x) = ( u0 (x) − W(x, 0). Let us estimate the norm t 0
(k)
Eβ |g(x, τ ; v(x, τ ))|2 dxdτ,
where Eβ(k) (x) is a step weight-function (I.1.16). We have t 0
(k)
≤c
Eβ (x)|g(x, τ ; v(x, τ ))|2 dxdτ t 0
+ + +
Eβ(k) (x)|8 f(x, τ )|2 dxdτ
t 0
0
t t 0
=c
t 0
Eβ(k) (x)|(v(x, τ ) · ∇)v(x, τ )|2 dxdτ Eβ(k) (x)|(V(x, τ ) · ∇)v(x, τ )|2 dxdτ (k) Eβ (x)|(v(x, τ ) · ∇)V(x, τ )|2 dxdτ
(k) Eβ (x)|8 f(x, τ )|2 dxdτ
+I
(1)
(t) + I
(2)
(t) + I
(3)
(t) .
(4.76)
L EMMA 4.2. Let v ∈ W22,1 (T ), T ∈ (0, ∞]. Then (v · ∇)v ∈ L2 (T ) and t 0
(k)
≤ε
Eβ (x)|(v(x, τ ) · ∇)v(x, τ )|2 dxdτ t 0
|α|≤2
+ cε B1
(k)
Eβ (x)|D α v(x, τ )|2 dτ
t 0
(k)
Eβ (x)|∇v(x, τ )|2 dx
2 dτ
∀t ∈ (0, T ],
(4.77)
where B1 is defined in (4.67) and the constant c is independent of k, t ∈ [0, T ] and T . P ROOF. Applying inequality (I.1.11) to v(x, t) and ∇v(x, t), we get (v(·, t) · ∇)v(·, t); L2 (ωj s )2 ≤ v(·, t); L4 (ωj s )2 ∇v(·, t); L4 (ωj s )2 ≤ cv(·, t); L2 (ωj s )v(·, t); W21 (ωj s )∇v(·, t); L2 (ωj s )∇v(·, t); W21 (ωj s ) ≤ cv(·, t); L2 (ωj s )∇v(·, t); L2 (ωj s )2 ∇v(·, t); W21 (ωj s ).
592
K. Pileckas
Therefore, due to (4.67) and Young inequality
t
(v(·, τ ) · ∇)v(·, τ ); L2 (ωj s )2 dτ
0
≤ c sup v(·, τ ); L2 (ωj s ) τ ∈(0,t)
t
× 0
1/2 ≤ cB1
t
≤ε 0
v(·, τ ); W22 (ωj s )∇v(·, τ ); L2 (ωj s )2 dτ
t 0
v(·, τ ); W22 (ωj s )∇v(·, τ ); L2 (ωj s )2 dτ
v(·, τ ); W22 (ωj s )2 dτ + cε B1
t
∇v(·, τ ); L2 (ωj s )4 dτ.
(4.78)
0
Obviously the constant cε in (4.78) does not depend on t, T and s. Multiplying (4.78) (k) by Eβ (s), summing the obtained inequalities over s from 0 to ∞ and using properties (I.1.14) of the function Eβ(k) (x), we obtain t 0
(k)
j
≤ε
Eβ (x)|(v(x, τ ) · ∇)v(x, τ )|2 dxdτ
t 0
j |α|≤2
+ cε B1 ≤ε
j |α|≤2
+ cε B1 ≤ε
0
j |α|≤2
+ cε B1
ωj s
|∇v(x, τ )| dx 2
∞
j
0
t
ωj s
s=0 ωj s
(k) Eβ (x)|∇v(x, τ )|2 dx
dτ
Eβ(k) (x)|D α v(x, τ )|2 dxdτ (k)
j
Eβ (x)|∇v(x, τ )|2 dx
Analogously,
" "∇v(x, τ )|2 dx dτ
(k)
t 0
(k)
Eβ (x)|∇v(x, τ )|2 dx
Eβ (x)|D α v(x, τ )|2 dxdτ
t 0
t
t ∞ 0 s=0
t 0
(k)
Eβ (x)|D α v(x, τ )|2 dxdτ
(v(x, τ ) · ∇)v(x, τ ); L2 ((2) )2 dτ
2 dτ.
(4.79)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
t
≤ε
v(·, τ ); W22 ((2) )2 dτ + cε B1
0
t
593
∇v(·, τ ); L2 ((2) )4 dτ. (4.80)
0
Inequality (4.77) follows from (4.79) and (4.80). L EMMA 4.3. Let v ∈ W22,1 (T ). Then (V · ∇)v ∈ L2 (T ), (v · ∇)V ∈ L2 (T ) and t 0
(k)
Eβ (x)|(V(x, τ ) · ∇)v(x, τ )|2 dxdτ
t
+ ≤ cε
(k)
Eβ (x)|(v(x, τ ) · ∇)V(x, τ )|2 dxdτ
0
0
|α|≤2
t
(k)
Eβ (x)|D α v(x, τ )|2 dxdτ
+ cε (A1 + 1)
t
(k)
Y (τ ) 0
Eβ (x)|∇v(x, τ )|2 dxdτ,
∀t ∈ (0, T ], (4.81)
where A1 is defined by (4.65), Y (t) =
J
U(j ) (·, t); W22 (σj )2 + W(·, t); W22 ((3) )2 ,
j =1
and the constant c is independent of k, t ∈ [0, T ] and T . P ROOF. Arguing as in Lemma 4.2, we derive the estimates
t
(V(·, τ ) · ∇)v(·, τ ); L2 (ωj s )2 dτ
0
t
≤
V(·, τ ); L4 (ωj s )2 ∇v(·, τ ); L4 (ωj s )2 dτ
0
t
≤c 0
t
≤ε 0
×
∇V(·, τ ); L2 (ωj s )2 ∇v(·, τ ); L2 (ωj s )∇v(·, τ ); W21 (ωj s ) dτ v(·, τ ); W22 (ωj s )2 dτ + cε sup ∇V(·, τ ); L2 (ωj s )2 τ ∈(0,t)
t
∇V(·, τ ); L2 (ωj s )2 ∇v(·, τ ); L2 (ωj s )2 dτ.
0
From (4.60) and (4.13) it follows that sup ∇V(·, τ ); L2 (ωj s )2
τ ∈(0,t)
594
K. Pileckas
≤c
&
sup U(j ) (·, τ ); W21 (ωj s )2 + sup ∇W(·, τ ); L2 (ωj s )2
τ ∈(0,t)
'
τ ∈(0,t)
≤ c U(j ) ; W22,1 (jT )2 + W(·, τ ); W22,1 (T(3) )2 ≤ cA1 ,
∇V(·, t); L2 (ωj s )2 ≤ V(·, t); W22 (ωj s )2 ≤ c U(j ) (·, t); W22 (σj )2 + W(·, t); W22 ((3) )2 = cYj (t), and, therefore,
t
(V(·, τ ) · ∇)v(·, τ ); L2 (ωj s )2 dτ
0
t
≤ε 0
v(·, τ ); W22 (ωj s )2 dτ
t
+ cε A1
Yj (τ )∇v(·, τ ); L2 (ωj s )2 dτ.
(4.82)
0 (k)
Multiply inequalities (4.82) by Eβ (s) and sum obtained relations over s. This gives: t 0
(k)
j
Eβ (x)|(V(x, τ ) · ∇)v(x, τ )|2 dxdτ
t
≤ε
0
j |α|≤2
+ cε A1
(k)
Eβ (x)|D α v(x, τ )|2 dxdτ
t
(k)
Yj (τ ) 0
j
Eβ (x)|∇v(x, τ )|2 dxdτ.
(4.83)
Consider now the second term on the left-hand side of (4.81). We have
t
(v(·, τ ) · ∇)V(·, τ ); L2 (ωj s )2 dτ
0
t
≤
v(·, τ ); L4 (ωj s )2 ∇V(·, τ ); L4 (ωj s )2 dτ
0
≤c
t
0
≤c
0
t
∇v(·, τ ); L2 (ωj s )2 ∇V(·, τ ); L2 (ωj s )∇V(·, τ ); W21 (ωj s ) dτ ∇v(·, τ ); L2 (ωj s )2 V(·, τ ); W22 (ωj s )2 dτ
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
≤c
t
595
Yj (τ )∇v(·, τ ); L2 (ωj s )2 dτ.
0 (k)
Multiplying these inequalities by Eβ (x) and summing over s, we obtain t 0
(k)
j
Eβ (x)|(v(x, τ ) · ∇)V(x, τ )|2 dxdτ
≤c
t
(k)
Yj (τ ) 0
j
Eβ (x)|∇v(x, τ )|2 dxdτ.
(4.84)
It can be proved analogously that
t
t
(V(·, τ ) · ∇v(·, τ ); L2 ((2) ) dτ + 2
0
(v(·, τ ) · ∇)V(·, τ ); L2 ((2) )2 dτ
0
t J
≤ cε A1 +ε
Yj (τ )∇v(·, τ ); L2 ((2) )2 dτ
0 j =1 t
v(·, τ ); W22 ((2) )2 dτ.
0
(4.85)
Estimate (4.81) follows from (4.83) and (4.84), (4.85).
Now we are in a position to prove the main a priori estimate for the solution v(x, t) of problem (4.62). ◦
L EMMA 4.4. Let ( u0 ∈ W 12 (), ( f ∈ L2 (T ), T ∈ (0, ∞], and let v ∈ W22,1 (T ) be a solution of (4.62). If the number γ∗ in inequality (I.1.143 ) for the weight function Eβ (x) is sufficiently small, then there holds the estimate t 0
(k) Eβ (x) |vτ (x, τ )|2 + |D α v(x, τ )|2 dxdτ
+
t 0
|α|≤2 (k)
(k)
Eβ (x)|∇ p 8(x, τ )|2 dxdτ ≤ cB2 ,
∀t ∈ [0, T ],
(4.86)
where B2 = (A3 + A1 + A21 )(1 + ec(B1 +A1 ) (B1 + A1 )2 ), " 2 T (k) "u0 | + |∇( u0 |2 )dx + 0 Eβ(k) (x)|( f(x, t)|2 dxdt. A(k) 3 = Eβ (x)( ( (k)
(k)
2
The constant c in (4.86) does not depend on k, t ∈ (0, T ] and T .
(4.87)
596
K. Pileckas
P ROOF. Consider the solution v(x, t) of problem (4.62) as a solution of the linear problem (4.75). Then there holds the estimate (see the proof of Theorem 4.2)
t 0
(k) Eβ (x)
+
≤c
|D v(x, τ )|
2
α
dxdτ
|α|≤2
t 0
+
|vτ (x, τ )| + 2
Eβ(k) (x)|∇ p 8(x, τ )|2 dxdτ
(k) Eβ (x) |8 u0 (x)|2 + |∇8 u0 (x)|2 dx
t 0
"2 " (k) Eβ (x)"g(x, τ ; v(x, τ ))" dxdτ
(4.88)
with the constant c independent of k. According to Lemma I.1.3,
(k) Eβ (x) |v(x, t)|2 + |∇v(x, t)|2 dx ≤c
t 0
(k) Eβ (x) |vτ (x, τ )|2 + |D α v(x, τ )|2 dxdτ.
(4.89)
|α|≤2
The constant in the last inequality is independent of t and k. Relations (4.88), (4.89) together with inequalities (4.76), (4.77), (4.81) yield
(k)
Eβ (x)|∇v(x, t)|2 dx +
0
+
t
(k) Eβ (x)
≤c
+ c∗ ε
|D v(x, τ )|
2
α
dxdτ
(k)
Eβ (x)|∇ p 8(x, τ )|2 dxdτ
(k) Eβ (x) |8 u0 (x)|2 t
+ cε B1
|α|≤2
t 0
|vτ (x, τ )| + 2
0
(k)
Eβ (x)
0
+ cε A1
0
t 0
|∇v(x, τ )| dx
(k) Eβ (x)|∇v(x, τ )|2 dx
(k) 2 Y (τ ) Eβ (x)|∇v(x, τ )| dx dτ.
(k) Eβ (x)|8 f(x, τ )|2 dxdτ
|D α v(x, τ )|2 dxdτ 2
t
|α|≤2
t
+ |∇8 u0 (x)| dx + 2
dτ
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
597
Fixing ε = c∗ /2, from the last inequality we find
(k)
Eβ (x)|∇v(x, t)|2 dx +
t 0
+
t 0
|α|≤2 (k)
(k) Eβ (x) |vτ (x, τ )|2 + |D α v(x, τ )|2 dxdτ Eβ (x)|∇ p 8(x, τ )|2 dxdτ
(k)
≤c
Eβ (x)(|8 u0 (x)|2 + |∇8 u0 (x)|2 ) dx +
+ c(B1 + A1 )
t 0
×
t 0
|∇v(x, τ )|2 dx + Y (τ )
(k) Eβ (x)|∇v(x, τ )|2 dx
(k) Eβ (x)|8 f(x, τ )|2 dxdτ
dτ.
(4.90)
Omitting the second and the third terms on the left-hand side of (4.90), we also get
(k)
Eβ (x)|∇v(x, t)|2 dx (k)
≤c
+
Eβ (x)(|8 u0 (x)|2 + |∇8 u0 (x)|2 ) dx
t 0
(k) Eβ (x)|8 f(x, τ )|2 dxdτ
+ c(B1 + A1 )
t 0
×
|∇v(x, τ )|2 dx + Y (τ )
(k) Eβ (x)|∇v(x, τ )|2 dx dτ.
(4.91)
Denote ∇v(·, t); L2 ()2 + Y (t) = Z(t),
t 0
(k)
Eβ (x)|∇v(x, τ )|2 Z(τ ) dτ = X(t).
Multiplying both sides of relation (4.91) by Z(t), we receive the inequality d X(t) ≤ c1 (B1 + A1 )Z(t)X(t) + c2 Z(t)(t), dt
598
K. Pileckas
where
(k)
(t) =
Eβ (x)(|8 u0 (x)|2 + |∇8 u0 (x)|2 ) dx +
t 0
(k) Eβ (x)|8 f(x, τ )|2 dxdτ.
Therefore, t t d X(t)e−c1 (B1 +A1 ) 0 Z(τ )dτ ≤ c2 e−c1 (B1 +A1 ) 0 Z(τ )dτ Z(t)(t). dt
(4.92)
Integrating (4.92) over t gives X(t) ≤ c2 ec1 (B1 +A1 )
t 0
Z(τ )dτ
t
e−c1 (B1 +A1 )
τ 0
Z(s)ds
Z(τ )(τ ) dτ
0
≤ c2 e
c1 (B1 +A1 )
t 0
Z(τ )dτ
t
(4.93)
Z(τ )(τ ) dτ. 0
Using the definition of Z(t) and inequalities (4.67), (4.60), (4.13), (4.64), (4.66), we get
t
Z(τ )dτ =
0
t
∇v(·, τ ); L2 () dτ + 2
0
+
t
j =1 0 t
0
J
U(j ) (·, τ ); W22 (σj )2 dτ
W(·, τ ); W22 ((3) )2 dτ ≤ c(B1 + A1 ),
(4.94)
t
Z(τ )(τ )dτ 0
≤
+
(k) Eβ (x) |8 u0 (x)|2 + |∇8 u0 (x)|2 dx
t 0
(k) Eβ (x)|8 f(x, τ )|2 dxdτ
t
Z(τ )dτ 0
(k)
≤ c(A3 + A1 + A21 )(B1 + A1 ). From (4.93)–(4.95) follows the estimate 2 X(t) ≤ cec(B1 +A1 ) (A(k) 3 + A1 + A1 )(B1 + A1 ) 2
and, therefore, (4.90) furnishes t 0
(k) Eβ (x) |vτ (x, τ )|2 + |D α v(x, τ )|2 dxdτ |α|≤2
(4.95)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
+
t 0
599
(k)
Eβ (x)|∇ p 8(x, τ )|2 dxdτ
(k) 2 ≤ c A3 + A1 + A21 1 + ec(B1 +A1 ) (B1 + A1 )2 . R EMARK 4.3. The idea of getting global a priori estimate for W22,1 -norm of solutions of two-dimensional time-dependent Navier–Stokes system in a bounded domain (i.e., the analogous to (4.86) estimate in spaces without weights) belong to Professor V.A. Solonnikov (private communication). R EMARK 4.4. Taking βj = 0, j = 1, . . . , J , from (4.86) we get the inequality
t
0
vτ (·, τ ); L2 ()2 + v(·, τ ); W22 ()2 dτ
t
+
∇ p 8(·, τ ); L2 ()2 dτ ≤ cB2 ,
∀t ∈ [0, T ],
0
(0)
where B2 = B2 . (j )
Let T = ∞. Consider the case of Fj (t) and f(j ) (x1 , t), j = 1, . . . , J , admitting the representations Fj (t) = Fj[1] + Fj[2] (t),
(j )
(j )
(j )
f(j ) (x1 , t) = f(j,1) (x1 ) + f(j,2) (x1 , t),
(4.96)
where Fj[1] are constants, f(j,1) (x1 ) are independent of t, f(j,1) ∈ L2 (σj ), Fj[2] ∈ W21 (0, ∞), f(j,2) ∈ L2 (σj × (0, ∞)). Then a priori estimate (4.86) holds for any finite time interval [0, T ]. However, for large |Fj[1] | and f(j,1) ; L2 (σj ) this estimate fails on an (j )
infinite time interval. Let us prove the global a priori estimate for sufficiently small |Fj[1] | and f(j,1) ; L2 (σj ). Assume that J
j =1
Fj[1] = 0
(4.97)
600
K. Pileckas
and let u0 (x1 ), Fj (t) satisfy compatibility conditions (4.59). If |Fj[1] | and f(j,1) ; L2 (σj ) are sufficiently small, then according to Theorem III.3.9, there exists a solution (u[1] (x), p [1] (x)) of steady Navier–Stokes problem (III.3.2): (j )
(j )
⎧ −νu[1] (x) + (u[1] (x) · ∇)u[1] (x) + ∇p [1] (x) = f[1] (x), ⎪ ⎪ ⎪ ⎪ ⎨ div u[1] (x) = 0, " u[1] (x)"∂ = 0, ⎪ ⎪ ⎪ ⎪ ⎩ [1] (j ) = F [1] , j = 1, . . . , J, j σj u (x) · n(x) dx
(4.98)
where f[1] (x) =
J
(j )
(j )
ζ (x2 )f(j,1) (x1 ).
j =1
Moreover, this solution (u[1] (x), p [1] (x)) admits representation (III.3.93) with U(j,1) ∈ 2 () of W22 (σj ), W[1] (x) having compact support and v[1] (x) belonging to the space W2,β exponentially vanishing functions (since ( f(x) = 0, this fact follows from Theorem III.3.4). There hold the estimates
[1] 2 v[1] ; W2,β ()2 ≤ cA[1] 1 (1 + A1 ),
J
U(j,1) ; W22 (σj )2 ≤ cA[1] 1 ,
j =1
(4.99) J J [1] 2 (j,1) ; L (σ )2 + where A[1] 2 j j =1 |Fj | ). 1 = ( j =1 f Let us represent in (4.9) the flux carrier V(x, t) and the pressure function P (x, t) in the form V(x, t) = u[1] (x) + V[2] (x, t),
P (x, t) = p [1] (x) + P [2] (x, t),
where V[2] (x, t) and P [2] (x, t) are the flux carrier and the pressure function corresponding (j ) (j,2) (j ) (j,2) (j ) (j,2) (j ) to the data Fj[2] (t), f(j,2) (x1 , t) = (0, f2 (x1 , t)) and u0 (x1 ) = (0, u02 (x1 )), (j,2)
(j )
(j )
(j )
(j,1)
(j )
8(x, t)) we get u02 (x1 ) = u02 (x1 ) − U02 (x1 ), j = 1, . . . , J . Then for (v(x, t), p (x, t) + f[2] problem (4.62) with the right-hand sides 8 f(x, t) = ( f(x, t) + f[2] (1) (2) (x, t) and
[2] 8 u0 (x) − W[1] (x) − v[1] (x) − W[2] (x, 0), where f[2] u0 (x) = ( (1) (x, t) and f(2) (x, t) are defined (j )
by the same formulas (4.63) with the Poiseuille solutions (U(j,2) (x1 , t), P (j,2) (x (j ) , t)) (j ) (j,2) (j ) corresponding to the data Fj[2] (t), f(j,2) (x1 , t), u0 (x1 ). Note that here we have used the fact that (u[1] (x), p [1] (x)) is the exact solution of the steady nonlinear Navier–Stokes problem (4.98) in the whole domain .
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
601
(j )
L EMMA 4.5. Assume that Fj (t) and f(j ) (x1 , t), j = 1, . . . , J , admit representations (4.96) with Fj[1] ∈ R, f(j,1) ∈ L2 (σj ), Fj[2] ∈ W21 (0, ∞), f(j,2) ∈ L2 (σj × (0, ∞)). Let ◦
◦
(j ) (j ) (j ) u0 ∈ W 12 (σj ), ( u0 ∈ W 12 (), ( f ∈ L2 ( × (0, ∞)), and let u0 (x1 ), Fj (t) satisfy compatibility conditions (4.59) and numbers Fj[1] satisfy condition (4.97). If |Fj[1] |, f(j,1) ; L2 (σj ), j = 1, . . . , J , and the number γ∗ in inequality (I.1.143 ) are sufficiently small, then for the solution v ∈ W22,1 ( × (0, ∞)) of problem (4.62) holds the estimate
t 0
(k) Eβ (x) |vτ (x, τ )|2 + |D α v(x, τ )|2 dxdτ
+
|α|≤2
t 0
8 , Eβ (x)|∇ p 8(x, τ )|2 dxdτ ≤ cB 3 (k)
(k)
∀t ∈ [0, ∞],
(4.100)
where [2] 81 + A[2] 2 , 81 + A 82 1 + ec(B81 +A1 )2 B 8(k) = A(k) + A B 1 1 2 3 $ [2] # 81 = 1 + A[2] ecA1 A2 + A 81 = A[1] + A[2] , 81 + A 82 , B A 1 1 1 1 (j,2) J 2 + u(j,2) ; W 1 (σ )2 f = ; L (σ × (0, T )) A[2] 2 j j =1 1 2 j 2 02 [2] 1 2 + Fj ; W2 (0, ∞) and the constant c does not depend on t and k. P ROOF. First, let us look over the proof of Lemma 4.1. Consider estimate (4.70). It contains now an additional term J3 (t) = (v(x, t) · ∇)u[1] (x) · v(x, t) dx. Arguing as in the proof of estimates for J1 (t) and J2 (t), using representation (III.3.93) for u[1] (x) and inequalities (4.99), we derive [1] |J3 (t)| ≤ cA[1] 1 1 + A1
|∇v(x, t)|2 dx.
[1] If cA[1] 1 (1 + A1 ) ≤ ν/4, then we receive again the estimate analogous to (4.71) with M(t) depending only on V[2] (x, t). Therefore, repeating the argument of Lemma 4.1, we obtain 81 instead of B1 . inequality (4.67) with B Further, using Sobolev embedding theorem (see Lemma I.1.2), Hölder inequality and (4.97), we get
t
(u[1] · ∇)v(·, τ ); L2 (ωj s )2 dτ +
0 [1]
≤ sup |u (x)| x∈ωj s
2 0
t
(v(·, τ ) · ∇)u[1] ; L2 (ωj s )2 dτ
0 t
∇v(·, τ ); L2 (ωj s )2 dτ
602
K. Pileckas
+ u[1] ; L4 (ωj s )2 ≤ cu[1] ; W22 (ωj s )2 [1] ≤ cA[1] 1 1 + A1
t
0
t
0 t
v(·, τ ); L4 (ωj s )2 dτ v(·, τ ); W22 (ωj s )2 dτ
0
v(·, τ ); W22 (ωj s )2 dτ.
Analogously,
t
(u
[1]
· ∇)v(·, τ ); L2 ((2) ) dτ + 2
0
[1] ≤ cA[1] 1 1 + A1
0
t
(v(·, τ ) · ∇)u[1] ; L2 ((2) )2 dτ
0 t
v(·, τ ); W22 ((2) )2 dτ.
Therefore, instead of (4.81) holds the estimate t 0
(k)
Eβ (x)|(V(x, τ ) · ∇)v(x, τ )|2 dxdτ
+
t 0
(k)
Eβ (x)|(v(x, τ ) · ∇)V(x, τ )|2 dxdτ
[1] ≤ c ε + A[1] 1 1 + A1 + cε A[2] 1
t 0
t 0
(k)
Eβ (x)
|D α v(x, τ )|2 dxdτ
|α|≤2
(k) Y (τ ) Eβ (x)|∇v(x, τ )|2 dx dτ.
Using this estimate in the proof of Lemma 4.4 and assuming that the number ε + A[1] 1 (1 + [1] A1 ) is sufficiently small, we obtain estimate (4.100). 4.4.3. Solvability of problem (4.62) Let (l) be a bounded domain with the boundary ∂(l) ∈ C 2 such that (l+1) ⊂ (l) ⊂ (l+2) . Consider in (l) the following problem ⎧ (l) vt (x, t) − νv(l) (x, t) + (v(l) (x, t) · ∇)v(l) (x, t) ⎪ ⎪ ⎪ ⎪ ⎨ + (V(x, t) · ∇)v(l) (x, t) + (v(l) (x, t) · ∇)V(x, t) + ∇ p 8(l) (x, t) =8 f(x, t), ⎪ ⎪ ⎪ ⎪ ⎩
" v(l) (x, t)"
div v(l) (x, t) = 0, ∂(l)
= 0,
(l)
v(l) (x, 0) = 8 u0 (x), (4.101)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem (l)
603
(l)
where 8 u0 (x) = ξ (l) (x)8 u0 (x) + 8 w0 (x), ξ (l) (x) is a smooth cut-off function equal to 1 in ◦ (l) (l) (l) (l) and equal to 0 in (l) \ (l+1) , 8 w0j (x) and 8 w0 (x) = Jj=1 8 w0j ∈ W 12 (ωj l ) are the solutions to the problems ⎧ (l) ⎨div 8 w0j (x) = −∇ξ (l) (x) · 8 u0 (x), " (4.102) (l) " ⎩ 8 w0j (x) ∂ω = 0. jl
Since
(j ) u0 (x) · n(x)(x)dx1 σj 8
= 0 and div8 u0 (x) = 0, we have
∇ξ (l) (x) · 8 u0 (x) = 0 ωj l (l)
and, by Lemma I.1.10, functions 8 w0j (x) exist and satisfy the estimates (l)
u0 ; L2 (ωj l ). ∇8 w0j ; L2 (ωj l ) ≤ c∇ξ (l) · 8
(4.103)
The cut-off functions ξ (l) (x) can be chosen so that |∇ξ (l) (x)| ≤ c, where c is independent of l. Therefore, from (4.103) follows that u0 ; L2 (ωj l ). ∇8 w(l) 0j ; L2 (ωj l ) ≤ c8
(4.104)
◦
(l) (l) (l) It is evident that div8 u0 (x) = 0, 8 u0 ∈ W 12 ((l) ) and 8 u0 − 8 u0 ; W21 () → 0 as l → ∞. (l) f(x, t) are fixed, we can reduce problem (4.101) to an operator Assuming that 8 u0 (x) and8 2,1 equation in the space W2 ((l)T ):
v(l) = S −1 N v(l) = Bv(l) ,
(4.105)
where S −1 : g ∈ L2 ((l)T ) → v(l) ∈ W22,1 ((l)T ) is the bounded inverse operator of the linear time-dependent Stokes problem ⎧ (l) ⎪ v (x, t) − νv(l) (x, t) + ∇ p 8(l) (x, t) = g(x, t), ⎪ ⎨ t div v(l) (x, t) = 0, (4.106) ⎪ " ⎪ (l) ⎩ = 0, v(l) (x, 0) = 8 u (x), v(l) (x, t)" ∂(l)
0
and f(x, t) − (v(l) (x, t) · ∇)v(l) (x, t) − (V(x, t) · ∇)v(l) (x, t) N v(l) (x, t) = 8 − (v(l) (x, t) · ∇)V(x, t).
(4.107)
L EMMA 4.6. For any bounded domain (l) and any T < ∞ the operator N : W22,1 ((l)T ) → L2 ((l)T ) is compact.
604
K. Pileckas
P ROOF. Let {um (x, t)} be a weakly convergent in W22,1 ((l)T ) sequence. Then there exists a constant A such that um ; W22,1 ((l)T )2 ≤ A, ∀m ≥ 1. Let us show that N um − N un ; L2 ((l)T ) ≤ (un · ∇)(um − un ); L2 ((l)T ) + ((um − un ) · ∇)um ; L2 ((l)T ) + (V · ∇)(um − un ); L2 ((l)T ) + ((um − un ) · ∇)V; L2 ((l)T ) tends to zero as m, n → ∞. Using inequalities (I.1.11) and (I.1.6), we get (un · ∇)(um − un ); L2 ((l)T )2 T ≤ un (·, t); L4 ((l) )2 ∇um (·, t) − ∇un (·, t); L4 ((l) )2 dt 0
T
≤c
∇un (·, t); L2 ((l) )2 ∇um (·, t) − ∇un (·, t); L4 ((l) )2
0
≤ c sup ∇un (·, t); L2 ((l) )2 t∈[0,T ]
≤ cA
T
∇um (·, t) − ∇un (·, t); L4 ((l) )2 dt
0
≤ cun ; W22,1 ((l)T )2
T
T
∇um (·, t) − ∇un (·, t); L4 ((l) )2 dt
0
∇um (·, t) − ∇un (·, t); L4 ((l) )2 dt
0
and ((um − un ) · ∇)um ; L2 ((l)T )2 T ≤ um (·, t) − un (·, t); L4 ((l) )2 ∇um (·, t); L4 ((l) )2 dt 0
T
≤c
um (·, t) − un (·, t); L4 ((l) )2 ∇um (·, t); L2 ((l) )
0
× ∇um (·, t); W21 ((l) ) dt T 1/2 um (·, t) − un (·, t); L4 ((l) )2 um (·, t); W22 ((l) ) dt ≤ cA ≤δ 0
0
T
um (·, t); W22 ((l) )2 dt
+ cδ A 0
T
um (·, t) − un (·, t); L4 ((l) )4 dt
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
T
≤ δA + cδ A
605
um (·, t) − un (·, t); L4 ((l) )4 dt.
0
Analogously, (V · ∇)(um − un ); L2 ((l)T )2 T ≤ cV; W22,1 ((l)T )2 ∇um (·, t) − ∇un (·, t); L4 ((l) )2 dt, 0
((um − un ) · ∇)V; L2 ((l)T )2 ≤ δV; W22,1 ((l)T )2
+ cδ V; W22,1 ((l)T )2
T
um (·, t) − un (·, t); L4 ((l) )4 dt.
0
Thus, we have proved that N um − N un ; L2 ((l)T )2 ≤ δ(V; W22,1 ((l)T )2 + A) + cδ V; W22,1 ((l)T )2 + A + c V; W22,1 ((l)T )2 + A
T
um (·, t) − un (·, t); L4 ((l) )4 dt
0 T
∇um (·, t) − ∇un (·, t); L4 ((l) )2 dt.
0
T By Lemmata I.1.8 and I.1.7, the integrals 0 um (·, t) − un (·, t); L4 ((l) )4 dt and T (l) 2 0 ∇um (·, t) − ∇un (·, t); L4 ( ) dt tend to zero as m, n → ∞. Therefore, for any given ε > 0 we can take δ=
ε 2(V; W22,1 ((l)T )2
+ A)
and then chose n, m so that c V; W22,1 ((l)T )2 + A
cδ V; W22,1 ((l)T )2 + A
T
∇um (·, t) − ∇un (·, t); L4 (G)2 dt < ε/4,
0
T
um (·, t) − un (·, t); L4 ((l) )4 dt < ε/4.
0
Thus, N um − N un ; L2 ((l)T ) → 0 as m, n → ∞ and, consequently, the operator N is compact.
606
K. Pileckas
R EMARK 4.5. The norm V; W22,1 ((l)T )2 grows linearly in l as l → ∞. However, in the proof of Lemma 4.6 the number l is fixed. ◦
1 (l) 8 (l)T ), T < ∞, problem (4.101) admits T HEOREM 4.6. For any 8 u(l) 0 ∈ W 2 ( ), f ∈ L2 ( at least one solution (v(l) (x, t), p (l) (x, t)) satisfying the estimate
v(l) ; W22,1 ((l)T )2 + ∇p (l) ; L2 ((l)T )2 ≤ cB2 .
(4.108)
The number B2 is defined in Remark 4.3 and the constant c is independent of l and T . P ROOF. The solvability of problem (4.101) is equivalent to the solvability of operator equation (4.105). Since the operator S −1 is bounded and the operator N is compact, we conclude that B is a compact operator and, according to Leray–Schauder theorem, (4.105) has a solution, if the norms v(l,λ) ; W22,1 ((l)T ) of all possible solutions to the equation v(l,λ) = λBv(l,λ) ,
λ ∈ [0, 1],
(4.109) (l)
are bounded by the same constant independent of λ. Using the fact that 8 u0 − 8 u0 ; W21 () → 0 as l → ∞ and repeating, almost word by word, the considerations of Section 4.4.2, we find that for any solution v(l,λ) ∈ W22,1 ((l)T ) of problem ⎧ (l,λ) ⎪ vt (x, t) − νv(l,λ) (x, t) + λ(v(l,λ) (x, t) · ∇)v(l,λ) (x, t) ⎪ ⎪ ⎪ ⎪ ⎪ 8(l,λ) (x, t) + λ(V(x, t) · ∇)v(l,λ) (x, t) + λ(v(l,λ) (x, t) · ∇)V(x, t) + ∇ p ⎪ ⎨ = λ8 f(x, t), ⎪ ⎪ (l,λ) ⎪ (x, t) = 0, div v ⎪ ⎪ ⎪ " ⎪ ⎩ = 0, v(l,λ) (x, 0) = λ8 u(l) (x), v(l,λ) (x, t)" 0
∂(l)
which is equivalent to operator equation (4.109), holds the estimate v(l,λ) ; W22,1 ((l)T )2 + ∇p (l,λ) ; L2 ((l)T )2 ≤ cB2 . It is easy also to verify that the constant in this estimate does not depend on λ and T . Thus, by Leray–Schauder theorem, operator equation (4.105) (or equivalently, system (4.101)) admits at least one solution (v(l) (x, t), p (l) (x, t)) and there holds estimate (4.108). T HEOREM 4.7. Let ∂ ∈ C 2 , Fj ∈ W21 (0, T ), T (0, ∞], and let the initial data u0 (x) and the external force f(x, t) are represented in the form (4.58) with (j )
◦
u02 ∈ W 12 (σj ), 1 ( u0 ∈ W2,β (),
(j )
f2
∈ L2 (jT ),
( f ∈ L2,β (T ),
βj ≥ 0,
j = 1, . . . , J.
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
607
Moreover, assume that there hold compatibility conditions (4.59). If the number γ∗ in (I.1.143 ) is sufficiently small, then problem (4.62) admits a unique solution v ∈ 2,1 W2,β (T ), ∇ p 8(x, t) ∈ L2,β (T ). There holds the estimate 2,1 v; W2,β (T )2 + ∇ p 8; L2,β (T )2 ≤ cB3 , (k)
(4.110)
(k)
where B3 = limk→∞ B2 (B2 is defined in (4.87)). (
P ROOF. Let us consider the solutions v(l) ∈ W22,1 ((l)T ) of problem (4.101) in bounded domains (l) , where T( = T for T < ∞ and T( is arbitrary finite number for T = ∞. By ◦ Lemma I.1.3, for almost all t ∈ [0, T(] hold the inclusions v(l) (·, t) ∈ W 12 ((l) ) and (
sup v(l) (·, t); W21 ((l) )2 ≤ cv(l) ; W22,1 ((l)T )2 .
