OXFORD 1,1,11 "I'l'I1I: SI:11I1-;S IN \Ir1'1'III'.`IATICS
f1\I) ITS AP1'LI(',VTI(1\S
10
Mathematical Topics in Fluid Mechanics \TOILillle 2
Compressible Models PIERRE-LOUIS LIONS
OXI-1()RD SCIENCE PuIII,I(: v1'I()NS
Oxford Lecture Series in Mathematics and its Applications 10 Series edilors John Ball
Dominic Welsh
OXFORD LECTURE SERIES IN MATHEMATICS AND ITS APPLICATIONS 1. J. C. Baez (ed.): Knots and quantum gravity 2. I. Fonseca and W. Gangbo: Degree theory in analysis and applications 3. P.-L. Lions: MathemIJtit'al topics injluid mechanics. Vol. J: Incompressible models 4. J. E. 8easley (ed.): Advances in linear and integer programming 5. L. W. Beineke and R. 1. Wil son (eds): Graph connections: Relationships between graph theory and other areas 0/mathematics
6. I. Anderson: Combinatorial designs and tournaments 7. G. David and S. W. Semmes: Fractured/ractals and broken dreams 8. Oliver Pretzel: Codes and algebraic curves 9. M. Karpinski and W. Rytter: Fast parallel algorithms/or graph matching problems
10. P.-L. Lions: Mathematicallopics in fluid mechanics. Vol. 2: Compressible models
Mathematical Topics in Fluid Mechanics Volume 2 Compressible Models Pierre-Louis Lions University Paris-Dauphine
and Ecole Polytechnique
CLARENDON PRESS OXFORD 1998
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Published in the United States by Oxford University Press, Inc., New York 0 Pierre-Louis Lions. 1998
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To Dorian and Lilo
PREFACE
Our goal in this series of books 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 and, in fact, all those contained in this second volume are new, even though some of them have been announced in various places. The first volume was essentially devoted to incompressible models, with an introductory chapter where the fundamental models to be studied were derived. This volume is entirely concerned with compressible equations. As we have already said, all the results are new and thus this book, in many respects, should most adequately be characterized as a research monograph which tries to cover a variety of mathematical issues associated with compressible equations. Of course, we shall give complete proofs which are essentially self-contained. We also tried not to assume from the reader too many technical prerequisites from (non-linear) partial differential equations
and (or) analysis. Finally, let us warn the reader who has looked at Volume 1 that the intended contents of Volume 2 which were presented in Volume 1 differ somewhat from the actual contents, which we shall describe below. The general organization is identical as far as chapters 5-8 (Part II) are concerned but each individual chapter is now much more developed since we incorporate quite a few recent and new results. In view of the length of these chapters, we decided not to incorporate the chapter (initially thought to be chapter 9 and Part III) on asymptotic limits. The corresponding material will be published later. Before we briefly describe the topics covered here, we wish to mention that this book, and this volume in particular, does not pretend to be a complete survey of the existing mathematical results on fluid mechanics equations even though we recall (or refer to) quite a few works. We have certainly omitted many relevant contributions to the field. We have tried to compensate for these omissions in the text by a rather extensive-but not exhaustive-bibliography (let us warn the reader that some of the references included there are not quoted in the text). Finally, we wish to make clear 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; stability analysis, attractors, inertial manifolds, etc.), reactive flows and combustion models, magnetohydrodynamics (MHD), geophysical flows, multiphase flows, free boundary problems, and so on, are not even touched on here. As we shall see, many basic open questions are left unanswered and we shall
viii
Preface
recall a large number of important open problems. In fact, we shall even add a few new ones! 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 though considerable progress has been (slowly) made. We only hope that these research notes will be a small contribution to the formidable task of the mathematical understanding of fluid mechanics models. Let us now describe the contents of this volume. The first chapter-namely
chapter 5, since we look at this volume as a self-contained continuation of Volume 1-is concerned with compactness results for compressible isentropic Navier-Stokes equations. At this stage it is worth explaining that one of the main goals in these notes is the construction of global weak solutions (in some situations, we shall be able to prove the existence of smooth solutions). The general strategy for such constructions is clear: by convenient approximations, one builds sequences of approximated solutions or, more precisely, sequences of solutions of approximated problems. Then, the key issue is the passage to the limit which, at least formally, should lead to solutions of our original problem. This passage to the limit always requires some form of compactness in order to be able to handle the non-linearities. This is precisely the topic we study in chapter 5 where we consider sequences of solutions of our original problem (the extension to approximated problems being straightforward at least if we choose the approximations carefully). We begin in section 5.1 with some preliminaries (motivation, definitions, and natural a priori bounds). Then, in section 5.2 we present our main compactness results together with some examples that show some of the difficulties which arise. The proofs are next detailed in sections 5.3 and 5.4 in a layered presentation. In the final two sections of this chapter, we deal with some extensions of the previous results and methods of proof: first, in section 5.5, we consider the case of general pressure laws, while in section 5.6 we treat various cases of boundary conditions (such as exterior domains, tubelike situations, non-vanishing data at infinity, etc.) which are slightly different from the prototypical ones studied in the previous sections-and induce minor additional difficulties. Chapter 6 is a rather long chapter devoted to stationary problems and time-
