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Mathematical Topics in Fluid Mechanics Volume 1 Incompressible Models Pierre-Louis Lions University Paris-Dauphine and Ecole Polytechnique
CLARENDON PRESS 1996
OXFORD
Oxford University Press, Walton Street, Oxford OX2 6DP Oxford New York Athens Auckland Bangkok Bombay Calcutta Cape Town Dares Salaam Delhi Florence Hong Kong Istanbul Karachi (Cttola Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto and associated companies in Berlin Ibadan
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A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data (Data available)
ISBN 0 19 851487 5 Typeset by the author in TEX Printed in Great Britain by Bookcraft (Bath) Ltd Midsomer Norton, Avon
PREFACE
Our goal in this book is to present various mathematical results on fluid mechanics models such as, for instance, Navier-Stokes equations both in the incompressible case and in the compressible case. Most of these results are new even if some have been announced in various places. For each of these recent results, we shall give complete proofs and we try to present as much as possible self-contained proofs that do not assume from the reader really
technical prerequisites other than a basic training in (nonlinear) partial differential equations. The book is divided into two volumes, the first one being essentially devoted to incompressible models while the second one is concerned with compressible models and asymptotic problems. Before we briefly describe the topics covered here, we wish to mention
that this book does not pretend to be a complete survey of the existing mathematical results even if we recall quite a few works on fluid mechanics equations. We selected what we consider to be the most significant results and in making such a biased selection we certainly omitted many relevant contributions to the field. We tried.to compensate for these omissions by a rather extensive bibliography (even if some of the references included there
are not quoted in the text). Let us also warn the reader that this book is concerned only with newtonian fluids and that many important subjects, such as the numerical approximation of the models we study, turbulence models, qualitative properties of solutions (bifurcation theories, attractors, inertial manifolds), reactive flows and combustion models, magnetohydrodynamics (MHD), multi-phase flows, and free boundary problems, are not even touched on here. Also, as we shall see, many basic open questions are left unanswered and we shall recall a large number of open problems.
More than two centuries after the introduction by L. Euler (and later by Navier) of the fluid mechanics equations, much remains to be understood mathematically even if considerable progress has been (slowly) made. We only hope that these notes will be a small contribution to the formidable task of the mathematical understanding of fluid mechanics models.
Let us now go a little bit into the contents of this book. We begin in chapter 1 by recalling the fundamental equations modelling newtonian fluids together with the basic approximated and simplified models-more are to be found in the following chapters. Much more could be said on the derivation of these models and we strongly advise the reader to consult such
vul
Preface
classical references as G.K. Batchelor [25], and L. Landau and E. Lifschitz [281]. Many references may be found in the bibliography here.
The rest of the book is divided into three parts. The first one (which concludes volume 1) is concerned with incompressible models and is divided into three chapters (2, 3, 4). We begin in chapter 2 with a study of the so-called density-dependent Navier-Stokes equations. We present in section 2.1 the most general existence results for such problems, namely global existence results of weak solutions in arbitrary dimensions with possibly vanishing density (i.e. a vacuum is allowed in some regions). These
results are due to R.J. DiPerna and the author-they were announced in [126], P.L. Lions [307]-and extend previous results due to various authors like S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov [17], S.N. Antontsev and A.Y. Kazhikov [16], J. Simon [436],[437]. Complete proofs are indicated in sections 2.3 and 2.4, while in section 2.2 we discuss regularity questions, stationary problems and mention various open questions like uniqueness, although we devote section 2.5 to partial "uniqueness results" showing that global weak solutions are equal to a strong one if it exists. The following chapter (3) deals with the classical Navier-Stokes equations for homogeneous, incompressible fluids. In section 3.1 we deduce from the analysis performed in chapter 2 the celebrated results of J. Leray [283],[284],[285] concerning the global existence of weak solutions, and
we recall various classical facts on Navier-Stokes equations that can be found in many existing books (see for instance P. Constantin and C. Foias [102], and the bibliography). Sections 3.2 and 3.3 are devoted to some recent "regularity" results in three dimensions consisting of variants or extensions of results shown by C. Guillope, C. Foias, R. Temam [203], L. Tartar [470], P. Constantin [98], R. Coifman, P.L. Lions, Y. Meyer and S. Semmes [95]. These results aim to make precise the regularity of the global weak solutions. Finally, in section 3.4, we consider briefly the so-called Rayleigh-Benard equations and we indicate existence results of global weak solutions.
Chapter 4 is the last chapter of this part on incompressible models. The first four sections are devoted to the classical Euler equations: we first recall (4.1) the state of the art on Euler equations. Let us mention at this stage that more details can be found in A. Madja [316] and J.Y. Chemin
[90]. We next (4.2) make a few remarks on the two-dimensional case, comparing the multiple notions of weak solutions and showing the existence and uniqueness of weak solutions for almost all initial conditions in V. We then briefly discuss in section 4.3 the fundamental open question of a priori
estimates in three dimensions, and we give some details about the simple examples introduced in R.J. DiPerna and P.L. Lions [126] showing the lack of "intermediate" a priori estimates. Finally, we introduce in section 4.4 the notion of "ultra weak" solutions, which we call dissipative solutions, whose
Preface
ix
only merits are their global existence and the fact that they coincide with classical solutions as long as they exist. Despite their weakness, they will turn out to be an extremely useful tool for recovering Euler equations from some compressible models in part III. The last two sections are devoted to other incompressible models, namely density-dependent Euler equations in section 4.5, and the models obtained via the hydrostatic approximation in section 4.6.
We then present in Appendices A-E various "technical" results used in chapters 2-4. The second volume will consist of parts II and III. The second part deals with compressible models. The first three chapters deal with the so-called (somewhat inappropriately) compressible isentropic Navier-Stokes equa-
tions and detail the results (and their proofs) announced by P.L. Lions [303],[304], [305]. The first chapter (5) deals with the compactness properties of sequences of solutions: we first explain (in section 5.1) difficulties encountered in such models which are due to the possible propagation of oscillations in densities (related to acoustic modes) and we state the available compactness results. Those results are shown in section 5.2, while the slightly more difficult case of Dirichlet boundary conditions is treated in section 5.3.
Chapter 6 deals with stationary (or time-discretized) problems associated with the compressible isentropic Navier-Stokes equations: we state our existence and regularity results in section 6.1 (in two and three space dimensions), we prove these results in section 6.2 and we treat the isothermal case in two dimensions in section 6.3. All these three sections are concerned with time-discretized problems. The final section of this chapter (6.4) is concerned with real stationary problems. Next, chapter 7 is concerned with global existence results for the Cauchy problem. Our main results are stated in section 7.1 and two "different" proofs are respectively given in sections 7.2 and 7.3 which rely crucially on the results and methods introduced in chapter 5. Finally, we consider in section 7.4 the case of other (and more realistic from the applications point of view) boundary conditions. Chapter 8 collects additional results and information on compressible models and related systems of equations. Section 8.1 is concerned with the exact compressible model and we present, in particular, some rather preliminary existence results. We come back in section 8.2 to isentropic and isothermal models and we present some global existence results of a different nature. Next, we investigate in sections 8.3 and 8.4 a shallow water model and we present some results essentially taken from P.L. Lions and P. Orenga [309]: section 8.3 is devoted to the global existence of weak solutions while section 8.4 is concerned with the global existence of
x
Preface
smooth solutions. Then we discuss in section 8.5 the compressible isentropic Euler equations (i.e. the inviscid case) in one space dimension, reviewing the results due to R.J. DiPerna [124], G.Q. Chen [92], P.L. Lions, B. Perthame
and E. Tadmor [311], and P.L. Lions, B. Perthame and P.E. Souganidis [310]. We also show in that section some new bounds (valid for all weak solutions and not only entropy solutions). Finally, we discuss in section 8.6 a tentative low Mach number model, directly inspired by A. Majda [315]. The final part of the book consists of a single chapter devoted to asymptotic limits. Sections 9.1, 9.2, and 9.3 are devoted to incompressible limits (small Mach number, density nearly constant) from the solutions of compressible, isentropic, Navier-Stokes equations built in chapter 7. In particular, we consider in section 9.1 the problem of recovering the global weak solutions (a la Leray) of the incompressible Navier-Stokes equations, and in section 9.2 we strengthen the convergence results in two space dimensions. Finally, in section 9.3, we obtain the solutions of the incompressible Euler equations in the smooth regime (globally in time in two space dimensions, and on the maximal time interval in three space dimensions). This analysis relies upon the notion of dissipative solutions of Euler equations introduced in section 4.4. Next, section 9.4 is devoted to the rigorous derivation of the linearized system (around a constant flow) and thus of a simple acoustics limit. Finally, section 9.5 is concerned with some asymptotic problems (of homogenization type) for the compressible, isentropic Navier-Stokes equations. P.L.L.
Paris December 1995
CONTENTS
1. Presentation of the models 1.1. Fundamental equations for newtonian fluids 1.2. Approximated and simplified models
.1 1
9
Part I: Incompressible Models 2. Density-dependent Navier-Stokes equations 2.1. Existence results 2.2. Regularity results and open problems 2.3. A priori estimates and compactness results 2.4. Existence proofs 2.5. Uniqueness: weak = strong 3. Navier-Stokes equations 3.1. A brief review of known results 3.2. Refined regularity of weak solutions via Hardy spaces 3.3. Second derivative estimates 3.4. Temperature and the Rayleigh-Benard equations
4. Euler equations and other incompressible models 4.1. A brief review of known results 4.2. Remarks on Euler equations in two dimensions 4.3. Estimates in three dimensions? 4.4. Dissipative solutions 4.5. Density-dependent Euler equations 4.6. Hydrostatic approximations Appendix A. Truncation of divergence-free vector fields in Sobolev spaces Appendix B. Two facts on D1"2(IR2) Appendix C. Compactness in time with values in weak topologies
19 19 31 35 64 75
79 79 92 98 110
124 125 136 150 153 158 160
165
173
177
xii
Contents
Appendix D. Weak L' estimates for solutions of the heat equation Appendix E. A short proof of the existence and uniqueness of renormalized solutions for parabolic equations
178 183
Bibliography of Volumes 1 and 2
196
Index
233
Contents
INTENDED CONTENTS OF VOLUME 2
Part II: Compressible Models 5. Compactness results for compressible isentropic Navier-Stokes equations
5.1. Propagation of oscillations and compactness results 5.2. Proofs of compactness results 5.3. The case of Dirichlet boundary conditions 6. Stationary problems
6.1. Existence and regularity results for discrete time problems 6.2. Proofs 6.3. The isothermal case in two dimensions
6.4. Stationary problems 7. Existence results 7.1. Global existence of weak solutions 7.2. Existence via regularization 7.3. Existence via time discretization 7.4. Other boundary conditions
8. Related questions 8.1. Remarks on the full compressible model 8.2. Remarks on the isentropic and isothermal models 8.3. Shallow water models: global existence of weak solutions 8.4. Shallow water models: regularity of solutions 8.5. One dimensional compressible isentropic Euler equations 8.6. On a low Mach number model
Part III: Asymptotic Limits 9. Asymptotic limits 9.1. The incompressible limit and the convergence to Leray's solutions
xiii
Contents
xiv
9.2. Further results in two dimensions 9.3. The incompressible limit and the convergence to Euler equations 9.4. Acoustics and the linearized system 9.5. A homogenization problem Index
1
PRESENTATION OF THE MODELS
1.1 Fundamental equations for newtonian fluids We recall here the standard derivation of the classical fluid dynamics equa-
tions in eulerian form. For the evolution of a fluid (or a gas) in N spatial dimensions (1V >-1)-,`Vthe description involves (N + 2) fields, namely the mass density, the velocity field and the energy. We shall derive the corresponding (N + 2) evolution equations in the case of a fluid filling the whole space, and we postpone the discussion of boundary conditions. Finally, let us mention that more details on the considerations which follow can be found in G.K. Batchelor [25], L. Landau and E. Lifschitz [281], and P. Germain [170] (see also the bibliography of this book). We begin with the evolution equation for the (mass) density p(= p(x, t)). Applying the principle of conservation of mass, we simply observe that for any volume 0-say, a smooth bounded open set of IRN-the variation of mass inside 0, i.e. fo 292 at dx, must be equal to the flux of mass on 80. Since particles of fluids are moving along the integral curves of X = u(X, t), this flux is clearly equal to fao p u n dS, where we denote by n the unit outward normal to a0. Therefore,
-
fdx = -J
ao and, if we apply Stokes' formula, we deduce from (1.1)
(1.1)
f{+div} = 0. Since 0 is arbitrary, we obtain finally + div(pu) = 0.
(1.2)
Presentation of the models
2
Another way (clearly equivalent) to derive (1.2) is to say that the transport of mass on a time interval (t, t+h) and the conservation of mass yield
p(X(t+h), t+h) J(h) = p(x, t) where X (t) = x, k (s) = u(X(s), s) and J(h) is the jacobian of the transformation (x H X(t+h, x)). Standard considerations on ordinary differential equations give: J(h) = 1 + h div u(x, t) + o(h); therefore, we obtain p(x, t) + h{ ap + u Vp + (div u) p} (x, t) + o(h) = p(x, t), and we recover (1.2).
We now turn to the evolution of the velocity field u = (ui (x, t), ... , UN(X, t)) or equivalently of the momentum pu. If we follow the first argument above and if we apply the principle of conservation of momentum, we find
f
pu(u n) dS + fO pf dx +180 F n dS.
(pu) dx =
(1.3)
faO
The two last terms of the right-hand side of (1.3) represent the forces acting on the fluid, namely volume forces corresponding to external forces (gravity, Coriolis, electromagnetic forces), and surface forces which, roughly
speaking, are due to the fact that we are dealing with a continuum and which can be (somewhat improperly) thought of as contact forces with, for instance, the fluid particles lying outside 0. Clearly, F is a tensor and if we
set a = -F, the tensor a is called the stress tensor. Classically, a fluid in motion is submitted to two kinds of stresses corresponding to compression effects and viscous effects, and one writes .
Q = -pl+T
(1.4)
where p, a scalar function, is the pressure and r is the viscous stress tensor. We denote by 1 the identity matrix (tensor), i.e. 1 = (8ij)ij. Therefore, using (1.3) and (1.4) and the Stokes formula, we deduce finally
(pu)+div(pu®u)-div(T)+Vp = pf
(1.5)
or (in a given orthonormal basis) a
at
(pui) + aj (puiuj -Tip+pbij) = p fi,
for
1 < i < N.
(1.6)
Fundamental equations for newtonian fluids
3
Here and below, we write equivalently 8j = a and systematically use the convention of implicit summation over repeated indices. Let us observe also that if we expand the derivatives of pu= and pusu3 in (1.6) and we use (1.2), we can also write (at least, if all functions are smooth)
p j +p(u-O)u-div(r)+Vp = pf.
(1.7)
Another way (again equivalent) to derive equation (1.7) is to use Newton's equations, noticing that the acceleration of a fluid particle at (x, t) is a {u(X (r), r)}I,.=t = (ai + (u V)u) (x, t). We then obtain
fp{+(u.v)u}dx
ffdx+fF.ndS o
=
and (1.7) follows easily.
We have now to describe the scalar unknown p and the unknown tensor r. Postponing the discussion of p, we recall a few facts concerning r. First of all, the conservation of angular momentum-which we shall not describe here-leads to the following fact: r is a symmetric tensor. Next, a classical fluid is a continuum where the constitutive law for r is of the following form
r = r(Du, p,T)
(1.8)
where the temperature T will be discussed later on. Then, if we postulate that r is a linear function of Du, is invariant under a change of reference frame (translation and rotation) and that the fluid is isotropic, we deduce from elementary algebraic manipulations that necessarily
r = \divul+2µd
(1.9)
where d is the so-called deformation tensor
d = 2 (Du + 'Du)
(1.10)
and A, p are the so-called Lame viscosity coefficients. Observe that in general A and p are functions of p and T in view of (1.8). The three assumptions mentioned above that lead to (1.9) correspond to the so-called newtonian fluids, the only case we shall consider in this book. In other words, we shall always assume here (1.9) and very often we shall consider A and p as fixed constants (i.e. independent of p and T). Furthermore, the kinetic theory of gases (monatomic gases) indicates that the Stokes relationship should hold, namely
=- 2µ. 3
Presentation of the models
4
This is the three-dimensional situation. For an "N-dimensional gas" we while of course we still have A = -28 for the evowould obtain A lution of a "3-dimensional gas" depending only on one or two coordinates. For most fluids and gases, experiments indicate that A + 3 is very small and this is why it is often set to 0 for common fluids, an assumption we shall not need in the rest of this book. In practice, and this is also crucial for the mathematical analysis of these models, we have
J /1>0, A+>-0.
(1.12)
Since T = 0 if A = p = 0, this case is called the inviscid (non-viscous) case, while if p > 0, A + p > 0, we have the viscous case. We finally derive the last equation which corresponds to the conservation of energy, which expresses the first law of thermodynamics. Before we discuss the equation in more detail, let us make a physical comment: the derivation of the energy equation relies upon the assumption that, in
a fluid in motion, the fluctuations around thermodynamic equilibria are sufficiently weak so that the classical thermodynamics results hold at every point and at all times. In particular, the thermodynamical state of the fluid is determined by the same state variables (thermodynamic pressure p, internal energy per unit mass e, thermodynamic temperature T, density p) as in classical thermodynamics, and these variables are determined by the same state equations.
Next, we observe that the total energy E is given by the sum of the kinetic energy pIu12/2 and of the internal energy pe. Then the conservation of energy reads d
df
p(lu2
o
r
+edx = -faO p(I 2
+pf udx+ Jo
2
r
(1.13)
ao
The two last integral terms of the right-hand side of (1.13) correspond to the work done by the forces and, assuming there are no sources of heat (we are dealing with non-reacting fluids), Q is simply the amount of heat received (or lost) across the boundary 80 characterized by a heat flux. In other words
Q=
fq.nds o
for some vector q to be determined. Using Stokes' formula, we deduce from (1.13) and (1.4)
at lpll22 +e)}+div{u[p`I22 +e)+`] -div(q)+pf - u. =
(1.14)
Fundamental equations for newtonian fluids
5
In order to close the system, it remains to describe p, e and q. If we choose as independent (thermodynamic) variables p and T (the thermodynamic temperature), then p and e are functions of p and T, i.e. obey some given state equations of the following form
p = p(p,T),
e = e(p,T).
(1.15)
We shall come back to this point later on. There remains to describe the heat conduction, or in other words to examine the dependence of q upon T (and p). If we suppose that the fluid is isotropic, we are led to
q = -k(p,T, (DTI) VT
(1.16)
for some scalar function k which, in most cases, is taken to be simply a function of p, T or even a constant called the thermal conduction coefficient. We wish now to recall some basic facts from thermodynamics that will
be useful later on. First of all, there exists a state variable called entropy (or entropy per unit mass), denoted by s, satisfying
ds = T {de+pd(1)},
(1.17)
and the basic assumption on thermodynamic equilibria yields the so-called Gibbs equation
1 {de +p d (1)} dt l p J dt = T dt ds
(1.18)
where At denotes the total time derivative, i.e. along fluid particle paths, namely (1.19)
Tt =
Then the entropy of a given volume 0 is given by fo ps dx. Applying the second law of thermodynamics, we find
qT ndS
f(ps)dx > - fa p
at
o
or
(ps) + div (ups) > -div ( ) .
(1.20)
On the other hand, we deduce from (1.18) and (1.2)
pds = dt
1
(p de +,pdivu).
T \
dt
(1.21)
Presentation of the models
6
In addition, multiplying (1.7) by u, we have p 8
22) +p(u-0) I2z
pf u-r.Du = pf
or, in view of (1.2),
pf -u--r-d.
(1.22)
Comparing (1.14) and (1.22), we find
5j (pe) + div (upe) + p div u = -div q + r d,
(1.23)
hence in view of (1.2)
p
de
+pdivu = -divq+r d,
and inserting this relation into (1.21), we find, using (1.2) once more
C (ps) + div (ups) _
-T div q + T r d.
(1.24)
In particular, we find, comparing (1.20) and (1.24) 0,
(1.25)
an inequality which must thus be satisfied for all (p, u, T). In particular, choosing u - 0, we deduce
-q . VT > 0, an inequality which, combined with (1.16), yields k _> 0, a fact which is consistent with experiments. Similarly, if we choose T to be constant, we deduce
r d > 0.
For a newtonian fluid, we deduce from (1.9) for all u
2µ1d2 + i1(divu)2 > 0
and this is easily seen to be equivalent to (1.12), namely the "natural" constraint on p and A.
Fundamental equations for newtonian fluids
7
At this stage there only remains to discuss the "thermodynamical constraints" on the state equations (1.15); therefore, we choose p and T as independent variables for the thermodynamic relations. Then we deduce from (1.17) as
as
1 ae
aT = T aT'
ap
=
1 lee
p 1
T ap - p2 J '
(1.26)
and, in particular, s can be deduced from the expressions (1.15). But (1.26) (T { also implies the following compatibility equation: p (TU8 Te ) _ }), i.e.
-
2P {p_T}
= aeP
(1.27)
This relationship -constrains possible laws for p and e.
Additional constraints can be derived from the second law of thermo-
dynamics. Choosing now (e, r) or (s, r), where r = 1, as independent variables, one deduces from the second law the following equivalent statements (1.28) s(e, r) is concave in (e, r) or
e(s,r) is convex in (s, r).
(1.29)
These statements are obviously equivalent since (1.28) is equivalent to the convexity of the set {(s, e, r) / s < s(e, r)} while (1.29) is equivalent to the convexity of the set {(s, e, r) / e(s, r) < e}, and those two sets are identical since (1!7) yields ae =' T1 > 0.
Let us mention two consequences of (1.28)-(1.29). First of all, p or r 82, being fixed, a' = T and thus = aT. Hence, (1.29) yields e(p, T) Similarly, s being fixed, aT p(p, T)
p(p,T)T
is increasing in T.
(1.30)
= -p, a = T and thus (1.28), (1.29) imply
is increasing in p for s fixed (s = s (p, T) ) is increasing in p for e fixed (e = e(p,T)).
(1.31)
In conclusion, the restrictions on the state equations (1.15) are given by (1.27) and (1.29), while (1.26) indicates how to derive s once (1.15) is given.
We wish now to conclude by giving two examples and we begin with the most common model, namely the case of an ideal gas. An ideal gas is a fluid which obeys the following laws: Mariotte's law, namely p = pf (T), and Joule's law, namely e = e(T) for some scalar functions f and e. Observe then that (1.27) implies that f (T) is linear and we deduce
p = RpT,
e = e(T)
(1.32)
Presentation of the models
8
where, because of (1.31), R > 0, e is increasing in T. The constant R is called the ideal gas constant. Writing C,, = e'(T) (> 0), CP = O _ R + e'(T) where h = e + P is the so-called enthalpy, we see that CP > C and we set -y = c . Common fluids and gases in "normal" conditions (i.e. not with high densities or at high temperatures) are often well described by the ideal gas model, which assumes furthermore that CP and C are constants, i.e. e= C, , > 0 and -y = 1+ cv > 1. We then deduce from (1.26) that, l i -1 ) = C log P It is possible to deduce in this case, s = R log (T P from the kinetic theory of gases that -y = N (for a monatomic gas) or (y 3 if N = 3. The most interesting region for physical applications is
Tr
5
\1 31As a mathematical exercise, it is worth looking at the case when we use
only Joule's law, namely e = e(T). Then, because of (1.27), we deduce
e = e(T),
p = pi(p)T
(1.33)
and because of (1.30) and (1.31), e(T) and pl (p) have to be increasing (or at least non-decreasing). In fact, these monotonicity conditions on e and p are, in this case, equivalent to (1.28) since, up to some irrelevant constant,
s = h(e) = P(r) where h'(e) = T (e = e(T)), P'(-r) = pi(T) Let us conclude by summarizing the model we derived for newtonian fluids
a
+ div (pu) = 0
a(pu=) + div (puus) - aj{µ(a;u; + a=uj)}
at
- as(A div u) + asp = pfi, a(pe)
for l < i <
(C)
+ div (pue) + pdiv u - a;(kajT) =
(au3 + a,us)2 + A(div 26)2
2 where A, y, p, e are functions of p and T; k is a non-negative function of p, T
and possibly IVTI; A, p satisfy (1.12); and (1.27), (1.29) hold. The most common case being the one when A, p, k are constants, we have p = RpT (R > 0), e = CT (C > 0) (the ideal gas) and we set -y = 1 + cv . Finally, s, defined (up to a constant) by (1.26), satisfies a(mps)
+ div (ups) (1.34)
_ +(div (kVT) + 2 (asuj + ajus)2 + A(divu)2).
Approximated and simplified models
9
When p > 0, A + N > 0, the system of equations (C) is called the "compressible Navier-Stokes equations".
1.2 Approximated and simplified models In this section we wish to discuss successively incompressible models, low Mach number expansions, ideal fluids and some simplified mathematical models. Many more approximations or simplifications are possible (shallow waters, quasi-geostrophic approximations, etc.), some of which will be studied in the rest of this book, in which case we will say a few words on their derivation. We thus begin with incompressible models or, more accurately, models for incompressible fluids, which are very important for applications since many common fluids (liquids) are incompressible or only very slightly compressible. Mathematically, the incompressibility means
div u = 0.
(1.35)
Indeed, if at fixed time t > 0, we consider the volume of an open set 0 filled with the fluid, then, at time t + h (h > 0), the corresponding fluid particles will fill the set
Oh = {x(t+h, y) / y E 0} where i(r) = u(x(r), r), x(t, y) = y. The volume of Oh is given by J0
J(h, y) dy
where J(h, y) is the jacobian of (y H x(t+h, y)). Finally, it is well known that we have J(r) = div u(x(r), r) J(r). Therefore, the volume of Oh is equal to the volume of 0 (for all 0, h > 0) if and only if (1.35) holds. Next, we may follow the same derivation of the equations as in the preceding section and we recover the conservation of mass (1.2) and the integral
form (1.3) of the conservation of momentum. Notice that since div u = 0, if p is constant initially it remains so, in which case we say the fluid is homogeneous (p - p E (0, oo)). Also, the stress tensor o has to preserve the incompressibility and we deduce from the hypothesis of a newtonian flow
0 = 2,2d - p l
(1.36)
Presentation of the models
10
where p > 0. The hydrostatic pressure, p, is in fact a Lagrange multiplier associated to the incompressibility constraint (notice that oi; aiuj = 0 for
all u such that div u = 0 if and only if 5i; = p bi; for some p). Notice finally that div d = a a; (ai ld; + a; ui) = a Dui since div u = 0, and we obtain finally a't
(pui) + div (puui) - a; (p(aiu; + a;uj)) + aip = pfi, for l < i < N, div u = 0,
(1.37)
or, if p is constant,
a (pui) + div (puui) - pAui + aip = pfi, at
(1.38)
for1
In conclusion, we look for the (p, u, p) solution of the following system of equations
5 + div (pu) = 0 a(pui)
at
(1.39)
+ div (puui) - a; (p(aiu; + a;ui)) + aiP = pfi
forl
J
If p > 0 (and is constant), i.e. if viscous effects are present, this system of equations is the so-called non-homogeneous incompressible Navier-Stokes equations (quite often, incompressible is omitted), and a particularly relevant case consists in choosing p - -p > 0, in which case (1.39) reduces to aui
p at + pdiv (uut) - pout + aiP = Pfi, forl
(1.40)
This system is called the homogeneous, incompressible Navier-Stokes equations, often simply called the Navier-Stokes equations(!). Replacing p by v = p , p by p , we see it is enough to study the case when P = 1. It remains to write down an equation for the internal energy e (= e(p, T)): following the procedure described in the preceding section and taking into account the incompressibility condition (1.35), we find a
jT- + div (upe) - div (kVT) = 2 (19,U; +
a;ui)2
(1.41)
where k > 0 may depend on p, T. The most common case corresponds to k > 0, k constant (i.e. independent of T) and, if we are in the homogeneous
Approximated and simplified models
11
situation, e = e(P,T) is only a function of T, for instance e =
for
some
We next discuss low Mach number expansions. We just want to present here some formal asymptotics, some of which allow us to go from the compressible models introduced in the preceding section to the incompressible models we stated above. First of all, let us recall the precise definition
Therefore, letting M go to 0 means that, keeping p,T of a typical size 1, we consider u of order e where E > 0 is a small parameter (going to zero). We also rescale the time variable,
of the Mach number: M =
n,(v) .
considering finally
tl
Pe_(x, t) = P (x, E J
ue (x, t)
_
( t E u \y' E ) 1
Te(x,t) =T(x, t)
(1.42)
E
where (p, u, T) solve (C) and \, µ, k depend upon the scaling parameter E in a way to be determined. We begin with the "traditional" derivation of the homogeneous, incompressible Navier-Stokes equations. We then take µ(e) = Eµ, A(e) = e,1, k(e) = ek, f (E) = elf, where A, µ, k may be functions of (p, T), and we deduce from (C)
J+div(uepc) = 0 8(PEue)
8t
+ div (peue 0 ue) - div (2yde + A div ue)
+v(' p,) =Pef
(1.43)
a(p.ee) + div (Pufee) - div (kVTe) +pe div ue 8t = E2 (88ue,a + 8jue,,)2 + A(divu,)2}.
l
2
The equation corresponding to the conservation of momentum indicates that pe should behave like P(t) + e2p + 0(E2) where P(t) is independent of x. One way to ensure this fact consists in assuming that pg and Tf are initially constant (up to terms of order e) in order to avoid initial layers, and that pe is initially constant up to terms of order e2 (this is clearly the case if pe and Te are also constant up to terms of order E2). Then, at least formally, if we set 0(E
Te = T + ETi + O(e2),
ue = u + O(E)
Presentation of the models
12
we deduce from (1.43)
p(-p, T) = P(t) > 0 is independent of x P
(p, T)P1 +
( p, T
)Ti =
P1 (T)
+ di v
a )
(1.44)
is inde pendent of x
(up) = 0
(1 . 45) (1 . 46)
- div (2µd + A div u) + V7r = T f
(1 . 47)
a(pe) + di v (Pd-) - div(kVT) + P(t) div u = 0
(1 . 48)
+ div (pu ®u )
where e = e(p,T), y = µ(p, T), k = k(p,T), it = lien
o+
(pE-P-EPi)
From now on, we assume, to simplify the calculations, the ideal gas laws:
p = RpT, e = CT. Then we deduce from (1.48) that j5-9 = 711 P(t) is independent of t; this, in fact, requires some boundary conditions like, for instance, if we consider the evolution in the whole space IR'i, u
and VT vanishing at infinity. Hence, P(t) = Po > 0. We then deduce from (1.48) and (1.46)
a (log T) + u V(log T) _ - (
+ u V) (log p)
div u =
- . div (WT). Y PO
Hence, if T is constant initially, T is equal to that constant for all (x, t). Therefore, p and T are positive constants and div u = 0. In conclusion, (1.46)-(1.48) reduce to p
pf,
5
where µ = µ(p, T), p > 0, T =
diva = 0
and we recover the homogeneous, incompressible Navier-Stokes equations. We next present some variants of the above formal derivation. First of all, we consider the case when p(E) = ep, A(e) = ea, k(e) = Eke and ke -> k > 0 as a --> 0+, f (E) = e2 f . We still assume the ideal gas laws (p = RpT, e = CT) and we now assume that p is initially constant up to terms of order E2 but we do not assume that p or T are initially constant (or equivalent to constants). With the same notation as above, we still recover (1.46)-(1.48) with k replaced by T. Exactly as before, we deduce
p
,
that P - Po > 0 and -y Po div u = div (kVT).
(1.49)
Approximated and simplified models
13
In particular, if k = 0, i.e. k(6) = o(e), div u = 0 and (1.46)-(1.48) become
+ div (up) = 0, a(Pu)
div u = 0,
+div(pu®u) -div(Wd)+0ir = pf,
i.e. the non-homogeneous, incompressible Navier-Stokes equations. Notice
that 1 = p (p , p) and that, if p is independent of p and T, div (mod) _ µAu. Next, if k > 0, we obtain the following new system of equations 6P -
5T
div (up) = 0,-
- -y
div u = div (kV
())
(1.50)
+ div(pu 0 u) - div (mod + a div u) + 0ir = p f . We would like to point out that we are not aware of any formal derivation of incompressible models including a temperature equation (or energy equation) like (1.41). One possible physical explanation for this apparent lack of consistency between compressible and incompressible models is the following assertion: compressible models are valid for gases and the incompressible limit yields particular incompressible models, while the general incompressible models aim to describe liquids where compression effects are neglected.
We next briefly make a few remarks on perfect fluids which correspond to the particular case when A - p - k - 0. The compressible models (also called compressible Euler equations in this case) take the following form C9 p
+ div (pu) = 0
aPu
+ div (pu (9 u) + Op(p, T) = pf a
(1.51)
Z
at(p(2 +e))+div{u{pi. ++pe+p}}
= pf - u
and in the ideal gas case p(p, T) = RpT, e = y = 1+R-. Furthermore, we might expect that the entropy equation (1.34) simply becomes aps
+div(ups) = 0 or pat
o.
However, this is the case only on regions where p, u, T are smooth. In fact, it is well known that shocks (discontinuities) appear in finite time and that
Presentation of the models
14
the entropy equation does not hold "on shocks". Instead, only the following inequality, which is deduced from (1.34), remains true: 8ps
at
+div (pus) > 0.
(1.52)
As long as shocks do not form, the entropy equation is valid and if initially s is constant, it remains constant: s_ so. In the ideal gas case, this leads to p = (Re'0/c0) p"r = co p7 (co > 0) and we obtain the so-called isentropic gas dynamics system
a 5 + div (pu) = 0
a)
(1.53)
+ div (pu ®u) + coVp'r = 0.
This system has clearly a somewhat limited physical validity in view of what we have just recalled but has recently received a lot of (mathematical)
attention. For a -general pressure law, s = so means p = p(p) (= &,T) with T determined by s = so) where P is increasing in p (see (1.31)). For a perfect fluid the incompressible models take the following form: in the homogeneous case, we obtain the classical Euler equations
at + div (u ®u) + Vp = 0, div u = 0
(1.54)
(the constant density p =_ p is scaled out by considering p/p instead of p). In the inhomogeneous case, we obtain the density-dependent Euler equations
ap
a + div (pu) = 0,
ate)
div u = 0 (1.55)
+ div (pu ®u) + Vp = 0.
We now turn to some models obtained by further simplifications or different (singular) asymptotic limits. We first consider the compressible system (C) with p, A, k constants (p, A > 0, k >_ 0) and we neglect the heating due to viscous dissipation, an approximation which is reasonable except for hypersonic gases. We then obtain for the temperature (or energy) equation
ape
+ div (pue) + div up - kzT = 0.
(1.56)
We also assume Joule's law: e = e(T), p = p(p)T (for instance the ideal gas law e = p = RpT) where P is increasing. Next, we take k = 0 (we
Approximated and simplified models
15
do not claim this is a relevant physical assumption!) obtaining an equation which is easily seen to be equivalent to the following entropy equation
as +pu.Vs = 0. (1.57) at In particular, if s is constant initially, we now expect to deduce s - so p
(we do not expect shocks since A, p > 0), and we obtain the compressible isentropic Navier-Stokes equations
a +div (pu) = 0 (1.58)
ate) + div (pu ®u) - µAu - (A+p)V divu + Vp(p) = pf where p(p) =p(pT) (and s(p, T) = so) is increasing in p because of (1.31). In particular, in the ideal gas case, p(p) = co p' with co = Re-1°' . Let us make at this stage a general remark, which is related to isentropic or barotropic pressure laws. Precise state equations like p = p(p, T) are not an easy matter nor are they fully understood. In particular, many semiempirical laws exist (depending on the physical phenomena to be studied). For instance, one can find the following pressure law for water in R. Courant
and K.O. Friedrichs [108] (p. 8): p = A(p/po)1 - B with the proposed values -y = 7, A = 3001, B = 3000 (atm.) and po is the density at 0°C. We finally describe some (mathematical) formal asymptotics. First of all, we take p = p(p)T, e = CT and we let C go to +oo (!). The temperature equation then yields aT 0 p at or
p at + pu VT - icLT = 0
if cv - ,c > 0. A particular solution is clearly T - To > 0. We then obtain
the compressible, isothermal Navier-Stokes equations
+div(pu) = 0 a
)+div(pu(9 u)-MAu-(A+p)Vdivu+ToVp(p) = 0
and p(p) = Rp in the ideal gas case (p(p) is increasing, as usual).
The other limit consists in letting k go to +oo. We deduce T
T(t)
(independent of x), and in the ideal gas case, the function T is determined by the conservation of energy
d
J
=
fpf.u
Presentation of the models
16
where M = f p (independent of t), or equivalently by the entropy
R
d dt J p log
p
=
Iaiui + ajuil2 + A(div U)21.
fT{ 2
We now conclude this section (and the chapter) with a brief discussion of initial and boundary conditions. Our main goal in this book is to study the Cauchy problems for the models derived and described above. In other words, we study the systems of equations for t > 0 prescribing the values of p, pu, pe at t = 0. Of course, for the models introduced in this section in which one does not write an energy equation or one takes p - p constant, we only prescribe p and pu, or u at t = 0. The question of boundary conditions is much more delicate and would require a detailed discussion. Our ambition in this book is somewhat limited since we shall consider problems set in a domain S2, with standard (mathematical) boundary conditions on 811 for (possibly) p, u, T. We shall essentially always restrict ourselves to three cases: (i) S2 is a smooth, bounded, connected, open set in IRN, and we impose Dirichlet-type boundary conditions on all on p and u while we impose on T Dirichlet, Neumann or mixed boundary conditions. A typical example would be the case of a homogeneous Dirichlet condition, i.e. u n = 0
on 812 or u = 0 on alt if viscous terms are included in the model, in which case no conditions on p on all are imposed. If we incorporate a temperature equation, a rare event unfortunately in this book, we can impose for instance homogeneous Neumann conditions: an = 0 on 812. Here and above, n denotes the unit outward normal to all.
(ii) 12 = U 1(0, Li) with Li > 0 (V 1 < i < N) and all functions are periodic in each xi of period Li, for all 1 < i < N. (iii) 1 = IRN and (p, u, T) "vanish at infinity" or are "constant at infinity"
PART I INCOMPRESSIBLE MODELS
2
DENSITY-DEPENDENT NAVIER-STOKES EQUATIONS This chapter is devoted to the so-called density-dependent Navier-Stokes equations or inhomogeneous, incompressible Navier-Stokes equations, namely ap
+ div (pu) = 0, p > 0, in St x (0, +oo) + div (pu ® u) - div (2µd) + Vp = pf, div u = 0,
(2.1)
(2.2)
in SZ x (0, +oo)
where d = 2 (a=uk + abut), f is given on S x (0, +oo) and µ = µ(p) is a continuous, positive function on (0, +oo). Of course, (2.1), (2.2) are to be complemented with boundary and initial conditions. The above model was derived in chapter 1 but we wish to add to that derivation the fact that it can be also seen, due to the possible dependence of µ on p, as a model for the evolution of a multi-phase flow consisting of several immiscible, incompressible fluids with constant densities and various viscosity coefficients.
2.1 Existence results Let us first describe the boundary and initial conditions for (2.1)-(2.2) which we shall consider here. We study only three model cases (recall that
N>2): (i) (Dirichlet case) SZ is a smooth, bounded, connected open subset of IRN and
u=0
on
5l.
(2.3)
Density-dependent Navier-Stokes equations
20
(0, L2) (Li > 0, d 1 < i < N) and (p, u) are periodic so that we can consider that (2.1)-(2.2) hold on IRN x (0, +oo) with (p, u) periodic in each xi of period Li, f o r all i E {1, ... , N}. (iii) (RN case) Sl = IRN and we want p to be bounded (for example) while u satisfies, in a sense to be made precise, (ii) (Periodic case) SZ = fl
u - u00
for all t>0
(2.4)
where uo, is fixed in IRN. We now discuss initial conditions. In view of (2.1)-(2.2), we need to
impose conditions on p and pu at t = 0. Observe that we cannot directly impose initial conditions on u in case p vanishes on some part of 12, i.e. if there is some vacuum. We then consider,- if fl- is bounded, Po > 0 a.e. in 0, Po E L°°(Sl), mo E L2(SZ)N, mo = 0 a.e. on {po = 0},
(2.5)
(mol2/Po E L1(fl) where we agree that Imo I2/Po = 0 a.e. on {po = 0}. We also want to impose
Pit=o = Po
on 12,
putt=o = mo on 12.
(2.6)
Of course, in the periodic case, we may extend po and mo periodically on IRN.
If Sl = IRN (case (iii) above), (2.5) is replaced (if u,,,. # 0) by Po > 0 a.e. in IRN, Po E L°°(IRN), mo - pour E L2(IRN), mo = 0 a.e. on {po = 0},
(2.7)
Imo - Pouool2/Po E L1(IRN)
For technical reasons, we need to assume, in the case when Sl = IRN, in addition to (2.7), one of the following three conditions (1/Po)1(po
for some bo > 0,
(2.8)
or
(P - Po)+ E LP(IRN),
for some p E (0, oo), p E
N
(7,00),
(2.9)
or
if N = 2,
if N > 3,
f
POP
<x>2(p-1) (log
<x>)' dx < oo
R2 for some p E (1, +oo], with r > 2p - 1
Po E LT
00 (IRN
where we define <x>= (1 + IxI )1/2
)
(2.10)
Existence results
21
Finally, we assume that the force f satisfies
f E L2(S2 x (0,T))N,
for all T E (0,00).
(2.11)
Since we shall always make these assumptions in the rest of the chapterunless explicitly mentioned-we shall not recall them. Our main existence result will state the global existence of weak solutions that could be called "solutions a la Leray" by analogy with the classical global existence results for the homogeneous, incompressible NavierStokes equations obtained by J. Leray [2831,[284},[285], and we wish now to define precisely what we mean by weak solutions. We look for solutions satisfying, for all T E (0, oo), R E (0, oo): p E L°°(11 x (0, oo)); u E L2(0,T; H o (S2))N (Dirichlet case (i)), u E L2(0, T; HPer)N (periodic case (ii)), u E L2(0, T; H' (BR))N-and u E L2 (0, T; LT (IRN)N if N > 3 (whole space case (iii)) ; plul2 E LO° (0, T; Ll (Sl)) if fl is bounded, pju - u°° 12 E LO° (0, oo; L1(IRN)) if g2 = N. Vu E L2(S2 X (0, T)); p E C([0, oo); L' (S2)) if D is bounded, p E C([0, oo); LP(BR)) if SZ = IRN
for all 1 < p < oo. Finally, if (2.8) or (2.9) hold, we require that u E L2(IRN x (0, T)) for all T E (0, oo).
Here and everywhere below, BR = {y E RN / I yI < R}, Hl (O) _ { f E L2(O) / ai f E L2 (0) }, Ho (S1) is the subspace of HI (fl) consisting of functions whose trace on 811 vanishes, Hpet = if E Hl (BR) for all R < 00, f is periodic in xi of period Li for all 1 < i < N}. Furthermore, (2.1) holds in the sense of distributions (for example) in 12 x (0, oo) (case (i)) or in IRN x (0, 00) (cases (ii) and (iii)) and the initial condition on p (contained in (2.6)) is meaningful since we require p to be continuous in time (with values
in LP or L0C). Of course, we require that diva vanishes as a distribution (on S2 x (0, oc) or IRN x (0, oo) in the periodic case or if n = JRN). In the periodic case (ii), p is periodic in xi of period Ls for all 1 < i < N, for all t E (0, oo). It only remains to explain the meaning of (2.2) and of the initial condition on pu (contained in (2.6)). We shall use a weak formulation based upon a class of smooth test functions, namely the class (D of 0 E C°°(IRN x [0, 00))N, 0 has compact support in f2 x [0, oo) in the Dirichlet case or if S2 = IRN, (0 is periodic in xi of period Li for all 1 _< i < N in the periodic case, div q5 = 0 on IRN x (0, cc). We want u to satisfy, for all 0 E 1),
r
rr
- J rno O(x, 0) dx +I
nx (o,oo)
80
Puiu; aio; (2.12)
1 +2A(aiuj +a?ui)(aioj+ajoi)-P.f'0 } dx dt = 0.
Let us also emphasize the fact that, in view of classical density resultssee, for instance, R. Temam [472], R. Dautray and J.L. Lions [115]-we
Density-dependent Navier-Stokes equations
22
could equivalently take test functions 0, say in the Dirichlet case, such that div 0 = 0, 0 E L2(0, T; Ho (1))N and at E L2(S2 x (0, T))N for all T E (0, oo), 0 vanishes on 1 for t large, VO E L2(0, T; LP(S))) (for instance)
where p > 2 if N = 2, p = N if N > 3. This last integrability requirement follows from the fact that, by Sobolev embeddings, we have, for all T E (0, oo), if N > 3 (for example) PIuI2 E L°° (0, T; L1(0)),
PIuI2 E L1
(O, T;
L7 (f))
(Q)) (V 1 < i, j < N). hence puiuj E L2 (0, T; L At least formally, we expect solutions of (2.1)-(2.2) to satisfy the energy identities:
T Jn
PIuI2 dx +
Jn
I
µ(aiuj + aju2)2 dx =
2pf u
if f is bounded, or
d fn
PIu-u.I2 dx
+ J µlauj + a3ui)2 dx = n
f 2Pf (u-u.) n
if SZ = IRN. These identities are obtained upon multiplying (2.2) by u (or by u - u<), observing that, because of (2.1), ( (pu) + div (pu 0 u)) u =
(p 2)+div(pu122)} and integrating by parts over 0. As it is often the case when dealing with global weak solutions of nonlinear partial differential equations, the global weak solutions we obtain satisfy the following energy inequalities dt
f
PIuI2 dx +
f
f pIu12 dx +
<,n
I
µl(guj + ajui)2 dx < 2 fn pf
u
in D'(0, oo), (2.13)
f f µ(aiuj + ajui)2 dx ds
oI
Po
t
dx + 2
fo fn p1 u dx ds
(2.14)
a.e. t E (0, oo),
if f is bounded, and if f = IRN dt
f
N
PIu-u.I2 dx +
r < 2 J N pf
JUN
µ(asuj + ajut)2 dx
l (2.15)
in D'(O, oo),
j
Existence results
23
IIr
I
0
]RN
J]RN
ImO-Pouoo12
p(S,uj +83u;)2dx ds
dx+2
PO
'
[[pf.(u_u,)dxds
(2.16)
Jo Jsz
a.e. t E (0, oo).
We may now state our main existence result.
Theorem 2.1. There exists a global weak solution (p, u) of (2.1)-(2.2) satisfying the boundary and initial conditions described above and the energy inequalities stated above. Furthermore, we have for all 0 < a < 8 < oo meas {x E RN / a < p(x, t) < ,Q} ---
(E [0, +oo])
is independent of t > 0.
(2.17)
In particular, if SZ = IRN and PO - p°O E LP(IRN) for some 1 < p < 00, p°° E [0, oo), p - p°O E C([0, oo); LP(]RN)) and IIP - P°°IILP( N) is independent of t > 0. Also, if pO = p on SZ for some p E [0, oo), then p - p on SZ x (0, oo).
Remarks 2.1. 1) Global existence results for non-homogeneous, incompressible Navier-Stokes equations were first obtained by A.V. Kazhikov [254]-see also A.V. Kazhikov and S.H. Smagulov [260], S.N. Antontsev and A.V. Kazhikov [16], S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov [17]-in the case when p is independent of p, po is bounded away from 0. These results were extended by various authors and in particular by J. Simon [435],[436],[437] allowing po to vanish but still with a constant A. The methods of proofs do not adapt to the situation treated here, namely p depending on p. Furthermore, the continuity in t of p (with values in LP) or properties like (2.17) were not known with the above generality. Particular cases of results similar to Theorem 2.1 were obtained by R.J. DiPerna and
the author relying on general results on "transport equations" shown in [128] and were announced by R.J. DiPerna and P.L. Lions in [126]. This is the approach we detail (and extend) here. 2) Many questions are left open; we shall come back to these in the next section. Let us only mention a rather technical matter: the weak formulation (2.12) yields the existence of a pressure field p (defined up to a "distribution in t only") which is essentially a distribution (one can analyse
its singularity in terms of negative Sobolev spaces, or sums of negative Sobolev spaces)-see J. Simon [437] for more details; however, we do not know if p (appropriately normalized) E Li °(SZ x (0, oo)). 3) The final statement of Theorem 2.1 is a trivial consequence of (2.17).
Obviously, if p - p > 0, u is a weak solution of the homogeneous, incompressible Navier-Stokes equations (1.40) (the classical Navier-Stokes equations!). We shall come back to this particular case in chapter 4.
Density-dependent Navier-Stokes equations
24
4) Let us also explain how (2.17) yields the statements on p- p°O: indeed,
(2.14) implies that the distribution function of p - p°° is independent of t > 0, therefore I I P- Poo II LP (U.N) is independent of t. Since we already know,
by definition of weak solutions, that p E C([0, oo); LP1"(IRN)), we deduce the fact stated in Theorem 2.1, namely p E C([0, oo); LP(IRN)), from the followin; classical observation from measure theory. Let wn E LP(RN) be such that w,a - w in L o°(IRN), and w1, and w are equimeasurable, i.e. have n
the same distribution function. Then w,a --+w in LP(IRN). Working with n
Iwn I sign wn instead of wn if p = 1, we see that it is enough to consider the case when 1 < p < oo. Then, obviously wn converges weakly in LP(IRN) to w while II wn I I LP = I I w I I LP . The strong convergence in Lp (IRN) follows. 5) At the end of this chapter (in section 2.4), we will briefly explain how
the proof of Theorem 2.1 yields even more general results where we are allowed to take p° E LP (f) if SZ is bounded, p° E L C (IRN) if SZ = ]RN and p > a if N > 3 while p > 1 if N = 2. In this case, we need to assume that (t H µ(t)) is bounded and bounded away from 0 on [0, oo) (0
case when u00 = 0. Indeed, if SZ = IRN and (p, u) solves (2.1)-(2.2), then (p(x+V(t),t), (u(x+V(t),t) v(t))) also solves (2.1)-(2.2) where
-
v E L1(0, T; IRN) (`d T E (0, oo)) and V (t) = f o" v(s) ds. In particular, if we
-
choose v(t) = u00, we only have to work with (p(x+u0t, t), u(x+u0t, t) U..) and thus we only need to study the case u00 = 0. 8) Let us finally mention another consequence of (2.17) in the whole
space case (SZ = IRN). We claim that if (2.8) or (2.9) hold then u E L2(IRN x (0,T)) for all T E (0, oo). Indeed, if (2.9) holds, (2.17) implies
that (2.9) holds with p(t) replacing p0. We then deduce from the Sobolev and Holder inequalities, for all t > 0, defining a = (p - p)T, that PII UII L2(IRN)
IIaIILP(13N) IIuIIL2(RY) IIDUII
(RiV)
where 2 + 1-2N -2 = pp 1. Observe that 9 E (0, 1). Our claim follows easily since Vu E L2(IRN x (0, T)) for all T E (0, oo). If (2.8) holds, we observe that (2.17) also implies the equality II(1/P(t)) 1(p(t)
Existence results
25
Then, u = u 1(p<6o) + u 1(p>6o) and u 1(p>6o) E LOO(O,T; L2(IR.N)) since Iu In addition, we have u11(p>6o) 7ro
Jul 1(p<6o) = WTIu!)
1
P)
1(p<60) E LOO(O,T; Ll(IRN)),
for all T E (0, oo). By Sobolev embeddings,, we also have, if N > 3: u E L2(0, T; L2N/(N-2)
(IRN)) (V T E (0,oo)) and thus Iul1(p<6o) E L2(IRN x (0,T)) (V T E (0, oo)), and even L242 (0, T; L2 (IRN)). If N = 2, we obtain the L2 bound using the results shown in Appendix B.
Finally, we would like to make a general comment on the initial condition imposed on pu and on the weak formulation (2.12). Indeed, we did not require any time continuity (with values in an arbitrarily negative Sobolev space) upon pu and, in fact, there is no reason why, in the sense of distributions, pu should converge to mo at t goes to 0+. The only information we can obtain is that, roughly speaking, pu converges to mo up to a "gradient-like" distribution. A convincing argument which shows that we cannot expect more is to consider the case when po > p > 0. Then, by Theorem 2.1, p > p on ) x [0, oo). If (pu)(t) --+ mo in L10C (S) as t -+ 0+, then, since p E C([0, oo); L O0), u(t) -+ uo = in L L(0) as t --+ 0+. In particular, we must have
div (uo) = 0
in
V'(SZ).
(2.18)
We did not impose such a condition on uo since, in general, uo does not exist when po vanishes.
This is clearly a delicate point that we wish to clarify-in particular because we shall encounter similar difficulties in the study of incompressible limits (chapter 9). In order to keep ideas clear and avoid unpleasant technicalities, we only state and prove the following result in the case when S2 = IR.'v and u0O = 0 even if similar results can be obtained in the case of Dirichlet boundary conditions-the periodic case or the case SZ = IRN, u0O 0 0 are easily adapted from the case we treat. In order to avoid (too) negative Sobolev spaces, we introduce some notation: we denote by Rt = (_A)-112 a (1 < i < N) the usual Riesz transform and we denote ate, respectively by RA, R and R the operators (-A)-1/2 rot , (-0)-1/20 and (-A)-1/2 div. We shall also need the following elementary variant of Hodge-de Rham decomposition:
--
Lemma 2.1. Let N > 2, p E LOO(IRN) such that p > p > 0 a.e. on IRN for some p E (0, oc). Then there exist two bounded operators PS, Q6 on
Density-dependent Navier-Stokes equations
26
L2(IRN)N (whose norms depend only on p and IIPIIL-(RN)) such that for
all m E L2(]RN)N, (p, q) = (Ppm,Qpm) is the unique solution in L2(IRN) of
m=p+q, R (lp) =0, RAq=O.