(4.111)
t∈[0,T(]
The constant in the last inequality is independent of l. To verify this fact, it is enough to use inequality (I.1.6) for each of the domains ωj k , j = 1, . . . , J ; k = 0, 1, . . . , l + 1, and for (1) and to sum the obtained relations. ◦
(
The functions v(l) (x, t) can be considered as elements of spaces W 21,1 ((l)T ). Extending ◦
(
them by zero to the whole domain , we get a sequence {v(l) } ⊂ W 21,1 (T ) such that ◦ {v(l) (·, t)} ⊂ W 12 () for almost all t ∈ [0, T(] (for extended functions we use the same notations). In virtue of (4.111), (4.108), (
sup v(l) (·, t); W21 ()2 + v(l) ; W21,1 (T )2 ≤ cB2 .
t∈[0,T(]
(4.112)
The constant c in (4.112) is independent of l. Therefore, there exists a subsequence ◦ ◦ ( ( {v(lm ) (x, t)} which converges weakly in W 21,1 (T ) to some v ∈ W 21,1 (T ) and for v(x, t) remains valid estimate (4.112). Each v(lm ) (x, t) satisfies the integral identity t 0
vτ(lm ) (x, τ ) · η(x, τ ) dxdτ
+ + =
t 0
0
t
t 0
+ν
0
∇v(lm ) (x, τ ) · ∇η(x, τ ) dxdτ
((V(x, τ ) + v(lm ) (x, τ )) · ∇)v(lm ) (x, τ ) · η(x, τ ) dxdτ (v(lm ) (x, τ ) · ∇)V(x, τ ) · η(x, τ ) dxdτ
8 f(x, τ ) · η(x, τ ) dxdτ
t
∀t ∈ [0, T(]
(4.113)
608
K. Pileckas ◦
(
for any divergence-free η ∈ W 21,0 (T ) with suppx η(x, t) ⊂ (lm ) . Let us fix η(x, t) with compact support and pass in (4.113) to a limit as lm → ∞. This gives t 0
t
vτ (x, τ ) · η(x, τ ) dxdτ + ν
+ + =
t 0
0
t
t 0
0
∇v(x, τ ) · ∇η(x, τ ) dxdτ
((V(x, τ ) + v(x, τ )) · ∇)v(x, τ ) · η(x, τ ) dxdτ (v(x, τ ) · ∇)V(x, τ ) · η(x, τ ) dxdτ
8 f(x, τ ) · η(x, τ ) dxdτ
∀t ∈ [0, T(].
(4.114)
Note that in passage to the limit in the third and fourth terms on the left-hand side of (4.113) we have used the same compactness arguments as in the proof of Lemma 4.6. This is possible because the integration is carried out over the bounded domain (supp η is compact). Moreover, it is obvious that v(x, 0) = 8 u0 (x). Let us show that (4.114) remains ◦ ( 1,0 T valid for any divergence-free η ∈ W 2 ( ). We have " t " " " " (v(x, τ ) · ∇)v(x, τ ) · η(x, τ ) dxdτ "" " 0 t ≤ v(·, τ ); L4 ()∇v(·, τ ); L2 ()η(·, τ ); L4 () dτ 0
≤c
t
∇v(·, τ ); L2 ()2 ∇η(·, τ ); L2 () dτ
0
t
≤ c sup ∇v(·, τ ); L2 () t∈[0,T(]
(∇v(·, τ ); L2 ()2
0
+ ∇η(·, τ ); L2 ()2 ) dτ. Further, |V(x, t)|2 |∇v(x, t)|2 dx ≤
J ∞
|V(x, t)|2 |∇v(x, t)|2 dx
j =1 s=0 ωj s
+
|V(x, t)|2 |∇v(x, t)|2 dx (3)
≤
J ∞
sup |V(x, t)|2
j =1 s=0 x∈ωj s
|∇v(x, t)|2 dx ωj s
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
609
+ sup |V(x, t)|2
|∇v(x, t)|2 dx (3)
x∈(3)
≤c
J ∞
V(·, t); W22 (ωj s )2
j =1 s=0
+ cV(·, t); W22 ((3) )2 ≤ cY (t)
|∇v(x, t)|2 dx ωj s
|∇v(x, t)|2 dx (3)
|∇v(x, t)|2 dx
(4.115)
and |v(x, t)| |∇V(x, t)| dx ≤ 2
2
J ∞
v(·, t); L4 (ωj s )2 ∇V(·, t); L4 (ωj s )2
j =1 s=0
+ v(·, t); L4 ((3) )2 ∇V(·, t); L4 ((3) )2 ≤c
J ∞
V(·, t); W22 (ωj s )2
j =1 s=0
|∇v(x, t)|2 dx ωj s
+ cV(·, t); W22 ((3) )2
|∇v(x, t)|2 dx (3)
|∇v(x, t)|2 dx,
≤ cY (t)
(4.116)
where Y (t) is defined in the formulation of Lemma 4.3. From (4.115) and (4.116) follow the estimates " t " " " " (V(x, τ ) · ∇)v(x, τ ) · η(x, τ ) dxdτ "" " 0
" t " " " " +" (v(x, τ ) · ∇)V(x, τ ) · η(x, τ ) dxdτ "" 0 t 1 ≤ |V(x, τ )|2 |∇v(x, τ )|2 + |v(x, τ )|2 |∇V(x, τ )|2 dxdτ 2 0 t + |η(x, τ )|2 dxdτ ≤c
0 t
|∇v(x, τ )|2 dxdτ +
Y (τ ) 0
≤ c sup ∇v(·, τ ); L2 () t∈[0,T(]
2 0
t 0
t
|η(x, τ )|2 dxdτ
Y (τ ) dτ +
t 0
|η(x, τ )|2 dxdτ
610
K. Pileckas
≤ cA1 sup ∇v(·, τ ); L2 ()2 + t∈[0,T(] ◦
t 0
|η(x, τ )|2 dxdτ.
(
Thus, for any η(x, t) ∈ W 21,0 (T ) and v(x, t) satisfying (4.112), all integrals in identity ◦
(
(4.114) are finite and, since smooth η(x, t) with compact supports are dense in W 21,0 (T ), ◦
(
we conclude that (4.114) is valid for all η(x, t) ∈ W 21,0 (T ). From this we get that for almost all t ∈ [0, T(] holds the integral inequality ν ∇v(x, t) · ∇η(x) dx + (v(x, t)) · ∇)v(x, t) · η(x) dx
H(x, t) · η(x) dx
=
◦
∀η ∈ W 12 (),
where H(x, t) =8 f(x, t) − vt (x, t) − (V(x, t) · ∇)v(x, t) − (v(x, t) · ∇)V(x, t). Therefore, ◦ v(x, t) ∈ W 12 () can be considered as a weak solution of steady Navier–Stokes problem (II.2.2) with the right-hand side H(x, t) and zero fluxes. Then for v(x, t) hold the estimates (see [28], estimate (36) in Chapter V) v(·, t); W22 (ωj s )2 ≤ c v(·, t); W21 (( ωj s )8 + H(·, t); L2 (( ωj s )2 ωj s )2 ≤ c v(·, t); W21 ()6 v(·, t); W21 (( + 8 f(·, t); L2 (( ωj s )2 + vt (·, t); L2 (( ωj s )2 + (V(·, t) · ∇)v(·, t); L2 (( ωj s )2 + (V(·, t) · ∇)v(·, t); L2 (( ωj s )2 . Summing these relations over s and using (4.115), (4.115), we get v(·, t); W22 (j )2 ≤ c v(·, t); W21 ()6 v(·, t); W21 (j )2 + 8 f(·, t); L2 (j )2 + vt (·, t); L2 (j )2 + (V(·, t) · ∇)v(·, t); L2 (j )2 + (V(·, t) · ∇)v(·, t); L2 (j )2
≤ c v(·, t); W21 ()6 v(·, t); W21 ()2 + 8 f(·, t); L2 ()2 + vt (·, t); L2 ()2 + Y (t)∇v(·, t); L2 ()2 . Integration of the last inequality with respect to t yields in virtue of (4.112) 0
T(
v(·, t); W22 (j )2 dt
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
≤ c sup v(·, t); W21 ()6 t∈[0,T(]
T(
+
T(
v(·, t); W21 ()2 dt
0
8 f(·, t); L2 ()2 dt +
0
T(
vt (·, t); L2 ()2 dt
0
+ c sup
t∈[0,T(]
v(·, t); W21 ()2
≤ c B24 + B2 (1 + A1 ) +
T(
611
T(
Y (t)dt 0
2 8 f(·, t); L2 () dt .
0
Analogously, we get that
T(
0
v(·, t); W22 ((3) )2 dt ≤ c B24 + B2 (1 + A1 ) +
T(
8 f(·, t); L2 ()2 dt .
0
(
Thus, v ∈ W22,1 (T ) and from (4.114) we conclude that equations (4.62) are satisfied almost everywhere in × (0, T(). By Lemma 4.4, v(x, t) obeys estimate (4.86), i.e.,
T(
0
(k) Eβ (x)
|vτ (x, τ )| + 2
|D v(x, τ )|
2
α
dxdτ
|α|≤2
T(
+ 0
(k)
(k)
Eβ (x)|∇ p 8(x, τ )|2 dxdτ ≤ cB2 ,
where 2 c(B1 +A1 ) (B + A )2 ), B2(k) = (A(k) 1 1 3 + A1 + A1 )(1 + e T( (k) (k) (k) A3 = Eβ (x)(|( u0 (x)|2 + |∇( u0 (x)|2 ) dx + Eβ (x)|( f(x, t)|2 dxdt 2
0
1 (), ( f ∈ L2,β (T ), and the constant c is independent of T( and k. Since ( u0 ∈ W2,β
Eβ(k) (x) |( u0 (x)|2 + |∇( u0 (x)|2 dx + 1 ()2 ≤ ( u0 ; W2,β
0
T(
Eβ(k) (x)|( f(x, t)|2 dxdt
+ ( f; L2,β (T )2 = A3
and 0
T(
(k) Eβ (x) |vτ (x, τ )|2 + |D α v(x, τ )|2 dxdτ |α|≤2
612
K. Pileckas
+ 0
T(
(k)
Eβ (x)|∇ p 8(x, τ )|2 dxdτ ≤ cB3 ,
(4.117)
2 with B3 = (A3 + A1 + A21 )(1 + ec(B1 +A1 ) (B1 + A1 )2 ). If T( = T < ∞, we can let k → ∞ in (4.117) and, thus, to we obtain (4.110). If T = ∞, then T( is arbitrary finite number. Therefore, equations (4.62) are valid almost everywhere in × (0, ∞). Moreover, the constant in (4.117) is independent of T(. Passing in (4.117) first k → ∞, and then T( → ∞, we get the inequality (4.110) for T = ∞. Let us prove the uniqueness of the solution of problem (4.62). Obviously, the difference of two solutions w(x, t) = v[1] (x, t) − v[2] (x, t), q(x, t) = p 8[1] (x, t) − p 8[2] (x, t) satisfies the equations
⎧ wt (x, t) − νw(x, t) + ∇q(x, t) = −(V(x, t) · ∇)w(x, t) ⎪ ⎪ ⎪ ⎪ [1] [2] ⎪ ⎪ ⎨−(w(x, t) · ∇)V(x, t) − (w(x, t) · ∇)v (x, t) − (v (x, t) · ∇)w(x, t), div w(x, t) = 0, " ⎪ ⎪ " ⎪ w(x, t) ∂ = 0, w(x, 0) = 0, ⎪ ⎪ ⎪ ⎩ w(x, t) · n(x) ds = 0, j = 1, . . . , J. σj
Then, for (w(x, t), q(x, t)) holds the estimate (see the proof of inequality (4.91)) (∇w(·, t); L2 ()2 + Y (t))
Eβ (x)|∇w(x, t)|2 dx
≤ c(B1 + A1 )(∇w(·, t); L2 ()2 t + Y (t)) (∇w(·, τ ); L2 ()2 + Y (τ )) Eβ (x)|∇w(x, τ )|2 dxdτ 0
from which we conclude that (∇w(·, t); L2 () + Y (t))
Eβ (x)|∇w(x, t)|2 dx = 0.
2
Thus, w(x, t) = 0 and from the Navier–Stokes equations it follows that ∇q(x, t) = 0.
In the case where the data admit representations (4.96) we argue analogously and we get the following (j )
T HEOREM 4.8. Assume that Fj (t) and f(j ) (x1 , t), j = 1, . . . , J , admit representations (4.96) with Fj[1] ∈ R,
f(j,1) ∈ L2 (σj ),
Fj[2] ∈ W21 (0, ∞),
f(j,2) ∈ L2 (σj × (0, ∞)).
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
613
Moreover, let ◦
(j )
1 ( u0 ∈ W2,β (),
u0 ∈ W 12 (σj ),
( f ∈ L2,β ( × (0, ∞)),
βj ≥ 0,
j = 1, . . . , J,
and let u0 (x1 ), Fj (t) satisfy compatibility conditions (4.59) and Fj[1] satisfy con(j )
(j )
dition (4.97). If |Fj[1] |, f(j,1) ; L2 (σj ), j = 1, . . . , J , and the number γ∗ in inequality (I.1.143 ) for the weight-function Eβ (x) are sufficiently small, then problem (4.62) ad2,1 ( × (0, ∞)), ∇ p 8(x, t) ∈ L2,β ( × (0, ∞)). There holds mits a unique solution v ∈ W2,β the estimate 2,1 83 . v; W2,β ( × (0, ∞))2 + ∇ p 8; L2,β ( × (0, ∞))2 ≤ cB
4.5. The Three-dimensional Navier–Stokes problem 4.5.1. Reduction of problem (1.2) to a problem with zero fluxes Let ∂ ∈ C 2 . Assume that the initial data u0 (x) and the external force f(x, t) are represented in the form (1.4), (1.5) with
2 (), ( u0 ∈ W2,β (j ) u0
◦
∈ W 12 (σj ),
( f ∈ L2,β (T ),
f(j ) ∈ L2 ( T ),
(j )
(j )
βj ≥ 0,
Fj ∈ W21 (0, T ),
j = 1, . . . , J, j = 1, . . . , J. (4.118)
Moreover, let Fj (t) and (u0 (x (j ) ), f3 (x (j ) , t)), j = 1, . . . , J , satisfy the compatibility conditions (2.3). Then in each cylinder #j = σj × (0, T ) there exists a generalized
Poiseuille solution (U(j ) (x (j ) , t), P (j ) (x (j ) , t)) having the form (II.2.112) and satisfying estimates (II.2.115) and (II.2.149). Let us define V(x, t) by formula (2.9) while U(x, t) and P (x, t) are given by (2.4). Consider Navier–Stokes problem (1.2). Looking for the solution (u(x, t), p(x, t)) of (1.2) in the form u(x, t) = v(x, t) + V(x, t),
p(x, t) = p 8(x, t) + P (x, t),
(4.119)
614
K. Pileckas
we obtain for (v(x, t), p 8(x, t)) problem (4.62) with8 f(x, t) =( f(x, t) + f(1) (x, t) + f(2) (x, t), 8 u0 (x) = ( u0 (x) − W(x, 0), (1) f(1) (x, t) = f(1) (x, t), f3 (x, t) ,
(j ) f(1) (x, t) = Jj=1 νζ
(x3 )U(j ) (x (j ) , t)
−ζ (x3 )ζ (x3 )U3 (x (j ) , t)U(j ) (x (j ) , t)
(j ) (j ) −ζ (x3 )(ζ (x3 ) − 1)(U(j ) (x (j ) , t) · ∇ )U(j ) (x (j ) , t) ,
(j ) (j ) f3(1) (x, t) = Jj=1 νζ
(x3 )U3 (x (j ) , t) (j )
(j )
(j )
(4.120)
−ζ (x3 )ζ (x3 )|U3 (x (j ) , t)|2 (j )
(j )
(j )
−ζ (x3 )(ζ (x3 ) − 1)(U(j ) (x (j ) , t) · ∇ )U3 (x (j ) , t) (j ) (j ) −ζ (x3 )x3 q (j ) (t) , (j )
(j )
(j )
f(2) (x, t) = −Wt (x, t) + νW(x, t) − (W(x, t) · ∇)W(x, t) −(U(x, t) · ∇)W(x, t) − (W(x, t) · ∇)U(x, t).
(4.121)
It easy to see that suppx (f(1) (x, t) + f(2) (x, t)) ⊂ (3) . Using Sobolev embedding inequalities (see Lemma I.1.2), estimates (II.2.115), (II.2.149)
for U(j ) (x (j ) , t) and estimate (4.13) for W(x, t), we obtain the inequalities t 0
|f(1) (x, τ )|2 dx dτ
≤c
J t
j =1 0
(|U(j ) (x (j ) , τ )|2 + |U(j ) (x (j ) , τ )|4 j 3
+ |U(j ) (x (j ) , τ )|2 |∇ U(j ) (x (j ) , τ )|2 + |q (j ) (τ )|2 ) dxdτ J t
≤c (|U(j ) (x (j ) , τ )|2 + |q (j ) (τ )|2 ) dx (j ) dτ j =1 0
+c
J
σj t
j =1 0 x
sup (|U(j ) (x (j ) , τ )|2 )U(j ) (·, τ ); W21 (σj )2 dτ
(j )
∈σ j
≤ c(A0 + A1 ) + c
t
× 0
J
sup U(j ) (·, τ ); W21 (σj )2
j =1 t∈[0,T ]
U(j ) (·, τ ); W22 (σj )2 dτ
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
≤ c(A0 + A1 )(1 + A0 + A1 ) := cA2 , t 0
615
(4.122)
|f(2) (x, τ )|2 dx dτ
≤c
t 0
(|Wt (x, τ )|2 + |W(x, τ )|2
(3)
+ (|W(x, τ )|2 + U(x, τ )|2 )|∇W(x, τ )|2 + |W(x, τ )|2 |∇U(x, τ )|2 ) dxdτ ≤ cA1 +c
t 0
% (|∇W|2 + |∇U|2 ) dx dτ
sup (|W(x, τ )|2 + |U(x, τ )|2 ) (3)
x∈(3)
≤ cA1 + c sup
t∈[0,T ]
W(·, t); W21 ((3) )2 +
J
U(j ) (·, t); W21 (σj )2
j =1
t J
2 2 (j ) 1 2 W(·, τ ); W2 ((3) ) + × U (·, t); W2 (σj ) dτ 0
j =1
≤ cA2 .
(4.123)
(j )
A0 depend only on Theorem II.2.9), A1 =
J
(j ) (j ) (j ) ; W 2,1 ( T )2 and j =1 A0 , where A0 are bounds for U j 2
(j ) u0 ; W21 (σj )2 and f(j ) ; L2 (jT )2 (see estimate (II.2.115) in
In (4.122), (4.123) A0 =
J
(j ) (j ) Fj ; W21 (0, T )2 + u03 ; W21 (σj )2 + f3 ; W21 (jT )2 . j =1
4.5.2. Estimates of the nonlinear terms 2,1 L EMMA 4.7. Let v ∈ W2,β (T ), βj ≥ 0, j = 1, . . . , J ,T ∈ (0, ∞]. Then (v · ∇)v ∈ L2,β (T ) and
t
(v(·, τ ) · ∇)v(·, τ ); L2,β ()2 dτ
0 2,1 ≤ c min{1, T 1/2 }v; W2,β ( × (0, t))4
∀t ∈ [0, T ],
(4.124)
where the constant c is independent of t ∈ [0, T ] and T . P ROOF. First of all we mention that the condition v ∈ W22,1 (T ) implies v(·, t) ∈ W21 () (see I.1.6). Denote Ij s (t) = (v(·, t) · ∇)v(·, t); L2 (ωj s )2 . Then by Hölder inequality and
616
K. Pileckas
by multiplicative inequalities (I.1.9), (I.1.10) Ij s (t) ≤ v(·, t); L6 (ωj s )2 ∇v(·, t); L3 (ωj s )2 ≤ cv(·, t); W21 (ωj s )2 ∇v(·, t); W21 (ωj s )∇v(·, t); L2 (ωj s ). (4.125) Therefore,
t
Ij s (τ ) dτ ≤ c sup v(·, τ ); W21 (ωj s )2 τ ∈(0,t)
0
≤ c sup v(·, τ ); W21 ()2 τ ∈(0,t)
≤
0
t
0 t
∇v(·, τ ); W21 (ωj s )2 dτ
v(·, τ ); W22 (ωj s )2 dτ
cv; W22,1 ( × (0, t))2 v; W22,1 (ωj s
× (0, t))2 .
(4.126)
On the other hand,
t
0
∇v(·, τ ); W21 (ωj s )∇v(·, τ ); L2 (ωj s ) dτ ≤ sup v(·, τ ); W21 (ωj s )t 1/2 τ ∈(0,t)
≤ cT
1/2
v; W22,1 (ωj s
× (0, t))
t 0
1/2 v(·, τ ); W22 (ωj s )2 dτ
2
and from (4.125) we derive
t
Ij s (τ ) dτ ≤ cT 1/2 sup v(·, τ ); W21 ()2 v; W22,1 (ωj s × (0, t))2 τ ∈(0,t)
0
≤ cT 1/2 v; W22,1 ( × (0, t))2 v; W22,1 (ωj s × (0, t))2 . (4.127) Inequalities (4.126) and (4.127) yield
t
(v(·, τ ) · ∇)v(·, τ ); L2 (ωj s )2 dτ
0
≤ c min{1, T 1/2 }v; W22,1 ( × (0, t))2 v; W22,1 (ωj s × (0, t))2 .
(4.128)
Obviously, the constant c in (4.128) does not depend on s. Multiplying inequalities (4.128) by Eβj (s) and summing obtained relations over s from 0 to ∞, we get in virtue of properties (I.1.14) of weight-function Eβ (x) 0
t
(v(·, τ ) · ∇)v(·, τ ); L2,βj (j )2 dτ
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
617
2,1 ≤ c min{1, T 1/2 }v; W22,1 ( × (0, t))2 v; W2,β (j × (0, t))2 j 2,1 ≤ c min{1, T 1/2 }v; W2,β ( × (0, t))4 .
(4.129)
Analogously, it can be proved that
t
(v(·, τ ) · ∇)v(·, τ ); L2 ((3) )2 dτ
0
≤ c min{1, T 1/2 }v; W22,1 ( × (0, t))2 v; W22,1 ((3) × (0, t))2 .
(4.130)
Inequality (4.124) follows from (4.129), (4.130).
2,1 (T ), βj ≥ 0, j = 1, . . . , J , T ∈ (0, ∞]. Then (u · ∇)v ∈ L EMMA 4.8. Let v, u ∈ W2,β T T L2,β ( ), (v · ∇)u ∈ L2,β ( ) and there holds the estimates
t
(u(·, τ ) · ∇)v(·, τ ); L2,β ()2 dτ
0
t
≤ cε 0
2,1 u; W2,β ( × (0, τ ))2 dτ
2,1 2,1 + εcv; W2,β ( × (0, t))4 u; W2,β ( × (0, t))2 ,
(4.131)
and
t
(u(·, τ ) · ∇)v(·, τ ); L2,β ()2 dτ
0
t
≤ cε 0
2,1 v; W2,β ( × (0, τ ))2 dτ
2,1 2,1 + εcu; W2,β ( × (0, t))4 v; W2,β ( × (0, t))2 .
(4.132)
Constants in (4.131), (4.132) are independent of t ∈ [0, T ] and T . P ROOF. As in estimate (4.125) we obtain that Nj s (t) =
t 0
≤c 0
|u(x, τ )|2 |∇v(x, τ )|2 dxdτ
ωj s t
u(·, τ ); W21 (ωj s )2 v(·, τ ); W21 (ωj s )v(·, τ ); W22 (ωj s )dτ.
Therefore, Nj s (t) ≤ c sup v(·, τ ); W21 (ωj s ) sup u(·, τ ); W21 (ωj s ) τ ∈(0,t)
τ ∈(0,t)
618
K. Pileckas
t
×
u(·, τ ); W21 (ωj s )v(·, τ ); W22 (ωj s ) dτ
0
≤ cv; W22,1 (T )u; W22,1 (ωj s × (0, t)) t 1/2 1 2 × u(·, τ ); W2 (ωj s ) dτ 0
t
× 0
1/2 v(·, τ ); W22 (ωj s )2 dτ
≤ cv; W22,1 (T )2 u; W22,1 (ωj s × (0, t)) t 1/2 × u; W22,1 (ωj s × (0, τ ))2 dτ 0
2,1 ≤ εcv; W2,β (T )4 u; W22,1 (ωj s × (0, t))2 t + cε u; W22,1 (ωj s × (0, τ ))2 dτ 0
and Nj s (t) ≤ c sup u(·, τ ); W21 (ωj s )2 τ ∈(0,t)
t
× 0
v(·, τ ); W21 (ωj s )v(·, τ ); W22 (ωj s )dτ
2,1 ≤ εcu; W2,β (T )4 v; W22,1 (ωj s × (0, t))2 t + cε v; W22,1 (ωj s × (0, τ ))2 dτ. 0
Multiplying these relations by Eβj (s) and then summing over s furnishes t 0
j
Eβj (x)|u(x, τ )|2 |∇v(x, τ )|2 dxdτ
2,1 2,1 (T )4 u; W2,β (j × (0, t))2 ≤ εcv; W2,β t 2,1 + cε u; W2,β (j × (0, τ ))2 dτ, 0
t 0
j
Eβj (x)|u(x, τ )|2 |∇v(x, τ )|2 dxdτ
2,1 2,1 ≤ εcu; W2,β (T )4 v; W2,β (j × (0, t))2
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
619
t 2,1 + cε v; W2,β (j × (0, τ ))2 dτ. 0
Analogously it can be proved that t 0
|u(x, τ )|2 |∇v(x, τ )|2 dxdτ
(3) 2,1 (T )4 u; W22,1 ((3) × (0, t))2 ≤ εcv; W2,β t + cε u; W22,1 ((3) × (0, τ ))2 dτ, 0
t 0
|u(x, τ )|2 |∇v(x, τ )|2 dxdτ
(3) 2,1 (T )4 v; W22,1 ((3) × (0, t))2 ≤ εcu; W2,β t + cε v; W22,1 ((3) × (0, τ ))2 dτ 0
and, therefore, we get (4.131), (4.132).
2,1 L EMMA 4.9. Let v ∈ W2,β (T ), T ∈ (0, ∞]. Then (V · ∇)v ∈ L2,β (T ), (v · ∇)V ∈ L2,β (T ) and
t
(V(·, τ ) · ∇)v(·, τ ); L2,β () dτ + 2
0
t
(v(·, τ ) · ∇)V(·, τ ); L2,β ()2 dτ
0 2,1 ≤ c min{1, T 1/2 }(A0 + A1 )v; W2,β ( × (0, t))2
where the constant c is independent of t ∈ [0, T ] and T . P ROOF. Arguing as in Lemma 4.7, we get for Jj s (t) = (V(·, t) · ∇)v(·, t); L2 (ωj s )2 the estimates t Jj s (τ ) dτ 0
≤ c sup V(·, τ ); W21 (ωj s )2 τ ∈(0,t)
× 0
t
∇v(·, τ ); W21 (ωj s )∇v(·, τ ); L2 (ωj s ) dτ
∀t ∈ [0, T ],
(4.133)
620
K. Pileckas
≤ c sup V(·, τ ); W21 (ωj s )2 v; W22,1 (ωj s × (0, t))2 τ ∈(0,t)
and
t
Jj s (τ ) dτ ≤ c sup V(·, τ ); W21 (ωj s )2 sup v(·, τ ); W21 (ωj s ) t 1/2 τ ∈(0,t)
0
t
× 0
≤ cT Therefore,
1/2
τ ∈(0,t)
1/2
∇v(·, τ ); W21 (ωj s )2 dτ
sup V(·, τ ); W21 (ωj s )2 v; W22,1 (ωj s × (0, t))2 .
τ ∈(0,t)
t
Jj s (τ ) dτ 0
≤ c min{1, T 1/2 } sup V(·, τ ); W21 (ωj s )2 v; W22,1 (ωj s × (0, t))2 . τ ∈(0,t)
(4.134) From (II.2.115), (II.2.149) and from (4.13) it follows that & sup V(·, τ ); W21 (ωj s )2 ≤ c sup U(j ) (·, τ ); W21 (ωj s )2 τ ∈(0,t)
τ ∈(0,t)
+ sup W(·, τ ); W21 (ωj s )2 ≤c
&
'
τ ∈(0,t)
sup U(j ) (·, τ ); W21 (σj )2
τ ∈(0,t)
+ sup W(·, τ ); W21 ((3) )2
'
τ ∈(0,t)
≤ c U(j ) ; W22,1 (σjT )2 + W; W22,1 (T(3) )2 ≤ c(A0 + A1 ). Thus, (4.134) yields
t
0
Jj s (τ ) dτ ≤ c min{1, T 1/2 }(A0 + A1 )v; W22,1 (ωj s × (0, t))2 .
Multiplying (4.135) by Eβj (s) and then summing over s, we derive 0
t
(V(·, τ ) · ∇)v(·, τ ); L2,βj (j )2 dτ
(4.135)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem 2,1 ≤ c min{1, T 1/2 }(A0 + A1 )v; W2,β (j × (0, t))2 . j
621
(4.136)
Analogously,
t
(V(·, τ ) · ∇)v(·, τ ); L2 ((3) )2 dτ
0
≤ c min{1, T 1/2 }(A0 + A1 )v; W22,1 ((3) × (0, t))2 .
(4.137)
From (4.136), (4.137) it follows that
t
(V(·, τ ) · ∇)v(·, τ ); L2,β ()2 dτ
0 2,1 ≤ c min{1, T 1/2 }(A0 + A1 )v; W2,β ( × (0, t))2 .
Consider now the integrals Kj s (t) = (v(·, t) · ∇)V(·, t); L2 (ωj s )2 . Using (I.1.10), we get Kj s (t) ≤ v(·, t); L∞ (ωj s )2 ∇V(·, t); L2 (ωj s )2 ≤ c∇v(·, t); W21 (ωj s )v(·, t); W21 (ωj s )∇V(·, t); L2 (ωj s )2 . Thus,
t
Kj s (τ ) dτ 0
≤ c sup ∇V(·, τ ); L2 (ωj s )2 τ ∈(0,t)
× 0
t
∇v(·, τ ); W21 (ωj s )v(·, τ ); W21 (ωj s ) dτ
t
≤ c(A0 + A1 ) 0
v(·, τ ); W22 (ωj s )2 dτ
≤ c(A0 + A1 )v; W22,1 (ωj s × (0, t))2 . On the other hand, 0
t
Kj s (τ ) dτ ≤ c(A0 + A1 ) sup v(·, τ ); W21 (ωj s ) × 0
τ ∈(0,t)
t
v(·, τ ); W22 (ωj s ) dτ
≤ c(A0 + A1 )T 1/2 v; W22,1 (ωj s × (0, t))T 1/2
622
K. Pileckas
t
× 0
1/2 v(·, τ ); W22 (ωj s )2 dτ
≤ c(A0 + A1 ) T 1/2 v; W22,1 (ωj s × (0, t))2 . Hence,
t
(v(·, τ ) · ∇)V(·, τ ); L2 (ωj s )2 dτ
0
≤ c(A0 + A1 ) min{1, T 1/2 }v; W22,1 (ωj s × (0, t))2 and, as above, we derive t (v(·, τ ) · ∇)V(·, τ ); L2,βj (j )2 dτ 0
2,1 ≤ c(A0 + A1 ) min{1, T 1/2 }v; W2,β (j × (0, t))2 . j
It can be proved analogously that
t
(v(·, τ ) · ∇)V(·, τ ); L2 ((3) )2 dτ
0
≤ c(A0 + A1 ) min{1, T 1/2 }v; W22,1 ((3) × (0, t))2 and, therefore, t (v(·, τ ) · ∇)V(·, τ ); L2,β ()2 dτ 0 2,1 ≤ c(A0 + A1 ) min{1, T 1/2 }v; W2,β ( × (0, t))2 .
By the considerations similar to those of Lemmata 4.8, 4.9, we prove also the following 2,1 (T ), T ∈ (0, ∞]. Then there holds the estimate L EMMA 4.10. Let v ∈ W2,β
t
(V(·, τ ) · ∇)v(·, τ ); L2,β ()2 dτ +
0
t
(v(·, τ ) · ∇)V(·, τ ); L2,β ()2 dτ
0 2,1 ≤ εc(A20 + A21 )v; W2,β ( × (0, t))2 t 2,1 + cε v; W2,β ( × (0, τ ))2 dτ, 0
where the constant c is independent of t ∈ [0, T ] and T .
(4.138)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
623
4.5.3. Solvability of problem (4.62) T HEOREM 4.9. Let ∂ ∈ C 2 , Fj ∈ W21 (0, T ), and let the initial data u0 (x) and the external force f(x, t) are represented in the form (4.4), (4.5) with ( f ∈ L2,β (T ),
2 (), ( u0 ∈ W2,β (j ) u0
◦
∈ W 12 (σj ),
βj ≥ 0,
j = 1, . . . , J,
f(j ) ∈ L2 (jT ),
j = 1, . . . , J.
Moreover, assume that there hold compatibility conditions (4.8) and that the number γ∗ in inequality (I.1.143 ) for the weight function Eβ (x) is sufficiently small (such that the conditions of Theorem 4.2 are valid). If c2 min{1, T 1/2 }(A0 + A1 ) < 1, 4c1 c2 (A0 + A1 + A2 + A3 ) min{1, T 1/2 } < (1 − c2 min{1, T 1/2 }(A0
+ A1
(4.139)
))2 ,
where A0 , A1 , A2 are defined in Section 4.5.1, 1 f; L2,β (T )2 + ( u0 ; W2,β ()2 , A3 = (
and c1 , c2 are absolute constants defined below, then problem (4.62) admits a unique so2,1 lution v ∈ W2,β (T ), ∇ p 8 ∈ L2,β (T ). There holds the estimate 2,1 (T )2 + ∇ p 8; L2,β (T )2 ≤ cr0 , v; W2,β
(4.140)
where r0 = α0 = c1
3
1 − α1 +
k=0 Ak ,
2α0 (1 − α1 )2 − 4α0 α2
,
(4.141)
α1 = c2 min{1, T 1/2 }(A0 + A1 ), α2 = c2 min{1, T 1/2 }.
P ROOF. We prove the existence of the solution to problem (4.62) by the method of successive approximations using the scheme proposed in by V.A. Solonnikov in [93]. Let us consider problem (4.62) as a linear time-dependent Stokes problem ⎧ vt (x, t) − νv(x, t) + ∇ p 8(x, t) = g(x, t; v(x, t)), ⎪ ⎪ ⎪ ⎪ ⎨ div v(x, t) = 0, " (4.142) v(x, t)"∂ = 0, v(x, 0) = 8 u0 (x), ⎪ ⎪ ⎪ ⎪ ⎩ v(x, t) · n(x) ds = 0, j = 1, . . . , J, σj
with g(x, t, v(x, t)) = 8 f(x, t) − (v(x, t) · ∇)v(x, t) − (V(x, t) · ∇)v(x, t)
624
K. Pileckas
− (v(x, t) · ∇)V(x, t). Let us put v(0) (x, t) = 0, p 8(0) (x, t) = 0, and define the successive approximations recurrently as solutions of linear problems ⎧ (l+1) vt (x, t) − νv(l+1) (x, t) + ∇ p 8(l+1) (x, t) = g(x, t; v(l) (x, t)), ⎪ ⎪ ⎪ ⎪ ⎨ div v(l+1) (x, t) = 0, " ⎪ v(l+1) (x, 0) = 8 u0 (x), v(l+1) (x, t)"∂ = 0, ⎪ ⎪ (l+1) ⎪ ⎩ (x, t) · n(x) ds = 0, j = 1, . . . , J. σj v
(4.143)
In virtue of (4.122), (4.123), (4.124) (4.133) and (4.13), the right-hand sides g(x, t, v(l) (x, t)) and 8 u0 (x) admit the estimates g(x, t; v(l) (x, t)); L2,β (T )2 ≤ c8 f; L2,β (T )2 2,1 2,1 + c min{1, T 1/2 } v(l) ; W2,β (T )4 + (A0 + A1 )v(l) ; W2,β (T )2 ≤ c ( f; L2,β (T )2 + f(1) ; L2,β (T )2 + f(2) ; L2,β (T )2 2,1 2,1 + c min{1, T 1/2 } v(l) ; W2,β (T )4 + (A0 + A1 )v(l) ; W2,β (T )2 ≤ c(A3 + A2 )
2,1 2,1 (T )4 + (A0 + A1 )v(l) ; W2,β (T )2 + c min{1, T 1/2 } v(l) ; W2,β
and 1 1 1 8 u0 ; W2,β ()2 ≤ ( u0 ; W2,β ()2 + W(·, 0); W2,β ()2 1 ≤ c ( u0 ; W2,β ()2 + W; W22,1 (T(3) )2
≤ c(A3 + A1 + A0 ). Note that we have used the fact that suppx (f(1) (x, t) + f(2) (x, t)) ⊂ (3) , suppx W(x, t) ⊂ (3) , and, therefore, f(1) + f(2) ; L2,β (T )2 ≤ cf(1) + f(2) ; L2 (T )2 . By Theorem 4.2, all approximations (v(l+1) (x, t), p 8(l+1) (x, t)) are well defined and satisfy the estimates 2,1 v(l+1) ; W2,β (T )2 + ∇ p 8(l+1) ; L2,β (T )2 1 ≤ c g(·, v(l) ); L2,β (T )2 + 8 u0 ; W2,β () ≤ c1 (A0 + A1 + A2 + A3 ) 2,1 2,1 + c2 min{1, T 1/2 } v(l) ; W2,β (T )4 + (A0 + A1 )v(l) ; W2,β (T )2 2,1 2,1 (T )2 + α2 v(l) ; W2,β (T )4 . = α0 + α1 v(l) ; W2,β
(4.144)
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
625
If α1 < 1,
4α0 α2 < (1 − α1 )2 ,
(4.145)
then the quadratic equation α2 ρ 2 + (α1 − 1)ρ + α0 = 0 has positive roots minimal r0 of which is given by formula (4.141) Conditions (4.145) are satisfied because of assumptions (4.139). From (4.144), (4.145) it follows that, if v(l) ; W22,1 (T )2 ≤ r0 , then also 2,1 v(l+1) ; W2,β (T )2 + ∇ p 8(l+1) ; L2,β (T )2 ≤ r0 .