discretized problems. The length is due to the fact that we are able to say substantially more about the mathematical structure of these stationary problems than for time-dependent problems. We decided to position this material right after compactness issues and before existence results and proofs, since one possible approximation procedure (for the actual construction of global weak solutions, as already explained above) consists precisely in a time discretization and we then need to solve those time-discretized problems. After some general preliminaries (section 6.1), we present in section 6.2 our results on the existence of solutions (of such time-discretized problems) and their regularity. The proofs, given in sections 6.3-6.5, involve three major steps detailed in each of those three sections. First of all, we derive a priori estimates, then we analyse the compactness of sequences of solutions and we conclude with the construction of solutions
Preface
ix
using ad hoc approximations. Section 6.6 is entirely devoted to the study of a particular case, namely the isothermal case in two dimensions, which turns out to be, from a mathematical viewpoint, a critical case that we study using the concentration-compactness method introduced by the author ([347],[348]). We then turn, in section 6.7, to the study of (really) stationary problems which possess, in general, families of solutions that we parametrize in several ways using the total mass or more mathematical parametrizations that help in elucidating the structure of the set of solutions. Section 6.8 is devoted to the analysis of other boundary value problems such as, for example, exterior problems which, in the context of stationary problems, involve some major new difficulties. We next (section 6.9) investigate the higher regularity of solutions and we show by some examples that this regularity is not possible in general in the presence of a vacuum. Then, if there is no vacuum, we prove some delicate regularity results for solutions of stationary problems. Section 6.10 is concerned with various extensions like general pressure laws. Finally, in section 6.11, we show how
the previous results and methods can be used or adapted to treat stationary problems for general compressible models (with a temperature equation). In chapter 7, we turn to the existence of global weak solutions. We begin in section 7.1 with some new (and somewhat delicate) a priori bounds on the density. In section 7.2, we state (and comment on) our main existence results and we mention some important open questions. We present in the next two sections two different constructions of solutions (through two types of approximation procedures) based upon series of regularizations of the problem in section 7.3 and time discretization in section 7.4. Next, we show some extensions or adaptations to general pressure laws (section 7.5) and other boundary value problems (section 7.6).
The last chapter, namely chapter 8, contains results (and their proofs) on various related problems. One of the main goals of this chapter is to understand some of the difficulties associated with models involving the temperature and to show the applications of our work on such models. We begin; in section 8.1, with a simplified model where the entropy is no longer constant but is simply transported by the flow. For a such model, the analysis made in the previous chapters can be adapted. We next consider other models (which look more like isentropic models even though all of them have been introduced for other applications like the study of shallow waters) for which our methods can be combined with new arguments in order to yield a rather complete analysis: more precisely, we look at a semistationary model in section 8.2, at a Stokes-like model in section 8.3, and at some shallow water models in section 8.4. Next, in section 8.5, we discuss the compactness properties of solutions for compressible models with a temperature (or an energy) equation. Unfortunately, we need to postulate some bounds (which are not known). This is why, in section 8.6, we prove some existence results only for some of these models: roughly speaking, we treat situations where the constitutive laws (say for the pressure) only differ from the ideal gas assumptions at very large densities. Such modifications allow us to obtain some
x
Preface
of the missing bounds and then to derive the existence of global weak solutions. Then, we briefly discuss the compressible Euler equations in section 8.7, making some rather wild speculations more or less motivated by our analysis of the compressible Navier-Stokes equations! Finally, section 8.8 is devoted to a brief discussion of a low Mach number model. After these chapters, we incorporate six (more technical) appendices needed
in the previous proofs. Then, the bibliography may be found: it is in fact the joint bibliography of the book, that is of Volumes 1 and 2. Next, we have inserted a small erratum which lists a few misprints we have spotted in Volume 1.