(2.19)
P
Furthermore, if pn E L°°(IRN), p < pn < p a.e. on IRN for some 0 < p < P < oo and pn converges a. e. to p, then (PP. Mn, QPn Mn) converges weakly in L2(IRN) to (Ppm, Qpm) whenever mn converges weakly to m.
Remarks 2.2. 1) Notice that we always have f EN (PPm) (Qpm) dx _ P 0 and thus Pp = P In other words, the above P decomposition simply corresponds to a change of metric induced by p in the usual decomposition. 2) If mn converges strongly in L2 to m, then Pp,. mn, Qp. pn also converge
strongly. Indeed, we have in view of the weak convergence stated above (recall that pn fn n pf, Pr fn n p f if fn n f weakly in L2) 1
I PP,, mn I2 JRN
= JRN { pn Pp ,. (PnPpn mn) }
Mn
d2
n fN { p Pp(pPpm) } m dX _ f N IPpmI2 dz. 0
Proof of Lemma 2.1. We first recall the usual "div-curl" decomposition, namely the Hodge decomposition
RAg1=0.
m=pi+q1,
(2.20)
Recall that p1 = Pim, q1 = Qjm define bounded operators on L2. Next, (2.19) is equivalent to finding a unique v E L2 (IRN) such that
p = pi+Rv,
R. (pp) = 0
or equivalently to finding a unique v E L2(IRN) such that
R P (Rv + pi) = 0
in IRN.
(2.21)
The unknowns p and q are then given by p = pi + Rv, q = qi - Rv. The equation (2.21) is an unusual way of writing the elliptic equation div
(p (VV + pi)) = 0
in
IRN
27
Existence results
where V = (-i)-112V. The only advantage of (2.21) is that N = 2 does not play any role there. In particular, the solution of (2.21) is the unique minimum over L2(IRN) of
1 IRvI2 + pl . Rvdx.
RN 2p
P
The bounds follow trivially since we have
r
1 IRvI2
JRN P
dz = -
1?1 Rv dx
J1RN P
hence
IIRVIIL2(RN) IiPIIL3(n,N) < (fIRN
IpP
1/2
(2
dX
(P)-1/2 IIp111L2(>RN) <- (p)-1/2 II mII La(1RN)
In order to prove that the weak limit of P.. mn is Pam, we use a standard "hilbertian strategy". We first observe that PP is bounded on L2(IRN) uniformly in n in view of the preceding bounds. Next, for any ip E L2(IRN), p - O in L2(IRN) (by Lebesgue's theorem) and thus it is enough to check
-
that
jN (Pm) V) dz. f1Ft'
Pn
But, we have, in view of Remarks 2.2 (1),
f
1 (Pm) . P= f mn
RN Pn
R,N
(P',b) dx Pn
and it is enough to show that Pp, .,V converges (strongly) in L2 (IRN)N to 1 °('' for all cp E L2(IRN)N. In fact, it is enough to show that Pp. (p converges weakly in L2(IRN)N to PPCO since we have then, using again Remarks 2.2 (1),
f
W12
MN Pn
I PP,,
dX = f
R.N Pn
n while
Pn
(PPn `P)
cp dx = fR,N (PP,, cc) (-s-) dx
f.N(P'V)'MdX
-
>RNPIPacPI2dx
converges weakly in L2(IRN)N to (PPCp).
Without loss of generality, we may assume that PP cp, Q, cp (extracting subsequences if necessary) converge weakly in L2(IRN) to some p, q E
28
Density- dependent Navier-Stokes equations
L2(IRN). Therefore, we have obviously: cp = p + q, R A q = 0. Next, 1 (Pp V) converges weakly to .1 p and we thus obtain R (p p) = 0. The uniqueness of the decomposition (2.19) completes the proof of Lemma 2.1. 0
We may now go back to the issue of the "time-continuity" of pu in Theorem 2.1. We shall use the following notation: f E C([0, T]; L2 (IRN) ) if f E LOO (0, T ; L2 (I RN)) and if f is continuous in t with values in L2 (IRN )
endowed with the weak topology (picking up a metric for this topology on a large ball of L2(IRN) containing the values of f (t) for t E [0,T]). Recall,
as explained above, that in the following theorem we consider the case St = IRN and u,,. = 0.
Theorem 2.2. Let (p, u) be a weak solution of (2.1)-(2.2) as in Theorem 2.1. Then, for all T E (0, oo); R A (pu) E C([O,T]; Lv,) with IRA (pu)}(0) = R A mo and ac {R A (pu)} E L2(0,T; H-1(IRN)) + X Where
X = LP(O,T;W-1.q(IRN)) with 1 < p < oo, q = p-2 if N > 3, if N = 2 and po is bounded away from 0, and 1 < p < 00, q = X = LP(0,T; Wj-1,q(lRN)) with 1 < p < oo, 1 < q < ppl if N = 2; we can also take X = L°O (0, T ; W -1(1N)) for any c > 0 if N = 2. addition, if we assume that po is bounded way from 0, then we have
In
for all T E (0, oo). 1) pu, fu, u E C([0, T]; 2) pu (respectively ,Ip- u, u) converge weakly in L2(IRN), as t goes to 0+, to Pporno (respectively po (PPOmo), po (Pp,,mo)). Furthermore, if div uo = 0 in D'(IRN) where uo = p , pu (respectively u) converge strongly in L2(IRN), as t goes to 0+, to mo (respectively Pouo, uo) 3) The energy inequality (2.11) holds for all t > 0. 0
We are going to prove Theorem 2.2, admitting temporarily Theorem 2.1 whose proof is given in the following sections.
Proof of Theorem 2.2. First of all, taking 0 = R A b (1 E Co (IRN x (0, oo)) in the weak formulation (2.12), we deduce easily that the following holds in the sense of distributions
5i {RA(pu)}+aj{RA(puju)}--aj{RA(µ(Duj+aju))} = RA(pf). (2.22) Recalling that Rk is bounded on LP(IRN) for all 1 < p < oo, 1 < k < N, we remark that for all T E (0, cc) R A (pf) E L2(IRN x (0,T)),
R A (µ(Vuj + aju)) E L 2(]RN x (0,T))
Existence results
29
since p, IL E L°°(IRN x (0,T)), f E L2(IRN x (0,T)), Vu E L2(IRN x (0,T)). Next, we observe that pu j u E L°O (O, T; L 1(IR )) (V T) and by Sobolev embeddings u E L2 (0, T; L441 (IR.N )) if N > 3, hence pup E L1(0, T; L (RN)) for all T E (0, oo). Therefore, if N > 3, pub u E LP (O, T; Lq (IRN) )
(V T) for 1 < p < oo, q =
Np-2 .
If N = 2, by the definition of a
weak solution, u E L2(0,T;H1', (IR2)), hence, by Sobolev embeddings, u E L2(0, T; LrOC(IR2)) for all 2 < r < oo and pu,u E L°O(0,T; L1(IRN)) n
LP(O,T; Ll °(IRN)) for 1 < p < oo, 1 p > 0 a.e. on IRN, then (2.17) yields p > p a.e. on 1R2 x (0, oo). Therefore, in this case, u E L°°(0,T; L2(IR2)) n L2(O,T; H1(IR2)) (V T) and, by the GagliardoNirenberg inequalities, u E LP (0, T ; LQ (IR2) )- for 1 < p < oo, q = 2 P P I (V T). Collecting all these bounds, we deduce from (2.21) the regularity of at {R A (pu)} stated in Theorem 2.2. This yields of course the continuity in t of R A (pu) with values, say, in Wi- ' 1(IRN) but, by definition, R A (pu) is bounded in L2 (IRN) on each interval (0, T) (for all T E (0, oo)). Therefore, R A (pu) E C([O, TJ; L2,,,) (`d T) and R n (pu)tt=o = mo because of (2.12). We now turn to the proofs of claims 1)-3) in Theorem 2.2 in the case
when po > p a.e. in IRN for some p > 0. As we just saw, this implies p > p > 0 a.e. on RN x (0, oo) and thus u E L°°(0, T; L2(IRN)) (V T). We next prove that pu E C([0, T]; L,2,) (V T). With the notation of Lemma 2.1, setting m = pu, we see that RA (Pim) = RAm E C([0, T]; (V T) while, by definition, R R. (Pim) = 0. By classical properties of such div-curl decompositions (ellipticity!), we deduce that P1m E C([O,T]; L2W)) (V T). Next, we claim that we have
m = PP(Pim).
(2.23)
If it is the case, we deduce the fact that m E C([0, T];
from Lemma
2.1. Now, (2.23) follows from the obvious properties
R n (Pim - m) = R Qim = 0
R. (m) = R u = 0
by definition of P1, Q1 since div u = 0.
The rest of claim 1) is an easy consequence of, on the one hand, the fact we have just proved, namely pu E C([0, T]; L2,)) (d T) and, on the other hand, the properties of p, namely p E C([0, oo); LP(BR)) (V R , V 1 < p < cc). Indeed observe that if to t > 0, in , , p(1 P(t)
n
a(tn)
tT)
n V7p(t)
LP(BR) (V R, V 1 < p < oc) and remain bounded in L°°(IR.N) uniformly in n, and 1) follows by observing that gnfn - g f weakly in L2, if fn f n
weakly in L2, gn - g in L1OC and gn is bounded in LOO uniformly in n.
30
Density-dependent Navier-Stokes equations
Claim 3) follows from 1): indeed, for all t > 0, we deduce from (2.14) the existence of t E (0, t), to - t such that n
to
(JRN plul2 d2 (tn) + f JAN gai1l; + a;?I.)2 dx ds
f.
r
to Imo12 N
P0
d2 +
J0
JRN
pf u dx ds.
We only have to let n go to +oo observing that
lim (fRN
dX (tn) ? (JRN PI UI2 dx
since pu E C([0, T]; Lv,) (V T). It remains to prove claim 2). First of all, in view of the properties of p, it is clearly enough to show that pu converges weakly in L2(IRN), as t goes = 0 in 1Y(IRN) (or R . PO (L) = 0), to 0+, to Pp.(mo) while, if div (mo) PO fpu converges strongly in L2(IRN), as t goes to 0+, to . Since m = pu E C([0,11; L2 ), we only have to show that m(0) = Ppomo in order to prove the weak convergence. But, in view of what we have already shown
RA(mo-m(0)) = 0, R (
0))
=
0
since u E C([0,1]; L2,). Hence, m(0) = PPa (mo).
Finally, if div W = 0, we see that m(0) = mo. Hence,
pu converges
weakly in L2(IRN), as t goes to 0+, to -"`p . But, we also have, since (2.4) holds for all t > 0 (claim 3) proved above), 11,
PuIIL2 (RN)
mo )
I
2
PO (LZ(]RN)
and the strong convergence in L2(IRN) is proven.
We conclude this section by mentioning that the first statement in Theorem 2.2 can be expressed, in fact, in terms of Pl (pu), i.e. of the projection, say in L2(IRN), on the subspace of divergence-free vector fields-we used this fact in the proof of Theorem 2.2. In fact, the orthogonal decomposition (Pi, Ql) is also possible in the case of Dirichlet boundary conditions (and in this case Pl is the projection onto the subspace of divergence-free vector fields u with u n = 0 on all) or in the periodic case. If 11= IRN or in the periodic case, we obtain immediately
Regularity results and open problems
31
Theorem 2.3. Let (p, u) be a weak solution of (2.1)-(2.2) as in Theorem 2.1. Then, for all T E (0, oo), PI (pu) E C([0, T]; L2, (n)) and Wt' {P1(pu) } E L2(0,T; H-1(0)) + X where X = LP(O,T; W-1,4(IRN)) with 1 < p < 00, q = p 2 if N > 3, 1 < p < oo, q = p p l if N = 2 and po is bounded away with 1 < p < oo, 1 < q < if from 0,pand X = LP (0, T; W-1-E,1(1RN)) for any N = 2. If N = 2, we can also take X = L°O(0,T;
e>0. In the case of Dirichlet boundary conditions, one must replace W-1,9(52) by V-1'9(11) = Vo'q(f )' where V0'q(SZ) is the closure for the W'.q norm of the space of functions cp in Co (Il) such that div cp = 0.
2.2 Regularity results and open problems We begin this section by discussing some open problems on (2.1)-(2.2). First of all, the uniqueness of weak solutions is completely open in all dimensions. Of course, we expect this to be the case if N > 3 since, in view of Theorem 2.1 (and Remark 2.1 (3)), a particular case of our weak solution is p - p E (0, oo) and, in that case, u is a weak solution
"a la Leray" of the homogeneous, incompressible Navier-Stokes equations (the classical Navier-Stokes equations), and, for this particular case, the uniqueness of weak solutions is still an open question. But, even in two dimensions, the uniqueness of weak solutions is not known for (2.1)-(2.2). We show in section 2.5 some partial "uniqueness" results indicating that any weak solution is equal to a strong one if the latter exists. Of course, the uniqueness of solutions is closely related to the regularity of solutions: "smooth enough" solutions are indeed unique-this is not
difficult to check and results in that direction can be found in [17] for example. For the same reason as for the uniqueness, we cannot expect full
regularity results (like po, uo E C°° yield C°° solutions for all t > 0) to be known since they would imply regularity results for the homogeneous Navier-Stokes equations. However, as we shall see and recall in chapter 4, regularity in the preceding sense holds for the homogeneous NavierStokes equations if N = 2 and if N = 3 various regularity results are available. If N = 3, we do not have any further regularity information on u different from what we stated in section 2.1 (or easy consequences of what we stated). In particular, as we already mentioned in Remark 2.1 (2), very little is known on the pressure field. But, even when N = 2, regularity does not seem to be available at least when g depends on p. However, in the very particular case when N = 2, p is independent of p, i.e. p is a positive constant and po is bounded away from
Density-dependent Navier-Stokes equations
32
0, it was shown by A.V. Kazhikov [254] (see also S.N. Antontsev and A.V. Kazhikov [16], S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov [17]) that one can obtain more regularity and therefore also uniqueness results. We do not want to treat this case in great detail and we refer the interested reader to [17] for complete proofs. But we wish to explain the main idea of the proof of one regularity result, from which further regularity can be deduced by classical arguments (one way is to differentiate the equation and apply again the argument we are going to present). More precisely, we wish to explain why u E L2(0, T; H2) n C([0, T]; H'),
a E L2 (0, T; L2) (V T) if po > p > 0 a.e. for some p E (0, oo), uo = p E
H1 and div uo = 0, N = 2, µ E (0, oo). We simply want to obtain, at least formally, a priori estimates on u and
corresponding to the claimed
-regularity, and we shall do so only in the case when St = 1R2, u". = 0 to simplify the presentation. First of all, as we mentioned several times before, p > p > 0 a.e. on IR2 x (0, oo). Next, using (2.1), we write (2.2) as
JIDu+Op = Pf,
P5T + div u = 0,
(2.24)
in IR2 x (0, oo).
We then multiply (2.24) by at , integrate (by parts) over IR2 and find for
all t>0
2
P fR21
cat
I
+' dt
IIPoIIL-(>R2) II
2
II IUI IVUI IIL2(>R2) + IIPOIILx(IR2) IIf
at
IIL2(IR2)
IIL2(R2) llOtL2(]R2)
Hence, using the Cauchy-Schwarz inequality repeatedly, we deduce c9u II
2
at IIL2(IR2)
+ dt
IIVuJIL2(IR.2)
- c{IIuIIL4(>R2)IIVUIIL4(IR2) +IIfIIL2(1a2)}
where C denotes various constants independent of u, t. Next, we recall that we already have a bound (deduced from (2.14)) on u E L°°(0,T; L2), Vu E L2(0, T; L2) and that the following inequality holds for all v E H1(IR2) (a particular case of the Gagliardo-Nirenberg inequalities) IIvIIL4(>R2) < CI VII
Therefore, JOT
IIVVIIL (]R2)
Iul4dxdt < CIIuIILc(o,T;L2(IR2))IID'uIIL2(O,T;L2(p.2)) < C.
Using (2.24) once more, we finally deduce that we have for all e > 0 II
at
IIL2(IR2)
<
+ dt
Co(t)
IDuILI(,R2)
I 1 + IIVUIIL2(IR2)) +6IID2UIIL20
(2.25) 2)
Regularity results and open problems
33
where Co > 0, fT Co(t) dt < C (for all T E (0, oo)). Next, we observe that we have for all t > 0 in view of (2.24):
II-µDu+VpIJL2(Ia2) < CflfIIL2(]R2) +Xat IIL2(R2) 11
+ CII IuI loin IIL2(n v).
Since div u = 0, we can use classical regularity results on (linear) Stokes equations-see for example R. Temam [472]-to deduce IJUIIH2(1R2) < C{IIuIIL2(IR2) + IIfIIL2(I.a)
+IL9tL2(lR2) + II ICIIVuI IIL2(aR2)} Exactly as above, this yields for all e' > 0 IIUIIH2(J.2) s
E Ci(t) + CII at
IiL2(fft2 + E'IIUIIH2(]R2)
(2.26)
where C1-> 0, fa Ci (t) dt < - C (for all T E (0, oo)). Hence, choosing C' = 1/2,
2
IIuIIH2(R2) < C2(t) +CI at
(2.27)
where C2 > 0, fa C2(t) dt < C (for all T E (0, oo)). Inserting (2.27) in (2.25) and choosing e = is , we deduce finally for all t > 0 2
at
11L2
(R2)
+ dt IlV
C3(t) (1 + IIDu1IL2(]R,))
where C3 > 0, fa C3 (t) dt < C (for all T E (0, oo)). The desired a priori estimates on at in L2 (IR2 x (0, T)), u in L°° (0, T; H1(IR2)) and thus u in L2(0,T; H2(IR2)) (V T) follow using Gronwall's inequality.
Remark 2.3. Further regularity on u(p, p) can be deduced from the regularity we just obtained. One way is to differentiate and apply similar arguments. Another way is to observe that, since u E L2(0,T;H2), u satisfies: Iu(21, t) - u(x2, t)I < C(t)Ixi -x21 Ilog{min Ix1 -x21, } I for some 2 C E L2 (0, T). This implies that, for each T, p is Holder continuous in (x, t) on [0, TJ and this implies that D2u is Holder continuous in (x, t).
We would like to mention another interesting open question: suppose that po = 1D for a smooth domain D (C S2), i.e. a patch of a homogeneous
Density-dependent Navier-Stokes equations
34
incompressible fluid "surrounded" by the vacuum (or a bubble of vacuum embedded in the fluid). Then, Theorem 2.1 yields at least one global weak
solution and (2.17) implies that, for all t > 0, p(t) = 1D(t) for some set such that vol(D(t)) = vol(D). In this case, (2.1)-(2.2) can be reformulated as a somewhat complicated free boundary problem. It is also very natural to ask whether the regularity of D is preserved by the time evolution. Finally, we conclude this section with a few remarks on stationary problems associated with (2.1)-(2.2), namely p > 0,
div (pu) = 0,
div u = Of
p(u 0)u-µ0u+Vp = pf in 12 , u E HH(fl)N, P E L' (n)
(2.28)
looking, for example, at the case of Dirichlet boundary conditions, and p E (0, oo) independent of p. Choosing for instance f E L2 (SZ) N, we claim that in general (2.28) has a "huge number of solutions". First of all, some "trivial" solutions are obtained by setting p = A E [0, oo) and solving the stationary, homogeneous, incompressible Navier-Stokes equations:
A(u V)u-µ0u+Vp = of
in
SZ,
uEHo(c)N, divu=0 inil,
(2.29)
and we know (see for example R. Temam [4721) that, for each A > 0, there exists a solution (at least one) u E H2(SZ) (p E H2(S2)) at least if N = 2 or 3. In addition, uniqueness holds for instance if N = 2 and A II f 11L2 is small enough (µ and 0 being fixed).
In fact, there are many more (stationary) solutions of (2.28) than the preceding ones. Indeed, take for example N = 2, S2 = B1, fi = x2g(r), .f2 = -x19(r) where r = (xi + 22)1/2, g E L2(Bi) (i.e. f0 g2(s) ds < cc). Then, we look for solutions of (2.28) having the following forms : p = p(r) > 0, ui = x27P(r), u2 = -x1o(r). Obviously, div u = div (pu) = 0. It is easy to check that p(u 0)u = V (rp02) and -Dui = -M, ((r1') + 0)' -x2(L" + r v,'), -Du2 = xi(W' + T v'). Therefore, if p > 0, p E L°° is is given, solving (2.28) amounts to solving -tk" - 0' = pg, i.e. determined by
T
fr iP (1)
= 0,
T3
sap(s) g(s) ds,
(2.30)
and thus for each p, we obtain one stationary solution (smooth if p and g are smooth) !
A similar example can be built in the periodic case: take fj = g (x2 + 2' where 9 is odd, periodic of period L2, p = p(x2+ 2 ) where p > 0, p is even,
A priori estimates and compactness results
35
periodic of period L2/2 (g E L°O) and solve -µu" = pg on IR, u periodic of period L2. Then, p, u = (u(x2), 0) solve (2.28). This indicates that the right way to formulate the stationary problem (2.28) might be to constrain (2.28) with an additional requirement on the distribution function of p, a direction that needs to be investigated in more detail. Let us finally mention that the regularity analysis of (2.28) follows closely
the known results on the steady-state homogeneous Navier-Stokes equations: in particular, if N < 4, u E H2(1) and u E W2'p(fl) if f E L"(SZ)
for any 2 < p < oo. In fact, since the case N = 4 does not seem to be well known, we shall come back to this point in chapter 4. However, even if f E C' (1), we cannot expect more regularity on u in view of the preceding
examples: this is due to the fact that p may not even be continuous.
2.3 A priori estimates and compactness results Let us first explain the organization of this section. We shall work mainly in the periodic case and after each proof we shall explain how to modify the preceding proofs in the Dirichlet case or in the case when 0 = IRN.
Next, we begin with a priori (formal) estimates and then we state and prove some general compactness results on sequences of solutions. These compactness results will play a fundamental role in the existence proofs since they will allow us to deduce the existence of the global weak solutions upon passing to the limit in conveniently approximated problems and using the compactness results shown in this section. We thus begin with a priori estimates. We first remark that (2.1) and
the incompressibility condition (div u = 0) immediately imply that the distribution function of p(t)-considered as a function of x-is independent of t. In other words, (2.17) holds. This is in fact nothing but the celebrated
Liouville's theorem. A direct formal proof consists in observing that if /3 E C1([O, co); IR), 0(p) satisfies
as(p)
+ div (u)3(p)) = V),3(p) = /3/(P){ `rat + u -
+ u Vp} = 0.
Therefore, integrating over f (periodic case or Dirichlet case), we find, using the boundary conditions, that d
f
dt n
3(P) dx = 0
or equivalently that (fn ,3(p) dx) is independent of t. In particular, choosing ,3 = gn E C1([0, oo[, IR.), 1 _> gn > 0 such that gn (t) = 0 if t 0 [a, 01 (where 0 < a < /3 < oo are fixed) and gn (t) = 1 if
Density-dependent Navier-Stokes equations
36
t E [a + 1 , 0 - .1] (take n > Q? ), we deduce (2.17) from the preceding fact upon letting n go to +oo. In particular (2.17) yields the following L°O a priori estimate (2.31) 0 <_ P(x, t) : IIPoIIL. a.e. (in fact IIP(t)IILo = IIPoIIL- for all t > 0!).
The other a priori estimate that we can obtain simply follows from the energy identity: indeed, we expect, at least formally, in view of (2.1), that (2.2) implies, multiplying by u and integrating by parts, that 2
T fPi---dx+f(8iu1+83ui)85udx = Jn Pf -udx or
f
2 dt f nPIuI2 dx + 2 n
µ(au; + a,u=)2 dx =
in
pf udx.
(2.32)
Next, the right-hand side of (2.32) is bounded, in view of (2.31), by
(JPfl2dx)
1/2
1/2
(fpJuI2dx)
< IIPoIIL= IllIILv IIv'
IIL2,
and, because of (2.31), µ = µ(p(x, t)) > µ = inf {µ(a) / 0 < J < IIPoIILc } > 0, and thus 1
jp(t9u1+ajuJ2 dx >
µ
(aiu1 +a1us)2 dx
J = µ f IVuI2+2aiu38,uidx 2
In addition, we find, integrating by parts,
In
aiu1 a;ui dx =
1 (aiui)2 dx
= 0.
In conclusion, we obtain for all T E (0, oo)
(fPIuI2dx)t+uf Vu2dxds 2 n o
IIPollL
f o
t
IIfJIL2 II\/ UIIL2 d3+ 1
2 n
Irol2 dx,
V t E [0,T].
PO
(2.33)
A priori estimates and compactness results
37
In the case of Dirichlet boundary conditions, using the Cauchy-Schwarz inequality, we thus deduce IIouIIL2(s2x(O,T)) < C
(2.34)
sup
(2.35)
0
IIPIul2IIL1(s2) < C
where C denotes various constants which depend only on T, Sl and bounds 2
on IIPOIIL0, IIfIIL2(nx(o,T)), IIPOIuoI2IIL1(n) - II
'ol
IILI(n)
In the case of Dirichlet boundary conditions, we then deduce from Poincare's inequality (2.36)
1IFIIL2(O,T,H1(f )) < C.
In the periodic case, we claim that (2.35) also holds. In order to see this, we introduce
< f > _ fn f dx
where
fn
=
meas(11) I
J
and we deduce from (2.35)
(fPdx)I12
_<
2f PIu12dx+2 in PIu-
I2dx
< C + 211POIIL" II u- IIL2
< C + CIIouII2L2
for all t E (0, T). Then, we can assume without loss of generality that PO $ 0 (otherwise the problem is trivial: p 0), in which case we deduce from the argument made above on p that we have for all t > 0
(fo
pdx)(t)= JnPOdx=MO>0.
Hence, we have
I
T
II2dt
(2.37)
which, combined with (2.34), yields (2.36). We conclude this brief discussion of a priori estimates by explaining the modifications of the preceding arguments needed to treat the whole space
case (fl = IRN). First of all, the derivation of (2.17) is simply identical. Next, instead of multiplying by u, we now multiply by u - u, and find in a similar way
2 dt fffN
P-2
-J
1RN
IVuI2dx <_
PoIII.f1IL2 Iv(uuco)IIL2,
38
Density-dependent Navier-Stokes equations
and we still obtain (2.34), while (2.35) is now replaced by SUP
0
11p1U
(2.38)
U0o1 1!L1(IRN) < C.
In particular, if p0 > p a.e. on IRN for some p > 0, then, by (2.37), we deduce p(x, t) > p > 0 a.e. on IRN x (0, oo). Therefore, (2.38) yields IIu-uooIIL2(O,T;H1(1RN)) < C,
(2.39)
and we deduce from Sobolev embeddings if N > 3 Itu-uo,I1Ls (o,T;L
r (1RN))
< C.
(2.40)
At least formally, we in fact deduce (2.40) from (2.34) if N > 3 in all cases
even without assuming that po is bounded away from 0. In particular, if N > 3, we obtain IIUIIL1(0,T;H1(BR)) < C,
(2.41)
for all R E (0, oo) (C depends now on R).
Finally, we claim that, even if N = 2, (2.41) holds. In order to prove this claim, we first observe that, assuming again that po 0 0, we can find for all T E (0, oo) fixed, some Ro E (0, oo) such that
p(x, t) dx > 0. inf mo = 0
(2.42)
In fact, as we shall see, mo and Ro depend only on po and on "a modulus of continuity in t of p in Li C" . Indeed, arguing by contradiction, if such an Ro does not exist, we find that for each n > 1, there exists t,i E [0, T] such that p(t,,) = 0 a.e. on B. Extracting a subsequence if necessary, we may assume that t, t E [0, T]. Then, since p E C([0, T]; Li C), we see
-
that p(t) - 0 a.e. on IRN. Then, by general uniqueness results shown later on in this section, this implies that p - 0 on IRN x [0, oo) and we reach a contradiction. This proof, however, does not yield uniform bounds, i.e. bounds independent of the solutions, and we re-prove (2.42) below by a different and more efficient argument. But, before we do so, we wish to explain how (2.42) yields (2.41). Indeed, we just have to copy the argument that led to (2.37) in order to obtain T
1
I R 2dt
for
R > Ro,
(2.43)
A priori estimates and compactness results
39
where R= fBR udx, and (2.43) combined with (2.34) yields (2.41). We finally give another proof of (2.42) that yields uniform bounds. Since the difficulty encountered here will be encountered many times in this book, we state and prove a general lemma which is more general than what we really need here.
Lemma 2.2. Let T E (0, oo), pn E C([O,T); L1(BR)) for all R E (0, oo).
We assume that Pn > 0 a.e. and that po = p'(0) satisfies, for some Ro E (0, oo) and for some v > 0 independent of n,
Po dx > v > 0.
(2.44)
JBRO
We also assume that pn satisfies 9pn
+ div (mn) = 0
in D'(IR2 x (0 , oo))
(2 . 45)
where Mn = Mn +m2, and we have, for some C > 0 independent of n, II
C,
m1IILl(R2x(o,T))
(2.46)
IIm2IIL2(JR2x(o,T)) < C.
Then there exist R > Ro and no > 1 such that for n > no we have inf
J
o<e
p" dx > v2
(2.47)
Before we prove this result, let us explain how we use this lemma in the above context: assume that (pn, un) is a sequence of solutions of (2.1)-(2.2) with the bounds already shown, which we assume to be uniform in n. As we have seen in Remark 2.1 (7), it is enough to treat the case when uca = 0, hence we have for all T E (0, oo) sup
0
JIR
pnlunl2dx < C,
2
IIP"IILx(1R2X(o,OO))
C.
We claim that if po = pn(O) satisfies (2.44), then (2.47) holds. Indeed, we
just have to check that (2.48) holds with m2 = Mn = p'2un, and this is obvious since ImnI2
Jfl2
dx < (fJR2 PnJunl2 dx IIeIILx(RR2).
Proof of Lemma 2.2. Without loss of generality, extracting subsequences if necessary, we may assume that Imi I and converge weakly in the Im2I2
Density-dependent Navier-Stokes equations
40
sense of measures to some bounded, non-negative measures on IR2 x [0, T]
denoted respectively by µl and µ2. Next, we choose cp E Co (IR2) such that cp - 1 on B1, 0 < cp < 1 on IR2, W(x) = 0 if Ixi > 2 and we define 1/2.
C1 = suPJR2 IvWI C2 = (J'0W I2 d2 We We then consider a > 0 such that C1a + C2a1/2T1/2 < v/2. Then, we 1
observe that there exists R > max (Ro,1) such that for i = 1, 2
Jfl2x[O,Tj
1(R-1
Then, for n large, we also have for i = 1, 2 T
rI2
dt
JI0 I
J
dx 1(R<JzJ<2R) µi < a
where 1 = Im1 I, µ2 = Im2 i2 We next multiply (2.45) by coR(x) = cp(R) and integrate over IR2 x [0, t] (for all t E [0, T]) to find in view of (2.44) ds
(fR2
(t)
f f T
>_ v -
dt
0
IR2
dx 1R
> v - R1 a - C2
o
Jo
dt
r J
1/2 2
i.
1R
> v - Cla - C2T1/2a1/2 > 2 in view of the choices of a and R. The proof of the lemma is then complete. O
We finally briefly explain some bounds in the case when Sl = IR`v. Let us first observe that (2.17) obviously implies that (2.8) and (2.9) hold uniformly in t > 0. Next, if N = 3 and (2.10) holds, p(t) E L4,1(RN) for all t > 0 and II p(t)IIL ,,x(JRN)=IIPoIIL#.,x(IRN) Furthermore, as explained in Remark 2.1 (8), if (2.8) or (2.9) hold, we obtain a priori estimates on u in L2(IRN x (0, T)) and thus in L 2 (0, T; H1 (IRN)) for all T E (0,00). We
now consider the case when (2.10) holds and N = 2: recall that we wish to show that this bound propagates (in t). In order to prove this claim, we wish to multiply (2.1) by w(x) =< x (log < x >)r. Of course, we need to justify the integration by parts to be performed but this point can >2(p-1)
A priori estimates and compactness results
41
be checked easily in view of the bounds that follow. We then obtain, since pP also satisfies (2.1), for some m E [2, oo) to be determined later on, PpW dx
dt
<
1
f
2
Pplul IocoI dx <
C1.
Iulm 2 (log
f
pp
pplul <x>-1 cp dx 2
<x>)-a
dx 1/m
cps <x>2T (log <x>)-T dx
< CXmII PIIix (fIR2 pPco{co r <x>
l
(log )T }dx
m dz)1/m
where a is chosen in (2 +1, oo) and Xm = (fez (log <x>)-Ot In view of Appendix B, Xm is bounded in L2(0,T) (V T E (0,oo)), and our claim follows if we choose m large enough so that m > 2p and thus cp
-LT < x > T (log < x >) ----LT is bounded on IR.2.
We now turn to the fundamental compactness results that we need in the existence proofs presented in the next section. We consider, for the reasons
explained above, the periodic case and we suppose that two sequences pn,u" are given satisfying: pn E C((0,TI;L1(BR)) (V R E (0, 00)), on > 0 is periodic in x= of period Li (V 1 < i < N), un E L2 (0, T; Hper)N where T E (0, oo) is fixed. We define po = pn (0) and we assume
0 < pn < C
a.e. on 11 x (0, T)
divun = 0 a.e. on SZ x(0,T),
IIunIIL2(o,T;H1(n)) <_ C
(2.48) (2.49)
n
div (pnun) = 0 at + PO n Po
in L' (11),
u"
nu
in
DI(lRN x (0, T))
weakly in L2(0, T; Hpe1),
(2.50) (2.51)
for some po which thus satisfies 0 < po < C a.e., and where C denotes various positive constants independent of n. Notice that, because of the bound (2.48), the convergence of po to po also holds in LP(92) for all 1 _< p < oo and that, because of (2.49), div u = 0 a.e. on f2 x (0, T).
Theorem 2.4. 1) With the above assumptions, pn converges in C([0, T]; LP(S2)) for all 1 < p < oo to the unique periodic solution p, bounded on f2 x (0, T), of
+ div (pu) = 0 p E C([0, T]; L' (11)),
in D'(IRN x (0,T)),
p(0) = po a.e. in D.
(2.52)
Density-dependent Navier-Stokes equations
42
2) We assume in addition that pn l un 12 is bounded in L°° (0, T; L' (91)) and that we have for some q E (1, oo), m > 1
<
(pn,Un)
,
>l
C IIcPIIL9(0,T;W'n,q(c))
(2.53)
for all cp E Lq(0,T; Wm,q(S2)) periodic such that divcp = 0 on RN x (0, T).
Then, for all 1 < i < N, pnui converges to pui in LP(0,T;L''(S2)) for IV# 2 < p < oo, 1 < r < Np 4, and ui converges to ui in Le(0, T; L (SZ)) is replaced by an arbitrary for 1 < 0 < 2 on the set {p > 0} (if N = 2, r in [l, oo)).
Remarks 2.4. 1) Part 1 is essentially contained in R.J. DiPerna and P.L. Lions [128] and we re-prove it for the reader's convenience. 2) It is possible to weaken the bounds on pn and un. For instance, if we keep (2.49), it is enough to assume instead of (2.48) that pn is bounded, uniformly in t E [0, T], in LP (n) where p > 2N -one can even treat the case when p = rr+2 if 'N > 3. In fact, if we consider renormalized solutions instead of solutions in the sense of distributions, the above result holds with p = 1! This is shown in R.J. DiPerna and P.L. Lions [128]. 3) The same result holds with some obvious adaptations in the case of Dirichlet boundary conditions replacing L2 (0, T; Hper) by L2(0, T; Ho (S2)),
and assuming that (2.50) holds in f2 x (0, T) and that cp E Co (0 x (0,T))N with div cp = 0 in S2 x (0, T) in (2.53).
Proof of part 1 of Theorem 2.4. The proof is divided into several steps. Without loss of generality, we may assume, extracting a subsequence if necessary, that pn converges weakly to some p in LP(1 x (0, T)) for all
1 < p < oo where p satisfies (2.52), p is periodic. In addition, since p' u' is bounded in L2 (0, T; Lq (fl)) with 1 < q < a2N2 (q < oo if N = 2), we deduce easily from (2.50) that pn converges to p in C([0, T], W -'.P(92)) for 1 < p < oo, m > 0; see for instance J.L. Lions [293] for very general compactness results of that sort. If we equip LP(f2) (1 < p < oo) with the weak topology and an associated distance over a large ball containing all values pn (t) (n > 1, t E [0, T]), we also deduce easily that pn converges to pin C([0,T]; and, in particular, p(0) = po a.e. in S2. Then, we first prove (step 1) that p uniquely solves (2.52). Next, we give a general regularization procedure for solutions of transport equations like (2.50) (step 2). In step 3, we complete the proof of part 1.
Step 1. In order to check that p solves (2.52), that is
a + div (pu) = 0
in
D'(IRN x (0,T)),
A priori estimates and compactness results
43
we have only to show that ptzufz converges to pu in D'(IRN x (0, T)). This is
in fact rather straightforward since p" converges to p in L2(0,T; H-1(BR)) for all R < oo, while u"cp converges weakly to ucp in L2 (0, T; Ho (BR)) for all cp E Co (IRN x (0,T)) supported, say, in BR x (0,T). Hence, T/
T
p"u" W dt dx =< p", UnW > n = o fIRN
fo J
N
pu cp dt dx,
and our claim is shown.
We next explain why p uniquely solves (2.52). More generally, if g E L°°(IRN x (0, T)), g periodic, g E C([0, T]; Lw(l)) (1 < p < oo) satisfies: g(O) = 0 a.e. on IRN, 099
+ div (ug) = 0
in D'(IRN x (0, T)),
then g =- 0. Indeed, we deduce from the regularization property proved below (step 2) that IgI also solves the same equation. Then, we simply integrate the equation in x using the periodicity to find d dt j' IgI dx
=
0
in
D'(O,T)
and fn IgI dx = m(t) E C([0,T]) satisfies m(0) = 0. Therefore, m = 0 and
g=0. Step 2. A general regularization for solutions of transport equations. This regularization is based upon the following classical lemma that we re-prove for the reader's convenience. We denote by w£ = Ev w (E) a smoothing sequence, i.e. w E Co (]RN), f RN w dx = 1, Support(w) C B1, w>O and CE (0,11.
Lemma 2.3. Let v E W 1,a (IRN), g E LA(]RN) with 1 < a,)3 < co, +
< 1. Then, we have Ildiv (vg) * wE - div (v(g * wf))IIL,(I.N)
(2.54)
< CIIvIIwI,a(J.N) II9IILP(1N)
=+
for some C > 0 independent of e, v and g and'Y is determined by It 13 In addition, div (vg) * w, - div {v(g * we)} converges to 0 in L'r(IRN) as e Ck
goes to 0if ry
Proof of Lemma 2.3. Once (2.54) is proven, the rest of Lemma 2.3 is clear using the density of Co (1RN) in W1t&(IRN) (if a < oo) or in LQ(]RN) (if ,Q < oo). Next, in order to prove (2.54), we define Cf =
Density-dependent Navier-Stokes equations
44
div (vg) * we - div (v(g * we)), which is nothing but a commutator, and we write CE = rE - (div v) (g * wE) where
f
rE
1
NE
(v(y)-v(x)) Vw(2E y) a 9(y) dy
Obviously, we have I(divv)(9 * wE)I
:5
V j IDvI I9 * wI.
On the other hand, we have, using Holder's inequality,
1/fB
}a
Ire:5 C
fB(x,e)
{ E I v(y) -v(x)I
Ift se (,)
I9It
where 1 < s, t < oo, ; +.1 = 1, 1 _< t _< _< s < a and C denotes various positive constants independent of E, v and g. Next, we write
/
-Eo 1
(v(y)-v(x))
<
JI
3 I
Vv(x + A(y-x))
IVv(x+A(y-x))I'
.
(y-z) dA
ly-xl'da. E
Therefore I
fdAJ
1(v(y) -v(x)) l ds <
B(z,e) E
IO v(x+Aew) ' l
I
< f dA o
JBl
I Vv(x+.ew) I' dw = IDvI' * Xf
)v-I _ 11 1B, E-'v (and where Xe(z) = .fo dA 1 (E 1Ba. (z) = 1v1 i ((z IRN Xe = meas(Bi)). Thus we obtain, defining Xe = (meas(Be))-1 1B,
ICel
<- C{IDvI l9 * wel + (IDvI, * XE)1/'(I9It * Xe)1/t
a.e. on IR`v, (2.55)
and we conclude easily since we have, by classical properties of convolutions,
for all6>0 II(IDvI"
119 * weIILA <_ II91ILO,
* Xe)I/$IIL-
lis
< IIDvIIL° IIXEIILl
11 (191' * Xe)I/tIILO :5 II9IIL13,
A priori estimates and compactness results
45
and IIXEIILI = meas(Bi).
In fact, the manipulations leading to (2.55) need to be justified and one way is to argue by density on (2.55). 0
In particular, we deduce from Lemma 2.3 the following fact: if v E L2(0,T;H1(IRN)), g E LOO(IRN x (0,T)), v and g are periodic, divv = 0 a.e. and (2.56) on D'(IRN x (0,T)), a + div (vg) = 0 then, for any 3 E C(IR; IR), /3(g) also solves (2.56). Indeed, in view of Lemma 2.3, we have age
Vg' at + v
re
in IRN x (0, T),
where gE = g * we, re - 0 in L2 (BR x (0, T)) (V R < oo). Hence, if /3 E C' (]R; IR), 0(ge) satisfies a'a (9e) + div(v/.3(ge))
at
_ (+v.v)/3(ge:) = 3'(ge)v, in RN x (0, T).
Then, our claim follows upon letting e go to 0, at least when 6 is C'. If ,Q is merely continuous, we simply approximate it (uniformly on [-IIgIILc, II9IILx]) by C' functions and pass to the limit in the sense of distributions.
Let us point out that we already used this fact (with /3(t) = ItI) in step 1 above. Let us also remark that this regularization allows us to show that g E C([0, T]; LP (n)) for all 1 < p < oo: indeed, we observe that, for all 77,e>0, dt
1/p'
1/p
d
0
(in r-r,7Idx
(j:n I9e-917 Ip d2
Hence, T
sup II9e-9,71ILp < II9(0)*we - 9(0)*w, LP + fo Ilre
-rnfILP(SZ)
dt.
(0,T1
Therefore, gE converges to g in C([0, T]; LP (n)).
Step 3. We have only to show that p" converges to p, say, in C([0, T]; L2())) (to deduce the convergence in C([0,T]; LP(SZ)), for all 1 < p < oo). We already know from step 2 that p E C([O, T]; L 2 (D)) and from the argument given before step 1 that pn converges to p in C([0, T]; L2 (S2)). Therefore, we have only to show that p"(t") converges in L2(c2) to p(t) if
Density-dependent Navier-Stokes equations
46
to (E [0, T]) converges to t, while we already know that p" (t,,) converges weakly in L2(f) to p(t). Hence, the proof of part 1 is complete if we show that we have for all
t>0: (Pn(t))2 dx =
-I
f(pn)2
o a = fp(t)2dx.
"
2
(2.57)
In fact, the convergence is obvious in view of (2.48) and thus we have only to check the fact that pn(t) (resp. p(t)) has the same L2 norm as po (resp. po). Next, in view of step 2, (p")2 (resp. p2) also solves (2.50) (resp. (2.52)) and the claimed conservations simply follow upon integrating these equations and using the periodicity- of all the functions considered. 0
Remark 2.5. As mentioned in Remark 2.4 (3), the proof of part 1 is easily modified if we replace the periodicity requirement on p", u" by Dirichlet boundary conditions, namely un = u = 0 on BSt, or in other words un, u E L2(0, T; Ho (Il)). Of course, in that case, all equations are set in St x (0, T). The only argument which needs some explanation is the "integration over c of div (pu)" where p E L°° (11 x (0, T) ), u E L2 (0, T; Ho (fl)). This is done by observing that, by classical Hardy-type inequalities, d E L2 (cl x (0, T) ) where d = dist (x, all). Then, we consider, for e small enough, WE E C0 (SZ)
such that 0 < 'pE < 1 W,, (x) = 0
in SZ,
if d(x) > e,
co (X) = 1
if d(x) <
IOwe! < C
2,
in S1
for some C > 0 independent of e. Then we have I< div(pu), cce >I = V I fa Pu <_ C
f
n
IuI
e
E
l(d<E)
and (f(d<e) d dx) --> 0 in L2(0,T) as c -> 0+.
dx < CJ
T1T1
d dz
0
Proof of part 2 of Theorem 2.4. We first prove that we have (2.58)
Indeed the condition (2.53) shows that at {P1(pnun)} is bounded in
LQ(0, T; W-7n,4(S ))N while, by assumption, pnun and thus Pl(pnun) are
A priori estimates and compactness results
47
bounded in L°° (0, T; L'(q) N. Hence, by classical compactness theorems (see for instance J.L. Lions [293], R. Temam [4721), P1(pnun) is compact in L2(0, T; H-1(cl))N. In particular, since pnun converges weakly to pu (step 1 of the proof of part 1 in LO°(0,T; L2(fl))N for the weak-* topology, P1(pnun) converges to P,(pu) in L2(0,T; H-1(SZ))N. Hence, we have
J dt Jndx pnlunl2
fT
'T
=
dt (pnun, u'')L2(n)
o
T
T
dt (Pl(pnun), u'a)L2(n) _ J dt H-1 XH1
=J T
dt H-1xH1
n r
=J
fT
T
dt
dt (pu, u)L2(n) =
/' T
dt (Pi(pu),u)L2(n)
r dx pIu12 .1s
o
where we use the fact that P1(un) = un, p, (U) = u since div un = div u = 0 in D'(IRN x (0, T)). In the case of Dirichlet boundary conditions, the passage to the limit for pn dun 12 is shown in exactly the same way, replacing H-'(n) by V-1"2(SZ). Once (2.58) is shown, we observe that pnun converges weakly in L2(fl x (0, T)) to f u. Indeed, in view of step 2 of the proof of part 1, pn also solves (2.50) and converges to f in C([0, T]; LP(SZ)) (1 < p < oo) because
of part 1. Then, applying step 1 of the proof of part 1, we deduce our claim, namely the weak convergence of pn Un to /u in L2(SZ x (0, T)). This weak convergence, combined with (2.58), yields the strong convergence in L2(S1 x (0, T)) of pnun to . / u. The convergence of pnun stated in part 2 of Theorem 2.4 then follows from the bounds we assumed on pn, un and pnun. The final statement of part 2 concerning the convergence of un to u on {p > 0} is shown if we show that un converges in
measure to u on {p > 0}. But we deduce from the fact just shown that, extracting subsequences if necessary,
pnun converges a.e. on SZ x (0, T) to ,fp-u. In addition, because of part 1), we may assume that pn converges
a.e. on f x (0, T) to lp-. Hence, on the set {p > 0}, un converges a.e. to u and the proof of Theorem 2.4 is complete.
0
We conclude this section by explaining how the preceding result, valid in the periodic case and in the case of Dirichlet boundary conditions, can
be adapted to the case SZ = IRN. We begin by stating conditions and assumptions on pn, u''; of course, we no longer require pn and un to be periodic and only request un to be bounded in L2(0, T; H' (BR))N for all R E (0, oo). We still assume (2.48) and (2.50) while (2.49) and (2.51) are now replaced by
div un = 0
a.e. on IRN x (0, T)
(2.59)
Density-dependent Navier-Stokes equations
48
Po n Po
in L1(BR),
u" - u weakly in L2 (0, T; H'(BR)), n
1+nx 1
,.>
for all R E (0, oo)
(2.60)
= Fn+F'n21
Fi (resp. F2) is bounded in L' (0, T; Ll (IRN))
(2.61)
(resp. L1(0, T; L°° (RN)))
for all S > 0. Since we shall deal mostly with situations where pn Iun (2 is bounded in L°° (0, T; L' (IRN)), we only wish to observe at this stage that such a bound obviously implies that un 1(pTM>b) is bounded in L°°(0, T; L2(IRN)) and thus (2.61) holds. _
Theorem 2.5. 1) Under the above conditions, pn converges in C([0, T]; LP(BR)) (for all 1 < p < oo, R E (0, oo)) to the unique bounded solution p of in D'(IRN x (0 , T) ) , + di v ( pu) = 0 (2.62) RN at = PO a.e. in Pit=o such that u 1 + 1xI
1 (IpI>5)
E L1(O , T ;
L1(IRN )) +L1 ( 0 , T , L°° ( IRN ))
(2 .63 )
2) We assume in addition that pnlunl2 is bounded in L°°(0, T; LI(IRN)),
Vu" is bounded in L2(IRN x (0, T)), if N > 3 that un is bounded in (IRT)) and that (2.53) holds with fl = IRN and for all cp E L4(O,T; Wm,9(IR.N)) such that div cp = 0 a.e. on IRN. Furthermore, we assume that either un is bounded in L2 (IRN x (0, T)) or N > 3, po E LI,= (IRN), po E L*.°°(IRN) or N = 2 and L2 (0, T; L
sup sup
0
n
r (Ply <x>2(p-1) (log <x>)'' dx <
00
R2
for some p E (1, oo]
(2.64)
with r > 2p - 1.
Then, for all 1 < i < N, pnu? converges to v/;5ui in LP (0, T; LP (BR)) for 2 < p < oo, 1 < r < Np 4, 0 < R < o0, and u; converges to ui in Le (0, T; L
(BR)) on the set {p > 0} for 1 < 0 < 2, 0 < R < oo.
Remark 2.6. 1) Similar extensions to those described in Remark 2.4 (2) are possible for the preceding result.
A priori estimates and compactness results
49
2) Part 1 of Theorem 2.5 allows us, in fact, to extend slightly some of the uniqueness results obtained by R.J. DiPerna and P.L. Lions in [128].
0 Proof of part 1 of Theorem 2.5. The proof is divided in three steps. Step 1. Truncations and consequences. We introduce e,6 = (pn - S)+ for b E (0, 1]. Obviously, (2.61) yields for all 5 > 0 Iunl
< X>
is bounded in
1 (p6 mo o)
L1(0, T;
(2.65)
L°° (IRN)) + L1(0, T; L1(IRN))
where we define <x>= (1 + We are going to show below (in step 3) that p converges in LP(BR) uniformly in t E [0, T] to some 7i6 > 0 (E L°° (IR x (0, T))) for all 1 < p < oo, T E (0, oo). This will be done using,. in particular, some general IxI2)1/2.
uniqueness results established in step 2 below that also show the uniqueness statement contained in part 1 of Theorem 2.5. We wish to show now why such a convergence of p6 yields the convergence of pn in C([0, TI; LP(BR)) (V 1 < p < oo, b R E (0, oo)) to some p which,
obviously, is bounded on IRN x (0, T) and solves (2.62). Then, we show why (2.63) holds in the limit. First, we observe that we have for n, m > 1 Ipn-prI
(pn-a)+
-
(pm-5)+I + 2b.
This is enough to ensure that (p'1)n is a Cauchy sequence in C([0,T]; LP(BR)) (V 1 < p < cc, V R E (0, oc)) and thus converges to some p. Obviously, p = lim6jo+ T P6 (and one can in fact deduce a posteriori from the uniqueness statement and its proof the fact that p6 = (p-5)+). Next, we show that (2.63) holds. To this end, we observe that in view of (2.61)
u
<x>
X6(pn)I
< M"(t)+Fn
(2.66)
where Mn _> 0, Mn is bounded in L1(0,T), Fn > 0 is bounded in L'(IRN x (0, T)) and X6 E C([0, cc) , [0, oo)) satisfies: 0 < X6:5 1 on [0, oo), X6(t) = 1
if t > 5, X6 (t) = 0 if t < 6/2. Obviously, for all R E (0, oo), X6(e) converges in C([O, T]; LP(BR)) (V 1 < p < oo) and is uniformly bounded on IRN x (0, T) while un converges to u, for example, weakly in L2(BR x (0, T)). Therefore, X6(pn) converges weakly in LP(BR x (0, T)) to > X6(p).
On the other hand, we may assume without loss of generality (extracting subsequences if necessary) that Mn converges weakly in the sense of measures to a non-negative bounded measure M on [0, T] while Fn converges
Density-dependent Navier-Stokes equations
50
weakly in the sense of measures to a non-negative bounded measure F on ]RN x [0,T]. Then, we deduce from (2.66) and all these convergences
u
xa(P)< M+F
<X>
(2.67)
Let us then denote by M and F, respectively, the absolutely continuous parts (with respect to the Lebesgue measure) of M and F.. Since u X6 (p) E L'(BR x (0, T)) (V R E (0, oo)), we deduce finally 1(a>6) -<
and (2.63) is shown.
I
x6(P)I <- M+F
0
Step 2. Uniqueness. We consider here fl, f2 bounded solutions in C([O,T]; L1(BR)) (d R E (0, oo)) of
+ div (vf) = 0
in D'(1RN x (0, T))
(2.68)
such that f, (0) = f2 (0) a.e. on IRN and
l(If:l>6) E L1(O,T; Ll(IRN)) + L'(0,T; LOO(RN)),
(2.69)
for all i = 1, 2, b > 0. Then, if divv = 0 a.e. on IRN x (0,T)-we could as well assume divv E L1(0, T; LOO (IRN)) as in [128]-and v E RN x (0,T). L1(O,T; W,1, (]RN)), fi = f2 a.e. on Let us first remark that f = f, - f2 satisfies the same properties as f, and f2 with, of course, f (0) - 0 a.e. on IR.N. Indeed, we just need to observe 1(Ifl-6)
1(Ihl-6/2) + 1(If2l>>6/2)-
Next, we use Lemma 2.3 and step 2 of the proof of Theorem 2.4 to deduce
that, for all b > 0, (If I - 6)+ satisfies exactly the same properties as f. In other words, we can assume without loss of generality that f > 0 a.e. on JRN x (0, T) and Iv
1(f>a) E L1(O,T;L1(IRN))+L'(O,T;Lao (]RN).