(4.146)
2,1 Since, obviously v(0) ; W2,β (T )2 ≤ r0 , we conclude that (4.146) is valid ∀l ≥ 0. Let us show now that the sequence {(v(l) (x, t), p 8(l) (x, t))} converges to the solution (v(x, t), p 8(x, t)) of problem (4.62). The differences
w(l) (x, t) = v(l+1) (x, t) − v(l) (x, t),
q (l) (x, t) = p 8(l+1) (x, t) − p 8(l) (x, t)
are the solutions of the following linear problems ⎧ (l) (l) ⎪ w(l) ⎪ t (x, t) − νw (x, t) + ∇q (x, t) ⎪ ⎪ ⎪ ⎪ = g(x, t; v(l+1) (x, t)) − g(x, t; v(l) (x, t)) ⎪ ⎪ ⎪ ⎪ (l) (l) ⎪ ⎪ ⎨ = −(V(x, t) · ∇)w (x, t) − (w (x, t) · ∇)V(x, t) −(w(l) (x, t) · ∇)v(l) (x, t) − (v(l) (x, t) · ∇)w(l) (x, t), ⎪ ⎪ ⎪ div w(l) (x, t) = 0, ⎪ ⎪ ⎪ " ⎪ (l) ⎪ w(l) (x, 0) = 0, w (x, t)"∂ = 0, ⎪ ⎪ ⎪ ⎪ ⎩ w(l) (x, t) · n(x) ds = 0, j = 1, . . . , J.
(4.147)
σj
2,1 Let X (l+1) (t) = w(l+1) ; W2,β ( × (0, t))2 + ∇q (l+1) ; L2,β ( × (0, t))2 . Using Theorem 4.6, Lemmata 4.8, 4.10 and estimate (4.146), we derive the inequality
X (l+1) (t) ≤ cg(·; v(l+1) ) − g(·; v(l) ); L2,β ( × (0, t))2 ≤ c (V · ∇)w(l) ; L2,β ( × (0, t))2 + (w(l) · ∇)V; L2,β ( × (0, t))2 + (w(l) · ∇)v(l) ; L2,β ( × (0, t))2
+ (v(l) · ∇)w(l) ; L2,β ( × (0, t))2 2,1 2,1 ≤ εc A20 + A21 + v(l) ; W2,β (T )4 w(l) ; W2,β ( × (0, t))2 t 2,1 + cε w(l) ; W2,β ( × (0, τ ))2 dτ 0
2,1 ≤ εc∗ A20 + A21 + r02 w(l) ; W2,β ( × (0, t))2
626
K. Pileckas
+ cε
t
0
Let us fix ε = M+1
1 2c∗ (A20 +A21 +r02 )
X (l) (t) ≤
m=2
2,1 w(l) ; W2,β ( × (0, τ ))2 dτ.
and sum relations (4.148) by l from 1 to M. This yields
t M M 1 (l) X (t) + c X (l) (τ ) dτ. 2 0 m=1
Setting Y (M) (t) =
M
m=1 X
(l) (t),
Y
(t) ≤ 2X
(1)
(4.149)
m=1
from (4.149) we get
(M)
(4.148)
(t) + c
t
Y (M) (τ ) dτ
0
and, by Gronwall inequality, Y (M) (t) ≤ 2ect X (1) (t). (l) (l+1) (x, t), Therefore, the series ∞ l=1 X (t) converges for any finite t and, consequently, {v 2,1 (l+1) T T ∇p 8 (x, t)} converges in the norm of the space W2,β ( ) × L2,β ( ) to the solution (v(x, t), p 8(x, t)) of problem (4.62). Obviously, for (v(x, t), p 8(x, t)) remains valid inequality (4.146). Let us prove the uniqueness of the solution to problem (4.62). Obviously, the difference 8[1] (x, t) − p 8[2] (x, t) satisfies of two solutions w(x, t) = v[1] (x, t) − v[2] (x, t), q(x, t) = p the equations ⎧ wt (x, t) − νw(x, t) + ∇q(x, t) = −(V(x, t) · ∇)w(x, t) ⎪ ⎪ ⎪ ⎪ [1] [2] ⎪ ⎪ ⎨−(w(x, t) · ∇)V(x, t) − (w(x, t) · ∇)v (x, t) − (v (x, t) · ∇)w(x, t), div w(x, t) = 0, " ⎪ ⎪ " ⎪ w(x, t) ∂ = 0, w(x, 0) = 0, ⎪ ⎪ ⎪ ⎩ w(x, t) · n(x) ds = 0, j = 1, . . . , J. σj
Therefore, for (w(x, t), q(x, t)) holds the estimate 2,1 ( × (0, t))2 + ∇q; L2,β ( × (0, t))2 w; W2,β t 2,1 ≤ c w; W2,β ( × (0, τ ))2 dτ 0
from which it follows that w(x, t) = 0, ∇q(x, t) = 0. R EMARK 4.6. If the data are sufficiently small, i.e., if
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
c2 (A0 + A1 ) < 1,
627
4c1 c2 (A0 + A1 + A2 + A3 ) < (1 − c2 (A0 + A1 ))2 , (4.150)
then from Theorem 4.9 follows that the solution (v(x, t), p 8(x, t)) of problem (4.62) exists for any finite time interval [0, T ]. Moreover, the constant in estimate (4.140) does not depend on T and, therefore, the solution exists also for the infinite time interval (0, ∞). If the data are “large” ((4.150) is not valid), then the solution of problem (4.62) exists for “small” time intervals [0, T ], where the bound for T is given by (4.139). In the case where the data admit representations (4.96) we argue analogously and we get the following (j )
T HEOREM 4.10. Assume that Fj (t) and f(j ) (x1 , t), j = 1, . . . , J , admit representations (4.96) with Fj[1] ∈ R,
f(j,1) ∈ L2 (σj ),
Fj[2] ∈ W21 (0, ∞),
f(j,2) ∈ L2 (σj × (0, ∞)).
Moreover, let (j )
◦
u0 ∈ W 12 (σj ),
1 ( u0 ∈ W2,β (),
( f ∈ L2,β ( × (0, ∞)),
βj ≥ 0,
j = 1, . . . , J,
and let u0 (x1 ), Fj (t) satisfy compatibility conditions (4.59), Fj[1] satisfy condi(j )
(j )
tion (4.97). If |Fj[1] |, f(j,1) ; L2 (σj ), j = 1, . . . , J , and the number γ∗ in inequality (I.1.143 ) for the weight-function Eβ (x) are sufficiently small, then problem (4.62) 2,1 admits a unique solution v ∈ W2,β ( × (0, ∞)), ∇ p 8(x, t) ∈ L2,β ( × (0, ∞)). There holds the estimate 2,1 83 , ( × (0, ∞))2 + ∇ p 8; L2,β ( × (0, ∞))2 ≤ cB v; W2,β
83 depends only on the data. where B 4.5.4. Problem (4.2) in a straight cylinder # Let us consider the Navier–Stokes problem (4.2) in a straight cylinder # ⊂ R3 assuming that the right-hand sides admit the representations f(x, t) = f0 (x , t) +( f(x, t),
u0 (x) = U0 (x ) + ( u0 (x).
(4.151)
We look for the solution of problem (1.2) in the form u(x, t) = U(x , t) + v(x, t),
p(x, t) = P (x , t) + p 8(x, t),
(4.152)
628
K. Pileckas
where (U(x , t), P (x , t)) is the generalized Poiseuille solution corresponding to f0 (x , t), U0 (x ) and the flux F (t). Then for (v(x, t), p 8(x, t)) we derive the following problem ⎧ vt (x, t) − νv(x, t) + (v(x, t) · ∇)v(x, t) + (U(x , t) · ∇)v(x, t) ⎪ ⎪ ⎪ ⎪ ⎪ 8(x, t) =( f(x, t), +(v(x, t) · ∇)U(x , t) + ∇ p ⎪ ⎨ div v(x, t) = 0, (4.153) ⎪ " ⎪ ⎪ v(x, t)"∂# = 0, v(x, 0) = ( u0 (x), ⎪ ⎪ ⎪ ⎩
σ v3 (x, t) dx = 0. In this section we prove global existence for problem (4.153) without assumption that f0 (x , t), U0 (x ) and the flux F (t) are small. In the book O.A. Ladyzhenskaya [28] it is mentioned that, if the Navier–Stokes problem has a solution for some data f0 (x, t), u0 (x), then the solution exists also for the data that are close to f0 (x, t), u0 (x). In our case we know the global existence of the generalized Poiseuille solution corresponding to f0 (x , t), U0 (x ), F (t) and we look for the solution v(x, t) of problem (4.153) in the neighf(x, t) and borhood of this Poiseuille solution U(x , t) assuming only that the norms of ( ( u0 (x) are “sufficiently small”. First, let us consider the linearized problem ⎧ vt (x, t) − νv(x, t) + (U(x , t) · ∇)v(x, t) ⎪ ⎪ ⎪ ⎪
⎪ 8(x, t) = h(x, t), ⎪ ⎨+(v(x, t) · ∇)U(x , t) + ∇ p div v(x, t) = 0, (4.154) " ⎪ ⎪ " = 0, ⎪ ( v(x, t) v(x, 0) = u (x), ⎪ 0 ⎪ ∂# ⎪ ⎩
= 0. v (x, t) dx 3 σ The existence of a solution (v(x, t), p 8(x, t)) to problem (4.154) will be proved using Galerkin approximations, following the scheme described in [28]. Let {wk (x)} be a fundamental system in the space H (#) consisting of all divergence-free vector-fields ◦ from W 12 (#). Without loss of generality we may assume that the system {wk (x)} is orthonormalized in L2 (#). We look for the approximate solution v(N ) (x, t) in the form v(N ) (x, t) =
N
(N )
αk (t)wk (x)
k=1
where αk (t) are found from the following equalities (N ) vt (x, t) · wk (x) dx + ν ∇v(N ) (x, t) · ∇wk (x) dx
+
(U(x , t) · ∇)v(N ) (x, t) · wk (x) dx
+
(v(N ) (x, t) · ∇)U(x , t) · wk (x) dx
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
629
=
h(x, t) · wk (x) dx,
k = 1, . . . , N,
(4.155)
(N ) and from the initial conditions αk (0) = γk , where ( u0 (x) = ∞ k=1 γk wk (x). Note that (N ) (4.155) is a linear system of ordinary differential equations for αk (t), k = 1, . . . , N . It is easy to verify that Galerkin approximations satisfy the following integral equalities 1 d |v(N ) (x, t)|2 dx + ν |∇v(N ) (x, t)|2 dx 2 dt # # = − (v(N ) (x, t) · ∇)U(x , t) · v(N ) (x, t) dx #
h(x, t) · v(N ) (x, t) dx
+
(4.156)
#
and
(N )
#
|vt
(x, t)|2 dx +
|∇v(N ) (x, t)|2 dx #
(U(x , t) · ∇)v(N ) (x, t) · vt
(x, t) dx
(v(N ) (x, t) · ∇)U(x , t) · vt
(x, t) dx
(N )
=− #
(N )
−
ν d 2 dt
#
(N )
+ #
h(x, t) · vt
(4.157)
(x, t) dx.
First, let us prove a priori estimates for Galerkin approximations v(N ) (x, t). ◦
L EMMA 4.11. Let ( u0 ∈ W 12 (#),U ∈ W22,1 ( T ), h ∈ L2 (#T ), T ∈ (0, ∞]. Then there hold the following a priori estimates |v
(N )
(x, t)| dx + ν 2
t 0
#
|∇v(N ) (x, τ )|2 dxdτ
#
≤ c(A0 ecA0 + 1)A1 (t) = cB1 (t),
∀t ∈ (0, T ],
(4.158)
and t 0
#
) 2 |v(N τ (x, τ )| dxdτ +
|∇v(N ) (x, t)|2 dx ≤ cB1 (t),
∀t ∈ (0, T ],
#
t where A1 (t) = # (|( u0 (x)|2 + |∇( u0 (x)|2 ) dx + 0 # |h(x, τ )|2 dx dτ , U; W22,1 ( T )2 and the constant c does not depend on t and N .
(4.159) A0 =
630
K. Pileckas
P ROOF. Using Hölder and Young inequalities and inequality (I.1.7), we get " " " " " (v(N ) (x, t) · ∇)U(x , t) · v(N ) (x, t) dx " " " # ∞ |v(N ) (x , x3 , t)|2 |∇U(x , t)| dx dx3 ≤ −∞ σ
≤
∞
−∞
≤c
1/2
σ
∞
−∞
|v(N ) (x , x3 , t)|2 dx
|∇ U(x , t)|2 dx
×
|∇ U(x , t)|2 dx
1/2 dx3
σ
σ
1/2
|∇ v(N ) (x , x3 , t)|2 dx
1/2
σ
1/2 dx3
σ
≤
|v(N ) (x , x3 , t)|4 dx
∞
−∞
c |v(N ) (x , x3 , t)|2 dx |∇ U(x , t)|2 dx σ
σ
ν + |∇ v(N ) (x , x3 , t)|2 dx dx3 4 σ ν (N ) 2 ≤ cM1 (t) |v (x, t)| dx + |∇v(N ) (x, t)|2 dx, 4 # # where M1 (t) = σ |∇ U(x , t)|2 dx. Therefore, from (4.156) follows the estimate d |v(N ) (x, t)|2 dx + ν |∇v(N ) (x, t)|2 dx ≤ c M1 (t) |v(N ) (x, t)|2 dx dt # # # |h(x, t)|2 dx. +c #
T
Since 0 M1 (t)dt ≤ U; W22,1 ( T )2 , the last relation yields, in view of Gronwall’s inequality, the estimate (4.158). Now, let us estimate the first two integrals at the right-hand side of (4.157). Arguing as before and using Sobolev embedding theorem (see Lemma I.1.2), we obtain " " " " ) " (v(N ) (x, t) · ∇)U(x , t) · v(N " t (x, t) dx " " #
≤c +
∞
−∞
1 4
#
|v
(N )
4
(x , x3 , t)| dx
1/2
(N )
|∇ U(x , t)| dx
σ
|vt
4
1/2 dx3
σ
(x, t)|2 dx
≤ cU(·, t); W22 (σ )2
(N )
|∇ v #
1 (x, t)| dx + 4
2
#
(N )
|vt
(x, t)|2 dx;
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
631
" " " " ) " (U(x , t) · ∇)v(N ) (x, t) · v(N " (x, t) dx t " " # 1 (N ) ≤ |v (x, t)|2 dx 4 # t |∇v(N ) (x, t)|2 dx + c sup |U(x , t)|2 x ∈σ
#
≤ cU(·, t); W22 (σ )2
|∇v(N ) (x, t)|2 dx + #
1 4
#
(N )
|vt
(x, t)|2 dx.
From latter displayed relations and from (4.157) it follows that d (N ) 2 |vt (x, t)| dx + |∇v(N ) (x, t)|2 dx dt # # |h(x, t)|2 dx, ≤ cM2 (t) |∇v(N ) (x, t)|2 dx + c #
(4.160)
#
T where M2 (t) = U(·, t); W22 (σ )2 . Obviously, 0 M2 (t) dt ≤ U; W22,1 ( T )2 . Therefore, by Gronwall inequality, relation (4.160) yields (4.159). ◦
◦
T HEOREM 4.11. Let ∂# ∈ C 2 , U0 ∈ W 12 (σ ), f0 ∈ L2 ( T ), ( u0 ∈ W 12 (#), h ∈ L2 (#T ), T ∈ (0, ∞]. Moreover, assume that there hold the compatibility conditions div U0 (x ) = 0, div( u0 (x) = 0, U03 (x ) dx = F (0). (4.161) σ
Then problem (4.154) admits a unique solution (v(x, t), p 8(x, t)) such that v ∈ W22,1 (#T ), T ∇p 8 ∈ L2 (# ). There holds the estimate v; W22,1 (# × (0, t))2 + ∇ p 8; L2 (# × (0, t))2 ≤ c(1 + A0 )(A0 ecA0 + 1)A1 (t),
(4.162)
where the constant c is independent of t ∈ [0, T ] and T . P ROOF. From estimates (4.158), (4.159) we conclude that there exists a subsequence {v(Nl ) (x, t)} of Galerkin approximations which is weakly convergent in the space ◦ 1,1 W 2 (#T ). Using standard arguments (e.g. [28]) it is easy to show that the limit vector◦
u0 (x) and the integral identity field v ∈ W 21,1 (#T ) satisfies the initial condition v(x, 0) = ( t 0
vτ (x, τ ) · η(x, τ ) dxdτ + ν
#
+
t 0
#
t 0
∇v(x, τ ) · ∇η(x, τ ) dxdτ #
(U(x , τ ) · ∇)v(x, τ ) · η(x, τ )dxdτ
(4.163)
632
K. Pileckas
+
t 0
=
#
t 0
(v(x, τ ) · ∇)U(x , τ ) · η(x, τ )dxdτ
h(x, τ ) · η(x, τ )dxdτ
# ◦
for every divergence-free η ∈ W 21,0 (#T ). Obviously, for v(x, t) remain valid estimates (4.158), (4.159). From (4.163) we conclude that for almost all t ∈ [0, T ] holds the identity ν ∇v(x, t) · ∇ξ (x) dx = − vt (x, t) · ξ (x) dx + h(x, t) · ξ (x)dx #
#
−
#
(U(x , t) · ∇)v(x, t)
#
+ (v(x, t) · ∇)U(x , t) · ξ (x) dx. Therefore, v(x, t) could be treated as a weak solution of the steady Stokes problem (II.2.1) with the right-hand side equal to vt (x, t) + h(x, t) − (U(x , t) · ∇)v(x, t) − (v(x, t) · ∇)U(x , t). According to Theorem II.2.10, v(·, t) ∈ W22 (#) and there exists a pressure function p 8(x, t) such that ∇ p 8(·, t) ∈ L2 (#). Moreover, (v(x, t), p 8(x, t)) satisfy system (II.2.1) almost everywhere in # and there holds the estimate v(·, t); W22 (#)2 + ∇ p 8(·, t); L2 (#)2 ≤ cvt (·, t) + h(·, t) − (U(·, t) · ∇)v(·, t) − (v(·, t) · ∇)U(·, t); L2 (#)2 . (4.164) It follows from considerations of Lemma 4.11 that (U(·, t) · ∇)v(·, t); L2 (#)2 + (v(·, t) · ∇)U(·, t); L2 (#)2 ≤ U(·, t); W22 (#)2 ∇v(·, t); L2 (#)2 . Integrating (4.164) over t and using the last estimate, we derive, in view of (4.158), (4.159), the inequality t t 2 2 v(·, τ ); W2 (#) dτ + ∇ p 8(·, τ ); L2 (#)2 dτ 0
0
t
≤c
vτ (·, τ ); L2 (#)2 dτ +
0
+ 0
t
h(·, τ ); L2 (#)2 dτ
0 t
U(·, τ ); W22 (#)2 ∇v(·, τ ); L2 (#)2 dτ
t U(·, τ ); W22 (#)2 dτ ≤ c B1 (t) + A1 (t) + sup ∇v(·, τ ); L2 (#)2 τ ∈(0,t)
0
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
≤ c(B1 (t) + A1 (t) + A0 B1 (t)) ≤ c(1 + A0 ) A0 ecA0 + 1 A1 (t).
633
(4.165)
Inequalities (4.158), (4.159) and (4.165) yield (4.162). The uniqueness of the solution could be proved just in the same way as in Theorem 4.9. ◦
u0 ∈ T HEOREM 4.12. Let ∂# ∈ C 2 , U0 ∈ W 12 (σ ), f0 ∈ L2 ( T ), h ∈ L2,β (#T ), ( ◦
1 (#) ∩ W2,β W 12 (#), β ≥ 0, and let there hold compatibility conditions (4.161). If the number γ∗ in the inequality (I.1.143 ) for the weight-function Eβ (x) is sufficiently small, then for the solution (v(x, t), p 8(x, t)) of problem (4.154) holds the estimate
2,1 v; W2,β (#T )2 + ∇ p 8; L2,β (#T )2 ≤ c(1 + A0 ) A0 ecA0 + 1 B2 ,
(4.166)
1 (#)2 + h; L T 2 where B2 = ( u0 ; W2,β 2,β (# ) .
P ROOF. We argue as in the proof of Theorem 4.2. Let us multiply equations (4.154) by (k) (k) Eβ (x3 )v(x, t) + W(k) (x, t), where Eβ (x3 ) is the “step” weight function (I.1.16) and W(k) (x, t) is a vector-field constructed in Lemma I.1.17. Integrating the by parts over #, we get 1 d 2 dt
#
(k) Eβ (x3 )|v(x, t)|2 dx
+ν
#
(k)
Eβ (x3 )|∇v(x, τ )|2 dx
vt (x, t) · W(k) (x, t)dx
=− #
−ν #
−
#
− #
+ #
(k) ∇v(x, t) · ∇Eβ (x3 )v(x, t) + ∇W(k) (x, t) dx
(k) (v(x, t) · ∇)U(x , t) · Eβ (x3 )v(x, t) + W(k) (x, t) dx (k) (U(x, t) · ∇)v(x , t) · Eβ (x3 )v(x, t) + W(k) (x, t) dx 6
(k) h(x, t) · Eβ (x3 )v(x, t) + W(k) (x, t) dx = Ij (t).
(4.167)
j =1
As in Theorem 4.2, we derive ν (k) + cγ∗ Eβ (x3 )|∇v(x, t)|2 dx |I1 (t)| + |I2 (t)| + |I3 (t)| + |I6 (t)| ≤ 8 # (k) Eβ (x3 )|vt (x, t)|2 dx + cγ∗
+c #
#
(k)
Eβ (x3 )|h(x, t)|2 dx.
634
K. Pileckas
Let us estimate the integrals I4 (t) and I5 (t). Using Lemma I.1.17 for the estimates of integrals containing the function W(k) (x, t), we obtain ∞ (k) |I4 (t)| ≤ Eβ (x3 ) |v(x, t)|2 |∇U(x , t)| dx dx3 −∞
σ
" " " " (k)
" + " (v(x, t) · ∇)W (x, t) · U(x , t)dx "" # ν (k) (k) ≤ cM1 (t) Eβ (x3 )|v(x, t)|2 dx + Eβ (x3 )|∇v(x, t)|2 dx 8 # # 1 (k) + sup |U(x , t)|2 Eβ (x3 )|v(x, t)|2 dx 2 x ∈σ # 1 (k) + E (x3 )|∇W(k) (x, t)|2 dx 2 # −β (k) ≤ c(M1 (t) + M2 (t)) Eβ (x3 )|v(x, t)|2 dx +
ν + cγ∗ 8
#
(k)
#
Eβ (x3 )|∇v(x, t)|2 dx
and, analogously,
(k)
|I5 (t)| ≤ c(M1 (t) + M2 (t))
ν + + cγ∗ 8
#
Eβ (x3 )|v(x, t)|2 dx
(k)
#
Eβ (x3 )|∇v(x, t)|2 dx.
Therefore, from (4.167) follows the relation 1 d (k) (k) 2 E (x3 )|v(x, t)| dx + ν Eβ (x3 )|∇v(x, τ )|2 dx 2 dt # β # (k) (k) Eβ (x3 )|vt (x, t)|2 dx ≤ c(M1 (t) + M2 (t)) Eβ (x3 )|v(x, t)|2 dx + cγ∗ + cγ∗ + +c #
3ν 8
#
#
#
(k)
Eβ (x3 )|∇v(x, t)|2 dx
(k)
Eβ (x3 )|h(x, t)|2 dx. (k)
(4.168) (k)
Now, multiplying (4.154) by Eβ (x3 )vt (x, t) + Wt (x, t) and integrating by parts over #, we receive by the same arguments the inequality ν d (k) (k) 2 Eβ (x3 )|vt (x, t)| dxd + E (x3 )|∇v(x, t)|2 dx 2 dt # β #
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
635
3 (k) (k) Eβ (x3 )|∇v(x, t)|2 dx + cγ∗ + Eβ (x3 )|vt (x, t)|2 dx 8 # # (k) (k) Eβ (x3 )|∇v(x, τ )|2 dx + c Eβ (x3 )|h(x, t)|2 dx. (4.169) + cγ∗
≤ cM2 (t)
#
#
Summing inequalities (4.168) and (4.169) yields 1 d 2 dt
(k)
#
Eβ (x3 )(|v(x, t)|2 + ν|∇v(x, t)|2 ) dx
(k)
+ #
Eβ (x3 )(|vt (x, t)|2 + ν|∇v(x, t)|2 ) dx
(k)
≤ cM(t) #
Eβ (x3 )(|v(x, t)|2 + |∇v(x, t)|2 ) dx
3 (k) + c 1 γ∗ + E (x3 )(|vt (x, t)|2 + |∇v(x, t)|2 ) dx 8 # β +c Eβ(k) (x3 )|h(x, t)|2 dx,
#
where M(t) = M1 (t) + M2 (t). If γ∗ ≤ 8c11 min{ν, 1}, then from the last inequality we conclude d (k) E (x3 )(|v(x, t)|2 + ν|∇v(x, t)|2 ) dx dt # β Eβ(k) (x3 )(|vt (x, t)|2 + ν|∇v(x, t)|2 ) dx + #
(k)
≤ cM(t) #
+c #
Eβ (x3 )(|v(x, t)|2 + |∇v(x, t)|2 ) dx
(k)
Eβ (x3 )|h(x, t)|2 dx.
Thus, the Gronwall inequality yields #
(k)
Eβ (x3 )(|v(x, t)|2 + ν|∇v(x, t)|2 ) dx +
t 0
#
(k)
Eβ (x3 )(|vτ (x, τ )|2 + ν|∇v(x, τ )|2 ) dxdτ
(k) cA0 ≤ c A0 e +1 Eβ (x3 )(|( u0 (x)|2 + |∇( u0 (x)|2 ) dx +
t 0
#
#
(k) Eβ (x3 )|h(x, τ )|2 dxdτ
≤ c A0 ecA0 + 1 B2 .
(4.170)
636
K. Pileckas
The right-hand side of (4.170) does not depend on k. Passing in (4.170) to a limit as k → ∞ gives 1,1 1 (#)2 + v; W2,β (#T )2 ≤ c A0 ecA0 + 1 B2 . v(·, t); W2,β
(4.171)
Now consider (v(x, t), p 8(x, t)) as a solution to the Stokes problem ⎧ −νv(x, t) + ∇ p 8(x, t) = h(x, t) − vt (x, t) − (v(x, t) · ∇)U(x , t) ⎪ ⎪ ⎪ ⎪ ⎪ −(U(x , t) · ∇)v(x, t), ⎪ ⎨ div v(x, t) = 0, " ⎪ ⎪ ⎪ v(x, t)"∂# = 0, ⎪ ⎪ ⎪ ⎩ σ v(x, t) · n(x) ds = 0. By Theorem III.3.2, there holds the estimate 2 (#)2 + ∇p(·, t); L2,β (#)2 v(·, t); W2,β
≤ c(h(·, t); L2,β (#)2 + vt (·, t); L2,β (#)2 + (v(·, t) · ∇)U(·, t); L2,β (#)2 + (U(·, t) · ∇)v(·, t); L2,β (#)2 ).
(4.172)
Integrating inequality (4.172) with respect to t and using (4.171), we derive 0
t
2 v(·, τ ); W2,β (#)2 dτ
+
t
∇ p 8(·, τ ); L2,β (#)2 dτ
0
t 1 U(·, τ ); W22 (σ )2 v(·, τ ); W2,β (#)2 dτ ≤ c A0 ecA0 + 1 B2 + 0
1 (#)2 ≤ c A0 ecA0 + 1 B2 + sup v(·, τ ); W2,β × 0
t∈[0,T ]
t
U(·, τ ); W22 (σ )2 dτ
≤ c(1 + A0 ) A0 ecA0 + 1 B2 .
The last estimate together with (4.171) is equivalent to (4.166).
Now we are in a position to prove the existence result for the nonlinear Navier–Stokes problem (4.153). T HEOREM 4.13. Let ∂# ∈ C 2 , F ∈ W21 (0, T ), and let the initial data u0 (x) and the ex◦
2 (#) ∩ ternal force f(x, t) are represented in the form (4.151) with ( u0 ∈ W2,β W 12 (#),
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
637
◦
( f ∈ L2,β (#T ), β ≥ 0, U0 ∈ W 12 (σ ), f0 ∈ L2 ( T ), T ∈ (0, ∞]. Moreover, assume that there hold compatibility conditions (4.161) and that the number γ∗ in inequality (I.1.143 ) for the weight function Eβ (x) is sufficiently small. If 2 1 f; L2,β (#T )2 + ( u0 ; W2,β (#)2 < 1, c1 c2 (1 + A0 )2 A0 ecA0 + 1 ( (4.173) where c1 , c2 are absolute constants defined below, then problem (4.153) admits a unique 2,1 (#T ), ∇ p 8 ∈ L2,β (#T ). There holds the estimate solution v ∈ W2,β 2,1 v; W2,β (#T )2 + ∇ p 8; L2,β (#T )2 1 ≤ c(1 + A0 ) A0 ecA0 + 1 ( f; L2,β (#T )2 + ( u0 ; W2,β (#)2 .
(4.174)
P ROOF. The arguments are similar to those of Theorem 4.9. We prove the existence of the solution to (4.153) by the method of successive approximations. Let us put v(0) (x, t) = 0, p 8(0) (x, t) = 0, and define the successive approximations recurrently as solutions of linear problems ⎧ (l+1) v (x, t) − νv(l+1) (x, t) + ∇ p 8(l+1) (x, t) + (U(x , t) · ∇)v(l+1) (x, t) ⎪ ⎪ ⎪ t ⎪ ⎪ f(x, t) − (v(l) (x, t) · ∇)v(l) (x, t), +(v(l+1) (x, t) · ∇)U(x , t) =( ⎪ ⎨ div v(l+1) (x, t) = 0, ⎪ " ⎪ ⎪ ⎪ v(l+1) (x, 0) = ( u0 (x), v(l+1) (x, t)"∂# = 0, ⎪ ⎪ (l+1) ⎩ (x, t) · n(x) ds = 0. σv In virtue of Lemma 4.7, we have 2,1 (v(l) · ∇)v(l) ; L2,β (# × (0, t))2 ≤ cv(l) ; W2,β (# × (0, t))4
∀t ∈ (0, T ].
8(l+1) (x, t)) are well Therefore, according to Theorem 4.12, approximations (v(l+1) (x, t), p defined for all l and satisfy the estimates 2,1 (#T )2 + ∇ p 8(l+1) ; L2,β (#T )2 v(l+1) ; W2,β 1 ≤ c1 (1 + A0 ) A0 ecA0 + 1 ( f; L2,β (#T )2 + ( u0 ; W2,β (#)2 2,1 (#T )4 + c2 (1 + A0 ) A0 ecA0 + 1 v(l) ; W2,β 2,1 =( α0 + ( α1 v(l) ; W2,β (#T )4 ,
(4.175)
638
K. Pileckas
1 (#)2 ), ( where ( α0 = c1 (1 + A0 )(A0 ecA0 + 1)(( f; L2,β (#T )2 + ( u0 ; W2,β α1 = c2 (1 + cA 2 0 A0 )(A0 e + 1). If 4( α0( α1 < 1, the quadratic equation ( α1 ρ − ρ + ( α0 = 0 has positive roots minimal r0 of which is given by the formula
r0 =
2( α0 . √ 1 + 1 − 4( α0( α1
Condition 4( α0( α1 < 1 is satisfied because of assumption (4.173). From (4.175) it follows 2,1 (#T )2 ≤ r0 , then also that, if v(l) ; W2,β 2,1 v(l+1) ; W2,β (#T )2 + ∇ p 8(l+1) ; L2,β (#T )2 ≤ r0 .
(4.176)
2,1 Since, obviously v(0) ; W2,β (#T )2 ≤ r0 , we conclude that (4.176) is valid ∀l ≥ 0. 8(l) (x, t))} to the solution (v(x, t), p 8(x, t)) The convergence of the sequence {(v(l) (x, t), p of problem (4.153) and the uniqueness of this solution could be proved just repeating the arguments of Theorem 4.9.
R EMARK 4.7. If F (t) and f0 (x , t) admit representations F (t) = F [1] + F [2] (t),
[2]
f0 (x , t) = f[1] 0 (x ) + f0 (x , t),
[2] ∈ W 1 (0, ∞), f[2] ∈ L (σ × (0, ∞)), then from Theowhere F ∈ R, f[1] 2 0 ∈ L2 (σ ), F 0 2 rem 4.13 follows only long time existence (i.e., for arbitrary finite time interval [0, T ]). However, if the steady part of the data is small (i.e., |F [1] | and f[1] 0 ; L2 (σ ) are sufficiently small), then the result of Theorem 4.13 is true also for T = ∞.
4.6. Uniqueness of the solution to problem (4.2) In Sections 4.4 and 4.5 we have proved the existence of a unique solution to problem (4.2) which has the asymptotic representation u(x, t) = v(x, t) + V(x, t),
p(x, t) = p 8(x, t) + P (x, t),
where (V(x, t), P (x, t)) coincides in each outlet to infinity j with the Poiseuille solution (U(j ) ((j ) , t), P (j ) (x (j ) , t)) corresponding to this outlet. Below we show that problem (4.2) cannot have other solutions in a class of functions that are “bounded at infinity”. In particular, from these results follows the uniqueness of the time-dependent Poiseuille flow in a straight cylinder. Note that for the steady case the uniqueness of Poiseuille flow is not known and perhaps is not true. Denote by W22,1 (T ) the space of functions having the finite norm 1/2 J
u; W22,1 (T ) = u; W22,1 (T(3) )2 + sup u; W22,1 (ωjTs )2 . j =1 s≥0
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
639
Obviously, solutions of problem (4.2) which were found in Sections 4.4 and 4.5 belong to W22,1 (T ). T HEOREM 4.14. Let ⊂ R3 . Problem (4.2) cannot have two different solutions in the space W22,1 (T ). P ROOF. Let (u[1] (x, t), p [1] (x, t)) and (u[2] (x, t), p [2] (x, t)) be two solutions to problem (4.1). The differences w(x, t) = u[1] (x, t) − u[2] (x, t), q(x, t) = p [1] (x, t) − p [2] (x, t) satisfy the linear system ⎧ wt − νw + ∇q = −(w · ∇)u[1] − (u[2] · ∇)w, ⎪ ⎪ ⎪ ⎪ ⎨ div w = 0, " (4.177) " w(x, t) ∂ = 0, w(x, 0) = 0, ⎪ ⎪ ⎪ ⎪ ⎩ w(x, t) · n(x) ds = 0, j = 1, . . . , J. σj
Denote 3 E−γ ∗ (x) =
x ∈ (0) ,
1, e
(j ) −γ∗ xn
,
(4.178)
x ∈ j , j = 1, . . . , J,
2,1 where γ ∗ = (γ∗ , . . . , γ∗ ), γ∗ > 0. It is evident that W22,1 (T ) ⊂ W2,−γ (T ). Therefore, ∗
2,1 we may treat w ∈ W2,−γ (T ) as a weak solution of linear problem (4.18) with the right ∗ hand side f(x, t) = −(u[2] (x, t) · ∇)w(x, t) − (w(x, t) · ∇)u[1] (x, t) and u0 (x) = 0. Let us show that f ∈ L2,−γ ∗ (T ). We have
t 0
E−γ ∗ (x)|u[2] (x, τ )|2 |∇w(x, τ )|2 dxdτ
t
≤
0
+
|u[2] (x, τ )|2 |∇w(x, τ )|2 dxdτ
(3) J ∞ t
j =1 s=0 0
(j )
e−γ∗ x3 |u[2] (x, τ )|2 |∇w(x, τ )|2 dxdτ.