Paris and Ajaccio December 1996
P.L.L.
CONTENTS
5
Compactness results for compressible isentropic NavierStokes equations 5.1 5.2 5.3 5.4 5.5 5.6
6
Preliminaries Compactness results and propagation of oscillations Proofs of compactness results in the whole space case Proofs of compactness results in the other cases General pressure laws Other boundary value problems
Stationary problems 6.1 6.2
6.3 6.4 6.5 6.6 6.7 6.8 6.9
Preliminaries problems A priori estimates Compactness Existence proofs The isothermal case in two dimensions Stationary problems Exterior problems and related questions Regularity of solutions
Existence results for Cauchy problems 7.1 7.2 7.3 7.4 7.5 7.6
8
1
7 15
30 36 39 49 49
Existence and regularity results for time-discretized
6.10 Related problems 6.11 General compressible models 7
1
A priori bounds Existence results Existence proofs via regularization Existence proofs via time discretization General pressure laws Other boundary-value problems
Related problems 8.1 8.2
Pure transport of entropy A semi-stationary model
8.3 8.4 8.5
A Stokes-like model On some shallow water models Compactness properties for compressible models with
temperature
51
57 80 84 97 112 128 144 158 162
172 172
180 182 197 205
209 213 213 224 236 251
254
Contents
xii
8.6
Global existence results for some compressible models
8.7 8.8
with temperature On compressible Euler equations On a low Mach number model
Appendix A: A few facts about some function spaces Appendix B: On a weakly continuous product Appendix C: A remark on the limiting case of Sobolev inequalities Appendix D: Continua and limits Appendix E: On sums of LP spaces Appendix F: A remark on parabolic equations Bibliography Errata (Volume 1) Index
CONTENTS LIST FOR VOLUME 1
1
Presentation of the models 1.1 1.2
I 2
2.4 2.5
3.3 3.4
4
Existence results Regularity results and open problems A priori estimates and compactness results Existence proofs Uniqueness: weak = strong
Navier-Stokes equations 3.1 3.2
Abrief review of known results Refined regularity of weak solutions via Hardy spaces Second derivative estimates Temperature and Rayleigh-Bernard equations
Euler equations and other incompressible models 4.1 4.2
4.3 4.4 4.5 4.6
9
INCOMPRESSIBLE MODELS
Density-dependent Navier-Stokes equations 2.1 2.2 2.3
3
Fundamental equations for newtonian fluids Approximated and simplified models
1 1
A brief review of known results Remarks on Euler equations in two dimensions Estimates in three dimensions? Dissipative solutions Density-dependent Euler equations Hydrostatic approximations
19 19 31 35
64 75
79 79 92 98 110
124 125 136 150 153 158 160
Appendix A: Truncation of divergence-free vector fields in Sobolev spaces
165
Appendix B: Two facts on D1'2(R2)
173
Appendix C: Compactness in time with values in weak topologies
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 Bibliography of Volumes 1 and 2 Index
177 178 183 196
233
xiv
Contents
Bibliography of Volumes 1 and 2 Index
196
233
5
COMPACTNESS RESULTS FOR COMPRESSIBLE ISENTROPIC NAVIER-STOKES EQUATIONS 5.1 Preliminaries Almost all of this chapter is devoted to the analysis of the Cauchy problem for the compressible isentropic Navier-Stokes equations. Namely we look for global
solutions (p, u)-as usual, p is a non-negative function that corresponds to the density of the fluid (or gas) and u is a vector-valued (in RN) function that corresponds to its velocity-of the following system
+ div (pu) = 0 a(p'ui) + div (pu uj) -plus
where p > 0, p +
-
(5.1)
= pfi, 1 < i < N (5.2) > 0, a > 0 and ry E (1, oo) and f = f (x, t) is a given 8i div u + ai (apry)
function corresponding to force terms on 12 x (0, T) for some fixed T E (0, oo). Exactly as in Volume 1, we shall mainly consider three situations : 1) the case when the equations are set in the whole space (called the whole space case or the