Indeed, observe that f - 0 follows from (If I - b)+ - 0 for all 5 > 0.
(2.70)
A priori estimates and compactness results
51
We deduce from (2.70) the following fact
(f>o) <- M t + G(x, t) a.e. in IRN x (0 T) GEL1(IRNx(0,T)), G>0, MEL'(0,T), M>0. IvI
1
(2.71)
Next, we consider cp E Co (IR), even, 0 _< cp < 1, V(x) = 1 if JxJ cp(x) = 0 if JxJ > 2, cp nonincreasing on [0, oo), and we multiply (2.68)
by cpn(ec(t) < x >) where tpn(x) = cp(n), n > 1, C(t) = fo M(s) ds. Integrating by parts over IRN x (0, t), we find for all t E (0, TJ f(t)co(eC(t)
JRN
<X>)dx
r
+ Jot I
dsdx (_.cd(eC(3)
{{M(s) <x> -v . <x> Jf =
))
0.
In view of (2.71) and the properties of cp, we deduce JRN
f (t)cp(e
<x> ) n
t
dx < C
LLN ds dx G
This is enough to show that f - 0 upon letting n go to +oo since the right-hand side goes to 0 as n goes to +oo. Step 3. Convergence of p6. Without loss of generality, we may assume that p6 and 6)2 converge respectively to some p6, 7.2 weakly in L°O (IRN.x (0, T)) - *. Furthermore, exactly as in the proof of Theorem 2.4, we know that P6r p satisfy the equation (2.62), belong to C([0, T]; L1(IR.N)-w) and satisfy 761t=o = (Po - 8)+, 761t=0 = (Po - 6)+2 = a.e. in IRN. In addition, by Lemma 2.2 and step 2 of the proof of Theorem 2.4, we see that (p6)2 also satisfies (2.62). 1p6>a) E L1(0, T; L°O(IRN)) Therefore, if we show that > 1(pa>a), (P6)2It=o
+L1(0, T; LI (IRN)) for all a > 0, we deduce from step 2 above that 7. = (p6)2. Hence, Pn converges to p6 in L2 (BR x (0, T)) and thus in LP(BR X (0, T)) for all R E (0, oc), 1 < p < co. Since the bounds on I'l 1(P6>a) and on 1(p6 > a) are proven in exactly the same way, we just show them for I'l
1(p,>a).
We first remark that (2.61) implies that we have
lunj
<x>
1(pn>6) < Mn(t) + Fn (x, t)
a.e. in IRN x (0, T)
where Mn, Fn > 0, Mn is bounded in L1(0,T), Fn is bounded in L1(IRN x (0, T)). In particular, we deduce for some C independent of n (and b) 1
<x>
Iu"Pn1
C
Fn(x, t)}
a.e. in F N x (0, T).
Density-dependent Navier-Stokes equations
52
Then, step 1 of the proof of Theorem 2.4 shows that u"pa converges to u7p-6 weakly, say, in L2 (BR x (0,T)) for all R E (0, oo), and we deduce as in step 1 above
a.e. in RN x (0, T)
lul
> p6 < C {M + F}
(2.72)
where M E L1 (0, T), F E L1(IRN x (0, T)) are respectively the absolutely continuous parts (with respect to Lebesgue measure) of the weak limits (in the sense of measures) of Mn, F. In particular, (2.72) yields the desired fact: lul 1(P6>a) E L1(0, T; LOO(IR.N)) + L1(0, T; L1(IRN)) for all a> 0. At this stage, we have shown the convergence of pb to p6 in LP(BR x (0, T)) for all 1 < p < oo, R E (0, oc). In order to show the convergence in C([0,T]; LP(BR)) for all R E (0, oo), 1 < p < oo, it is enough to consider the case when p = 2 for instance. Then, we fix R in (0, oo) and we choose cp E Co (IRN), cp 0, cp . 1 on BR. We claim that we have, for all n > 1 and for all t E [0, T],
JAN
-6)+I Z dx
P(t) 2 dx =JRN
I
( 2.73)
t
drds {(pb )2u'L vW2
+ 0
JRN
IVP6(t)I2dx =
IR.N
fWN
Ico(Po-b)+I2dx
t+
ff
(2.74)
dads {(p6)2u Vcp2}.
RN
Indeed, using step 2 of the proof of Theorem 2.4 once more, (pb) 2 and (p6)2 satisfy respectively the same equations as pa and p6 and belong to C([0, T]; LP(BR,) - w) for all 1 < p < oo, R' E (0, oo). Then, (2.73), (2.74) follow upon multiplication by cp2, and show, by the way, that 7i6 E
R' E (0, oo), 1 < p < oo. In addition, we
C([0, TI; Lp (BR') - w) for all
check as in the proof of Theorem 2.4 that pa converges to j56 in C([0, T]; L2(BR-)-w) for all R'(0, oc) and thus, in particular, cppb converges to ccp6 in C([0, T]; L2(IRN)-w). This fact, combined with (2.73) and (2.74), shows that V p6 converges to V p6 in C([0, T]; L2 (IRIv)) provided we show
ff
dx ds { (Pa) tun N
n ffR
.
V.21 (2.75)
dads {(p6)2u V2},
uniformly on [0,T].
A priori estimates and compactness results
53
This is easy since (pb )2 converges to (p6)2 in L2 (BR, x (0, T)) ('d R E (0, oo)) and un converges to u weakly in L2(BR, x (0, T)) (V R' E (0, oo). Therefore, cpp6 converges to 0-p,6 in C([0, T]; L2(IRN)) and thus p6 converges to pb in C([0, T]; L2(BR)) for all R E (0, oo), and the proof of part 1 is complete. 0
Proof of part 2 of Theorem 2.5. Let us first observe that step 1 implies easily that pn un converges weakly to /u in L2 (IRN x (0, T) ) (for instance). Thus, exactly as in the proof of Theorem 2.4, we have only to show that we have for all R E (0, cc)
0 Rn 0 T
dtf
J
T
/
dx pnlun12
dtj
dx pIuj2.
(2.76)
BR
The proof of (2.76) is divided into four steps. First, we show that P(pnun) converges to P(pu) in L2 (BR x (0, T)) (V R E (0, oo)) where we denote by P the orthogonal projection in L2(IRN)N onto divergence-free vector fields
(P = Pi with the notation of section 2.1). Then (step 2), we show that (2.76) holds in the case when un is bounded in L2(IRN x (0, T)). Step 3 is devoted to the proof of (2.76) in the case when N > 3, p E L T '°°(lRN), and we treat the case N = 2 with the condition (2.64) in step 4.
Step 1. Compactness of P(pnun) in L2(BR x (0,T)) (V R E (0, oo)). Since P = Id + O (-A) - l div is bounded on each Sobolev space W'n,q (IR )
(m > 0, 1 < q < oo), we deduce from (2.53) that A P(pnun) is bounded in Lq(0,T;W-'n,q'(IRN)). On the other hand, by assumption, P(pnun) is bounded in L°O(0,T; L2(IRN)). Hence, Appendix C implies that P(pnun) converges to P(pu) in C([0, T]; L2(IRN - w). We then wish to show that, for each fixed R E (0, oo),
rT J
o
r dt J BR
T f dx (P(pnun)I2 -' I dt n
0
dx IP(pu)I2 d2.
(2.77)
R
We then write cp = 1 BR . Then, we take as in Lemma 2.3 a regularizing kernel we (E E (0, 1]) and we observe that {cpP(pnun)} * We converges to {cpP(pu)} * we in C([O,T]; L2(1RN)). Therefore, we have for all E E (0, 1] T
dx {cpP(pnun)} ({P(pnun)} *WE)
1dt J IRN
fT
Jo
r
dt f N dx {cpP(pu)} ({P(pu)} * wE),
since P(pnun) converges weakly in L2(IRN x (0, T)) to P(pu).
Density-dependent Navier-Stokes equations
54
In view of this convergence, it only remains to show that the following integral can be made arbitrarily small for a small enough uniformly in n T
J0
dtJR dx
P(p"un) *we)
N
rT J 0
dt JRN
dx P(vP(Pu)) {(Pun) - (Pun) * W.}.
First of all, we remark that P(p"un) is bounded in L°°0,T; L2(]RN)) and thus P(ccP(pnu")) is also bounded in L°°(O,T; L2(IRNN)). Furthermore, from the definition of P, we deduce easily the following bound IP(cP(P"u")i <
C
a.e.
IXTN
IxI>R+1, tE(O,T).
These two facts imply that it is enough to show lim sup IIPnun-(Pnun) * WeIILI(O,T;L2(B,,M)) = 0, e-0 "
(2.78)
for all M E (O, oo). To this end, we drop the superscript n and write
(Pu) * we-Pu = (P * we)u - pu
+
Vu(x+A(y-x)) . (y-x) P(y) w,(x-y) dy dA.
JO'j.N
Next, we remark that we have
fo ifIRN
Vu(x+.1(y-x)) (y-x) p(y)we(x-y)dydA
ff 1
IIPIIL-(]RN) e
0
IVu(x+A(y-x))I w,(x-y) dy dA
]RN
and if we take the L2 norm on ]RN of the last integral, we can estimate its square, using the Cauchy-Schwarz inequality, by
f f 1
dA
RN
0
=
dy
f d\ J
we(Y_x) dx
I
1RJN
we(z) dz
f
dxlVu(x+Az)2 =
J
IDu(x)I2 dx.
A priori estimates and compactness results
55
On the other hand, we have by Sobolev embeddings for some p = p(N) E (1,00) II I(pn * we) - pnI Iunl II L2(B,j) < CIIpn * we - P'eII L2(o,T;Ln(BM)).
This is enough to prove (2.78) and to conclude since the compactness shown in part 1 yields lim sup I) pn * we - pall L2(0,T;LP(Bxl)) C-0 n>1
0.
Step 2. The case when un is bounded in L2. We complete here the proof of part 2 in the case when we assume that un is bounded in L2 (IRN x (0, T)). By definition of. P, we know there exists 7r n E L2(0,T;D1'2(IRN)) (D1'2(IR2) if N = 2-see Appendices A and B for more details on these spaces) such that (2.79)
pnun = P(pnun) + V7rn. Since un is divergence free, we deduce for all b > 0 1
un =
7r6 , 9r6 E L2(0, T; D1,2(1RN))
7rn = 7rb
if N = 2)
(L2(0, T; Dl'2(]R2))
div (pn 1
(2.80)
{P(pnun) + bun + V7rn}
pn
+b div (pn 1
{D7ra + P(pnun)}) = 0
+S
{V* + Sun}) = 0
in in
D'
(2.81)
(2.82) (2.83)
and we write with obvious notation similar decompositions for u, p that involve 7r, 7r6 and r6.
We next remark that we have
Sun+Vfr I2 I
JRN
bun+Dirb I2 dx
pn +S a2
Iuni2dx s 5IIunIIL21]RN) JRN pn + b
since we deduce easily from (2.83) (using the density of Co (IRN) into Dl'2(IRN) or D1'2(IR2) if N = 2; see Appendix A) JRN p n+S
{O*
I2+5u''.V*a}dx
= 0.
Density-dependent Navier-Stokes equations
56
In particular, we obtain lim sup 6-'0 n>1I I
"
'
(P(Pnun) + O7rb) } 1IL2(]RNx(O,T)) = 0 1 un -V/F{ pn+b (2.84)
and similarly lim
Vpu -
{ p+b
(P(pu)+0ir6)}
L2 (IRN x (0,T))
= 0.
(2.85)
Therefore, in order to complete the proof, we need only to show that for
each 6>0 {P(pnun) + 07r6 } n
`pn+b)
in L2(BR x (0,T)),
(p+b) {P(pu) + Vir6}
V R E (0, 00).
But we know from part 1 that pn and thus
Pn, pn+6 converge respectively
to p, ,I-p, P+6 in C([0, T]; LP (BR)) (V 1 < p < oo, V R E (0, oo)). In addition, we also know from step 1 above that P(pnun) converges to P(pu) in L2 (BR x (0, T)), V R E (0, oo). Therefore, it only remains to show that Deb converges to 0ir6 in L2(BR x (0, T)). As seen from the following result, the above convergence is, in fact, a consequence of (2.82) and the convergences we just recalled.
Lemma 2.4.
Let hn be bounded in L2(IRN)N, let (a i)1
bounded in L°O(IRN). We assume N
3v>0, Vn>1,a.e. inxEIRN, V
EIRN,
?Ci j > vl
l2,
i,j=1 anti
-n
- aij in L'(BR) for all R E (0, oo), n
hn +h
in L1(BR)
for all R E (0, oo).
(2.86) (2.87) (2.88)
We consider the unique solution fn E D1,2(IRN) (if N = 2, f n E D1,2 (IR2)
with fB. fn dx = 0) of N
i,j=1
(= 0 n
axi
in D'(IRN)
(2.89)
and we denote by f the solution of (2.89) with a'., h' replaced by ail , h. Then, fn converges in H1(BR) to f for all R 'E (0, oo).
A priori estimates and compactness results
57
Proof of Lemma 2.4. First, we observe that (by the density of Co (IRN) in D1'2(IRN), or Co (IR2) in D1'2(IR2)-see the argument of Appendix A) D1,2(IR2)) and converges weakly in fn is bounded in D1,2(RN) (resp. D12(IRN) (D1,2(1R2) if N = 2) to f since h' converges to h weakly in L2(IRN)N. In particular we deduce, from the Rellich-Kondrakov theorem,
that fn converges to f in L2(BR) for all R E (0, oc). Then, for R E (0, cc) fixed, we consider cp E Co (IRN) satisfying: cp > 0 on IRN, cc = 1 on BR; and we multiply (2.89) by W fn (or cp f). We then obtain n
N
a fn a fn
f,N E a.. axi ax;
dx
i,.i=1
N
_
an 7jRK
,
=,9=1
N
ftN
aij i ,J=1
LN =,7=1 .
.
ofn
N
axi
fn dx -
a`p
a nh' LN
i,J=1
axi
(pfn) dx
of of axi ax; aij
S°
a f app
ax; axi
f dx -
aij h s,9=1
a
axi
( cd)
dx.
We then claim that the right-hand side converges, as n goes to +oo, to the right-hand side of (2.91). Indeed, a is, uniformly in n, bounded and converges in LP(Supp cc) for 1 < p < oo while fn and hn converge in L2 (Supp cp) therefore a a fn, a h converge in L2 (Supp cp) to, respecn tively, aij a f , aij h; . In addition, and as: (cp f n) converge weakly in L2 (IR.N) to, respectively, a and at; (cp f ), and this shows our claim. Hence, the right-hand side of (2.90) converges to the right-hand side of (2.91). But we also have V
1.N IV(fn-f)I2dx <
f
,N .,,=1
axi
.13
N
of )
ax,
(afn ax,
fn of " dx + fN axz ax,
- ax, af) dx N
ant, o JjRJNr
J =1
f
N a N
s,,=1
(afn of
i ,9 =1
v v an -:? axi ax;
of
n axi axi ax; + of ax;
dx,
dx
Density-dependent Navier-Stokes equations
58
and the lemma is shown if we prove that this upper bound goes to 0 as n goes to +oo. This is the case since we just proved that the first term goes dx, while the second term converges obviously to to fJRNEi"" =1 a13 4 the same quantity in view of the properties of a ! . Finally since a aL, or respectively (by Lebesgue's converges in L2(IRN) to a13 a ,ate a lemma), the last integral converges to 2 -1 ash -L L dx, and we
a
conclude.
O
Step 3. The case when N _> 3, pn, p E L ,°° (IRN). Let us first observe that the proof of part 1 of Theorem 2.5 and the proof of step 2 of Theorem 2.4 immediately yield the fact that meas {p' > A}, meas {p > A} are inde-pendent of t E [0, T] for all A > 0 and thus pn, p E L°°(0,T; L r,°O(IRN)). We are going to use the results of Appendix A and more precisely Theorem 2. To this end, we introduce the solutions uR,E. UR,e for a.e. t E (0,T), R E (0, oo), e E (0, 1] of, respectively
- DuR,e +
pnuR,e + OpnR,E _ -Aun + 1 pnun in D'(BR),
e uR,e E Ho (BR),
- AUR,, +
1
div uR,E = 0
PUR,e + VpR,e =
UR,e E Ho (BR),
(2.92)
a.e. in BR;
div uR,e = 0
-Du + e pu in D'(BR),
(2-93)
a.e. in BR.
We may then apply Theorem 2 in Appendix A with f n = pn, the assumptions required in Theorem 2 being satisfied in our case in particular because of part 1 of Theorem 2.5 proven above, and we obtain, for all e E (0,1],
'BR P14R,12 dx + JBR I D4R,e I2 dx
<
f puun dx + R
BR
Vun uRdx,
(2.94)
a.e. t E (0, T), _.n
.
_.n
Sup n
_
Ilun
__ r2/t% m_,n1.2/vnNx-%
- 4R,. II L2 (BA, X (o,T)) ' 0
__ n -> +oo,
(2.95)
as R -* +oo, (2.96)
for all M E (0, oo). Finally, we also have
uR, - uR,E weakly in L2(0,T; H1(BR)), for all R E (0, oo), E E (0, 1].
(2.97)
A priori estimates and compactness results
59
Indeed, we deduce H'(BR) bounds on u"R,e from (2.94)-recall that pnlunl2 is bounded in L°° (0, T; L1(IR.N )) by assumption. Then, (2.97) follows from the uniqueness of the equation (2.93) passing to the limit in (2.92) (recall that, by part 1, pn converges to p in C([0, T]; LP(BR)) for all 1 < p < oo). We then write, for all M E (0, oo) fixed II V Fun _
pull L2 (Bti, x (O,T)) : C Sup Ilun - UR,e iI L2 (BA, x (o,T))
n>1
UR
+ IIV/'
-
L2(B,vjx(0,T))Y
where C denotes various positive constants independent of n, R, e. Using (2.95) and (2.96), we deduce IIvun
- V'T UIIL2(Bj&rx(OO-,T))
11N/74, - N III L2(B&,x(O,T)) +we(R)
(2.98)
where we denote by we (R) various positive constants that depend only on e and R, and such that we(R) -- 0 as R -+ +oo, for each e E (0,T]. Next, we remark that (2.94) yields
1T1 R
pIuI2 dx dt
pnuudx dt + Cc
(2.99)
R
wh i le we have T
pu uR,e dx dt fO"fBR
,
UR,e12 dx dt
BR pI
J0
T PIu12
0
(2.100)
dx dt
IRN
as R goes to +oo, for all c E (0, 1]. N
Indeed, p E L°° (0, T; L (IRN)) and by the (sharp in Lorentz spaces) Sobolev embeddings (2.95) implies that uR,e converges to u in L2 (0, T;
L7 (IRN)). We then claim that we have for all R E (0, oc), e E (0, 1]
j p"un uR dx dt - J J
rT r
rT I 0
BR
n
0
pu uR,e dx dt.
(2.101)
BR
Indeed, since uR a and UR,e are divergence-free vector fields and vanish outside BR, each integral can be rewritten as an integral over IRN (or BR) of P(pnun) uR e, P(pu) uR,e respectively. Then, in view of step 1 above,
Density-dependent Navier-Stokes equations
60
P(pnun) converges to P(pu) in L2 (BR x (0, T)) while 4R ,E converges weakly to UR, , in L' (BR x (0, T)) because of (2.97) as n goes to +oo. This proves the claim (2.101). Combining (2.99), (2.100) and (2.101), we obtain II v 7uR,E - fUR,EIIL2(BRx(OAT)) T
+
dt 0
< Ce +w (R)
r
JBR
and
- VFun,E - v" UIIL2(BRx(o,T)) < Ce+we(R)
(2.102)
rim II n
since u,E pn converges strongly to VP-R converges weakly to uR,E while Adding_ up (2.98) and (2.102), we finally deduce lim II n
un - VUII L2(Bmx(o,T)) <- Ce+wf(R),
and we conclude upon letting first R go to +oo and then e go to 0.
Step 4. The case when N = 2, (2.64) holds. We first remark that, because of part 1r of Theorem 2.5, (2.64) yields
p" 2(p-1) (log <x>)'' dx < oo.
sup
0
(2.103)
We are going to show in that case that converges to ,/u in L2(IR2 x (0, T)). To this end, we use the notation of Appendix A. We Vrnun
observe first that TR(un) is bounded in L2(0, T; D1"2(1R2)) uniformly in n, R, TR(u) is bounded 'in L2(0, T; D1,2(IR2)) and that TR(un) converges weakly in L2(0, T; D1"2(IR2)) (or L2(0, T; Ho (BR))) to TR(u) as n goes to +oo (V R E (0, oo)). In particular, Theorem 1 of Appendix A yields for all Ro E (0, oo)
sup n
U
T pn l un
I2
- pnun TR (un) dx dt
fT
r J
Bas
(2.104)
-4 0
R - +co. Next, we claim that we have dX
s up
+
f I
x I>R o
xI>Ro
x
dx
dt pnlUnl2 - PnIu I ITR(un)
f
T
dt pIuI2
0
as
- pIuI ITR(u)I - ' 0
R-++oo.
I
(2.105)
A priori estimates and compactness results
61
Indeed, we choose s E (2p-1, r) and we use Holder's inequality to obtain
fxo
IT dt flvllw)
dx
l>tt
0
f l
<_
1/p
dz fp
<x>2(p-1) (log
<x>)s
1,12!R0
jTdt IR2
JR
J0
2
_P'
< (log
1+R0)
(log <x>)-
dx <X>22
J dt J dx r
J]R2
wl2p'
<X>2 (log <x>)I
s
;r
dx fp <x>2(p-1) (log )r
p
IItIIL2(0,T;D1,2(1R2)) IIWIIL2(0,T;D1.2(R2))
where f = p", p, v = u",u, w = u", TR(u"), U, TR(u). The limit (2.105) is shown using the bounds (2.64) and (2.103) on p", p and the bounds recalled above on TR(u"), TR(u). Finally, we notice that we have, since TR(u't), TR(u) are divergence-free and vanish outside BR
1T1 p' u TR(un) dt dx R2
jj
Tr
rT r
JO
ff
/
oBP(P"u'1)
= ff P(Pu) TR(u) dt dx R
. TR(un) dt dx
R
P(pu) TR (u) dt dx =
T
n Jf T
P(P'a)
TR(u) dt dx
pu TR (u) dt dx,
J0
for all R E (0, oo).
J
The convergence is a consequence of the strong convergence in L2(BR
X
(0, T)) of P(p"u") towards P(pu) as shown in step 1 above, and of the weak convergence in L2 (BR x (0, T)) of TR(u") to TR(u) recalled above. Combining the preceding convergence with (2.104) and (2.105), we deduce easily
ff J0
R2
ff T
'T
p"Iu"I2dxdt
"
0
and we conclude the proof of Theorem 2.5.
pIul2dxdt
Il3.2
0
Remarks 2.7. 1) It is plausible that the additional bounds assumed upon u" on p" in part 2 of Theorem 2.5 are superfluous. They are used to allow
Density-dependent Navier-Stokes equations
62
us to pass to the limit in pn I un I2 and if this passage to the limit were true (without these extra conditions), more general existence results than Theorem 2.1 in the case when SZ = IRN would be possible. We indicate in
the next two remarks different arguments that can be used to pass to the limit but which, unfortunately, do not give better results than the methods introduced above.
2) We begin with a method taken (and adapted) from [16] (see also the references therein). This argument requires un to be bounded in L2(O,T; H1(IRN))N, N = 2 or 3 and some conditions on p explained below. We just sketch the argument, letting (pn, un) be a sequence of (weak) solutions of (2.1)-(2.2) satisfying, uniformly in n, the a priori estimates described at the beginning of this section. The method of proof consists in integrating (2.2) in time between t and t + h and multiplying by un(t+h) un(t): this yields, using the a priori estimates,
-
(e(t+h)u"(t+h) -
fi.N
pn(t),fn(t))
(u"(t+h) - u"(t)) dx
< Chl12(Ilun(t+h)IIH1(1RN) + Ilun(t)IIH1(IR.N))
rt+h
+C
J
ds
f r
1/2
dx V7I'!ln I4
.2
t
(IIVut+hIJL2N ) + IIDun (t) IIL2(iaN))
Next,ifN=2orN=3
f t
t+h ds fR
I N
t /'t+h
<J
rt
da /IunI IUnI3 1/2
ds JAN pnlunl2 dx
1/2 /' JAN Iunls d.2
t+h
ds IIunIIH (RN) <
Chl/4.
Hence, we find for all h E (0,1) T
J0 dt
(Pn(t+h)un(t+h)
- Pn(t)un(t)) . (un(t+h) - un(t)) dx < Chl/4.
It is in fact possible to extend this argument to N > 4 multiplying by un(t+h) * w£ - un(t) * wE (instead of un(t+h) - un(t)). This leads to a bound like (Ch112 + -C h + CE) = Chl/(N+1) if e = h1/(N+1) using the fact that Ilun
-
un * WC 11 L2
(IRN x (o,T)) < Ce.
A priori estimates and compactness results
63
Next, we deduce from the preceding inequality
fTf dt
N
Ipn (t+h)un (t+h) T
< Ch'14 + I dt
JU
<
-
Ch1/4
T
f
+ f dt Q
jN
7(t)u"(t) I2 dx
dx u"(t) u"(t+h) I V/7-(t)
-
pn(t+h)I2
f
RN I un(t)I lun(t+h)I I Pn(t) - Pn(t+h)[
< w(h)
where w(h) --> 0 as h - 0+ and w(h) does not depend on n. The last inequality requires some assumptions on pn like for instance po = p + fo with p > 0, fo E LQ(1RN) for some q E (2 , oo), fo - fo in L4(IRN). This condition as explained above yields L2 (0, T; H') bounds and one deduces easily from part 1 of Theorem 2.5 (and its proof) that pn = 75+f n where f' converges in C([0, T]; LQ(IRN)). This is enough to yield the above bound. The above "time-continuity in L2s of pnun allows us to obtain compactness in L2(BR x (0,T)) of pnun, using the compactness of p" (see
above and part 1 of Theorem 2.5) and the fact that un is bounded in L2(O,T; H1(IRN)) 3) Another method of proof consists of using some particular projections: we introduce PR, the projection from L2(IRN)N onto {v E L2(IRN)N, v =
0 a.e. on BR, dive = 0 in D'(IRN)}; notice that necessarily v . n = 0 on aBR by trace theorems. Then, if we consider pn, un as in the preceding remark, it is not difficult to check that for all Ro E (1, oo), there exists R, E (Ro, Ro+1), such that A {PRn (pnun)} is bounded in Lq(0,T; W'1,9(BRn))
for some q > 1. This is enough to ensure that PRn(p"un) is relatively compact in L2 (BRn x (0, T)) (for instance) and converges to PR (pu) if R, (or a subsequence) converges to some R E [Ro, Ro + 1]. On the one hand, we have then
P"un = PRn (pnun) +
C,rn
PIuni2 = PR..(P"u")un +div
in (un1r")
BRn ,
in BRn
for some rrn E L°° (0, T; H1(BRn)) (which we can normalize by fBR rrn dx = n
0), and, if we let n go to +oo, we obtain p J U12 = PR(pu) u + div (µ)
in BR
where IuI2 is the weak limit of Iu"I2 and µ is the weak limit of u"rrn. Notice that g E L2(0,T;LT'1(BR)) (in fact, it is bounded in that space
Density- dependent Navier-Stokes equations
64
uniformly in Ro) if u" is bounded in L2 (1RN x (0, T)) or if p" is bounded in L4',O0(1RN) (assume for instance N > 3). On the other hand, we also have PIu12
= PR(pu) u +div (µ)
in BR
with the same bounds on µ. In particular, we deduce, upon letting R go to +00,
div(µ) = p(TU -
Iu12)
E L°°(0,T;Ll(IRN)) ? 0
L719'Y-71,1 (IRN)). It is then easy to conclude that where µ - µ E L2 (0, T; µ = 0 and thus p"u" converges to pu in L2(BR x (0, T)) (V R E (0, oo)). Let us finally observe that -this method of proof-which requires many rather technical justifications that we leave to the reader-requires either u" to be bounded in L2(IRN x (0, T)) or p" to be bounded in L°° (0, T; L4,00(1RN)), i.e. po to be bounded in L4'°0(IRN). 0
2.4 Existence proofs In this section, we give complete proofs of the existence part of Theorem 2.1. We split the argument into three steps. In the first one, we solve an approxi-
mated problem and thus construct approximated solutions. Next, in step 2, we use the a priori estimates and the compactness results obtained in the preceding section to pass to the limit and build solutions of (2.1)-(2.2), and this will prove Theorem 2.1 in the cases when 12 is bounded, namely the periodic case or the case of Dirichlet boundary conditions. Finally, in a third step, we treat (and deduce) the case when 12 = IRN.
Step 1. Construction of approximated solutions. Our goal here is to construct solutions of the following approximated system a8tp
a
+ div (usp) = 0
in D'
+ div (pus ® u) - div (2µ£d) + Vp = P.fs
(2.106)
(2.107)
in D', div u=0inV. If we consider the periodic case, then (2.106)-(2.107) hold in D'(IRN x (0, oo)) and all unknowns are assumed to be periodic of period Ti > 0 in x2, for each i E {1, ... , N}. Let us recall that we define in this case 12 = fN 1(0, Ti), while, if we treat the case of Dirichlet conditions, (2.106)(2.107) hold in D'(12 x (0, oc)) and we require u to vanish on 812 x (0, oo).
Existence proofs
65
We now have to explain the real meaning of (2.106), (2.107) or, in other words, the precise definition of ue and pE which are regularizations of u and µ(p) respectively, depending upon a parameter e E (0, 1]. In the periodic case, we simply take uE = u * we, pe = p` (p) * we7 fe = f * we, where we is a
regularizing kernel as in the preceding section and pe is defined below. In fact, for technical reasons, we take fe = (f *we)(e(t) where (e E COO ([0, TI), C, (t) = 1 if t > 2e, 0 < C, (t) < lift a [0, T], (e = 0 if t <_ e. Observe that we still have div ue = 0. In the case of Dirichlet boundary conditions, we set µ(p) = pe(p) in Sl, = 1 in 0' and define µE = /ie(p) *we In. The definition of uE in that case is a bit more delicate since we wish to smoothe u, while keeping the Dirichlet conditions and the divergence-free property. One possible (explicit) way is the following: if u E L2(0, T; Ho (Sl)) (for example) where T E (0, oo)
is fixed, we set uE to be the truncation in Sle of u (extended by 0 to fl) as defined in Appendix A and we define uE by uE * we/2. Clearly, uE, which vanishes near 90, is smooth in x (recall from Appendix A that ue E L2(0, T; Ha (Sl))) and satisfies div ue = 0 in IR.N. Finally, we set fe = Ce(f 1(d>2e)) * we where d = dist (x, c9S1).
We would like also to make a simplification on since, anyway, all values of solutions p remain uniformly bounded (typically in an interval [0, IIPoI1L=]), we can assume without loss of generality that p(t) is constant
for t > 0 large and in particular that (t ' - p(t)) is bounded on [0, +oo). Then, µe is a C°O ([0, oo [) function, bounded away from 0, which is constant for t > 0 large and such that sup(o,..) 1µe - Al < E.
We now discuss the initial conditions associated to (2.106)-(2.107), namely
PIt=o = po,
pult=o = me on f
(2.108)
where pe = (po)e + e, me is defined below using M0 = (rnopo 1/2)e(plo/2)e. In the periodic case, for f = po, p01/2, moPo 1/2, we define fe = f * we. In the case of Dirichlet conditions, (po)E = p"o * wEIn, (P1o/2)e = wEIst and (mopo 1/2)e = (moPo 1/2 1(d>2e)) * we where Po = Po on Sl, = 1 on c and d = dist (x, 8Sl). Obviously, pe E Coo(?!), M'0 E Co (Q). Let us immediately remark that we have for some Co > 0 independent of e
< A < Co Po
Po
in
LP(Sl)
mo - mo in L2(f ),
(2.109)
(1 < p < oo),
mo (Po)-112 -' mopo
1/2
in L2(Sl).
The last convergence in (2.110) is easily deduced from the following facts: (Pp/2)e < (Po)e12 in D, ('i` s) a '1` a in L2(Sz). po
po
Density-dependent Navier-Stokes equations
66
We finally build mo. First of all, we decompose, as in section 2.2, o in the following way
mo = Pouo + V qo, div uo = 0 in lt,
To, qo E Coo (S2),
uo n = 0 on ast
(2.111)
(denoting by n the unit outward normal to 8Sl). Let us observe that qo is determined, up to an additive constant, by the equation (Vq- M.6) } = 0 in 11,
div {
2go
= 0 on all,
(2.112)
PO
and we finally set
mo = Pouo + Oqo, 1Iuo-U'011L2(0) < e,
where ua E Co (S2), divuo = 0 in 0.
(2.113)
We then deduce from (2.109) and (2.110)
mo - mo in L2 (lt),
mo (Po)
-1'2
-' moPo
Observe that we have mo = M06 + po(uo-Uo),
1/2
m0e
in L2 (S2).
(2.114)
(Po)-1/2
(PO)1/2(uo-o)
In fact, as we explained in section 2.1, (2.108) is not really meaningful since (2.107) shows that pu is determined "up to a gradient" and thus the initial condition, contained in (2.108), on putt=o really means an equality modulo a gradient. Since po satisfies (2.109), and div uo = 0 in S2, we may-see also section 2.1-impose
PIt=o = Po
in 12,
ujt=o = uo
in St.
(2.115)
We then state and prove the following existence result.
Theorem 2.6. With the above notation and assumptions, there exists a solution (p, u) of (2.106)-(2.107) and (2.115) such that p, u E Ct (IRN x (0, oo)), p, u periodic in the periodic case; p, u E C°° (32 x (0, oo)), u = 0 on 8St x [0, oo).
Remarks 2.8. 1) The regularization procedure we are using is directly inspired by J. Leray's original work on (homogeneous) incompressible NavierStokes equations ([284]). 2) Obviously, we have e < p < Co on SZ x [0, oo) (since div u£ = 0 in n x (0, oo), and uE is periodic or vanishes on 8g).
Existence proofs
67
3) It is in fact possible to prove the uniqueness of (p, u), using for instance the type of arguments developed in section 2.5.
Proof of Theorem 2.6. We are going to show the existence of a solution by a fixed point argument. In fact, this fixed point argument will yield a solution (p, u) with the following regularity: p E C(12 x [0, oo)), u E L2(O,T; H2(n) ) n C([0, T]; Ho (cl)), at E L2(St x (0, T)) for all T E (0, oo) in the case of Dirichlet boundary conditions and a similar regularity in the periodic case. To limit the length of the proof, we only treat the case of Dirichlet boundary conditions: the proof in the periodic case follows the same line of arguments and is in fact much simpler. Finally, we fix T E (0, oo) and work on [0, T]. We now define the mapping whose fixed point will yield a solution. Let C be the convex set in C(? x [0, T]) x L2 (0, T; Ho (1)) defined by
C = { (p, u) E C(S1 x [0, T]) x L2(0, T; Ho (S2) /
s < Ti < Co in SI x [0, T] , div is = 0 a.e. on 1 x (0, T), IIuIIL2(0,T;Ho(0)) < Ro} where Ro > 0 is to be determined.
We define a map F from C into itself as follows: F(p, i) = (p, u) as defined below. First of all, we solve
5 + div (zcf p) = 0
in f x (0, T), p1 t=o = pa in fl,
(2.116)
where uE is constructed from a as uE was from u above. Observe in particular that ii E L2 (0, T; Ck (S2)) for all k > 0, div uE = 0 in Sl x (0, T), UE vanishes near asl (a.e. t E (0, T)). The solution of (2.116) by classical (and elementary) considerations on (divergence-free) transport equations is given by a simple integration along "particle paths", i.e. solutions of the following ordinary differential equation dX
ds= uE (X, S),
X (s; x, t) = x, x E U, t E [0, T].
(2.117)
In view of the properties of uE., there exists aunique solution X of (2.117), continuous in (s, t) E [0, TJ2, smooth in x E Sl such that a.X E C([0, T] x Sl x [0, TJ) for all cY and X (s; x, t) E 92 for all (s, t) E [0, T]2, x E Sl. Then, we have
P(x,t) = P0, (X(0;x,t)), Obviously, E < p
Vx E SZ , Vt E [0,TJ.
(2.118)
Co in S2 x [0, T], p E C([0, T]; Ck(S2)) for all k > 0 and in view of (2.116) at E L2 (0, T; Ck (fl)) for all k > 0. Furthermore, p and
Density-dependent Navier-Stokes equations
68
7 are bounded in these spaces uniformly in (p, u) E C. In particular, the set of p built in this way is clearly compact in C(Sl x [0, T]). We now build u: first of all, we set It,, = µ(p), with the same construction as above and we wish to solve the following problem p
au
+ pue Du -div (2p ,d) + V p = pfE in fl x (0, T),
div u = O
in SZ x (0, T),
uE
ult_o = uo
in
(2.119)
S2,
L2 (0, T; H2(SZ)) n C([0, T]; Ho (fl));
Op,
(2.120)
E L2(St X (0, T)).
This is nothing but an inhomogeneous (linear) Stokes equation with rather smooth coefficients: the regularity of p, uE has been discussed above, µ, E Coo (S2 x [0, TI), f£ E Co (12 x (0, T]) and uo E COO (0), div uo = 0 in fl.
We postpone the discussion of this problem and admit temporarily that there exists a unique solution u of (2.119)-(2.120) (depending continuously on data). This fact is established in Proposition 2.1 below. Then, when (p, u) E C, u is bounded in L2(0, T; H2(SZ)) while is bounded in L2(11 x (0, T)). Therefore, u is compact in L2(0, T; Ho (SZ)). This shows that the mapping F is compact on C. Hence, if we wish to use the Schauder theorem in order to conclude the existence of a fixed point, we have only to choose Ro in such a way that IIUIIL2(o,T;Ho(n)) <- Ro. To this end, we multiply (2.119) by u, integrate by
parts using (2.17) and obtain easily (all manipulations are justified by the regularity of p and u) for all t E [0, T]
f p 122 (x,t)dx+u
Jn
J0t
jst
IvuI2(x,s)dxds rc
COIIf IIL2(nX(O,T))
J0 Jft Iu12(x, s) dxds
1/2 ,
hence, SUP tE [O,T]
IIu(t)IIL3(o) + IIUIIL2(0,T;Ho(g)) < C1
(2.121)
where C1 depends only on µ, f, C but not on Ro, p, U. We then choose Ro > C1.
In order to conclude, we still have to show that a fixed point (p, u) is in fact smooth. This is easily done by a bootstrap argument that we only sketch. First of all, we observe that uE E C([0, T]; Ck (Sl)), p"4, E CO,1/2([O, T]; Ck(SZ)) for all k > 0, and using LP-theory (see V.A. Solon-
nikov [444], [445] for instance), or direct proofs similar to the proof of
Existence proofs
69
Proposition 2.1, we deduce from (2.119)-(2.120) that u E LP(O,T; W2'p(St)), Ft E LP(St x (0, T)) for all 1 < p < oo. With this regularity on u, we can bootstrap and gain more time regularity on uE then p and thus more regularity (in (x, t)) on u. Before stating Proposition 2.1 which fills the only gap left in the above proof, we first observe that (2.119) may be written as
aui
+ bi . Vu - aAui + p = gi in SZ x (0,T), 1 < i < N, a div u = 0
in SZ x (0, T),
u{t=o = u°
(2.122)
in Q7
where g E L' (!Q x (0, T)), c E LOO (11 x (0, T)), a E L°°(0, T; W""°° (St)), ai E-L1(O, T; L= (D)), b E L2(0, T; L' (D)), c > 8, a > S a.e. on St x (0, T) for some S > 0, u0 E Ho (SZ).
Proposition 2.1. There exists a unique solution u of (2.120)-(2.122).
Proof of Proposition 2.1. We only prove that the a priori estimates contained in (2.120) hold. The proof will show the uniqueness of solutions, and the existence follows in a straightforward way from a priori estimates by standard arguments that we leave to the reader.
Next, in order to prove a priori estimates, we multiply (2.122) by at , sum over i and integrate (by parts) over Sl to find for almost all t E (0, T)
Sf atl`2 dx+2Jnadt{Vu12dx l
<_
f I1 l TI +{b{{ou{IatI +{ValIVul
1'5i9u I
dx.
Hence, using the Cauchy-Schwarz inequality, we find 6
2
z
f l at
dx + l
1d 2 dt
j a{Vu{2 dx
)
< C 1 + IibIIL.(R) + Iloa{{Lx(n) +
MIL-(j Jcc alVu{2 dx.
Since b c- L2 (0, T; L°° (SZ)), Va E L2 (0, T; L°° (S2)),
iE
L1(0, T; L°° (S2)),
we deduce from Gronwall's inequality an a priori estimate on u in L'(0, T; Ho '(Q)) and on ai in L2(fl x (0, T)) depending only on the data c, a, b, g, uo. In particular, we may then write (2.122) as
-aLu + Vp = h
in St,
div u = 0
inn,
u E Ho (Sl)
(2.123)
for almost all t E (0, T). In addition, h E L2(S2 x (0, T)) and its norm is bounded in terms of the data. From here on, whenever we say bounded, it
Density- dependent Nauier-Stokes equations
70
means that the bound depends only on the data. From the previous bound on u, we deduce in particular that
Vp = and thus is bounded in L'(0, T; H-1(11)). Therefore, if we normalize p by imposing
in
pdx = 0,
a.e.
t E (0,T),
we deduce that p is bounded in L2(11 x (0, T)) (see [4721 for instance). Then, we write (2.123) as a usual Stokes problem, namely
-Au + VP = h inn, div u = 0 in fl,
where h = a
-
u E Ho (fl),
(2.124)
is bounded in L2(ft x (0, T)), a, and we con-a clude that u is bounded in L2(0,T;H2(fl)) by classical regularity results 2--
on Stokes equation (see [472] for example). This completes the proof of Proposition 2.1 and of Theorem 2.6.
0
Step 2. Passage to the limit. First of all, we collect a priori estimates and follow the arguments of section 2.3. Since div uE = 0, we immediately obtain for all f3 E C(IR, IR)
in
Q(pe) dx =
fI3(p)dx
for all t E (0, oo).
(2.125)
Here and below, we denote by (ue, pE) the solution built in step I (observe that uE is the regularization of uE, namely (7)E *CJe/2 in the case of Dirichlet boundary conditions).
Next, exactly as in section 2.3, we obtain the analogue of (2.34)-(2.36), namely IIuEIIL2(0,T;H1(tt))
sup
0
:5 C,
IIpCIILEI2IIL1(n) < C
(2.126)
(2.127)
where C denotes various positive constants independent of E.
Because of (2.110), we may then apply part 1 of Theorem 2.4 to deduce that pE converges, up to the extraction of subsequences, to some p in C([0,T]; LP(0)) (V 1 < p < co, d T E (0, oo)) which is bounded, satisfies (2.125) with pE and po replaced respectively by p and p0i and thus satisfies (2.17). Furthermore, p satisfies (2.1) (and is periodic in the periodic case) where u is a weak limit in L2(0,T; H1(ft)) (V T E (0, oo)) of uE. Of course, u is periodic in the periodic case and u E L2 (0, T; H0 (Il)) in the case of Dirichlet boundary conditions.
Existence proofs
71
In particular, this convergence implies, in view of the construction of µE, that we have pE --> µ(P)
in
C([0, T); Lp(12))
(2.128)
(Vi < p < oo , V T E (0,00)), as c--+0, P' f, -+ p f in L2(S2 x (0, T))
(V T E (0, oo)), as a - 0.
(2.129)
In addition, from the results shown in Appendix A, uE is also bounded in L2 (0, T; H1 (0)) and uE converges weakly in L2 (0, T; H1 (0)) to u-this is obvious in the periodic case. These bounds imply that (2.53) holds with q = 2, m = max (2 -1, 1): indeed, pedt is bounded in L2(Sl x (0, T)) while pEuE 0 uE is bounded in L2 (0, T; LP(S2)) with p E [1, 2) if N = 2, p = N 1 if N > 3, and thus is bounded in L2 (0, T; H-- (SZ)) with s > 0 if N = 2, s = a - 1 if N > 3.
We then deduce from part 2 of Theorem 2.4 thatu6 converges to and thus pEuE put in LP(O,T; L''(S )) for 2 < p < oo, 1 < r < N converges to put in LP(0, T; LT(S2)) for the same (p, r). These convergences allow us to recover (2.2) from (2.107) upon letting e go to 0. In fact, we recover (2.12) (the weak formulation of (2.2)) provided we show in the case of Dirichlet boundary conditions that
in
PCuo . O dx
E
in
as
0 = 0. This is clear in view of (2.113)-
(2.114) since we have
fPu.0dx=fm.0dx -fmo.c5dx
as
for all 0EL2(f2)Nwith div0=0in D'(S2),0-u=0on 8S2. The only fact left in order to complete the proof of Theorem 2.1 is the energy inequalities (2.13)-(2.14). This is in fact relatively easy since (pE, uE) also satisfies some energy identities obtained as in section 2.3 by multiplying (2.107) by uE and integrating over S2, using (2.106) and the boundary conditions. We find then for all t > 0
dt
j
f pe(8tuj+8jui)2dx = 2 f ptf
ut dx.
(2.130)
We have seen above that pEuE converges in L2(SZ x (0, T)) (in particular) to pu, µE converges in C([0, T]; LP(S2)) (V 1 < p < oo) and is uniformly
bounded on 11 x (0,ec), while fE converges to f in L2(S2 x (0, T)), for
Density-dependent Navier-Stokes equations
72
all T E (0, oo). This is enough to imply (2.13) provided we show for all cpECo (0,oo),cp>0 00
lim J C
dt
Jn
o
dx cp(t) me(PE)(asu + ajuz )2 (2.131)
fdt J dx 00
>
(t) p(p)(au+ aju2)2.
o
In order to show (2.131), we observe that we have
f f 00
0<
dt
dx W (t) pE(a=(u; -u;) + a; (u= -U,))2
n
o
r
fdt Jndx cp(t) µ(au+ +
f dt f
+ 8ju)2 dx
dx cp(t)
n
- fdt 00
dx W(t) pe(81uj + aju=)(a=u; + aju4)n
Since cp(t)(aluj + a;u1)2 E L' (n x (0, oo)) and pe is uniformly bounded and converges in measure on SZ x Supp (cp) to µ(p), we deduce easily that WiLe(aiuj+8ju,)2 and cp1/2µC(,9=uj +a;u;) converge, respectively, to cpp(p) (81uj + ajul)2 in L' (S2 x (0, oo)) and to cp1/2p(p)(81uj + 8ju1) in L2(S2 x (0, oo)). In addition, cp1/2 (ayuj, + ajui) converges weakly in L2(f x (0, cc)) to cp1/2(aluj + 8ju1). Therefore, the two last integrals converge, as c goes
to 0, to f' dt fn dx W(t) p(p) µ(p)(aluj + ajul)2. This implies (2.131). Next, in order to prove (2.14), we first integrate (2.130) between 0 and t to find t P eIueI2dx(t)+
fn
t
2
f dsf
n
o
PoIuoI2 dx
f
n
dsJ
J
n
(2.132)
Pefe uedz+Po1uodx fn
1
e
I MO
P5
- Ogo I2 dx
f I met oI + nA met n
µE ((9tuu+ajui)2dx
Pe(0
I
0pgol e2
2 (Pouo+Vq) Vqa dx 2
PO
0
2
e
E
0eI2 PO
dx.
Existence proofs
73
(in the case of Dirichlet boundary conditions) and
Since uo = 0 on 8
div uo = 0, we finally obtain
in Since
Po)
PoIuoI2 dx +
IVgol2 dx =
.In
Po
converges in L2 (n) to
i in
Imol2 odx PE
we deduce (2.14) from (2.132) PO
exactly as before.
Remark 2.9. In fact, it is often possible to sharpen a little the energy inequality (2.14), replacing 1O by poluol2 for some uo to be determined satisfying div uo = 0. However, we cannot do it in full generality and we have to make some assumptions on po. The first case we can treat is when infessn po > 0. Then, exactly as in section 2.1, we can check that ua converges in L2(&) to uo = PP0(mo); in the case of Dirichlet boundary conditions, V qo = mo - uO is determined by the elliptic equation div
(Vo_mo)
in c,
Vqo E L2(Sz),
(2.134)
PO
(Vqo-mo) n = 0 on aci, and it is clear that (2.14) holds (in fact for all t > 0 since u E C([O, T]; for all T E (0, oo), see section 2.1) with PO replaced by poluol2. The second case allows po to vanish. For instance, we assume that SZ is
connected, po = 0 a.e. on n - w, po > 8 > 0 a.e. in w where U E 11, w is smooth, and we only consider the case of Dirichlet boundary conditions. First of all, we observe that IVgoI = (p.) i2 (pa)l/2 is bounded in L2(SZ).
Next, we can normalize qa in such a way that fan qo dS = 0. Therefore, if we extract subsequences if necessary, qo converges weakly in H1(SZ) to qo satisfying f a qodS = 0, Vqo = 0 on n - w and thus qo = 0 on n - w. Hence, qo E H (w). In addition, uo is bounded in Li jw) and we may assume that uo converges weakly in L2 (K) (V K compact C w) to some
uo E L?,() such that
f
Po I uo I2 dx
+
r
poI uo12
IVgol2
E L1(w), and we have
dx
PO
mo = Pouo + V qo in w,
f div
ImoI2 dx
=
Po
(vo_mo) = 0
ImoI2
dx
(2 . 135)
Po
in w.
(2.136)
PO
We next claim that there exists a unique qo E Ha (w) which satisfies (2.136)
(assuming that mo E L2(w),
ao E L2(w)), and we have f. I '
12 dx
=
Density-dependent Navier-Stokes equations
74
fw °2 dx. If this claim were established, we would deduce that (2.135)
i z
is in fact an equality and thus poIu0I2,
POI
opo 2 ,
converge in L1(11) to poIuoI2,
1
I°-
respectively, where we extend these functions to IZ by 0 outside w.
This is enough to conclude that (2.14) holds with po I uo I2 replacing 1"`0
In order to show the above claim, we have only to show that for any solution qo E Ho (w) of div(Og0-mo)
PO
= 0
in w
J
2
we have fw Vao dx = f,'720-m' dx. Then, we multiply the equation by goC(E) where d = dist (x, 8w), e > 0, ( E Coo ([0, oo)), C(t) = 0 if t < 1/2,
C(t) = 1 if t > 1, 0 < C(t) < 1 on [0, oo), and we obtain
f
(Vqo-mo) Vgo,(d)dx+ f Vqo-rno Vd
w
e
PO
w
e (I (d)godx e
PO
= 0.
It only remains to show that the second integral goes to 0 as a goes to 0. Indeed, we have
f
Vqo-mo O, E
Po
<
(e)g0
C
Oqo-mo
1/2
1/2
II
Po
qo IIL2(w)
f(O
dx
1/2
0.
In fact, the above analysis can be extended to the case when p > a d(x)7, p <- 8 d(x)7 a.e. in w for some a, Q > 0, y > 0. This is possible in view of the following variant of Hardy's inequality
i
2
d2+7
dx
- C JL
0
Idly
2
dx.
This inequality follows easily from the following computation: we have for all f E Co (0, oo)
hence f°°
dx < (7+1)r f °D 1 dx.
O
Remark 2.10. We observe here without proof that the existence and compactness results can be extended to the case when po (> 0) is assumed
Uniqueness: weak = strong
75
to be in LP (11) where p > 1 if N = 2 and p = 2 if N > 3. We still assume E L1(SZ) (mo = 0 a.e. on {po = 0}). Then weak solutions are that defined exactly as in section 2.1 except that p E C([O, oo); LP(SZ)), and one
can adapt the preceding proofs to show that Theorem 2.1 holds in that case
D
Step 3. Existence in the case when fl = IRN. We use the existence results we just proved with ) = BR and Dirichlet boundary conditions on OBR. We then obtain approximated solutions (PR, UR) and we let R go to +oo. More precisely, we denote by (PR, uR) a global weak solution of (2.1)-(2.2) in BR where R E (0, oo) with the boundary condition (2.6), restricting of course po, mo to BR. Recall that, as explained in Remark 2.1 (7), we may assume without loss of generality that u., = 0. Next, we observe that all the estimates shown in section 2.3 in the case when SZ = IRN hold uniformly in R large. We may then apply Theorem 2.5 to deduce the relative compactness of pR, PRUR and PRUR in
2T];LP(BM)) (1 < p < oo), LP(O,T;L'(BM)) (2 < p < oo, 1 < r < ) respectively (for all T E (0, oo), M E (0, oo)). Finally, this compactness allows us to prove the existence of solutions in IRN letting R go to +oo and using the same arguments as in step 2 above. 0
2.5 Uniqueness: weak = strong In this section, we show that any global weak solution coincides with a more regular solution as long as such a "strong" solution exists. More precisely,
we prove that a weak solution is equal to a strong solution whenever the latter exists. It is not difficult to check that smooth solutions exist for a
certain time interval-at least if po does not vanish-and the result that we are going to present then implies that any weak solution is equal to the smooth one on this time interval.
In order to simplify the presentation, we only treat the periodic case and the case of Dirichlet boundary conditions even if similar results can be obtained in the case when fl = IRN by convenient adaptations of the arguments below. We then consider a global weak solution u of (2.1)(2.2) and (2.6) as built in Theorem 2.1. We assume (for instance) that f E L2(O,T; L= (f?)) and fix T E (0, 00). We next assume that there exists a solution p, u E C(SZ x [0, T]) (resp., in the periodic case, C(IRN x [0, TJ ) periodic) of (2.1)-(2.2) in S (resp. in IRN) with Vu E L2(0,T;L°O(SZ)), vP E L2(0,T; L=()), a E L2(0,T; L°O(SZ))) and with u = 0 on 8S2 x (0, T). Furthermore, we assume that p is locally Lipschitz on [0, 00) and
Density-dependent Navier-Stokes equations
76
that p, u satisfy Plt=o = po
in ft,
pint=o = mo in ft.