ωj s
In virtue of Lemma I.1.5 (see inequality (I.1.10)), there hold the estimates t 0
(j )
e−γ∗ x3 |u[2] (x, τ )|2 |∇w(x, τ )|2 dxdτ
ωj s
≤c
t
0
≤c
0
t
e−γ∗ s u[2] (·, τ ); L∞ (ωj s )2 ∇w(·, τ ); L2 (ωj s )2 dτ u[2] (·, τ ); W21 (ωj s )u[2] (·, τ ); W22 (ωj s )
(4.179)
640
K. Pileckas
× e−
γ∗ (j ) 2 x3
∇w(·, τ ); L2 (ωj s )2 dτ γ∗ (j ) x3
≤ c sup u[2] (·, τ ); W21 (ωj s ) sup e− 2 τ ∈[0,t]
t
× 0
τ ∈[0,t]
γ∗ (j ) x3
u[2] (·, τ ); W22 (ωj s )e− 2
≤ ε sup u[2] (·, τ ); W21 (ωj s )2 τ ∈[0,t]
t
+ cε
t
∇w(·, τ ); L2 (ωj s ) dτ
e−
γ∗ (j ) 2 x3
∇w(·, τ ); L2 (ωj s )2 dτ
0
u[2] (·, τ ); W22 (ωj s )2 dτ sup e− τ ∈[0,t]
0
∇w(·, τ ); L2 (ωj s )
γ∗ (j ) 2 x3
∇w(·, τ ); L2 (ωj s )2 .
Therefore, ∞ t
s=0 0
(j )
e−γ∗ x3 |u[2] (x, τ )|2 |∇w(x, τ )|2 dxdτ ωj s
≤ ε sup( sup u[2] (·, τ ); W21 (ωj s )2 ) s≥0 τ ∈[0,T ]
×
∞
s=0 0
t
e−
t
+ cε sup s≥0
×
∞
0
γ∗ (j ) 2 x3
∇w(·, τ ); L2 (ωj s )2 dτ
u[2] (·, τ ); W22 (ωj s )2 dτ
sup e−
γ∗ (j ) 2 x3
s=0 τ ∈[0,t]
≤ cεu
[2]
; W22,1 (T )2
∇w(·, τ ); L2 (ωj s )2
t
e−
γ∗ (j ) 2 x3
∞
e−
γ∗ (j ) 2 x3
|u[2] (x, τ )|2 |∇w(x, τ )|2 dxdτ (3)
γ∗ (j ) 2 x3
w; W22,1 (ωj s × (0, t))2
∇w(·, τ ); L2 (j )2 dτ
0
Analogously, we get that
0
e−
s=0 t
+ cε u[2] ; W22,1 ( × (0, t))2 e−
t
∇w(·, τ ); L2 (j )2 dτ
0
+ cε u[2] ; W22,1 ( × (0, t))2 ≤ cεu[2] ; W22,1 (T )2
γ∗ (j ) 2 x3
w; W22,1 (j × (0, t))2 .
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
≤ cεu[2] ; W22,1 (T )2 + cε u
[2]
t
641
∇w(·, τ ); L2 ((3) )2 dτ
0
; W22,1 ( × (0, t))2 w; W22,1 ((3)
× (0, t))2 .
(4.180)
Inequalities (4.179)–(4.180) yield (u[2] · ∇) · w; L2,−γ ∗ ( × (0, t))2 ≤ cεu[2] ; W22,1 (T )2 e−
γ∗ (j ) 2 x3
w; W22,1 ( × (0, t))2
+ cε u[2] ; W22,1 ( × (0, t))2 e−
γ∗ (j ) 2 x3
w; W22,1 ( × (0, t))2 .
(4.181)
Consider now the term (w(x, t) · ∇)u[1] (x, t). Using inequality (I.1.9) instead of (I.1.10) and arguing as above, we obtain (w · ∇) · u[1] ; L2,−γ ∗ ( × (0, t))2 ≤ cεu[1] ; W22,1 (T )2 e−
γ∗ (j ) 2 x3
w; W22,1 ( × (0, t))2
+ cε u[1] ; W22,1 ( × (0, t))2 e−
γ∗ (j ) 2 x3
w; W22,1 ( × (0, t))2 .
(4.182)
Thus, f ∈ L2,−γ ∗ (T ) and, if γ∗ is sufficiently small (i.e., γ∗ satisfies the conditions of Theorem 4.2), then, according to Theorem 4.2 and Remark 4.2, for w(x, t) holds estimate (4.23) which together with (4.181), (4.182) yields 2,1 w; W2,−γ ( × (0, t))2 ∗
≤ cf; L2,−γ ∗ ( × (0, t))2 2,1 ≤ c1 ε u[1] ; W22,1 (T )2 + u[2] ; W22,1 (T )2 w; W2,−γ ( × (0, t))2 ∗ + cε u[1] ; W22,1 ( × (0, t))2 2,1 + u[2] ; W22,1 ( × (0, t))2 )w; W2,−γ ( × (0, t))2 . ∗
Fixing in the last inequality ε≤
1 2c1 (u[1] ; W22,1 (T )2
+ u[2] ; W22,1 (T )2 )
furnishes 2,1 ( × (0, t))2 w; W2,−γ ∗ ≤ c2 u[1] ; W22,1 ( × (0, t))2
2,1 + u[2] ; W22,1 ( × (0, t))2 w; W2,−γ ( × (0, t))2 . ∗
(4.183)
642
K. Pileckas
Let t1 be such that c2 (u[1] ; W22,1 ( × (0, t))2 + u[2] ; W22,1 ( × (0, t))2 ) < 1. Then 2,1 relation (4.183) yields w; W2,−γ ( × (0, t1 ))2 = 0 and, hence, w(x, t) = 0 in × ∗
[0, t1 ]. If t1 < T , we repeat our considerations for t ∈ [t1 , t2 ], where c2 (u[1] ; W22,1 ( × (t1 , t2 ))2 + u[1] ; W22,1 ( × (t1 , t2 ))2 ) < 1, and get that w(x, t) = 0 in × [0, t2 ], and so on. By finite number of steps we deduce that w(x, t) = 0 in × [0, T ] for any T < ∞. Analogously, using Lemma I.1.6 instead of Lemma I.1.5 we prove T HEOREM 4.15. Let ⊂ R2 . Problem (4.2) cannot have two different solutions in the space W22,1 (T ).
4.7. Remarks on weak Hopf’s solution The global unique solvability of problem (4.62) is proved in Section 4.5 for a threedimensional domain assuming that data are sufficiently small (see conditions (4.139)). On the other hand, it could be proved (using the methods similar to [28]) that for arbitrary ◦ data there exists a weak global Hopf’s solution vH (x, t) such that vH ∈ W 21,0 ( × (0, ∞)) and vH (·, t); L2 () = ϕ(t) ∈ L∞ (0, ∞). Therefore, for any ε > 0 there exists such t∗ = t∗ (ε) > 0 that ◦
vH (·, t∗ ); W 12 () ≤ ε. If t∗ is sufficiently large the norm 8 f; L2 ( × (t∗ , ∞) also becomes small. Moreover, since U(j ) ∈ W22,1 (σj × (0, ∞)), W ∈ W22,1 ( × (0, ∞)), we get J
U(j ) ; W22,1 (σj × (t∗ , ∞)) + W; W22,1 ( × (t∗ , ∞)) ≤ ε,
j =1 ◦
if t∗ is sufficiently large. Considering vH (·, t∗ ) ∈ W 12 () as an initial data for problem (4.62), we see that for sufficiently large t∗ there holds the condition (4.150) and, therefore, by Theorem 4.9 on interval (t∗ , ∞) this problem admits a unique solution v ∈ W22,1 ( × (t∗ , ∞)). Comparing the regular solution v(x, t) with the weak Hopf’s solution vH (x, t), it is possible to prove that they coincide on interval [t∗ , ∞). Thus, we may conclude that in a three-dimensional domain with cylindrical outlets to infinity also for large data there exists a weak solution to Navier–Stokes problem having the form u(x, t) = V(x, t) + vH (x, t) (it may be not unique) and that all such solutions tend in outlets j to the corresponding Poiseuille flows in the sense that 0
∞
(|vH (x, t)|2 + |∇vH (x, t)|2 ) dxdt < ∞.
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
643
Moreover, for t ≥ t∗ , these solutions become regular and
∞ 9 9 ∂v
t∗
9 9
H (·, t)
∂t
92 9 2 2 dxdt < ∞. + v (·, t); W () ; L2 ()9 H 2 9
Note that we cannot prove better decay of vH (x, t) for large t, i.e., we cannot prove that for t ≥ t∗ the solution vH (x, t) belongs to certain weighted space of vanishing at infinity function. Such proof requires a regularity of the solution vH (x, t). We do know that vH ∈ ◦ W22,1 ( × (t∗ , ∞)). However, we do not know whether the initial data vH (·, t∗ ) ∈ W 12 (), which we use to obtain the regularity of the weak solution, belongs to some weighted space 1 () with β > 0, j = 1, . . . , J . W2,β j Acknowledgments I would like to express my cordial gratitude to Professors H. Beirão da Veiga, G.P. Galdi and A.M. Robertson for providing me with preprints of [9] and [22] before their publication and, particularly, to Professor V.A. Solonnikov for informing me (private communication) about an elegant method which allows to obtain global a priori estimate for W22,1 -norm of solutions to two-dimensional time-dependent Navier–Stokes system. These ideas of Professor V.A. Solonnikov are used in Section 4.4 of Chapter IV.
References [1] F. Abergel and J.L. Bona, A mathematical theory for viscous free–surface flows over a perturbed plane, Arch. Rational Mech. Anal. 118 (1992) 71–93. [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying generalboundary conditions II, Commun. Pure Appl. Math. 17 (1964) 35– 92. [3] R.A. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London (1975). [4] C.J. Amick, Steady solutions of the Navier–Stokes equations in unbounded channels and pipes, Ann. Scuola Norm. Sup. Pisa 4 (1977) 473–513. [5] C.J. Amick, Properties of steady Navier–Stokes solutions for certain unbounded channels and pipes, Nonlinear analysis, Theory, Meth., Appl. 2 (1978) 689–720. [6] C.J. Amick and L.E. Fraenkel, Steady solutions of the Navier–Stokes equations representing plane flow in channels of various types, Acta Math. 144 (1980) 81–152. [7] A.V. Babin, Asymptotic expansion at infinity of a strongly perturbed Poiseuille flow, Advances in Soviet Mathematics 10 (1992) 1–83. [8] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge (2002). [9] H. Beirao da Veiga, On time-periodic solutions of the Navier–Stokes equations in an unbounded cylindrical domains. Leray’s problem for periodic flows, Arch. Rat. Mech. Anal. 178 (3) (2005) 301–325. [10] S. Blazy, S.A. Nazarov and M. Specovius–Neugebauer, Artificial boundary conditions of pressure type for viscous flow in a system f pipes J. Math. Fluid Mech. (to appear). [11] M.E. Bogovskii, Solutions of some problems of vector analysis related to operators div and grad, Proc. Semin. S.L. Sobolev 1 (1980) 5–40 (in Russian). [12] W. Borchers and K. Pileckas, Existence, uniqueness and asymptotics of steady jets, Arch. Rational Mech. Anal. 120 (1992) 1–49.
644
K. Pileckas
[13] W. Borchers, G.P. Galdi and K. Pileckas, On the uniqueness of Leray–Hopf solutions for the flow through an aperture, Arch. Rational Mech. Anal. 122 (1993) 19–33. [14] H. Chang, The steady Navier–Stokes problem for low Reynolds number viscous jets, Proceedings of the conference “The Navier–Stokes equations: Theory and Numerical Methods,” Oberwolfach 1991, Lecture Notes in Math. 1530, J.G. Heywood, K. Masuda, R. Rautmann and V.A. Solonnikov, eds. (1992), 85–96. [15] V. Coscia and M.C. Patria, Existence, uniqueness and asymptotic decay of steady flow of an incompressible fluid in a half space, Stability Appl. Anal. Cont. Media 2 (1992) 101–127. [16] L.E. Fraenkel, Laminar flow in symmetrical channels with slightly curved walls. I. On the Jeffery–Hamel solutions for flow between plane walls, Proc. R. Soc. London, Ser. A. 267 (1962) 119–138. [17] L.E. Fraenkel, Laminar flow in symmetrical channels with slightly curved walls. II. An asymptotic series for the stream function, Proc. R. Soc. London, Ser. A. 272 (1963) 406–428. [18] L.E. Fraenkel, On the theory of laminar flow in channels of certain class, Proc. Camb. Phil. Soc. 73 (1973) 361–390. [19] L.E. Fraenkel and P.M. Eagles, On the theory of of laminar flow in channels of certain class, Math. Proc. Camb. Phil. Soc. 77 (1975) 199–224. [20] G.P. Galdi, An Introduction to the Mathematical Theory of Navier–Stokes equation, Vol. I and II, Springer Tracts in Nat. Ph. 38, 39 (1994). [21] G.P. Galdi, M. Padula and V.A. Solonnikov, Existence, uniqueness and asymptotic behavior of solutions of steady-state Navier–Stokes equations in a plane aperture domain, Indiana Univ. Math. J. 45 (4) (1996) 961–995. [22] G.P. Galdi and A.M. Robertson, The relation between flow rate and axial pressure gradient for timeperiodic Poiseuille flow in a pipe, J. Math. Fluid Mech. 7 (supplement No. 2) (2005) 215–223. [23] J.G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math. 136 (1976) 61–102. [24] L.V. Kapitanskii, Stationary solutions of the Navier–Stokes equations in periodic tubes, Zapiski Nauch. Semin. LOMI 115 (1982) 104–113. English Transl.: J. Sov. Math. 28 (1983) 689–695. [25] L.V. Kapitanskii and K. Pileckas, On spaces of solenoidal vector fields and boundary value problems for the Navier–Stokes equations in domains with noncompact boundaries, Trudy Mat. Inst. Steklov 159 (1983) 5–36. English Transl.: Proc. Math. Inst. Steklov 159 (2) (1984) 3–34. [26] L.V. Kapitanskii and K. Pileckas, Certain problems of vector analysis, Zapiski Nauchn. Sem. LOMI 138 (1984) 65–85. English Transl.: J. Sov. Math. 32 (5) (1986) 469–483. [27] O.A. Ladyzhenskaya, Stationary motion of a viscous incompressible fluid in a pipe, Dokl. Akad. Nauk SSSR 124 (1959) 551–553 (in Russian). [28] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, London, Paris (1969). [29] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translation of Mathematical Monographs 23, AMS, Providence (1968). [30] O.A. Ladyzhenskaya and V.A. Solonnikov, On some problems of vector analysis and generalized formulations of boundary value problems for the Navier–Stokes equations, Zapiski Nauchn. Sem. LOMI 59 (1976) 81–116. English Transl.: J. Sov. Math. 10 (2) (1978) 257–285. [31] O.A. Ladyzhenskaya and V.A. Solonnikov, On the solvability of boundary value problems for the Navier– Stokes equations in regions with noncompact boundaries, Vestnik Leningrad. Univ. 13 (Ser. Mat. Mekh. Astr. vyp. 3) (1977) 39–47. English Transl.: Vestnik Leningrad Univ. Math. 10 (1982) 271–280. [32] O.A. Ladyzhenskaya and V.A. Solonnikov, Determination of the solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral, Zapiski Nauchn. Sem. LOMI 96 (1980) 117–160. English Transl.: J. Sov. Math. 21 (5) (1983) 728–761. [33] O.A. Ladyzhenskaya and V.A. Solonnikov, On initial-boundary value problem for the linearized Navier– Stokes system in domains with noncompact boundaries, Trudy Mat. Inst. Steklov 159 (1983). English Transl.: Proc. Math. Inst. Steklov 159 (2) (1984) 35–40. [34] O.A. Ladyzhenskaya, V.A. Solonnnikov and H. True, Resolution des equations de Stokes et Navier–Stokes dans des tuyaux infinis, C. R. Acad. Sci. Paris 292, ser. I (1981) 251–254. [35] L.D. Landau and E.M. Lifschitz, Electrodynamics of Continuous Media, Pergamon (1960). [36] L.D. Landau and E.M. Lifschitz, Hydromechanik, Akademie-Verlag, Berlin (1971). [37] C. Lin, Spatial decay estimates and energy bounds for the Stokes flow equation, Stability Appl. Anal. Contin. Media 2 (1992) 249–264.
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
645
[38] C. Lin and L. Payne, Spatial decay bounds in the channel flow of an incompressible viscous fluid, Math. Models and Meth. in Applied Sciences 14 (6) (2004) 795–818. [39] S.G. Mikhlin, Integral Equations, Gostekhizdat, Moscow (1947) (in Russian). [40] S.A. Nazarov, Structure of solutions of elliptic boundary value problems in thin domains, Vestnik Leningrad. Univ., Ser. Mat. Mekh. Astr. 7 (1982) 65–68. English Transl.: Vestnik Leningrad Univ. Math. 15 (1983) 99–104. [41] S.A. Nazarov, Asymptotic solution of the Navier–Stokes problem on the flow of a thin layer of fluid, Siberian Math. J. 31 (2) (1990) 296–307. [42] S.A. Nazarov, On the two-dimensional aperture problem for Navier–Stokes equations, C. R. Acad. Sci. Paris, Ser. I. Math. (1996) 699–703. [43] S.A. Nazarov and K. Pileckas, On the behaviour of solutions of the Stokes and Navier–Stokes systems in domains with periodically varying section, Trudy Mat. Inst. Akad. Nauk SSSR 159 (1983) 95–102. English Transl.: Proc. Math. Inst. Steklov 159 (2) (1984) 141–154. [44] S.A. Nazarov and K. Pileckas, The Reynolds flow of a fluid in a thin three-dimesional channel, Litovskii Mat. Sb. 30 (1990) 772–783. English Transl.: Lithuanian Math. J. 30 (1990). [45] S.A. Nazarov and K. Pileckas, On noncompact free boundary problems for the plane stationary Navier– Stokes equations, J. für reine und angewandte Math. 438 (1993) 103–141. [46] S.A. Nazarov and K. Pileckas, The asymptotics of solutions to steady Stokes and Navier–Stokes equations in domains with noncompact boundaries, Rend. Sem. Math. Univ. Padova 99 (1998) 33–76. [47] S.A. Nazarov and K. Pileckas, On the solvability of the Stokes and Navier–Stokes problems in domains that are layer-like at infinity, J. Math. Fluid Mech. 1 (1) (1999) 78–116. [48] S.A. Nazarov and K. Pileckas, The asymptotic properties of the solutions to the Stokes problem in domains that are layer-like at infinity, J. Math. Fluid Mech. 1 (2) (1999) 131–167. [49] S.A. Nazarov and K. Pileckas, Asymptotic conditions at infinity for the Stokes and Navier–Stokes problems in domains with cylindrical outlets to infinity, Quaderni di Matematica, Advances in Fluid Dynamics 6 (1999) 32–132. [50] S.A. Nazarov and K. Pileckas, On the Fredholm property of the Stokes operator in a layer-like domain, ZAA 20 (1) (2001) 155–182. [51] S.A. Nazarov and B.A. Plamenevskii, Elliptic boundary value problems in domains with piecewise smooth boundaries, Walter de Gruyter and Co, Berlin (1994). [52] S.A. Nazarov, A. Sequeira and J.H. Videman, Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution, J. Math. Pures Appl. 80 (10) (2001) 1069–1098. [53] S.A. Nazarov, M. Specovius–Neugebauer and G. Thäter, Full steady Stokes system in domains with cylindrical outlets, Math. Annalen 314 (1999) 729–762. [54] S.A. Nazarov, M. Specovius–Neugebauer and G. Thäter, Quiet flows for Stokes and Navier–Stokes problems in domains with cylindrical outlets to infinity, Kyushu J. of Math. 53 (2) (1999) 369–394. [55] A. Quarteroni, Mathematical Modelling of the cardiovascular system, Proceedings of the ICM, Beijing 3 (2002) 839–850. [56] K. Pileckas, Three-dimensional solenoidal vectors, Zapiski Nauchn. Sem. LOMI 96 (1980) 237–239. English Transl.: J. Sov. Math. 21 (1983) 821–823. [57] K. Pileckas, On spaces of solenoidal vectors, Trudy Mat. Inst. Steklov 159 (1983) 137–149. English Transl.: Proc. Math. Inst. Steklov 159 (2) (1984) 141–154. [58] K. Pileckas, Existence of solutions for the Navier–Stokes equations having an infinite dissipation of energy, in a class of domains with noncompact boundaries, Zapiski Nauchn. Sem. LOMI 110 (1981) 180–202. English Transl.: J. Sov. Math. 25 (1) (1984) 932–947. [59] K. Pileckas, On the problem of motion of heavy viscous incompressible fluid with noncompact free boundary, Litovskii Mat. Sb. 28 (1988) 315–333. English Transl.: Lithuanian Math. J. 28 (2) (1988). [60] K. Pileckas, The example of nonuniqueness of the solutions to a noncompact free boundary problem for the Navier–Stokes system, Differential Equations and their Application, Vilnius 42 (1988) 59–65 (in Russian). [61] K. Pileckas, On plane motion of a viscous incompressible cappilary liquid with a noncompact free boundary, Arch. Mech. 41 (2–3) (1989) 329–342. [62] K. Pileckas, Weighted Lq -solvability of the steady Stokes system in domains with noncompact boundaries, Math. Models and Methods in Applied Sciences 6 (1) (1996) 97–136.
646
K. Pileckas
[63] K. Pileckas, Classical solvability and uniform estimates for the steady Stokes system in domains with noncompact boundaries, Math. Models and Methods in Applied Sciences 6 (2) (1996) 149–167. [64] K. Pileckas, Strong solutions of the steady nonlinear Navier–Stokes system in domains with exits to infinity, Rend. Sem. Mat. Univ. Padova 97 (1996) 72–107. [65] K. Pileckas, Weighted Lq -theory and pointwise estimates for steady Stokes and Navier–Stokes equations in domains with exits to infinity, Proceedings of Third International Conference on Navier–Stokes Equations and Related Nonlinear Problems, A. Sequeira, ed., Plenum press (1995) 33–53. [66] K. Pileckas, Recent advances in the theory of Stokes and Navier–Stokes equations in domains with noncompact boundaries, Pitman Notes in Math. 354 (1996) 30–85. [67] K. Pileckas, On asymptotics of solutions to steady Navier–Stokes equations in a layer-like domain, Mat. Sbornik 193 (12) (2002) 69–104. [68] K. Pileckas, On the behavior of the nonstationary Poiseuille solution as t → ∞, Siberian Math. J. 46 (4) (2005) 890–900. [69] K. Pileckas, On the nonstationary linearized Navier–Stokes problem in domains with cylindrical outlets to infinity, Math. Annalen 332 (2005) 395–419. [70] K. Pileckas, Existence of solutions with the prescribed flux of the Navier–Stokes system in an infinite pipe, J. Math. Fluid. Mech., First online (2005). [71] K. Pileckas, On nonstationary two-dimensional Leray’s problem for Poiseuille flow, Advances in Mathematical Sciences and Applications 16 (2006). [72] K. Pileckas, Global solvability in W22,1 -weighted spaces of the two-dimensional Navier–Stokes problem in domains with strip-like outlets to infinity, J. Math. Fluid. Mech. (2006) (to appear). [73] K. Pileckas, Solvability in weighted spaces of the three-dimensional Navier–Stokes problem in domains with cylindrical outlets to infinity, Topological Methods in Nonlinear Analysis (2006) (submitted). [74] K. Pileckas and V. Keblikas, On the existence of nonstationary Poiseuille solution, Siberian Math. J. 46 (3) (2005) 514–526. [75] K. Pileckas and J. Socolowsky, Analysis of two linearized problems modeling viscous two-layer flow, Math. Nachrichten 245 (2002) 129–166. [76] K. Pileckas and J. Socolowsky, Viscous two-fluid flows in perturbed unbounded domains, Math. Nachrichten 278 (5) (2005) 511–623. [77] K. Pileckas and V.A. Solonnikov, On stationary Stokes and Navier–Stokes systems in an open infinite channel. I., Litovskii Mat. Sb. 29 (1) (1989) 90–108. English Transl.: Lithuanian Math. J. 29 (1) (1989); II., Litovskii Mat. Sb. 29 (2) (1989) 347–367. English Transl.: Lithuanian Math. J. 29 (2) (1989). [78] K. Pileckas and M. Specovius–Neugebauer, Solvability of a noncompact free boundary problem for the stationary Navier–Stokes system. I., Litovskii Mat. Sb. 29 (3) (1989) 532–547. English Transl.: Lithuanian Math. J. 29 (3) (1989); II., Litovskii Mat. Sb. 29 (4) (1989) 773–784 (in Russian). [79] K. Pileckas and W. Zaj¸aczkowski, Global solvability for a large flux of a three-dimensional time dependent Navier–Stokes problem in a straight cylinder, Studia Mathematica (2006) (submitted). [80] K. Pileckas and L. Zaleskis, On steady three-dimensional noncompact free boundary problem for the Navier–Stokes equations, Zapiski POMI 306 (2003) 134–164. [81] A.I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, Basel (1999). [82] P.J. Rabier, Invertability of the Poiseuille linearization for stationary two-dimensional channel flows: symmetric case, J. Math. Fluid Mech. 4 (4) (2002) 327–350. [83] P.J. Rabier, Invertability of the Poiseuille linearization for stationary two-dimensional channel flows: nonsymmetric case, J. Math. Fluid Mech. 4 (4) (2002) 351–373. [84] L. Rivkind and V.A. Solonnikov, Jeffery-Hamel asymptotics for steady Navier–Stokes flow in domains with sector-like outlets to infinity, J. Math. Fluid Mech. 2 (4) (2000) 325–352. [85] V.A. Solonnikov, On the boundary value problems for systems elliptic in the sence of A. Douglis, L. Nirenberg, I, Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964) 665–706; II, Trudy Mat. Inst. Steklov 92 (1966) 233–297. English Transl.: I, Amer. Math. Soc. Transl. 56 (2) (1966) 192–232, II, Proc. Steklov Inst. Math. 92 (1966) 269–333. [86] V.A. Solonnikov, Estimates of the solutions of linearized Navier–Stokes system, Trudy Mat. Inst. Steklov 70 (1964) 213–317. English Transl.: Proc. Math. Inst. Steklov 70.
The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem
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[87] V.A. Solonnikov, On boundary value problems for linear parabolic systems of general form, Trudy Mat. Inst. Steklov 83 (1965) 1–162. English Transl.: Proc. Math. Inst. Steklov 83 (1965). [88] V.A. Solonnikov, On the solvability of boundary and initial-boundary value problems for the Navier– Stokes system in domains with noncompact boundaries, Pacific J. Math. 93 (2) (1981) 443–458. [89] V.A. Solonnikov, On solutions of stationary Navier–Stokes equations with an infinite Dirichlet integral, Zapiski Nauchn. Sem. LOMI 115 (1982) 257–263. English Transl.: J. Sov. Math. 28 (5) (1985) 792–799. [90] V.A. Solonnikov, Stokes and Navier–Stokes equations in domains with noncompact boundaries, Nonlinear Partial Differential Equations and Their Applications. Pitmann Notes in Math., College de France Seminar 3 (1983) 240–349. [91] V.A. Solonnikov, Solvability of the problem of effluence of a viscous incompressible fluid into an open basin, Trudy Mat. Inst. Steklov 179 (1989) 193–225. English Transl.: Proc. Math. Inst. Steklov 179 (2) (1989) 193–225. [92] V.A. Solonnikov, Boundary and initial-boundary value problems for the Navier–Stokes equations in domains with noncompact boundaries, Math. Topics in Fluid Mechanics, Pitman Research Notes in Mathematics Series, J.F. Rodriques, A. Sequeira, eds, 274 (1991) 117–162. [93] V.A. Solonnikov, On problems for hydrodynamics of viscous flow in domains with noncompact boundaries, Algebra i Analiz 4 (6) (1992) 28–53. English Transl.: St. Petersburg Math. J. 4 (6) (1992). [94] V.A. Solonnikov and K. Pileckas, Certain spaces of solenoidal vectors and the solvability of the boundary value problem for the Navier–Stokes system of equations in domains with noncompact boundaries, Zapiski Nauchn. Sem. LOMI 73 (1977) 136–151. English Transl.: J. Sov. Math. 34 (6) (1986) 2101–2111. [95] J.S. Song, Improved decay estimates in time-dependent Stokes flow, J. Math. Anal. Appl. 288 (2) (2003) 505–517. [96] M. Specovius-Neugebauer, Approximation of the Stokes Dirichlet problem in domains with cylindrical outlets, SIAM J. Math. An. 30 (1999) 645–677. [97] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press (1970). [98] R. Temam, Navier–Stokes equations, Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam (1984). [99] F.G. Tricomi, Integral Equations, Intersience, New York (1957). [100] O. Vejvoda, L. Herrmann, V. Lovicar, M. Sova, I. Straškraba and M. Št˘edrý, Partial Differential Equations: Time-Periodic Solutions, Martinus Nijhoff Publishers, The Hague (1981).
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CHAPTER 9
Periodic Homogenization Problems in Incompressible Fluid Equations∗ C. Conca Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, and Centro de Modelamiento Matemático, UMI 2807 CNRS-UChile, Casilla 170/3-Correo 3, Santiago, Chile; e-mail: [email protected]
M. Vanninathan TIFR Center, IISc. Campus, P.O. Box 1234, Bangalore-560012, India; e-mail: [email protected]
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . 2. Fluid flows in porous media . . . . . . . . . . . 2.1. Darcy’s law . . . . . . . . . . . . . . . . . 2.2. Fourier boundary conditions . . . . . . . . 2.3. The Stokes sieve problem . . . . . . . . . . 3. Advection and diffusion of passive scalars . . . 3.1. Case I (a) (α = −1/2, β = 3/2) . . . . . . 3.2. Case I (b) (α = −1, β = 2) . . . . . . . . . 3.3. Case I (c) (α = −1/2, β = 1) . . . . . . . . 3.4. Case I (d) (α = −1/2, β = 2) . . . . . . . 3.5. Case II . . . . . . . . . . . . . . . . . . . . 3.6. Case III . . . . . . . . . . . . . . . . . . . . 3.7. Estimates on the effective diffusion tensor 4. Convection effects . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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651 652 652 666 671 676 678 680 681 681 682 684 687 689 696 696
Abstract This write-up is a review of some old and some recent developments in the homogenization of fluid flows modelled by Stokes, Navier–Stokes, Euler, Advection-Diffusion equations. Oscillations may enter these systems through domains, the coefficients or through the initial data. ∗ This work has been partially supported by FONDAP through its Programme on Mathematical-Mechanics.
HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME IV Edited by S.J. Friedlander and D. Serre © 2007 Elsevier B.V. All rights reserved. 649
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Various phenomena involved in the homogenization process are highlighted. Even though the details of proof are omitted, the main ideas behind them are presented.
Keywords: homogenization, periodic structures, porous media, fluid mechanics, advection, diffusion, convection MSC: 35B27, 76M50, 76S05, 76R05, 76B99, 76D05
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1. Introduction The aim is to review some of the developments which took place in the field of applications of homogenization ideas to describe the macroscopic behavior of fluids. This is of course a huge area encompassing various subjects including Mathematics, Physics, Engineering, robust body of technology, and so on. To remain within reasonable goals, we have decided to focus on the mathematical aspects concerning problems which involve periodic structures and incompressible Newtonian single phase fluids. For the same reason, neither random nor numerical aspects of homogenization problems are discussed. Each one of the topics mentioned above will require a separate survey paper. As a consequence, we will be considering mainly fluids flows governed by the system of Stokes, Navier–Stokes, Euler equations (steady or unsteady). This is not a comprehensive review and consequently due to the limitations of space and time and the capabilities of the authors, it is clear that this cannot include everything relevant and therefore forcibly important contributions of several authors could not be reviewed. We apologize in advance for this lack of completeness. Nevertheless, it is our feeling that this write-up can be considered as a first step towards a more exhaustive survey on this topic. In the systems treated in this review, the periodic structures and thereby oscillations enters in different ways: through domains, coefficients, and initial data. It can also appear through the boundary conditions, a subject which is extensively already reviewed in Volume III of this Handbook (see [22]). As is well known, periodic homogenization limit is concerned with the asymptotic analysis as the period goes to zero. In this process, oscillations appear in the fluid system and the homogenization theory attempts to describe the macroscopic behavior of the fluid. The limiting equations are known as the homogenized system. As far as we know, some of the fundamental issues of the field are open. For instance, there is no tidy justification of turbulent models currently in use in fluid mechanics literature [33,46]. The intention here is not to present all possible scenarios but rather make a sample of important problems and highlight various phenomena involved in the process of homogenization. Following sections include discussion on various transport processes involved in the fluid equations: diffusion, advection/convection, motion due to the pressure gradient etc and their interaction with homogenization process. Let us now mention the fundamental model problems discussed in this paper. The starting point is fluid flows in periodic media wherein we discuss Darcy’s law, Brinkman’s law and other important linear and nonlinear versions. These developments are included in Section 2. Apart from these, Section 2 also describes the macroscopic behavior of fluids flows through porous media with Fourier boundary conditions and through sieves. Section 3 discusses advection and dissipation of passive scalars by fluid flows whose velocities are assumed to be periodically oscillating. Examples of passive scalars are temperature, concentration of tracer particles and so on. How periodic oscillations in the initial data evolve through convection is the main point of discussion of Section 4. We indicate several open problems along the way. The sections can be read independently. The usual summation convention with respect to repeated indices is adapted throughout.
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2. Fluid flows in porous media 2.1. Darcy’s law Since the appearance of works by A. Bensoussan, J.L. Lions and G. Papanicolaou [12], and E. Sánchez-Palencia [53], the homogenization method has been successfully applied to several problems of Fluid Mechanics and it has been able to explain some of its main phenomena. Perhaps the most glaring example is the classical Darcy’s law which describes the macroscopic behavior of fluids in porous media. The starting point is Stokes flow in a porous medium modeled by a periodically perforated domain in RN , N = 2, 3. Let Y denote the cube ]0, 1]N in RN and H ⊂ Y be a connected open subset representing a generic perforation. Let Y ∗ = Y \ H¯ . We will periodically distribute the above structure throughout RN with period unity: def
Yk = Y + k,
def
Hk = H + k,
∀k ∈ ZN .