case when St = RN), namely (5.1)-(5.2) are required to hold on RN x (0, T) for some fixed given T E (0, oo) ; 2) the case of Dirichlet boundary conditions where (5.1)-(5.2) hold in SZ x (0, T) and u = 0 on 8 St x (0, T)-in that case, we assume for simplicity that St is a bounded connected open smooth domain in RN; 3) the periodic case where (5.1)-(5.2) hold in RN x (0, T) and we require that all the data and unknowns are periodic in each xi (1 < i < N) with period Ti E (0, oo). These assumptions (and this terminology) will not be repeated in what follows. As we saw in Volume 1, global existence results often follow from the analysis of the convergence of sequences of solutions and of the passage to the limits inside
the equations. Existence results for the system (5.1)-(5.2) of equations will be shown in chapter 7 and their proofs will indeed rely upon such convergence issues. This is why we concentrate in this chapter on the analysis of the convergence and of the compactness of sequences of solutions. Precise results are given in the next
Compactness results
2
section together with some examples showing that the issue is a bit complicated and requires a careful analysis. We then present complete proofs of these results in the sections that follow. We thus consider a sequence of solutions of (5.1)-(5.2) (ptz, un); of course, we are in one of the three cases mentioned above. The notion of solutions that we use has to be made precise but we want to emphasize that the results stated in Section 5.2 below are relevant even for smooth solutions solving (5.1)-(5.2) in a classical sense. Let us finally mention that the methodology consisting of building (or trying to build) global weak solutions of non-linear evolution equations by making first an analysis of the stability (and compactness properties) of sequences of solutions is by now a classical approach but our inspiration for such an approach is certainly L. Tartar's work on oscillations and compactness issues (see in particular L. Tartar [530], [531], [532]). We wish to emphasize here this point since we shall stress later on similarities and differences between the phenomena arising for systems like (5.1)-(5.2) and those occurring in non-linear hyperbolic conservation laws. These differences will require some rather new arguments to prove some specific compactness properties, which rely on compactifying properties of some commutators which are intimately connected to the presence in (5.1)-(5.2) of the convective derivative (9 + u - V). If we compare with L. Tartar's work on scalar conservation laws (in one space dimension) [530], [531] the results are going to be different in nature and the proofs will use different compacteness properties even if, as shown by R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes [110] the compensated-compactness results in [401] (such as the div-curl lemma) can be recast in terms of compactifying commutators. Another common feature will be the systematic use of non-linear expressions to test the stability and convergence properties of solutions as was done in [531] with entropies and in R.J.
DiPerna and P.-L. Lions [154] (for linear transport equations) with arbitrary "renormalizations".
Let us conclude this introductory section by defining precisely the "weak solutions" we are going to build and use. First of all, in all cases (of boundary conditions), we require p and u to satisfy p E L°°(O,T;Ly(S2))nC([O,T] ; L"(1))
for 1 < p < -y, p > 0 a.e. (5.3)
Vu E L2(0, T; L2(T )) , p Iu12 E L°°(0,T; L1(St))
(5.4)
p u E C([0,T] ; L2y/(y+1) _ w).
(5.5)
Recall that, as in Volume 1, we denote by C ([0, T] ; X - w) the space of continuous functions with values in a closed ball of X endowed with the weak topology where X is an arbitrary separable Banach space, for instance.