(2.137)
Notice that this equality implies in fact that mo = pou(O) with div u(0) = 0 in St. Let us notice, that, of course, (2.2) holds with some pressure field p that belongs to L1(O,T; L°°(SZ)) + L2(0,T; W-1'°°(St)).
Theorem 2.7. Assume in addition that p # 0. Then we have u - u a.e. in SZ x (0, T).
Proof of Theorem 2.7. We first recall that we have for (almost) all t E (0,T) rf pIu2 dx +
<
Jf
Ltf µ(P)(8su +a3u,)2 dx ds
dsf
I mo l2
p f u dx ds + 2 Po
(2.14)
Next, we remark that, in view of the regularity of u, we deduce from the weak formulation (2.12) of (2.2) the following equality
ff t
pu
J
udx +.21
= J rrtio u(0) dx +J J pf n on
µ(p)(a1uj+ajui)(aiu?+a?uj) dxds
u dx ds +J J pu
on
{
+u
ds (2.138)
a.e. t E (0, T). Then we write p
6U--
Pf + (P-70( a +U. Du) + p(u-u) Du - div(2(µ(P)-µ(P))d)(2.139)
If we first multiply (2.139) by u and integrate over St x (0, t), we find t
f µ(P)(8=uj+a;ui)(atuj+ajui) dy ds
=J
Jo n
t
pf u+(P-P)(
o
+
1
2
f
(2.140)
+U-VU)
L
Jo n (µ(P) - Ft(P))(aiu,i+a,Ui)(aiuj +a,ui)
dx ds.
Uniqueness: weak = strong
77
Combining (2.138) and (2.140), and using (2.137), we obtain for almost all
t E (0,T)
rt J
oJn t
j I'J2o12 Jst Po + fo Jc
+ (p-P) (
+
u(P)(ajuj+ajuj)(ajUj+ajj)dxds
Pf u+pf u +u
Vi) u+ p(u-u) Vu u dx ds
ftf) (ju(P) - p(P))(aiij+ajUi)(aiuj+ajui) dads.
21
Finally, we multiply (2.139) by u and integrate over S2 x (0, t) to find t
1 f PIul2 dx + 1 ff
ft
ImaI2 2
fn
Po
dxds
+
o
fn Pf u
(2.142)
+(p-p)(.+ u- Vu) u+p(u-u) 1(u(P) - µ(P))(aiuj+ajui)((9iuj+ajui) dxds Then, if we add up (2.14) and (2.142) and substract (2.104), we obtain
1 fPIu_l2dx+ <
on
2
f fr f (u-u)(P-P) dads t
o
+
i
21
t
f
t
t
+ fo in
(P(P)-l4(P))(aiUj+ajUi)(ai(uj-ui)+ai(us-ui))dxds
au
5 + u Vu) (u-u) - p(u-u) V (u-u) dx ds.
Hence, we deduce from the assumptions made upon u that we have for almost all t E (0, T) and for all e > 0
f plu-u12dx+ J f IV(u-u)I2dxds potst
n
(2.143)
t
<
ff
C(s)PIu-uI2 + eIu-uI2
+ CE(s)I P-7jI2 dx ds,
Density-dependent Navier-Stokes equations
78
where C, CE denote various non-negative measurable functions in L1(0,T). Next, we wish to estimate IIP - pII L2 (n) We write a
(p-p) + div {u(p-p)} = (u-u) Vp
and deduce easily (see section 2.3 for related arguments) for all t E [0, T] t plu-uI2
inn
+ IP-pl2dx+ 10fn IV (u-u)I2dxds t
<
fds C(s)
f
f f t
plu-uI2 + IP-7;12 + E
n
ds
o
Iu_ I2 dX.
n
(2.144)
Next, we observe that there exists e > 0 such that we have for all v E H1(SZ) and for all p E L°° (SZ) such that fn p dx = fn Po dx > 0, IIPII L- (n) < IIPOIILc(n)
f
e
Ivi2 dx <
2 f IvvI2 dx + 2
IpIvi2dx.
Indeed, if this were not the case, we would find v, pn satisfying
fiVVnH2+fPntVnI2dx
flvnidx=17 t
n
0,
Po>O,
pn-'Pw-L°°(SZ)-*,
fPdx=fPodx.
Hence vn converges to 1 in H' (f2), and pn I vn 12 --+ p w - L1(11). The conn tradiction proves our claim. Inserting the above inequality in (2.144), we then conclude that u = u, p = p a.e. in 11 x (0, T), by applying Gronwall's inequality.
Remark 2.11. Modifying a little the above proof (using Sobolev's inequality), one can extend the preceding result to the case when Vp E L2(0,T; LP(SZ)), Vi E L2(O,T; L' (Q)) where p = N if N >_ 3, p > 2
ifN=2.
3
NAVIER-STOKES EQUATIONS This chapter is devoted to the classical Navier-Stokes equations in the homogeneous, incompressible case. The system, described in section 1.2, can be deduced from (2.1)-(2.2) by setting p - p where p is a positive constant and by introducing the kinematic viscosity v = 4(p)/p and a reduced pressure field p/p. We then obtain
f, div u = 0
in
fl x (0, T)
(3.1)
where T > 0 is fixed and f is given on Il x (0, T). Of course, (3.1) is complemented with boundary conditions (the same as in chapter 2) and an initial condition in S2. (3.2) ult=o = uo
Without loss of generality-otherwise we simply subtract a gradient term from u-we may always assume that we have
div uo = 0
in D.
(3.3)
3.1 A brief review of known results We begin with the celebrated results due to J. Leray [283] (see also [472], [293] and the bibliography for more references on the subject) concerning the global existence of weak solutions. In order to simplify the presentation and notation, we denote by H' (H3, W'n'P) the usual Sobolev space H'(IRN) in the case when St = IRN, or Hper = {u E H11.,r u periodic}
in the periodic case and by H-1 the dual space (H-3, W-',P'). In the results which follow, we assume
uo E L2(fl),
f E L2(0, T; H-1)
(3.4)
Navier-Stokes equations
80
In the case of Dirichlet boundary conditions, we assume in addition
u° n = 0
on
(3.5)
811.
Recall that (3.5) is meaningful since (3.3) holds and uo E L2(Q) (hence uo n E H-1/2(aSZ)). Again this is not a restrictive assumption since we can always decompose (uniquely and continuously) any iio E LP(SZ) into a gradient term (in LP(SI)) and a divergence-free vector field in LP(SZ) satisfying (3.5) (for all 1 < p < oo). In the case of Dirichlet boundary conditions, we need to introduce some functional spaces for rather delicate reasons to which we shall come back in detail later on. We set for 1 < p < oo, V°1P(SZ)
= {u E LP(SZ), divu = 0 in SZ, u n = 0 on
V"(SZ) = {u E Wo'P(SZ)
,
aSZ}
div u = 0 in SZ},
and we recall that D(l) = {cp E Co (SZ), div cp = 0 in S2} is dense in V°>P(SZ), V1"P(SZ) respectively for the LP, W",P norms. Finally, we denote by V-1,P the dual space of V1"P where -1 + P, = 1 (1 < p < oo).
Next, we recall the weak formulation of (3.1) as given in chapter 2 for a more general system without checking that all terms written below make sense, since this point will be a straightforward consequence of the reg-
ularity we assume for weak solutions: we request that we have for all So E C°°(1 x [0, T]) such that div cp = 0 and with compact support in SZ x [0, T)
frT J
o
=
fdtdx1vVu.vco-uuj0coj-u.-j
J
c2
I
T
H-1xH°idt+
r
J
n
divu=0 inD'(Slx(0,T)). (3.6)
In the periodic case (or in the case SI = IRN) we replace Ho by H' and w is then assumed to satisfy: cp E C°° (IRN x [0, T]), div cp = 0, cp is periodic in xfor all tE [0,T]. This formulation implies that (3.1) holds in the sense of distributions for some pressure field which is a distribution. Observe also that the term uiujajcpj can be written as -ujajuticpz (as soon as Vu E L2(SI x (0,T)), u E L' (92 x (0, T)) for example). In the case of Dirichlet boundary conditions, (3.6) is also equivalent to a more abstract formulation involving the spaces V 1,P (SZ) (and V -1'P) . As we shall see, the weak solutions satisfy: u E L2(0, T; V 1,2 (SZ)), U E L°° (0, T; L2 (SZ)), Ju12 E L2
and thus (3.6) implies that
A brief review of known results
81
au should be written u' since it is considered as a (time) derivative of a function with values in some Banach space), and (3.6) is then equivalent to
F E L2(0, T; V- 1071--4T) (in fact
u + vAu + B(u, u) = f where f, Au E L2(O,T;V-1"2), B(u, u) E L2(O,T;V-1'T) are defined by
=H-1xHo, VvE V"2
rn Du Vvdx,
=
=
VVE
V1,2
_-J u=u;8iv;dx, VvE V"
.
n
n
Observe finally that, since S2 is bounded, V 1'N C V"2 and we deduce that V-1,2 C
V-"7J T (identifying L2 with its dual as usual).
We now state some global existence results of weak solutions: the first two results concern the case when SZ = IRN (or the periodic case) in two dimensions (N = 2) and in dimensions N > 3 respectively, while the next two results are devoted to Dirichlet boundary conditions with N = 2 or N > 3 respectively.
Theorem 3.1. (N = 2, f = IR2 or the periodic case). There exists a unique weak solution u of (3.1)-(3.2) with the following properties: u E L2(O, T; H') f C([0, T]; L2), a E L2(0, T; H-1). Furthermore, there exists a unique p E L2(BR x (0, T)) (for all R E (0, oo)) such that Vp E L2(O,T; H-1), fQ pda = 0 a.e. t E (O,T) where Q = Bi for example if S2 = 1R2 or Q = SZ in the periodic case, and such that (3.1) holds in the sense of distributions. We have for all t E [0, T] 1
2
/ Iu(x,t)I2dx+v n
1
_ 2Jst -/
f/ t
o
sz
(3.7)
t
dx +
Jo
<.f (s), u(s) > H-1 XHl ds.
Theorem 3.2. (N > 3, St = IRN or the periodic case). There exists a weak solution u of (3.1)-(3.2) and a pressure field p such that (3.1) holds in the sense of distributions, and the following properties hold: u E L'(0, T; H1)nC([0, T]; Ly,)nC([0, T]; L°(BR)) (b' 1 < s < 2, d R E (0, oo)), at E L2(0, T; H-1) + (L3 (O, T; W-1' n LQ(O, T; L'')) for 1 < s < oo,
Navier-Stokes equations
82
1 < q < 2 and r =
N Nq+g--21
p E L2(BR x (0, T)) + L' (0, T; Lam) for
1 <_ s < oo, R E (0, oo), Vp E L2(0, T; H-1) + Lq (0, T; L'') for 1 < q < 2 ; and we have and r = 1
Iu(x,t)I2dx+vJ
tr IVul2dxds
Jc < 2 J Iuol2 dx+ ItH_lxHlds,
2 Jn
o
Jo
n
d ( ' f Iu(x,t)I2dx +v
dt
(3.8)
<H-1xHI inD'(O,T).
2 n
(3.9)
Furthermore, if N = 3, there exists a solution satisfying in addition 2 at (IuI 2
1
22 +p} )
- VA 122 + VIVuI2 < u f
(3.10)
in D'.
Theorem 3.3. (N = 2, Dirichlet boundary conditions). There exists a unique weak solution u of (3.1)-(3.2) which satisfies: u E L2(0,T; Ho (ft)) V-1,2), (or equivalently u E L2(0,T; V1"2)), u E C([0, T]; L2), ai E L2(0,T; and (3.7) holds.
Theorem 3.4. (N > 3, Dirichlet boundary conditions). There exists a weak solution u of (3.1)-(3.2) satisfying (3.8),(3.9) and such that u E n C([0, T]; L' (f1)) for all 1 < s < 2, L2(0,T; Ho (11)), u E C([0, T];
i 1
EL2(0,T;V-1"2)+(L'(0,T;V-1'
2)nL9 (0,T;L'')) fort <s
Remarks 3.1. 1) We do not claim any originality in the results presented above except the slightly unusual presentation and, maybe, some regularity (or partial regularity) information on i and p. The results presented in Theorems 3.1 and 3.2 are essentially contained in J. Leray [283],[284],[285]
and further references can be found in R. Temam [472], J.L. Lions [293] and the bibliography for instance. 2) Let us remark that u{ Iu12 +p} makes sense in (3.10) (since we know a that, for instance, IuI2 E L3/2(0, T; L9/5), p E L3/2(0, T; L9/5)+L2(0, T; L2) while u E L3 (0, T; L1815) n LOO (0, T; L2) (by Sobolev embeddings) ands +
= i5 < 1. The meaning of u f in (3.10) has also to be clarified: since f E L2 (0, T ; H-1) , u E L2 (0, T; H1), u f is the distribution defined by 1a
< u f , cp > = < f, ucp > observing that uc E H1 for smooth test functions W.
A brief review of known results
83
3) We shall see below-see also sections 3.2 and 3.3-further regularity properties of weak solutions. Let us recall however that uniqueness of weak solutions is an outstanding open problem (even for more regular data f and
uo). It is clearly related to regularity issues: in particular, if we postulate more regularity on the weak solutions, the uniqueness follows. It is possible to show (this result is due to J. Serrin [428]) that if there exists a more regular solution then the weak solution coincides with this one. These results of the type "weak = strong" are very much in the same spirit as the one shown in section 2.5. However, the optimal-possibly the fullregularity of global weak solutions is not known: for instance, if N = 3, SZ = IR3, f = 0, uo is smooth, results due to J. Leray [283],[284],[285], L. Caffarelli, R.V. Kohn and L. Nirenberg [77] show that weak solutions are smooth except for "small sets" (zero--one-dimensional Hausdorff -measurein [77]) containing the possible singularities. The solution is also known to be smooth for t small and for t large, and for all t if uo and f are small in appropriate spaces. Of course, if N = 2, much more is known: for instance, if uo E Ho(SZ) (or H'(IR2)), f E L2 (n x (0,T)) then u E L2(0,T; H2(0)) n C([0, T]; Ho '(D)), ai E L2(S2 x (0, T)) and p E L2(O,T; H1(0)), and if f is smooth, u is smooth for t > 0. Another topic which has been extensively studied concerns the space of
initial conditions uo (take f = 0, 0 = IR3 for example) in which there exists a (unique) solution for t small or a global small solution. The most general result in that direction is probably the result of M. Cannone and Y. Meyer [80] which states that if N = 3, S2 = IR3, uo E L3, f = 0, uo is small in then there exists a global solution u E C([0, oo); L3), and the solution is then automatically C°° for t > 0 as we shall see below. Let us finally mention that some marginal improvements of the regularity of ut and p shall be given in section 3.2 below, particularly in the case of Dirichlet boundary conditions. 4) We wish now to explain some of the difficulties encountered in the case of Dirichlet boundary conditions. First of all, let us remark that the information contained in Theorems 2.3-2.4 on at does not say much if we insist upon looking at au as a distribution. Indeed, if we can check that V-1"P(f2) is W-1'P(f2)/{Vq/q E LP(fl)}, then Taut- E L2(O,T; V-1,2(SZ)) (say if N = 2) does not imply E L2 (0, T; H-1(11)). In fact, even if B.1"2'6
N = 2, we do not know if ai E L2(0,T;H-1(11)). Of course, there is a distribution ir such that at - Vir E L2 (0, T; H-1(SZ)), and it is possible to choose r harmonic (in x): indeed, we can write 8u
at
= -AU + Dir,
div U = 0
in SZ,
U=0
The regularity stated in Theorems 3.3 and 3.4 yields:
on BSZ.
(3.11)
Navier-Stokes equations
84
U E L'(0, T; Ho (Sl)) + (L3(O,T; K '
(12)) n Lq(O,T; W 2°r(Sl))) (3.12)
for 1 < s < oo, 1 < q < 2, and r =
From this we deduce that au -Vir E (S2))(1Lq(0,T;Lq(1))) Ft and we finally observe that (3.11) implies A r = 0. In other words, up to a gradient harmonic distribution, ai has the regularity we expect. From N L2(0,T;H-1(S2))+(Ls(0,T;W-1,
(3.1) (which holds for a certain distribution p), we deduce that Vp has also
the same properties as at ("natural regularity" up to the gradient of an harmonic function). This question does not arise when 11 = ]RN or in the periodic case since
if 7r is harmonic and Vir E H-1 (and periodic in the periodic case) then Vp - 0. A related issue is the possibility of projecting (3.1) on divergencefree vector fields. Let us recall that we denote by P the projection (in L2 say) onto divergence-free vector fields, i.e.
P = Id + V(-A)-ldiv. Both in the periodic case and in the case when 11 = IRN, P commutes with translations and thus with derivatives so it is bounded (and a projection)
from any H' into H' (s E IR). In fact, since P is bounded in LP (1 < p < oo), P is also bounded in any H3'p (s E IR, 1 < p < oo). Then, we can write in those two cases
aui at
+ P(( u V)u) : - vAu ; = P(f) s
s
(3 . 13)
s
or equivalently Clui
+ as (P(ujui )) - v Lui = P(f)
i
(3 . 14 )
On the other hand, when we consider the case of Dirichlet boundary conditions, it does not seem to be possible to write (3.1) in such a concise form. The natural replacement for P is the orthogonal projection (in L2) onto divergence-free vector fields v E L2(S2)N such that v n = 0 on recall that if v E L2(S2)N, dive E L2(SZ) then v n E H-1/2(a1). Then P is also bounded in LP(S2) for 1 < p < oo but it no longer commutes with derivatives. In fact, it is easy to check that P is bounded in W1"1'(SZ)
for 1 < p < oo but P does not leave WJ"(0) invariant (we only obtain P(u) n = 0 on 811 and not Pu = 0 on 812) . In particular, we cannot deduce (by duality) any information on P in W-1"r(S2) including a definition.
The fact that P(u) does not make sense if u E H-1(0) (say) can be seen from the fact that P(-Du) does not make sense if u E Ho (S2) even if
A brief review of known results
85
div u = 0 in SZ. Indeed, if u E H2(SZ) n Ho (l1), then P(-Du) E L2(SZ) and
we claim that P(-Du) is not bounded in H-1 if u is bounded in H01(0) even if div u = 0 in SZ. To prove this claim, we argue by contradiction and thus assume that P(-Lu) extends by continuity to a continuous map from Ho (SZ) into H-1(SZ). Then, let u E H2 (SZ) n Ho '(Q) be such that div u = 0, 0 on 8SZ (Du E L2()), div (Du) = 0 in SZ so that makes sense
in H-112(&2)); examples of such a u are not difficult to build. We next choose un E Co (SZ) such that un converges to u in Ho (SZ) and div un = 0
in SZ. Clearly, P(-Dun) = -Dun converges in H-1(SZ) to -Au while, by assumption, it should also converge to P(-Du), and we reach the desired
contradiction since Du n # 0 on 8f and thus P(-Au) # -Du. This argument shows that P(-Du) defined on H2(Sl) n V1,2(Q) cannot be extended by continuity to V1"2(SZ). Of course, even if it does not seem a natural thing to try, we might attempt to define P(-Au) on V1,2(n) in a different manner using the orthonormal basis of V°'2(S2) composed of the eigenfunctions wi (i > 1) of the Stokes operator, namely
-Awi + Vii = Ai wi,
Ai E IR,
wi E V 1,2 (!Q)
where Xi (> 0) are the eigenvalues. The set {wi / i > 1} is an orthogonal basis of V1,2 (P) and we have 00
00
U=
uiwi,
1
i=1
(u12 dx
Ivil2
=
for all u E V°'2(f ),
i=1
00
IVu12 dx
=
A2IuiJ2
for all u E V1"2(St),
i=1
where ui = fn u wi dx.
Then, a possible attempt to define P(-Lu) is to consider the limit (if it exists) of P(-A (EN, uiwi)) as N goes to +oo or in other words the limit of EN1)iuiwi since P(-Owi) = Aiwi. This is not possible: indeed, arguing by contradiction, if EN 1 aiuiwi converges as N goes to +co in some space, say H-'(!Q) (we could also assume that it stays bounded in H-1(SZ) and with a little more work we would reach a similar contradiction) for all u E V1'2(SZ), then EN1 Aiuiwi converges to T(u) where T is a linear
mapping continuous from V"2(f) into H-1(0). We claim that T(u) _ P(-Au) if u E H2 (SZ) n V1,2 (0): if this claim is proven, we conclude easily in view of the fact shown above. Then, if u E H2(D) n V1>2(SZ), Du E L2(SZ)
Navier-Stokes equations
86 and thus
N
N
N
wi J u) wi dx _
a;uiwi = i=i
r
(-Au)wi dx
i=1
i=1
_
r
wi
wiJ P(-Au)wi dx -- P(-Du) i=1
n
in L2(Q) as N goes to +oo. The specific difficulties encountered in the case of Dirichlet boundary conditions are intimately related to the simple observation already mentioned above that there exist non-trivial (non-constant) harmonic functions in L2! Indeed, there exists h E L2, h is harmonic in Sl: hence, Vh E H-1(1) and div (Vh) = 0. In the periodic case we immediately see that h is con-
stant and thus V h = 0. If c = IRN, T E H-1, curl T = 0, divT = 0 in 1Y(IRN) then we also obtain T - 0.
5) The energy inequality (3.8) shows that u(t) converges to uo in L2(Q) as t goes to 0+. We now sketch the
Proof of Theorems 3.1-3.4. First of all, the existence of weak solutions is a particular case of Theorem 2.1 taking po - 1 and thus, by (2.17), p = 1. Notice that Theorem 2.1 also yields the energy inequalities (3.9) and (3.8) for almost all t > 0. The fact that (3.8) in fact holds for all t > 0 is then a simple consequence of the continuity in time (with values in LL,) of u. Let us remark that the fact that u E C([0, T]; L,2,) is a consequence of Theorem 2.3 since P(u) = u. Also the continuity in t with values in L oC for p < 2 (or LP in the periodic case or in the case of Dirichlet boundary conditions) follows upon decomposing u into ui + U2 where ui solves:
- vLui +Vpi = f, divui = 0, ui E C([0,T];L2),
uo
and where u2 solves: at - vlu2 + Vp2 = - (u V)u, div u2 = 0, u2I t=o = 0, U2 E W2,1, 4' (f x (0, T)) (see V.A. Solonnikov [444],[445] and section 3.3 for such estimates), therefore u2 E C ([0, T]; W ' + (11)). It only remains to explain the additional regularity information on , Vp (step
1), the uniqueness statements if N = 2 (step 2), and the local energy inequality (3.10) (step 3).
There is however one more point to clarify: what we claimed above about the applications of Theorem 2.1 is not entirely correct since we need to assume that f E L2 (SZ x (0, T)) in order to apply Theorem 2.1 while the results above only require that f E L2(0, T; H-1(f1)). The reason why we neglect this technical point is the following: when p is constant, say p - 1, then all the a priori estimates and passages to the
A brief review of known results
87
limit are valid if we only assume that f E L2(0, T; H-1(1)) and thus the proofs already given easily adapt to that case. Another way to argue is to approximate f in L2(0,T; H-1(0)) by fn E L2(f2 x (0,T)). We then apply Theorems 2.1 and 2.3 and obtain weak solutions un, which as we shall prove below satisfy the properties listed in Theorems 3.1-3.4. Finally, we recover the desired results passing to the limit as n goes to +00.
Step 1. Regularity information on ae , Vp. In the periodic case or if f2 = IRN, we simply use (3.13) or (3.14) (which are easily deduced from the definition of weak solutions). If N = 2, we recall that u E L4() x (0, T)) and thus P(u,ui) (d i, j) E L2 (n x (0, T)), hence ac E L2(O, T; H-1). Of course,
this yields the regularity statements made upon p and the continuity of u in time with values in L2. This also allows us to justify (3.7). If N > 3, we remark that u E L°O (0, T; L2) n L2 (0, T; L) (by Sobolev embeddings) while Vu E L2 (1 x (0, T)). Therefore, u Vu E L9 (0, T; L') for 1 < q < 2, )for 1 < s < oo since while u Vu= (d i) E L(0, T; W-1 ° r=
u Vui = div (uui). The regularity for at then follows from (3.13) and (3.14).
The regularity of p stated in Theorem 3.2 is deduced from (3.1) in the following way: we take the divergence of (3.1) and we find
-/p = ai(ujajui) - divf = aiuj ajui - divf = ai.7 (uiu2) - div f
in IRN
(3.15)
and in the periodic case p is periodic-p can be normalized by requesting that fQ pdx = 0 where Q is the periodic cube or Q = B1 if SZ = IRN. The regularity of p then follows from elliptic regularity and from the bounds
on uiuj, (u - V)ui (1 < i,j < N) obtained above. Of course, we could also obtain the regularity of Vp from equation (3.1) in view of the bounds shown above on (u . V)u and at . In the case of Dirichlet boundary conditions, the argument for ae is the same except that, for reasons detailed above, we have to replace W-' by V-1,p.
Step 2. Uniqueness if N = 2. We only need to observe that if u, v C LOO (O, T; L2) n L'(0, T; H1) and thus u, v E L4(f2 x (0, T)) and div v = 0 a.e. in 11 x (0, T), we have for all t E [0, TJ
Navier-Stokes equations
88 rt
r
ds [(u V)u - (v V)v] (u-v)
dx fn
Io rt
r
=
=
Jn
dx
io
ds [(u-v) Vu + (v V)(u-v)] (u-v) t
f dxf ds [(u-v) Vu] (u-7)) > -CO
f
f
t
IIVUIIL2 IIu-vIIi4 ds
t
IIVu1IL2 1iu-vIIL2
IIV(u-v)IIL2 dS
0
for some Co > 0 independent of u, v. Then, if u, v are solutions of (3.1)(3.2) as in Theorem 3.1 or 3.3, we deduce easily from the above inequality (and the regularity of u, v, a , at) for all t > 0 2 I1u-vIIL2 (t) + v
< CO
f
f
t
IIV(u-v)IIL2 ds
0
IIVuIIL2 IIv-vIIL2 IIV(u-v)IIL2 ds
t
<
2
j
2
II(V(u-v)IIi2 ds +
v
ft IIVUIIL2 IIu-vIIL2 ds,
and the uniqueness follows from Gronwall's inequality.
Step 3. The local energy inequality (3.10). In order to show (3.10), we go back to the construction of weak solutions performed in section 2.4
in our special case, namely p = 1. In other words, we consider uE E C°O(IRN x [0, T]) (vanishing at infinity if S = IRN, periodic in the periodic
case) as a solution of 8
, + (u, * wg) - VU, - VAU, + VPe = fE
div u, = 0
in
in IRN x [0, T],
(3.16)
IRN x [0, T] UE I t=0 = UO * WE
in
IRN
(3.17)
where fE. E C°O(IRN x (0, T)) converges to f in L2(0,T; H-1), fE van-
ishes near t = 0 (V x), ff is periodic in the periodic case and fE E Co (IRN x (0, T)) if S = IRN. Let us emphasize that this is essentially the original approximation of (3.1) introduced in J. Leray [283]. We also know that u. converges weakly-extracting subsequences if necessary in L2 (0, T; H')-to a weak solution u satisfying the conditions listed in Theorem 3.2. In addition, uE converges to u in Ls(0, T; L9(BR)) for 2 < s < oo, q < N3 4 and for all R E (0, oo). In particular, converges to Iui2 in T2, L'(0, T; L4(BR)) for 1 < r < oo, 1 < q < N R E (0, oc).
A brief review of known results
89
If N = 3, we deduce that luel2 converges to 1u12 in LI(O,T;Lq(BR)) for q < 2 while uE converges to u in L3(0, T; Lq (BR)) for q < 5 (V R E (0, oo)). Hence, (uE*C. a)Iuel2 converges to ulul2 in L1(BR x (0,T)) for all R E (0,00). In addition, the bounds obtained in step 1 on as and p are easily shown to hold for (3.16) and are uniform in E E (0, 1). In particular, pE converges
weakly to p in L2 (0, T; Lq (BR)) for q < 2 while ue converges to u in L2(0,T; Lq(BR)) for q < 6. Next, we multiply (3.16) by ue and we obtain on IRN x [0, T] 2
8 t-
2
+vIVUeI2 = uefe (1Iuel2)+div((uE*w,){IuEi2+pe})-v0Iu£ 2
Without loss of generality, we may assume that I Due l2 converges weakly (in the sense of measures) to a bounded non-negative measure D on IRN x [0, TI. By standard functional analysis considerations, we deduce that D > VIVU12. This fact, together with the convergences established above, allows us to pass to the limit as c goes to 0 in the above equality to recover the inequality (3.10), thus concluding our proof.
We conclude this section with an observation on the regularity of solu-
tions of Navier-Stokes equations if N > 3: we postulate the existence of weak solutions u of (3.1)-(3.2) with the properties listed in the above existence results and such that u E C([0,T]; LN(ft)). The result that follows shows that if f is smooth then u is smooth for t > 0. More precisely, we have the following classical result
Theorem 3.5. Let N > 3, let f E L2 (Q x (0, T)) r1L'(fl x (0, T)) for some r E [2, N). Let u be a weak solution of (3.1)-(3.2) as given by Theorem 3.2 or 3.4. We assume that u E C([O,T]; LN(S2)). Then, for each e > 0, u E Lq(e,T;W2,q(Sl)), p E Lq(e,T;W1,q(SZ)) and t E L"(e,T;Lq(fl)) for
2
Remarks 3.2. 1) If uo is smooth enough (in the appropriate Besov space), the argument below shows that we can take e = 0 in the preceding result. 2) If f is smooth, then it is straightforward to deduce from the above result the fact that u is smooth for t > 0. 3) The proof presented below can be adapted to the case when u is assumed to satisfy u E L(0, T; LO(O)) where 2 < a < oc, Q = 2 (or even more generally L«1(0, T; L,01 (0)) + L12 (0, T; L132 (S2)) + C([0, T]; LN(SZ))
where 2 < al, a2 < oo, ,3z = N'`, i = 1, 2). The above result remains valid.
4) It is possible to extend slightly the above result by requiring u to satisfy the following property: for all E > 0, u = u1 +u2 where Hu1 ILL-(o,T;LN,x (f2)) < E, U2 E L' (D x (0, T)).
Navier-Stokes equations
90
Proof of Theorem 3.5. We first observe that for all e > 0 there exist (ui, u2) such that u = U1 +u2,
U2 E L' (fl x (0,T)),
1Iu1IIL°D(0,T;LN(0)) < E,
(3.18)
and we may even assume u2 to be smooth on S2 x [0, T] (periodic in the periodic case or in Co (SZ x (0, T)) in the other cases). We next wish to make a few remarks on the following linear equations
-jT+U.VV-VAV+VP = 9, div v = 0 in St x (0, T),
3.19
vlt=o = 0
in St
with the same boundary conditions for v as for u. We first claim that if g E L'' (St x (0, T)) n L2 (St x (0, T)) for some r E [2, N) then there exists a unique solution v E Lq (0, T; W 2'q (11) ),
i E Lq (0, T; Lq (11)), Vp E LIT (0, T; Lq (St) )
of (3.19) for all q E [2, r]. The existence (and uniqueness) follows from the a priori estimates we explain now. First of all, we have (multiplying by v) a priori estimates in L2 (0, T; H') n L°° (0, T; L2). Next, we remark that we have by Sobolev embeddings and because of (3.18) flu, ' VvIIL9(nx(0,T)) < CIIVvilLq (o,T,L
(n))
<
CEIID2VIIL9(nx(O,T)),
11u2 ' VVIIL9(nx(O,T)) < CEIIVVIIL9(n)
cEllvlli9(nx(O,T)) IID2vlliq(nx(O,T)) 1-B 9/2
CEllvll
-
(nx(O,T))
IlV ll
1/2
nx(O,T)) IID2vll (nx(O,T)) (L 1-7
8v !EIIVIIL2(nx(O,T)){II
at
IILq(nx(O,T)) + IID2VIIL9(nx(O,T))}
where, above and below, C denotes various positive constants independent of v, c, q in [2, r], CE denotes various positive constants independent of v, q
1]1 1 -= 1q,s= in (2, r], 'y= 2 E (0, 2, 2+ B
N+2 q? , s = +oo if (ifq< 2
N+2-2q
q > 2 , s arbitrary in (2, oo) if q = 2 ). Then, we use Lq (S2 x (0, T)) estimates for linear (Stokes) equations due
to V.A. Solonnikov [444],[445] (which are in fact valid in all dimensions) and we deduce
IT IIL9(nx(O,T)) + IIDPIIL9(0x(0,T))
IIVIIL9(O,T;W2.9(n)) + (I
C7t
< CEIID2VIIL9(nx(O,T)) + CEIIVIIL2(nx(O,T)) 1--y
11
(3t
11
L9(nx(O,T))
+ IID2VIIL9(nx(0,T))
+ CII9IIL9(nx(0,T))
A brief review of known results
91
Since we already have a priori estimates on v in L2 (0 x (0, T)), the desired a priori estimates are shown. The next step consists in showing that there exists a unique weak solution (E L2 (0, T; Hl) n LOO (0, T; L2)) of (3.19) or in other words that if q = 0
then v - 0. To this end, we first observe that u Vv E L2 (0, T; L M ) and thus ac E L2(0, T; H-1) (or L2(0, T; V-1,2) in the case of Dirichlet boundary conditions). Therefore, v E C([0, T]; L2) and we multiply (3.19) by v to obtain for all t E [0, TJ
f
tn
1Iv(t)I2dx+v r f lVv12dxds 2
o
t
-1.t n
on
the last computations being easy to justify since u E C([0, T]; LN), u . Vv E L2 (0, T; L ), v E L2 (0, T; L) (argue by density on v for instance). We may now complete the proof of Theorem 3.5 by observing that vi =
to (pi = tp) solves (3.19) with g = tf +u E L"(11 x (0,T))nL2(12 x (0, T)) with ri = min(p, 4) hence vi E Lq(0, T; W2,q(SZ)), Vp1 E Lq(SZ x (0, T)) for 2 < q < r1. If p < 4, we conclude, while if p > 4 (hence N > 5), we observe
4)
that v2 = t2u solves (3.19) with g = t2f + 2vi E L''2(SZ x (0,T)) n L2 (!n x (,0 T)) while r2 = min (p, if N > 7, r2 = p if N = 5 or 6 (since
ui E L4 1 (P x (0, T)) by the regularity just established). If p <- 4 N we conclude.
If p > 4 N (hence N > 11), we consider v3 = t3u and
reiterating the preceding argument, we prove Theorem 3.5.
0
The same type of technique can be used to prove the regularity of weak solutions of stationary Navier-Stokes equations if N = 4 (such results are classical if N = 2 or 3). More precisely, we consider stationary weak solutions of Navier-Stokes equations, in the case of Dirichlet boundary conditions to fix ideas, namely solutions of
- vAu + (u . V)u + Vp = f uEHo(SZ),
divu=0
in 0.
in
S2,
(3.20)
Then, if f E H-1(S2), there exists at least one solution u of (3.20) (see for instance R. Tern am [472]). We claim that if N = 4, any such solution belongs to W24q(f) if f E Lq(1) and q E [N+2, N), and in addition Vp E Lq(SZ). In particular, if this claim is shown, then f E L°°(S2) implies u E W2'q(11) for q < N and thus u E Ca(f)) for all a E (0,1). By standard regularity results, we then deduce u _E W2,q (Sl) for all q < oo, and if f E Co (S2) (a E (0, 1)) then u E C2'a(? ).
Navier-Stokes equations
92
In order to prove the above claim, we argue as in the proof of Theorem 3.5 and we remark that, by Sobolev embeddings, u E L4(12) (recall that N = 4) hence for all E > 0 U = u1 + U2,
IIUI IIL4(f2) < E.
U2 E L°°(c1),
Next, for any g E Lr(SZ) n L414 (f2), there exists a unique solution v of g V E Ho (S2),
in
11,
div v = 0 in S2
(3.21)
and V E W2(f) for q E_[_N+ , r], where r E [ N+ , oo). _
The proof of this claim relies upon the above decomposition of u and follows the same lines as the corresponding argument in the proof of Theorem
3.5. In particular, taking g = f, r = q, u has to be this unique solution and thus has the claimed regularity.
3.2 Refined regularity of weak solutions via Hardy spaces In this section, we review some results due to R. Coifman, P.L. Lions, Y. Meyer and S. Semmes [95] which concern some (marginal) improvements of the known regularity of weak solutions. They rely upon multi-dimensional Hardy spaces and they are valid in the periodic case or in the case when 12 = IRN. We shall discuss corresponding results in the case of Dirichlet boundary conditions in the next section. In order to simplify the presentation, we only consider the case St = RN since the adaptations to the periodic case are straightforward. We first recall the definition and some of the main properties of Hardy spaces introduced by E. Stein and G. Weiss [455] (for more facts on these spaces see C. Fefferman and E. Stein [149], R. Coifman and G. Weiss [961). The Hardy space, denoted by 7-i'(IRN) to avoid confusion with Sobolev spaces, is a closed subspace of LI (IRN) defined by x1(IRN)
_ {f
E
L1(IRN) / suplht * fI E L1(IRN)} t>o
(3.22)
where ht = . h(t ), h E Co (IRN), h > 0, Supp h C B(0,1); in fact, it can be shown that this space is independent of the choice of h. Also, 7-0 can be characterized in terms of Riesz transforms R3 as ?-L1(IRN)
= If E
L1(IRN) / V 1 < j:5 N, R? f E L1(IRN)}.
(3.23)
Refined regularity of weak solutions via Hardy spaces
93
In addition, we have (3.24) Rj is bounded from H1 into H1, provided we equip x1 with a norm taken to be, for instance, II suet>o I ht * V is a separable Banach space whose dual is BMO(IRN) fIIILI(IRN). and which is the dual of VMO(IRN)-the "completion of Co (IRN) for the BMO norm" (supQ fQ Jb- fQ bi where the supremum is taken over all cubes of IR.N).
In [95] it was shown that Hardy spaces can be used to analyse the regularity of the various nonlinear quantities identified by the compensated compactness theory due to L. Tartar [468], [469] and F. Murat [349], [350], [351] . In particular, it was shown that E - B E ?{1(IRN) if E, B E L2(IRN), curl E = div B = 0 in D'(IRN) and we have for some C > 0 independent of E, B IIE . BII-H1(jFtN) < CIIEIIL2(jRN) IIBIIL2(]RN).
(3.25)
If u E H1(IRN)N, div u = 0 a.e. in IRN, we can use this result to deduce II(u - V)ujjI l(]RN) < CIIuIIL2(jRN) IIVuiIlL2(jRN),
foralll
for all 1 < i, k < N.
}
(3.26)
(3.27)
1
We then consider a weak solution u of the Navier-Stokes equations (3.1)-
(3.2) (in the case 12 = IRN) and we recall that we always assume uo E L2(]RN), f E L2(0, T; H-1(IR.N)) (at least) while u satisfies the conditions listed in Theorems 3.1-3.2. Let us also observe that we can assume without loss of generality that the force term f satisfies
div f = 0
in D'(lRN x (0,T)).
(3.28)
Indeed, we can always decompose f = f1 + V-7r where div f, = 0, fl, V7r E
L2(0,T; H-1(IRN)), incorporate 7r to the pressure p and replace f by fl. Notice that for "most" functional spaces X = LP(0,T; H3'q(IRN)) (s E IR, q E (1, oo), p E [1, oo]) if f E X then f, E X. With this normalization of f, we have Theorem 3.6. The following properties hold Oijp E L1(O,T; R1(IRN))
Vp E L1(0, T; p E L1(0,T; L7°
N
(1 < i, j < N),
L=-'1(IRN) )
'1(]RN))
pE L 1 (0, T; Co (IRN))
N
(3.29)
if N > 3,
if N=2,
Navier-Stokes equations
94
(u . 0)u, Vp E L2(0, T; ll(IRN))N, u E L1(0,T; Co(IRN))
fE
(3.30)
if N = 3 and (3.31)
L1(O,T; L''1(1R3)),
If Duo is a bounded measure on IRN,
and if D f is a bounded measure on RN x (0, T)
(3.32)
then Du E L°° (0, T; L1(lRN)).
Remarks 3.3. 1) The case N = 2 of (3.29) for the regularity of p is due to L. Tartar [470]. The regularity shown in (3.31) was first obtained by L. Tartar (unpublished), and C. Foias, C. Guillope and R. Temam [152] by different methods. Finally, (3.32) is a slight improvement of a result originally proven by P. Constantin [99], where it was shown that curl u E L00(0,T;L1(IRN)) if N = 3.
2) If N > 4, (3.31) may be replaced by u E L1(0, T; L7.1 (RN)) with a similar proof.
3) Since p is defined up to a constant, (3.29) really means that we normalize p in such a way that p goes to 0 as jxj goes to +oo (in L7"1(IRN) sense if N > 3). 4) If we go back to the above modification of f (f = f j + Vir; div f, = 0; f1,Vir E L2(O,T; H-1(IRN))), let us observe that (3.32) holds if Dfi is a bounded measure on IRN x [0, T). 5) Theorem 3.6 is stated in [95] but the proofs of (3.31)-(3.32) are only sketched there. This is why we give below a complete proof of this result. O
Proof of Theorem 3.6 Step 1. Proof of (3.29)-(3.30). Recall that p satisfies (3.15) and that f satisfies (3.28). Then, using (3.26),(3.27) and the known regularity of u, we deduce immediately that (u 0)u E L2(0,T;f1(IRN))N, Ap E L1 (0, T; rill (IRN)) and Op = 9ihi in IRN where hi E L2(0, T; ?-l1(IRN))N (1 _< i _< N). Since 9ijp = RtiRR(-Op), &p = -RtRkhk (1 < i,j < N), we deduce from (3.24) that 8t3p E L1(O,T;rf1(IRN)), V p E L2(0, T; 7-L1(IRN))N (1 < i, j < N). Hence, (3.30) is proven while the rest of (3.29) follows from the regularity of D2p and Sobolev embeddings.
Step 2. Proof of (3.31). If N = 3, f E L1(O,T; L1,1(IR.3)), we claim that we have for each i E {1, ... , N} au=
- uAu- E L1(0 T 4
at
,
-,
IL,1(IR3))
(3 33)
Refined regularity of weak solutions via Hardy spaces
95
Indeed, in view of (3.29), we have only to show that (u . V)ui E Ll (0, T; L Y "(R3)), and this follows from Sobolev embeddings since they imply that u E L'(0, T; L6'2(1R3))3 while we have Vui E L2(0, T; L2(IR3))3. Next, we remark that the solution iii of aili 8t
- v0ui = 0 in IR3 x (0,T),
is given by ui(t) = uP *
((4ivt)-3/2
'ui l t=o = u°
in IR3
Hence, IIui(t)IIL-(]R3) <
C- I4vL ).
2nv8-3/4
and we deduce easily that ui E LP (0, T; Co (R3)) for all 1 < p < 4/3. Therefore, (3.31) follows from IIu°IIL2(1R3)
Lemma 3.1. Let N > 3, let g E L1(O, T; L # 1(1RN )) and let v be the "
solution of 8v - vAv = g in IRN x (0, T), 8t
vlt=o = 0
in ]RN.
(3.34)
Then, for almost all t E (0, T), v(t) E Co(IR3) and v E L' (0, T; Co (R3)).
Proof of Lemma 3.1. By density, it is enough to show that we can estimate IIvlIL1(o,T;L0c(1.3)) in terms of IIgIIL1(0,T;L4,1(].N))' Using the den-
sity of functions piecewise constant in t, we see that it is enough to show j(IRN)
such an estimate when g = 1(o,ta) h where to E (0,T), h E L Then, rescaling (t, x) (i.e. considering v (_, o) ), it is enough to consider the case when to = 1 provided we obtain an estimate of v in L1(0, oo; L°°(IRN)). In addition, replacing h by ha, the Schwarz spherical decreasing rearrangement, we increase, for all t > 0, IIv(t)IIL-(i.N) (see, for instance, C. Bandle [20], A. Alvino, P.L. Lions and G. Trombetti [81,[91)
and thus, without loss of generality, we may assume that v and h are non-negative, spherically symmetric and nonincreasing with respect to IxI. Then, f O° II v(t) II L (]RN) dt = fo v(0, t) dt. On the other hand, V (x) = f o' v (x, t) dt solves
-vAV = h in RN or equivalently V = and V E Co(IRN) since h E LIr"(IRN), 1x,1_2 E L V(0) = IIVIIL-(1R.N) s CIIhIIL
and the proof of the lemma is complete.
,1(FtN)
0
N
IxJ1N_2
* h,
'°°(lRN). Hence,
with C = v I
Navier-Stokes equations
96
Step 3. Proof of (3.32). We essentially follow an argument introduced by P. Constantin [98]. Formally, we differentiate (3.1) and we obtain for all l < i, k < N on IRN x (0,T)
at
(akui) + (u - V)(akUi) - v0(akui) = akfi - akaiP - akuj ajui.
In view of the assumptions made upon f , (3.29) and the fact that u E L2 (0, T; H1(IRN)), the right-hand side of the above equation is a bounded measure on IRN x [0, T). Still arguing formally, we deduce
-5t IakUil + (U o)lakUil - Volakuil < m where in is a bounded non-negative measure on IRN x [0, T), and integrating over IRN x [0, t] we obtain a uniform (in t) bound on ]RN I akui (x, t) I dx. It only remains to justify the above argument for any weak solution. To this end, we consider, for h E (0, 1], i, k E {1, ... , N} fixed, vh (x, t)
(ui(x+hek, t) - ui(x, t)). We have obviously on IRN x (0, T) ayh
1
+ u Vvh - LOvh = h (fi(. + at h
(3.35)
Exactly as above, we deduce that the right-hand side, denoted by mh, is bounded in L1(IRN x (0, T)) uniformly in h E (0,1]. Since u E L2 (0, T; H1(IRN)) and vh E L°O(0,T;L2(IRN)) n L2(0,T;L7(IRN)), we deduce from Lemma 2.3 (section 2.3) that we have at
(vh * WE) + u 0(vh * WE) - v0(vh * WE) = mh * WE + rh
(IRN)) for each h > 0 where r" ->, 0 in L2(0, T; L' (IRN)) n L1(0, T; L fixed. Then, we write, recalling the classical convexity inequality (-AI If I <
(-0 f) sign f in D'), 49
ivh*W,I+(U'c)Ivh*W,I-Voivh*wEI
-< Imh*W,I+Irhl,
and we recover, letting e go to 0+,
a I
Ivhl
+ (u V)IvhI - VOIVhI < Imhl
t-o
= wh,
Ivhl E
in RN x (0,T),
L2(0,T;H'(IRN) nL°°(0,T;L2(]RN))
(3.36)
Refined regularity of weak solutions via Hardy spaces
97
u°I E L1(IRN) n L2(IRN) and is bounded in L1(IRN) uniformly in h E (0,1]. Next, we multiply (3.36) by cp (n) where W E Co (IRN), So = 1 on B1, W(x) = 0 if lxi _> 2, 0 < cp < 1 on IRN and we find, integrating over IRN x [0, t) for all t E [0, T], denoting by C various non-negative constants independent of t, n and h,
where wh = h
f
N
ffjf
Ivh(xt)I `) do < C+
N
IVhl (-0`p(n)) dads
t
+n
sup [0,T]
f
Iul lvhl 1
N
f
V`n)
T
I vh i dx
(Ixl
+
n
/ f0
Ivh I dx ds
(n
- IIullL'°(O,T;L2) IIvhIIL-(O,T;L2)
or
T
sup [O,TJ
f(IxI
Ivh I dx < (c+ C(h)) + n
n
ff 0
lvh l dx ds. (3.37)
(n
In particular, we have
<+
sup f(lxl
C
(fTj
1/2
Ivh l2 dx ds n)
N/2n
If N < 4, we deduce that sup[o,T] fIRN Ivh I dx <_ C. If N > 5, we deduce
that sup [O,T)
f
lvhl dx
< C + C(h) n=
IxI
and we insert this bound in the right-hand side of (3.37) to obtain sup [o,T)
f
(Izl
IVhI dx < C n
n2
If N < 7, we obtain the same bound as before. If N > 8, we go back to (3.37).
Nattier-Stokes equations
98
In conclusion, we have shown
sup [p,T]
JIRN
hvhI dx <_ C.
(3.38)
We deduce (3.32) from (3.38) letting h go to 0+: indeed, since u E L2(0,T; H1(1RN))N, vh converges to 8kuj in L2(IRN x (0,T)) as h goes to 0+ and (3.32) follows.
0
3.3 Second derivative estimates In this section we want to present various a priori estimates on weak solutions of Navier-Stokes equations and their second derivatives in x. This will yield similar estimates upon W"'. We first consider the case when 11 =1R.N (and the periodic case) and next we consider the case of Dirichlet boundary conditions which presents specific difficulties already mentioned in section 3.1.
We thus begin with 11 = IRN and we mention without further detail that all the results we state and prove below are also valid mutatis mutandis in the periodic case. Also, as explained in the preceding section, we always normalize the force term in such a way that (3.28) (unless explicitly mentioned) holds. Then we observe that in view of the results shown in the preceding section, we have for any weak solution as built in Theorems 2.1-2.2
(u V)u,
Vp E La(0,T; LQ(IRN),
L2(0,T; 71'(]RN)),
L' (o, T; LlP-r+1(]RN))
}
(3.39)
whenever 1 < a < 2, 7 = 1 --ya . In particular, we can choose a =,3 = N+1, i.e. 4 if N = 3. Next, any weak solution u is the sum of ul and u2 where u1i u2 are respectively solutions of 8u1
aU2 j -t-
- iAu = 0 zLu2 = g
in IRN x (0 T) u I
= If
in 1RN
(3 401
in IRN x (0, T), u2It_o = 0
in IRN
(3.41)
where g = f - (u V)u - Vp. Of course, ul is smooth for t > 0 and its global regularity on IRN x [0, T] depends on the properties of uo, and, in view of (3.39), g E La(0,T; LA(IRN)) if f E La(0,T; LQ(lRN)). Notice that the reduction to (3.28) leaves invariant this property together with
Second derivative estimates
99
the L2 (0, T; 7-fl (IRN)) or the L1 (0, T; L W 1(IRN )) regularity. Therefore,
by classical LP (Lq) estimates on heat equations (see also Appendix D), ate ,
D2Xu2 E L°(0, T; La(JRN))
if
f E La(0,T; LO(JRN))
(3.42)
with 1 < a < 2, p = 1 - -0 . In conclusion, we find ,
DZU E L(0,
La(jpN)) if f E L°`(0,T; LO(IRN)),
1
-1- Na
(3 . 43)
if uo belongs to an appropriate (Besov) space and in particular to
W''A(IRN) where s = 2 aa1 (we still assume that uo E L2(]RN)), while if uo E LA(IRN) (nL2(IRN)), we obtain for all e > 0
8
,
D2u E La(e,T; DO(IRN)) if f E L(O,T; L,6 (RN)),
1
1
-
= 1- 2-a
(3.44)
Na
Of course, these results really mean that we have a priori estimates in these norms for weak solutions in addition to such regularity information. The borderline cases are more delicate: in particular, we do not know if (3.42) holds with 'LQ(O,T;LQ(IRN)) replaced by L2(0,T;711(IRN))_ this does not seem to be known for the heat equation! As we shall see later on, we can prove that as , D2u E LP(0, T; L1(IRN)) for all p E (1, 2). A similar difficulty occurs with L1 (0, T; LN'1(IRN)) : however, in that case, using the results of Appendix D, we can deduce conclusions similar to (3.43)-(3.44) replacing at , D2,u E L1(0, T; L1(]RN)) by a , D2u E LV'1(IRN) a.e. t E (0, T) (we already know they belong to LP(1RN) for p < N 1) and ai, D 2 u E L1'°O(O,T;LT'1(1RN)) where a E Ll,'(O,T;X) means meas{jIa(t)Ijx > Al < a for all A > 0, for some C E [0, oo).
At this stage, we obtain the conclusions (3.43)-(3.44) and if we insist upon having a = Q we find a = ,Q = -NN++2-1. However, as shown by P. Constantin (98], one can obtain a better integrability on D2u by a different argument: in [98] it was shown that D 2 u E LP(IRN x (0, T)) if p < 4/3, N = 3 (under appropriate conditions on f, uo). We shall show below that Dzu E L3'°°(lRN x (0, T)) (for all N > 2) by a variant of the argument in [89]. Before we even state a precise result, we would like to explain
Navier-Stokes equations
100
formally the origin of this exponent 3. Differentiating (3.1) (and taking f = 0 to simplify) we obtain
j (akuz) + (u - V) akuz - u/ (akut) = -(8ku, a,ut + azkp) and we remark that the right-hand side belongs to L' (IRN x (0, T)) since u E L2(0,T; H1(IRN))N by the definition of a weak solution and by (3.29)we could, as in [98], avoid the use of (3.29) by taking the curl of (3.1). The maximal regularity we can deduce from this fact and the preceding equation is D2(akui) E Ll(IRN x (0,T)). In fact, even if the convection term (u-V) were not creating additional difficulties, this is not correct since the L1 maximal regularity is not true for the heat equation, but let us ignore this borderline problem for the sake of the argument. Then we recall that akui E L2(IRN x (0, T)) and thus, by interpolation, Dx(aku{) E LP(IRN X (0, T)) where = (a) + 1 = 4 and we recover the claim Dzu E L4/3 (IRN x 2 shall 2see below, this formal (and false!) argument can be (0,T)).pAs we almost justified and the only price to be paid is the replacement of L4 by Lf,OO! Let us also observe by the way that "Dzu E L1" is a good guess since it also yields "Dzu E L1(0, T; LT,1(IRN))" (Sobolev's embeddings)
which is the borderline case of (3.43) (and (3.39)) taking a = 0. Also, a-
,
D3u E L1(IRN x (0, T)) implies that Dzu E L'(0, T; L1(IRN)) which
is the other borderline case of (3.43) taking a = 2. Notice however that these two (equally false!) deductions do not use the L2 integrability of Du which seems to be the key to the improvement of the LP(IRN x (0, T)) integrability of Dzu from p = N+1 to p = 3 (or almost). Notice finally that only in the case N = 2 (the nice case), N+1 coincides with 3. We now state precisely this integrability result; as in the preceding section and above, u is any weak solution as built in Theorems 3.1-3.2 and f has been normalized to have zero divergence.