Let G = RN \ ∪{Hk | k ∈ ZN }. Given a small parameter ε > 0, we then re-scale the above periodic structure by ε. The result is a periodically perforated structure in RN with ε period. The holes obtained in this way are denoted {εHk | k ∈ ZN }. Now, let be a bounded domain in RN and let ε = \ ∪{εHk | k ∈ ZN and εYk ⊂ } denote the part occupied by the fluid which obeys Stokes system. It is worth remarking that the domain ε is obtained by removing from the non-perforated domain only those tiny holes εHk which are included in a cell εYk completely contained in . This has the effect that no holes can meet the boundary ∂ and various mathematical technicalities involving boundary conditions, a priori estimates, and odd possible geometrical situations are avoid from the very beginning. ε ) and the pressure scalar-field p ε satisfy the The velocity vector-field v ε = (v1ε , . . . , vN following constitutive law: σ ε = −p ε I + 2νe(v ε ), where ν is the fluid viscosity, σ ε is the stress tensor and e(v ε ) is the strain rate tensor: e(v ε ) =
1 ε t ε ∇v + ∇v . 2
If f ∈ L2 ()N is a force field acting on such a fluid, then the resulting motion is governed by the Stokes system: Find v ε ∈ H01 (ε )N and p ε ∈ L2 (ε )/R satisfying:
−νv ε + ∇p ε = f div v ε = 0
in ε , in ε .
Since v ε will be of order ε 2 (cf. Poincaré inequality in ε stated below), a re-scaling is called for. Let us therefore make the following change of variables: v ε = ε 2 uε and write down the re-scaled equation: −νε 2 uε + ∇p ε = f in ε div uε = 0 in ε .
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Fig. 1. Classical homogenization process.
We impose the no-slip boundary condition on the boundary of holes: uε = 0 on ∂ε . This will act as a break on the fluid motion. At the same time, since the viscosity is O(ε 2 ), the fluid feels little resistance to flow as ε → 0. Thus there will be a limiting behavior as ε → 0 as a balance between these two aspects which we seek to describe (see Figure 1). The goal is to study the behavior of (uε , p ε ) as ε → 0. One way of doing it is to apply the method of multiscale asymptotic expansion to the above system (see [12], [53] for details). The method proposes the following ansatz (with y = xε ): uε (x) = u0 (x, y) + εu1 (x, y) + · · · p ε (x) = p 0 (x, y) + εp 1 (x, y) + · · · , where uj , p j are functions defined on × G and Y -periodic with respect to the second variable y. Classically, x ∈ is called the slow variable and the periodic variable y is known as the fast variable. In such an ansatz, it is an implicit assumption that there is a scale separation in the variation of the solution for ε small. In the sequel, we focus our discussion only on the first term of the above expansion, namely (u0 , p 0 ). Substituting the above expansion into the Stokes system and treating (x, y) as independent variables, we are led to the following system of equations: ⎧ ∇y p 0 = 0, ⎪ ⎪ ⎨ −νy u0 + ∇y p 1 = f − ∇x p 0 , divy u0 = 0, ⎪ ⎪ ⎩ 0 u|∂G = 0. In order to solve the above Stokes system for (u0 , p 1 ), we introduce spaces "
def 1 ¯ " ϕ is Y -periodic H#1 (Y ∗ ) = ϕ ∈ Hloc (G) " def V# = v ∈ H#1 (Y ∗ )N " divy v = 0 in G, v = 0
on ∂G .
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The cell test functions {χ i | i = 1, . . . n} are defined via the following variational formulation in the space V# : ⎧ i ⎪ ⎪ ν ∇χ · ∇vdy = vi dy ∀v ∈ V# , ⎪ ⎪ ⎨ ∗ ∗ Y Y def 1 ⎪ i i ⎪ χ ∈ V# , χ = χ i dy = 0. ⎪ ⎪ ⎩ |Y |
(1)
Y
The first term u0 is then given by ∂p 0 u0 (x, y) = fi (x) − (x) χ i (y), ∂xi where the usual summation convention is adopted. To obtain the equation for p 0 , we average the above relation with respect to y ∈ Y ∗ . Denoting the average of a function g as in (1) by g, we have u0j (x) = Kij
∂p 0 fi (x) − (x) ∂xi
∀j = 1, . . . , N,
where the permeability matrix (Kij ) is defined by Kij = χji ∀i, j = 1, . . . , N . The above relation is classically known as Darcy’s law. It says that the limiting flow is proportional to the total force acting on the fluid, i.e., the external force and the pressure field. The proportionality tensor (Kij ) depends only on the geometry of the reference hole. The divergence with respect to x of the above relation yields the following equation for p 0 Kij
∂ 2 p0 ∂fi = Kij ∂xi ∂xj ∂xj
because we will have divx u0 (x) = 0. For the same reason, u0 will satisfy the boundary condition u0 (x) · n = 0 on ∂. This, in turn, implies the following Neumann boundary condition on the pressure Kij
∂p 0 fi (x) − (x) nj = 0 ∂xi
on ∂.
Since (Kij ) can be shown to be a symmetric positive definite matrix (see [53]), the above set of conditions constitute a nice elliptic boundary value problem for the pressure p 0 . Our next goal is to present some elements of mathematical justification of the Darcy’s law. The appendix of the book by E. Sánchez-Palencia [53] contains the celebrated proof of Tartar given in late 1970’s.
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T HEOREM 1. Let (uε , p ε ) and (u0 , p 0 ) be as above. If we extend uε by zero inside the holes and we denote the extension by uε again, then uε ! u0 (x)
in L2 ()-weak.
Further, there is an extension operator Qε : L2 (ε )/R −→ L2 ()/R such that Qε p ε −→ p 0
in L2 ()/R strong.
The proof of the above result is divided into three main steps: Step (1) (A priori estimates) The starting point is Poincaré inequality in ε (see [57]): |v|2 dx ≤ Cε 2 |∇v|2 dx ∀v ∈ H01 (ε ). ε
ε
Multiplying the Stokes system by uε and using Poincaré inequality, we obtain ∇uε L2 (ε )N×N ≤ Cε −1 uε L2 (ε ) ≤ C. As a consequence, we have uε ! u∗ in L2 ()-weak along a subsequence. Estimates on the pressure are not straightforward. We require an extension operator. First, we define a restriction operator and then make use of duality. L EMMA 1. There exists a restriction operator R ε : H01 ()N −→ H01 (ε )N with the following properties: (i) (Restriction property) w ∈ H01 (ε )N
⇒
R ε w˜ = w
where w˜ = w in ε and w˜ = 0 inside holes. (ii) (Incompressibility preservation) w ∈ H01 ()N ,
div w = 0 in
⇒
div R ε w = 0
in ε .
(iii) (Boundedness)
R ε wL2 (ε )N ≤ C wL2 ()N + ε∇wL2 ()N×N
∇R ε wL2 (ε )N×N ≤ C ε −1 wL2 ()N + ∇wL2 ()N×N .
Admitting the above lemma, let us proceed to complete the proof of Theorem 1. Thanks to the above lemma, we can consider (R ε )∗ ∇p ε as an extension of ∇p ε from ε to . Because R ε preserves incompressibility condition, it is easily seen that (R ε )∗ ∇p ε is
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the gradient of a function in , denoted by Qε p ε . More precisely, define F ε ∈ H −1 ()N by F ε , w = ∇p ε , R ε wε
∀w ∈ H01 ()N .
From the restriction property of R ε , it follows that F ε = ∇p ε
in ε .
Further, we can see that F ε is given by the gradient of an L2 () function. Indeed, if div w = 0 in then F ε , w = 0 and so the classical De Rham’s Theorem implies the desired result. The boundedness properties of R ε yield the following estimates: ⎧
⎨ |∇Qε p ε , w | ≤ C w 2 N + ε∇w 2 N×N , L () L () (2) ⎩ ∇Qε p ε H −1 ()N ≤ C. Since gL2 () ≈ | g| + ∇gH −1 ()N , it follows that {Qε p ε } is bounded in L2 ()/R. Therefore, for a subsequence, we will have Qε p ε ! p ∗ in L2 ()/R-weak ε ε ∗ ∇Q p ! ∇p in H −1 ()N -weak. Next, we show that the convergence of ∇p ε is strong in H −1 ()N . To this end, let us simply note that we can pass to the limit in the duality bracket ∇Qε p ε , w ε against an arbitrary weakly convergent sequence {w ε } ⊂ H01 ()N using (2). Hence
Qε p ε → p ∗ ∇Qε p ε → ∇p ∗
in L2 ()/R strong, and in H −1 ()N strong.
Step (2) (Oscillating test functions) The differential interpretation of the cell problem (1) is the following: ⎧ ⎪ −νy χ i + ∇y π i = ei in G, ⎪ ⎨ in G, divy χ i = 0 i =0 ⎪ on ∂G, χ ⎪ ⎩ i i are Y -periodic. {χ , π } In the sequel, we consider the sequences χ i ( xε ) and π i ( xε ) and the system satisfied by them. Step (3) (Convergence) We follow the method of oscillating test functions to pass to the limit in homogenization problems introduced by F. Murat and L. Tartar [47]. Taking φ ∈ D(), the procedure consists of multiplying by φuε the system satisfied by
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(χ i ( xε ), π i ( xε )) and multiplying by φχ i ( xε ) the system satisfied by (uε , p ε ). Subtracting the resulting relations, we see that all trouble shooting terms get cancelled and we can easily pass to the limit. At the limit, we find the so-called Darcy relation between u∗ and p∗ : u∗j (x) = Kij
∂p ∗ fi (x) − (x) ∂xi
∀j = 1, . . . , N.
On the other hand, it is clear that div u∗ = 0 in and u∗ · n = 0 on ∂ (since div uε = 0 in and uε = 0 on ∂ε ). As a consequence, we see that p ∗ is the unique solution (modulo constants) of the boundary-value problem: ⎧ ∂ 2 p∗ ∂fi ⎪ ⎪ = Kij ⎨ Kij ∂xi ∂xj ∂xj ⎪ ∂p ∗ ⎪ ⎩ Kij nj = 0 ∂xi
in , on ∂.
These relations imply p ∗ = p 0 and hence u∗ (x) = u0 (x). The proof of Theorem 1 is finished. Proof of Lemma 1 Let us now discuss the proof of Lemma 1, which was fundamental in the extension of the pressure. Since we want R ε w to vanish in the holes, a first attempt would be to use a suitable cut-off function associated with the holes. This does not work as this operation does not preserve incompressibility. Let us therefore proceed differently. To simplify matters, let us assume that the hole H satisfies H ⊂⊂ Y . (More general cases have been treated by G. Allaire [1]). Choose an open set W which is a strip around H such that H ⊂⊂ H¯ ∪ W ⊂⊂ Y . We denote by ∂H , the boundary of H and by γ , the boundary of H¯ ∪ W . The operator R ε is then an ε-rescaled version of an operator R which we construct on the unit scale as follows: Given w ∈ H 1 (Y ), we put 3 w(y) Rw(y) =
v(y) 0
if y ∈ Y \ (H ∪ W ) if y ∈ W if y ∈ H .
where v is the solution to the following non-homogeneous Stokes system in the strip W : ⎧ −νv + ∇q = −νw ⎪ ⎪ 1 ⎪ ⎨ div v = div w + div w dy |W | ⎪ H ⎪ ⎪ ⎩v = w v=0
in W, in W, on γ , on ∂H.
It is straight forward to verify that the above construction satisfies all the requirements of the Lemma.
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2.1.1. Time-dependent system We will now consider time-dependent Stokes system in the porous medium ε : ⎧ ε ∂u ⎪ ⎪ − νε 2 uε + ∇p ε = f ⎪ ⎪ ⎨ ∂t div uε = 0 ⎪ ⎪ ⎪ uε = 0 ⎪ ⎩ ε u (x, 0) = uε0 (x)
in ε × (0, T ), in ε × (0, T ), on ∂ε × (0, T ), in ε .
In the above system, f ∈ L2 (0, T ; L2 ())N is the given external force acting on the fluid. The initial data uε0 is given in the space V ε = v ε ∈ H01 (ε )N | div v ε = 0
in ε .
By means of suitable assumptions on the initial data, the goal is to capture the asymptotic behavior of (uε , p ε ) as ε → 0. We will be led to a modified Darcy’s law which incorporates memory effects. To this end, we will use the concept of two-scale convergence (see [5], [48]). The starting point is to assume that the initial data satisfies:
u˜ ε0 L2 () + ε∇ u˜ ε0 L2 ()N ≤ C, where we recall that ˜ denotes the extension by zero inside the holes. From the compactness property of two-scale convergence, we know that a subsequence of u˜ ε0 will admit a twoscale limit. We assume something stronger:
The entire sequence u˜ ε0 converges to v0 (x, y) ∈ L2 (; L2# (Y )) in the sense of two-scales.
The advantage with the two-scale convergence is that it is easy to pass to the limit and we can write down what is known as the two-scale homogenized system which contains the usual homogenized system. Indeed the usual homogenized system is obtained by taking average with respect to the fast variable. Thus, we see that the two-scale homogenized system is a closed system describing not only the macroscopic behavior but also the oscillations around it. In our case, the two-scale homogenized system is as follows:
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T HEOREM 2. There is an extension of the solution, still denoted by (uε , p ε ) which twoscale converges to (u0 (x, y, t), p(x, t)) which is the unique solution of ⎧ ∂u0 ⎪ ⎪ (x, y, t) + ∇y p1 (x, y, t) + ∇x p(x, t) ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ −νy u0 (x, y, t) = f (x, t) ⎪ ⎪ ⎪ ∇y u0(x, y, t) = 0 ⎪ ⎪ ⎪ ⎨ ∇x · u0 (x, y, t)dy = 0 Y ⎪ ⎪ u0 (x, y, t) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u0 (x, y, t)dy · n = 0 ⎪ ⎪ ⎪ Y ⎪ ⎪ ⎪ ⎩ u0 , p1 u0 (x, y, 0) = v0 (x, y)
in × G × (0, T ), in × G × (0, T ), in × (0, T ), in × H × (0, T ), on ∂ × (0, T ), are Y -periodic in y-variable, in × G.
We notice that there are two pressure gradient terms which correspond to the two incompressibility conditions present in the above system. As in the Darcy’s law of the steady state case, p is the limit of p ε and p1 provides a first order corrector for p ε . In order to get the homogenized system, we first observe that the solution (uε , p ε ) satisfies the following apriori estimates: 9 ε9 9 ∂u 9 ε 9 9 + ε∇u + ≤ C, ∞ 2 ε ∞ 2 ε 9 ∂t 9 2 L (0,T ;L ( )) L (0,T ;L ( )) L (0,T ;L2 (ε )) ⎪ ⎩ p ε 2 ≤ C. L (0,T ;L2 (ε )/R) ⎧ ⎪ ⎨ uε
In order to extract convergent subsequences, we extend uε by zero in the holes and p ε is extended in the same way as in stationary Darcy’s law. We keep the same notation for the extended quantities. Of course, the above estimates are valid for them too. Thus, we have the following convergences for a subsequence, still denoted by ε: uε ! u in L∞ (0, T ; L2 ())-weak *, pε ! p
in L2 (0, T ; L2 (/R))-weak.
The homogenized problem consists of identifying these limits. To this end, we follow the standard procedure which consists of averaging the two-scale system with respect to y. Before doing that, let us decompose the solution (u0 , p1 , p) as (u0 , p1 , p) = (w1 , q1 , 0) + (w2 , q2 , p),
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where the right hand side terms are the unique solutions characterized by ⎧ ∂w1 ⎪ ⎪ (x, y, t) + ∇y q1 (x, y, t) in × G × (0, T ), ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ − νy w1 (x, y, t) = 0 in × G × (0, T ), ∇y · w1 (x, y, t) = 0 ⎪ ⎪ (x, y, t) = 0 in × H × (0, T ), w 1 ⎪ ⎪ ⎪ ⎪ , q are Y -periodic with respect to y, w 1 1 ⎪ ⎩ in × G. w1 (x, y, 0) = v0 (x, y) ⎧ ∂w2 ⎪ ⎪ (x, y, t) + ∇y q2 (x, y, t) + ∇x p(x, t) in × G × (0, T ), ⎪ ⎪ ⎪ ∂t ⎪ ⎪ −ν ⎪ y w2 (x, y, t) = f (x, t) ⎪ ⎪ ⎪ · w (x, y, t) = 0 in × G × (0, T ), ∇ ⎪ y 2 ⎪ ⎪ ⎪ (x, y, t) = 0 in × H × (0, T ), w ⎪ 2 ⎪ ⎪ ⎪ , q are Y -periodic with respect to y, w ⎪ 2 2 ⎪ ⎪ ⎪ (x, y, 0) = 0 in × G, w ⎨ 2 ∇x · w2 (x, y, t)dy in × (0, T ), ⎪ ⎪ ⎪ Y ⎪ ⎪ ⎪ ⎪ ⎪ = −∇x · w1 (x, y, t)dy ⎪ ⎪ ⎪ Y ⎪ ⎪ ⎪ ⎪ ⎪ w (x, y, t)dy ·n on ∂ × (0, T ). ⎪ ⎪ Y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = − w1 (x, y, t)dy · n Y
Note that the contributions from the external force f and the macroscopic pressure p are included in the second system written above and not in the first system. The first system contains contributions from initial data. Further, the second system contains nonhomogeneous divergence condition and boundary condition on ∂ which depend on the solution of the first system. Surprisingly, the second system can be explicitly solved via the introduction of cell test functions (X i , #i ) satisfying ⎧ ∂X i ⎪ ⎪ − νy X i + ∇y #i = ei ⎪ ⎪ ∂t ⎪ ⎨ ∇y · X i = 0 ⎪ Xi = 0 ⎪ ⎪ i i ⎪ ⎪ ⎩X ,# i X (y, 0) = 0
in G × (0, T ), in G × (0, T ), in H × (0, T ), are Y -periodic in y-variable, in G.
Indeed, w2 satisfying the above conditions (except the last two which will be imposed later) is given by t ∂X i ∂p fi − (x, s) w2 (x, y, t) = (y, t − s)ds. ∂xi ∂t 0
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By differentiating the above expression with respect to t, it is straight forward to check that it satisfies the above system with q2 defined by t q2 (x, y, t) =
∂#i ∂p ∂p fi − (x, s) (x, t)#i (y, 0). (y, t − s)ds + fi − ∂xi ∂t ∂xi
0
The next step in the two-scale convergence method is to take average of the systems defining w1 and w2 . The weak limit of uε (which was denoted by u) is equal to the sum of averages of w1 and w2 . Using the explicit expression for w2 , we can compute its average. We obtain u(x, t) = w1 (x, ·, t) + w2 (x, ·, , t) 1 = w1 (x, ·, t) + ν
t A(t − s)(f − ∇p)(x, s)ds, 0
where the entries of the matrix A(t) are defined by Aij (t) = ν
∂X i j · e ∀i, j = 1, . . . , N. ∂t
On the other hand, the weak limit obviously satisfies
divx u(x, t) = 0 in × (0, T ), u(x, t) · n = 0 on ∂ × (0, T ).
The above three relations constitute the homogenized problem for the unknown p in the present case. The first relation is known as Darcy’s law with memory. The matrix A(t), called permeability matrix, is symmetric, positive definite, and decays exponentially for large times. The homogenized problem is an integro-differential equation of convolution type. Let us now address the issue of existence and uniqueness of solutions to the homogenized problem. To simplify matters, let us consider the case where A(t) is a scalar matrix a(t)I . For the general case, see Lions’ course at Beijing [36], pp. 168–171. In this case, the homogenized system can be rewritten as follows: t a(t, s)p(x, s)ds = g(x, t)
in × (0, T ),
0
t a(t, s) 0
∂p (x, s)ds = h(x, t) ∂n
on ∂ × (0, T ),
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where the right hand side are known entities. To solve these, let us differentiate them with respect to t. We obtain t a(0)p(x, t) +
a (t − s)p(x, s)ds =
∂g ∂t
in × (0, T ),
0
∂p a(0) (x, t) + ∂n
t
a (t − s)
∂h ∂p (x, s)ds = ∂n ∂t
on ∂ × (0, T ).
0
These are Volterra integral equations of the second kind for the unknowns (p(x, ·), ∂p
∂n (x, ·)). If the kernel a (t − s) is continuous, it is classical that the above equations are uniquely solved (see [32], p. 33). Once (p(x, ·), ∂p ∂n (x, ·)) are known, it is easy to recover p(x, t) by solving the corresponding boundary-value problem.
2.1.2. Effects of nonlinearity Since we have already isolated the effects of time dependence, we now consider stationary Navier–Stokes system in the perforated domain ε :
−νε 2 uε + ε γ (uε · ∇)uε + ∇p ε = f
in ε ,
div uε = 0
in ε ,
uε = 0
on ∂ε .
We have introduced a parameter γ to measure nonlinear effects. It turns out that γ = 1 is the critical value. If γ < 1 then the nonlinearities dominate and the homogenization problem is essentially open. If γ > 1 then the above system is close to (linear) Stokes system which has been already treated. Thus the case γ = 1 is the most interesting one. Rigorous homogenization results were first established in the case N = 2: T HEOREM 3. Let γ = 1 and N = 2. We assume that the external forcing f ∈ D() such that f H 3 () is sufficiently small. Then there is an extension of (uε , p ε ) from ε to , denoted by the same notation, such that 3
uε (x) − u0 (x, xε )L2 ()2 ≤ Cε, pε → p
in Lq ()
∀1 < q < 2,
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where (u0 , p, p1 ) is the unique solution of the two-scale homogenized system: ⎧ −νy u0 + (u0 · ∇y )u0 + ∇x p(x) ⎪ ⎪ ⎪ ⎪ +∇y p1 = f (x) ⎪ ⎪ ⎪ 0 ⎪ div ⎪ yu =0 ⎪ ⎪ ⎪ ⎪ ∇ · u0 dy = 0 ⎪ ⎨ x Y ⎪ ⎪ ⎪ u0 (x, y) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ u0 dy · n = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩Y0 u , p1
in × G, in × G, in , in × H, on ∂ are periodic with respect to y.
The derivation of the above system via two-scale expansion was given by E. SánchezPalencia [53]. For a rigorous proof, see A. Mikeli´c [44]. The smallness of force is used to show the existence and uniqueness of a solution for the above stationary system. For large forcing, there may be bifurcation branches and the two-scale asymptotic expansion is not suitable. Unlike the previous cases, we cannot average out the y-variable in the above system and write the homogenized system separately. Thus, Darcy’s law remains implicit, non-local and nonlinear in the present case. We could capture these features due to the two-scale convergence concept. R EMARK 1. Let us now consider the evolution Navier–Stokes system in ε with γ = 1. Formal asymptotic expansion (see E. Sánchez-Palencia [53]) shows that the corresponding homogenized system is the same as the one written above. In particular, it is a stationary system with t appearing as a parameter coming from the external forcing. Thus the limiting system is well-posed provided the external forcing is small. If not, the system may exhibit bifurcation and consequently the ansatz is not appropriate. If N = 3, a derivation of a nonlinear filtration law including inertia effects via homogenization has been obtained in 2000 by E. Maruši´c-Paloka and A. Mikeli´c [41]. 2.1.3. Brinkman’s law Here we are concerned with flows in periodic porous media with holes of smaller size. Homogenization of elliptic problem in such domains has been studied in the pioneering work by D. Cioranescu and F. Murat [15], where one can also find a brief history of the problem along with the relevant references. This has been generalized to fluid flows in a series of papers [2–4,34,35,54]; see also Allaire’s review paper in [24]. The method followed in the above papers is the so-called oscillating test functions method introduced in [47]. We begin with some notations and basic assumptions. We follow the earlier notations with one change. The size of the hole inside Yiε is now a ε instead of ε. More precisely, we denote by Oεi the hole inside Yiε obtained rescaling the unit hole H to the size a ε i.e., Oεi = a ε H , up to a translation by i ∈ ZN . These holes are smaller than the period: a ε ! ε ε i.e., lim aε = 0. Previous cases considered correspond to a ε = ε. ε→0
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CN (ε) Let ε = \ i=1 Oεi be the part occupied by the fluid. Here N (ε) is the number of holes which have non-empty intersection with . The goal of this section is to describe the macroscopic behaviour of the problem 3
−νuε + ∇p ε = f div uε = 0 uε = 0
in ε , in ε , on ∂ε ,
where we assume that f ∈ L2 ()N . To this end, we introduce a new parameter r ε which governs the macroscopic behaviour: ⎧ ⎪ ⎪ ⎪ ⎨
1 2 εN ε N −2 (a ) rε = " ε "1/2 ⎪ " a "" ⎪ ⎪ ⎩ ε "" log ε "
if N ≥ 3, if N = 2.
The Brinkman’s law which describes the macroscopic behaviour is stated in terms of a permeability matrix which we introduce next. P ROPOSITION 2. (a) Let N ≥ 3. For 1 ≤ i ≤ N , the following the cell problems are well-posed: (Formally, these are obtained by rescaling the cell Yiε to the size (a ε )−1 ): ⎧ i i ⎪ ⎪ −νw + ∇π = 0 ⎨ i div w = 0 i ⎪ ⎪w = 0 ⎩ w i → ei
in RN \ H, in RN \ H, in H, as |x| → ∞.
Then the matrix M defined by Mij =
∇w i · ∇w j dy
∀i, j = 1, . . . , N
RN \H
(equivalently, M can also be defined by Mei =
∂w i i − π n dy ∂n
∀i = 1, . . . , N )
∂H
represents the drag forces exerted on the obstacle placed at H by the surrounding fluid. (b) Let N = 2. In this case, the cell problems differ by the following requirement at ∞: w i (x) ∼ ei log |x|
as |x| → ∞.
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The definition of M remains the same as in (a). Furthermore, whatever be the shape or size of the hole H , the matrix M is always given by M = 4π
if N = 2.
R EMARK 2. The above result in the case N = 2 is somewhat surprising and it is related to the well-known Stokes paradox which asserts that there is no solution to the cell problem with the asymptotic condition w i → ei as |x| → ∞. Using the matrix M, we are now in a position to state the main result which describes the macroscopic behavior. The result is stated in terms of the extension of the solution (uε , p ε ) from ε to . This construction is analogous to the one we have seen in the previous section. T HEOREM 4. (a) If the hole is too small, i.e., lim r ε = ∞, then there exists an extension again denoted ε→0
by (uε , p ε ) such that (uε , p ε ) → (u, p)
in H01 ()N × L2 ()/R,
where (u, p) is the unique solution of the Stokes equations: 3
−νu + ∇p = f div u = 0 u=0
in , in , on ∂.
(b) If the hole is of critical size, i.e., lim r ε = r > 0, then there exists an extension again ε→0
denoted by (uε , p ε ) such that (uε , p ε ) → (u, p)
in H01 ()N × L2 ()/R,
where (u, p) is the unique solution of the Brinkman’s law: 3
−νu + ∇p + div u = 0 u=0
ν Mu = f r2
in , in , on ∂.
(c) If the hole is too big, i.e., lim r ε = 0, then there exists an extension, again denoted by (uε , p ε ) such that
ε→∞
uε ε , p → (u, p) (r ε )2
in L2 () × L2 ()/R,
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where (u, p) is the unique solution of Darcy’s law: ⎧ ⎪ ⎨ u = 1 M −1 (f − ∇p) in , ν in , ⎪ ⎩ div u = 0 u·n=0 on ∂. Idea of the proof It is similar to the idea used to derive Darcy’s law. It consists of multiplying the Stokes system by suitable test functions, integrating by parts and passing to the limit. The extra difficulty here is that the test function must vanish on the boundary of the holes. And this requires the construction of boundary layers {w i,ε | 1 ≤ i ≤ N } around the holes. This is carried on by cell solutions w i to the scale a ε . The required Nrescaling i,ε multiplier is then of the form i=1 φi w where φi ∈ D(), 1 ≤ i ≤ N . R EMARK 3. (Nonlinear effects) Consider the stationary Navier–Stokes system in ε : 3
−νuε + (uε · ∇)uε + ∇p ε = f div uε = 0 uε = 0
in ε , in ε , on ∂ε .
Note that the factor ε γ is not present in the nonlinear term because holes are small. It is well-known that a unique solution exists for small values of f L2 ()N for N = 2, 3. Its asymptotic behaviour can be described following the three cases listed in the above theorem. In the sub-critical case (a), the limiting system is the Navier–Stokes system in . In the critical case (b), the homogenized problem is Navier–Stokes–Brinkman’s law: 3
−νu + (u · ∇)u + ∇p + div u = 0 u=0
ν Mu = f r2
in , in , on ∂.
In the supercritical case, the limit problem is given by linear Darcy’s law.
2.2. Fourier boundary conditions In this section, we study the homogenization problem of Stokes and Navier–Stokes systems in perforated domain with Fourier boundary conditions on the boundary of the holes. Such model appears in flows around arrays of tubes inside nuclear reactors and heat exchangers. To avoid technicalities, we assume that the domain ⊂ RN is rectangular. We consider a subsequence ε → 0 such that is a finite union of the re-scaled cells {Yiε }. (For more general cases, the reader is referred to C. Conca [17], T. Levy and E. SánchezPalencia [35]).
Periodic homogenization problems in incompressible fluid equations
As in § 2.1, let ε denote the perforated domain with period ε and the hole εH translation: D
ε = \
667 1
up to
εHk .
k∈ZN
We need to distinguish between the external boundary ∂ and the boundary of holes ε which, by our assumption, are disjoint. With these notations, we consider the following Stokes system: ⎧ −νuε + ∇p ε = f ⎪ ⎪ ⎪ ⎨ div uε = 0 uε = 0 ⎪ ⎪ ⎪ στε + εατ uετ = 0 ⎩ σnε + εαn uεn = 0
in ε , in ε , on ∂, on ε , on ε ,
where the coefficients ατ , αn are constants such that ατ ≥ 0, αn ≥ 0. Here στε , σnε are respectively tangential and normal components of the Cauchy force σ ε n where σ ε is the stress tensor: ε σn = σijε nj ni , στε = σijε nj τi . In a similar way, uετ , uεn are defined by: uεn = uεi ni ,
uετ = uεi τi .
By a simple application of Korn’s inequality, one can easily see the well-posedness of the above problem. Our aim is to study the behavior of the solution as ε → 0. We start by presenting the homogenized system: 3 Lu = u=0
|Y ∗ | f |Y |
in , on ∂.
The homogenized operator L is of the form L = (L1 , L2 , . . . , LN ) with Li u = −aij kl
∂ (τ ) (n) ek& (u) + dij uj + dij uj , ∂xj
1 In this section, the generic perforation H is assumed to be compactly included in the reference cell Y ; this additional hypothesis is required for the existence of various extension operators we introduce in what follows.
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where the homogenized coefficients are defined by % ⎧ δij 1 k& k& ⎪ ⎪ aij kl = 2ν δ(i,j );(k,&) + eij (χ )dy − η dy , ⎪ ⎪ |Y | |Y | ⎪ ⎪ ∗ ∗ ⎪ Y Y ⎪ ⎪ ⎨ (τ ) ατ τi τj ds, dij = |Y | ⎪ ⎪ ∂H ⎪ ⎪ ⎪ αn ⎪ (n) ⎪ ⎪ = ni nj ds. d ⎪ ⎩ ij |Y | ∂H
The auxiliary functions (χ k& , ηk& ) are the unique solutions to the cell problems: ⎧ −νy χ k& + ∇y ηk& = 0 ⎪ ⎪ ⎪ divy χ k& = −δk& ⎨
−στ (χ k& , ηk& ) = ν(n& τk + nk τ& ) ⎪ k& k& ⎪ ⎪ ⎩ −σn (χ , η ) = 2νnk n& k& k& χ ,η
in G, in G, on ∂G, on ∂G, are Y -periodic.
The homogenized tensor enjoys the following properties: (i) aij k& = ak&ij = aij &k ∀i, j, k, &. (ii) ∃δ1 > 0 such that aij k& ξij ξk& ≥ δ1 ξij ξij for all symmetric matrices (ξij ). (τ ) (τ ) (n) (n) (iii) dij = dj i ≥ 0, dij = dj i ≥ 0 ∀i, j . Let us note the features of the homogenized problem which are different from the earlier ones. It does not involve pressure and the incompressibility condition does not figure. Extension operators For the extension of velocity field, a well-known procedure introduced in [16] is adapted. In particular, the extension operator P ε satisfies the estimate: ∇P ε uε L2 () ≤ C∇uε L2 (ε ) . On the other hand, pressure field extension is not classical but it is more simple than the previous cases. More precisely, it is extended as a global constant in the holes which is uniquely determined by the property that the extended pressure has zero mean value in . We denote it by Qε (see [17] for details on its construction). The homogenization result is as follows: T HEOREM 5. We have P ε uε ! u
in H01 ()N -weak,
where u is the unique solution of the homogenized system. With regard to the pressure field, we have Qε p ε ! bk& ek& (u)
in L2 ()-weak.
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669
Idea of the proof As remarked earlier, the absence of the pressure term in the homogenized equation is the novel feature here on which we focus. We know that the original system implies that σ ε satisfies ⎧ ⎨ −div σ ε = f σ ε + εατ uετ = 0 ⎩ τε σn + εαn uεn = 0
in ε , on ε , on ε .
As a first step, we multiply this system by an arbitrary test function v ∈ H01 ()N . We see the appearance of boundary integrals on ε . These terms are converted to integrals over ε by means of a suitable smooth extension of the normal and tangential fields on ε to the whole of ε . Passing to the limit, we obtain −div σ =
|Y ∗ | f − (d (n) + d (τ ) )u |Y |
in ,
where σ is the L2 ()-weak limit of σ ε . In the next step, we play the usual game with equations satisfied by (uε , p ε ) and (χ k& ( xε ), ηk& ( xε )) in which we use the extended fields of normal and tangent. In this process, the pressure term p ε disappears because div y χ k& is constant and the average of Qε p ε over vanishes. Thus the limiting relation involves only (σ, u, f ) and the homogenized coefficients. Eliminating f from the two relations obtained above, we get −σij +
|Y ∗ | aij k& ek& (u) = 0 in ∀i, j. |Y |
Substituting σ into the previous relation will yield the required homogenized equation for u. To obtain the macroscopic behaviour of the pressure, one has to use the extended constitutive relation in : Qε σ ε = 2νχε e(uε ) − Qε p ε I. The difficulty lies with the first term on the right-hand side of the above relation which involves the product of two weakly convergent sequences. The idea to overcome this is to use the first order corrector for ∇uε , namely
χε ∇uεi = χε ∇ui + ek& (u)∇χik& + riε with riε → 0, for 1 ≤ i ≤ N .
in L2 (),
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Let us now consider the Navier–Stokes system with a Fourier boundary conditions, namely ⎧ −νuε + (uε · ∇)uε + ∇p ε = f ⎪ ⎪ ⎪ ⎨ div uε = 0 uε = 0 ⎪ ⎪ ⎪ στε + εατ uετ = 0 ⎩ σnε + εαn uεn = 0
in ε , in ε , on ∂, on ε , on ε .
It is well-known that the above system is well-posed provided f is small. The homogenized system in this case is the following one:
Lu + (u · ∇)u + Mu = u=0
|Y ∗ | |Y | f
in , on ∂,
where M is a nonlinear operator M = (M1 , . . . , MN ) with Mi u = fij k& uj ek& (u) and ∂χik& 1 dy. fij k& = |Y | ∂yj Y∗
The convergence result in this case can be stated as follows: T HEOREM 6. Let N = 2, 3. Assume that f L2 ()N is small. Denoting by P ε uε , the same extension of uε as before, we have P ε uε ! u
in H01 ()N ,
where u is the unique solution satisfying the homogenized boundary value problem written above. As in the linear case, the homogenized problem does not contain pressure force. However, there is no known result describing the macroscopic behaviour of the pressure in the nonlinear case. Another difference with the linear case is the appearance of the strange term Mu. Further, it is somewhat surprising that this term is of the same type as the inertial term (u · ∇)u. We did not meet such a phenomenon in Brinkman’s law, where the strange term was of order zero. These are some of the effects of Fourier boundary conditions. Idea of the proof The nonlinear term is treated as a compact perturbation in H −1 ()N by pushing it to the right hand side. Thus the analysis is reduced to the linear case where the external forcing on the right hand side is strongly convergent in H −1 ()N . This can be carried out in a manner analogous to the one discussed above.