Preliminaries
3
Next, in the case of Dirichlet boundary conditions, we ask that (5.1)-(5.2) hold in the sense of distributions and u E L2 (0, T ; Ho (11)) . In the case when SZ = RN, (5.1)-(5.2) hold in RN x (0, T) and u E L2 ((0, T; H1 (B)) for any ball B C RN, u E L2(0, T; L2N/(N-2) (RN)) if N _> 3 and, finally in the periodic case, we simply add to the preceding requirements that p and u are periodic in each xi with period Ti (for all 1 < i < N). In view of (5.3) and (5.5), we can specify initial conditions, namely
PLO = Po
in SZ
putt=. = mo in SZ
,
(5.6)
where we always assume that po, mo satisfy Po ? 0 a.e. in Sl, po E L1 n L11(0), mo E L2y/(-r+1)Al mo = 0 a.e. on {po = 0} ImoI2/po (defined to be 0 on {po = 0}) E L1(12) and po 0.
(5.7)
And of course, in the periodic case, po, mo given on RN are assumed to be periodic. As we can see, the above notion of weak solutions is a very natural one provided
one explains the origin of the requirements (5.3) and (5.4). Obviously, they correspond to a priori estimates and in fact, they simply "follow" from the (global) conservation of mass and the energy identity. Indeed, (5.1) yields that p remains
non-negative for all t > 0 and that fn p(t) dx is independent of t > 0. In addition, multiplying (5.2) by u and using (5.1), at least formally, we deduce U12 /2
P
+ pu
eI
+t;(divu)2 +
2
V 8 5,
1
2
- p0
ry-1 (_a
pry
12I2
- e div(u div u) + pI DuI2
+ div u
a^f
7-1
(5.8)
p,
=pu
Here and below, we are using the usual convention of implicit summation over repeated indices. Integrating (5.8) with respect to x, we deduce d dt
Jn
p
a
{iJ-+u2
2
dx + in p IDuI2 + t; (div u)2dx
ry- 1
n
(5.9)
pu fdx fro,
and obviously f n(div u)2 dx = fn aiuj 88ui dx < fn IDuI2 dx. Since p > 0, p+t; > 0, we deduce from (5.9) that we have for some v > 0 d dt
Jp--- +
a
1u12
If f satisfies
-f
-1
p,
1
dx + v
f
n
IDuI2 dx
i is bounded in L°° (O,T; Li n Lry(11)) , (pi, Iun12)n>i is bounded in L°°(O,T; LI(fl)), (Vun)n>1
is bounded in L2 (SZ x (0,T)), and in the case of Dirichlet boundary conditions that (u")n>1 is bounded in L2 (0, T; Ho (0)), in the periodic case or in the case when fl = Rn that (un)n>1 is bounded in L2 (0, T; Hi (BR)) (for
all R E (0, oo)) and finally that, if h = RN and N > 3, un is bounded in L2 (0,T;
L2N/(N-2) (RN))
. All these bounds are the natural ones as we saw in the preceding section. Finally, for the force terms f", there are various rather technical conditions we can impose and this is why, in order to keep ideas clear, we shall simply assume that (f n)n>i is bounded in L' (fl x (0, T)) n LOO (SZ x (0,T)), f n converges weakly in L°° (fl x (0, T)) (weak-*) to f E L1(11 x (0, T)) n LOO (fl x (0,T)) ,
(5.18)
and we shall indicate later on the precise assumptions which are really needed.