Theorem 3.7. We assume that Duo, D f are bounded measures respectively on IRN and IRN x [0, T]. Then, we have
Dyu E L °°(IRN x (0,T)) and
JJTN 1
<
( 3.45 )
s,k=1
Remarks 3.4. 1) Taking the curl instead of an arbitrary first derivative of (3.2) in the proof below, we obtain (3.45) with ID2uI2 l(IDul
Second derivative estimates
101
2) As we shall see below, the proof of Theorem 3.7 also shows that
AV E M,(IRN), v E L2(IRN) = Dv E L4'°°(IRN) 5T
(3.46)
- 2AV E Mb(IRN x [0,T]), vlt=0 E Mb(RN),
(3.47)
x (0,T)).
v e L2(IRN x (0,T)) = Dv E
Here and below Mb denotes the space of bounded measures (with a norm denoted by II II,,,). Even though we do not know if in Theorem 3.7 L4'°O can be replaced by L413, we can check that such an extension is not true for (3.46) or (3.47). Indeed, we claim that Dv E L4/3 (IRN) is not true in general since, -if-it -were,-we would have estimates of Dv in L4/3 in
-
terms of HAVIIr + IIvuiL2 or 11 ai
2AVIJm + IIv(0)IIm + IIVIIL2, and, if we
choose for (3.46), N = 4, v,,(x) = min (I-
E Co (IRN),
,
((x) = 1 if IxI _< 1, e E (0, 1), we check easily that IIAVEII,n, is bounded while 4/31 log EI + C)3/4, IIvEIIL2 = (IS31 I log El + C)112 and IIVVeIIL4/3 = (IS312
and we reach a contradiction. In the case of (3.47), we choose ve = (27r(t+ 2
E))-N/2exp-2'x' )for tE[0,T),xEIRN,N=2,TE(0,oo),EE(0,1). Obviously, a - 2AV, _= 0, I)v,(0)II,n = 1 and live IIL2(>R2x(0,T)) =
f
T
1/2
(47r)-1(t+E)-2(t+E)dt)
C0
=Clog (1+-)}
1/2
On the other hand, we have T IIDveIIL4(]R2X(0,T))
=
Cf
and we reach a contradiction.
(t+E)-1
dtl
3/4
= C (log (I +
'j'e) )3/4
/J
0
Proof of Theorem 3.7 Step 1. We prove
fT IDakUgi2 1(Iaku:I0 IRNJ0 Sup
for all
(3.48)
1 < i, k < N.
Let us first explain formally the proof of this estimate (directly related to the idea of renormalized solutions for elliptic and parabolic equations-see for instance R.J. DiPerna and P.L. Lions [1281, P.L. Lions and F. Murat
Navier-Stokes equations
102
[308] and Appendix E). First of all, we differentiate (3.1) and we find for
all 1
- akip - akuj ajui
(3.49)
In view of Theorem 3.6, of the properties of weak solutions and of the assumption made upon f, we see that the right-hand side is a bounded measure on IRN x (0, T). Then, we multiply (3.49) by TR(akui) where
TR(z) = z if IzI < R, = R if z > R, = -R if z < -R, and we find, integrating by parts over IRN,
T JlR N SR(akui) dx + Jfft N (U 0) SR(akui) dx +v
f
N
Ioakuil2 1(Iaku,l
lFt
since ITR(z)I < R for all z E IR, where SR(z) = 2 if IzI < R, RIzj - 22 if I z I > R. Hence, T
II
7S
J
N
IVakui121(1a,,u:1
N
1,
SR(akui(0)) dx < CR,
-
since 0 < SR(z) < RIzi for all z E IR. In order to justify this computation, we argue as in the proof of Theorem 3.6 and we find, defining vh(x, t) = (ui(x+hek, t) - ui(x, t)) (i, k fixed in h {1, ... , N}, h E (0,1)), avh
at
+ u Vvh - VAVh = gh
vhIt_o = V°h
bounded in L1(1RN x (0,T)),
bounded in L1(IRN)
}
We then multiply by TR(Vh) justifying the computations exactly as in the proof of Theorem 3.6 and we obtain finally T
f 0
T
jRN
IVTR(vh)I2dxdt
= fO jR N
IVvh121(Iv,,l
for some C > 0 independent of R E (0, oo) and h E (0, 1). The only new point to check is the fact that we have for each h, R
f 0
Tu
IuI ISR(vh)I 1(n
as
n
+oo,
Second derivative estimates
103
and this is immediate since u, vh E L2(IRN x (0, T)).
Once (3.50) is established, we conclude easily upon letting h go to 0. Indeed, we then deduce that TR(akui) E L2(0, T; H1(IRN)) and
fT 0 o
f
I DTR(akui)I2 dxdt < CR,
>ftN
R E(0,oo), 1 < i,k < N..
for all
Since, as shown above, D2u E Ll C(IRN x (0, T)) for some q > 1 (observe indeed that since f E Mb (RN x (0,T)) and f E L2(0,T;H-1(IRN)), f E L"(O,T; L4/3(IRN)) for all p < 4/3), we conclude since VTR(akui) _ Dakui 1(Iaku,I
Step 2. We prove that D2 u E L1700(IRN x (0, T)). For each (i, k) E fl, ... , N12 fixed, we set v = ak ui and we recall that v E L2 ((IR N x (0, T) ) while, by step 1, SLIP
R
0
I
d.
RN
J
dt OvI21(Ivl
We then decompose Vv as follows for each R E (0, oo) IovI = IDvI 1(IvIR), II
v V 1(IvI
(3.51)
< C R 1/2 La(RN x(o,T))
Next, we wish to estimate IVvI 1(IvI>R). In order to do so, we write
fT I
IN
Vv11(IvI>R) dxdt
fT
00
I
j=0
VvI 1(R2i
IR.N
00
E C(R2' )1/2 meas {(x, t) E IRNx (0, T) / R2j < I v (x, t) I < R2j+1
11/2
< C (R2i)2 meas{(x, t) E IRNx (0, T) / j=o 00
1/2
R2' < Iv(x, t) I
1/2
(E(R2j)-'
j=o
<
CR-1/2
r
1/2
rT
JJ
1RN 0
1(R
<
CR-1/2.
Navier-Stokes equations
104
This estimate, combined with (3.51), yields Iovl = d1 + d2,
IIdlIIL2(u.Nx(o,T)) !5 CR1'2,
IId2IIL1(IRNx(o,T)) < CR-1/2
This is enough to complete the proof of (3.45) and of Theorem 3.7. Indeed, we have for all A > 0
meas {IVvl > A} < meas {di > 2 } + meas {d2 > 2 }
< C(R +
R11 2A)
=
Ca-4/3
if we choose R =
,\2/3.
C]
The estimate (3.45) and the argument used in step 2 above can be used to derive some regularity information on at and D2u in L"(O,T; L1(IRN)) for all p E [1, 2) which correspond, roughly speaking, to the regularity of
(u V)u and Vp contained in (3.39), namely (u V)u, Vp E L2(0, T; More precisely, we have
Theorem 3.8. We assume that Duo, D f are bounded measures on IRN, IRN x (0,T) respectively and that f E LP(0,T;L1(IRN)) for all p E [1,2). Then, we have
D2 u, 5 E Lp(0,T; L1(IRN))
for all p E [1, 2).
(3.52)
Remark 3.5. If we do not normalize f (to have zero divergence), we need to assume that V (- 0) -1 div f is a bounded measure on IRN x [0, T) in addition to the assumptions made in the above theorem.
Proof of Theorem 3.8. We use the estimate shown in step 1 of the proof of Theorem 3.7 and the fact (shown in Theorem 3.6) that Du E LOO (0, T; L i (IRN)) while, of course, Du E L2 (IRN x (0,T)). Therefore, Du E Lq (0, T; L T (IRN)) for 2 < q < oo. Next, we write with the notation of the proof of step 2 of Theorem 3.6
I
IRN
IVvl dx = O L IVvI 7=o
p
dx + J
(ivl
IVvI dx
Second derivative estimates
105
where
fi(t) =
lVv121(2i
L(t)
-
1/2
(fZRN IVvI21(1,1<1) dx
aj (t) = meas {x E IRN / 2j < lv(x, t) I < 2j+' j
for all j>0, a.e.tE (0,T). In view of (3.45), we have for some C > 0 independent of j, t II fj I1 L2(O,T) < C22,
(3.53)
II.f Il L2 (o,T) < C.
On the other hand, we recalled above that v E L4 (0, T; L 2 < q < oo. Hence, choosing q < oo (q > 2), we deduce aj 22
T
(]RN)) for
E LQ-1(0,T).
(3.54)
j=o
Using (3.53) and (3.54), we may then write 00
f NIVvIdx <
f+(00f2-i
) 1/2
\ -o ._
and f E L2(0, T), (Ej=0 aj 2i
)1/2
00
_ j-o
)1/2 E L2(q-1) (0, T), while
f2
2-i)1/2 E L2(0,T) since we obviously have, because of (3.53), 00
dt < C E 2j j-o
2-2
-
00
= C E2-7'-' = j-o
C(1
-
9
(0, T; L1(IRN)). Since We have thus shown that Vv, and thus D2u, E L2 q is arbitrary in (2, oo), we have proven Theorem 3.8 for D2u. The claim for ai then follows from equation (3.1) and (3.30). 0
Remark 3.6. The method introduced above can be used (and modified slightly) to build global weak solutions for two-dimensional Navier-Stokes equations (N = 2, S2 = IRN for example or the periodic case) when the initial data uo satisfies: uo E L2'°°(I ,2), wo = curluo E Mb(IR2). Indeed, one
Navier-Stokes equations
106
only needs to obtain appropriate a priori estimates and the existence follows upon regularizing uO and passing to the limit. These a priori estimates can be obtained as follows: first of all, w = curl u should solve
5 + u Vw - vEw = 0
in IR2 x (0, oo),
w1t=0 = wo
in IR2
(3.55)
(we take f = 0 to simplify the presentation). Indeed, if N = 2, curl ((u __ 0)u) = (u V) curl u since div u = 0. Next, using symmetrization results due to A. Alvino, P.L. Lions and G. Trombetti [8],[9], we deduce that for each t > 0, w(t) is "dominated" by U(t) = wa * e 4wvt (4irvt)-1 where wa is the (Schwarz) spherically symmetric decreasing rearrangement of wo and
domination means in particular that all LP and Orlicz norms of w(t) are less than i(t). Therefore, w E L2,oo(0, oo; L2(IR2)) fl L' (0, oo; L1(]R2)) and 1lw(t)IIL-(R2) _< 4n t tIwolIM6. In particular, we deduce that u E L°°(0, oo; L2'°'(IR2)). In addition, (3.55) yields as in the proof of Theorem 3.7
J
00 0
dt J dx IVwI2 1(I,
v
I
mOllmv
for all R > 0.
(3.56)
Next, we estimate, in a manner similar to the proof of Theorem 3.7, for all
A>0,R>0
meal {l Dwi > A, (wj > R} 00
_ 1: meas {iDwI > A, 2jR < jwl < 23+1R} j=o °o
< Emeas {IDwj > A,
jwj < 22+1R}1/2 meas {Iwl > 2aR}1/2
2=0 00
< CE
(R2?+1)1/2 A
(R2j)-1 <
CR-1/2A-1
j=0
in view of (3.56) and the fact that w E L2,O0(IR2 x (0, oo)). Therefore, we deduce for all A > 0 meas { l Dw i > A}
= meal { I DwI > A, jwj < R} + meas { I DwI > A, iwl > R}
f
R + 2
1
AR1/2
=C
A-4/3
choosing R =
A1/2.
Hence, Vw E L1,00(IR2 x (0, oo)) and thus D2u E L3'°°(IR2 x (0, oo)). Even if more estimates can be derived for u, Du, D2u, at , p, the estimates listed
Second derivative estimates
107
above are already enough to pass to the limit in regularized problems and build a solution.
We now conclude this section by considering in more detail the case of Dirichlet boundary conditions. First of all, we take a E (1,2) and set 1 - N' . If we assume that f E La(O,T; LA(S1))--and we always normalize it to satisfy div f = _0 in D'(SZ x (0, T))-then by LP-regularity results for linear Stokes equations (see V.A. Solonnikov [444],[445] for instance) and extensions due to Y. Giga and H. Sohr [183] about Li (Ly) regularity, we deduce immediately
j-r- Dyu, Vp E L1(E,T;LQ(SZ))
(3.57)
for all c > 0 and we can take e = 0 if uo E W2(1- * ) ,p(11). We next decompose p as follows: p = po + P1 where Po, P1 satisfy
- Apo = 0
in SIP
In
-Opt =aju;0jut
in
po dx = 0 S1,
Pl = 0
a.e. t E (0, T),
(3.58)
on OS1.
Observe that since (u V)u E L01 (0,T; LQ(S1)) (and aju., 9jui = div ((u V)u)), Vp1 E La(0,T; LQ(S2)) and thus Vpo E La(E,T; La(S2)). From now on, in order to simplify the presentation we take e = 0. Therefore, we deduce from (3.58) and using the arguments developed in the preceding section
Vp1 E La(0,T; LQ(f )),
Vpo E La(0,T; L13(S1)),
(3.59)
D2P1 E L1(0, 7; xl (n)), Vp1 E L1 (0,T;LT71(c)) n L2(O, T;1-fl (0)),
P1 EL
1
N (O,T;L1
1
I
(S1)),
(u V)u E L2(0,T; N'(52)) n L' (0, T; L
(3.60)
(IRN)),
(if N = 2, we replaceL7'1(S1) by C(S2))
P0 E La(0,T; Hoc(S2))
for all
k > 0.
(3.61)
Here and everywhere below, whenever we write LP(Xtoc(f2)) for some func-
tion space X we mean LP(X(K)) for any relatively compact subdomain K of SZ such that K C S2.
Navier-Stokes equations
108
Next, we claim that if D f, Duo are bounded measures respectively on fl x [0, T), 12, then we have
Du E L°°(0,T;L1 ,:(S2)),
D2u E L'(O,T;Li°°(1)) sup
1
(3.62)
for all p < 2,
(3.63)
N
T
dt J dx E IV8kuiI2 1(1akUij
R>O RJ0
K
i,k=1
for all compact sets K C S2,
}
D2u E Lf°°°O1T;L °(S2))
(3.64)
(3.65)
Let us recall before explaining the proof of all these estimates that (3.62) is the analogue of the estimate (3.32) (Theorem 3.6), while (3.64)-(3.65) correspond to Theorem 3.7. Finally, (3.63) is the analogue of (3.52); we do not know if a similar estimate holds for 'Ft. If we go back to the proofs
made in the case when 0 = IRN, we see that we considered in each case vh = (ui(x+hek, t) - ui(x, t)) for h E (0, ho) (where x E 1 h0 = {x E h (x, all) > ho}, t E [0, T)) which satisfies S2, dist
avh
at
+ (u V)vh - VAvh = mh
' L1(O,T; Li C(S2))
(3.66)
aat I + (u' V)Ivhl - vAIvhl < Imhl.
(3.67)
mh is bounded in
and as in the proof of Theorem 3.6
Then, if we fix a compact set K C S2, we choose cp E Co (11) and ho in such a way that 0 < cp < 1 in 0, co - 1 on K, Supp cp C S2ho. We may next multiply (3.67) by cp and we find
f
n
T
IvhlWdx(t)
and (3.62) follows.
T11u12+IVul2dxdt p
Second derivative estimates
109
Similarly, (3.64) follows upon multiplying (3.66) by cpTR(Vh). We then obtain as in the proof of Theorem 3.7 cojVvh121(Iv,,1
< CR +
f
o
dxdt
Tr / VVhTR(Vh) Vcp dx dt n
rT r
< CR + f
o
/ SR(vh)Ocp dx dt + R
in
+ JTJ Iul IVi ISR(vhdxdt
f f lullVWlIvhldxdt o
sz
< CR. Once (3.64) is established, (3.65) follows exactly as in the proof of Theorem 3.7.
(a)), then
Finally, we claim that if f E L1 (0, T; L10 D2 U,
19U
5t
E L1,00 (0, T; Ll7A -1
u E L1(0, T; Cloy (52))
(3.68)
if N = 3.
(3.69)
Indeed, we observe that we have, in view of (3.60), au
- vIu = 9 E
74a-TL111(S2))
L1(0, T;
-
In particular, we deduce
U8 vA u
2vVu VV - uA
and gcp - 2vVu Vco - uLc' E L1(O,T; L
in IRN X 0 T)
'1(IRN)). Therefore, (3.68)
follows from the result shown in Appendix D while (3.69) is a consequence of Lemma 3.1.
Let us summarize the regularity information we have obtained with the
Theorem 3.9. Let f E L2 (0, T; H-1(SZ)) n La (0, T; LO (SZ)) n L1(0, T;
L T"(IRN)) for 1 < a < 2 (with I = 1
- Nom) satisfy: div f = 0
in D'(5l x (0,T)), Df is a bounded measure on n x [0, T). Let uo E tiY2(1-a) Q(SZ)nL2(SZ) for 1 < a < 2 satisfy: divuo = 0 in D'(f ), 0 on 190, Duo is a bounded measure on n.- Then, there exists a weak solution
u (or (u, p)) as in Theorems 3.3-3.4 satisfying in addition (3.57) (with = 0), (3.59)-(3.61), (3.62)-(3.65), (3.68)-(3.69) and the local energy inequality (3.10) if N = 3.
Navier-Stokes equations
110
The only new information is the local inequality (3.10) if N = 3 which was first obtained by L. Caffarelli, R. Kohn and L. Nirenberg [771. Let us emphasize that contrary to all the other information listed in Theorem 3.9, we do not know if any weak solution satisfies (3.10). But we can build at least one that satisfies (3.10). In fact, the existence procedure taken from chapter 2 and recalled in section 3.1 (see (3.16)-(3.17) in the case S2 = IRN) yields such a solution. Indeed, one can check without any difficulty that all the estimates derived above hold uniformly: in particular, Vpe is bounded in L: (Sl x (0, T)) (take a = 4 and thus /3 = 4) and this is enough to derive (3.10). Indeed, if we compare with the proof of Theorem 3.2 (step 3), we have only to explain how to pass to the limit in uepe. The preceding bound
shows that-normalizing pe such that fn pe dx = 0 a.e. t E (0,T)-pE converges weakly to some p in L l (0, T; Li5/7(1l)) while we already know and that ue converges to u (a weak solution) in L5(0,T;L9(S2)) for q <
we conclude since 30 + s = s < 1.
3.4 Temperature and the Rayleigh-Benard equations In this section, we study the Navier-Stokes equations ((3.1)-(3.2)) complemented with an equation for the internal energy e (or equivalently for the temperature T), namely (1.41) written under the assumption that p is constant, namely
+ div (ue) - div (M) = 2 (au3 + a8 ut)2
in 92 x (0, T).
(3.70)
In order to simplify the presentation, we shall consider here only the case when we have (3.71) e = Cog, Co > 0, k E (0, oo).
In that case, (3.70) reduces to 80
+ div (u9) - aDO =
2G,o (asu; + a;u=)2
in SZ x (0, T)
(3.72)
where a = ca E (0, oo). In the case of Dirichlet boundary conditions (for the Navier-Stokes part of the system of equations), various boundary conditions are possible like, for instance, Neumann boundary conditions a9
an
=0
on 60 x (0, T).
(3.73)
Recall that n denotes the unit outward normal to aSZ. In the periodic case, we simply require 0 to be periodic.
Temperature and the Rayleigh-Benard equations
111
At this stage, let us mention that if most of this section is devoted to the model described above (and a variant of it), we also discuss at the end of the section an interesting (both physically and mathematically) variant of the classical homogeneous, incompressible Navier-Stokes equations, namely a model for a homogeneous incompressible flow with internal degrees of freedom taken from S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov (17}.
Let us first observe that, at least formally, that is when solutions are smooth, (3.1) and (3.72) are equivalent to (3.1) and 2
{I
2
2
2
+C0B}+div(u{I 2 +C 0+p}) -vA in 11x(0,T),
I
-kAO
2
(3.74)
which is nothing but the "total energy" equation. Recall also that we have
-Op = 8zuj 8jui = div ((u V)u)
in
fl x (0, T)
(3.75)
at least if we normalize f to satisfy, as we can always do as explained in the preceding sections,
div f = 0
in f2 x (0, T),
(3.76)
an assumption we always make from now on without recalling it. Jul' = 0 on 811 x (0,T) in Observe that we have, at least formally, the case of Dirichlet boundary conditions and thus we deduce from (3.74) Iu(t)I2
in
2
+ Co8(t) dx = fn
luol2
t
r
+ CCOo dx + J dsJ dx f u (3.77)
2
o
sz
denoting by Bo the initial condition for 6, i.e. 61t=o = Bo
in 0.
(3.78)
Of course, in the periodic case, we assume that Bo is given on IRN and periodic, and we always assume from now on that Bo E L1(SZ).
From the above considerations, we see that there are two ways to look at this system of equations (often called the Rayleigh-Benard equations). Either we decouple the two parts and solve first (3.1)-(3.2), then, given a weak solution u of (3.1)-(3.2), we attempt to solve (3.72)-(3.73) (with the initial condition (3.78)). The other possible approach is to build simultaneously (u, T) solving (3.1)-(3.2), (3.74) and (3.78). The reason why these two approaches might not yield the same solutions is the fact that we do
Navier-Stokes equations
112
not know if any (or even some) weak solution of (3.1)-(3.2) satisfies the energy identity
a et
(uu
2
) + div I u
\
) - v0
luI 2
lul2 2
+ VIvul2 = fu
in Sl x (0,T).
If it were the case, then both approaches could be reconciled but this is an open problem that can be solved only if N = 2. Indeed, in the twodimensional case, the regularity and uniqueness of weak solutions allow us to compare the two approaches presented below but we shall not discuss this point further here. We begin with the decoupled approach in which u is a given weak solution of (3.1)-(3.2) as available from Theorems 3.1-3.4. Then, we wish to solve (3.72)-(3.73) and (3.78). Let us write D = - (8=u3 + 8ju;)2; obviously,
D E L' (n x (0, T)). Therefore, from heat equation considerations, we cannot expect a better integrability for 0 than: 0 E L°O(0,T; L1(Sl)) n L 1(0, T; LQ (St)) for all q < N 2 . Hence, uO E Ll if N = 2 or N = 3. This explains why solving (3.72) in the sense of distributions is not adapted to the problem in hand. Instead, we use the notion of renormalized solutions which is more flexible (and also more precise)-see R.J. DiPerna and P.L. Lions [128], P.L. Lions and F. Murat [308]. We recall the definition: we shall say that T is a renormalized solution of (3.72), (3.73) and (3.78) if 0 E C([O,T];L'(f )) n L1(O,T; Lq(S )) for all
q < N2
satisfies
TR(0) E L2(0, T; H1(SZ))
for all R > O and (3.79)
[dx!T
lim
1T1
dt IVTR(0)12 = 0
1
R
ff 0
S
[i3(9)
+
{
f
+ u vW+ dx f3(0o) co(0)
t
- I3"(0) IV0I2+
= fdx/3(0(T))co(T) (3.80)
for all Q E C2 (IR) such that 8' has compact support and for all cp E C°O (. X [0, T]) (periodic in the periodic case, with compact support in IRN x [0, T] if Sl = ]RN). Some explanations are necessary: indeed-see [308] for more
details and the preceding section for a related argument-(3.79) yields the fact that V O E L3(11 x (0, T)) for all s < NN 1. In particular, V9 E L1 and VTR(0) = V01(le1
Temperature and the Rayleigh-Benard equations
113
weak formulation of the following equation and conditions
(
+ u V - aA) ,8(8) + aQ"(8)IV912 = D[3'(8)
/3(9) = 0 T,3(0)=O
on 8SZ x (0,T),
p(e)lt=o = /3(8o)
in SZ x (0, T) in St.
The equation for 3(9) follows formally from (3.72) and the notion recalled above simply consists in requesting that these natural changes of variables are indeed possible. Finally, the condition on TR (8) follows, at least formally, from (3.72) and the integrability of D and Bo since we deduce from (3.72) upon multiplying by TR(8) T
J
1
fTf
IVTR(0)12
dx DTR(8) + jdx SR(0o).
Next, we observe that iR TR(9) is bounded by 1 and converges a.e. to 0 while 4 SR(8o) is bounded by 9o and converges a.e. to 0, and (3.79) follows. Let us finally recall a few facts from [308]: if l uJ8 E L1C and 8 is a renormalized solution of (3.72) then 0 satisfies (3.72) in the sense of distributions. On the other hand, using the fact that u E L2(0, T; H1(&))), it is not difficult to check that if 9 satisfies (3.72) in the sense of distributions and 8 E L2 (St x (0, T)) then 9 is a renormalized solution. With this notion, the following result proved in Appendix E holds.
Theorem 3.10. There exists a unique renormalized solution of (3.72), (3.73) and (3.78).
Remarks 3.7. 1) Recall that u is any weak solution of (3.1)-(3.2) as given by Theorems 3.1-3.4. 2) Using the results of R.J. DiPerna and P.L. Lions [128], we immediately
see that this result also holds if k = 0 (a = 0). Then, we only know that 8 E C([0,T];L1(IR3)).
3) If fl = IRN, we can take Oo to be in L1(IRN) + L°° (IRN) (changing appropriately the spaces to which 9 belongs). 4) One can show that infessXEc 8(x, t) (E [-oc, +oc)) is a nondecreasing function of t. This is a simple consequence of the fact that D > 0 a.e. 5) If u satisfies (3.10) and lute E L C(SZ x (0,T))-these two facts hold if N = 3-then we see that we have 2
9
at
(l 2
2
+ Cob) + div
(ul l 2 f - u + vajuj 0 ui
2
+ Cob + p}) - vA
l2
in V'(S2 x (0, T)),
- kOT
(3.81)
Navier-Stokes equations
114
and in all cases, we only obtain
u(x, t) I2 2
uo
+ Co9(x, t) dx
+CoOodx+
1
2
and
d dt
f
u2 2
f ds H-1xH1 t
2
(3.82)
o
+CoOdx << f,u>H-1XH1
in D'(0,T).
(3.83)
In -particular if f = 0 the total energy is known to be conserved if and only if u satisfies some energy identity instead of an energy inequality, an open problem if N > 3 as we saw in section 3.1. p
We now turn to the second approach where we solve simultaneously (3.1) and (3.74). This will lead to weak solutions (u, 9) for which the total energy is conserved (when f = 0), a fact which is physically expected of course. The price to be paid for this "improvement" is the requirement that N = 2 or N = 3 because of integrability requirements for Jul3 or tub9. The case when N = 2 being straightforward in view of the regularity of weak solutions, we consider only the case when N = 3, and we begin with the case when ft =1R3, the periodic case being completely similar. Theorem 3.11. Let 9o E L1(1R3). Then there exists (u, 9) such that u is a weak solution of the Navier-Stokes equations (3.1)-(3.2) (as in Theorem 3.2) satisfying (3.8)-(3.10), 9 E L°° (0, T; L1(1R3)) n LS'°°(1R3 x (0, T)) n L1(0,T; Lq(1R3)) for all q < 3, supR>o R f jR3dd f"dt I09l211e1
and (u, 9) solves (3.74), (3.78) in a weak form, namely we have for all P E Co (IR3 x [0, T)) JR dxJO dt
{(- 22 +co9) L'P + u (l 22
+CoO+p) . VW
2
+ V lu
2
ff
0w + k90cp + f - ucp + va=u; a;usW }
+J.3dx{
-12
+Co9o}W(x,0)
(3.84)
= 0. 0
In the case of Dirichlet boundary conditions, we have the
Theorem 3.12. Let 9o E L1(fl), let uo E W2/5,5/4(Q) n L2(0) be such that div uo = 0 in 0 and uo n = 0 on t91, let f E L5/4(f x (0, T)) be
Temperature and the Rayleigh-Benard equations
115
such that div f = 0 in D' (Q x (0, T)). Then there exists (u, 0) such that u E L514 (0, T; W2, 4 (St)), , Vp E L5"4 (S2 x (0, T)), u is a weak solution of (3.1)-(3.2) as in Theorem 3.4 satisfying (3.8)-(3.10), 0 E L°°(O, T; L' (S2)) n L1(0,T; Lq(St)) for all q < N 2, SUPR>O fndx fo dt IV012 1(1e1
Remarks 3.8. 1) Remark 3.7 (3) also holds for Theorems 3.11 and 3.12. 2) Remark 3.7 (4) holds in the contexts of Theorems 3.11 and 3.12. 3) We deduce from the above results the following facts d
"
IuJ2
+CoOdx =
ff.udx
(=H-1xH1)
+ div (u0) - a/ 8 > C' (a1uj + 8iu1)2
20
in
D'(11 x (0, T)).
4) From the weak formulation, one deduces easily that 2 + Cog is continuous in t with values into Mb(Si) endowed with the weak * topology (weak topology of measures).
5) In Theorem 3.12, the weak formulation incorporates the Neumann boundary condition (3.73) together with the observation already made above, namely: ea'2 = 0 on &Q x (0, T) since u = 0 on ail x (0, T) (at least formally). 6) The only (new) term in (3.84) whose meaning has to be explained is the term u0. Since u E L°°(0,T;L2) n L2(O,T; Ls) (Sobolev embeddings), u E LV while 0 E Lq for all q < Hence 0u E L1(S2 x (0,T)). p 3.
The proof of Theorem 3.11 being similar (and in fact simpler) to the one of Theorem 3.12, we only present the latter.
Proof of Theorem 3.12. With the notaticn of section 2.4, we consider the solution uE of au-,
at
+uE . VuE-vL uE+Vpe = f E in JL x (O,T),
uEI t=o = uo
in St,
uE E C2 (Si x [0, T] ),
u=0 on all x (0, T), div uE = 0
(3.85)
in Q x (0, T).
We already know, extracting subsequences if necessary, that, as a goes to 0+, uE, uE converge weakly in L'(0, T; Ho (f)) n LOO (0, T; L2 (Sl)) (weak
*) to a weak solution u of (3.1)-(3.2) satisfying (3.8), (3.9). In addition,
f E and uE Vu' (same proof as for u Vu) being bounded in L5/4(Sl x
Navier-Stokes equations
116
(0, T)), we deduce as in the proof of Theorem 3.9 that uE is bounded in L5/4 (0, T; W2' 1 (0)), Oug and Op' are bounded in L 4 (SZ x (0, T)): in x (0,T)) and (3.10) particular, Dyut, ai , Vpe are bounded in Finally, let us recall that uE converges to u as a goes to 0+ in holds. L5/4(.Q
LP(0,T; L2(11)) for all p E [1, oo), and in L2(0,T; Lq(S))) for all 1 < q < 6. Next, we introduce the solution 9E of E
+ uE 09E - a09' = 2Co (azu? + ajuz )2 in fl x (0, T) a9E
an
= 0 on aSZ x (0, T),
9E It=o = 9o
in SZ
where 90 ' E Ca (SZ), 90 6 converges to 9o in L'(11) as e goes to 0+.
Since uE is smooth and (8 u + a;u:)2 E C'(St x (0,T)) (for example), this is nothing but a standard linear parabolic problem and _we know there E C(SZ x (0, T)). exists 9, in, say, C2,1 (?j x (0, T)), i.e. u, Dyu, Dyu, Since (8=u,6 + 83ui)2 is bounded in L1(SZ x (0,T)), we deduce when 0 = IRN in the periodic case from estimates on solutions of heat equations (via, for instance, symmetrization results due to [8],[9]) that BE is bounded in C([O,T]; L' (n)) n L'°°(SZ x (0,T)) n L1(0,T; Lq(SZ)) for all q < 3. In
the case of Dirichlet boundary conditions (SZ # IR ), we deduce from the results shown in Appendix E that 9E is bounded in C([0,T]; L' (Q)) n L 1(O, T; Lq (S2)) for all q < 3. Finally, as in the proof of Theorem 3.10, we deduce that is bounded in L2(0,T; H1(Sl)) uniformly in TR(9E)R-112
R, E. Therefore, without loss of generality we may assume that 9E converges weakly in L° (f x (0, T)) for all a E (1, to some 9 E L°° (0, T; L1(Sl)) n 3) L1(O,T; Lq(SZ)) (V q < 3). Also, as in the proof of Remark 3.6 (see also
P.L. Lions and F. Murat [308]), V9E is bounded in L"(SZ x (0, T)) and thus VO E L"(SZ x (0, T)) for all r < 45-.
Next, we deduce from (3.85) and (3.86) that we have
a
(" 2
- v0
+ Co9E) + div {u,
I,uEi2
2
a
(IuEI2
an
2
(" 2 + Co9E) + UEpe }
- kLOE = f E . uE + v a5u? ajui ) on ac . + Co9E = 0
in S2 x (0, T),
(3.87)
In addition, we know that Lu2l, 9E are bounded in C([0, T]; L1(SZ)) n L1(0, T; Lq(S))) (V
q < 3) and that V, V0E are bounded in L''(SZ x
(0,T)) (V r < ). From these bounds and equation (3.85), observing that 4
Temperature and the Rayleigh-Benard equations
117
aju; a;u: = at((ue V)u;) and using classical compactness theorems, we deduce easily that + Co 9E converges to L2 + Co O in L'' (0, T; L1(f)) n L1(0, T; LQ (S2)) (for all 1 < r < oo, 1 < q < 3). Therefore, 9E converges to 9 in LT (0, T; L2 (D)) fl L1 (O, T; LQ (SZ)) (for all 1 < r < oo, 1 < q < 3). Then,
deducing (3.84) from (3.87) is an easy exercise using these convergences and the bounds collected above.
Remark 3.9. Combining the methods developed in this chapter and those introduced in chapter 2, it is possible to study density-dependent models with temperature such as
a
+div(pu)=0, divu=0
a(Puz)
+div(puui)- 2 ai(it (P,0)(aiuj+a,u:))+Vp=P1
Co a(a O)
+ Co div (pug) - div (k (p, 9)V9) _
(Z' 9) (a: u; +a;ui)
(3.88)
where p, k e C([O, oo) x IR), inf {p(t, s) / Itl < R, s E ]R}, inf {k(t, s) / It) < R, s E IR} > 0 for all R > 0 (for instance). However, we shall not attempt to present here precise results on such a system of equations.
Finally, we conclude this section and this chapter with a model for an incompressible, homogeneous, newtonian fluid taking into account internal degrees of freedom (for more details see S.N. Antontsev, A.V. Kazhikov and
V.M. Monakhov [17]). We only describe the three-dimensional situation with Dirichlet boundary conditions and we look for u(x, t), w(x, t) E 1R3 solutions of
at
+(u . V)u-vLu+Vp = f +(w x u),
divu = O
at In
in Sl x (0, T),
u=0 on aSZ x (0, T
+ div (uw) + F(p)w = m in SZ x (0, T), p dx = 0
(3.89)
in (O, T)
where F is a continuous, non-negative function on IR satisfying
IF(t)l < C(1+Itlc')
on IR., for some n- E [0,
(3.90)
and for some non-negative constant C. Finally, we keep the initial condition (3.2) where uo E L2(f2) satisfies (3.3) and uo n = 0 on 9S , and we add an
Navier-Stokes equations
118
initial condition for w
wlt=o = wo
in
(3.91)
SZ,
and we assume that in E LO°(SZ x (0,T))3, wo E L°°(Sl)3, uo E W',4(SZ) n L2(SZ) (div uo = 0 in SZ, uo n = 0 on all), f E L2(0, T; H-1(SZ))3 n L4 (SZ x (0, T))3. Then, we can prove
Theorem 3.13. There exists a solution (u, p, w) of (3.89) (in the sense of distributions), (3.2) and (3.91) such that u E L2(0, T; Ho (ft)) n C([0, T]; 5
,2
is
'S
LT-4 LW (cl)) n C([0, T]; L' (fl)) n L"S (0, T; u'2'+(cl)), i E p E L4 (0, T; W1, 4 (ft)), w E L°°(SZ x (0,T)), w E C([0, T]; LQ(SZ)) for all
1
f 3(w(x, t)) dx +J ds
n
n
=
Jn
dxF(p)w Vf(w)
Q(w°)dx + J ds o
ln
dx m V,3(w)
for all t E 10, T]. 0
Proof of Theorem 3.13. Following the arguments developed in chapter 2 (the situation being somewhat easier here), we introduce the following approximated system of equations
au, VuE-vOue+Vp' = fE+(we x uE), at +uE div uE = 0 in SZ x (0, T) E
+div (uEwC)+F(pE)wC = mE
in SZ x (0, T),
(3.92)
L with uw = 0 on an x (0,T), uElt=o = uo, w£It=o = wo, where uE, f=, uo have
been defined previously (see chapter 2 in particular) and wo E CO' (c), me E Co (fZ x (0, T)), wo,mE are bounded uniformly ins respectively in L00 (fl), LOO (SZ x (0, T)) and wo, me converge respectively to wo, m a.e. and in LQ(SZ), L4(SZ x (0,T)) for all 1 < q < oo. The existence of smooth (on SZ x [0, T]) solutions (ut, pE, wE) of (3.92) is
an easy adaptation of the argument introduced in section 2.4 and of the
Temperature and the Rayleigh-Benard equations
119
bounds we obtain now. First of all, multiplying the equation satisfied by uE by itself, we find the usual energy identity valid for all t E [0, T]
f
2Iue(x,t)I2dx+
n
f dsJ dxvI VueI2(x,S) o
=
n
(3.93)
1
2
in
which yields a bound (uniform in e) on uE in C([0,TJ; L2(SZ)) n L2(0,T; Ho (f )). Next, using the equation satisfied by uE, we obtain easily for all t E [0, T], for all ,8 E Ci (IR3; IR)
f
/B(w'(x, t)) dx + J / F(pE)w' V,(3(w`) dads 0
=
f()dx+ffmE.V/3(cf)dxds. 3wo
In particular choosing ,13(x) = IxiImxi form > 0, i = 1, 2, 3, we obtain sup sup f IwEl4 dx < oo e>0 tE[O,T[
for all
1 < q < oo
(3.95)
Z
and keeping track of the precise bounds as q - +oo (or applying directly the maximum principle), we deduce sup sup { 1W- 'j / x E Q, t E [0, T] } < oo.
(3.96)
E>o
Then, going back to the equation satisfied by uE, we deduce using the preceding bounds (and Sobolev embeddings) that f' + wE x uE - (uE V)uc is bounded in L 1(0, T; L+ (fi)) uniformly in e. Therefore, u-, as , , pE are bounded uniformly in e respectively in L I (0, T; W2, iz (Q)), L 5 (0, T; L+ (SZ)), L (0,T; W1,i4 (SZ)). From these bounds, we deduce easily that, extracting subsequences if necessary, uE converges to some u E C([0, T]; Lv,(f )) n L2(0, T; Ho (S2)) n C([0, T]; L1(S2)) n L-s(O, T; W2,+ (S2)) and the convergence is a weak convergence in LO° (0, T; L2(S2)) (weak *) nL2 (0, T; Ho (St)) n L 3 (0, T; W2, + (SZ)) and a strong convergence in C([0, T]; LP (S2) )
(V 1 _< p < 2), in Lq(0,T; W11q(f2)) (V 1 < q < 2) and in L'(0, T; Lq(c)) (V 1 < q < 6). Similarly, 3-t , Vpe converge weakly respectively in 5 L (0, T; L (S2)), L (0, T; W1, (S2)) to t and Vp for some p which satisfies: fn p dx = 0 in (0, T). In addition, we may assume that We converges weakly in L°° (SZ x (0, T)) (weak *) and strongly in C([O, T); W-3"P(f2))
Navier-Stokes equations
120
(V s > 0, V 1 < p < oo) to some w E L°°(1 x (0,T) n C([0, T]; Lu,(Sz)) (V q < oo) which satisfies wlt=o = wo on Q. Observe indeed that, because of (3.90), F(pt) is bounded in L" (SZ x (0, T)) where r = as . Finally, we assume without loss of generality that F(pe) converges weakly in L'' (S2 x (0, T)) to
some F > 0. Obviously, we can pass to the limit in the equation satisfied by uE. We also recover the energy inequalities (3.8)-(3.10) mentioned in Remark 3.10 + div (uE from (3.93) and its local variant, namely at + uepe) v,6L "Y + v Due 2 = E . uE. In order to complete the proof of Theorem f 2 3.13, it only remains to pass into the limit in the equation satisfied by wE and to show that w E Q0, T]; Lq(S2)) for all 1 < q < oo. In fact, we are first- going- to -show that wE converges to w in C([0, T]; Lq(SZ)) for all 1 < q < oo and that w satisfies the desired equation with, however, F(p) replaced by F. Then we shall show that F = F(p). The second step is easy: indeed, once we know that wE converges to win, "212
1
I
-
I
say, Lq(S2 x (0, T)) for all 1 < q < oo, then we deduce from the convergences
of tE, Vu', f-' listed above that f-' + wE x nE - (uC V)uC converges to f + w x u - (u V)u in Lg1(0, T; LQ2 (S2)) for all 1 < ql < 3, 1 < Q2 < 14. Hence, using the results of Y. Giga and H. Sohr [183] on Stokes equations, we deduce that uE converges to u in Lq(0,T;W2'q(S2)), at converges to at in Lql (0, T; LQ2 (SZ)) and VpE converges to Vp in Lql (0, T; LQ2 (S2)) for all ql < 3, 1 < q2 < i4 . Since we normalize pE and p to satisfy fn pE dx = fn p dx _ 0 on (0, T), we deduce that pE converges to pin LQ1(0, T; W.2(1)) for all qj < 3, 1 < q2 < 14 and in particular in Lq(S2 x (0,T)) for all 1 < q < 5/3. Since F satisfies (3.90) and F is continuous, we deduce easily that F(jt) converges in L" (S2 x (0, T)) to F(p) for 1 < r' < r = s« . Hence F(p) and we conclude. Finally, the above claim on wE is proven by a convenient adaptation of the method introduced in steps 1-3 of the proof of part 1 of Theorem 2.4 (chap-
ter 2, section 2.3). More precisely, we claim that if F(p6)w6, F(p6)Iw612 converge weakly in L'' (52 x (0, T)) respectively to Fw and Fw2-where w2 is the weak * L°° (S2 x (0, T)) limit of IcE I2-then the convergence of wE to w in C([0, T]; L2 (fl)) and thus in C({0, T]; Lq(S2)) for all 1 < q < oo follows easily. Indeed, if this claim is shown, then w and w2 solve respectively: W, w2 E L°° (SZ x (0, T)) n C([0, T]; Lw (12)) (V 1 < q < oo)
at
+div (uw) + Fw = m in S2 x (0, T), wIt=o = wo
awe at
+ div (u
2
w2 I t=o = Iwo 12
) + 2Fw2 = 2-rn . , , in Q.
in S2
in S2 x (0 T) ,
,
}
(3.97)
(3.98)
Temperature and the Rayleigh-Benard equations
121
In addition, the proof of Theorem 2.4 mentioned above adapts easily to show that IwI2 also solves (3.98) and that w2 = IwI2 (uniqueness of transport equations, recall that div u = 0 and F > 0). Hence, wE converges (strongly) to w in L2(SZ x (0, T)). Finally, the convergence in C([O, TI; L2 (R)) of W-'
also follows from the adaptation of the arguments of section 2.3: indeed, we deduce from (3.97) (and the uniqueness) that w E Q0, TI; L2(SZ)) and from (3.98) that we have for all t E [0, T] 2FI_wl2
fcttIw (x, t)I2 dx +
dx ds =
fn
Iwo(x)I2 dx,
rs
Jn
I wE (x, S) 12 dx +
J Jn 2F(pC) IwC 12 dx do = o
n
Iwo (2) I2 dx.
Then, if s,, t (s, E (0, T]) and c,,, --n 0, we already know that We- (s") n converges weakly in L2(0) to w(t). The above equalities together with the fact that, as claimed above, f o " fn 2F(pE^) l w--- I2 dx dv n fa fn 2Fw2 dx dv n
fo ff 2FIw12 dx do, show that and we conclude.
=
(sn) converges (strongly) in L2 (SZ) to w(t),
The only claim remaining to prove is the weak convergence of F(p6)w-', F(pe) lwe I2 respectively to Fw, Fw2. Since the proofs are entirely similar, we only detail them in the case of F(pe)we. First of all, since pE is bounded
in L (0,T;W1,4 (n)), there exists, f o r all b > 0, p6 E L I N (for instance) such that p6 is bounded in LI (0, T; WI, (SZ)) uniformly in e, 8 > 0 and IIPE - P IIL5/3(12x(O,T)) < 6-
(3.99)
Then, we introduce Fn E Cb (IR, IR) (Fn and F,', are bounded and continuous on IR) such that (3.90) holds uniformly in n with F replaced by F,,, and Fn converges to F uniformly on compact sets of IR. Obviously, Fn(p6) is bounded in L''(SZ x (0, T)) uniformly in e, S, n and without loss of generality, we may assume that F'(pa) converges weakly to F-6 as a goes to 0. Next, we estimate Fn (p56) - F(pE) in L1(Sl x (0, T)) and we have for
Navier-Stokes equations
122
all R E (0, oo), y E (0,1), T
J0
I.
dt
dx I
F(pE)
jTj
<
{F(p) + F(p)} l1p1>_R
dt
+ C{ su p IFn(s)-F(s)I} + IsIR < C{ sup I
JdtJ dx IF(i)-F(pe)
Fn(s)-F(s)I }
IsI
+C
+C
J0
J
It 0
f
dt r d2 n
a+1) - (1[P>R + 1Ip6-p-I>7
r dtJ dx (IPIa+1) (1lpl>R + 1lp6-pcI>7 l
/
n
+ C sup {IF(x) -F(y)I / Ix-yI < y, IxI, Iyl <- R} < En (R) +WR(y) < En (R)
+CR-(a-a)
+WR('Y)+CR-(4-a)
+Cy-(4-a) IIp6 +Cy-aa)
pEII°-a
L 1 (n x (O,T)
83-a
where we used (3.90) and (3.99) and en(R) - n 0 for R > 0 fixed, cR(y) --> 0
as y -- 0+ for R > 0 fixed. Hence, letting first n go to +oo and b go to 0, then y go to 0 and finally R go to +oo, we deduce urn E>p
IIFn(p) - F(pE)IIL1(nx(O,T)) = 0.
In particular, we deduce that F converges in L1(S2 x (0, T)) to F as n goes to +oo and S goes to 0. Therefore, we have only to show that Fn(pb)wE converges weakly in L3 (Q x (0, T)) (say) to F6 w. But this means
we can now assume without loss of generality that F(p) is replaced by Fn(p6) which is bounded on SZ x (0,T) and satisfies: Fn(pb) is bounded in L3 (0, T; C1(Sl)). In other words, we may assume that F(pc) = FE is bounded in L' (0 x (0,T)) n L3(0,T; C1(? )). Repeating the above argument, i.e. approximating FE (in L 3 (0, T; C1(SZ)) ), we may in fact assume that FE is bounded in L°° (S2 x (0, T)) n L3 (0,T; Ck(Sl)) for an arbitrary k _> 0 and thus in particular FE is bounded in Lq(0,T;C2(Sl)) for any q E [1, oo).
Next, from the equation satisfied by wE, we deduce that awe
is bounded in
L''(S2 x (0,T)) + L°°(O,T; H-1(fl)).
(3.100)
Temperature and the Rayleigh-Benard equations
123
We then write (in the sense of distributions) FEwe
=
a
at
/'t wE
J0
Fe ds -
awE
(It FE ds
and we conclude easily since we can use the above bounds on w- and on FE to deduce that (fo FE ds) converges uniformly on SZ x [0, T] and in LQ(0,T;C'(S2)) to (fo Fds). 0
4
EULER EQUATIONS AND OTHER INCOMPRESSIBLE MODELS This chapter is essentially devoted to the study of incompressible (homogeneous) Euler equations, namely ,ou
diva=0 in St x (0,oo);
0,
int=o = uo
in
St
in
Q.
(4.1)
with uo given on St satisfying
div uo = 0
(Substracting a gradient term from uo, we can always make such an assumption.) Of course, we have to prescribe boundary conditions (unless fl = IRN or in the periodic case) which take here the following form
u-n = 0
on
ast x (0,oo)
and we assume that uo satisfies
uo n = 0
on
&2.
Recall that we assume that St, in the case of "Dirichlet boundary conditions" (4.4), is a bounded, smooth, connected open set of IRN (N > 2) and n denotes the unit outward normal to ast. Let us also mention that we could as well consider extensions of (4.1) with a right-hand side (a force term) but we shall not do so here to simplify the presentation. In fact, sections- 4.1-4.4 are devoted to the above system of equations while two variants are considered in the final two sections of this chapter (sections 4.5-4.6).
A brief review of known results
125
4.1 A brief review of known results The situation is completely different in two dimensions (i.e. N = 2) and in dimensions N > 3. This is due to the following fact: if N = 2 (and
only if N = 2), w = curl u (a scalar if N = 2, w = a - a ) satisfies the following equation, deduced from (4.1) by taking the curl of the equation and observing that if N = 2, curl [(u V)u] = u V (curl u) when div u = 0:
aw + (u 0)w = 0.
(4.6)
(This fact was also used in chapter 3 in the context of Navier-Stokes equations.) - When-N-> 3, the only results which are available concern the existence and uniqueness of smooth solutions (say continuous in t with values in H3 for s > N + 1, or in C1,1 for a E (0,1) in the case of a bounded domain) on a maximal time interval [0, To) where To E (0, +oo] and if To < oo the
solution's norm blows up as t goes to To_. In fact, it is even known-see J.T. Beale, T. Kato and A. Majda [28], G. Ponce [391]-that 11w(t)JJLhas to blow up (at a "certain integral rate") when t goes to To. It is not known whether To can be finite or in other words if smooth solutions become singular in finite time. We shall come back to this fundamental issue in sections 4.3 and 4.4. If N = 2, the Cauchy problem for incompressible Euler equations is much
better understood and we refer the reader to various existing surveys on the question: see A. Majda [316], J.Y. Chemin [90]. Before we state results on the above problem, let us first define precisely what we mean by solution of (4.1)-(4.2) ((4.4) in the case of Dirichlet boundary conditions): we consider u E L°°(0, oo; L2(fl))N, satisfying div u = 0 in D'(SZ x (0, oo)) and u n = 0 on an x (0, oo), such that we have for all cp E C°° (52 x [0, oo))N (for instance) vanishing on S2 for t large 0.
J
(4.7)
Let us recall that we denote by P the projection on divergence-free vector
fields (div (Pp) = 0 in 5l, (Pcp) n = 0 on 852 in the case of Dirichlet boundary conditions, curl Pcp = curl cp in f2) that we used several times in chapters 2 and 3. If 52 = IRN, the above formulation is replaced by the (equivalent) usual weak formulation of (4.1), namely
fdtfdxu.{+(u.V)cc}+fdxtzo.w(x0)=0, for all cp E Co (52 x [0, oo)),
div cp = 0
i n 11 x (0, oo),
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Euler equations and other incompressible models
and in the periodic case, we impose (4.7) for all cp E COO (IRN x [0, oo)),
periodic in x, vanishing on IRN for t large and satisfying div W = 0 in IRN x [0, oo). In general, (4.8) is contained in (4.7) but the converse might not be always true (in the case of Dirichlet boundary conditions). We may now state a few typical results that are available when N = 2.
Theorem 4.1. Let uo E L2(1)2 satisfy (4.3) (and (4.5) in the case of Dirichlet boundary conditions). We assume that curl uo E L" (SZ) for some r E (1, +oo]. Then, there exists u E C([0, oo); W1,') in the periodic or in the case of Dirichlet boundary conditions, u E uo+C([0, oo); L4nW1,'') with q = max (1, +2) if S1 = IR2, u E C([0, oo); L2 n W1'1) for all s E (1, +oo) if r = +oo and 11 is bounded, u E C([0, oo); L2 n Wil,) for all s E (1, +oo) if r = +oo and fl = IR2, curl u E L°° (1 x (0, oo)) if r = +oo.
Furthermore, such a solution is unique when r = +oo, and if uo E Wk,P(fl) where k E IN, 1 < p < oo, k > 1 + 2/p, resp. uo E Ck,a(S2) where k E IN, k >_ 1, a E (0, 1), then u E C([0, oo); WkP), ut E C([0, oo); Wk-1,P), resp. U E C([0, oo); Ck,a), ut E C([0, oo); Ck-1,a).
Remarks 4.1. 1) We shall see below (Corollary 4.1) additional properties of solutions when r E (1, oo). 2) It is possible to consider cases when 11 = IR2 and curl uo E L''(IK.2) (1 < r < oo) but we shall not do so here. We shall come back to the specific case S2 = JR2 in the next section. 3) Also, in the next section, we shall discuss the important borderline case r = 1, different (and more precise) formulations of the equation (including the vorticity equation (4.6)). 4) When 1 < r < oo, the uniqueness of the above solutions is not known. We shall see in the next section that, for "generic" uo E L2, there exists a unique solution u E C([0, oo); L2).