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2.3. The Stokes sieve problem It is a classical technique to generate turbulence by passing the fluid through a fine sieve with high velocity. The general problem of describing oscillations of the velocity field in such a situation is open. Here we consider a simpler case. More precisely, we are interested in the behavior of the fluid flow governed by Stokes system across a periodic sieve (see Figure 2). We begin by describing the geometrical configuration (see Figure 3). + Let =]0, L1 [× · · · ×]0, LN −1 [ be a rectangle in RN −1 and let = ×]L− N , LN [ be − + N the rectangular fluid container in R . We assume that LN < 0 and LN > 0 and we set − LN = L + N − LN . We define also + = ×]0, L+ N [, − = ×]L− N , 0[, + = × {L+ N }, − = × {L− N }. & respectively. On , we place a The lateral part of the boundary of ± are denoted ± periodic sieve with period ε and the size of the holes is also of order ε. The reference cell is Y ⊂ RN −1 and the hole inside Y is denoted by H . Let = Y \ H . We consider a subsequence ε → 0 such that is a finite union of re-scaled cells {Yi ε }. (For more general cases and complicated geometries, the reader is referred to C. Conca [18]).
Fig. 2. A Stokes flow across a periodic sieve.
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Fig. 3. Geometrical configuration for the Stokes sieve problem.
The hole inside Yi ε is denoted as Hi ε . Let ε = \ ∪ Hi ε denote the solid part of the i
sieve. The fluid flows through ∪ Hi ε . The region occupied by the fluid is i
ε = + ∪ − ∪ H ε
with H ε =
D i
Hi ε .
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673
The flow of the fluid is governed by the Stokes system: ⎧ −νuε + ∇p ε = f ⎪ ⎪ ⎪ div uε = 0 ⎪ ⎪ ⎨ uε = 0 uε = b+ ⎪ ⎪ ⎪ ε − ⎪ ⎪ ⎩ uε = b u =0
in ε , in ε , on γ ε , on + , on − , l . on ±
We assume b± are smooth in ± and have compact support in ± respectively. The problem of homogenization here is to study the asymptotic behavior of the above system as ε → 0. A crucial parameter in the analysis is the effective flux across the holes of the sieve:
uεN (x , 0)dx
F= H ε
which is equal to F=
+
b · nds = −
+
b− · nds
−
due to the incompressibility of the fluid. The macroscopic behaviour depends on whether ε ) for the restriction of F vanishes or otherwise. In the sequel, we use the notation (uε± , p± ε ε ± ε (u , p ) to the domains respectively. We choose p uniquely such that its average over ε vanishes. We set ⎧ ⎪ ε ⎪ ⎨ p¯ ± = ⎪ ⎪ ⎩ ◦ε p
1 |± |
ε ± = p±
p ε dx, ±
ε. − p¯ ±
T HEOREM 7. Let F = 0. We consider the solution (uε , p ε ) where p ε is determined uniquely with the condition that it has zero mean value in ε . Then we have ⎧ ε → u± ⎪ ⎨ u± 1 ε →0 ε 2 p¯ ± ⎪ ◦ ⎩ ◦ε p± →p±
in H 1 (± ), in R, in L2 (± ).
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Here (u+ , p+ ) and (u− , p− ) are respectively: ⎧ −νu± + ∇p± = f ⎪ ⎪ ⎪ ⎪ ⎨ div u± = 0 u± = 0 ⎪ ± ⎪ u ⎪ ±=b ⎪ ⎩ u± = 0
solutions of the (decoupled) Stokes problem in ± in ± , in ± , on , on ± , l . on ±
Furthermore, we have the trace estimate 1
uε L2 ()N ≤ Cε 2 .
As one can see, the average of the pressure has a singular behavior as ε → 0 and this is due to the presence of the solid part of the sieve. When F = 0, this singular behavior worsens. The trace estimate on gives rise to u± = 0 on which allows the decoupling of the limit problem. T HEOREM 8. Let F = 0. Then we have 3 1 in H 1 ()N , ε 2 uε → 0 1 ◦ε ε 2 p ± → 0 in L2 (± ). The above result begs the following questions: Can one describe the oscillations in uε ◦ε ε? and p ± ?, and What is the behaviour of p¯ ± To this end, we introduce test functions which capture the oscillations due to the sieve. Let B be the infinite vertical cylindrical domain: B = (Y × R) \ H . We consider the Stokes-type problem in B where we impose no-slip boundary condition on : ⎧ −νχ + ∇η = 0 in B, ⎪ ⎪ ⎪ in B, ⎪ ⎨ div χ = 0 χ =0 on = Y \ H , ⎪ (χ, η) are periodic with respect to y , ⎪ ⎪ ⎪ ⎩χ → 1 as |y N | → ∞. N The last requirement allows a matching between the flow near the sieve and far from it. It can be shown that there are constants η±∞ such that lim
yN →±∞
η = η±∞ .
We define η∗ = η−∞ − η+∞ .
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T HEOREM 9 (Description of oscillations near the sieve). Let F = 0. Then (i) There exists a reminder r ε ∈ H 1 ()N such that x F χ + r ε (x) in ε , uε (x) = || ε rε → 0 in L2 ()N , 1 ε rε 2 r → 0 in H 1 ()N . ε ∈ L2 (± ) such that (ii) There exist s± ◦ε
p ± (x) = 1
x F −1 ◦ ε η + s± (x) ε ± || ε
ε →0 ε 2 s±
in ± , in L2 (± ).
ε ∈ R such that (iii) There exist t± ⎧ F |∓ | ∗ −1 ⎨ ε ε η ε + t± , p¯ ± = ∓ |||| ⎩ ε εt± → 0.
(iv) As a consequence, we deduce uε !
F N e ||
in L2 ()N .
From the above theorem, one can show that the macroscopic behaviour in + and − decouples. More precisely, T HEOREM 10 (Behavior away from the sieve). Let F = 0. We have then the following convergences: uε± → u± ◦ε ◦ p ± →p±
in L2 (± ), in H −1 (± ),
where (u± , p± ) are characterized as the solution of (decoupled) Stokes problems: ⎧ −νu± + ∇p± = f ⎪ ⎪ ⎪ ⎪ div u± = 0 ⎪ ⎨ F N e u± = ⎪ || ⎪ ⎪ ⎪ ⎪ ⎩ u± = 0 u± = b±
in ± , in ± , on , l , on ± on ± .
We can interpret the above result as follows: As the sieve becomes finer and finer, two zones appear. A boundary layer zone near the sieve where the fluid is forced by the noslip boundary condition on the solid part of the sieve. Because of this, a periodic pattern
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of the flow develops and is described by the test functions (χ, η). Outside this boundary layer zone lies a second zone in which the flow is slow and adjusts itself from the periodic pattern to the boundary conditions imposed on top-bottom boundaries. Regarding the proofs, we remark that method of oscillating test functions is followed with (χ, η) as test functions. 3. Advection and diffusion of passive scalars This section is devoted to the application of homogenization techniques to advectiondiffusion equation. To this end, let us consider a passive scalar field T = T (x, t) (temperature, concentration of tracer particles) advected by an oscillating velocity field and diffused in a fluid flow. This is an old problem, some aspects of which have been reviewed in [46]. We focus here on some of the recent developments, in particular on the presence of a mean flow and its effects on the homogenization process. Let u = u(x, t) denote the velocity field of the flow which is assumed to be incompressible: divx u = 0. Above situation is modeled by the equation 3
∂T + u · ∇T = T ∂t T (x, 0) = Tin (x)
in RN × R+ , in RN .
We assume that u is a superposition of a large scale mean flow V and a small scale periodic fluctuation v of mean zero. We suppose both V and v are incompressible. We introduce the space-time average over the periodic cell: def
1
v =
v(y, τ )dydτ. 0 [0,1]N
By our hypothesis v = 0. We will also have occasion to use space and time averages as well: def vy (τ ) = v(y, τ )dy, [0,1]N def
1
vτ (y) =
v(y, τ )dτ. 0
A simple nondimensionalization procedure applied to the above equation puts it in the following form: ⎧ % x t aε ⎨ ∂T (x, t) + V (x, t) + av , · ∇T (x, t) = T (x, t), S ε η P ⎩ ∂t T (x, 0) = Tin (x).
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Here a, ε, η represent respectively the ratios of the small scale velocity magnitude, length scale and time scale to their large scale counterparts. The basic assumption of the homogenization method is that there is scale separation both in space and time: ε ! 1, η ! 1. Two other nondimensional parameters S and P enter the picture: P , the Péclet number associated to the small scale velocity, roughly measures the ratio of the corresponding advection to the molecular diffusion. The parameter S is known as Strouhal number measures the ratio of the sweeping time associated with the mean flow to its characteristic time. In the same vein, one can also introduce the Péclet number Pg associated to the large scale flow and the Strouhal number Sl associated with the small scale flow. Since we have already assumed relationship between large and small scale flow quantities via (a, ε, δ), following relations result: S=
ηa Sl , ε
Pg =
1 P. aε
In the sequel, we assume that (P , S) are fixed parameters independent of (ε, η). We simplify further by taking S = 1. On the other hand, we make (a, η) depend on ε via a = εα
η = εβ
for some β > 0 and α ∈ R. Thus, we have the following equation: % ⎧ ∂T ε x t ⎪ ⎪ ∇T ε (x, t) in RN × R+ , (x, t) + V (x, t) + ε α v , β ⎨ ∂t ε ε ε α+1 ε ⎪ ⎪ ⎩ = P ∇T (x, t) T ε (x, 0) = Tin (x)in RN . Our objective is to analyze the behaviour of T ε as ε → 0. The progress in this problem has been reviewed in [39], [38]. For recent advances, see [51]. We have several cases to consider. Broadly they can be categorized as follows: Case I. a > 1 or equivalently α < 0. This case signifies that the strength of the mean flow is weak compared to that of the fluctuating part. Case II. a = 1 or equivalently α = 0. In this case, both mean flow and the fluctuations have comparable strength. Case III. a < 1 or equivalently α < 0. Here mean flow dominates over the fluctuations. As we shall see, there are several subcases to consider inside each case. To arrive at them, let us bring out the multiscale structure of the basic operator of interest which is as follows: % ∂ x t ε α+1 ε α L ≡ · ∇x − + V (x, t) + ε v , β x . ∂t ε ε P To this end, we follow multiscale asymptotic expansion [12] by introducing the fast periodic variables: y = xε , τ = εtβ . The method treats these variables independently along with the slow variable (x, t). By chain rule, Lε can be expressed in terms of (x, y, t, τ ): Lε = ε −β R0 + ε α−1 R1 + ε −1 R2 + ε α R3 + R4 + ε α+1 R5 , def
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where def
R0 =
∂ ∂τ
def
R1 = v(y, τ ) · ∇y −
1 y P
def
R2 = V (x, t) · ∇y def
R3 = v(y, τ ) · ∇x −
2 ∇x · ∇y P
∂ + V (x, t) · ∇x ∂t 1 def R5 = − x . P def
R4 =
Note the nonuniform way in which various powers of ε appear in Lε . Viewed in this manner, the case β = 1 − α is special in the sense that the first two terms containing R0 and R1 combine into one single term. It is then natural to consider subcases: β = 1 − α,
β < 1 − α,
β > 1 − α.
We will not present complete analysis of all cases. We will merely consider some examples in each case and highlight the phenomena involved. For full details, the reader may refer to the literature cited above. 3.1. Case I (a) (α = −1/2, β = 3/2) In this case, Lε takes the form Lε = ε −3/2 (R0 + R1 ) + ε −1 R2 + ε −1/2 R3 + R4 + ε 1/2 R5 def
wherein we notice the structure of a geometric series with respect to ε with ratio ε 1/2 . Thus it is reasonable to propose the following ansatz (which is a power series in ε 1/2 ) for the solution of Lε T ε = 0: T ε (x) = T0 + ε 1/2 T1 + εT2 + · · · . Here each Tj = Tj (x, y, t, τ ) is a function of fast and slow variables and periodic with respect to fast variables (y, τ ). Following [12], it is now straight forward to carry out the computations to find Tj . We merely remark that the basic operator (or, pivot operator) for which we apply Fredholm alternative to solve the equation at each level is R0 + R1 . We describe below the first three terms: T0 , T1 are independent of (y, τ ),
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T0 is unique solution of ∂T0 + V (x, t) · ∇x T0 = 0, ∂t
T0 (x, 0) = Tin (x).
T1 is characterized as the solution of ∂T1 1 ¯ + V (x, t) · ∇x T1 = ∇x · {(I + K).∇ x T0 }, ∂t P T1 (x, 0) = 0, T2 (x, y, t, τ ) = χj (y, τ )
∂T0 (x, t), ∂xj
where {χj , j = 1 . . . N} are solutions of cell problems: ∂χj 1 + v · ∇y χj − ∇y χj = −vj , ∂τ P χj is periodic in (y, τ ). Combining the first two terms T¯ = T0 + ε 1/2 T1 , we arrive at an approximation to T ε : T ε ∼ T¯ and T¯ is the solution of ⎧ ⎨ ∂ T¯ ε 1/2 + V · T¯ = ∇ · (K ∗ ∇ T¯ ) P ⎩ ¯∂t T (x, 0) = Tin (x)
in RN × (0, ∞), in RN .
In the above equation, we see the appearance of the matrix K ∗ called the effective diffusion tensor. It is given by K ∗ = I + K¯ where K¯ is the enhanced diffusion tensor defined by K¯ ij = −P vi χj
∀i, j = 1, . . . N,
The approximation T¯ is called the effective/homogenized solution and the equation satisfied by T¯ is called the homogenized equation. Comments Let us make several observations on the above result. 1. Justification of the above asymptotic expansion is by now standard in homogenization. 2. Comparing the equations satisfied by T ε and T¯ , we see that the main phenomenon is the replacement of the periodically oscillating advection term by the additional diffusivity term. To see this quickly, let us substitute the multiscale asymptotic expansion in the advection term and average with respect to (y, τ ): we obtain
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ε
−1/2
x t x t ε −1/2 v ∇x T = ε Tε ∇x · v , , ε ε 3/2 ε ε 3/2 x t = ε −1/2 ∇x · v , 3/2 T0 ε ε x t + ∇x · v , 3/2 T1 ε ε x t 1/2 T2 + o(ε 1/2 ). + ε ∇x · v , ε ε 3/2
Since T0 and T1 are independent of (y, τ ) and v = 0, the first two terms vanish. The third term is ε 1/2 vi χj
∂ 2 T0 ε 1/2 ¯ ∂ 2 T0 =− Kij ∂xi ∂xj P ∂xi ∂xj
which gives the enhanced diffusion term. 3. The cell problems, the enhanced and the effective diffusion tensors depend only on the fluctuation and not on the mean flow. This is to be expected in the present case which supposes that the mean flow is small compared to the fluctuations. 4. The effective diffusion tensor is in general not symmetric. However, because it is constant, only its symmetric part will appear on the right hand side of the homogenized equation. It can easily be checked that the symmetric part is given by 1 Sij = (K¯ ij + K¯ j i ) = ∇y χi , ∇y χj 2 and hence nonnegative definite. Consequently, the diffusion is enhanced. This is due to the incompressibility of the fluid. This cannot be guaranteed in other situations. For instance, if V ≡ 0 and v is in given by a periodic potential (hence compressible) then the effective diffusivity is depleted in the sense that the additional diffusion tensor K¯ ≤ 0 [52]. 1 5. The global Péclet number is given by Pg = ε −(α+1) P = ε − 2 P and this explains the coefficient on the right hand side of the homogenized equation. On the other hand, the local Strouhal number Sl = 1. Hence time variation with respect to τ , local advection and the local diffusion are of same order and that is why they all contribute in the cell problem. 3.2. Case I (b) (α = −1, β = 2) In this case, we have Lε = ε −2 (R0 + R1 ) + ε −1 (R2 + R3 ) + (R4 + R5 ). def
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This case is called the diffusive limit case. We note easily that Lε has a structure similar to the one considered in Case I (a). Hence the same computations and conclusions remain valid here too. Computations are made somewhat simpler here because the powers of ε are integers and not fractions. We singled out this case separately mainly because it is widely studied in the literature [38,39]. 3.3. Case I (c) (α = −1/2, β = 1) We see that the operator Lε is of the form Lε = ε −3/2 R1 + ε −1 (R0 + R2 ) + ε −1/2 R3 + R4 + ε −1/2 R5 . def
Though Lε has the structure of geometric series with respect to ε, the pivot operator changes from (R0 + R1 ) to R1 . This has consequences and differences with Case I (a) which we want to highlight. The main point is that the local Strouhal number is Sl = ε 1/2 and hence small. Temporal fluctuations are then too slow and so they are not expected to play the same role as in the Case I (a). Indeed, the cell problem changes to the following one where τ is only a parameter: 1 y χj = −(vj − vj y ), P χi is periodic with respect to y.
v · ∇y χj −
Here, we recall vj y denotes the average of vj with respect to the variable y. Other conclusions remain the same as in Case I (a) and hence will not be repeated. 3.4. Case I (d) (α = −1/2, β = 2) The basic operator takes the form Lε = ε −2 R0 + ε −3/2 R1 + ε −1 R2 + ε −1/2 R3 + R4 + ε 1/2 R5 . def
The pivot operator has changed from (R0 + R1 ) to R0 when we compare with Case I (a). The local Strouhal number Sl = ε −1/2 and hence is large which signifies that the time oscillations in v are fast and hence they get averaged in the definition of the cell problem which is now the following: vτ · ∇y χj −
1 y χj = −vj τ , P χj is periodic in y.
We see that χj is independent of τ . The ansatz of Case I (a) works fine here too wherein we incorporate the new cell problem defined above.
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Other cases Let us now pass a few comments concerning other subcases of Case I not considered above. First assume that α = −1/2 and β > 0 is arbitrary. In this case, the operator is as follows: 1
Lε = ε −β R0 + ε −3/2 R1 + ε −1 R2 + ε −1/2 R3 + R4 + ε 2 R5 . def
When β ∈ { 12 , 1, 32 , 2}, we see the loss of geometric series structure of Lε . A similar situation has already been encountered in the literature. See [12], pp. 263–266. The main point is that the ansatz proposed in Case I (a) is insufficient and we need to supplement it with other terms. Computations with such an expansion and the corresponding justification can be carried out along the lines of [12]. In fact, the above idea can be adapted to treat the cases β < 1 − α and β > 1 − α (even if α = − 12 ) provided we know how to treat the case β = 1 − α. In view of this, it remains to consider the case (α = − 12 , β = 1 − α). The structure of Lε is then more complicated. To our knowledge, there is no published version of the analysis of this case. However, some progress is reported in the thesis of G. Pavliotis [51].
3.5. Case II In this section, we consider the situation where mean flow and oscillating part are of equal strength i.e. α = 0. Further, we discuss only the case β = 1 − α = 1. Other cases may be treated in the manner indicated previously. Thus our equation governing the passive scalars is % ⎧ ∂T ε x t ⎪ ⎪ · ∇T ε (x, t) in RN × R+ , ⎪ ⎨ ∂t (x, t) + V (x, t) + v ε , ε ε = T ε (x, t) ⎪ ⎪ ⎪ P ⎩ ε T (x, 0) = Tin (x) in RN . Next, we give the statement describing the average behaviour of T ε : T HEOREM 11 (Main Result). The solution T ε is approximated by T 1,ε whose average T¯ = T 1,ε is characterized by the homogenized equation: ⎧ ε ⎨ ∂ T¯ (x, t) + V (x, t) · ∇ T¯ (x, t) = ∇ · (K ∗ (x, t)∇ T¯ (x, t)) ∂t P ⎩ T¯ (x, 0) = Tin (x) Here the effective diffusion tensor is given by Kij∗ (x, t) = δij + K¯ ij (x, t), K¯ ij (x, t) = −P vi (y, τ )χj (x, t, y, τ ).
in RN × R+ in RN .
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The corresponding cell problem for χ = (χ1 , χ2 . . . χN ) is as follows: 3
1 ∂χ + {V (x, t) + v(y, τ )} · ∇y χ − y χ = −v, ∂τ P χ = χ(x, t, y, τ ) is periodic in (y, τ ).
The above result is similar to the corresponding main result for weak mean flows (in Case I) except for the new feature that the cell problems this time depend on the mean flow and consequently the effective diffusion matrix K ∗ depends on (x, t). This has an interesting consequence, namely that right hand side of the homogenized equation is not purely diffusive; there is also an advection. More precisely we can rewrite it in the form Kij∗ (x, t)
∂ 2 T¯ ε − U · ∇ T¯ , ∂xi ∂xj P
where def
Ui (x, t) = −
∂Kij∗ ∂xj
,
i = 1 . . . N.
The advection term of the right hand side can be combined with the one already present in the left hand side. Secondly, the matrix (Kij∗ ) on the right hand side can be replaced by its symmetric part (Sij ). We finally obtain ∂ 2 T¯ ∂ T¯ εU ε . + V+ · ∇ T¯ = Sij ∂t P P ∂xi ∂xj The important feature is the appearance of both the symmetric and the antisymmetric part of K ∗ in the new advection term. This was not the case with weak mean flows of Case I. Next, the presence of ε in the right hand side of the homogenized equation should not surprise us because the global Péclet number in the present case is Pg = ε −1 P . Finally, we note another feature of the homogenized equation: it is easily verified that the contribution of the antisymmetric part of K ∗ to the additional advection velocity is divergence free. The same cannot be said of that of the symmetric part of K ∗ . Thus even though we started with incompressible flows, the homogenized equation contains an advection term in which the flow is general compressible. The proof of the main result is via multiscale asymptotic expansion. Introducting fast variables y = xε , τ = εt , the operator under consideration can be expanded as Lε = ε −1 L0 + L1 + εL2 , def
with def
L0 =
∂ 1 + (V + v) · ∇y + y ∂τ P
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∂ 2 + (V + v) · ∇x − ∇x · ∇y ∂t P 1 def L2 = − x . P def
L1 =
Thus the following ansatz is natural: T ε = T0 + εT1 + ε 2 T2 + · · · where each Tj is a function of (x, t, y, τ ) and periodic in (y, τ ). Substitution into the equation yields as usual the following results: T0
is independent of (y, τ ),
∂T0 + V · T0 = 0 in RN × R+ ∂t T0 (x, 0) = Tin (x)
in RN ,
T1 (x, t, y, τ ) = χ(x, t, y, τ ) · ∇x T0 (x, t) + T¯1 (x, t), where T¯1 is the solution of ⎧ 1 ⎨ ∂ T¯1 + V · ∇ T¯1 = ∇ · (K ∗ (x, t)∇T0 ) ∂t P ⎩ T¯1 (x, 0) = 0
in RN × R+ , in RN .
Thanks to above results, we can take T 1,ε = T0 + εT1 and this provides the required approximation of T ε . Justification of these steps is done in a classical manner.
3.6. Case III We consider the case α > 0. The general difficulty of dealing with advection by mean flow and molecular diffusion is due to the difference between the time scales associated with them. This difficulty gets worse when V is large. Because of this, there are many unsolved issues in the homogenization problem. In particular, the method of asymptotic expansion used earlier does not succeed fully. We now employ the two-scale convergence [5]. It is easy to get the first term T0 via two-scale convergence for the range 0 < α < 1. Indeed, the standard energy estimates show that for any fixed time τ ∗ , we have T ε L2 (0,τ ∗ ,L2 (RN )) + ε
1+α 2
∇T ε L2 (0,τ ∗ ,L2 (RN )) ≤ C.
By compactness of two-scale convergence, it follows that 2
T ε ! T0 (x, t)
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1+α 2
685
2
∇T ε ! ∇y T1 (x, y, t),
for some T0 independent of y and T1 ∈ L2 (0, τ, H#1 (Y )/R) where the symbol # signifies periodicity. Using this information, it is easy to pass to the limit in the equation for T ε in the sense of two-scales. This shows that T0 is the solution of ∂T0 + V · ∇T0 = 0, ∂t
T0 (x, 0) = Tin (x).
To proceed further and get the second term, the idea in the thesis of G. Pavliotis [51] is to introduce the quantity T 1,ε = ε −(α+1) (T ε − T0 ) and consider its two-scale limit T1 . The difficulty lies in deriving suitable estimates on T 1,ε . The source of the difficulty seems to be the resonant interaction between the mean flow and the molecular diffusion. This effect has been seen numerically by the following line of investigation: Fixing the relative magnitude a independent of ε, the corresponding problem can be homogenized as in Case II since the mean flow and the fluctuation are of same order. The cell problem will depend upon a. Its asymptotic behaviour (as a consequence, that of the effective diffusion tensor) for small a, can be investigated, for instance, numerically. As a third approach, let us present analytical results in a special case. Example (Pure advection equation). Here, we consider the limit case where Péclet number P = ∞ and thus the molecular diffusion is completely absent. On the other hand, the advecting velocity field u, which is assumed to be stationary, is more general and need not be in the form V (x) + ε α v( xε ) with α > 0. Let us describe the main result from [19] and [26] which is proved using two-scale convergence. The reader can find results of this type also in [21] where different arguments are used. Consider x ∂T ε + u x, · ∇T ε = 0, T ε (x, 0) = Tin (x) in RN . ∂t ε Assume as before u = u(x, y) is smooth, periodic with respect to y and satisfies divx u(x, y) = 0
and
divy u(x, y) = 0.
Assume further that Tin ∈ L2 (RN ). It follows easily that T ε L∞ (R + ;L2 (RN )) ≤ C. It can be proved that the entire sequence T ε converges to T0 = T0 (x, y, t) ∈ L2loc (R+ , L2 (RN ) × L2# (Y )) in the sense of two-scales. Further T0 is characterized as the unique solution of the following homogenized system: u(x, y) · ∇y T0 (x, y, t) = 0,
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R+ RN Y
% ∂φ + u · ∇x φ T0 dxdydt + φ(x, y, 0)Tin (x)dxdy = 0, ∂t RN Y
for all test functions satisfying 3 u(x, y) · ∇ φ(x, y, t) = 0, y φ = φ(x, y, t) is smooth, periodic with respect to y compactly supported in (x, t).
and
Once again, we have a system in which the mean quantities are coupled with the oscillations. A situation where they can be decoupled is the following: Assume that the space dimension is two and the vector field u(x, y) does not have any stagnation point: u(x, y) = 0 ∀ x, y. Consider the average u(x) = Y u(x, y)dy. Suppose that the components of the vector u(x) are not commensurate for x a.e. Under this hypothesis, one can show [19] that T0 is independent of y ∂T0 + u(x) · ∇x T0 = 0, ∂t
T0 (x, 0) = Tin (x).
The above requirement on u(x, y) says that the flow defined by the vector field u(x) on the two-torus is ergodic for x a.e. Under this hypothesis, we note that there is no addition to the diffusivity and it remains zero in the homogenized equation as well. The important conclusion which emerges from the above discussion is that the flow properties of the mean velocity field play a crucial role in homogenization of pure advection problem. We can heuristically argue then that the same is true for our initial problem if P is finite but large. Numerical results reported in [51] seem to support this conclusion. More precisely, let us now go back to our advection-diffusion equation and assume that the mean velocity field V is constant and the space is two-dimensional. If the components of V are not commensurate then there is no enhancement of diffusivity in the limit V % v. If we suppose the contrary, then there is enhancement of diffusion in the limit V % v and P % 1. Further this enhancement is independent of the mean flow. Thus in Case (III), a greater variability of the effective diffusivity can occur, depending upon the specific properties of the mean flow: from no enhancement in the diffusivity to the appearance of resonant enhanced diffusion. As far as our knowledge goes, there is no general homogenization result validating above observations and covering a wide range of mean flows. As a closing remark in this direction, let us mention that if the fluctuations are assumed to be random with sufficient mixing properties (instead of being deterministic and periodic) then the above difficulties disappear surprisingly and one can prove a homogenization result [31] and it is shown that the effective diffusion tensor is computed through the so-called Green–Kubo formula. Since the random case is not discussed in this review, we will not pass any further comments in this direction. Next, we would like to mention the
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paper [13] which deals with the construction of boundary and initial layers in the solution of the advection-diffusion equation in an infinite strip. However the model in [13] incorporates higher molecular diffusivity than the one considered in this review. Consequently, the difficulties invoked in Case (III) above do not seem to appear in [13]. Let us also mention that [8,9] deal with some other aspects of the problem of homogenization of passive scalars. One can also add nonlinear terms representing chemical reaction to the advectiondiffusion equation. Majda and Souganidis have studied homogenization of such systems in a series of publications. See [40] for instance. 3.7. Estimates on the effective diffusion tensor While our discussion so far was devoted to obtaining the effective diffusion tensor K ∗ associated with the advection-diffusion equation, this paragraph is about the estimates on K ∗ in terms of the nondimensional parameters, namely the Péclet number P and the Strouhal number S appearing in the equation. We review some of the results in this direction. Most of these results are concerned with two dimensional cases in which mean flow V ≡ 0 or V ≡ constant. Further, they assume that P is large and S is fixed. General situations remain largely unexplored. Accordingly, we confine ourselves to the cases mentioned above. The kind of issues addressed here, methods followed and the results obtained may be compared to those in the case of estimation of heat conductivities for two-phase composites [45]. First consider the case V ≡ 0. Let us begin with an instructive special case which is somewhat analogous to layered composites [45]. Example (Oscillating shear flow). Let us place ourselves in the Case I (b) discussed above with V ≡ 0. Assume that v is independent of τ and is in the form v(y) = (0, v2 (y1 )). This vector field defines a shear in y2 -direction. The corresponding cell problem can be explicitly solved. Indeed, we get χ1 ≡ 0, since v1 ≡ 0. The solution χ2 depends only on y1 since v2 does so. Thus the equation for χ2 is reduced to an ODE with constant coefficients which can be solved via Fourier decomposition. We find ∗ K11 = 1, ∗ = 1 + P 2 v2 2H −1 , K22 ∗ ∗ = K21 = 0. K12
The above example can be generalized and bounds on the effective diffusivity K ∗ = K ∗ (P ) can be deduced using the Stieltjes integral representation [7] or (primal/dual) variational principles for K ∗ [7,20]. At this point, let us mention an important tool used to obtain variational principles is the so-called stream matrix (y) related to the stationary oscillating velocity field v(y) via the relation: ∇y · (y) = v(y).
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Since v is incompressible, we can choose to be skew-symmetric. Introduction of the stream matrix enables us to connect advection-diffusion operator with the following parabolic operator: ∂t = ∇ · {(I + )∇}. Consequently, variational principles ([45], [30], [49]) for the homogenized matrix associated with the later can be exploited. As an application, it is shown that for every unit vector e, ˆ K ∗ eˆ · eˆ
is increasing with respect to P ,
C1 ≤ K ∗ eˆ · eˆ ≤ C2 P 2 , where C1 , C2 are independent of P . The question of attainment of these lower and upper bounds motivates one to introduce the following concepts: We have maximally enhanced diffusion in a given direction eˆ if K ∗ eˆ · eˆ scales like P 2 for P large. Similarly, if K ∗ eˆ · eˆ remains bounded then we have minimally enhanced diffusion in the direction e. ˆ The next natural step is to investigate the above concepts in a given flow configuration. Various two-dimensional examples are studied in [38,39]. Typically, two types of flows are considered for the oscillating part v: shear flows which model hyperbolic flows and cellular flows modelling elliptic flows. For steady shear flows, diffusion is maximally enhanced along all directions except the one orthogonal to the shear direction. In the orthogonal direction, we have minimally enhanced diffusion. This can be heuristically explained as follows: In the absence of molecular diffusion, tracer particles would move in the shear direction at a ballistic rate (meaning that the distance traveled grows linearly in time). The addition of molecular diffusion knocks the tracer off its path, destroying the ballistic motion and producing a diffusive transport behaviour instead. This explains why K ∗ scales like P 2 in nonorthogonal directions. Contrary to our impression, molecular diffusion acts an impediment to the effective transport in this case. In the case of steady cellular flows, the growth of K ∗ is like P 1/2 as P → ∞ and hence it is neither minimally nor maximally enhanced diffusive in any direction. This comes about in a manner explained below: Tracer particle is rapidly transported across any given cell. Molecular diffusion is needed to take it across the boundary to an adjacent cell. However, there is a sort of boundary layer whose width is P −1/2 . Flux across the boundary is determined by the sharp gradient of the passive scalar near the boundary of the cell. From these elements, we can easily determine that K ∗ ∼ O(P 1/2 ). In this case, molecular diffusion helps in the large scale transport. The case of time dependent velocity fields is treated in [43]. The next question is to know the effects of the addition of a constant mean flow V on the above picture of minimally/maximally enhanced diffusion. In this case, there is a drastic change in the behaviour depending on the number theoretic properties of V . Mean flows V with irrational ratios and without any stagnation point (and satisfying some technical conditions) lead to minimally enhanced diffusion of in all directions. In contrast, flows V with rational ratios will generically induce maximally enhanced diffusion in all directions other than the one perpendicular to V , along which the diffusion will be minimally enhanced.
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Number theoretic connection in this context is somewhat reminiscent of KAM theorem in dynamical systems. For these assertions and more, see [39]. Let us close this section by mentioning a topic of relatively recent research. It is concerned with the extension of the above analysis to cover the case of advecting velocity field which is a superposition of infinitely many scales (even continuum of scales) without separation. This is a difficult problem. We have seen that the incompressible (resp. potential) vector field enhances (resp. depletes) diffusion coefficient of passive scalars. In the presence of several scales, we may therefore expect anomalous (sub/super) diffusion. For instance, in the case of enhancement, one may expect to see the characteristics of turbulent diffusion. Apart from the estimates on the effective diffusion tensor, such an analysis requires a suitable renormalization scheme. While there is an important amount of literature in the random case (see [39], Section 3.4 and [6], and the works cited therein), the deterministic case is also interesting and needs more attention. Let us finish this section by citing a few papers in this direction [10,11,50].
4. Convection effects In the previous section, we addressed the homogenization problem of advection with a known velocity field and therefore the equation was linear. In the present section, we consider convection models in which velocity field is one of the unknowns. More precisely, we address the problem of homogenization of Euler equations and Navier–Stokes equations governing incompressible fluid flows. There have been very recent developments in this area by Hou and his collaborators [25,27–29] which we would like to describe. We will be focused mainly on the Euler equation in two dimension. Three dimensional and viscous effects will be mentioned below referring the reader to the cited papers for details. The problem of homogenization that we want to study in this section can be stated as follows: The initial velocity field is supposed to be a superimposition of two parts: a slowly varying part U (x) and a rapidly oscillating part. We assume that oscillations are periodic with small period ε, an assumption with which we are working throughout this review. Smallness of ε signifies the scale separation between slow and fast motions. How such an profile evolves with time obeying Euler equations? Do the scales remain separated? If so, what are the evolution equations governing them? These are some of the issues addressed below. The basic tool to analyze this multiscale behaviour is the asymptotic expansion of [12]. This tool needs significant modifications because of the various new features of the present problem: nonlinear, nonlocal, generation of new scales etc. If one were to propose a naive expansion scheme, it will lead to an infinite number of scales without separation and the method breaks down. A quick glance through the literature of Fluid Mechanics will reveal a variety of turbulent models derived in adhoc manner using heuristic arguments [33]. Further, a given model need not be valid for all flow configurations. Based on its numerical performance and on comparison with experiments and observations, the model is typically evaluated for its virtues and drawbacks. For a mathematically oriented researcher, it is not easy to understand this type of analysis. A general unresolved issue has been to develop a way to understand these models and the associated numerical techniques (such as LES, subgrid
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modelling · · ·). Once it is developed, it will hopefully lead to better and more universal models and more efficient fluid computations. One of the first works to address this problem with a mathematical view point is [42]. Their idea of introducing a new phase function in the expansion to describe the evolution of small scales is quite insightful. Such an idea is classical in the study of propagation of high frequency waves using linear/nonlinear geometrical optics method [12], Chapter 4. But its introduction in the present context is new. However, the assumption in [42] that the small scales are convected by the mean flow turned out to be not fully correct and because of this, its authors ended up in a homogenized system and a cell problem whose well-posedness is not clear. By making more appropriate choices of the phase function, the works of Hou and his collaborators are geared toward removing the above defect. However, the task is not completed. As far as we know, there are lot of issues to be settled: convergence of the asymptotics, the well-posedness of the two-scale homogenized system, time of validity of the approximation, demonstration of its better stability properties, computational issues etc. The above contributions can only be regarded as a first step in this direction. In including the present discussion in this review, our goal has merely been to invite the attention of wider audience toward such problems. As mentioned above, the ideas will be presented for the following model: ⎧ ∂uε ⎪ + (uε · ∇)uε + ∇p ε = 0 in RN × R+ , ⎨ ∂t uε = 0 in RN × R+ , ⎪ ⎩ div x ε in RN , u (x, 0) = U (x) + W (x, ε ) where U (x) and W (x, y) are smooth vector fields W (x, y) being periodic with respect to y with zero mean: Let us denote by f , the average of a function f with respect to y variable over the periodic cell. The domain for variables is x ∈ R2 , t > 0. Further, we assume velocity fields vanish at infinity. Thus boundary effects and infinity effects are eliminated and we have only convection effects. Our objective is to obtain suitable ansatz for the solution (uε , p ε ) of the above system which reveals its multiscale nature and enables us to answer the questions raised above. The study starts by replacing the critical assumption of [42] by the following one: small scales are convected by the full velocity field uε . To justify it, let us recall a classical fact, namely vorticity ωε = ∇ ⊥ · uε = ∂x2 uε1 − ∂x1 uε2 is preserved along Lagrangian map. This means that ωε (x, t) = ωin (θ ε (x, t), ε −1 θ ε (x, t)), where θ ε is defined by ∂t θ ε + (uε · ∇)θ ε = 0,
θ ε (x, 0) = x.