Without loss of generality-extracting subsequences if necessary-we can assume that pn, pn, un, pn fn, Pn Un, pnun, pnui uj converge weakly respect
Compactness results
8
ively to some p, 7, u, 77 , v, m, e1 (b'1 < i, j < N) in the appropriate weak topologies. To be more precise, let us detail these weak convergences in the case of Dirichlet boundary conditions for instance: pn n p weakly in L7 (11 x (0, T)) u weakly in L2 (0, T; HH(f )), and p E Loo (0,T; L1 n Ly(S1)) (p > 0 a.e.), un v weakly in L2 (Sl x (0, T) ) pn weakly in L21(0 x (0, T)), pn un and v E L°O (0, T; L2())), pnun -n& m weakly in L2y/(y+1) (11 x (0, T)) and m E ei3 in the sense of measures on fl x (0, T) LOO (0, T; L2y/(y+1) (St)) , pnu= ujn and ezj is a bounded measure on 11 for almost all t E (0, T) which is bounded uniformly in t E (0, T); and f n - T f weakly in Ly (11 x (0, T)) say. Next, we are going to postulate some a priori bounds on pn. Of course, when
-
-
e
we address the existence question in chapter 7, we shall have to prove these bounds on pn (or at least in some cases). We thus assume that for some fixed
q>2, q>y, s>N/2
(pn)n>1 is bounded in Lq (Sl x (0,T)) n L°O (0, T; L(11))
in the periodic case or if Sl = RN, N > 3 (pn)n>i is bounded in Lq (0, T; Lq(K)) n LOO (0, T; L'(11)) in the case of Dirichlet boundary conditions or if 11 = R2
(5.19)
where K denotes an arbitrary compact set included in 11. Of course, this means
that p has the same Lq integrability. In addition, these bounds imply that (pnui u )n>, are bounded in various spaces (and thus e13 belong to these spaces) namely L°°(O,T; L1) n L1(O,T; LO) where 1/Q = 1/s + (N - 2)/N if N > 3 and
1 < Q < s if N = 2; this follows easily from the Sobolev inequalities and the Holder inequalities using the fact that un is bounded in L2 (0, T; H') and pn Jul' is bounded in L°° (0, T; L1). Let us observe at this point that when y > N/2 (this is automatic for N = 2 !), we already know that pn is bounded in L°° (0, T; Ly (11))
and thus the L°° (0, T; L') bound is satisfied. On the other hand, the Lq bound is of a different nature and will have to be proved in order to be able to use the following result.
We may now state our main result.
Theorem 5.1 (1) We always have : v = it, m E pu, eij = puiuj a.e. in Sl x (0, T) (V 1 < i, j < N). (2) If, in addition to the above assumptions, we assume that po converges in L1(Q) to po then (p, u) is a weak solution of (5.1)-(5.2) satisfying the initial condition (5.6) and we have
pn n pin C ([0,T]; L" (0)) n L- (Ki x (0,T))
forall 10, Q
where p > 0 and pIt=o = po in RN.
We then consider p"(x, t) = p(nx, t), un(x, t) = (1/n)u(nx, t) and we check easily that (pn, u") solve (5.1)-(5.2) with the initial conditions (5.23) and with f replaced by fn given by
f"(x, t) = i ( au + (u 0)u
na
(nx, t)
on RN x (0, oo).
(5.25)
Obviously, (5.18) and (5.19) hold. However, (5.20) does not hold (as soon as p is not constant in x which is the case as soon as po is not a constant). In other words, oscillations can persist. Related phenomena are presented in D. Serre [4871.
It is possible to study much more complex situations where po = po (x, nx), mo = mo(x)+(1/n)mi(x, nx) and po(x, y), mi(x, y), mo(x) are periodic in (x, y) and in x respectively. Let us only mention that if, for instance, f " - 0 then the solution (pn, u") of (5.1)-(5.2) corresponding to the initial conditions (5.24) does indeed behave like p(nx, t), (1/n)u(nx, t). In this example, the weak limits (p, u) are very simple: u - 0, p - fQ pody. And it turns out that (p, u) is a solution of (5.1)-(5.2). However, we shall explain in the next remark why it is not always the case.
Compactness results and propagation of oscillations
11
Remark 5.9 (Weak limits are not in general solutions) . We wish to explain here why weak limits are not in general solutions. In fact, we are going to show that, roughly speaking, if the weak limit (p, u) is a solution of (5.1)-(5.2) in the case when 11 = RN then necessarily po converges strongly to po. More precisely, we assume that (p, u) is a solution of (5.1)-(5.2) and that we have sup n>1
f po (x) dx - 0
as R tends to + oo.