5) The existence and regularity properties of the pressure are discussed
below in the case when r E (1, cc). If uo E Wk,P (resp. Ck,a) then the pressure lies in C([0, oo); Wk,P) (resp. C[0, cc);
Ck,a)).
6) In the above result, one could add in the existence of solutions the conservation of energy, namely the fact that fn I u(t)12 dx is independent of t.
7) The growth of high order estimates of solutions as t goes to +oc is an interesting open problem: for instance, if uo E H3(IR22), how does the H3 norm of u(t) behaves as t goes to +co ? Only an upper bound of the form Ce ee is known. 8) We shall briefly sketch below parts of the above result leaving aside the regularity results which follow in a direct way from the L°O bound on curl u and the uniqueness in the case r = +oo originally shown in V.I. Yudovich [494] and extensively studied (among other topics) in J.Y. Chemin [90]. Again, the crucial bound is the L°O bound on curl u which then implies
A brief review of known results
127
that u is, uniformly in t, "almost" Lipschitz (i.e. admits a tj log tj modulus of continuity).
Corollary 4.1. Under the assumptions of Theorem 4.1 and if p < +oo, for any solution satisfying the properties listed in Theorem 4.1, there exists
pEC([O,oo);Lq)with 1 2 (replacing L°° by Co if r = +oo) such that (4.1) holds (in the sense of distributions). In addition, in the periodic case or in the case of Dirichlet boundary conditions, we have: at E C([O, oo); Lq), p E C([O, 00); W 1,q)
where q = r ifr > 2, q E [1, r) ifr = 2, q = 4 TT if 3 _< r < 2 ;
in
W-lea),
the periodic case, p E C([0, oo); W2,7) if r > 2 ; at E C((0, oo); p E C([O, oo); Lq) where q = 2rr if 1 < r < 3 . Finally;-if-11 = IR2, we have: ai E C([O, oo); LI), p E C([O, oo); W l,'?)
where +2 < q < r ifr > 2, 1 _< q < r ifr = 2, 1 < q < 4 rT if 3 < r < 2, D2p E C([O, oo); L1) if r = 2, p E C([O, oo); W2,r/2) if r > 2 au E C([0 00)> ; 1,q), p E C([0 0c)> ; Lq) where 1 < q r if 1 < r < 43 Ft - 2-r >
>
>
and W-"A=IT ES'/(-0)-1/2TELq}.
Remarks 4.2. 1) The proof of Corollary 4.1, given below after the proof of Theorem 4.1, shows in fact a bit more. When r = , q = 4 TT becomes 1 and we may replace L1 by the Hardy space ?..(1 (see ssection 3.2 in chapter 3); similarly, we may replace W1,1 by {p E L1 , Vp E xl}. The same remark holds when r = 2 or when q = 1, St = IR2 replacing L1 by '}{1, W1,1 by If E L1, V f E 7-(1 } or W2,1 by If E W", D2 f E l1 I. Finally, using the results of R. Coifman, P.L. Lions, Y. Meyer and S. Semmes [95] in the proofs below, we shall see that, in the case when St = IR2 (or in the periodic case), ai (and thus u - uo) E C([O, 00); ?Jq) where +2 < q < 42'
if r < 2 where ?{q = Lq if q > 1-indeed observe that at = -P((u 'P)u), (u V)u E C([O, oo);7-1) and P maps 1-c' into ?{q (if q > 3)-and D2p E C([O, 00); ?-(r/2), Dp E C([O, oo); ?-(q) where +2 < q _< r if r < 2. This last observation on D2p follows from the fact that we have
-Op = det (D20) where u = V
(4.9) aX1
and D20 E C([O, oo); Lr). Indeed, by the results of [95], det (D20) E C([O, 00); ?-(r/2) and thus a
_
(-p)-1 a2 d:t (D2) E C((O, 00); xr/2),
since r/2 > 1/2. 2) The proof of Corollary 4.1 also shows that i E C([0, cc); W-1'q)
when Ii=IR2and1 2. We can even replace W-14 by If E S' / (-A)-1/2 f E C°`} where a = 1 2 in the case when r > 2.
-
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Euler equations and other incompressible models
Proof of Theorem 4.1. The proof is divided into several steps. Let us recall that, for the reasons mentioned in Remark 4.1 (8)), we only prove here the case when 1 < r < oo.
Step 1. Fundamental a priori bounds. We first wish to explain the heart of the matter. First of all, multiplying (4.1) by u, we expect u to satisfy the following local form of the energy identity (or energy conservation), namely 2
(2 )
2
+div u [ l 2 + p]
=0
(4.10)
which implies, at least formally, the conservation of energy, i.e. the fact
that fn Iul2 dx should be independent of t and thus a bound on u in C([0, oo); L2).
Next, using (4.6), we deduce formally
at
{Iwlr} + div {ulwlr} = 0
(4.11)
hence integrating in x
fn
lwlr dx
is independent of t.
(4.12)
In particular, this yields a bound on w in C([0, oo); Lr) hence a bound on Vu in C([0, oo); L2): indeed observe that div u = 0 and u n = 0 in the case of Dirichlet boundary conditions. This bound, combined with the C([0, oo); L2) bound on u, yields a bound on u in C([0, co); W',r) in the periodic case, in the case of Dirichlet boundary conditions or if n = IR2, r > 2. Finally, if 11 = IR.2, we observe that (4.1) yields
+ P[(u . 0)u] = 0.
(4.13)
Since (u - 0)u is bounded in C([0, oo); L r+) in view of the preceding bounds, at is also bounded in C([0, co); L7) and thus u - ua is bounded in C([0,TJ; Lr+) for all T E (0, 00).
Step 2. Navier-Stokes approximation. In the periodic case or in the case when n = IR2, there is no difficulty and we simply approximate (4.1) by the Navier-Stokes equation
j
0,
divu=0
keeping the same initial condition (4.2), where v > 0.
(4.14)
A brief review of known results
129
However, in the case of Dirichlet boundary conditions, a fundamental difficulty arises with boundary conditions. It is of course tempting to use again (4.14) with Dirichlet boundary conditions, namely: u = 0 on all x (0, oo). The difficulty which then appears is emphasized in the following remark.
Remark 4.3. (2D Navier-Stokes - 2D Euler, with Dirichlet boundary conditions?). We consider (4.14) in the case of Dirichlet boundary condi-
tions: u = 0 on all x (0, oo). It is quite clear that a boundary layer will form since solutions of Euler equations only satisfy u n = 0 on 992 and there is no reason why the tangential part of u should vanish (and they do not in general!). It is not known if the "L2 strength" of this "layer" goes to 0 and, more precisely, the limit as. r goes to 0 of solutions of NavierStokes equations with Dirichlet boundary conditions to solutions of Euler equations is an important open problem. Equivalent formulations of this problem can be found in T. Kato [238]. 0 To circumvent that difficulty, we modify the Dirichlet boundary conditions associated to (4.14) in the following way
u-n=0, w=0
on ac x (0, oo)
(4.15)
.
Then (4.14)-(4.15) is equivalent to
5 + (u
V) w
- vLw = 0
in 0 x (0 , oo) ,
w = 0 on all x (0, oo), curl u = w, div u = 0 in f2 x (0, oo), u n = 0 on ac2 x (0, oo).
}
(4.16)
Notice also that since we are in two dimensions
Du = V div u + V1 curl u
/
where O1 = (
a a
_axi
hence, if div u = 0, Du n = (n V1) curl u and we deduce from (4.15)
Ju n = 0
on ac x (0, oo),
(4.17)
and we may rewrite (4.14) as
j + P((u V)u) - v0u = 0
in S2 x (0, oo).
(4.18)
We then claim that the system (4.14)-(4.15) (or equivalently (4-16)) can be solved exactly as the usual Navier-Stokes equations (i.e. with the
Euler equations and other incompressible models
130
usual full Dirichlet boundary conditions)-see section 3.1-and thus that we obtain (in all cases) for each v > 0 a unique solution u E L2 (0, T; H1) n
E L2(0, T; H-1) (for all T E (0, oo)). All these facts follow from a modified energy identity that we briefly sketch (and from xgtx)- Multiplying by u, (4.18) to obtain the regularity information C([O, oo); L2), O"u
0 on 80 x (0, oo),
integrating by parts over 8SZ and using the fact that we obtain
Iu12dx+vJ IVul2dx =
2 dt f
V
f
n a n au
n
-udS.
Using the boundary conditions, namely u n = 0, w = 0 on a Q, it is easy u = rcIuI2 on 90, where x is the curvature of 811 . Hence, to check that we obtain for some Co > 0 that depends only on St 2 dt
fIuI2dx+vjIVuI2dx
_< Cov
f
IuI2 dS.
an
(Recall that in the periodic case, or if n = IR2, or in the usual Dirichlet case we obtain an equality with a right-hand side that vanishes.) Then, using a classical trace inequality, we deduce from the preceding inequality 1 d
2 dt
1/2
f
1u12 dx + v
f IVu12 dx < CV f IDu12 dz
1/2
f Iu12 dx
n
where, here and below, C denotes various positive constants that depend only on 11. Hence
d jIuI2dx+vfIVuI2dx < Cv
f
IuI2dx,
(4.19)
and our claim is shown. We have obtained in fact the following bound J
I u(x, t)12 dx < ec"t f Iuol2 dx
for all t > 0.
(4.20)
Let us finally observe that, for each v > 0, u, is in fact smooth for t > 0: for instance, multiplying (4.16) by tw and integrating by parts, we immediately obtain for all T E (0, oo):
f
t
ds < C(T, v)
for
0
o
or
tjjuv11Hl +
f
0
t
ds < C(T, v)
for
0
A brief review of known results
131
Step 3. Existence when r > 2. We treat here the case when r > 2 and we first consider either periodic or Dirichlet boundary conditions. Then, we explain how to modify the proof when SZ = IR2. First of all, we obtain an estimate on w, in C([0, oo); Lr). We simply multiply (4.16) by Iw, Ir-2 w and obtain in all cases (either periodic or Dirichlet boundary conditions)
fS=
for allt>0 1
t
lwl2dx
Iwy(t)Irdx+v(r-1) jf
(4.21)
1 r nf Icurl uoI r dx.
This identity yields the desired bound. -Of course, the preceding calculation has to be justified. This is not difficult and we leave it to the reader (multiply (4.16) by ITR(wv)Ir-2 where max(min(wR), -R) for R > 0 and let R go to +oo, observing that for instance wL E L2(0, 00; H1) fl C([0, oo); L1)). Observe also that Vv-w is bounded in L2(0, oo; H1) (take r = 2 in (3.21)).
The bound on w yields a bound on u in C([0, oo); W 1,r) exactly as in step 1. Then, from the equation (4.16), we deduce that at is bounded in L2(0, T; W-14) for all T E (0, oo), 1 < q < 2 for instance. This is enough to ensure that w is relatively compact in C([0, T]; Lr - w) (recall that Lr - w means Lr endowed with the weak topology) by the observation detailed in Appendix C. Hence, u is relatively compact in C([0, T]; W l "r - w) by the same argument as in step 1 and u is relatively compact in C([0, T]; Lq) for all 1 < q < oo, T E (0, oo) by the Rellich-Kondrakov theorem. Extracting subsequences if necessary, we may thus assume that, as v goes to 0, u converges to some u in C([0, T]; W l,r w) fl C([0, TJ; L9) for all 1 < q < 00 while w, converges to some w in C([0, T]; Lr - w) (for all T E (0, 00)) - In particular, we have on passing to the limit: u(0) = uo, w(0) = curl uo in Sl,
-
fC2 I u(t)I2 dx < fn Iuo12 dx for all t > 0 in view of (4.20), u E C([0, T]; L2) ,
div u = 0
,
curl u = w
+ div (uw) = 0
in
in S2 x (0, oo)
S2 x (0, oo)
(4.22) (4.23)
and u n = 0 on &I x (0, oo) in the case of Dirichlet boundary conditions. Furthermore, we may pass to the limit in (4.18) and we obtain
t a + P((u - V)u) = 0
in SZ x (0, oo).
(4.24)
We claim that this yields (4.7) and we explain why (4.7) holds in the case of Dirichlet boundary conditions: indeed, Vu E L°° (0, 00; Lr), u E
Euler equations and other incompressible models
132
L°° (0, oo; W 1,r) and thus we have for all cp E Coo (SZ x [0, oo)) (vanishing for t large) 00
0=
dt
Jo
n
dx{-u
r
r OO
= f dtJ dx f -u .o
n
r°o
= -J
-
+ (u V)u
Pcp} - J dx uo cp(x, 0) n r
+ div(u ®u) - Pcp}
r
dt1 dx{u a`p + u (u V)[PcP] } -
o
at
n
dx uo cp(x, 0)
f
n
dx uo cp(x, 0)
since divu = 0 and u n = 0 on 8S2 x (0, oo). In order to complete the proof of Theorem 4.1 in this case, it only remains
to show that w E C([0, oc); Lr) and thus u E C([0, oo); W1"2). -This fact is a consequence of (4.23). Indeed, by general results due to R.J. DiPerna and P.L. Lions [128] and recalled in section 2.3 on transport equations with divergence-free vector fields, we see that w is necessarily equal to the unique renormalized solution of (4.23) which satisfies w E C([0, oo); Lr) (and fn 1W (t) I r dx is independent oft > 0). We have used here the fact that r > 2 and thus, in particular, u E C([0, cc); W1,2) while w E C([0, oo); L2).
Remarks 4.4. 1) Let us observe that since w E C([0, oo); Lr) and fn Iw(t)Ir dx is independent of t, the identity (4.21) immediately yields the convergence of w tow in C([0, T]; L') and thus of u to u in C([0, T]; W1,r) for all T E (0, oo). 2) (4.24) shows that E C([0, oo); Lq) for all 1 < q < r, and, multiplying (4.24) by u, one can easily justify the energy conservation since we
have for allt>0 r
fn
Iu(t)I2 dx
- Jn
rt
IuoI2 dx
r
= -2J dsJ dx [(u V)u] Pu
ff2 n
o
t
= -2 f dsrdx[(u V)u] u o
_
1n
= 0.
This verification only requires Iu12IVUI to be integrable and this is the case as soon as u E L' (0, cc; W 1,r) with r > since, in this case, Iu12 E L°°(0, oo; L3) by Sobolev inequalities.
We now briefly explain how to modify the above proof in the case when fl = IR.2. All the steps of the proof are easily adapted, replacing C([0, T]; LQ) or C([0, T]; L2) by C([0, T]; L oc ) or C([0, T]; L2")°): in particular we obtain u E LO°(0, oo; L2) n C([0, oc); LO C), Vu E L°°(0, oo; Lr), U E C([0, oo);
A brief review of known results
133
WI .) satisfying u(0) = uo in IR2, div u = 0 in IR2 and ,jT
+ P((u V)u) = 0
in JR2 x (0, co).
(4.25)
In particular, i E L°° (0, oo; L --';-I) hence u - uo E C ([0, oc); L). This yields the fact that u E C([0, oo); L2 n W 1,r) since +2 < 2 < r.
Remark 4.5. The proofs "given" in Remark 4.4 are easily adapted to the case when S2 = ]R2 and yield the conservation of energy together with the convergence (up to the extraction of subsequences) of u in C([0, T]; L2 n W 1,r) for all T E (0, oo).
Step 4. Existence when 1 < r < 2. We now treat the case when 1 < r < 2 and we begin again by excluding the case when c = IR.2 and thus considering periodic or Dirichlet boundary conditions. In fact, we shall give the proof in the case of Dirichlet boundary conditions, the periodic case being similar and even somewhat simpler.
We shall deduce the existence of a solution when 1 < r < 2 from the case we just treated. We then introduce uo E Cl (f) satisfying uo n = 0 in S2, div uo = 0 in f2 and such that uo converges in W 1°r to uo as e goes to 0+: the existence of uo can be obtained by considering Puo where uo E C°°(0) converges to uo in W""r. Next, we denote by uE a solution of the Euler equation provided by the preceding steps (it is in fact unique) and by we = curl uE. We have the following information: uC E C([0, oo); W 1.4), w' E C([0, oo); LQ) for all 1 < q < oo. (4.7), (4.13) and (4.6) hold with u, w replaced respectively by u, w. Also, we have
fIwe(t)dx
=
f
Iwofrdx E'
fIwordx
defining wo = curl uo, wo = curl uo. In particular, as in step 1, we see that uC E L°° (0, oc; W 1,r) and without loss of generality we may assume that uE converges weakly to some u E L°0(0, oo; W1,r) which satisfies u n = 0 on 812 x (0, oc), div u = 0 in f x (0, oo). Then we invoke the compactness results shown in R.J. DiPerna and P.L. Lions [128] essentially recalled and proved in section 2.3-only the L°° case was established there but the general Lr case follows as well by considering TR(wc) instead of w£'-and we deduce that wE converges in C([0, T]; Lr) (for all T E (0, oc)) to the unique renormalized solution w of (4.6) satisfying wl t=o = wo in P. Here, as in [128], renormalized solution means that we have for all /3 E Cb ° (IR; ]R) a/3(w)
at
+ div (uAA(w)) = 0
in 1)'(O x (0 I 00))
(4 26)
134
Euler equations and other incompressible models
Therefore, by the same argument as in step 1, uE converges in C([O, T];
W1,') (for all T E (0, oo)) to u and we can pass to the limit in (4.7) completing the proof of Theorem 4.1 in this case. Notice, by the way, that, in particular, uE converges to u in C([0, T]; L2) and thus Remark 4.1 (6) is deduced in that case from Remark 4.4 (2). In the case when 1 = IR2, the same proof as above applies (we now simply regularize uo by convolution) and shows that wE converges to w = curl u in C([O, T]; Lr) (for all T E (0, oo)) which is the unique renormalized so-
lution of (4.6) in IR2 x (0, oo) satisfying wlt=o = wo in IR2. Since it is bounded in L°O(O, oo; L2), we deduce that Dut converges to Vu in C([0, T]; L'), uE converges to u in C([0, T]; Wor) for all T E (0, oo) where u E C([0, oo); W1l,) n L°O(0, oo; L2), Vu E C([O, oo); Lr). It only remains to show that u-' converges to u in C([O,T]; L2) (for all T E (0, oo)) and that u-uo E C([0, oo); L') when 1 < r < 2. In order to do so, we follow Remark
4.2 (1): (uE D)uE is bounded, in view of [95], in C([0, cc), fl + ). Since at = -P((ut D)ut) and r2+2 > 3 if r > 1, we deduce that aueCIF is bounded in C([0, oc); f -2 ) and thus it - uo is bounded in C([0, T]; f +-,) for all T E (0, oo). On the other hand, since D(ue -uo) converges to D(u-uo) in C([0, T]; Lr) and 1 < r < 2, we deduce from Sobolev embeddings that uE
- uo converges to u - uo in C ([O, T]; L
for all T E (0, oo) . Hence, by interpolation, uE-uo converges to u - uo in C([0,T];L9) for 1 < q < 2 T and our claims are shown since uo converges to uo in L2 by construction.
0 Remark 4.6. It is possible to give a much more elementary proof of the convergence of uE to u in C([O,T]; L2) (V T E (0, oo)) which also yields the fact that u E uo + C([0, oo); LQ n W1,1) for +1 < q < r (but does not reach L1!) when 1 < r < 2. One simply observes that 2
E
-
at and
-E a P(ueue) .7=1
9
is bounded in C([0, oo); L' n L'). Hence, we have 2
uE-uo
=E j=1
a aX.7 (f; ),
v(uE-uo) =
gE
where fl is bounded in C([0, T]; LQ) for all T E (0, oo), 1 < q < 2-'r, while
9,' is bounded in C([0, cc); Lr). These two facts imply easily that u`-u' is bounded in C([0, T]; Ls) for r+1 < s _< 2? r, 0 < T < oo. Observe that UE = (-0)-1/2(uE-u0 ) is bounded in C([0, T]; L9) while -DUE is bounded
A brief review of known results
135
in C([0, oo); Lr), and this is enough to show our claims following the end of the proof of Theorem 4.1. We now turn to the
Proof of Corollary 4.1. We begin with the existence of p which is a straightforward consequence of (4.7). Indeed, we have < ai + div (u u), cp >D'xv= 0 for all cp E Co (S x (0, oo)) such that div cp = 0. This implies the existence of a distribution p such that (4.1) holds in the sense of distributions. Next, we need to show the integrability requirements indicated in Theorem 4.1. Since P is bounded from Wo'q into W1,4 for all 1 < q < oo, we deduce from the weak formulation (4.7) (using PV for if r < 2, arbitrary smooth cc) that ai E C([0, oo); W-1,q) for 1 < q 5
12sinceuEC([0,oo);L2nL rr)ifr< 2 (L2 n Lq if r = 2 for 2 < q < oc, L2 n Lm if r > 2) by Sobolev embeddings.
The integrability of p then follows since Vp = - ai - div (u 0 u). In the periodic case or when 1 = 1R2, the argument is a bit simpler: we just write au
2
/_P(uiu), E ; ;_1
Vp = - at
- div (u ®u).
All the claims listed in Corollary 4.1 follow easily from the bounds on
u, Sobolev embeddings, the fact that a = -P((u V)u) when r > 3 and that we have
-op = div {(u V)u} =
aui au; _ ax; axi
a2
axt ax; (uiuj) (4.27)
Remarks 4.7. 1) The proof we gave of Theorem 4.1 shows a few additional
properties of at least one weak solution. The first one is the fact that w = curl u is a renormalized solution of (4.6), that is it satisfies (4.26). This fact can be recovered a posteriori when r > 2 using a regularization technique as in Lemma 2.3 (section 2.3, chapter 2) for the equation (4.6) which is satisfied in the sense of distributions, but it is not clear that this can be done when 1 < r < 2. The second property we want to mention is the local form of the conservation of energy, namely equation (4.10): our existence proof shows it holds for at least one weak solution if r > 5. Indeed, in that case we obtain the convergence of u£, p£ in C([0, T]; L3), C([0, T]; L3/2) respectively for all T E (0, oo) and we can recover (4.10).
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Euler equations and other incompressible models
2) Concerning the convergence of solutions of Navier-Stokes equations to solutions of Euler equations (with the boundary conditions modification introduced in step 2 of the proof of Theorem 4.1 in the case of Dirichlet
boundary conditions), let us mention that when 1 < r < 2 it is possible to show that w converges to w in C([O, T]; L') and thus Du converges to Vu in C([O, T]; L") for all T E (0, oo) using the (duality) method of the last section of R.J. DiPerna and P.L. Lions [128]. 3) In the case of Dirichlet boundary conditions, it is possible to say a bit more about the regularity of p when r > 2. Indeed, we have (4.27) and thus Ap E C([0, oo); Ln/2) (L' being replaced by fl' when r = 2). In addition, one can show by the proof of Theorem 4.1 that there exists a weak solution (u, p) such that, denoting by n the curvature of all, we have ap = -icIu12 an
on
all x (0, oo).
(4.28)
This is what we expect from (4.1) since (u V )u n = - (u V )n . u (u n = 0 on as2 and thus (u V)(u n) = 0 on 8Q). Notice that if 11 C 1RN, N > 2, iclul2 is replaced by the "curvature quadratic form" applied to u. Combining (4.27) and (4.28), it is not difficult to show by elliptic regularity that D2p E C([0, oo); Ln12(f2)).
4.2 Remarks on Euler equations in two dimensions This section is essentially devoted to a discussion of the Euler equation in two dimensions when the initial condition uo only lies in L2(Sl) or belongs to L2 and is such that curl uo is a bounded measure. Roughly speaking,
this corresponds to the case when r = 1 in the preceding section, a case which was of course excluded from our analysis. This borderline situation is not only very interesting mathematically but also corresponds to various relevant physical situations. We refer the reader to the fundamental series of works by R.J. DiPerna and A. Majda [129],[130] on this subject for a more complete discussion of the background of this issue (and of `vortex sheets"). Only at the end of this section shall we leave this issue to mention a few other questions of interest on Euler equations in two dimensions. Let us now describe what we discuss below. First of all, if uo E L2, div uo = 0 (and uo - n = 0 on (9!Q in the case of Dirichlet boundary conditions), the existence, uniqueness and stability of solutions are completely open. However, using the regularity which is available for smooth uo (and a few simple tricks), we shall see that there exists a Gb set of initial conditions in L2 (that is a countable intersection of dense open sets in L2) for which there exists a unique solution of (4.1)-(4.4) in C([0, oo); L2) with a
Remarks on Euler equations in two dimensions
137
conserved energy (i.e. fn I u(t) I2 dx is independent of t). As we shall see this is a "cheap" result whose only merit is to indicate that the problem is well posed for most initial conditions in L2.
The other angle of attack that we discuss in this section consists in pushing as much as we can towards L' the arguments developed in the previous section, which are obviously based upon the transport equation (4.6). Since (4.6) involves a divergence-free vector field we expect solutions
of (4.6) to preserve the initial distributions function (or in other words, we expect the decreasing rearrangement of solutions to be independent of t)-and this is precisely the case with renormalized solutions. This will lead to two different kinds of results which are essentially optimal for this type of approach. However, we shall remain rather "far" from L1 or bounded measures. We shall not address here in detail the problem of vortex sheets (uo E L2, curl uo is a bounded measure) and we refer instead to R.J. DiPerna and A. Majda [129],[130] for a discussion of the possible phenomena involved-see also the presentation of their results in L.C. Evans [141]. Let us also mention the existence of global weak solutions in the case when uo E L2, curluo is a bounded measure such that (curl uo)+
(or (curl uo)-) E L1 which was obtained by J.M. Delort [118]; a simpler proof was proposed by A. Majda [318]. We now begin with our generic result. We introduce the Hilbert space H (for the L2 scalar product) defined by: H = {uo E L2(S2)2, div uo = 0 in D'(Sl), uo - n = 0 on 8S1}. In the case of Dirichlet boundary conditions, H = {uo E L2(1R2)2 , divuo = 0 in D'(1R2)} if SZ = 1R2, H = {uo E L C(1R2)2 , uo is periodic , divuo = 0 in D'(IR2)} in the periodic case.
Theorem 4.2. There exists a decreasing sequence of dense open sets O, in H such that, for any uo E 'n>1 On, there exists a unique solution u E C([0, oo); L2)2 of (4.1)-(4.2) (and (4.4) in the case of Dirichlet boundary conditions) such that fn Iu(t)I2 dx is independent of t > 0. Furthermore, for any weak solution u E L°° (0, oo; L2)2 n C([0, oo); L2 - w)2 of (4.1)(4.2) (and (4.4) in the case of Dirichlet boundary conditions) such that I
fIiz(t)I2dX 0, then ii -uin S2x (0,oo). Proof of Theorem 4.2. The proof is based upon the fact that if uo E L2 n C""a for some fixed a E (0, 1), then, see Theorem 4.1, there exists a unique solution u E C([0, oo); L2 n C1"1) of (4.1),(4.2) with uo replaced by
uo (and (4.4)) such that, in particular, IIoulIL-(nx(o,T)) <- C(uo,T) for all T E (0, oo), and we can always assume that C(uo,T) > 0 is nondecreasing with respect to T. The second ingredient which is basic for our proofs is the following (essentially classical) observation. If a is any weak solution of the Euler equation as in Theorem 4.2 then we claim that we have for all t E [0, T] 01(t)
- U(042 < ec("0,T )t 11 uo
- Uo 11 L2.
(4.29)
Euler equations and other incompressible models
138
Indeed, on the one hand we have for all t > 0
fn
lu(t)12 dx <
jIi(t)I2dx
Jdx,
= fIiioI2dx
(4.30)
and on the other hand we deduce from (4.7) using cp = u (a choice that can be justified by a simple approximation argument, take cpn = P(cpn) where cpn converges in Cl,* to u)
ju(t)1(t)dx_fdsfu.{+(uV)ii}dx
=
Jn
uo uo dx
or equivalently using the equation satisfied by u (and the fact that div u = 0) t
liz
u(t) u(t) d x -
fo
dsfu
O]u} dx =
I
n
f uo uo dx.
Next, we observe that f 0t ds fn u {[(u - tu) 0]u} dx = fo ds fn (u - u) ) dx = 0 using the fact that div u = div u = 0 (and (u- u) . n = 0 in V( the case of Dirichlet boundary conditions), and thus we obtain for all t > 0
in
u(t) u(t) dx =
f
n
t
uo uo dx
r
fo ds ndx (u-u) Vu (u-u). (4.31)
Combining (4.30) and (4.31), we deduce finally for t E [0, TI r
t
I u(t) -u(t)12 dx < in luo -uo 12 dx + 2C(uo, T) f ds J lu(s) -u(s)12 dx 0 o n In (4.32)
and we deduce (4.29) from (4.32) using Gronwall's inequality. We then introduce, for n > 1, the following open set On
On = UoEH/3-ffoEL2 n Cl,a
,
IIu0-UOIIL2 <
1 n
e-C(uo,n)
(4.33)
and we wish to check the statements listed in Theorem 4.2 for this choice of on. Since On contains L2 nC1"a(nH), it is not difficult to check that On is dense in H : indeed, (L2 n C1,a)2 is dense in (L2)2 and P((L2 n Wl,a)2) (L2 n Ci,a)2 n H. Next, if uo E nn>1 On, it is not difficult to show the uniqueness part of Theorem 4.2. Indeed, if uo E i ln> 1 On, there exists, for each n > 1, some
o E L2 n C""a n H such that IIuo-uo IIL2 < n e-c( on)n. Then, if ui, u2
Remarks on Euler equations in two dimensions
139
are two weak solutions as in Theorem 4.2, (4.29) implies suptE[o,n] IIu1(t),u2 (t) I I L2 < n , and the uniqueness is proven. The existence part also follows from (4.29). It is clearly enough to show that u is a Cauchy sequence in C([0, T]; L2) (for all T E (0, oo)). Then, if n, m > T, we deduce from (4.29) that we have
-u
Sup IIu (t)
tE[O,T]
(t) II L2
f
< min (eC( o,n)T
eC(
u
,m)T
l
} II 0 - 0 II L2
min{eC o,n)T ec("° ,m)T} {lI-uoIIL2 + I1u0 <
<
min{eC(uo,n)T
- o IIL2]
eC( o m)T } [2_Crn)n +, 1 -C( 0 ,m)m' m
1+ 1. n m
Our claim is shown, thus completing the proof of Theorem 4.2.
We next discuss some other existence results based upon the fact that we expect w = curl u to satisfy (4.6), i.e. a transport equation with a divergence-free vector field (namely u). Thus, the distributions function of w(t), namely the function µw(t) on (0, oo) defined by meas {x / I w(x, t) I > .1} for .1 > 0, should be independent of t. Notice that this is precisely the case with renormalized solutions of (4.6). Indeed, (4.26) yields the fact (integrating in x) that fn /3(w(t)) dx is independent
of t for al". From now on, in order to avoid unnecessary technicalities, we restrict ourselves to the case of Dirichlet boundary conditions (or to the periodic case).
The first type of result we wish to discuss consists in pushing the proof of Theorem 4.1 towards L'. Looking carefully at the proof of Theorem 4.1, we see that we only need a bound on u in W"1. But, if we introduce the stream function, i.e. the solution of -AV) = w
in ci,
'r/.' = 0
on aci
(4.34)
in the case of Dirichlet boundary conditions for instance assuming that i is simply connected (to simplify the presentation), then u Therefore, Du E L1 if and only if D2ip E L', and by classical elliptic theory, this is the case in two dimensions as soon as w E L1, k I log Iw(t)II dx < oo. Observe in addition that fn I w(t) I I log 1w (t) I I is, at least formally, inI
dependent of t. Once these observations are made, it is not hard to copy
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Euler equations and other incompressible models
the proof of Theorem 4.1 and to show that if uo E L2(S2), div uo = 0 in 52, uo n = 0 on 852, wo = curl uo E Ll (0) and fn Iwo I I log wo I dx < oo, then there exists a solution u E C([0,oo);W1°1(S2)) of (4.1), (4.2), (4.4) such that fn Iu(t)I2 dx is independent of t, w E C([0, oo); L1(11)) (and even "L1 log L1i) is a renormalized solution of (4.6), i.e. satisfies (4.26). Indeed, regularizing wo, we obtain a sequence of solutions of (4.1) (ue,We)
and, by the results of R.J. DiPerna and P.L. Lions [128], we check that we converges in C([0, T]; L1) (d T E (0, oo)) to some w E C([0, oo); L1(52). In addition, we I log Iwe I also converges in C([0, T]; L') (V T E (0, oo)) to wI log wI. Hence, ue converges to some u in C([0, T]; L2) (V T E (0, oo)) and we conclude. However, if we follow this argument, we can ask for less information on I
w and we might simply try to deduce from the invariance in t of µ4J. (t) (= µ41o) some compactness in L2 (or C([0,T]; L2)) for all T E (0, oo) of uE. This leads to the following question: find the optimal distributions (or rearrangement) invariant class for w such that the corresponding velocity field u belongs to L2(Sl) and then prove the existence of solutions of (4.1) with such initial conditions. This question can be solved completely using some symmetrization techniques. In order to do so, we need some notation. First of all, if w E L' (Q), we denote by w* the decreasing rearrangement which is the inverse function of 44J: in other words, w* is the unique nondecreasing function in L'(0, ISZI) (ISlI = meas(f )) such that w* E L1(0, IS2I)
and µ4,. = µ4J a.e. In two dimensions, we denote by wl the Schwarz symmetrization of w (or spherically symmetric decreasing rearrangement), i.e. the unique spherically symmetric function in V(0) which is nondecreasing with respect to r = IxI such that µ4,r = µ4, a.e., where Sla is the ball centred at 0 with the same volume as Q (or in other words with a radius Ro given by (u)112). Obviously, wO(x) = w*(7rIXI2) a.e. in f . From now on, we restrict ourselves (to simplify the presentation) to the case of Dirichlet boundary conditions, assuming in addition that 1 is simply connected (even though the results below hold in general), and we introduce the stream function, i.e. the solution of (4.34). We then recall the following general comparison result due to G. Talenti [462]
a.e. in
n Io
i2 dx <
(4.35)
Sia
Io11I2
dx
(4.36)
r
where %P is the solution of (4.34) with Sl, w replaced respectively by 520, wa, namely
-D'Y = wa
in
Sla,
41 = 0
on
9Q4.
(4.37)
J0
Remarks on Euler equations in two dimensions
Recall that T is given by the explicit formula
f
Ro
1r
f
F'(Ixl) _ - 1 IxlO sw)(s) ds.
r swa(s) ds,
1XI
141
(4.38)
(- a ), we see that the optimal rearrange-
Then, if we observe that u =
ment invariant class ensuring that u E L2(11) is given by
wEL1(SZ)/f IVWI2dz
.
a
But we have easily I0jpj2 dx = 21r Ror(W'(r))2 dr
r-
Ro
r
Ro l
1
IflI
47r
fo
t
J0
Jo
r swa(s) ds
2
dr
2
dr
sw*(xs2) ds
r J ifr2 jRo1 r o
1
r
Jo
fo
= 21r
1
= 2ir
2
* (s) ds 2
w*(s) ds
dr
t dt
In conclusion, the optimal class is given by the space L2 (S2)
=
fIc2I (ft w E L1(S2) /
w* (s)
)2dt
< oo
(4.39)
For instance, in view of the observation made by A.B. Mergulis [341], it contains the Orlicz class of functions w in L' (0) such that fn IwI log Iw1I1/2 dx < oo.
We then have
Theorem 4.3. Let wo = curl uo E L1(11) (with div uo = 0 in 0, uo - n = 0 on aS2). Then there exists a solution u E C([0, oo); L2(f )2) of (4.1), (4.2), (4.4) such that f0 I u(t)12 dx is independent oft > 0. Remarks 4.8. 1) The results due to J.M. Delort [118], that we mentioned above, show the existence of a weaker solution of (4.1), (4.2), (4.4) when wo E L'(SZ), uo E L2(SZ) since the conservation of energy is not known in that case and u E Lm (0, oo; L'(11)). I_log IxI I-', then 2) If S2 = 110 (is a ball!) and if we consider w(x) one can check that w E L1 if and only if cx > 1, w E L2 if and only if a > 2
while D2ib (= D2IF) E L1(f) if and only if a > 2.
Euler equations and other incompressible models
142
Proof of Theorem 4.3. Let us define wo = Tn(wo) for n > 1 and let uo be the unique element of L2(SZ)2 such that: curl U0 = wo , div uo = 0 and
uo n = 0 on all. In view of Theorem 4.1, there exists a unique solution un E C([0, oo); L2(SZ)2) of (4.1), (4.2) and (4.4) with uo replaced by uo, such that curl un E L°° (SZ x (0, oo)). Of course, we wish to recover the existence result stated in Theorem 4.3 by passing to the limit as n goes to +oo. To this end we recall that tt,,,.(t) = j.t for all t > 0, n _> 1 and thus we have, writing by wt = n(t), (4.40)
(wt ) * = (wu) * = wo A n.
In particular, in view of the derivation above of L2'(0), ua is bounded in L2(S )2, and since Iun(t)IL2 = Jun IL2, we finally deduce that un is bounded in C([0, oo); L2(SZ)2).
Next, because of (4.7) and since P maps {cp E C"(1)2, cp = 0 on all} into Cl"a(?Z)2 for any 0 < a < 1, at is bounded in C([0, oo); Xa) for any 0 < a < 1 where Xa is the closure of Co (SZ)2 in Cl,a(Si)2 (= {v E Cl,a(S1)2 , v = Vv = 0 on 8SZ}). Then, we deduce from Appendix C that, extracting a subsequence if necessary, we may assume that u' converges in C([0,T]; L2(SZ)2-w) (for all T E (0, oo)) to some u E C([0, oo); L2(Sl)2-w) satisfying (4.2) and (4.4). In addition, we have
J
I u(t) I2 dx _<
lim
J
r;::; -00 n
{2
tun (t) I2 dx = lira
f
n-oo n Iuo 12 dx.
(4.41)
We are going to show below that we have
un(tn) -+u(t) in
L2(SZ)2
n
if 0 < to -> t > 0. n
(4.42)
If this claim were proved, we would deduce on the one hand that uo converges in L2(1)2 to uo and on the other hand that for all t > 0
I
f=
I u(t) 12 dx = 1im inf f n
(t) I2 dx
fn
lim inf n
L
Iuo I2 dx =
Iuo12 dx.
fn
Hence, in particular, u E C([0, oc); L2(SZ)2), and this fact combined with (4.42) would show that un converges to u in C([0, T]; L2(St)2) for all T E (0, oo). Once this convergence is shown, Theorem 4.3 follows easily.
Therefore, we have only to prove (4.42). Let us first remark that we already know that un(tn) converges to u(t) weakly in L2(St2). In addition, since Oo (tn) is bounded in L'(SZ) (and Dtn(tn) is bounded in L2(0)2),
Remarks on Euler equations in two dimensions
143
we deduce from elliptic regularity that Von(tn) and thus un(tn) converge in LP(S2)2 for all 1 < p < 2. Finally, let us observe that (4.40) yields for all T E (0, oo) wn (s) * ds 2 dA
sup sup
n>1tE(OT) 0 _
-> 0
as
a --* 0+.
(4.43)
A
0
In view of these facts, (4.42) is deduced from the following
Lemma 4.1. Let wn be bounded in L1(SZ) and satisfy Icl
t
< 00,
([wds)2 J0
If:
t
rE
sup
wri ds
n>1J0
0
(4.44)
t-0
2 dt
as
a - 0+.
Then, wn is weakly relatively compact in L1(S2) and, extracting a subsequence if necessary, we may assume that wn converges weakly in L1(S2) to some w E L'2(f2). Denoting by on the solution of (4.34) (with w replaced by wn), we then have i1' wn
n bw weakly in L1(S2),
z
in Ho(S2).
(4.45)
Proof of Lemma 4.1. The weak compactness in L' follows from the following observations Inl
fe
rt
Jo
weds
and
2 dt
wn ds
t
log
for all
s > 0,
C
1
w;,, ds
sup f Iwn I dx,
IAI<e A
for all e > 0.
(4.46)
Hence, we may assume that wn converges weakly in L'(0) to some w and, similarly, that wn converges weakly in L'(00) to some w. Since wn is radially increasing, we may assume without loss of generality that wn converges tow a.e. in S and thus wn converges to w in L'(f2a). In particular, one deduces easily that w E L2(Q) and, since fo w* ds < limn fo wn ds = fo w* ds, w E LI(Q). Therefore, as shown above when we introduced L2 (S)), wnOn is bounded in L1(0), wii E L' (S2) (observe indeed that fn I wol dx < fog w101 dx < ffa wOTO dx)
fvbnI2dx
= fwnhndx,
fIVbI2dx =
f wl dx
(4.47)
Euler equations and other incompressible models
144
fIVInI2th'
=
fhIndx
J,iVWl2 dx
fiw yp dx
(4.48)
with th e notation introduced before Theorem 4.3 (and obvious adaptations).
We first claim that wnWn, Din converge, respectively, in L'(SZa), L2(SZa)2 to wW, Vjj. Since wn,'yn are radially nonincreasing, non-negative and V'yn = -x fIxl swn(s) ds, we have only to show in view of (4.48) sup
f
0
I VTn I2 dx
a-->0+.
as
n>1 IxI <e
But this is immediate in view of (4.44) since we check as in the definition of L2(SZ)
f
Il<e x
I0'pn I2 dx =
47r
r
"E2
1
J
J0
o
t
w,n ds
2 dt
t.
Next, we check that Vnwn is weakly relatively compact in L1(SZ). Indeed, for any measurable set A
IA
'cnwn dx < L V)1u)n dx <
f
Wnwn dx a
and our claim is shown since Wnwn converges in V(04). It only remains to show that iPnwn converges weakly in Ll (1) to 'raw. Recalling that On converges a.e. to 1/i, we see that for all R E (0, oo), TR(0,,)wn converges weakly in L1(SZ) to TR(i)w since wn converges weakly in L1(Sl) to w. We can then conclude observing that
fo
bn-TR(bn)I Iwni dx = f(kbnHR)+InI dx l
:5f
( n-R)+wndx <
<
Wnwn 1(`F,,>R) dx
(fin-R)+wndx
no
and thus, since Wn converges in L1(SZa) (in H1(QZ) in fact), Wnwn converges
in L' (no), these integrals can be made, uniformly inn > 1, arbitrarily small letting R go to +oo. 0
Remark 4.9. Let us mention in passing the so-called "vortex-patches" problem which was settled recently. One considers the case of an initial vorticity (in the case when SZ = IR2 for instance) wo which is constant (say
Remarks on Euler equations in two dimensions
145
A) and supported on a bounded, smooth domain D. Then, the corresponding solution of (4.1), (4.2), (4.4) satisfies for all t > 0: w(t) = A 1D(t) for some measurable set D(t). It is proved in J.Y. Chemin [88] (a simplified
proof can be found in A. Bertozzi and P. Constantin [58], see also Ph. Serfati [419]) that D(t) is in fact a bounded, smooth domain for all t > 0.
0 We conclude this (two-dimensional) section with a different topic which concerns the case when S1 = IR2 and when we no longer assume that uo E L2(1R2)2. The first systematic treatment of that question seems to be given in D. Benedetto, C. Marchioro and M. Pulvirenti [51] where the case of an initial condition uo satisfying ((4.3) and), for some C > 0,
Iuo(x)I < C(1+IxJQ),
0 < a < min (1) 2),
wo = curl uo E LP n L°O (IR2)
(4.49)
for some 1 < p < oo,
is treated. We wish to extend this analysis here to the case when 1 < p < 00 and uo E Wioc (IR2),
divuo = 0 a.e. in IR2.
Duo E LP(IR2),
(4.50)
We shall be mainly using the vorticity formulation of (4.1)
at +div (uw) = 0
in D'(IR2 x (0, oo))
(4.51)
requiring u to belong to C([0, oo); Wj1, (IR2)) (i.e. u E C([0, oo); W1"P(BR))
for all R E (0, oo)), to satisfy (4.1) (in the weak sense defined in section 4.1) and Du E C([0, oo); LP(IR2)); div u = 0, (4.52) curl u = w a.e. in R2 x (0, 00). Let us mention that (4.51), as in section 4.1, holds if p > 2 and, if p < 2, (4.51) holds in the renormalized sense, i.e. (4.51) holds with w replaced by ,Q(w) for all 0 E Cb(IR; IR).
We may now state our main existence result.
Theorem 4.4. Let uo satisfy (4.50). Then, there exists u E C([0, oo); WiOC (IR2))2,
a weak solution of (4.1)-(4.2), satisfying (4.51)-(4.52).
Remarks 4.10. 1) If p < 2, the proof below shows that one can build u E C([0, oo); Lq(IR2))2 with q = LQ (IR2).
,
choosing uo (up to a constant) in
Euler equations and other incompressible models
146
2) It is possible to extend the above result to the case when Duo E LPl + 1,P2 where 1 < p1, p2 < 00-
3) If curluo belongs, in addition to the assumption (4.50), to LO°(IR2), then one can check (by standard arguments) the uniqueness of solutions (normalized by requesting for instance fBi u dx = fBi uo dx, see the proof of Theorem 4.4 below for more details). 4) If curl uO only belongs to L°° (IR2), we do not know if the above result (or an appropriate modification of it) holds.
5) Let us mention that when p > 2, u grows at infinity at most like Ix)1-21P. More precisely, we have for some C > 0
sup J R2
Iu(x)Il, (L+Ixi2)T-P
C{IlouIILP(J.2) + IIUIILP(B,)}.
(4.53)
Indeed, if u E Wo (IR2), Vu E LP(IR2), then we have, setting a = 1 - , < COIIVUIILP(1R2) sup lu(x)-u(x')I Ix-xila x#xI
-
(4.54)
for some CO > 0 independent of u. Therefore, we have iu(x)I <- COIIVUIILP(jR2)IXI" + lu(0)I
< COIIVUIILP(IR2)Ixla + Iu(0)-1-1/PIIUIILP(B,)I
+ 7r-1/PIIUIILP(B,) COII VUIILP(JR2) (1+IxIa) + 7r-1/PIIuIILP(B,)
using (4.54). The growth IxIa is essentially optimal since u = (1+IxI2)0/2 satisfies Vu E LP(1R2) for all 0 <,3 < a. 0
Before we present the proof of Theorem 4.4, let us first make some preliminary remarks. We introduce the Banach space D1"P(IR2) = {u E Wlap(IR2), Vu E LP(1R2)2} equipped with one of the equivalent norms IIouIILP(i.2) + IIUIILP(B,) or IIVuIILP(1R.2) + I fBl uI (we could as well re-
place B1 by any ball BR for 0 < R < oo) and we restrict our attention, as in Theorem 4.4, to the case when 1 < p < oo. When 1 < p < 2, by Sobolev embeddings, we see that V 1'P (IR2) = IR + {u E LQ (IR2 ), Vu E LP (IR2)2} where q = 2- p . We shall need, in the course of proving Theorem 4.4, some
technical results on this space given by
Lemma 4.2. Let u E D1,P(TR2)2 satisfy: div u = 0 a.e. in IR2. (i) We have au 7
a V-L curlu 7
a a
where V1-
ax,
(4.55)
Remarks on Euler equations in two dimensions
147
or in other words ay is given in terms of curl u by a singular integral convolution operator whose kernel is 2L '9 ( ) and x1 = (_Xi) . (ii) There exists u'n E Co (IR2)2 such that div un = 0 on 1R2, Dun converges to Vu in Lp(IR2) and un converges to u in LP(BR) (for instance for all R E (0, oo) if p > 2, while there exists c E IR2 such that un converges
to u - c in L9(1R2)2 when 1
Proof of Lemma 4.2. i) The proof of (4.55) is straightforward since one
a v1 curlu E Lp(1R2) (using the checks easily that fj =- " + (-Q)-1 axe fact that the singular integral which is a composition of appropriate Riesz transforms is bounded on Lp(IR2)) and curl fj = 0, div fj = 0 in D'(IR2). Hence, f j is harmonic on 1R2 (harmonic gradient in fact) and thus fj = 0 a.e. in IR2.
(ii) We have already shown in Appendix A the case when p = 2. When
1 < p < 2, the proof is rather easy since u = c + v where c E IR2, v E E Lp(IR2), dive = 0 a.e. in JR2 (q = 2). Then, by "standard" density results (or proofs), we can build vn E Co (1R2) such that div vn = 0 in IR2, vn converges in L4 (1R2) to v, Vvn converges in Lq(1R2)2, Vv
Lp(IR2) to Vv and the proof is complete in this case. One possible proof in the case when p > 2 consists in several layers of
approximations. First of all, we truncate and regularize w = curl u and obtain wk E Co(IR2) which converges, as k goes to +oo, to w in Lp(1R2). Next, we consider uk defined by uk E L''(lR2)2 for all r > 2, uk E Cr (IR2)2
and decays at infinity like s , div uk = 0 in J2, curl uk = wk in IR2 (uk = 2. * * wk). Using, as in (i) above, the boundedness in Lp(IR2) of Riesz transforms, we deduce that Vuk E L'(IR2) for 1 < r < oo and that Vuk converges in Lp(1R2) to -(-A)-1O01w = Vu in view of (4.55). Therefore, if we set ck = fBl u - uk dx, uk + ck converges in Dl"p(IR2) to u as k goes to +oc. Next (using for instance the case p = 2 already treated in Appendix A), there exists uk,m E Co (IR2) 2 such that div uk,m = 0 in IR2, uk,m converges in Di"p(IR2)2 to uk as m goes to +oo, for each fixed k > 1. Finally, we choose for each k > 1 some cpk E Co (1R2)2 such that div Wk = 0 in 1R2, cpk(0) = ck and we set Wk,m = cpk(m) for m > 1. Observe that for k > 1 fixed, W k,m converges to ck in Dl,p(1R2)2 since we have
f2IvmPdx
= m2-p I
dx --4 0
as m -> +oo.
z
In conclusion, we have shown that uk,m + Vk,m converges in D1"7'(IR2) as in goes to +oo to uk + ck which, in turn, converges to u in Dl"p(IR2) as k goes to +oc, and this completes the proof of Lemma 4.2. 0
148
Euler equations and other incompressible models
Remark 4.11. It is possible to give a different proof of part (i) of Lemma 4.2 (when p > 2) using the classical approach to density results, the fact that the range of the "divergence-map" from D1,p(IR2) into Lp(IR2) is Lp(IR2) and the density of Co (1R2) in D1'p(IR2). This last fact, however, requires some justification: we can either adapt the argument given above or argue in a slightly more direct way as follows. First, approximating if necessary u by TR(u)-observe that TR(u) converges to u in D1,p(IR2) as R goes to +oo-we can assume without loss of generality that u E LOO n Di'p(IR2). Then, we consider un = cp(n)u where tp E Co (IR2), 0 <- p < 1, V =- 1 on B1, Supp W C B2. If p > 2, un, converges to u in 151,p(IR2) since we have
Dun - Vu =
[fE)]Vu+.!-Vcp()u n
n
n
and
f!v()urdx I1n
< Cn2-p11u11L=
Then, smoothing u,, by convolution allows to conclude. If p = 2, we build in the above way un E Co (1R2) such that un converges to u weakly in D1'2(IR2) and this is enough to conclude. 0
Proof of Theorem 4.4 Step 1. The case when 1 < p < 2. This is in fact the easy case since there exists c E 1R2 such that uo - c E L (IR2)2, and we observe that because of the galilean invariance of the Euler equation we can always take
c to be 0: indeed, u is a solution of (4.1)-(4.2) if and only if v defined by v(x, t) = u(x + ct, t) - c on IR2 x [0, 00) is a solution of (4.1)-(4.2) corresponding to uo - c. Therefore, without loss of generality, we may assume that uo E LP (IR2)2, Vuo E Lp(IR2)2. Then, as in Lemma 4.2 (and its proof), we introduce uo E Co (IR.2)2 such that divuo = 0 in IR2, uo converges to uo in L- (IR2) and Duo converges to Duo in Lp(IR2) as n goes to +oo. In view of Theorem 4.1, we can solve (4.1)-(4.2) with uo replaced by uo and we find a smooth solution un on IR2 x [0, oo) such that wn = curl un solves (uniquely) 8wn
un Own = 0 at +
wn lc=o
= wo (= curl uo)
in
IR2 X (0, 00),
in
R2.
(4.56)
Since, for each n > 1, un is bounded on IR2 x [0, T1, we deduce that wn E Co (IR2 x [0, T]) for all T E (0, oo). In addition, because of (4.12), wn
Remarks on Euler equations in two dimensions
149
is bounded in C([0, oo); LP(IR2)) and thus Dun is bounded in C([0, 00); LP(IR2)) by Sobolev inequalities. This implies that un is bounded in C([0, oo); L3 (IR.2)). Extracting a subsequence if necessary, we may as(IR2 x (0, T)) (V T E sume that un converges weakly to some u in L (0, oo)). Since wo converges in LP(IR2) to wo =curl uo, we can now use the general convergence results of R.J. DiPerna and P.L. Lions [128]-see section 2.2 in chapter 2 for results of a similar nature-to deduce that wn converges in C([0, T]; LP(IR2)) (V T E (0, oo)) to the unique renormalized solution w of (4.51) and of course w = curl u. Then exactly as above we deduce that un, Dun converge respectively to u, Vu in C([0, T]; L4 (IR2)), C([0, T]; LP(1R2)) for all T E (0, oo). This is enough to conclude.