This simple fact shows that the multiscale structure in the initial vorticity is conserved provided we change the phase to θ ε . This suggests two things: we work with the Lagrangian description of the flow treating α = θ ε (x, t) as an independent variable and work with
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vorticity-stream function formulation of the system. These are some of the key ideas of the approach. A few words about the later formulation: The stream function ψ ε is associated to the flow via the relation uε = −∇ ⊥ ψ ε . It is well-known that Euler evolution for (uε , p ε ) becomes a coupled system between a transport equation for ωε and an elliptic equation for ψ ε : ∂t ωε + uε · ∇ωε = 0, ωε (x, 0) = ωin (x, xε ), −ψ ε = ωε . Here ωin (x, y) is obtained from uε (x, 0): ωin (x, y) = ε −1 σ0 (x, y) + σ1 (x, y) + ρ(x), where σ0 (x, y) = ∇y⊥ · W (x, y), σ1 (x, y) = ∇x⊥ · W (x, y) and ρ(x) = ∇x⊥ · U (x). Advantages of the new formulation are clear: multiscale analysis in elliptic equation is a classical stuff [12] and that in the transport equation was reviewed in the previous section. The only difference is that the multiscale structure of uε is unknown which results in a coupling of multiscale structures. As suggested above, let us pass to Lagrangian description. This means a change of variables (x, t) → (α, t) with α = θ ε (x, t). Inverse of this is given by x = X ε (α, t) where X ε (t, α) is the so-called flow map: ∂t X ε (t, α) = uε (X ε (α, t), t), X ε (α, 0) = α. In terms of the variables (α, t), our coupled system takes the following form: ⎧ T
⎨ −∇α⊥ Dα X ε (Dα X ε )∇α⊥ ψ ε = ωin (α, αε ), ε + {∇α⊥ ψ ε · ∇α }X ε = 0, ⎩ ∂t X ε X (α, 0) = α. One word about the notation used. For the derivatives of scalar functions, ∇α is used whereas Dα denotes the derivatives on vector functions. In this new description, we do not have evolution law for the vorticity because it is constant along Lagrangian trajectory. We have, instead, an evolution equation for the trajectory itself. Since the newly appearing phase function is taken as an independent variable, a straight forward multiscale expansion is expected to work. Accordingly, we propose
ψ ε = ψ (0) (α, t, τ ) + ε ψ¯ (1) (α, t, τ ) + ψ (1) (α, t, y, τ ) + · · ·
X ε = X (0) (α, t, τ ) + ε X¯ (1) (α, t, τ ) + X (1) (α, t, y, τ ) + · · · where y = ε −1 α and τ = ε −1 t. We assume that {ψ (k) , X (k) ; k ≥ 1} are periodic functions with respect to y with zero means. Note that we do not require periodicity with respect to τ .
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Under these conditions, we have the following result which presents a multiscale homogenized system in which equations for (ψ (0) , χ (0) ) are coupled with cell problems for (ψ (1) , X (1) ): T HEOREM 12 (Homogenization in Lagrangian form). (a) ψ (0) , X (0) are independent of τ . (b) For (ψ (0) , X (0) ), we have 3
X (0) = α at t = 0, ∂t X (0) + ∇α⊥ ψ (0) · ∇α X (0) = 0,
F T (1) G −∇α⊥ · Dα X (0) Dα X (0) ∇α⊥ ψ (0) − ∇α⊥ · A(0) ∇y⊥ ψ0 = ρ(α). (1)
Here ψ0 is defined in terms of (ψ (0) , X (0) , ψ (1) , X (1) ) via the relation T −1 (1) ψ (1) = ψ0 + ∇α ψ (0) Dα X (0) X (1) . Further, A(0) is the matrix (Dα X (0) + Dy X (1) )T (Dα X (0) + Dy X (1) ). We have also (1) that A(0) ∇y⊥ ψ0 is independent of τ . (c) The pair (ψ (1) , X (1) ) satisfies the following cell problems with periodic boundary conditions: ⎧ (1) ⊥ (1) (1) ⊥ (1) (0) ⎪ ⎨ ∂τ X + ∇y ψ0 · ∇y X + ∇y ψ0 · ∇α X = 0, X (1) = 0 at τ = 0 ⎪ ⎩ −∇y⊥ · A(0) ∇y⊥ ψ0(1) = σ0 (α, y). (d) The leading order term u(0) of the velocity field is as follows: (1) u(0) (α, t, y, τ ) = −Dα X (0) ∇α⊥ ψ (0) − Dα X (0) + Dy X (1) ∇y⊥ ψ0 .
Comments Let us now make several comments on the above result. 1. The homogenized system couples mean quantities with oscillations. We are familiar with such features from [5] under the name two-scale homogenized system. Here, we see a coupled system of four equations in four unknowns (ψ (0) , X (0) , ψ (1) , X (1) ), transport equations for (X (0) , X (1) ) and elliptic equations for (ψ (0) , ψ (1) ). It can be shown that the matrices of coefficients appearing in the elliptic equations are symmetric and positive definite and so the system comprises of well-posed individual equations. The nature of their coupling remains to be investigated. In general, oscillations cannot be averaged out and we do not have a closed system for the mean quantities. 2. The coupling in the homogenized system is not arbitrary. It is “triangular” in the following sense: the equation for X (0) contains only ψ (0) , the equation for ψ (0) involves (1) (1) (X (0) , ψ0 ), the equation of ψ0 contains (X (0) , X (1) ) and finally the quantities ap(1) pearing in the equation for X (1) are (ψ0 , X (0) ). This structure is not present in the
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6. 7. 8.
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original systems where the coupling is “full”. Thus the complexity due to coupling is reduced in the homogenized system. Compared with the original system, the homogenized system is larger both in the number of independent and dependent variables. Unlike the original system, the homogenized system does not contain oscillating forcing term on the right hand side. It should be remarked that (α, t) are just parameters in the equation for X (1) where the independent variables are (y, τ ). Likewise, except for y, all other variables play (1) the role of parameters in the equation for ψ0 . It is worth observing that the homogenized system involves only (σ0 , ρ) and inde(1) pendent of σ1 . Further, only ψ0 appears in it and not ψ (1) . As in classical cases, the quantities (X¯ (1) , ψ¯ (1) ) do not appear in the homogenized system and they do not contribute to the leading order term u(0) either. Let us consider the homogeneous situation where the initial datum does not contain oscillations at all, i.e., W (x, y) ≡ 0. In such a case, it is easy to see that the homogenized system is reduced to the following one for (X (0) , ψ (0) ): 3
0 ⊥ (0) (0) X = 0, ∇ ψ · ∇ ∂t X (0) + α α X = α at t = 0, T −∇α⊥ · Dα X (0) Dα X (0) ∇α⊥ ψ (0) = ρ(α), (1)
because we have ψ0 ≡ 0 and X (1) ≡ 0. In the nonhomogeneous case (i.e. σ0 ≡ 0) oscillations do not vanish (i.e. ψ0(1) ≡ 0, X (1) ≡ 0) and they contribute the following new term to the homogenized system of the homogeneous case: −∇α⊥ · A(0) ∇y⊥ ψ0 . (1)
9. Finally, let us mention another relevant issue with respect to the above expansion, namely its secular or non-secular nature as time evolves. Hou and his co-workers propose a projection method to avoid secular terms being generated. Evidently, this important issue needs further investigation. For engineering applications, it will be more convenient if we have an ansatz for (uε , p ε ) in Eulerian coordinates and not the one in the Lagrangian coordinates. As an intermediate step, let us replace (α, t) by (x, t) as independent variables but still work with the stream function ψ ε . However, we replace X ε by θ ε . The corresponding system of equations to consider is therefore the following:
θ ε (x, 0) = x, ∂t θ ε − ∇x⊥ ψ ε · ∇x θ ε = 0, −x ψ ε = ωin (θ ε (x, t), ε −1 θ ε (x, t)).
To this end, let α = θ (x, t) be the inverse map of x = X (0) (α, t), where X (0) is the first term of the previous expansion. Then if
x = X (0) (θ ε , t) + ε X¯ (1) (θ ε , t) + X (1) (θ ε , t, ε −1 θ ε , ε −1 t) + · · ·
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then since θ ε = θ (X (0) (θ ε , t), t), we have
θ ε = θ (x, t) − εDx θ (x, t) X¯ (1) (θ ε , t) + X (1) (θ ε , t, ε −1 θ ε , ε −1 t) + · · · . Thus it is reasonable to consider the following expansion for the system under consideration:
ψ ε = ψ (0) (x, t) + ε ψ¯ (1) (x, t, τ ) + ψ (1) (θ, t, y, τ ) + · · ·
θ ε = θ (x, t) + ε θ¯ (1) (x, t) + θ (1) (θ, t, y, τ ) + · · · where y = ε −1 θ ε and τ = ε −1 t. As usual, periodicity with respect to y is required. Note that the above expansion is implicit since θ ε occurs on both sides. This is important for otherwise it is not clear how to get a well-posed cell problem. Observe that the above ansatz combines the features of multiscale expansion with the ideas from geometrical optics. Thus, we have two different phase functions: slow one θ and the fast scale phase function θ ε . One has then a homogenized system for unknowns (θ, θ (1) , ψ (0) , ψ (1) ) for which we refer the reader to [27]. Using the above experience, we are in a position to pass to the velocity pressure formulation and propose uε (x, t) = u(x, t, τ ) + w(θ (x, t), t, y, τ ) + εu(1) (θ (x, t), t, y, τ ) + · · · p ε (x, t) = p(x, t, τ ) + q(θ (x, t), t, y, τ ) + εp (1) (θ (x, t), t, y, τ ) + · · · with y = ε −1 θ ε and τ = ε −1 t. We keep the previous expansion for θ ε . Periodicity with respect to y is required and further (w, q) have zero mean with respect to y. The corresponding homogenized system is given by T HEOREM 13 (Homogenization in Eulerian form). (a) For (θ, θ (1) ), we have ∂t θ + (u · ∇x )θ = 0, ∂t θ (1) + (w.∇x )θ = 0,
θ =x θ (1) = 0
at t = 0, at τ = 0.
(b) The oscillating part (w, q) is the solution of the cell problem: ⎧ T ⎨ ∂τ w + B (0) ∇y q = 0, (0)T ∇ ) · w = 0, y ⎩ (B w = W (x, y) at τ = 0. Here the matrix B (0) is computed from (θ, θ (1) ): B (0) = (I − Dy θ (1) )−1 ∇x θ.
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(c) Finally, (u, p) satisfying the following: 3 ∂t u + (u · ∇x )u + ∇x w ⊗ w + ∇x p = 0, ∇x · u = 0, u = U at t = 0. Comments 1. The striking difference between the above result and that of [42] lies in the cell problem. Note that it contains no convection term. Thus there is no difficulty in its wellposedness. 2. The homogenized system derived here has a structure similar to the Large Eddy Simulation model with the Reynolds stress term ! w ⊗ w % whereas the one derived in [42] was similar to (k − ε) model. 3. The ansatz and the consequent homogenization result would have been totally strange, had we not passed through Lagrangian formulation for vorticity-stream function. 4. In [27], we see other expansions in which the oscillating θ ε is replaced by its slowly varying mean θ¯ as the fast phase function. This is achieved by passing through what are called semi-Lagrangian variables. In general, changing phase function is a delicate affaire as it implies errors which can accumulate as time progresses. When the expansion with θ¯ is applied to the velocity-pressure formulation, it leads to a cell problem containing a convection term. Its global well-posedness is not clear even though it is locally well-posed. 5. We conclude by pointing out some extensions. Hou and his collaborators have been able to extend the above ideas to include various important additional effects. We mention some of them: Cases where there are infinitely many scales without separation, three dimensional Euler equations where there is vortex stretching effect, Navier–Stokes equations where there are viscous effects. For these developments and their use in numerical implementation, we encourage the reader to refer to their works already cited. 6. This is the appropriate place to say a few words about the effects of compressibility and viscosity and make a few remarks which will enable the reader to compare the situations in the compressible and the incompressible cases. If the fluid is viscous and compressible then it can be modeled, for instance, by a barotropic system which couples mass conservation (hyperbolic) equation for the density with momentum conservation (parabolic) equation for velocities and a nonlinear algebraic state equation. Such a system admits density oscillations. The new physical effect is the appearence of vacuum, that is, places where the density vanishes. It turns out that the concept of renormalized solutions (see P.L. Lions [37]) is appropriate to deal with such situations. Further, one can describe propagation of (not necessarily periodic) oscillations in such solutions using the concept of Young Measures (see L. Tartar [56]) and obtain an “inflated homogenized system”. One of the difficulties here is to pass to the limit in the nonlinear state equation and here is where Young Measures come into play. It is important to point out that under certain conditions the limit system is globally well-posed ([55], [23]), a result which may be interpreted by saying that density oscillations may persist indefinitely in the original system.
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Acknowledgments The authors warmly thank some of the colleagues in the homogenization community for their help in various ways. One part of the preparation was done when the first author was visiting TIFR Center, Bangalore. He thanks the hosts for the warm hospitality. Financial support that he received from TIFR Center, and CMM, UMI 2071 CNRS, University of Chile is gratefully acknowledged.
References [1] G. Allaire, Homogenization of the Stokes flow in a connected porous medium, Asymp. Anal. 2 (1989) 203– 222. [2] G. Allaire, Continuity of the Darcy’s law in the low-volume fraction limit, Ann. Scuola Norm. Sup. Pisa 18 (1991) 475–499. [3] G. Allaire, Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes, Part I: Abstract framework, a volume distribution of holes, Arch. Rat. Mech. Anal. 113 (1991) 209–259. [4] G. Allaire, Homogenization of the Navier–Stokes equations with a slip boundary condition, Comm. Pure Appl. Math. 44 (1991) 605–641. [5] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (6) (1992) 1482–1518. [6] M. Avellaneda, Homogenization and renormalization: The mathematics of multi-scale random media and turbulent diffusion, in: Dynamical Systems and Probabilistic Methods in Partial Differential Equations, E.P. Deift et al., ed., Lectures in Applied Mathematics 31, Providence, Rhode Island, American Mathematical Society (1996), 251–268. [7] M. Avellaneda and A.J. Majda, An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows, Comm. Math. Phys. 138 (1991) 339–391. [8] R. Battacharya, A hierarchy of gaussian and non-gaussian asymptotics of a class of Fokker–Planck equation with multiple scales, Nonlinear Analysis TMA 30 (1) (1997). [9] R. Battacharya, Multiscale diffusion process with periodic coefficients and an application to solute transport in porous media, Annals Appl. Prob. 9 (4) (1999) 951–1020. [10] G. Ben Arous and H. Owhadi, Super-diffusivity in a shear flow model from perpetual homogenization, Comm. Math. Phys. 227 (2002) 281–302. [11] G. Ben Arous and H. Owhadi, Multiscale homogenization with bounded ratios and anomalous slow diffusion, Comm. Pure Appl. Math. 56 (2003) 80–113. [12] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, NorthHolland, Amsterdam (1978). [13] A. Bourgeat, M. Jurak and A.L. Piatnitski, Averaging a transport equation with small diffusion and oscillating velociy, Math. Methods Appl. Sci. 26 (2) (2003) 95–117. [14] A. Cherkaev et al., eds, Topics in the Mathematical Modeling of Composite Materials, in: Progress in Nonlinear Differential Equations and their Applications 31, Birkhäuser, Boston (1997). [15] D. Cioranescu and F. Murat, A strange term coming from nowhere, in: Cherkaev et al. [14], pp. 45–93. [16] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl. 71 (1979) 590–607. [17] C. Conca, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl. 64 (9) (1985) 31–75. [18] C. Conca, Etude d’un fluide traversant une paroi perforée. I. Comportement limite près de la paroi, J. Math. Pures Appl. 66 (9) (1987) 1–43. [19] W. E, Homogenization of linear and nonlinear transport equations, Comm. Pur. Appl. Math. 45 (1992) 301– 326. [20] A. Fannjiang and G.C. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math. 54 (1994) 333–408.
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[21] F. Golse, Moyennisation des champs de vecteurs et EDP, in: Journées “Équations aux Dérivées Partielles” (Saint Jean de Monts, 1990), École Polytechnique de Paris, Palaiseau (1990), pp. Exp. No. XVI, 17. [22] E. Grenier, Boundary layers, in: Handbook of Mathematical Fluid Dynamics, Vol. III, S. Friedlander et al., eds, North-Holland, Amsterdam (2004), 245–309. [23] M. Hillairet, Propagation of density oscillations in solutions to the barotropic compressible Navier–Stokes system, Preprint. [24] U. Hornung, Homogenization and Porous Media, Interdisciplinary Applied Mathematics 6, Springer-Verlag, New-York (1997). [25] T.Y. Hou, Multiscale modelling and computation of incompressible flow, in: Applied Mathematics, Entering the 21st Century, Invited talks from ICIAM 2003 Congress, J.M. Hill and R. Moore, eds, SIAM Publications (2005), 177–210. [26] T.Y. Hou and X. Xin, Homogenization of linear transport equations with oscillatory vector fields, SIAM J. Appl. Math. 52 (1) (1992) 34–45. [27] T.Y. Hou, D. Yang and H. Ran, Multiscale analysis in Lagrangian formulation for the 2D incompressible Euler equation, Discrete Contin. Dyn. Syst. 12 (2005) 1–36. [28] T.Y. Hou, D. Yang and K. Wang, Homogenization of incompressible Euler equations, J. Comput. Math. 22 (2) (2004) 220–229. [29] T.Y. Hou, D.P. Yang and H. Ran, Multiscale analysis for the 3D incompressible Euler equations, submitted paper (2005). [30] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin (1994). [31] H. Kesten and G.C. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys. 65 (1979) 97–128. [32] R. Kress, Linear Integral Equations, in: Applied Mathematical Sciences 82, Springer-Verlag, Berlin (1989). [33] B.E. Launder and D.B. Spalding, Mathematical Models of Turbulence, Academic Press, New York (1972). [34] T. Levy, Fluid flow through an array of fixed particles, Int. J. Engrg. Sci. 21 (1983) 11–23. [35] T. Levy and E. Sánchez-Palencia, Suspension of solid particles in a Newtonian fluid, J. Non-Newtonian Fluid Mech. 13 (1983) 63–78. [36] J.-L. Lions, Some Methods in the Mathematical Analysis of Systems and Their Control, Science Press, Beijing and Gordon & Breach, Science Publishers Inc., New-York (1981). [37] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Volumes I and II, Oxford Lecture Series in Mathematics, Oxford University Press, Oxford, New York (1996). [38] A. Majda, Lectures in Turbulent Diffusion, Lectures Notes at Princeton University, Princeton (1990). [39] A. Majda and P.R. Kramer, Simplified models for turbulent diffusion, Phys. Rep. 314 (1999) 237–574. [40] A.J. Majda and P. Souganidis, The effect of turbulence on mixing in prototype reaction-diffusion systems, Comm. Pure Appl. Math. 53 (2000) 1284–1304. [41] E. Maruši´c-Paloka and A. Mikeli´c, The derivation of a nonlinear filtration law including the inertia effects via homogenization, Nonlinear Anal. Ser. A: Theory Methods 42 (1) (2000) 97–137. [42] D.W. McLaughlin, G.C. Papanicolaou and O. Pironneau, Convection of microstructure and related problems, SIAM J. Appl. Math. 45 (1985) 780–797. [43] I. Mezic, J.F. Brady and S. Wiggins, Maximal effective diffusivity for time-periodic incompressible fluid flows, SIAM J. Appl. Math. 56 (1996) 40–56. [44] A. Mikeli´c, Effects inertiels pour un écoulement stationnaire visqueux incompressible dans un milieu poreux, C. R. Acad. Sci. Paris, Série I Math. 320 (1995) 1289–1294. [45] G.W. Milton, The Theory of Composites, Cambridge University Press, New York (2002). [46] B. Mohammadi and O. Pironneau, Analysis of the K-Epsilon Turbulence Model, in: Research in Applied Mathematics 31, J. Wiley, New-York & Masson, Paris (1994). [47] F. Murat and L. Tartar, H-convergence, in: Cherkaev et al. [14], pp. 21–44. [48] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989) 608–623. [49] J.R. Norris, Long time behaviour of the heat flow: Global estimates and exact asymptotics, Arch. Rat. Mech. Anal. 140 (1997) 161–195. [50] H. Owhadi, Averaging versus chaos in turbulent transport?, Comm. Math. Phys. 247 (3) (2004) 553–599.
698
C. Conca and M. Vanninathan
[51] G. Pavliotis, Homogenization Theory for Advection-Diffusion Equation with Mean Flow, Ph.D. Thesis (2002). [52] G. Pavliotis and A.M. Stuart, Periodic homogenization for inertial particles, Physica D 204 (2005) 161–187. [53] E. Sánchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag, Berlin (1980). [54] E. Sánchez-Palencia, On the asymptotics of a fluid flow past an array of fixed obstacles, Int. J. Engrg. Sci. 20 (1982) 1291–1301. [55] D. Serre, Variations de grande amplitude par la densité d’un fluide visqueux compressible, Physica D 48 (1) (1991) 113–129. [56] L. Tartar, Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium IV, R.J. Knops, ed., Pitman Research Notes in Mathematics 39, Boston, Pitman (1979), 136–212. [57] M. Vanninathan, Homogenization of eigenvalue problems in perforate domains, Proc. Indian Acad. Sci. Math. 90 (1981) 239–271.
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).
Abergel, F. 449, 643 [1] Abraham, R. 5–7, 10, 24, 34 [1] Adams, R.A. 351, 402 [1]; 454, 643 [3] Aebischer, B. 12, 34 [2] Agarwal, R.K. 52, 119 [1] Agmon, S. 541, 643 [2] Alazard, T. 153, 154, 155 [Ala05]; 155 [Ala06]; 279, 327 [1] Alì, G. 151, 155 [Alì03] Allaire, G. 657, 658, 663, 684, 692, 696 [1]; 696 [2]; 696 [3]; 696 [4]; 696 [5] Amann, H. 354, 379, 402 [2] Amick, C.J. 448, 449, 528, 643 [4]; 643 [5]; 643 [6] Amrouche, Ch. 434, 442 [1] Anand, M. 413, 442 [2] Andrade, C. 411, 442 [3] Arnold, L. 337, 400, 401, 402 [3]; 402 [4]; 402 [5] Arnold, V. 3, 5, 7–10, 16, 32, 34 [3]; 34 [4]; 34 [5]; 34 [6]; 34 [7] Artola, M. 100, 119 [2] Asano, K. 151, 155 [Asa87] Asimov, D. 18, 34 [8] Audin, M. 18, 19, 34 [9] Avellaneda, M. 687, 689, 696 [6]; 696 [7]
Beirão da Veiga, H. 138, 155 [BdV94]; 155 [BdV95]; 162, 198 [1]; 451, 467, 643, 643 [9] Ben Arous, G. 689, 696 [10]; 696 [11] Ben-Dor, G. 43, 96, 119 [4]; 119 [5] Bendler, J.T. 412, 442 [7] Bennequin, D. 13, 34 [10] Bensoussan, A. 652, 653, 677, 678, 682, 689–691, 696 [12] Benzoni-Gavage, S. 41, 120 [6] Berger, M. 16, 34 [11] Besicovitch, A.S. 382, 402 [8] Besov, O.V. 170, 198 [2] Birman, J. 22, 34 [12] Blank, A. 99, 101, 121 [49] Blazy, S. 530, 643 [10] Bleakney, W. 67, 120 [7] Bogovski˘ı, M.E. 434, 442 [8]; 461, 464, 643 [11] Bona, J.L. 449, 643 [1] Borchers, W. 449, 643 [12]; 644 [13] Borer, M. 12, 34 [2] Botta, N. 154, 155, 156 [KBS+ 01] Bourgeat, A. 687, 696 [13] Bourgeois, F. 30, 34 [13] Boussinesq, J. 414, 442 [9] Brady, J.F. 688, 697 [43] Bresch, D. 155, 155 [BDGL02]; 261, 279, 280, 322, 327 [6]; 327 [7]; 327 [8]; 327 [9] Bridgman, P.W. 411, 424, 442 [10] Brio, M. 43, 51, 52, 66, 120 [8]; 121 [45] Brower, D. 400, 402 [9] Browning, G. 136, 155 [BK82]; 156 [BKK80] Bugrov, Ya.S. 170, 198 [3] Bulíˇcek, M. 418, 427, 434, 437, 442 [11]; 442 [12] Busuioc, A.V. 392, 402 [10]
Babin, A.V. 261, 262, 327 [2]; 327 [3]; 327 [4]; 334, 400, 402 [6]; 402 [7]; 567, 643 [7] Bair, S. 412, 442 [4]; 442 [5] Bardos, C. 261, 327 [5] Bargmann, V. 67, 119 [3] Barlow, A.J. 412, 442 [23] Barus, C. 412, 417, 442 [6] Batchelor, G.K. 467, 468, 643 [8] Battacharya, R. 687, 696 [8]; 696 [9]
Caffarelli, L. 161, 162, 176, 178, 180, 181, 198 [4] 699
700
Author Index
Cameron, R. 412, 442 [24] ˇ c, S. 43, 52, 120 [9]; 120 [10]; 120 [11]; Cani´ 120 [12]; 120 [13]; 120 [14]; 120 [15]; 120 [16]; 120 [17]; 120 [18] Cao, C. 399, 402 [11] Casasayas, J. 8, 16, 34 [14] Chae, D. 196, 198 [5] Chandrasekhar, S. 10, 24, 25, 34 [15]; 35 [16] Chang, H. 449, 644 [14] Charve, F. 262, 327 [10]; 327 [11] Chemin, J.-Y. 238, 252, 261, 262, 277, 327 [12]; 327 [13] Chen, G.Q. 42, 47, 96, 100, 120 [19]; 120 [20]; 120 [22]; 120 [23]; 122 [80]; 122 [81] Chen, Z.-M. 168, 181, 198 [6]; 198 [7] Cherkaev, A. 696, 696 [14]; 697 Chern, S. 15, 35 [17] Chicone, C. 15, 23, 35 [18]; 35 [19] Childress, S. 399–401, 403 [24] Choe, H.L. 174, 181, 198 [8]; 198 [9] Chorin, A. 4, 35 [20]; 136, 156 [Cho67] Chueshov, I. 336, 392, 394, 402 [13] Ciarlet, P.G. 339, 402 [12] Cioranescu, D. 663, 668, 696 [15]; 696 [16] Clemence, M. 400, 402 [9] Cockburn, B. 397, 398, 400, 404 [66] Colin, V. 35 [21] Collela, P. 62, 96, 120 [27]; 122 [62] Collins, J.P. 96, 122 [63] Conca, C. 666, 668, 671, 696 [17]; 696 [18] Constantin, P. 194, 195, 198 [10]; 340, 341, 349, 350, 354, 355, 360, 365, 376, 378, 393, 402 [14] Coscia, V. 449, 644 [15] Courant, R. 61, 120 [28] Cutland, N.J. 392, 401, 402 [15]; 402 [16] Cutler, W.G. 412, 442 [13] Dafermos, C. 42, 47, 50, 101, 103, 120 [29]; 120 [30] Dai, H.-H. 10, 23, 36 [67] Danchin, R. 125, 138, 155, 156 [Dan02]; 262, 293, 327 [14]; 327 [15] Dechun Tan 122 [70]; 122 [71] Dendzik, Z. 412, 443 [35] Desjardins, B. 155, 155 [BDGL02]; 156 [DG99]; 156 [DGLM99]; 238, 252, 261, 262, 277, 280, 322, 327 [6]; 327 [7]; 327 [12]; 327 [13]; 327 [16]; 327 [17]; 327 [18] Dombre, T. 9, 10, 23, 35 [22] Douglis, A. 541, 643 [2] Dowson, D. 412, 442 [14] Drazin, P. 24, 25, 35 [23] Duncan, M. 400, 404 [57]
Dutrifoy, A. 155, 156 [DH03] Dymnikov, V.P. 399, 403 [17] E, W. 685, 686, 696 [19] Eagles, P.M. 449, 644 [19] Ebin, D. 136, 151, 156 [Ebi77]; 156 [Ebi82] Eliashberg, Y. 13, 14, 19, 23, 28, 30, 35 [24]; 35 [25]; 35 [26]; 35 [27]; 35 [28]; 35 [29] Escauriaza, L. 162, 180, 190–192, 194, 198 [11]; 198 [12]; 198 [13]; 198 [14]; 198 [15] Etnyre, J. 14, 15, 17, 18, 21–23, 27, 28, 31, 33, 34, 35 [30]; 35 [31]; 35 [32]; 35 [33]; 35 [34]; 35 [35]; 35 [36]; 35 [37] Fannjiang, A. 687, 696 [20] Feireisl, E. 422, 442 [15] Feldman, M. 100, 120 [22] Fernández, F.J. 192, 198 [11] Filatov, A.N. 399, 403 [17] Fink, A.M. 382, 403 [18] Flandoli, F. 391, 392, 401, 403 [19] Foias, C. 24, 35 [38]; 334, 336, 340, 341, 349, 350, 354, 355, 360, 365, 376, 378, 391–393, 402 [14]; 403 [20] Fomenko, A. 16, 35 [39] Fontanella, J.J. 412, 442 [7] Fraenkel, L.E. 449, 643 [6]; 644 [16]; 644 [17]; 644 [18]; 644 [19] Franks, J. 16, 35 [40] Franta, M. 417, 418, 434, 442 [16] Freedman, M. 16, 35 [41] Frehse, J. 196, 198 [16]; 198 [17]; 198 [18] Friedlander, S. 10, 23, 25, 35 [42]; 35 [43]; 36 [44]; 36 [45]; 36 [46]; 162, 198 [19] Friedrichs, K.O. 61, 120 [28] Frisch, U. 9, 10, 23, 35 [22] Fuglister, F.C. 209, 327 [19] Fujita, H. 238, 327 [20]; 354, 379, 403 [21] Galdi, G.P. 178, 198 [20]; 350, 403 [22]; 447, 449, 451, 452, 467, 520, 643, 644 [13]; 644 [20]; 644 [21]; 644 [22] Gallagher, I. 137, 154, 156 [Gal00]; 156 [Gal98]; 238, 242, 252, 261, 262, 277, 279, 293, 295, 300, 303, 327 [12]; 327 [13]; 327 [21]; 327 [22]; 328 [23]; 328 [24]; 328 [25] Gamba, I.M. 52, 66, 121 [32] Gautero, J.-L. 9, 10, 23, 36 [47] Gazzola, F. 417, 430, 442 [17]; 442 [18] Gel’fand, I.M. 121 [33] Gérard-Varet, D. 261, 279, 327 [7]; 327 [8] Ghil, M. 335, 381, 397, 399–401, 403 [23]; 403 [24]; 403 [38]; 405 [78]