(5.26)
(IxI>R)
Then, we claim that po converges to po in LP(RN) for 1 < p < 7. Indeed, in view of part 1) of Theorem 5.1, the fact that (p, u) solves (5.1)(5.2) yields the convergence of V (p" )l' to V pry in D' (RN x (0,T)). Recalling that (pn)" is bounded in L°° (0, T; L1(RN)) and that p"r E L°° (0, T; L1(RN)), we deduce easily that (pn)7 converges to p7 weakly in the sense of measures
on RN x (0, T). Next, since 7 > 1 and thus (t '- try) is strictly convex on [0, oo), we deduce from standard functional analysis results that pn converges to p in L" (K x [0, T)) for any compact set K C RN. In particular, extracting a subsequence if necessary, we can find to E (T/2, T) converging to some t E [T/2, T] such that pn(tn) converges in L C(RN) to p(i). In addition, as we shall see in the proof of Theorem 5.1, (5.26) yields sup sup n>1
J
pn (x, t) dx ---> 0
as R tends to + oo.
(5.27)
(IxI>R)
Hence, pn (tn) converges to p (0 in LP (RN) for 1 < p < 7.
We then introduce p'n(x, t) = pn(x, to - t), un(x, t) = -u(x, to - t) for t E [0, tn], p(x, t) = p(x, t - t), u(x, t) = -u(x, t - t) for 0 < t < ) . We deduce from the above that pro = p`"It=o converges to po = Alt=o in LP(RN) f1 L 10C(RN)
for 1 < p < 7, Vun converges weakly in L2 to Du, and pn converges in V to p. Finally, we have obviously
ap 8t
+ div(lbn`n) = 0 in RN x (0, tn).
(5.28)
These facts imply our claim, namely An converges (uniformly in t) in LP(RN) to p for 1 < p < 7 and thus in particular po = pn(tn) converges in LP(RN) to po = p( ). This uniform convergence will be in fact proved at the end of the proof of Theorem 5.1 where we deduce "the uniform convergence of pn from the convergence in (x, t)" .
Remark 5.10 In the case when SZ = RN, it is possible to relax slightly the convergence of po to po in L1(f) replacing it by a convergence in Ll C(cl). Then, all convergence in (5.20)-(5.21) become local ones. In other words, the condition (5.26) is not needed neither for part 2) of Theorem 5.1, nor for Remark 5.4. This minor extension will be clear from the proof of Theorem 5.1.
Compactness results
12
Remark 5.11 The convergence result stated in Theorem 5.1 is precisely what was needed by D. Hoff [251] in order to extend the existence result of global solutions (close to an equilibrium) to more general barotropic laws p(p) (for instance p(p) = ap7 for a > 0, -y > 1) than p(p) = p. In fact, as indicated in the next remark, very general laws p(p) can be treated as well. 13
Remark 5.12 As we shall see after the proof of Theorem 5.1, Theorem 5.1 can be extended to general barotropic laws p = p(p) (instead of p = ap"). Our main assumption is that p is increasing on [0, oo): a natural assumption from the physical viewpoint-see chapter 1.
Remark 5.13 As we shall see in the next section, where we prove part (2) of Theorem 5.1, one of the key ingredients is the following fact (established in the next section)
Q(p") {(p + l;)div u" - a(p")^1}
- Q {(µ + t;)div u - ap}
in D'
(5.29)
for any continuous functions 3 on [0, oo) such that /3(t)t(q-,') and j3(t)t(g/2) goes to 0 as t goes to +oo, where we denote by p, 0 the weak limits (in the appropriate L'' spaces) of respectively. This property was shown by D. Serre [487] in the one-dimensional case, allowing a rather complete description of the propagation of oscillations (see also WE [167]). In [487], this descripsion of the propagation of oscillations was also obtained in higher dimensions postulating (5.29) (and various other more technical conditions). We shall come back to this interesting question in chapter 9.
We now turn to the proof of part (1) of Theorem 5.1. It is in fact a simple consequence of the following general lemma.
Lemma 5.1 Let g", h" converge weakly tog, h respectively in L°' (0, T; Lr2 (11)), L°1 (0, T; L'2 (51)) where 1 < pl, p2 < +oo,
-+-=-+-=1. pi 1
1
1
1
Qi
P2
Q2
We assume in addition that agn is bounded in Ll (0, T; W"1 (52)) for some m > 0 independent of n
8t
-
11h" - h"( +e,t)1ILgl(o,T;Lg2(n)) -* 0 as ICI -+ 0, uniformly in n.