Step 2. The case when p > 2. We use Lemma 4.2 to introduce uo E Co (1R2)2 such that uo converges to uo in D"P(IR2)2 as n goes to +oo. We
then follow the argument given in step 1 and obtain un, wn, as in step 1, which satisfy: Vun, wn are bounded in C([0, oo; LP(IR2)). We then set Cn(t)
fal
(un(t) - uo) dx
and define un by un (x, t) = un (x + ft cn (s) ds , t) - cn (t). We next set n = Pn + di . X, con = wn (x + fot cn (s) ds, t). Then, one checks that (un, fn) solves (4.1)-(4.2) with uo replaced by uo and that (4.56) holds for G!n with un replaced by fin. Let us also remark that Dun and Con are still bounded in C([0, oo); LP(IR2)), and in addition we have now, because of the choice of cn(t): for all t > 0, AB, 4Ln(t) dx = fBl uo dx fBluodx. n
Therefore, un is bounded in C([0, 00); D1"P(IR2)), and extracting a subsequence if necessary, we may always assume that un converges to some u E L°O(0, 00; D1,P(IR2)) in LQ(0, R; W1"P(BR)) weakly for all R E (0, oo),
1 < q < oo (and Dun converges weakly to Du in L4(O,T;LP(1R2)) for all T E (0, oo), 1 < q < oo, and JBl u(x, t) dx =1B uo(x) dx for all t > 0). Next, as in step 1, we wish to deduce the convergence of ion to w curl u) in C([0, T]; LP(IR2)) (V T E (0, oo)) from (4.56) where w is the unique solution of (4.51) (in C([0, T]; LP(IR2))). In order to apply the convergence results due to R.J. DiPerna and P.L. Lions [128] (see also section 2.2 in chapter 2), we simply need to check that we have for all T E (0, oo)
lul(1+lxl)-1 E L1(0, T; L1(IR.2)) + L°°(IR.2 x (0,T)).
(4.57)
Once this is checked, we conclude as in step 1 that Vun converges to Vu in C([O,T]; LP(IR2)) for all T E (0, oo) and thus u' converges to u in
Euler equations and other incompressible models
150
C([0, T]; Dl,p (IR2)) for all T E (0, co), as n goes to +oo. These convergences then yield the conclusions of Theorem 4.4.
When p > 2, (4.57) follows immediately from the fact that u LOO (0, oo; D1'p(1R2)) and (4.53): indeed, Jul
(1+JxI)-0'
E
E LO°(IR2 x (0, oo))
for some 0 < a < 1. When p = 2, we write u = u1 +u2 where ui, u2 are defined by
div ul = div u2 = 0,
curl ui = w 10W151),
curl u2 = w 1(1,1>1)
a.e. in IR2 x (0, oo),
u1dx=IB uodx, fB u2dx=0
lB1 ur, u2 E -L°°(0, oo; D1,2(IR2)).
Clearly, w 1(1,,,1:5 1) E LOO (0, oo; L2nL0°(IR2)), hence, as we have done several 151"p(1R2)) for 2 < p < oo and, as we just saw
times before, ui E L' (0, oo; above, i+lzl E L°°(IR2 x (0, oo)).
On the other hand, since w E Loo (0, oo; L2(IR2)), w 1(1w1>1) E L°°(0, oo;
L' n L2(IR2)) and thus u2 E L°O(0, oo; Di'p(lR2))2 for 1 < p < 2. In particular, as seen above, u2 = c(t) + to where to E L°°(0, oo; LQ(IR2))2 for 2 < q < oo and thus c(t) = JB1(u2 - w)(x, t) dx = f81 w(x; t) dx E L°°(0, oo). Therefore, u2 satisfies (4.57) and we conclude.
O
4.3 Estimates in three dimensions? The incompressible (homogeneous) Euler equations (4.1)-(4.2) in three di-
mensions (N = 3) are far from being understood. Indeed, the only information that is available concerns short time existence. More precisely,
in the case of Dirichlet boundary conditions or in the periodic case (or when SZ = IRN), it is known that if uo is smooth enough (uo E X where X = H' with s > 2 , Cl" with 0 < a < 1), then there exists a maximal time interval [0, T*) (T* < +oo) such that there exists a unique solution u E C([0,T];X) (V T E (0,T*)) of (4.1)-(4.2) (and with the appropriate boundary conditions of course) and if T* < oo, II2c(t)Ilx goes to +oo as t goes to T* -. Of course, the crucial question which is still completely open is to decide
whether T* < 00 or not. J.T. Beale, T. Kato and A. Majda [28] (see also G. Ponce [391] for a variant involving the deformation tensor instead of the vorticity) have established a fine criterion for the finite time blow-up of smooth solutions: let T E (0, oo); if there exists a (unique) solution u E C([0,T);X) such that fo Ilcurl u(t)IILeC dt < oo, then T* > T. Or in
Estimates in three dimensions?
151
LT
other words, if T. < oo, then ' (lcurlu(t)IILc dt = +oo. We do not wish to re-prove this statement here. Let us simply mention that the main idea behind this criterion is the following: if fo II curl u (t) II Lx dt < oo then one can bound II u(t) II x uniformly on [0, T) and, using the short time existence result, one can continue the solution on a larger interval than [0, T]. The appearance of singularities (i.e. the breakdown of smooth solutions) in finite time is an outstanding open mathematical problem, whose solution would have a serious impact on three-dimensional incompressible fluid mechanics. After many years of intensive numerical simulations which were inconclusive, two recent independent numerical experiments by R. Grauer and T. Sideris [193] and later by R. Kerr [262] indicate possible breakdowns of smooth solutions in finite time (and certainly violent growths
of Ilcurlu(t)IIL' that, for all practical mathematical purposes, make it difficult to believe in a priori estimates on curl u). The striking difference between the cases N = 2 and N = 3 is also illustrated by the so-called "2 + 1/2" dimensional flows, used in R.J. DiPerna and A. Majda [129] to provide examples of weakly convergent sequences of solutions of 3D Euler equations whose weak limits do not satisfy (4.1), namely solutions of (4.1) say in the periodic case (for instance) such that u is independent of x3. In that case, (u1, u2) solve the Euler equation (4.1) in two dimensions (in the periodic case) while w = u3 simply solves the following transport equation
+ div (uw) = 0 in IR2 x (0, oo),
wlt_o = wo
in IR2
(4.58)
and w, wo are periodic in x1, 22 (of periods T1, T2 > 0). The decoupling between (u1, u2) and w allows us to solve first for (ul, u2), using the results of sections 4.1-4.2, and then to solve (4.58) in a classical
way when u is smooth (or Lipschitz, or almost Lipschitz). When wo = curl uo (_ u1(0) - ayl U2(0)) E LP(S2) (Q = (0, T1) x (0,T2)) with p > 1, resp. wo I log wo I E Ll (St), then we have seen that there exists at least one solution of the Euler equation u E C([0, oo); W1,P(St)), resp. W1"101) In this case, there exists, as soon as wo E L9(S2) (1 < q < oo), a unique solution of (4.58) which belongs to C([0, oo); L4 (S2)) if q < 00 (and to C([0, oo); L''(Sl)) for all 1 < r < oo and to L°°(SZ x (0, oo)) if q = +oo)-see R.J. DiPerna and P.L. Lions [128]. These flows provide examples of global weak solutions of the 3D Euler equations which are smooth (and thus unique) if the initial conditions are smooth. But these flows, as was observed by R.J. DiPerna and the author [124], also show that solutions (even smooth ones) of the 3D Euler equations cannot be estimated in W 1,P, for 1 < p < oo, on any time interval (0, h) if the initial data is only assumed to be bounded in W1,P. In
Euler equations and other incompressible models
152
other words, intermediate a priori estimates or, more precisely, W 1,P a pri-
ori estimates do not hold in three dimensions-let us emphasize the fact that they do hold in two dimensions, see sections 4.1-4.2 above. Indeed, we choose for (u1, u2) stationary flows of the 2D Euler equations namely (u1(x2),0) where ui is smooth, periodic in x2 (of period T2). Then, the solution of (4.58) is simply given by wo(:r1--tui (x2), x2). The lack of a priori estimates in W1," is then clear choosing for instance u1j(x2) to behave like (E2 + x2)e/2 near x2 =_O (and smooth elsewhere uniformly in e) and
near 0 (and smooth elsewhere wo(x1,x2) to behave like (e2 + uniformly in e) where e E (0,1], 9 E (0,1), and we take 9 # . , otherwise log modifications have to be made. Then, obviously, (ui, wo) are bounded in W", uniformly in E, for 1 < p < 110 while for q > 1 we have 1x12)°-1/2
l awE(t) 1 q
axe
dx1 dX2
aC9W C
t'
= tq >
x0
q
(x1-tu1 (x2), x2) I
ax0 (yl, x2)Iq
"12
l
(t1)'(x2)lq dyl
1x1(9
2
1
dr2
1x214
q
JB6 ( e2 + l x l2 )(3-O)q
(
e2
+ x22) (1-3)q
dxi d22
for some 6 > 0 sufficiently small, independent of e, where v = 129 -11q 9q > 0. Since we have for some constant C(9, q) > 0 (if 1 < q < 1198 ) b
11lq
1x1(3
Ba
26)q
x21(1-9)q
dxi dx2 = C(9, q) f 73(1
1
B)q
r dr,
we deduce that 8" is bounded in Lq uniformly in a (for any fixed t > 0) if and only if q < 3(12 9) , and in particular it is not bounded if 3(12 9) < q < 1
11-81
This construction shows that, for each 1 < p < oo, t E (0, oo), E E (0, 1), S E (0, 1), there exists a smooth (periodic) solution u of Euler equations in < e and ll u(t) II W I,p > a . This also three dimensions such that ll u(0)1l w2,p
gives examples of smooth flows such that l l curl u (t) l l L-
> t ll curl u(0) l) L-
Let us conclude by mentioning that there are other known smooth regimes for three-dimensional Euler equations like the axisymmetric case without swirl (see A. Majda [316] and Ph. Serfati [420] for more details). But, even in that case, J.M. Delort [119] observed that the "vortex sheet" problem (see section 4.2 for a slightly more detailed presentation of this problem) is mathematically quite different from the pure two-dimensional case.
Dissipative solutions
153
Finally, let us observe that the lack of intermediate a priori estimates and more specifically of bounds that yield the compactness of appropriate sequences of solutions (or approximated solutions) has made impossible until now the construction of weak solutions in C([0, oo); L2) or even LO°(0, oo; L2) satisfying (4.1) in the sense of distributions. Weaker notions of solutions are considered in the next section.
4.4 Dissipative solutions As we have seen in the preceding section, even the global existence of weak
solutions of the Euler equations is not known in three dimensions. On the other hand, an obvious bound in C([0, oo); L2) follows trivially from the (formal) conservation of energy. It is then natural to attempt building up solutions in a weaker sense than in the distributions sense. A very weak notion (relying on relaxed Young measures or relaxed measure-
valued solutions) is proposed by R.J. DiPerna and A. Majda [129] but the relevance of this notion is not entirely clear since it is not known that "solutions" in the sense of [129] coincide with smooth solutions as long as the latter do exist. We shall propose here a different notion of "very weak" solution that we call dissipative solutions, for reasons we shall explain later on. This notion seems to be new and the idea behind the notion appears in P.L. Lions [306] in the context of Boltzmann's equation. We wish to emphasize immediately that we are not convinced that such a notion is neither relevant nor useful. Its only merits are: 1) such solutions exist, 2) as long as a "smooth" solution exists with the same initial condition, any such dissipative solution coincides with it, 3) we shall use it in some small Mach number limits in chapter 9 to pass to the limit from some compressible models to the Euler equations. From now on, we take N > 3 and we only consider the periodic case
in order to simplify the presentation and keep the ideas clear, although everything we do below can be extended or adapted to the case when St = IRN or to the case of Dirichlet boundary conditions. This is why all the functions considered below are always assumed to be periodic. The initial condition uOr e 4.2) is always assumed to belong to L2(&1)N and to satisfy (4.3) in Let us first explain the formal idea of this new notion. Let us consider a smooth test function v on IRN x (0, 00) such that div z: = 0 on IRN x [0, oo). We define
E(v)
at - P((v V)v)
(4.59)
Euler equations and other incompressible models
154
(recall that P is the projection onto periodic divergence-free vector fields), and we write vo = vlt=o If u is a solution of (4.1)-(4.2), then we can write
a
U
+u
v) (u-v) + (u-v) Vv + oir = E(v)
in
IRN x (0, 00)
for some scalar function ir. Then, multiplying this equality by u - v and integrating (over the period), we find
d fnIu-vI2 dx = -2.Jdl/s (d(u-v), u-v) dx + J 2E(v) (u-v) dx (4.60) s2
z
where d(= d(v)) = (2 (88vj + 8jvi)) _j:- We then set (for each t-> 0)
suP -( u K1=1 and we deduce from (4.60) dt
in
Iu-vI2 dx < 211d- II00 in I u-vI2 dx + 2J E(v) (u-v) dx.
(4.61)
Hence, in particular, we have for all t'> 0
f Iu-vI2 (x, t) dx < exp n
+2
rt
Jo
ds
f
n
(21t II
d- II00 ds
f
Iuo -vol2 dx
sz
(ft
dx exp
IHdII. dv E(v) (u-v).
The definition of dissipative solutions of (4.1)-(4.2) given below consists precisely in requesting (4.62) to hold for an appropriate class of v. The reason why we call these solutions dissipative solutions is the fol-
lowing: if we take v - 0 then obviously E(v) - 0, d _< 0 and (4.60) is nothing but the (formal) conservation of energy while (4.61) and (4.62) are standard relaxed energy inequalities which allow for a possible loss (dissipation) of energy though various losses of L2 compactness (via oscillations, concentrations, etc.). We may now give the precise Definition 4.1. Let u E LO( 0, oo; L2)N n C([O, oo); L2 - w) N. Then u is a dissipative solution of (4.1)-(4.2) if u(0) = uo, div u = 0 in D'(1RN x (0, 00))
and (4.62) holds for all v E C([0, oo); L2)N such that d E L1(0, T; L°°), E(v) E L1(O, T; L2) (d T E (0, oo)) and divv = 0 in D'(1RN x (0, oo)).
Dissipative solutions
155
Remark 4.12. 1) It is worth pointing out that the main regularity requirement for v in the above formulation, namely d E L1(0, T; L°°), is the same as in the blow-up criterion obtained by G. Ponce [391]-and, in fact, for similar reasons. 2) If d E LP(0, T; LOO) (or curl v E LP(0, T; L°°)) for some p E [1, ooJ then Dv E LP(O, T; L9) for all 1 < q < oo and Dv E LP(O, T; BMO): in particular, there exists A(t) > 0 E LP(O, T) such that if Ix - yl < 1/2 jv(x, t) - v(y, t)I
< A(t)jx-yjI log lx-yII
(4.63)
3) When E(v) = 0 (for instance) and Dvo E LQ for some 1 < q < oo then d E L1(0, T; L°°) implies formally that Dv E L°° (0, T; L4). Indeed, we observe that for i ',O j U a + v V) (ajvi
- aivj)
= 8,Vk 8kvi - aivk akvj = 28jvkdik-28ivkdjk.
Therefore, we deduce
dt llcurlvlliq < CoglldilL- IIDv11Lq Ilcurlvlligi <- C(q) IIdIIL- Ilcurlvlli4 and our claim follows easily.
4) Let us observe that if d E L' (O, T; L°°) then E(v) E L' (0, T; L2) if and only if Ft E L1(0, T; L2). Indeed, Wt + E(v) = -P((v 0)v) and Dv E L1 (0, T; LQ) for all 1 < q < oo. In particular, (v 0)v and thus P((v . V)v) E L1(0, T; Lt) for all 1 < r < 2 since v E LOO (0, T; L2). Furthermore, for each 1 < i < N, we have
(v V)vi = v; 8jvi = vj(ajvi + aivj) - ai (2 ivl2) 2d v - 81
Iv12).
Therefore, P((v 0)v) = 2P (d v) E L1(0, T; L2) since d E L'(0, T; LOO) and v E LOO (0, T; L2).
O
An obvious consequence of the definition is the following
Proposition 4.1. If there exists a solution v E C([0, T]; L2) of (4.1)-(4.2) on IRN x (0, T) satisfying d E L1(0, T; LOO) for some T E (0, oo), then any
dissipative solution u of (4.1)-(4.2) is equal to v on IRS' x [0, T].
0
Euler equations and other incompressible models
156
Indeed, E(v) = 0 and we conclude!
Before we give and prove our existence result for dissipative solutions, let us mention that, in Definition 4.1, we can replace the regularity requirements on v by much stronger ones. In other words, it suffices to check (4.62) for smooth test functions v. Indeed, if (4.62) holds for smooth v, then we can check it also holds for the class of v described in Definition 4.1 and in Remark 4.12 (4) by a straightforward regularization procedure. More precisely, let pE _ - p(E ), p E C0, (IRN ), f'N p dx = 1, Supp p C B1. We set of = v*pE. Obviously of converges to v in C([O,TJ;L2), a'U E L'(0,T;Cb ) for all k > 0, div vE ='O in IRN x (0, oo), vE E C([0, T]; Cb) for all k > 0, T E (0, oo) and
E(v,) = =
P((v, O)ve) l [- 8v - P[(v . O)v]] * pE + [P[(v at
.
= E(v)*pE+2[[P(d.v)] *pE - P(de =
V)v] * Pr - P((vE V)vE)] ve)]
in view of Remark 4.12 (4). Obviously, E(v) * pE converges to E(v), as e goes to 0+, in L1(0, T; L2) (V T E (0, oo)) and so does E(vf) provided we show that (d v) * pE - dE vE converges to 0, as c goes to 0+, in L1(0, T; L2) (d T E (0, oo)). Since d E L1(0, T; L°°), v E C([0, T]; L2) (V T E (0, oo)), this is straightforward. Observe in addition that jl df ll oo < Ild 11oo for all c > 0, a.e. t > 0 and that Ildf 11. converges to Ild 11. in L1(0, T)
(V T E (0, oc)). We then apply (4.62) with v replaced by vE and we conclude letting a go to 0 in view of the convergences collected above. Let us observe that by a second layer of approximation (regularizing in t), we can take v smooth on IRN x [0, oo) in (4.62). We can prove now the existence of dissipative solutions.
Proposition 4.2. There exists at least one dissipative solution of (4.1)(4.2).
Proof of Proposition 4.2. We consider u,,, a weak solution of the NavierStokes equation (see section 3.1),
+ (u,, D)u - vAu + Op = 0, div u, = 0 in IRN x (0, oo) (4.64) satisfying (4.2) and the energy inequality
df 2
s
IVU
, dx < 0
in D'(0, oo).
(4.65)
Dissipative solutions
157
Recall that uY E L2(0, T; Hl) (V T E (0, oo)) fl L°°(0, oo; L2) n C([0, oo); goes to uo in L2(SZ) as t goes to 0+). L2 w) (in addition, We next consider v as in Definition 4.1, and, as shown above just before the statement of Proposition 4.2, we can take v arbitrarily smooth on IRN x [0, oo). Then, multiplying (4.64) by v we find
-
dt fn
u v dx = nu at + (ut 0)v u dx
+v1
0
n
in D'(0,oo),
and thus, by definition of E(v) and since div u = div v = 0, we deduce d
I u v dx + u E(v) dx J J
in
Wt-
+ vJ Vu, Vv dx = 0
dx l
V]v
n
n
l
in D'(0, oo).
n
In addition, we have for all t > 0 d
1v12
dt,fn
2
Jv
dx
dx
n
e
J
v[-E(v) - (v V)v] dx (4.67)
- Jn v E(v) dx.
Combining (4.65), (4.66) and (4.67), we deduce
-
t
[!Iut,_v12dx 2= -Jf[(ut,_v).v]v.(ut,_v)dx n
+ vJ Du, Vv dx + fn E (v) n
dx
or
d
dt Jn
dx = -2
r
uv-v) dx
n
+ 2v fn Vu Vv + 2J E(v)
dx 1/2
< 21ld-fl Jn
+
fn 21E(v)
(u,, -v) dx
IVuI2dx
Euler equations and other incompressible models
158
and finally for all T E (0, oo) and for all t E [0, Tj
I
Iuv-vl2(x,t) dx <
e2fo Ild-I1,o da
tds 00
dx e2
f
(4.68)
_
e
+2
luo-voI2dx
IId
ilx do E(v) . (u,,-v) + CTV1/2,
S
for some positive constant CT which is independent of v. Here, we used the fact that (4.65) yields a bound, uniform in v, on v f °O dt f dx I Duy I2. Finally, we observe that as = -ajP(u,,,ju) v1/2ajP(v1/2aju) and
-
thus
is bounded in L2 (0, oo; H-1) + L°° (0, oo; W - (1+s),1) for all s > 0.
Extracting a subsequence if necessary (see Appendix C for more details) we may assume that u converges to some u weakly in L°° (0, oo; L2) -
and weakly in L2 uniformly in t E [0, T] for all T E (0, oo). And u E L°° (0, oo; L2) n C([0, oo); L2
- w), divu = 0 in D'(IRN x (0, oo)), ult=o =U0
in IRN and passing to the limit in (4.68), as v goes to 0+, we recover (4.62) and Proposition 4.2 is shown.
4.5 Density-dependent Euler equations In this section we briefly review the "state of the art" concerning the density-dependent Euler equations or, in other words, the inhomogeneous incompressible Euler equations.
We look for a non-negative scalar function p(x, t) (the density of the fluid) and for a divergence-free vector field u(x, t) (the velocity of the fluid) which are solutions of
5 + div (pu) = 0, div u = 0 aput
(4.69)
+ div (puu 1 ) + Op = 0 ,
for
1
for some scalar function p, the so-called hydrostatic pressure. We consider
as usual the periodic case (the unknowns are periodic and the equations hold in RN x (0, oo)), the case when SZ = RN or the case of Dirichlet boundary conditions where the equations are set in a smooth, bounded, open connected set in IRN and u n = 0 on all where n denotes the unit outward normal. Of course, we complement (4.69) with initial conditions PIt=o = P0,
Pul t=o = mo
(4.70)
where p0 > 0. In fact, exactly as in chapter 2 for the inhomogeneous incompressible Navier-Stokes equations, the precise meaning of the initial
Density-dependent Euler equations
159
condition for pu has to be interpreted correctly but in the simple case when we assume that po satisfies for some 0 < a < Q < 00
a < Po < Q then, we may set uo = P and we require uo to satisfy (4.3) (in
(4.71) 1RN
or in
S1 in the case of Dirichlet boundary conditions). Obviously, (4.69) contains the usual incompressible Euler equations as a particular case: take p = 1 in (4.69). Therefore, we cannot expect to know
more about this system of equations than for the incompressible Euler equations and in particular the case when N > 3 seems rather hopeless as far as the understanding of global solutions is concerned. But even if we concentrate -on 'N = 2 in the rest of this short - section, -this will not help much since very little is known even in this case. Of course, (4.69) is well posed locally in t provided (4.71) (and (4.3)) holds and we choose po, uo to be smooth enough but, even when N = 2, it is not known whether smooth solutions persist for all t > 0 or break down in finite time. Again, there seems to be some numerical evidence of finite time breakdown but this has yet to be confirmed systematically. There are very few known a priori estimates. Of course, (4.69) implies, at least formally, that we have for all t > 0 and for all 0 < a < b < oo:
meas {x / a < p(x, t) < b} = meas {x / a < po(x) < b}.
(4.72)
In particular, if po satisfies (4.71) then p(t) also satisfies (4.71) for all t >_ 0. The conservation of energy is obtained by multiplying the second equa-
tion of (4.69) by u and integrating by parts using the first equation, and reads
d
dt
fPIuI2dx = 0
for t > 0.
(4.73)
When (4.71) holds, this yields an estimate on u in C((0, oo); L2).
To the best of our knowledge, these are the only known a priori estimates even when N = 2. It is also worth remarking that the failure of intermediate a priori estimates that we showed in section 4.3 on the incompressible Euler equations when N > 3 can be established for (4.69) when N = 2 using in fact/ the same examples: u1 U&2), u2 0, p 0 and P(x, t) = Po(x1-tul (x2), x2) Let us conclude by mentioning a remarkable identity which however does
not help-or at least does not seem to help-to analyse (4.65) mathematically. Still for N = 2, we consider the vorticity w = a2u1 - a1u2 and we write at least formally
Euler equations and other incompressible models
160
hence, taking the curl (and using the fact that div u = 0)
a
pa2P = 0.
(4.74)
In particular, we deduce for all smooth function ,Q from IR into IR
at
{w,3(P)} + (u V){w,Q(P)} = Q'(P){a1(!)82(p) Q1(P)
{a2Paip
P2
- 82(p)alp}
- aiPa2P} = a2['Y(P)]aip - ai['Y(P)]a2P
= a2 {7(P)a1P} - ai {'y(P)a2p}
where Y = A e . Then, we integrate the resulting equation and, at least when f = IRN or in the periodic case, we deduce immediately dt
wQ(P) dx = 0
for all t>0
(4.75)
for any function ,Q from lit into IR..
4.6 Hydrostatic approximations There exists a huge literature on the so-called hydrostatic approximations of incompressible models and a considerable number of models have been proposed (see for instance J. Pedlosky [386] on geophysical flows, and P. Constantin, A. Majda and E.G. Tabak [103], J.L. Lions, R. Temam and S. Wang [296],[297],[298]) with applications to oceanography, meteorology, geophysical flows and the huge variety of quasi-geostrophic models. Some models have been analysed mathematically or implemented numerically but, to the best of our knowledge, very little seems to be known on the model we discuss in this section which is the inviscid version of a very classical model. Our motivation for restricting our attention to this model is mostly mathematical since we hope it could help to understand some of the singular features of the classical three-dimensional incompressible Euler equations.
Let us first present this "hydrostatic" inviscid model. We consider a three dimensional strip S = {(x1i X2, z) E lR3 / 0 < z < 1} and we look, to simplify the presentation and the notation, at a situation where all unknown functions are required to be periodic in x1 and in x2 (of periods
respectively T1 and T2 > 0). We could consider as well the case when
Hydrostatic approximations
161
(x1, x2) E w where w is a smooth bounded connected open set in IR2 and then we impose "Dirichlet boundary conditions" on aw x [0,1] but we shall not do so here. From now on, all differential operators V, div, A, curl refer to the two-
dimensional operators (acting on x1i x2) and we also define al =
a2=,a3=azaZ
We look for a velocity field (u, v) = (ul(xl, x2, z), U2(X1, X2i z), v(xl, x2, z)) (E IR3 for all (x1i x2, z) = (x, z) E S) and for a scalar function p (the
pressure as usual) satisfying
a
z
div u + a = 0
u+vp = 0
LP
az-
=0 (4 . 76)
in S x (0, oo)
vIZ-o = vIz=1 = 0
on
(4.77)
IR.2
(recall that all functions are required to be periodic in xl and in x2). Observe that (4.72) is nothing but the system of incompressible Euler equations in S "with v = U3" satisfying Dirichlet conditions on {z = 0, 1}, where the third equation on v is replaced by = 0. In other words, we neglect in the third equation for v the term 37 and simply write = 0 (the so-called hydrostatic approximation; in the presence of gravity
terms we can simply replace a = 0 by a = a for some fixed constant a but this does not modify (4.72) since we can then consider p - az). Let us immediately mention that (4.76)-(4.77) contains as a particular case the usual two-dimensional incompressible Euler equations: indeed, take v = 0; then u = u(xl, x2) solves the 2D Euler equation. Let us also observe that the energy fnx (0,1) 1uI2 dx dz is conserved (at least formally): indeed, multiply (4.76) by u and integrate over Sl x (0,1) to find
dt
f
2
x01
2
12 dx dz +
fn
(L2 + p) v (x, l) -
2
(L
2
+ p) v (x, 0) dx = 0
and our claim is proved in view of (4.77). There are various equivalent formulations of (4.77). Let us mention at least one. We define 7 = 7(x) = fo" cp(x, z) dz for an arbitrary function cp (periodic in x) on S. Then, if we integrate (4.76) in z from 0 to 1, we find using (4.77)
-
au-c
+ div u ®u + Vp = 0, div u = 0 in
IR2
(4.78)
Euler equations and other incompressible models
162
(recall that p = p(x, y) in view of (4.76)). In particular, we deduce taking the divergence of (4.78)
-Op = a? (utuj)
in
(4-79)
IR'2
and p (which is periodic) can be normalized to satisfy fn p dx = 0 for all t > 0. Then, it is possible to rewrite (4.76)-(4.77) as follows: u = u(x, z), p = p(x) solve i§F
(fZdivu(x)de)u + (u - 0)u -+Vp = 0 insx (O o0)
div u = 0 in IR2,
pdx=0
for
-1p = 8 (uluj)
in IR2,
t>0. (4.80)
Indeed, we have checked above that a solution of (4.76)-(4.77) satisfies (4.80). Conversely, we set v = - fo div u(x, ) dd, and obviously (4.76) holds, vlz=o = 0, and it just remains to check that vlz=1 = 0 in order to prove our claim. But, vlz=1 = - fo div u(x, C) 4 = -div u = 0 in 1R2, and we conclude.
In fact, it is possible to simplify (4.80) slightly. First of all, we prescribe initial conditions
ult=o = uo
in S
(4.81)
where uo (periodic in x) satisfies
div Uo = 0
in
(4.82)
1R2.
Then, we claim that (4.80) is equivalent to
a5T+(u-V)u- ( fo zdivu(x,
)d
uz+Vp = 0 in Sx (0,oo)
- Op = e (u{uj) in 1R2, 1 pdx = 0 fort > 0. n
} (4.83)
Indeed, we just have to check that v = - fo div u(x, ) ds satisfies: vlz=1 = 0. In order to do so, we write the first equation of (4.83) in conserved form
at +div(u®u)+ez(vu)+Vp = 0
in
S x (0, oo).
Then, we integrate this equation in z from z = 0 to z = 1, we take its divergence and we obtain in view of (4.79)
8t (div
i) + div {v(x,1) u(x,1)} = 0
in 1R2 x (0, oo)
Hydrostatic approximations
163
or, in other words, setting w(x, t) = v(x, 1, t) = -(div9)(x, t)
a - div {u(x,1, t)w} = 0
in IR2 x (0, oo).
(4.84)
Then, (4.82) shows that wlt=o - 0 on JR2 . This fact combined with the transport equation (4.84) allows us to conclude: v(x, 1, t) = w(x, t) = 0 on R2 x (0, oo). We continue our formal discussion of this model to propose a heuristic derivation of it from the three-dimensional Euler equations. We feel that any rigorous justification of this derivation would be a useful contribution. This derivation is similar to the one proposed by 0. Besson and M.R. Laydi (61] in some viscous situations. We consider the usual three-dimensional incompressible Euler equations in a shallow strip SE = {X E IR3 / X = (x, z) E R2 x (0, E) } where e > 0:
a+ U V x U+ V x P
0
diva (U) = 0
in S£ x (0, 00),
(4.85)
in SE x (0, oo)
with the following boundary conditions (4.86)
UIZ=o = UIZ=e = 0
and, for instance, periodic boundary conditions in x. We then assume that the initial conditions take the following form
UIt=o = (uO(x,),EvO(x.)) = Uo
in Se
(4.87)
where uo, vo are given functions on S with values respectively in 1R2, JR.
Then, requesting that Uo satisfies the boundary conditions (4.86) and divX Uo = 0 in Se amounts to requiring divuo + a9vo = 0 in S,
voIZ=o = voIZ=i = 0
in JR2.
(4.88)
Next, if Ue is a solution of (4.85)-(4.86) corresponding to the initial condition Uo, we may try to write: Ue (x, z, t) = (ue (x, E , t) , Eve (x, E , t)), Pe (x, z, t) = pE (x, A, t) in Sc x (0, co) where uE, ve are now defined on S x (0, oo) with values respectively in ]R2, IR. In that case, (4.85) and (4.86) become
at
+ (uE V)ue + veaaue + Op' = 0
Fe
divuE+
in
S x (0, oo), )
=0 (4.89)
(Ue V )V-E2
at +
.
div ue + az = 0
+ veazve) + a9pe = 0 in S x (0, oo),
in
S x (0, oo
164
Euler equations and other incompressible models
Velz=o = vEIz=1 = 0
in IR.2 x (0,00).
(4.90)
Then, at least formally, we expect that, as e goes to 0+, (uc, vE, pE) "converges" to (u, VIP) solving (4.76)-(4.77). This is why we think that the study of the model (4.76)-(4.77) might shed some light on the threedimensional incompressible Euler equations.
APPENDIX A Truncation of Divergence-free Vector Fields in Sobolev Spaces We sketch here a general procedure to approximate in W1,P (1 < p < oo) divergence-free vector fields, vanishing on the boundary in the case of a bounded region, by compactly supported divergence-free vector fields. More precisely, we consider u E Wo'P(S2) (resp. W1,P(IRN)) where St is a bounded, connected, Lipschitz domain in IRN (N > 2). We assume
divu = 0
a.e. in 9
(resp. IRN).
(A.1)
We then set 06 = {x E St, dist (x, 8S2) > 6} (resp. B1,,6 = {x E IRN / IxI < 1/6}) for 6 > 0 small enough in case SZ is smooth, otherwise we choose f26 to be a smooth, connected domain satisfying {x E S2, dist (x, 8S2) > 6} C
S26CSl6CQ. Next, we solve the following (linear) Stokes problem in 06
- Dub + u6 + Vp6 = -Au + u U6 E Wo'P(SZ6),
div u6 = 0
in
SZ6,
a.e. in Q.
(A.2)
If 12 is bounded, we can of course skip the zero-order terms u and u6 in equation (A.2). In view of classical results on Stokes problems (see (4721 for instance), there exists a unique solution (u6i p6) of (A.2) in W01'P(S26) X
(LP(16)/IR). If we request 06 to be "Lipschitz uniformly in 6", which is the case if we simply take S26 = {x E f , dist (x, 8S2) > 6} or B116i we can in fact normalize p6 in such a way that IIp611LP(c26)
: CIIuIIW1,P(s26) .
(A.3)
On the other hand, we always have IIUoIIW1-P(C26) < CIIuflw1.P(sa6) ,
where C denotes various positive constants independent of 6 > 0, u.
(A.4)
Appendix A
166
In fact, if p = 2, (A.4) takes a simpler form since, multiplying (A.1) by u6, we obtain
f IVU6I2 + Iu612 dx _< j Vu Dub + uu6 dx
(A.5)
6
6
and thus n Iou6I2 + Iu512 dx < 6
Jn 6
(DuI2 + IuI2 dx.
(A.6)
We next claim that, as 6 goes to 0+, u6 converges to u in Wo'P(S2) (resp. W1,P(IRN)) In order to make this claim meaningful, we have to extend u6 to St (resp. IRN) by 0: in doing so, we preserve the nullity of divu6 now in St (resp. in IRN). We next prove the (strong) convergence in Wo'P(S2) by two slightly different arguments, the first one in the case p = 2 where we use the simple relation (A.6) while the second one will be valid for all
1 < p < oo. First of all, we observe that in all cases, u6 converges to u weakly in Wo'P(S2) (resp. W 1,P(IRN)): indeed, extracting a subsequence if necessary, we may assume that u6 converges to u weakly in Wo'P (fl) (resp. W1'P(IRN)) while p6 converges to some 7r weakly in LP (.Q) (resp. LP(IRN)). Then we have
-0(u-u) + (u-u) - V7r = 0
in
1
(resp. in IRN),
u-u E WJ'P(c) (resp. W1,P(IRN)), div (u-u) = 0 a.e. in RN,
(A.7) (A.8)
and this implies, by the uniqueness for the Stokes problem, u = u, it = 0 (recall that 7r E LP) if SZ = IRN, zr is a constant otherwise.
Then, if p = 2, (A.6) implies the strong convergence ! In the general case, using the linearity of the construction and the bound (A.4), we deduce that it is enough to prove the strong convergence whenever u E Wo'4(f ) (resp.
u E W 1,q(1N) n W1,r(1N)) for some q E (p, +oo) (resp. q E
(p, oo), r E (1, p), r < 2 < q). We use here the density of "smoother" divergence-free vector fields in W1'P7 although it is possible to give slightly more complicated proofs of the strong convergence which do not use this fact. We first claim that us-u converges to 0 in C1, say, on compact subsets of 12. Indeed, taking the divergence of the equation (A.2), we obtain
-1p6 = 0
in
116
(A.9)
and we have already shown that p6 converges weakly to a constant (resp. 0) in LP. Hence, Vp6 converges to 0 in C1, say, on compact subsets of Q. Since -A (us - u) + u6 - u = -V p6 and u6 - u converges weakly to 0 in
Appendix A
167
W 1,P,
we deduce easily from the regularity results on Laplace's equation the convergence of u6 -u to 0 in C1, say, on compact subsets of fl.
On the other hand, u6 is bounded in Wo'q(S2) (resp. W1,q(IRN) n W 1 (RN)) Then, if SZ is bounded, we write, for 6 < So, using Holder's inequality and the W1,q bound IV(u6-u)IP +Iu6-uIP dx
J
< meas (SZ6o) sup(IV(u6-u)IP + Iu6-u1') + Cmeas (SZ-St6o)1/(q/P)' c 6o
and we conclude letting first b go to 0+ then 5o go to 0+. If SZ = IR.*1 we first observe that there exists µ,-- nonclecreasing, Lips-
chitz on [0, oo), such that p > 1 on [0, oo), p(t) -4 +oo as t - +oo and &N µ(IxI)(JVuJ2 + Iu12)dx < oo. Then, multiplying (A.2) by u(Ixl)u6i we easily deduce that we have
in
(A.10)
/(IxI) (Iou6I2 +,u6I2) < C. 6
Then, we write for all R E (0, oo), e E (0, oo), b E (0,1/R), defining f6 =
JI N
Iv(u6-u)IP + Iu6-uIP,
f6 dx < meas (BR) SUP f6 + CEP-r + BR
CEq-P
+ fe
where we used the W l,r n W 1,q bound. Hence, we have, denoting by CE various positive constants depending only on c and p
LN
f6 dx < Cep''' + Cc" + meas(BR) sup f6 BR
+ CE fBc IV(u6-u)I2 + Iu6-ul2dx R
< CEP-'' + Ce" + meas(BR) sup f6 +
Ce
A(R)-1
BR
and we conclude upon letting first 6 go to 0+, then c go to 0+ and finally R go to +oo.
Remark A.1. The above procedure yields a constructive approximation of u by Co (S2) (resp. Co (IRN)): indeed, one just needs to smooth u6 by convolution.
In chapter 2, we use a variant of the preceding truncation in IRN that we describe now. First of all, let D1, 2(IRN) = {u E L (IRN), Vu E
Appendix A
168
L2(IRN)} if N > 3. (D1,2(IRN) is a Hilbert space for the scalar product Ovdx.) Then let D1'2(IR2) _ {u E Ho" (IR2), Du E L2(IR2)}. D1,2(IR2) is a Hilbert space for the scalar product f'2 Du Vv + 1B1 uv dx and an equivalent norm is given by II V u I I L2 (1R2) + I fBl u dx ` . These facts on D1,2(R2) are easy consequences of standard inequalities like 1/2
(faR
IuI2 d2
1/2
C CR CR
(fBR
IDuI2 + 1B1 IuI2 d2
IIVuIIL2(BR) +
ul
`
Bfl
where CR > 0 only depends on R which is arbitrary in [1, +oo). We denote by III . III the norm in D1,2(RN) or V1'2(IR2).Next, we introduce essentially as before for u E D1"2(IR1NV) if N > 3, u E D1,2(R2) if N = 2 the solution UR of
in BR
- DUR + VPR = -Au UR E Hp (BR),
diV uR = 0
(A.11)
a.e. in BR
where R E [1, +oo). We shall also use, when N = 2, the following variant
- ZuR + 1B1 UR + VpR = -AU + 1B1 U UR E Ho (BR), div uR = 0 a.e. in BR.
in BR,
(A.12)
If N > 3, we introduce the linear map TR(u) = UR and, if N = 2, we consider two linear maps
TR(u) = UR + fB ( u-uR),
TR(U) = UR.
(A.13)
l
Theorem A.1. Assume that (A.1) holds. 1) TR(u) converges to u, as R goes to +oo, in
D1,2(IRN)
if N > 3 while
TR(u) and TR(u) converge to u, as R goes to +oo, in D1"2(IR2). 2) We have for all Ro E (0, oo)
sup IIu - TR(u)IIL2(BRO) - 0
as R ---* +oc
(A.14)
as R --4 +oc.
(A.15)
IIIUIII<1
SUP flu - TR(u)IIL2(BRO) Illulll<1
-a 0
Proof of Theorem A.1. First of all, we observe that, exactly as for (A.5), we obtain easily JIRN IDTR(u)12 dx <
fIVul2dx
(A.16)
Appendix A
169
and
J1R2 IoTR(u)12 + 1B1 ITR(u)I2 dx < JI2 IVU12 + 1B, IUI2 dx
(A.17)
where, of course, we extend UR and uR by 0 to IRN. Therefore, TR(u), TR(u) are bounded, uniformly in R > 1, in D1,2(IRN), D1"2(IR2) respectively. Since, if N = 2, h, TR(u) dx = fBl u dx, we have only to show the weak convergence to u in view of (A.16) and (A.17) or, in other words, extracting subsequences if necessary, that u = u if u is the weak limit in D1,2(IRN) (or D1,2(IR2)) of TR(U), TR(u) as R goes to +oo. We begin with TR(u) and we observe that, if we normalize pR by fBRpRdx = 0, we have in D'(BR), aiPR = aj fR" (A.18) IBR
PRdX=0,
IIfR'IIL2(BR) <
for some C > 0 independent of R (depending only on IIIuIII). This yields (A.19)
IIPR II L2 (BR) <- C
and we may thus assume, extracting a subsequence if necessary, that PR converges weakly in L2 (IRN) to some p. We then obtain, setting w = u - u,
- Aw + Vp = 0 in D' (IRN) , p E div w = 0
L2 (IRN ),
a.e. in IR N
(A . 20)
J
wED1,2(IRN)if N>3,wED"2(IR2)if N=2and then fBlwdx=0. and Taking the divergence of (A.20), we immediately deduce that p = 0 and
thus w-0if N>3,w-constant if N=2. Finally, w-0if N=2since fB wdx = 0. The proof for TR(u) is a bit more delicate. Multiplying (A.12) by wR, where 1 < R' _< R, w = u-u and WR is the solution of (A.11) corresponding to w (instead of u), we obtain
V(R(u)-u) VWR, dx + IBR
wR, dx = 0 l
and, letting R go to +oo,
JVW.VWRIdX+JW.WRIdX = 0. 1
(A.21)
Appendix A
170
If we show that fBI w dx = 0, we see that (A.21) holds with WR replaced by TR, (w) and thus, letting R' go to +oo, we deduce
flvwl2dx+jlwl2dx
=0
hence w = 0. In order to show our claim on fB, w dx, we take the curl of (A.12) and we find
-A{curl (TR(u)-u)} + curl (1B1(TR(u)-u)) = 0
in D'(BR)
and, letting R go to +oo, we obtain
-A curlw + curl (1B1w) = 0
in D'(IR2), curlw E L2(1R2).
Hence
curlw = 21 JB 1Y1
1
x2
2ir
IxI2
wi(y) dy IZ
(L,
- 2ir lB ly- 2 w2(y) dy 1
1
wi
dy + L I2I2
w2 dy + B1
v TX-1
as IxI - +oo.
Since curlw E L2(IR2), we deduce easily that fBl wdy = 0, and this completes the proof of part 1. Since the embedding of D1,2(IRN), D1'2(IR2) into L2(BRa) is compact,
we have only to show, in order to prove part 2, that TR. (u,,), TR (u,) converge weakly in D1,2(IRN), D1"2(IR2) respectively to u whenever u,t converges weakly to u in these spaces, and this fact is shown exactly as in part 1.
We conclude this appendix with the introduction and the study of some related truncations. To this end, we consider f E L°O (IRN) n L t,00 (IRN ),
f > 0, f # 0 and we assume in all that follows that N > 3. If u E D1'2(IRN), R E (0, cc), we define OR(u) = UR to be the solution of
- 4UR + f UR + VpR = -zu + f u UR E HD (BR),
div'UR = 0
in V'(BR),
a.e. in BR,
and we have
Theorem A.2. Assume that (A.1) holds. 1) eR(u) converges to u in D1'2(IRN) as R goes to +oo.
(A.22)
Appendix A
171
2) Let un converge weakly in D1'2(]RN) to some u. Let fn > 0, fn E
Lrr''°O (IRN ), fn 0 0 be bounded in L°° (IRN) and assume that v_ dun is
bounded in L2(IRN) and that fn converges to f in L1(BM) for all M E (0, oo). We denote by an(on) the solution of (A.22) with u, f replaced by if', fn respectively. Then, 6R(un) converges weakly to un in D1.2(IRN) uniformly in n as R goes to +oo and, in particular, we have
as R -> +oo.
Sllp Ilun - eR(un)IIL2(BRa) --> 0 n
(A.23)
Proof of Theorem A.2. 1) We have (extending as usual UR = 6R(u) by 0)
uRN
IVURI2 + f IuRI2 dx
=f
(A.24)
f::1Vu12
f IuI2 dx.
L,°°(]RN), IuI2 E L'1 (]RN). It Notice that fIuI2 E L1(IRN) since f E is thus enough to show that if UR converges weakly to some a in D1,2(IRN) then u = u. Writing w = -A(u-u) + f(u-u) E H-1(IRN), we deduce from (A.22) that we have
<w, 0>= 0,
for all 0 E CO' (IRN), div q5 = 0 on IRN.
(A.25)
Therefore, by classical results, there exists p E L'2(]RN)+L2(IRN) such that w = -Vp: once we know that to = -Vp, the integrability of p is easily seen by Fourier transforms. Indeed P = (1+1 IfI ,Z p E L2(IRN) and thus 1(IfI?1) E L2(]RN), while P 1(IfI<1) =
1(ICI<1)
1+1i
I
i/s
P EL
2(g],N)
since th E LN'°O(IRN) Furthermore, we have clearly
-Ap = divw = div(f(u-u))
in D'(IR.N)
and f (u-u) E L2 (]RN) since f E LN,°O(IRN), (u-u) E L2(IRN). Hence, we obtain
p E L2(IRN),
Vp E L+f'2(IRN).
(A.26)
We then set z = (u-u) and multiply to by zcpn where 'p,, = cp(n n > 1 and cp E Co (IRN), Supp cp C B2, 0 < cp < 1, cp = 1 on B1. We obtain
Appendix A
172
easily
f
N
(JVz12 + f IzI2) Pn
R,
Izl2(2
AWn)
C2
<
+pz DWndx
nJ
Izl2 dx +
C2
<
IpI l zl dx
(n
(n<JxJ <2n)
Iz12dx+C
x1/2n Izl2dx
(JRN
.
In view of (A.26), we conclude easily that z - 0 once we observe
n2 J(n<JxJ<2n)
r
2
IzI2 dx <
,I
CJ (n<Jx1 <2n)
Izdx
N-2
(nN)*
Izldx--*0. n
2) Since, for each n > 1, OnR(un) converges in D1,2(IRN) to u'n in view of the proof of part 1, we have only to show that 2' = ORn (un) converges weakly in D12(IRN) to u as n and Rn go to +oo, and (A.23) then follows in view of the Rellich-Kondrakov compactness theorems. Observe first that because of (A.23) and the assumptions made, fin is bounded in D1,2(IRN) and thus, without loss of generality, we may assume that un converges weakly in D1,2(1RN) to some u. Next, we simply observe
that f nun, f nun converge respectively in Li (and even L' for 1 < p < NN2) to fu, fu since fn, icn and un converge in Li c. At this stage, we conclude that u - u exactly as in part 1. 0 Let us remark that part 2 of Theorem A.2 yields the following variant of (A.23)
/
Sup{IIun-eR(un)IIL2(BRO)! 1+11Un112
-0
U,112
l/
1/2
j
as
(A.27)
for all un E D1,2(IRN).
Indeed, we just apply part 2 with un(1+
IIunJID1,2(RN) +II Vf-n un1IL2(]RN))-1/2 instead of un.
APPENDIX B Two Facts on 11'2(1R2) Recall (see Appendix A) that D1,2(IR2) = {u E Hloc(IR2) We begin with a remark.
,
Vu E L2(IR2)}.
Lemma B.1. We have D1,2(IR2) n (L'(I .2) + L2(IR2)) y H1(IR2), and, more precisely, there exists a constant C > 0 such that for all u E D1'2(IR2) satisfying u = ul + U2 with ul E L1(IR2), U2 E L2(IR2) IIUIIL2(jR2) <_
(2
{Ilu1ll
2)
(
(B.1)
Iloulli 1R2) +
Proof of Lemma B.1. We first remark that Jul = ul + u2 with ul = min(Iuil, Jul) E L1(IR2). 0 < u2 < 1u21 and u2 E L2(IR2). Next, we also V1'2(IR2), and have: I u 1 < 2w + Iu11(,,uI <2) where w = (lul - 1) + E
fR2
Iu12 1(:u;<2) dx < 2
L2
< 4 fIR2
ui 1(u1,52) dx + 2 uldx+2J
JIR2
u2 dx
Z u2dx
hence
IMu 1(1u1<2;''L2(IR2) < 2(IIu11Ii (1R2) +
Ilu2llL2(]R.2)).
(B.2)
On the other hand, we also have
1R2
wdx <
J
;u; 1(juj>i) dx
111/2
<
+Ilu2 IL2(1R2)
< IIu1IIL1(IR+ 1u2!1L2(1R2)
meas lug 2 +211u21IL2(IR2)} {v'ilu1lIR2)
2 J 1/2J
Appendix B
174
and we deduce from Gagliardo-Nirenberg type inequalities CIlwIIL IIWIIL2(IR2)
<_
(JR2)
IIowII
(1R2)
-
CIIwII
(]R2) IIOUIIi
c1R2)
therefore we find C11VU111/2 IIWIIL2(IF(2)
<_
(J
2){IlulllL (]F,2) 11U1111/4
+ IIu21Ii
(f2) + IIu2IIL2(R2) }
or
IIWIIL2(1R2)
< CIIVUII
(B.3)
(RR,2){IIuiIIh/2 1.Z) + U211L2(]R2)} 11
Combining (B.2) and (B.3) we obtain IIUIIL2(Ft2)
< C{IIu21IL2(jR2) + IIVuII (j,2)IIu1II
+
1/
C11 U1 Il
(jR2)}
1/2 (1R2) + CIIVuIIL (jR2)11u211L2(jR2).
Then, replacing u by \u and thus U1, U2 by Au1 i Au2 where A > 0 is arbitrary, we also deduce IIuIIL2(R2)
< C{IIu211L2(jR2) + IlouIIL C +C_ + V/,\IIU1IIL1(1.2)
X1,2)}
llVulli (1,2) IIu211L2(1R2).
L(am2) Since this inequality holds for all A > 0, we choose A = IIu1111/2
IIu2IIL2(1,2) and we find IIUIIL2(1,2) < C{IIu211L2(1.2)
+
IIVUIIL2(1.2)IIu111L1(1,2)}
+CUu11IL1(1,2) Iloulli (1,2)IIu2I1 (1,2),
and this inequality yields (B.1).
Remark B.1. A similar (and simpler) proof shows a similar result in D1,2(IRN). More precisely, if N > 3, u E D1"2(IRN) with u = u1 + u2i U1 E L1(IRN), U2 E L2(IRN) then u E L2(IRN) (and thus H1(IRN)) and we have for some C > 0 depending only on N 1Iu11L2(JRN) < C{IIul1IL1(1,N) IIVUIIL2(1,N) + I1u2I1L2(IR,N)}
(B.5)
Appendix B
175
where 0 = N+2 2 Recall We next turn to weighted LP bounds for elements of (see Appendix A) that we choose the scalar product (f 2 Du Vv + 1B1 u v dx) on D1'2(1R2) for which this space is a Hilbert space, and we denote by IIIuill the corresponding norm.
Theorem B.1. For all m E j2, +oo), there exists a positive constant C such that we have for all u E D1'2(R ) 1/M
<X>2 (log < x >) -e dz UR2 lulm
(B.s)
< CIIIUIII
if0E(2 +1,00). Proof of Theorem B.1. We begin with the case m = 2 and prove in fact 1/2
1/2
<2
Iu(rw)12 dw
B
VSI
+11VU111/2
(B.7)
(JR2)(log 2x)1/2,
IU12 dx
for all r > 1, where we write x = rw, w E S1, r = IxI, and the case m = 2 of (B.5) follows easily upon integrating in St. In order to show (B.7), we write for all r > 0 1/2
d
1/2
2
dr (f1 Iu(rw)12 dw
f 18r (rcv)
(
dw
(B.8)
1/2
(fIvu(rwlI2dw)
<
1
1
and observe that there exists TO E [ ,1] such that a 1/2
1/2
(fS1
<2
1u(row)12 dw
Iu12 dx B1
Hence, integrating (B.8) from ro to r > 1, we deduce 1/2
(fS1 Iu(rw)12
<
&1 1/2
2(f
+
Iu12dx
fro
1
1/2
fr
2
Iu2 dx
1/2
r
+
f 81 IDu(tw)12dw
(ff
I Du(tw)I2 t dw dt 1
1/2 dt
Clog
To
)
Appendix B
176
and (B.7) follows. The above calculations are obvious if u is smooth and (B.7) is thus shown for all u E D1'2(IR2) by density. Let us also remark that (B.7) implies for all R > 1 ul2 dx < C R2log 2R I IIUII I2
(B.9)
ul dx < CR2 (log 2R)112 Illuil1.
(B.10)
I
LR
'BR I
We next want to show (B.6) for m E (2, oo). We observe that we have for all R > 1
(IBR
1/m
I u- tr
Ulm d2
BR 1
<
C(
Iu -
fBR
/m
-
.R?
IVul2 dz
u12 dX BR
(B.11)
(LR
This is one form of the Gagliardo-Nirenberg inequalities (the fact that C does not depend on R can be easily deduced from a simple scaling argument). Combining (B.9), (B.10) and (B.11), we deduce for all R > 2
um dx < CR2 (log 2R)m/2 I IIuII I.