Author Index Ghrist, R. 15–18, 21–23, 27, 31, 33, 34, 35 [31]; 35 [32]; 35 [33]; 35 [34]; 35 [35]; 35 [36]; 36 [48]; 36 [49] Giga, Y. 164, 198 [21] Gilbert, A. 10, 23, 35 [43] Gildor, H. 400, 403 [25]; 403 [26] Gill, A.E. 204, 216, 328 [27]; 328 [28] Ginzburg, V. 15, 36 [50] Girault, V. 434, 442 [1] Giroux, E. 14, 31, 35 [21]; 36 [51] Givental, A. 30, 35 [27] Glaz, H.M. 96, 121 [34]; 122 [63] Golse, F. 261, 262, 327 [5]; 328 [29]; 685, 697 [21] Gonzalo, J. 13, 36 [52] Goriely, A. 10, 23, 36 [54] Goursat, E. 10, 36 [53] Greene, J. 9, 10, 23, 35 [22] Greenspan, H.P. 328 [30] Greenwood, J.A. 412, 442 [25] Grenier, E. 141, 155, 155 [BDGL02]; 156 [DG99]; 156 [DGLM99]; 156 [Gre97]; 238, 252, 261, 262, 277, 279, 280, 327 [8]; 327 [12]; 327 [13]; 327 [16]; 327 [17]; 327 [18]; 328 [31]; 328 [32]; 651, 697 [22] Griest, E.M. 412, 442 [19] Griffies, S.M. 333, 399, 400, 403 [27] Grisvard, P. 350, 403 [28] Gromov, M. 13, 36 [55] Guckenheimer, J. 22, 23, 36 [56]; 36 [57] Guderley, K.G. 66, 121 [35]; 121 [36] Gustafsson, B. 151, 156 [GS91] Hagstrom, T. 155, 156 [HL02]; 156 [HL98] Haitao Fan 121 [31] Hale, J.K. 336, 379, 388, 391, 403 [29] Halt, D.W. 52, 119 [1] Hameiri, E. 25, 37 [73] Hamilton, R. 15, 35 [17] Harabetian, E. 103, 121 [37] Harten, A. 50, 121 [38] He, Z.-X. 16, 35 [41] Heibig, A. 105, 121 [39] Henderson, L.F. 61, 62, 96, 120 [27]; 121 [40]; 121 [41]; 121 [42]; 122 [62] Hénon, M. 9, 10, 23, 35 [22]; 36 [58] Henry, D. 25, 33, 36 [59] Herrmann, L. 521, 647 [100] Heywood, J.G. 447, 448, 644 [23] Higginson, G.R. 412, 442 [14] Hillairet, M. 695, 697 [23] Hmidi, T. 155, 156 [DH03] Hofer, H. 18–20, 27, 30, 35 [27]; 36 [60]; 36 [61]; 36 [62]
701
Hoff, D. 155, 156 [Hof98] Hoffmann, L. 154, 155, 156 [KBS+ 01] Holmes, P. 16, 21, 23, 36 [49]; 36 [56]; 36 [63]; 36 [64]; 36 [65] Honda, K. 14, 28, 35 [21]; 35 [37]; 36 [66] Hopf, E. 178, 198 [22] Hornung, H. 43, 96, 121 [43] Hornung, U. 663, 697 [24] Hou, T.Y. 685, 689, 694, 695, 697 [25]; 697 [26]; 697 [27]; 697 [28]; 697 [29] Hron, J. 417, 418, 435, 439, 442 [20]; 442 [21] Huang, D.-B. 10, 23, 36 [67] Huang, K.-L. 10, 23, 37 [101] Hunter, J.K. 43, 51, 52, 62, 66, 88, 101, 103, 120 [8]; 121 [44]; 121 [45]; 121 [46]; 121 [47]; 122 [73] Huy, N.D. 434, 442 [22] Iftimie, D. 336, 338, 339, 391, 392, 394, 402 [10]; 403 [30]; 403 [31]; 403 [32] Iguchi, T. 151, 156 [Igu97] Il’in, V.P. 170, 198 [2] Imbrie, J. 334, 400, 403 [33] Imbrie, K.P. 334, 400, 403 [33] Imkeller, P. 337, 400, 401, 402 [5] Irwing, J.B. 412, 442 [23] Isozaki, H. 151, 156 [Iso87a]; 156 [Iso87b]; 156 [Iso89] Jikov, V.V. 688, 697 [30] Johnson, K.L. 412, 442 [24]; 442 [25]; 442 [26] Johnson, R.A. 386, 403 [34] Joly, J.-L. 130, 141, 154, 156 [JMR95]; 156 [JMR98]; 262, 328 [33]; 328 [34] Jouguet, E. 50, 121 [48] Jurak, M. 687, 696 [13] Kählin, M. 12, 34 [2] Kanda, Y. 14, 31, 36 [68] Kang, K. 174, 199 [30] Kannan, K. 417, 442 [27] Kapitanskii, L.V. 447–449, 462, 465, 467, 472, 644 [24]; 644 [25]; 644 [26] Kasahara, A. 136, 156 [BKK80] Kato, T. 238, 327 [20]; 354, 379, 403 [21] Katz, N.H. 162, 199 [31] Keblikas, V. 451, 467, 646 [74] Keisler, H.J. 392, 401, 402 [16] Keller, J.B. 99, 101, 103, 121 [46]; 121 [47]; 121 [49] Kendall, P. 10, 35 [16] Kerr, R.M. 33, 36 [69] Kesten, H. 686, 697 [31]
702
Author Index
Keyfitz, B.L. 43, 52, 120 [9]; 120 [10]; 120 [11]; 120 [12]; 120 [13]; 120 [14]; 120 [15]; 120 [16]; 120 [17] Khesin, B. 5, 7–10, 15, 16, 32, 34 [6]; 34 [7]; 36 [50] Kim, E.H. 52, 120 [12]; 120 [13]; 120 [14]; 120 [15] Kim, H. 168, 199 [32] Klainerman, S. 136, 137, 141, 154, 156 [KM81]; 157 [KM82] Klein, R. 154, 155, 156 [KBS+ 01]; 156 [Kle95]; 230, 328 [35] Koens, G. 411, 415, 443 [48] Kohn, R.-V. 161, 162, 176, 178, 180, 181, 198 [4] Komendarczyk, R. 34, 36 [70] Kottke, P. 412, 442 [4] Kozlov, S.M. 688, 697 [30] Kozono, H. 168, 199 [32]; 199 [33] Kramer, P.R. 677, 681, 688, 689, 697 [39] Kreiss, H.-O. 136, 155 [BK82]; 156 [BKK80]; 157 [Kre80] Kress, R. 662, 697 [32] Kruzkov, S. 50, 121 [50] Kukavica, I. 336, 394, 403 [35] Kuperberg, G. 17, 36 [71] Kuperberg, K. 17, 36 [72] Kwek, K.-H. 10, 23, 37 [101] Ladyzhenskaya, O.A. 161, 162, 167, 169, 178, 180, 181, 189, 198 [23]; 198 [24]; 198 [25]; 198 [26]; 199 [27]; 350, 354, 391, 403 [36]; 403 [37]; 447–450, 454, 455, 461, 470, 473, 489, 497, 498, 500, 511, 528, 529, 539, 548, 555, 565, 566, 572, 577, 580, 610, 628, 631, 642, 644 [27]; 644 [28]; 644 [29]; 644 [30]; 644 [31]; 644 [32]; 644 [33]; 644 [34] Lafontaine, J. 18, 19, 34 [9] Landau, L.D. 467, 468, 534, 644 [35]; 644 [36] Launder, B.E. 651, 689, 697 [33] Lax, P.D. 50, 121 [38]; 121 [51]; 127, 157 [Lax73] Le Treut, H. 335, 400, 401, 403 [38] Lee, J. 196, 198 [5] Leray, J. 178, 199 [28]; 236, 238, 328 [36]; 328 [37] Leuenberger, Ch. 12, 34 [2] Levermore, C.D. 50, 121 [38] Levy, T. 663, 666, 697 [34]; 697 [35] Lewis, J.L. 181, 198 [9] Li, J.-B. 10, 23, 37 [101] Li Ta-tsien 84, 85, 88, 98, 122 [67] Lieberman, G. 52, 120 [16] Lifschitz, E.M. 467, 468, 534, 644 [35]; 644 [36] Lighthill, M.J. 104, 121 [52]
Lin, C. 24, 25, 37 [74]; 567, 644 [37]; 645 [38] Lin, C.-K. 155, 155 [BDGL02]; 157 [Lin95] Lin, F.-H. 178, 180, 181, 199 [29] Ling Hsiao 120 [21] Lions, J.-L. 328 [38]; 340, 344, 347, 354, 399, 403 [39]; 403 [40]; 403 [41]; 403 [42]; 652, 653, 661, 677, 678, 682, 689–691, 696 [12]; 697 [36] Lions, P.-L. 125, 155, 156 [DGLM99]; 157 [LM98]; 157 [LM99]; 157 [Lio96]; 157 [Lio98]; 262, 279, 322, 327 [18]; 328 [39]; 328 [40]; 328 [41]; 695, 697 [37] Lipton-Lifschitz, A. 25, 36 [44]; 37 [73] Longuet-Higgins, M.S. 328 [28] Lorenz, J. 155, 156 [HL02]; 156 [HL98] Loss, M. 10, 37 [75] Lovicar, V. 521, 647 [100] Ludloff, H.F. 122 [76] Lunardi, A. 340, 349, 351, 380, 403 [43] Lutz, R. 12, 37 [76] Mach, E. 42, 83, 121 [53] Magenes, E. 344, 347, 403 [40] Mahalov, A. 261, 262, 327 [2]; 327 [3]; 327 [4]; 327 [5]; 334, 400, 402 [6]; 402 [7] Majda, A. 58, 79, 100, 119 [2]; 121 [54]; 121 [55]; 136, 137, 141, 143, 144, 154, 156 [KM81]; 157 [KM82]; 157 [Maj84]; 230, 328 [35]; 328 [42]; 399, 403 [44]; 677, 681, 687–689, 696 [7]; 697 [38]; 697 [39]; 697 [40] Majdoub, M. 276, 279, 328 [43] Málek, J. 162, 199 [34]; 409, 410, 413, 414, 417, 418, 421, 422, 425, 430–432, 434, 435, 437, 439, 442 [12]; 442 [15]; 442 [16]; 442 [20]; 442 [21]; 443 [28]; 443 [29]; 443 [30]; 443 [31]; 443 [32]; 443 [33] Maremonti, P. 164, 199 [35] Margolin, L.G. 399, 400, 404 [70] Markus, L. 401, 403 [45] Marsden, J. 4–7, 10, 24, 34 [1]; 35 [20] Marsh, G. 10, 37 [77] Martinet, J. 12, 37 [78] Martinez Alfaro, J. 8, 16, 34 [14] Maruši´c-Paloka, E. 663, 697 [41] Masmoudi, N. 155, 156 [DGLM99]; 157 [LM98]; 157 [LM99]; 261, 262, 279, 293, 327 [18]; 328 [32]; 328 [40]; 328 [41]; 328 [44]; 328 [45] McDuff, D. 12, 15, 19, 37 [79] McLaughlin, D.W. 10, 37 [80]; 690, 695, 697 [42] McMickle, R.H. 412, 442 [13] Mehr, A. 9, 10, 23, 35 [22] Meister, A. 154, 155, 156 [KBS+ 01]; 157 [Mei00] Mellet, A. 280, 328 [46] Menikoff, R. 61, 121 [42]
Author Index Métivier, G. 130, 141, 153–155, 156 [JMR95]; 156 [JMR98]; 157 [MS01]; 157 [MS03]; 262, 279, 327 [9]; 328 [33]; 328 [34]; 328 [47] Meyer, K.R. 401, 403 [45] Mezic, I. 688, 697 [43] Mikeli´c, A. 663, 697 [41]; 697 [44] Mikhlin, S.G. 478, 503, 645 [39] Milankovitch, M. 334, 336, 400, 404 [46] Milton, G.W. 687, 688, 697 [45] Mingione, G. 434, 443 [28]; 443 [29] Mirkovi´c, D. 52, 120 [18] Mishachev, N. 13, 35 [28] Moffatt, H. 3, 16, 21, 37 [81]; 37 [82]; 37 [83]; 37 [84] Mohammadi, B. 651, 676, 697 [46] Montgomery, D. 67, 119 [3] Morawetz, C. 51, 52, 64, 121 [56] Morgan, J. 18, 37 [85] Morokoff, W. 50, 121 [38] Moser, J. 381, 404 [47]; 404 [48] Munz, C.D. 154, 155, 156 [KBS+ 01] Murat, F. 656, 663, 696 [15]; 697 [47] Navier, C.L.M.H. 338, 404 [49] Nazarov, S.A. 449, 475, 527, 530, 531, 579, 643 [10]; 645 [40]; 645 [41]; 645 [42]; 645 [43]; 645 [44]; 645 [45]; 645 [46]; 645 [47]; 645 [48]; 645 [49]; 645 [50]; 645 [51]; 645 [52]; 645 [53]; 645 [54] Neˇcas, J. 162, 196, 197, 199 [34]; 199 [36]; 199 [37]; 417, 418, 430–432, 435, 442 [20]; 443 [30] Neustupa, J. 196, 199 [36]; 199 [38]; 199 [39] Newton, C.W. 208, 328 [49] Nguetseng, G. 658, 697 [48] Nguyen, T.-Z. 16, 35 [39] Nicolaenko, B. 261, 262, 327 [2]; 327 [3]; 327 [4]; 327 [5]; 334, 400, 402 [6]; 402 [7] Nikol’skii, S.M. 170, 198 [2] Nirenberg, L. 161, 162, 176, 178, 180, 181, 198 [4]; 541, 643 [2] Noll, W. 415, 444 [57] Norman, D. 354, 391, 392, 404 [50] Norris, J.R. 688, 697 [49] Nunes, A. 8, 16, 34 [14] Oberbeck, A. 414, 443 [34] Ohyama, T. 167, 199 [40] Oleinik, O.A. 688, 697 [30] Orlovsky, D.G. 477, 646 [81] Owhadi, H. 689, 696 [10]; 696 [11]; 697 [50] Padula, M. 449, 644 [21] Paicu, M. 276, 277, 279, 328 [43]; 328 [48]
703
Palmén, E. 208, 328 [49] Palmer, K.J. 386, 403 [34] Paluch, M. 412, 443 [35] Papanicolaou, G.C. 652, 653, 677, 678, 682, 686, 687, 689–691, 695, 696 [12]; 696 [20]; 697 [31]; 697 [42] Patria, M.C. 449, 644 [15] Pavliotis, G. 677, 680, 682, 685, 686, 698 [51]; 698 [52] Pavlovich, N. 162, 198 [19]; 199 [31] Payne, L. 567, 645 [38] Pedlosky, J. 204, 207, 213, 328 [50] Pelz, R. 33, 37 [86]; 37 [87] Penel, P. 196, 199 [38] Philander, S. 328 [51] Piatnitski, A.L. 687, 696 [13] Pileckas, K. 447–449, 451, 461, 462, 465, 467, 475, 527, 529–531, 568, 643 [12]; 644 [13]; 644 [25]; 644 [26]; 645 [43]; 645 [44]; 645 [45]; 645 [46]; 645 [47]; 645 [48]; 645 [49]; 645 [50]; 645 [56]; 645 [57]; 645 [58]; 645 [59]; 645 [60]; 645 [61]; 645 [62]; 646 [63]; 646 [64]; 646 [65]; 646 [66]; 646 [67]; 646 [68]; 646 [69]; 646 [70]; 646 [71]; 646 [72]; 646 [73]; 646 [74]; 646 [75]; 646 [76]; 646 [77]; 646 [78]; 646 [79]; 646 [80]; 647 [94] Pironneau, O. 10, 37 [80]; 651, 676, 690, 695, 697 [42]; 697 [46] Plamenevskii, B.A. 530, 579, 645 [51] Pliss, V.A. 332–337, 379, 381, 385, 386, 395–401, 404 [51]; 404 [52]; 404 [53]; 404 [54]; 404 [55] Pokorny, M. 196, 199 [39] Price, W.G. 168, 181, 198 [6]; 198 [7] Prilepko, A.I. 477, 646 [81] Protter, M.H. 109, 122 [60] Quarteroni, A. 450, 645 [55] Quinlan, G.D. 400, 404 [56] Quinn, T.R. 400, 404 [57] Rabier, P.J. 566, 646 [82]; 646 [83] Rajagopal, K.R. 409–411, 413, 414, 416–418, 420, 421, 423, 425, 430–432, 434, 435, 437, 439, 442 [2]; 442 [12]; 442 [16]; 442 [20]; 442 [21]; 442 [27]; 443 [30]; 443 [31]; 443 [32]; 443 [33]; 443 [36]; 443 [37]; 443 [38]; 443 [39]; 443 [40]; 443 [41]; 443 [42]; 443 [43]; 443 [44]; 444 [58] Ran, H. 689, 694, 695, 697 [27]; 697 [29] Ratiu, T. 5–7, 10, 24, 34 [1] Rauch, J. 130, 141, 154, 156 [JMR95]; 156 [JMR98]; 262, 328 [33]; 328 [34]
704
Author Index
Raugel, G. 336, 338, 339, 379, 386, 391, 392, 394, 395, 402 [13]; 403 [31]; 403 [32]; 404 [58]; 404 [59]; 404 [60]; 404 [61] Reed, M. 275, 329 [52] Reid, W. 24, 25, 35 [23] Rekalo, A. 336, 392, 394, 402 [13] Renardy, M. 417, 429, 437, 438, 443 [45]; 443 [46] Ricca, R. 16, 37 [88] Riemann, H. 12, 34 [2] Rivkind, L. 449, 646 [84] Robertson, A.M. 451, 467, 520, 643, 644 [22] Robinson, C. 23, 27, 37 [89]; 37 [90] Roelands, C.J.A. 412, 443 [47] Rokyta, M. 162, 199 [34] Rolfsen, D. 13, 37 [91] Roller, S. 154, 155, 156 [KBS+ 01] Rosales, R.R. 52, 65, 66, 121 [32]; 122 [68] Rousset, F. 261, 329 [53] Runnegar, B. 335, 381, 397, 400, 405 [78] R˚užiˇcka, M. 162, 196, 197; 198 [16]; 198 [17]; 198 [18]; 199 [34]; 199 [37]; 414, 443 [40] Rzoska, S.J. 412, 443 [35] Saal, R.N.J. 411, 415, 443 [48] Sacker, R.J. 386, 388, 404 [62]; 404 [63] Saint Jean Paulin, J. 668, 696 [16] Saint-Raymond, L. 242, 262, 279, 295, 300, 303, 328 [24]; 328 [25]; 328 [29] Sakurai, A. 96, 122 [61]; 122 [62] Salamon, D. 12, 15, 19, 37 [79] Sánchez-Palencia, E. 652–654, 663, 666, 697 [35]; 698 [53]; 698 [54] Sanders, J.A. 149, 157 [SV85] Saut, J. 24, 37 [92] Šˇcadilov, V.E. 338, 339, 341, 404 [71] Schaeffer, D.G. 432, 443 [49] Scheffer, V. 161, 176, 199 [41]; 199 [42]; 199 [43]; 199 [44] Schiessler, R.W. 412, 442 [13]; 442 [19] Schiffer, M. 136, 157 [Sch60] Schlessinger, M.F. 412, 442 [7] Schmalfuss, B. 391, 392, 401, 403 [19] Schneider, T. 154, 155, 156 [KBS+ 01] Schochet, S. 132, 136, 137, 141, 150–155, 157 [MS01]; 157 [MS03]; 157 [Sch05]; 157 [Sch86]; 157 [Sch87]; 157 [Sch88]; 157 [Sch94]; 157 [Sch94b]; 262, 279, 328 [47]; 329 [54] Schultz-Rinne, C.W. 96, 122 [63] Schwarz, M. 28, 37 [93] Secchi, P. 151, 157 [Sec00]; 417, 430, 442 [18] Sell, G.R. 332–342, 349, 352, 354, 355, 358, 360, 365, 368, 369, 376, 378–381, 385–401,
403 [32]; 403 [34]; 404 [51]; 404 [52]; 404 [53]; 404 [54]; 404 [55]; 404 [59]; 404 [60]; 404 [61]; 404 [62]; 404 [63]; 404 [64]; 404 [65]; 404 [66]; 404 [67] Sequeira, A. 449, 645 [52] Seregin, G.A. 161, 162, 166, 167, 169, 174, 178, 180–182, 184, 189, 190–194, 198 [12]; 198 [13]; 198 [14]; 198 [15]; 198 [26]; 199 [45]; 199 [46]; 199 [47]; 199 [48]; 199 [49]; 199 [50]; 199 [51]; 199 [52]; 199 [53]; 200 [54] Serre, D. 41, 47, 50, 62, 105, 106, 108, 110, 111, 120 [6]; 121 [39]; 122 [64]; 122 [65]; 122 [66]; 695, 698 [55] Serrin, J. 162, 167, 176, 200 [55]; 350, 354, 404 [68]; 413, 443 [50] Shen, W. 382, 386, 401, 404 [69] Shilkin, T.N. 182, 199 [52] Shuli Yang 122 [79]; 122 [80]; 122 [81] Shuxing Chen 101, 120 [24]; 120 [25]; 120 [26] Sideris, T. 155, 157 [Sid91] Simon, B. 275, 329 [52] Sirovich, L. 136, 157 [Sir67] Slemrod, M. 121 [31] Smolarkiewicz, P.K. 399, 400, 404 [70] Socolowsky, J. 449, 646 [75]; 646 [76] Sohr, H. 164, 198 [21] Solonnikov, V.A. 162, 164, 165, 174, 182, 199 [27]; 199 [35]; 199 [52]; 200 [56]; 200 [57]; 200 [58]; 338, 339, 341, 404 [71]; 437, 443 [51]; 447–449, 454, 461, 489, 497, 498, 511, 528–530, 541, 546, 547, 553, 566, 567, 578, 580, 623, 644 [21]; 644 [29]; 644 [30]; 644 [31]; 644 [32]; 644 [33]; 644 [34]; 646 [77]; 646 [84]; 646 [85]; 646 [86]; 647 [87]; 647 [88]; 647 [89]; 647 [90]; 647 [91]; 647 [92]; 647 [93]; 647 [94] Sonar, T. 154, 155, 156 [KBS+ 01] Song, J.S. 567, 647 [95] Souganidis, P. 687, 697 [40] Sova, M. 521, 647 [100] Soward, A. 9, 10, 23, 35 [22] Spalding, D.B. 651, 689, 697 [33] Specovius-Neugebauer, M. 449, 530, 643 [10]; 645 [53]; 645 [54]; 646 [78]; 647 [96] Spencer, A.J.M. 419, 443 [52] Srinivasa, A.R. 414, 420, 421, 443 [40]; 443 [41]; 443 [42] Stará, J. 434, 442 [22]; 443 [28]; 443 [29] Št˘edrý, M. 521, 647 [100] Stein, E.M. 534, 647 [97] Stokes, G.G. 411, 444 [53] Stoor, H. 151, 156 [GS91] Straškraba, I. 521, 647 [100]
Author Index Struwe, M. 162, 167, 168, 196, 200 [59]; 200 [60] Stuart, A.M. 680, 698 [52] Sullivan, D. 15, 37 [94] Sullivan, M. 16, 21, 23, 36 [49] Šverák, V. 162, 180, 181, 190–192, 194, 197, 198 [12]; 198 [13]; 198 [14]; 198 [15]; 199 [37]; 199 [53]; 200 [54] Szeri, A.Z. 412, 413, 443 [43]; 444 [54] Tabak, E.G. 52, 65, 66, 121 [32]; 122 [68] Tabor, M. 10, 23, 36 [54] Tadmor, E. 50, 106, 122 [69] Takahashi, S. 168, 174, 200 [61]; 200 [62]; 200 [63] Takayama, K. 96, 122 [62] Tao, L. 423, 443 [44] Tartar, L. 50, 122 [72]; 656, 663, 695, 697 [47]; 698 [56] Taub, A.H. 67, 120 [7] Taylor, M. 141, 143, 144, 157 [Tay96] Temam, R. 24, 35 [38]; 37 [92]; 136, 157 [Tem77]; 230, 328 [38]; 329 [56]; 334, 336, 340, 349, 350, 354, 391–394, 399, 403 [20]; 403 [41]; 403 [42]; 404 [72]; 404 [73]; 404 [74]; 405 [75]; 405 [76]; 447, 647 [98] Tesdall, A.M. 43, 52, 62, 66, 122 [73] Teshukov, V.M. 42, 76, 80, 88, 122 [74]; 122 [75] Tevaarwerk, J.L. 412, 442 [26] Thäter, G. 530, 645 [53]; 645 [54] Thomson (Lord Kelvin), W. 3, 37 [95]; 329 [55] Thurston, W. 13, 35 [29]; 37 [96] Tian, G. 181, 200 [64] Ting, F. 122 [76] Titi, E.S. 399, 402 [11] Tong Zhang 96, 120 [19]; 120 [20]; 120 [21]; 122 [70]; 122 [71]; 122 [79]; 122 [80]; 122 [81]; 122 [82] Tremaine, S. 400, 404 [57] Tricomi, F.G. 478, 503, 647 [99] True, H. 448, 566, 644 [34] Truesdell, C.A. 409, 413–415, 420, 444 [55]; 444 [56]; 444 [57] Tsai, T.-P. 162, 200 [65] Tziperman, E. 400, 403 [25]; 403 [26] Uhlenbeck, K. 25–27, 37 [102] Ukai, S. 151, 157 [Uka86] Uralt’seva, N.N. 162, 199 [27]; 454, 489, 497, 498, 511, 644 [29] Valero, J. 392, 405 [77] Van Dyke, M. 136, 157 [VD64] Vanninathan, M. 655, 698 [57] Varadi, F. 335, 381, 397, 400, 405 [78]
705
Vasin, I.A. 477, 646 [81] Vasseur, A. 280, 328 [46] Vasudevaiah, M. 417, 437, 444 [58] Vejvoda, O. 521, 647 [100] Verfürth, R. 338, 339, 405 [79] Verhulst, F. 149, 157 [SV85] Videman, J.H. 449, 645 [52] Vishik, M. 10, 23, 25, 35 [43]; 36 [45] Vogel, H. 411, 444 [59] von Neumann, J. 42, 61, 121 [57]; 121 [58]; 122 [59] von Wahl, W. 340, 350, 354, 405 [80] Wada, M. 18, 37 [97] Wagner, D.H. 43, 52, 120 [17] Waldhausen, F. 8, 37 [98]; 37 [99] Walenta, Z. 96, 122 [62] Wang, D. 42, 47, 120 [23] Wang, K. 689, 697 [28] Wang, S. 328 [38]; 399, 403 [41]; 403 [42] Webb, W. 412, 442 [13]; 442 [19] Weinberger, H. 109, 122 [60] Weyl, H. 122 [77] Whitham, G.B. 122 [78] Wiggins, S. 688, 697 [43] Williams, R. 16, 22, 23, 34 [12]; 35 [40]; 36 [57]; 36 [65]; 37 [100] Winer, W.O. 412, 442 [5] Winkelnkemper, H. 13, 37 [96] Wu, Y. 337, 400, 401, 402 [5] Wysocki, K. 18, 20, 27, 36 [61]; 36 [62] Wyszogrodzki, A.A. 399, 400, 404 [70] Xin, X. 685, 697 [26] Xin, Z. 181, 200 [64] Yang, D.P. 689, 694, 695, 697 [27]; 697 [28]; 697 [29] Yau, H.-T. 10, 37 [75] Yi, Y. 382, 386, 401, 402, 404 [69]; 405 [81] You, Y. 336, 340–342, 349, 352, 354, 355, 358, 360, 365, 368, 369, 376, 378, 380, 386–393, 404 [67] Yu Wen-ci 84, 85, 88, 98, 122 [67] Yudovich, V. 25, 36 [46] Zaj¸aczkowski, W. 451, 568, 646 [79] Zaleskis, L. 449, 646 [80] Zehnder, E. 18, 20, 27, 36 [61]; 36 [62] Zhao, X.-H. 10, 23, 36 [67]; 37 [101] Zheng, Y. 53, 65, 100, 122 [82]; 122 [83]; 122 [84]; 122 [85]; 122 [86]; 122 [87] Ziane, M. 230, 329 [56]; 336, 394, 399, 403 [35]; 404 [74]; 405 [75]; 405 [76]
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Subject Index Brinkman’s law 651, 663, 665, 670 Bubnov–Galerkin – approximation 360–365, 368, 369, 375, 376, 393, 396, 400 – equation 361, 363, 376 – method 376 buoyancy force, frequency 217, 334 Burger number 206 Burgers equation 52
ABC fields, Beltrami field 9 acoustic wave 75 adiabatic constant 41 ADN-elliptic problems 541 advection-diffusion equation 676, 686, 687 allowable perturbation 395, 397–400 almost periodic 383, 401 analytic semigroup 341, 347 aperture domain 447, 449 artificial boundary conditions 530 aspect ratio 206, 395 asymptotically stable 391 attractor 391, 392 averaging 149 axisymmetric flows 196
Caffarelli–Kohn–Nirenberg type theory 162 cell-problems 679, 680, 683 cell-test functions 654, 660 chaos 391 Chaplygin gas 70 characteristic – curve 47 – foliation 12 Class LH 366, 380 Clausius–Duhem inequality 422 climate 334, 395, 399 coercive Ls,n -estimates 163 compactness method 141 compatibility conditions 486, 488, 496, 497, 528, 531, 569, 578–580, 582, 583, 585, 607, 627, 631, 633, 637 compensated compactness 239, 265, 315 constitutive theory, explicit, implicit 413, 416 contact discontinuity (CD) 54 contact form 10 contact homology 30 Coriolis force, operator, parameter, term 205, 210, 237, 262, 281, 334, 385, 396 Couette–Taylor 396 cylindrical outlets 448, 450, 452, 457, 462, 528, 530, 642
backward uniqueness 190, 194 balance of energy, linear and angular momentum, mass 421 Banach contraction principle 530, 546 barotropic gas 76 Bernoulli – invariant 46, 92 – pseudo- 92 betaplane approximation 280 – equatorial 232 – mid-latitude 232 blow up pressure, velocity 193 bottom friction 223 boundary condition – Navier’s slip 428 – no slip 428 – spatially periodic 428 boundary conditions – homogeneous 343 – inhomogeneous 342 – physical 354 boundary force 337, 347, 401 boundary layer 417, 676 – Ekman 261 – Munk 280 boundary regularity 161, 166
Darboux Theorem 12 Darcy’s law 651, 654, 658, 659, 661, 663, 666 De Rham’s Theorem 656 deep ocean 342, 389 density, pressure dependent 412 707
708 diffracted shock 95 diffraction 71 diffusive limit case 681 Dirichlet integral 448, 528, 530, 579 Dirichlet norm 472 drag forces 664 effective diffusion tensor 679, 682, 685, 689 Ekman – layer 227, 261 – number 222 – pumping 223, 261 – transport 223 El Niño 333, 385, 396–398 elastic scattering 221 elastohydrodynamics 411 elliptic orbit 25 elliptic point 4 embedding 552 – operator 455 energy, specific; energy estimate, closed; energy norm 24, 41, 141, 143 energy-dissipation inequality 423 enhanced diffusion tensor 679 enthalpy 45 entropy, convex, inequality, mathematical, production 41, 49, 50, 56, 422 equation of state, complete, incomplete 41 equatorial – deformation radius 215 – trapping 216 Ergodic Theorem 382, 386 Euler equations 689, 695 Euler system – full 41 – isentropic 44 Evans function 85 extension operator 668 f-plane approximation 231 filtering operator 254, 289 flow – barotropic 44, 76, 125 – isentropic 44 – isochoric 415 – isothermal 44 – pseudo-steady 42 – steady 42 – upstream 58 fluid – anisotropic 420 – compressible 125 – definition 409
Subject Index – incompressible 125, 415, 420, 426 – inhomogeneous 414 – Navier–Stokes 409 – Newtonian 409 – nonisentropic 151–154 – non-Newtonian 410 fluid flows 649, 651, 652, 663, 672, 689 flux, carrier, general, time-dependent 447, 451, 467–469, 475, 476, 528, 529, 532, 534, 546, 549, 550, 554, 568, 569, 582, 583, 586 fluxes 531, 567, 569, 572, 582, 585, 613 force-free fields 10 Fourier boundary conditions 651, 666, 670 frequency vector 382, 384, 387, 396, 398, 401 Galerkin approximations 500, 503, 508, 628, 629, 631 gas, Chaplygin, ideal, perfect, van der Waals 41, 62, 70, 72 GCB (Great Conveyor Belt) 343, 400 geometrical optics 690, 694 geostrophic, approximation, constraint, motion 211, 284, 285, 299 global attractor 334, 336, 391, 393, 395, 401 global energy inequality 177 Global Regularity Problem 379, 392 gravity force 334, 337, 354, 384, 396, 397 Gray’s Theorem 12 Green–Kubo formula 686 Gulf Stream 400 Hardy’s inequality 537 Hausdorff dimension 162, 181 heat equation 162, 467, 496 heat transfer 399, 400 Helmholtz potential 422 Helmholtz–Leray projection 339, 350 Herculean Theorem 389 Hermite – functions 284 – polynomials 285 homogenization 344–347, 354 – process 650, 651, 676 homogenized coefficients 668, 669 homogenized system 651, 658, 659, 663, 690, 693, 694 Hopf fibration, field, link 5, 11, 17 Hugoniot curve 57, 72 hydrostatic – approximation 211 – assumption 399, 400 – law 205 hyperbolic orbit 25
Subject Index incident shock 77 incompressibility 414 incompressible gas 68 incompressible limit, fast, formal, history, justification for barotropic flow 125, 128–151 induced diffusion 221 inertial frequency 210 infinite layer 449, 527 infinite pipe 468 infinite strip 585 Interface Model 335, 337, 399, 401 interior regularity 161 invariant set 388–392 inverse parabolic problem 451, 467 inverse problem 476, 477, 498, 511, 521, 528 J -holomorphic curve
19
KAM theory 381 Kelvin waves 214, 291 knot 16 Korn inequality 339, 667 Krakatau eruption 401 L3,∞ -solution 162, 190 Ladyzhenskaya inequalities 455 Ladyzhenskaya–Prodi–Serrin condition 162, 167, 173, 190 Lagrangian description 690 Lagrangian map 690 Lagrangian turbulence 9 Large Eddy Simulation 695 latent energy 363 Lax shock 58 layer-like, outlet 448, 530 layered composites 687 Leray equations 196 Leray–Schauder theorem 474, 548, 606 Leray’s problem 450, 451, 528, 567 local energy inequality 177, 179, 183, 185 local regularity theory 163, 555, 565 LPS-condition 167, 168 LRP (Little Regularity Problem) 393 lubrication 412 Lyapunov stable 391 Mach number, absolute, relative; Mach Reflection, complex (CMR), double (DMR), Guderley, irregular, single (SMR), transitional (TMR) 42, 43, 62, 67, 73, 74, 125 Mach stem 43, 83 macroscopic behaviour 664, 665, 669, 670, 675
709
mass density 41 mass flux 53 memory effects 658 mid-latitudes approximation 214 mixed Poincaré–Rossby waves 217, 291 Morse homology 28 Moser–Weinstein Theorem 12 multiple time scales 129–131 Navier boundary conditions 338 Navier–Stokes – equations 125, 166, 168, 169, 173, 176, 178, 183, 184, 354, 358, 360, 369, 379, 392, 394, 689, 695 – nonlinear 451, 636 – problem 546 – steady 447, 448, 450, 451, 468, 469, 472, 527, 528, 544, 600, 610 – system 161, 662, 663, 666, 670 – three-dimensional 613 – time dependent 448, 451, 567 – two-dimensional 499, 567, 585 Neumann boundary condition 654 Neumann problem 578 no-slip boundary condition 227, 653, 674, 675 non-hydrostatic model 399 non-linear creep 410, 426 nonautonomous force 337 nondegenerate orbit 25 nonlinear filtration law 663 normal shock 75, 80 oblique shock 71 oceanic dynamics 385–394, 399 oceanic equations 333, 336, 353, 358, 360, 366, 369, 376, 378, 379, 386, 389, 394 oscillating shear flow 687 parabolic-like, domain, structure of outlets 448, 449, 527, 530 passive scalars 651, 682, 687, 689 Péclet number 677, 680, 683, 685, 687 perfect gas 41 permeability matrix 654, 661, 664 Poincaré waves 212, 290 Poiseuille, time dependent, flow, generalized, steady, time dependent, time-periodic, unidirectional 448, 450–452, 467, 469, 471, 475, 511, 512, 518–520, 528–530, 532–534, 559, 566, 567, 569, 583, 585, 601, 613, 628, 638, 642 Poiseuille linearization 566 Poisson equation 467, 468, 483, 485, 509, 518
710
Subject Index
Prandtl–Meyer variation 67 pressure, drop, thickening, thinning 41, 334, 350, 426, 447, 451, 467, 472, 475, 529 primitive equations 399 Quasi Periodic Ansatz 335, 347, 354, 381, 382, 384–386, 389, 390, 393, 396 quasigeostrophic 214, 291 Rankine–Hugoniot relation 53 rate of dissipation 422 – maximization 425 Reeb field 14 reflected shock 78 reflection – Mach (MR) 42, 67 – normal 80 – Regular (RR) 42, 67, 78, 93 – – strong 42 – – supersonic 80, 95 – – transonic 80, 95 – – weak 42, 88 – strong 79 – symmetric 94 – von Neumann (vNR) 43 – weak 79 Regularity A 347, 354, 359 regularity theory 161, 176, 179, 182 resonances 221, 255, 283, 295 resonant enhanced diffusion 686 Reynolds stress 695 Rossby, deformation radius, number, waves 206, 214, 290 Saint-Venant 280, 567 – principle 448 salinity equation 343 second law of thermodynamics 422 self-similar solutions 162, 197 SEM (Sun–Earth–Moon) 385 Serrin’s example 176 shock, diffracted, incident, infinitesimal, normal, reflected, small, strong, wave, compressive, Lax, oblique, supersonic 54, 56–58, 71, 75–78, 80, 87, 95 side-wall friction layer 227 sieve 671–675 skew product, dynamics, semiflow 381–394 solar system 396 solution, mild, strong, weak 365–380, 386, 389, 390, 395, 422 sonic line 93, 99 sound speed 47
specific energy, volume 41, 72 spin-down 223 stability – dynamical 88 stagnation point 686, 688 Standing Hypotheses 354, 385 stochastic event 401 Stokes 449, 511, 529–531, 534, 577, 636 – linear 511 – paradox 665 – problem 539, 543 – system 162, 163, 174, 182, 652, 653, 655, 657, 662, 666, 667, 671, 673 – steady 451, 527, 528, 581, 632 – time dependent 451, 511, 528, 567, 568, 572, 580, 582, 583, 591 stream function 4, 691, 693 stress relaxation 410, 426 Strichartz estimates 246 strip-like outlets 449 Strouhal number 677, 680, 681, 687 subsonic, pseudo-, relatively 49, 57, 93 supersonic, pseudo-, relatively 48, 57, 93 symmetric hyperbolic system 141 symplectic form 18 Tambora eruption 401 Taylor columns 211 Taylor–Proudman 210, 236, 239 temperature 41 – equation 342 thermal winds 210 thin domain dynamics 394, 395 transonic reflection 80 triple shock 59 turbulence, turbulent diffusion, turbulent models, turbulent viscosity 223, 651, 671, 689 twist flow 386, 388 two-scale convergence 658, 661, 684, 685 Uniqueness Problem
374
velocity, field, pseudo- 91, 676, 686, 689 viscosity, bulk, pressure, density and temperature dependent, shear, shear rate dependent 410–412, 426 Voltaire Hypothesis 333 Volterra integral equations 662 von Neumann paradox 62, 66 von Neumann Reflection 62 vortex sheet 54, 100 vortical singularity 100, 114 vorticity equation 7, 67, 100, 162, 167, 195, 690
Subject Index wave – acoustic 75 waveguide 215, 292 weak Hopf’s solution 642
711
weak Leray–Hopf solution 178 weighted Poincaré inequality 574 zero-entropy knots
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