(B.12)
'BR
Next, by a simple integration by parts, we obtain for all R > 2 m
fBR
<2>)-g
<x>2 (log
d2 1+R2)-e
fB Iulm d2 (1+R2)-1(log R
+
f
R
o
JB r Iulm dx l (1+r2)2
+ 1+r2 (log
< cIIIulllm 1 +
< CIIIulll'
if6> m+1. 0
1+r2)-(6i-1)
1+r2)-B
(log
1+r2
dr
JR 1 +r2
( log
1+r2)--e
dr
APPENDIX C Compactness in Time with Values in Weak Topologies -Let X be a separable reflexive Banach space and let fl be bounded in L°° (0, T; X) for some T E (0, oo) . We assume that f n E C([0, T] ; Y) where
Y is a Banach space such that X -* Y, Y' is separable and dense in X'. Furthermore, we assume that for all cp E Y', < cp, f n (t) >Y' xY is uniformly continuous in t E [0, T] uniformly in n > 1. We then choose a closed ball BR,, of X containing all the values of fn(t) for t E [0, T], n > 1. The weak topology of X, since X' is separable, makes
B& a compact metric space, and we denote by C([0, T], X -w) the space of continuous functions on [0, T] with values in BRo C X equipped with the weak topology.
Lemma C.1. Under the above conditions, fn is relatively compact in C([0,T]; X-w).
Proof of Lemma C.1. 'Let (cpk)k>1 be a dense sequence in Y'. We consider the "weak topology" distance on BRo given by d(f, 9)
I<1Pk,f-9>Y'xYI
1
- L, 2k
1+1 < ck , f -9 >Y' xY I
Using the Ascoli-Arzela theorem, we have only to show
supd(fn(t), fn (S)) -} 0
as
t,s E [0,T], It-s! - 0,
n>1
and this is obvious since, by assumption, we have for each k > 1 sup <92k, fn(t)-fn(s)>Y'xY
-+ 0
as
It-sl-->0.
n>1
Hence, for all k > 1 d(J
(t), f' (s)) < sup Sip 1<jk
I <SOj ,
fn(t)- fn(S)>Y'xY1 +
2k
and we conclude upon letting first It-si go to 0 and then k go to +oo.
13
APPENDIX D Weak Ll Estimates for Solutions of the Heat Equation We wish to prove in this appendix a result concerning solutions of the heat equation in IRN that we used in section 3.3. The main result we want to present in this direction is given by
Theorem D.I. Let f E L' (0, T; Lq,, (IRN)) where 1 < q < oo, 1 < r < oo. Let u E C([O,T];Lq,r(IR.N)) be the solution of
au
-
1
2
Du = f in IRN x (0,T),
ult=o = O
in
RN
(D.1)
Then, we have
a
,
Dz2 u E Lq°'.(]RN)
meal t E (0,T) / IIWIIL9.r(jRN) > a
a.e.
t E (0, T)
< Of
(D.2)
IILI(o,T;La.*(IRN))a-1
for all A>0 for some C > 0 independent of f , where V = at
,
D.
(D.3)
Remarks D.1. 1) Here and below, Lq,''(IRN) denotes the usual Lorentz spaces (Lq,q(IRN) = L9(IRN)).
2) The existence of the solution u of (D.1) and its continuity in t with values in Lq,r(IRN) follows easily from the representation formula u(x, t) =
f f t ds
dy f (y, s) exp
a.e. in IRN,
(- 2(t-s) )(21r(t_s)yN12
D.4
V t E [0, T].
3) The normalization constant z in front of 0 is irrelevant and the result is valid if we replace - 2 0 by -vA with v > 0 or, more generally, by an arbitrary uniformly elliptic second-order operator with bounded uniformly continuous coefficients.
4) If we take r = q, the estimate (D.3) together with the classical Lq(IRN x (0, T)) result for D2u and a, shows, by interpolation, that
Appendix D
179
F , D2 u E LP (0, T; L4 (IRN)) whenever 1 < q < 00,1 < p < q. Then, by an
easy duality argument, we see that this result is also valid if 1 < q, p < 00 and we recover the "classical" LP(L9) regularity result for solutions of the heat equation, a result that we used in section 4.3.
Proof of Theorem D.1. We first use and recall the following variant of the fundamental "covering lemma" due to A.P. Calderon and A. Zygmund [78] (see also A. Zygmund [497] and E. Stein [454]) contained in L. Hormander [222] (Lemma 3.1, observe that the proof in [222] immediately extends to LQ,7.(IRN) spaces): for each M > 0, we can find g, hk, disjoint intervals Ik in [0, T] such that 00
f = +Ehk, k=1 00
I I9II L1(O,T;L9,r(IRN)) +J] I I hk I I L1(O,T;L9,r(IRN)) < 3I1 f II L1(O,T;LQ.r(1N))
J (D.5) (D.6)
k=1
II9IIL9,r(1.N) < 2M
I
t E (0, T)
a.e.
T
hk(t,x)dt = 0
hk (t, x) = 0
a.e. in IRN,
(D.7)
if t ¢ Ik, b'k > 1 ,
00
E meal (Ik) <
II f II L1(o,T;L9.r(I,N)).
k=1
Then, we write, defining t
27r
IcI
exp
- f = f dsJ Ndy ° (t-s, y)9(s, y) +
t
Co
ds
k=1f 0
fgt
N
dy
(t-s, y) hk (s, y)
(D.8)
and denote by v and wk the first term and the generic term in the series of the right-hand side. We also write V(t) = IIv(t, )IIL9,r(IRN), Wk(t) = Iiwk(t,')IIL9.r(RN) a.e. t E (0, T). We are going to show the estimate (D.3) for at or equivalently for cp = ai - f , the proof being exactly the same for Dyu (or using elliptic regularity and the equation (D.1)). To this end, we estimate v and wk in various ways. From the classical LP(IRN x (0, T)) estimates for solutions of the heat equation (and by interpolation) we obtain, denoting by C various constants independent of f and M, V E L4°''(0, T; Lq,r(IRN)) and II V II L9.r(o,T) <- CII GII L9,r(o,T)
(D.9)
180
Appendix D
where we define G(t) =
II9IILq,r(I,N) a.e. on (0, T). Notice that (D.6) means
0 < G < 2M
t E (0,T)
a.e.
(D.10)
while (D.5) yields, normalizing without loss of generality IIf IIL1(o,T;LQ.r(RN))
to be 1, IIGIIL1(0,T)
(D.11)
3.
<_
Then, II GII Lq,r(0,T) < C(f °O µ(t)r/qtr-1 dt)1/r where µ(t) = meas IS E
(0,T) / G(s) > t} (0 < µ < T). Notice that µ(t) = 0 if t > 2M (by (D.10)) and that f oO0 µ(t) dt = IIGIILI(o,T) <_ 3 by (D.11). Hence, if 1 < r < q, we deduce, using Holder's inequality,
fdt 00
IIGIIL4.r(o,T) < C
1/4
(12M
µ
t'1/-'dt
< CM1-1/q Now, if r > q, we write 2M
tr(1- 9) dt
II9II LQ.r(O,T) < C f
(f
1/r
0
J0
f
1/r Ml-11q
µ dt
[SUP IA(t)t}].1_1 r Q
t> O 1/q
µdt
M1-1/9 <
CM1-11q
In both cases we have shown II GII Lq,r(0,T) < C
M1-1/q
(D.12)
and thus, in particular, we deduce from (D.9) that we have for all A > 0
meas It E [0, T] / V (t) > Al < C Mq-1A-q.
(D.13)
Next, denoting by Ik the interval with the same centre as Ik and with
a double length (if Ik = (tk, tk+l), Ik =
(3tk 2 k+1
3tk+2-tk
)), we set
I = Uk>1 I. We remark that (D.8) implies
meas {I} = 2 E 00 meas (Ik) < 2/M. k=1
(D.14)
Appendix D
181
Then, we claim that we have
dtWk(t) = IIwkIIL1(Ik:Lq,r(IR
)) :5 CIIhkJILI(O,T;Lq,*(]RN))
.
(D.15)
Admitting temporarily this claim, we complete the proof of the theorem. Indeed, if t j' I cc
where W(t) _ EWk(t)
IIkIILq,r(JRN) < V(t) + W(t)
k=1
hence, for all A > 0, we deduce using (D.14), (D.13), (D.15) and (D.5) meas {t E (0,T)_/ II(PIIL9,*(]RN) > a}
< meas(I) + meas {t ¢ I / II
WIILq,,-(1R.N) >
M+meas{t¢ I/W(t)> 2}+meas{tE(0,T)/V(t)>
<
<
2 + 2f MA 2
M 2
+
C 00
2
2}
W(t)dt+CMq-1A-q
,a
cWk(t)dt+CMq-1A-q
k=1 (I1)
+
_C a
00 CMq-1 A-q
k=1 IIhkIIL1(o,T;Lq,*(]RN)) + M+T+CiMq-1A-q
and we conclude choosing M = A.
The only point remaining to check is the claim (D.15). We recall that we have on IRN Wk(t)
if t ¢ 1, t < tk;
0p
(t-s) * h, (s) ds
Wk(t)
if t
I, t > tk+1
(D.16)
Ik
and because of (D.7) if t > tk+l Wk (t)
L at
(t - s) -
(t -tk ), * hk (s) ds.
(D.17)
We then wish to estimate the norm of Pt (t-s) - at (t-tk) as a multiplier in Lq,,(IRN) (denoting by . IIp,fq,r this norm) and we shall prove below that II
I
at
-
at
(t
-tk) II Mq,*
-< C t- S)tk2
.
(D.18)
Appendix D
182
If this is the case, we complete the proof of (D.15) easily: indeed, we have Wk (t) dt
T 3tk+1-tk
3 -
dt
[ Ik
4
T
C <
-
tk+
ds
dtL
2
f
(t-s) -
(t-tk))
L9,-(RN)
s -tk t-s)2 +00
tk+1
S-tk 3tk 1-tk (t-s)2
ds Iihk(S)IIL9.r(]RN)
tk tk+1
< C ftk
* hk(s) ds
-S-tk
/
3tk+1 -tic-2s l
ds ilhk(S)IIL9,*(]RN) {
tk+1
_<
since
Cftk
ds Ilhk(S)IIL9.*(]R 1) = CII hkll L1(o,T;L9,*(RN))
S-tk
<
3tk+1-tk -2S
tk+1 -tk 3(tk+1-tk)
-
1 3
if s E Ik
Next, in order to prove the estimate (D.18), we compute the Fourier IfI2
-
transform of t a(t-s) (t-k, namely {e-(t s) '- e`0 -e-(1+) We observe that t?, {e-ICIZ defines a multiplier which has the same Mq,' norm by the dilations invariance of this norm. Therefore, (1+'\)ICI2) (e-ICI2 (D.18) is proven if we check that _e e-I,I2 (1 _ e-a1f12) defines a multiplier whose M4 norm (and thus any M4,1 norm by
interpolation) is bounded by CA for all A and for all q E (1, oo). This is straightforward since we have for all H = m > 0 1
m
< CE (1+112)
e-I,IZ a(l+112)"
j=o < CA(1+IeI2)m+2 a-1 12 < C(m).,
and we conclude using classical results on Fourier multipliers.
APPENDIX E A Short Proof of the Existence and Uniqueness of Renormalized Solutions for Parabolic Equations The goal of this appendix is a short and direct proof of Theorem 3.10. We thus consider solutions of
-0f = F
of .f It=o = fo
in
SZ,
in n x (0,T)
of = 0 on
811 x (0, T)
(E.1)
(E.2)
where fo E L1(Sl), F E L' (11 x (0, T)) and SZ is, say, a bounded, smooth open set in IR (N > 2); we could as well treat the case when St = IRN, or the periodic case or the case when the Neumann boundary condition is replaced by a Dirichlet boundary condition. Finally, we assume that u satisfies
uEL2(SZx (0,T))N,
diva=0
in
D'(SZ x (0, T)),
on aSZx(0,T)
(E.3)
(recall that u n E L2(0,T; H-1/2(SZ))). More general conditions on u (and div u) are possible that yield the same result as below but these conditions are more than enough for the application we have in mind, namely Theorem 3.10. For the same reason, we shall assume
F>0
a.e. in cZ x (0, T)
(E.4)
even if it is not necessary for the result to follow; however, this condition allows us to give rather simple proofs. As explained in chapter 3, section 3.4, we cannot simply use distributions theory to solve (E.1)-(E.2) since we would have to define the product u f (writing u V f as div (u f) since div u = 0) ; and since we only assume u to be in L2(c x (0,T))N, we would need to know that f E L2(SZ x (0,T)). However, we cannot expect f to be in L2 (S2 x (0, T)) since F and f only belong to L1.
Appendix E
184
As we saw in section 3.4, we expect f (and we can obtain corresponding formal a priori estimates) to satisfy: f E C([0, T]; L1(SZ)) n L1(0, T; LQ (SZ))
TR(f) E L2 (0, T; H1(SZ))
for all q E [1,NN2) (E.5)
for all R E (0, oo),
T
lira
(E.6)
J dt I VTR (f) I2 = 0, 1J [dxl R n
o
where TR(t) = max (min (t, R), -R) for t E IR, R E (0, oo), V denotes the spatial gradient (in x) and
V, f E L' (Q x (0, T))
for all r E
11,
NN 1) .
(E.7)
In particular, we know that VTR(f) = V f 1(1 fI
In order to solve (E.1)-(E.2), we shall use the notion of renormalized solutions introduced by R.J. DiPerna and P.L. Lions [125] in the context of the Fokker-Planck-Boltzmann equations, see also L. Boccardo, I. Diaz, D. Giachetti and F. Murat [67], P.L. Lions and F. Murat [308] for nonlinear elliptic equations, D. Blanchard [65], D. Blanchard and F. Murat [66]
(and the references therein) for parabolic equations. The idea is simple: we write down the equation satisfied by 8(f) where ,6 E Co (IR.) using (E.1); let us just mention that many equivalent formulations are possible. First of all, we notice that /3(f) E C([0,T]; V(11)) n L'(1 x (0,T)) since Q is bounded, ,3(f) E L2 (0, T; H' (SZ)) since V;(3(f) = ,(3'(f )V f a.e. and 1,3'(f ) Vf I < sups,1,Q' (t) I I VTR (f) I where R > 0 is chosen in a such a way that Supp ()6') C [-R, R]. Similarly, '3"(f )I V f 12 E L1(SZ x (0, T)). Then, formally, we obtain
aA(f) +div{u,6(f)} - o/3(f) +Q"(f)IVf12 8t
= 8'(f) F in 11 x (0,T). (E.8)
Obviously, 3(f) should satisfy (E.2) with fo replaced by '3(fo), and this combined with (E.8) yields the following weak formulation: we have for all ,3 E Co (IR) and for all E C' (SZ x [0, T]) (for instance)
f
Q(f(x,t))cc(x,t)dx+ J Q(f){- at n +VQ(f). Vcp + /3"(f)lVf I2y- /3'(f)Fcpdx = 0 in D'(0,T), A(f) E C([0, T]; L1(f )), /3(f) It=o = i3(fo) a.e. in SZ.
(E.9)
Appendix E
185
In conclusion, we say that f is a renormalized solution of (E.1)-(E.2) if it satisfies (E.5)-(E.7) and (E.9). Theorem E.1. There exists a unique renormalized solution of (E.1)-(E.2).
Remarks E.1. 1) If we know that u E LP(SZ x (0,T)) for some p > 2 and that f E LQ(S2 x (0, T)) for some q E [1, +oo] such that .1 + 1 < 1 (thus lug f E L1(11 x (0, T))), then (E.9) combined with (E.6) implies that f is a "standard weak" solution of (E.1)-(E.2); we could as well treat cases where u, f have different integrabilities in x and in t. Before proving this claim, let us observe that the integrability of f can be estimated in terms of integrability requirements upon F and fo. In order to prove the above fact, we use (E.8) with,3 = ,dn as given in the proof of Theorem E.1 below and we let n go to +oo using (E.6) to -show that ,Q;' (f) I D f 12 converges to 0 in L1(S2 x (0,T)). 2) The proof of Theorem E.1 below-also shows that the unique renormalized solution of (E.1)-(E.2) is the limit (in C([O,T]; L1 (S2)) for instance) of the solutions of regularized problems (regularize u, F or fo). 3) The uniqueness proof given below also yields the expected fact that if fi and f2 are renormalized solutions of (E.1)-(E.2) with F, fo replaced respectively by F1, F2, A, 1, fo,2 and if A,µ E IR then Afi + µf2 is the
renormalized solution of (E.1)-(E.2) with F, fo replaced by )F1 + µF2, Afo, i + µfo,2 respectively. This is of course a very natural fact since (E.1)(E.2) is a linear problem; however, the notion of renormalized solutions is a nonlinear one! E3
Proof of Theorem E.1 Step 1. Uniqueness. Let fl, f2 be two renormalized solutions of (E.1)(E.2). We shall write equations like (E.8) for various quantities involving fi and f2 that satisfy the Neumann boundary condition contained in (E.2); whenever we do so, we really mean that the weak formulation like (E.9) holds. With this convention, we have by assumption a {,C3(fl) -,3(f2)} + div [u{,3(fi) -,3(f2)}]
+,3"(fl)IVf112
- op(fl)_/3(f2)]
- /3"(f2)IVf212 = F[,8'(f1)-O'(f2)].
Notice also that since divu = 0, u E L2(S2 x (0,T)), 3(f l) -#(f2) E L2(0,T; H'(SZ)) the second-term div [u{,3(fi) - ,3(f2)}] may be rewritten as u V (/(f 1) - ,3 (f2)) (one simply needs to argue by density on u). Next, let -y E Co (1R); g ='Y(Q(fi)-,Q(f2)) E C([0,T];L1(n))nL2(0,T; H1(SZ)) satisfies glt=o = 0 in f and
Appendix E
186
+ div (ug) - Og + {p"(fi)IVf112-p"(f2)IVf2J2}7'($(fi) +'r"()3(fl)-p(f2))Iop(fl)-V13(f2)12 -I1(f2))
(E.10)
= F[p'(fl)--p'(f2)]7'(p(fl)-/3(f2)) This computation is obvious formally: to justify it, we simply observe that we have as above
div (ug) = u Vg = u V[8(fi)-p(f2)J
-/3(f2))
while if ai - Oh E L' (f2 x (0, T)), h E L2 (0, T; H1(S2)), -1(h) satisfies
ate)
- 07(h) + 7"(h)
I Vhl2
= { at
- Ah} y' (h).
This last fact requires some explanations: first of all, regularizing h in t by convolution, we see it is enough to prove the above claim when h is smooth in t (h E C1([0,T]; Hl(f2))) since we then easily pass to the limit and prove the claim. When h is smooth in t, we just have to show that if -Oh E L1(S2), h E H1(f2) then -&y(h) = y"(h)JVhI2 + -y'(h)(-Oh). In order to check this fact, we use the weak formulation (incorporating the Neumann boundary condition), namely
L Vh Vc dx =
j(-/.h)wdx
for all cp E Cl(i).
This equality holds by density for cp E H'(0) n LOO (Q) (approximate such a cp by cpn E Cl (?) such that cpn, --+ cp in H1 (S2) n L' (S2) for all 1 < p < oo, n
cp bounded in L' (D)). Then, we take cp = y'(h)& where 0 E C'(11) and we find
fn (-Oh)y'(h),O dx = f Vh {7'(h)V1+7"(h)Vh?G} dx =
f
n
V y(h) VV) +'y"(h)JVh120dx,
and this completes the proof of (E.10).
We next use (E.10) with p replaced by on (t) = np1(n) (n > 1) where /31 E Co (IR), pi(t) = t if Itl < 1, 01(t) = 0 if ItJ > 2 and -y replaced by 7n(t) = -yo(t) C(t/n) where ( E Co (IR) > 0, C (t) = 1 if Jtl < 1, ((t) = 0 if Itl > 2, yo E Ca (IR), yo > 0 and YO' > 0 near 0, Yo(0) = 0, '0(0) = 0. Writing gn = yn(On(fl)-pn(f2)), we deduce from (E.10)
+div (ugn)-Ogn < C
{1v11121(If11<2n)
I
+ Vf2l21(If 1<2n)}
+ F[pn(fl)-pn(f2)]'yn'(pn(fl) - 3n(f2))
Appendix E where we used the fact that I/3n
<_
,
187
%' = 70/1( + yo(' (n) + n yo S" (n)
>
n independent of n > 1. n - n and where C denotes various positive constants
We integrate this inequality over f? (using the weak formulation of it with cp = 1!) and we deduce for all t E [0, T] and for all n > 1 1 7'n(/3n(f1)-Qn(f2))(x, t) dx f2
<
fTdtf C
da {I
n
TdtJr
+ 0
Vfl121(IfjI<2n) +
IVf2I21(If21<2n) 1
dx I F[Qn(f1)-In(f2)] 7yn(f3n(fl)-Qn(f2))I
f2
IT
J
o
[dxlCI Vf1I21(Ifj
Jn
<2n) + IVf2I21(If21<2n)J
[dt[dxIF1IQi(n)-,81(L2)1. n
Because of (E.6) the first term of the right-hand side goes to 0 as n goes to +oo, and the second term also goes to 0 since ,O' (IL), ,0' (LL) converge a.e. in 11 x (0, T) to 1 and are bounded uniformly in n. From the definition of yn and ,3n (and the fact that fl, f2 E C([0, T]; L'(11))) one checks easily that 7'n(on(fl) - /3 (f2)) converges to yo(f1- f2) in C([0, T]; L' (11)) as n goes to +oo. Hence, yo (f l - f 2) = 0 and fl-f2 - 0 a.e. in St x (0,T) since, by construction, yo(t) > 0 if Iti > 0.
Step 2. Approximation, bounds and convergence in C([0,T];L1(S2)). We next want to show the existence of a renormalized solution of (E.1)(E.2). To this end, we consider the following regularized problem aP at af£
+ uE - O fC
- O f e = F--
on aSZ x (0, T), an = 0
in SZ x (0, T), (E.11)
f
E It=o
= fo in
St
where - E (0,1], Fe E Co (Sl x (0, T)), FC - F in L1(SZ x (0, T)), FE > 0 a.e., foe E Co (SZ) , fo -' fo in L1 (S2), uE E Co (St x (0, T)), div -ue = 0
in S2 x (0, T), UC -+ u in L2 (S2 x (0, T)). The equation (E.11) is now a E standard parabolic problem that admits a unique smooth solution f c (say [0,T]))__ C2(? x One easily checks that fl is bounded, uniformly in e E (0,1], in C([0, T]; L1(12)). We next claim that (E.6) holds for f6 uniformly in e, from which bound we deduce a bound in L1(0, T; L9 (12)) (V q < N 2) by Theorem
Appendix E
188
E.2 below and a bound upon V f in L'(S1 x (0, T)) for r E [1, N) by the arguments developed in chapter 4. Next, in order to prove our claim, we multiply (E.11) by TR(f e), integrate by parts over SZ and find T
Sup IISR(fC(t))IIL1(c) + fo dt f dxlvTR(fe)12 n
tE[O,T]
r
jdxSR(f)+J n
where SR(t) =
T
0
r dt / dx Ff TR(fE)
(E.12)
n
if ItI < R, = RItI - 2Z if ItI > R. Since SR(t) < RItI
2t2
on IR and I TR (f e) I < R we deduce that TR (f e) is bounded uniformly in e
in L2(O,T; H1 (Q)) for each R E (0, oo). In addition, (E.12) yields for all
ME(0,oo)
RjT dt 1
dxIVTR(fe)I2 T
< fn R
SR(fo) dx +
J
dtf dx IF--I ITR(fc)I
< R meas(c) +(Ifosl>M) fIfo I dx +
R IIFdIIL1(nx(o,T))
T
+
I odtfn dx IFeI 1(If°I>M)
Our claim then follows since on the one hand fo converges to fo in L'(SZ)
and thus f(Ifo I>M) Ifo I dx goes to 0 as M goes to +oo uniformly in E E (0,1], and on the other hand FE converges to F in L' (SZ), meas { (x, t) E n x (0,T) / I fe(x,t)I > M} < T,r SUPOM) goes to 0 as M goes to +oo uniformly in E E (0,1]. In conclusion, we have shown since R SR(t) > 2 Itl 1It1>R 1
fTdtf
R
dxlyfCI21(I fdI
0 (E.13)
as R -; +oo, uniformly in E E (0, 11. sup tE[O
Ilfe(t) 1If1'(t)i_RII
(E.14)
as R -- +oo, uniformly in e E (0, T1.
We finally show that f E is a Cauchy sequence in Q0, TI; L'(2)). To this end, we follow the uniqueness proof with fl, f2 replaced by f E, f'7 for
Appendix E
189
E, 77 E (0,1], and we obtain, setting gn = -yn(Qn(fE) - Qn(f')),
+ div (ucgn) - Ogn
{,vfei2 1(IfeI<2n) +
<
+
CIuE-uni
IVf,7i21(1fnl<2n)J
iof''I 1(1f171 <2n)
+ [FEpn(fE) - F7)3n(f7)] 'yn(,3n(f6) -,8n(f''))
<
n
{IDfE121(If-1<2n)
+ IOf'7i21(If'7I<2n)} + Cnluc-uni2
+ CI FE -FI + CIF'' -Fi + CIFI 1(If°I>n) + CIF)1(Ifnl>n) .
Hence, integrating in x and t, we deduce using (E.13) and the definitions of uE, FE, foe
sup II-/n{,3n(fE)-Qn(f7)III L1(n) < bn +Wn(E,rl)
(E.15)
tE[0,T]
where 5n -> 0, wn (E, 77) --* 0 as E, r) -* 0+ for each n >- 1 fixed. We also used n
to derive (E.15) the following observations T
sup fo dt J ldx IFi 1(If`I>n)
£E(0,1J
S
-'
0
as
n -> +oo
since supE E (o,1] measz,t (I fe I > n) --* 0 as n - +oo, and
Cf0-f0I
0 <- 1'n(On(fo)-On(f0)) Next, we observe that we have
IfE-ffI <- lan(fe)-an(f'')I + (IfEl+lPl)(1(IfeI>n) +1(Ifnl>n)) and that for each 5 E (0, 1), there exists C5 > 0 such that ItI < S+Cb -yn(t)+ ItI 1111>n for all t E IR. Hence,
IfE-PI
S+Ca'yn(0n(fE)-)3n(f")) + I Qn(fE)-Qn(f'')1 1(1On(f')-On(fn)I>n)
+ (If1I+If''I)
(1(,fCI>fl) + 1(Ifn1>n))
< 6 +Cnn(On(fE)-On(f17))
+
C(IfEl+If17I) (1(Ifi
(fg)(>) +
1(An(fn)I>-))
+ (IfEI+If'7I) (1(fe,>fl) + 11ftI>n))
Appendix E
190
Since 1,8n(t)I > 2 implies 01(*) > 2 and t > 2, we obtain finally
+C(IfdI+If7'I) ('Of-1>0 +1(ifi>3))). Combining (E.15) and (E.16), we deduce IIfC(t)
sup tE [0,T]
- f''(t) IIL1(n)
< S + C,Sn + C6w(E, r) + yn
(E.17)
where -yn - 0, and we used the following facts n
sup measx (Ife(t)I > 2) -> 0 tE [0,T]
as n goes to +oo, uniformly in e E (0,1].
tE
Op
f
I f E(t)I
dx < M meas(A) + sup f C(t)1Q [0,1
O
and the last quantity goes to 0 as R goes to +oo uniformly in e E (0,1] in view of (E.14).
Letting e, 77 go to +oo and finally 6 to 0+, we deduce from (E.17) that f' converges uniformly in t E [0,T] in L1(11) and in L' (0, T; L4 (SZ)) to some f E C([0, T]; L1(SZ)) fl L1(0, T; Lq(SZ)) for all 1
Step 3. Convergence in L2 (0, T; HI (11)) of the truncations. We first observe that it is enough to show that TR(f £) converges to TR(f) (as e -4 0+) in L2(0,T;HI(11)) since 'all (fe)IVfdi2 ='0 (p) . IVTR(fe)12 for R large enough and aebe --' ab in L1(1) if aE - a, bC --> b in L' (SZ) and bE is e C bounded in L'(11). Next, we remark that we have for all S E (0, T) C.
fdtfdxIVTR(f`)12dz
_<
Jo
dt
n dxRIFeI +
n
RIf£(6)-fdz
and thus this quantity is small if S is small, uniformly in e E (0,1]. Hence, we only have to show that TR (f f) converges in L2 (S, T; H 1 (St)) to TR (f) .
Appendix E
191
Next, we write f E = gE + hE where gE, hE are respectively solutions of 89E
at
+ ,uE . OnC - AnC = 0
E
in St x (0 T) 7( 1
a(
0 on asl x (0, T), an = E
+ UE ahE
an
VhC
-
gE It=0 = J 0
in St x (0, T),
LihC = FE
=0 on act x (0, T),
(E.18)
n in SZ,
{{
hdI t=o = 0
in SZ.
Notice that hE > 0 in SZ x [0, T] and that everything we did in step 2 above applies to hE and gE: in particular, hE_ and gE converge in C([0, T]; L1(f )) and in L1(0, T; Lq (f)) for all 1 < q < N 2 to some h and g respectively. In addition, we claim that for each S > 0, gE is bounded in L°° (SZ x (b, T) )
uniformly in E E (0,1]. Indeed, we first observe that for each m > 1, gn1 = tinge solves agE,,,
in
+ uE - Vgm - Eg;n = mgm-1
1 x (0, T 'j
E
=0 on aS2 x (0, T),
9 ;,, I t=o = fo
in S2.
Of course, go = gE is bounded in Lq0 (Sl x (0, T)) for 1 < qo < NN 2 . Next, if gm-1 E Lqm-1 for some Qm-1 _> qo (m _> 1), we multiply the equation for g; , by Ig ;,, l q--1-2 gm and we obtain easily a bound on gem
in L2(0 x (0,T)), and in C([0,T]; Lqm-1(SZ)) and on O [Igml 9m-1 thus in L2 (0, T; H1 (11)) on Igmi 2 -1 gm. Using Sobolev (and GagliardoNirenberg if N = 2) inequalities and Holder inequalities, we deduce that
g; t is bounded in Lqm (SZ x (0, T)) with qm = NN 2 qm-1. Therefore, for m large (m > log (1 + E Lqm-1(SZ x (0, T)) where [log (1+ N]-1+1), 2) N2 2 qm_ 1 > To simplify notation, from now on, we write g in place of
gm and g in place of gm-1 and q in place of qm_ 1. Then, for p > 1, we multiply the equation satisfied by g by IgIP-2g and we obtain easily (where C denotes various constants independent of g, e, Sup Iig(t)IILp(n) + IIoI9IP/2IIL2(nX(0,T))
tE(0,T]
f
< CpII9IIL4(nx(O,T))
T
./
0
hence, exactly as above, we deduce
a1
dtJ[dx I91717(Pn
-
p-1
II9IILp2(Ox(0,?,))
< (Cp)l/P II9IILP_(nX(0,T)) IIg-IIL (nX(o,T))
Appendix E
192
We then observe that
N2 2
= A > 1 and we rewrite the preceding
inequality as (Cp)l'eP
11911LAP(nx(o,T)) <
e'°
11911LP(nx(o,T)) II9II
for all p >
1/0'
O= 4 -1 ,
(nx(0,T))
q
hence
max {119IILXP(Ox(O,T)), IIgIIL4(1x(O,T))}
< (Cp)'"°" max(II9IILP(nx(O,T)), IIgIIL4(nx(O,T)))
for allp > 1/0.
If we choose p > 1/0 and we iterate this inequality with p, Ap, ... , we find for all k > 1
Ak-1p
II9IILakP(nx(O,T))
<
(Cp) ip-
eXP
Op
log A
Ak-kA- k-1
A2-1
Ak
}
max { IIgIILP(nx (O,T)), IIgIIL4(nx (o,T)) }
and letting k go to +oo, we finally deduce II9IIL-(Ox(O,T)) < CIIgIILq(nx(o,T))
In particular, gE is bounded in L°O (0 x (0, T)) uniformly in e E (0,1] and
9,(5) converges in L2(fZ) to g(5). Furthermore, uE Vgf = div (ucgc) and siege converges in L2 (SZ x (0, T)) to ug. Therefore, gC converges to g in view
of (E.18) in L2(5,T; HI (fl)). We then claim that the proof of Theorem E.1 is complete as soon as we show that TR(hf) converges in L2(0, T; H1 (0)) to TR(h) for all R E (0, oo). Indeed, we have on (8,T) x SZ for each S > 0 fixed
IVTR(fe)12 = 1(IfgI
where R' = R + C6, IgE I _< C6 on I8, T] x SZ. The quantity in brackets converges in L1(SZ x (8, T)) to I Vg + VTR, (h) 12 (= ICf 12 if If I < R). Hence
jTj.
T
l oJo
VTR(f£)12dxdt In'
I(
n 1( IfI<_R)IVg+VTR,
I
T
=
J
n
IVTR(f)I2dxdt
(h)I2dxdt
Appendix E
193
using Lebesgue's lemma (and its converse), and we conclude.
Step 4. Convergence in L2(0, T; Ho (SZ)) of TR(he). It only remains to show that TR(hE) converges to TR(h) in L2(0, T; H'(1)). We observe that he > 0 since he I t=0 - 0 in S1 and FE > 0 in SZ x (0, T). Next, we claim it is enough to show that, for instance, (1+h) -1/2 converges to (1+h)-1/2 in L2(O,T; H1(11)). Indeed, if it is the case, extracting a subsequence if necessary, there exists C E L1(SZ x (0, T)) such that Iohsl2
(1+he)s V h'
Vh
a.e. in
St x (0 , T) ,
a.e. in SZ x (0, T).
In particular, IVTR(he)I2 < C(1+R)3, VTR(he) and we conclude.
f
VTR(h) a.e. in Stx (0, T)
Next, we set ,C3n(t) = 1+t/n fort > 0, n > 1, 'yn(t) =
for
1_(1
0 _< t < (1 - nn > 1. Let us remark that /3n = ryn o x(31, On is concave Next, we notice that we on [0, oo) while 'yn is convex on [0, (1 - n) have for all k > 1, E E (0,1]. 3k(h') + div {uE,3k(hM)} - O,3k(hM)
= Q'(hE)FE + (-A%(he)IVhMI2)
an
13k (
hE) = 0 on 8SZ x (0, T),
in n x (O, T)
Qk(M)I t-o = 0
in(.
Letting e go to 0, we deduce easily that there exists a bounded nonnegative measure µk on SZ x [0, T) (V k > 1) such that ,3k(h) + div {u,Qk(h)}
- 0,8k(h)
(E.20)
_ )3k(h)F + (-)3k (h) I ohE I2) + uk
in SZ x (O, T). 8IV(1+he)-1/212
and converges weakly in L2(0,T; H1 (n)) to (1+h)-1/2 in view of the convergences already shown. We next claim that t1k > pi in SZ x [0, T]. Formally, this is straightforward since we deduce from the equation satisfied by 31 (h) Indeed, -0k(hE)IOhEI2 = (i+h`
a
IohMI2
=
(1+hM)-1/2
,Qk(h) + div {u,Qk(h)} - D,Qk(h)
_
('yk o ,31(h)) + div {uyk o 01(h)j- o(-yk o /31(h))
'yk(Qi(h)) {,3i(h)F-,3i (h)IVhf2+,i1} =,6k
- ak (h)I ohI E + yk ($i (h))µi
- yk(Qi(h))(Qi(h))2IVhI2
Appendix E
194
Since ,'k(/31(h)) = (1 > 1, we deduce that µk > µ1. To justify these computations and the conclusion, we have only to show that we have at ,3k (h) + div {u,Qk(h)} - O,Qk(h)
> ,Qk(h)F-,3k(h)lVhl2+µl in 11 x (0,T). This is done exactly as in step 1 above, the final step consisting in showing
that if
-iH = G+m in S2, H E L°°(S2)nH'(O), a = 0
-
where IIHIIL-(c) < (1 kH bounded measure on St, then
> 0 a.e. in 52, C E L'(0), m > 0 is a
- O7k (H) > -rk (H)G - 7k (H) 8n
-Yk
(H) = 0
on
on 852, (E.21)
I VH12
+m
in
SZ,
(E.22)
all.
To this end, we use the weak formulation of (E.21) with ryk(Ha)cp where cp is arbitrary in C1(S2), Ha E C' (?Y) for a > 0, sups II Ha II Lc (f) < (1 -.0-1,
Ha > 0 in 52, Ha converges to H in H' (S2) and a.e. in S2 as a goes to 0. We then obtain if cp > 0 in S2
VH- {7'(Ha)VHacP+-yk(Ha)Vcp}dx > in
J
'k(Ha)G+mcpdx
and we recover (E.22) letting cx go to 0. Therefore, we deduce from the weak formulation of (E.20) T
,Qk(h(x,T)) dx > fo dt
j
dx01(h)F+ pi(52 x [0,T)),
R
T
Jn
h(x,T)dx _>
dtJ dxF+µ1(Sx
On the other hand, (E.19) yields
r
T
he (x, T) dx Jn
jdtjdx
[0,T))1.
Appendix E
195
hence, upon letting a go to 0+, fn h(x, T) dx = fo dt fn dx F. Therefore, converges to V(1+h)-1/2 in µ1 - 0 and this implies that V(1+hc)-1/2
L2(SZ x (0, T)), and the proof of Theorem E.1 is complete.
We conclude this appendix with a simple observation.
Theorem E.2. Let p E [1, oo) and let a E (0, p). We assume that f E LP (n x (0, T)), V f E L1(11 x (0, T)) (for instance) and that f satisfies for
all R>1
T
p
dtJ dx IofIP 1(Ifl
(E.23)
(1+I fI2)3 P E LP(0,T;W1"P(S1)).
(E.24)
n for some C > 0. Then, for all ,3 E (a, p) o
Remarks E.2. 1) Many variants and extensions are possible (less restrictive conditions on f , unbounded domains, time-dependent f, etc.); we skip these since the argument given below is extremely simple and can be easily adapted to various situations. 2) Of course, the norm in LP(0,T; W1"P(11)) is estimated in terms of the constant C in (E.24) and a bound on f in LP(S1 x (0, T)), say. 3) Using Sobolev inequalities we deduce easily (at least if p < N) from
(E.24) that f E LP-,3 (0,T; Lq(Sl)) where q = ( '°
.
Proof of Theorem E.2. It is enough to show that we have T
r
dtJ dxIVfIP(1+I.fI2) fn
< C.
Then, we write T
L
dt
f
dx IV f IP
n
<
(1+If12)-c
1T1
+
(1+If12)_1(2n
f dt f dxIVfIP 1(Ifk<2n+1) T
<
C+E2-n'o
n>0
0
SE
2-np 2na < C.
O
(E.25)
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INDEX a priori bounds 128 a priori estimates
density-dependent Navier-Stokes equations 35-41, 64, 69, 70 Euler equations 152, 153 density-dependent 159
weak solutions of Navier-Stokes equations 98-107 acoustics limit x almost Lipschitz 127, 151 approximation approximated solutions
for
density-
dependent Navier-Stokes equations
35, 64-7 renormalized solutions for parabolic equations 187-90 approximated models vii, 9-16 Ascoli-Arzela theorem 177 asymptotic limits vii, x, 14-16 asymptotic problems vii axisymmetric case 152 barotropic pressure laws 15 Besov space 99 blow-up 150, 155 boundary conditions ix, 1, 16, 19-20
compressible isentropic Navier-Stokes equations ix, 15 compressible isothermal Navier-Stokes equations 15 compressible models vii, ix-x, 13, 153 compressible Navier-Stokes equations 8-9 concentrations 154 conservation of energy 4 Euler equations 126, 132, 133, 135, 141 density-dependent 159 dissipative solutions 153, 154 conservation of mass 1-2, 9 conservation of momentum 2, 9, 11 convection term 100 convergence
density-dependent Navier-Stokes equations 41-61 Euler equations 149-50, 156 renormalized solutions equations 187-95 covering lemma 179
for parabolic
D1'2(IR2)
density-dependent Navier-Stokes equations 35, 41-64
55, 57, 168-70, 173-6 decreasing rearrangement 140 see also Schwartz spherically symmetric decreasing rearrangement deformation tensor 3, 150 degrees of freedom, internal 117 density 1-2, 158 density-dependent Euler equations ix, 14, 158-60 density-dependent models 117 density-dependent Navier-Stokes equations viii, 10, 12-13, 19-78 a priori estimates 35-41, 64, 69, 70 compactness results 35, 41-64 existence proofs 35, 41, 64-75 existence results viii, 19-31
Euler equations 133, 140, 143-4, 153, 154
regularity results and open problems
see also Dirichlet boundary conditions; Neumann boundary conditions; periodic case; whole space case breakdown 151, 159 Cauchy problems ix, 16, 125 compactness 117 compensated compactness theory 93 compressible isentropic Navier-Stokes equations ix
in time with values in weak topologies 177
compensated compactness theory 93 compressible Euler equations 13 compressible isentropic Euler equations x
31-5 uniqueness see uniqueness Dirichlet boundary conditions ix, 16, 183
density-dependent Navier-Stokes equations 19, 35
Index
234
Dirichlet boundary conditions (cont.) density-dependent Navier-Stokes equations (cont.) a priori estimates 37 compactness results 42, 46, 47 existence proofs 64, 65, 67, 70-1, 73 regularity results 25, 30, 31 uniqueness 75 Euler equations 124, 125, 127, 128, 136, 153
fundamental difficulty 129-30 two-dimensional 139 Navier-Stokes equations 80, 81, 82, 86, 91, 92 difficulties encountered 83-6 second derivative estimates 107-10
temperature and Rayleigh-Benard equations 110, 111, 114-15, 116, 117 dissipative solutions viii-ix, x, 153-8 distributions
density-dependent Navier-Stokes solutions compactness results 42, 45 existence results 23, 24, 25, 28 Euler equations 127, 135, 153 distributions function 137, 139 Navier-Stokes equations 80, 81, 83, 112, 113, 183 divergence-free vector fields
density-dependent Navier-Stokes equations 30, 53, 59 Euler equations 125, 132, 137, 139, 154 density-dependent 158 Navier-Stokes equations 84-6 truncation in Sobolev spaces 165-72 elliptic equations 101, 184 elliptic regularity 143, 179 energy 1 conservation see conservation of energy internal 4, 10-11, 110-23 kinetic 4 total 4, 111, 114 energy identities 71 Euler equations 128, 130 Navier-Stokes equations 112, 114, 119 energy inequalities 154
density-dependent Navier-Stokes equations 22-3, 73-4 Navier-Stokes equations 86, 114, 118, 120
local energy inequality 82, 88-9, 109-10 entropy 5, 13-14 Euler equations viii-ix, 14, 124-64 density-dependent ix, 14, 158-60 dissipative solutions viii-ix, x, 153-8
hydrostatic approximations ix, 160-4 review of known results 125-36 three-dimensional viii, 150-3, 163-4 two-dimensional viii, 125, 136-50, 161 eulerian form 1 existence of solutions density-dependent Navier-Stokes equations proofs 35, 41, 64-75 results viii, 19-31 Euler equations 125, 126, 131-4 dissipative solutions 156-8 two-dimensional 136-50 Navier-Stokes equations 21, 86-7, 106, 110, 118-19 renormalized solutions for parabolic equations 187-90 results for compressible models ix-x
first law of thermodynamics 4 fixed point 67-70 Fourier multipliers 182 free boundary problem 34
Gagliardo-Nirenberg inequalities 32, 174, 176, 191
galilean invariance 24 Euler equation 148 geophysical flows 160 Gibbs equation 5 global weak solutions viii, ix
density-dependent Navier-Stokes equations 21-5, 34, 35 equal to strong solution 31, 75-8 Euler equations 137, 151-2 Navier-Stokes equations viii, 21, 79-82, 86-9, 105-7 Hardy inequalities 46, 74 Hardy spaces 127 refined regularity of weak solutions 92-8 harmonic functions 83-4, 86, 147 heat equations 99, 112, 116 weak L' estimates 178-82 hilbertian strategy 27 Hodge-de Rham decomposition 25-6 homogeneous fluids 9 homogeneous incompressible Navier-
Stokes equations see Navier-Stokes
equations homogenization x hydrostatic approximations ix, 160-4 hydrostatic pressure 2, 10, 126, 158
ideal fluids 9, 13-14 ideal gases 7-8, 12, 14, 15 incompressible fluids 9
Index incompressible limits 25 incompressible models vii, viii-ix, 9-11, 13
inhomogeneous incompressible (density-
dependent) Euler equations ix,
14,
158-60 inhomogeneous incompressible NavierStokes equations see density-
dependent Navier-Stokes equations initial conditions 16, 20-1, 25, 83, 162 internal degrees of freedom 117 internal energy 4, 10-11, 110-23 inviscid (non-viscous) case x, 4 inviscid model 160-4 RN case see whole space case isentropic gas dynamics 14 isentropic pressure laws 15 isothermal case ix Joule's law 7, 8, 14 kinematic viscosity 79 kinetic energy 4 kinetic theory 3, 8 Lame viscosity coefficients 3 Leray, J. viii, 21, 66, 79 Liouville's theorem 35 Lipschitz, almost 127, 151 Lorentz spaces 59, 178
Mach number x, 153 low Mach number expansions 9, 11-13 Mariotte's law 7 mass, conservation of 1-2, 9 measure theory 24 measure-valued solutions 153 momentum, conservation of 2, 9, 11 multi-phase flow 19
Navier-Stokes equations vii, viii, 23, 31, 66, 79-123, 125, 156 density-dependent see density-dependent Navier-Stokes equations fundamental equations 10, 11-12 global weak solutions viii, 21, 79-82, 869, 105-7 refined regularity of weak solutions via Hardy spaces 92-8 review of known results 79-92 second derivative estimates 98-110 stationary 35, 91-2 temperature and Rayleigh-Benard equations 110-23 Theorem 4.1 128-9, 136
235
Neumann boundary conditions 16, 110, 115, 183, 186 newtonian fluids vii fundamental equations vii-viii, 1-9 model of incompressible, homogeneous newtonian fluid 117-23 non-linear partial differential equations 22 numerical simulations 151
12 = ffi.N see whole space case open problems viii, 31-5 ordinary differential equation 67 oscillations 154 propagation ix
parabolic equations 101 existence and uniqueness of renormalized solutions 183-95 particle paths 5, 67 passing to the limit 134 density-dependent Navier-Stokes equations 35, 64, 70-5
Navier-Stokes equations 106-7, 120 perfect fluids see ideal fluids; ideal gases periodic case 16 density-dependent Navier-Stokes equations 20, 25, 30-1 a priori estimates 35, 37 compactness 41-7 existence proofs 64, 65, 66, 67, 70 stationary problems 34-5 uniqueness 75 Euler equations 126, 127, 128, 130, 153 Navier-Stokes equations 79, 80, 81-2, 84, 86, 98, 114 pressure 2, 10, 126, 158 pressure field density-dependent Navier-Stokes equations 23, 31 Navier-Stokes equations 79, 80, 81-2 pressure laws 14, 15 propagation of oscillations ix
Rayleigh-Benard equations viii, 110-23 rearrangement 137, 140-1 regularity ix density-dependent Navier-Stokes equations viii, 31-5 existence proofs 67-70 elliptic 143, 179 Euler equations 126, 130, 136 dissipative solutions 155, 156 Navier-Stokes equations viii, 80, 82, 83, 84, 86, 87, 89-92, 112
Index
236
Navier-Stokes equations (cont.) refined regularity of weak solutions via Hardy spaces 92-8 second derivative estimates 98-9, 100, 109-10
regularization 135, 147, 156, 187-8
existence proofs for density-dependent Navier-Stokes equations 65, 66, 70 solutions of transport equations 43-5 Rellich-Kondrakov theorems 57, 131, 172 Remark(s) 2.1 23-5 Remark(s) 2.2 26 Remark(s) 2.3 33 Remark(s) 2.4 42 Remark(s) 2.5 46 Remark(s) 2.6 48-9 Remark(s) 2.7 61-4 Remark(s) 2.8 66-7 Remark(s) 2.9 73-4 Remark(s) 2.10 74-5 Remark(s) 2.11 78 Remark(s) 3.1 82-6 Remark(s) 3.2 89 Remark(s) 3.3 94 Remark(s) 3.4 100-1 Remark(s) 3.5 104 Remark(s) 3.6 105-7 Remark(s) 3.7 113-14, 115 Remark(s) 3.8 115 Remark(s) 3.9 117 Remark(s) 3.10 118 Remark(s) 4.1 126-7 Remark(s) 4.2 127 Remark(s) 4.3 129 Remark(s) 4.4 132 Remark(s) 4.5 133 Remark(s) 4.6 134-5 Remark(s) 4.7 135-6 Remark(s) 4.8 141 Remark(s) 4.9 144-5 Remark(s) 4.10 145-6 Remark(s) 4.11 148 Remark(s) 4.12 155 Remark(s) A.1 167-8 Remark(s) B.1 174-5 Remark(s) D.1 178-9 Remark(s) E.1 185 Remark(s) E.2 195 renormalized solutions 42, 101
Euler equations 132, 133-4, 135,
137,
140, 145
Navier-Stokes equations 112-13 proof of existence for parabolic equations 183-95
Riesz transforms 25, 92, 147
Schwarz spherically symmetric decreasing rearrangement 95, 106, 140 second derivative estimates 98-110 second law of thermodynamics 5, 7 sequences of solutions ix, 35, 39, 140 shallow water model ix shocks (discontinuities) 13-14 short time existence 150 simplified models vii, 9-16 singular integral 147 singularities 83, 151 Sobolev embeddings density-dependent Navier-Stokes equations 22, 25, 29, 38, 55, 59 Euler equations 134, 135 Navier-Stokes equations 82, 87, 92, 100 Rayleigh-Benard equations-115, 119 refined regularity of weak solutions 94-5 Sobolev inequalities 24, 78, 132, 149, 191, 195
Sobolev spaces 23, 25, 53, 79, 92 truncation of divergence-free vector fields 165-72 spherically symmetric decreasing arrangement 95, 106, 140 stability 136
re-
stationary Navier-Stokes equations 35, 91-2 stationary problems ix density-dependent Navier-Stokes equations viii, 34-5 Stokes equations 33, 68, 70, 90, 107, 120 Stokes problem 165-6 Stokes relationship 3-4 stream function 140 stress tensor 2, 9 viscous stress tensor 2 strong solution viii, 75-8 symmetrization 106, 116, 140
temperature 5, 110-23 Theorem 2.1 23, 86 Theorem 2.2 28, 30 Theorem 2.3 31 Theorem 2.4 41-2 Theorem 2.5 48 Theorem 2.6 66 Theorem 2.7 76 Theorem 3.1 81 Theorem 3.2 81-2 Theorem 3.3 82 Theorem 3.4 82 Theorem 3.5 89 Theorem 3.6 93-4 Theorem 3.7 100
Index Theorem 3.8 104 Theorem 3.9 109-10 Theorem 3.10 113, 183 Theorem 3.11 114, 115 Theorem 3.12 114-15 Theorem 3.13 118 Theorem 4.1 126, 136 modification 132-5 Theorem 4.2 137 Theorem 4.3 141 Theorem 4.4 145 Theorem A.1 60, 168 Theorem A.2 58, 170-1, 172 Theorem B.1 175 Theorem D.1 178 Theorem E.1 185 Theorem E.2 195
thermodynamic temperature 5, 110-23 thermodynamics, laws of 4, 5, 7 three-dimensional case Euler equations viii, 150-3, 163-4 Navier-Stokes equations 117-23 time compactness in 177 short time existence 150 time continuity 25, 28, 63 time-discretized problems see stationary problems total energy 4, 111, 114 transport equations 23, 67, 151, 163
with divergence-free vector fields 132, 137, 139
general regularization for solutions 43-5 uniqueness 121
truncations convergence in renormalized solutions for parabolic equations 190-3
divergence-free vector fields in Sobolev space 165-72 two-dimensional case Euler equations viii, 125, 136-50, 161 Navier-Stokes equations 105-7, 112
uniformly elliptic second-order operator 178
uniqueness
density-dependent Navier-Stokes equations viii, 31, 38 compactness results 49, 50-1
equality of weak solution to strong solution 75-8 existence proofs 67, 69-70 Euler equations viii, 125, 126, 151
237
two-dimensional 136, 138-9, 146 Navier-Stokes equations 82, 83, 86, 87-8, 92
and Rayleigh-Benard equations 112, 113 renormalized solutions for parabolic equations 185-7 transport equations 121
temperature
vacuum viii, 20, 34 velocity 158 velocity field 1, 2-4, 140, 161 viscous case 4 viscous stress tensor 2 vortex sheets 136, 137, 144-5, 152 vorticity 126,-144-5, 150 weak formulation 125
density-dependent Navier-Stokes equations 21, 25, 28 Navier-Stokes equations 112-13, 115 renormalized solutions for parabolic equations 184, 185, 186, 194 weak limit 70 weak solutions ix, 185
density-dependent Navier-Stokes equations viii, 21-5, 34, 35 equal to strong solution 31, 75-8 Euler equations viii, 135 global 137, 151-2 three-dimensional 151-2, 153
two-dimensional 145-50 Navier-Stokes equations viii, 79-89, 156 global weak solutions viii, 21, 79-82, 86-9, 105-7 Rayleigh-Benard equations 111-15 refined regularity via Hardy spaces 92-8 regularity 83, 89-92 second derivative estimates 98-110 weak topologies 28, 42, 131 compactness in time 177 weak topology of measures 115 whole space case (S2 = IR'v) 20, 153 density-dependent Navier-Stokes equations 24-5, 25, 30-1
a priori estimates 37-8, 40-1 compactness 47-8 existence proofs 64, 75 Navier-Stokes equations 79, 80, 81-2, 84 refined regularity of weak solutions 92-8 second derivative estimates 98-107 Young measures 153