(5.30) (5.31)
Then, gnh" converges to gh (in the sense of distributions on 52 x (0,T)).
Remark 5.14 In the statement above, weak convergences are weak-* convergences whenever some of the exponents are infinite. In (5.31), hn(. + C, t) is not really defined on fl x (0, T) but we can either extend h" to R" by 0 or restrict the T,a3 norm to 1' = fx E 0 /dist (x,852) > Ihl}.
Compactness results and propagation of oscillations
13
Proof of Lemma 5.1. Let e > 0. We are going to prove the above claim in fZEO x (0,T) where 11,0 = {x E 0/dist (x,811) > eo } and we simply denote from now on Sl instead of ftE. Then, we consider he = f hn(y, t) rcE(x - y) dy is a regularizing kernel: is > 0, Supp fore E (0, co) where r., = ,c c Bi, fRN rc dz = 1. The condition (5.31) ensures that we have 11h n
-
hE I I L91
(o,T;L42 (ft)) w
0 as a ---> 0, uniformly in n.
In addition, hn converges weakly to hE = h*rKE as n goes to +oo and hE converges strongly to h in L91(0, T; Lq2 (f2)) as a goes to 0. Hence, writing gnhn = gn(hn hE) + g"hE, we deduce easily that it is enough to show that gnhE converges in
-
D' to gh,. In other words, we may assume without loss of generality that hn is bounded in LQ1(0, T; Wk,q2 (Il)) for all k > 0 and converges weakly to h in Lg1(0, T; Wk,g2 (ft)).
Next, we set H" = fo hn(x, s) ds. The bound on hn when qi > 1 implies easily that Hn converges to H = fo h(s) ds in Wk,q2 (el) uniformly in t E [0, TJ for all k > 0 at least when 12 is bounded (otherwise the same convergence holds on compact subsets). The same claim holds when qi = 1. In fact, for any multiindex
a, Di hn is clearly bounded on ft x (0, T) and for all t, s E [0, T], for all x E 1 I Di Hn (x,
t) - DiHn (x, s) I < Cb + I
k
11
f DaH' x (y, t) dy a(x,a)
f Dz Hn (y, s) dy a(z,S)
t
dy
< CE -}I
BaI
(x,d)
dTDz hn(y, o) a
0.
-
It is then easy to conclude writing gnhn = 8(g"H")/at (ag"/at)H" (this equality of course requires to be justified but there is no difficulty here in view of the spaces to which belong gn, H", agn/at, h"). Indeed, g"H" obviously converges to gH (in fact weakly in Lp1(0, T; LP2 (f2)) while (8gn/8t)Hn converges in the sense of distributions to (8g/8t)H since Hn converges, in particular, to H in C' MI) uniformly in t E [0, T] . We conclude observing that
gh = a(gH)/at - (ag/at)H. 0 Proof of part (1) of Theorem 5.1. We begin by proving that m = pu, e = pu ®u. First of all, we recall that un is bounded in L2 (0, T; Hl). In this section, we do not specify on which domain we are working in order to avoid recalling each time the various cases we consider: let us simply indicate that in the proofs
Compactness results
14
that follow all function spaces concern a spatial domain which is 91 in the case of Dirichlet boundary conditions, the periodic cube in the case of periodic boundary
conditions or a ball of an arbitrary radius R E (0, oo) in the whole space case. Therefore, (5.31) holds with hn = un, q1 = 2, q2 arbitrary in [2, 2N(N - 2) (Q2 arbitrary in (2, oo) if N = 2).
We then wish to apply Lemma 5.1 with g' = pn and gn = pnun. If this is possible, we conclude: m = pu and e = m 0 u = pu 0 u. The bounds on 8pn/8t, 8pun/8t are straightforward since we have for instance in view of (5.1), (5.2) and the bounds on pn, un 8pn
W-1,27/0(+1))
is bounded in L°O (0 , T;
C L1 (0 , T;
W-1,1)
apnun is bounded in L' (0, T; W-1'1) + L2 (0, T; H-1) c L' (0, T;
W-1,1 )
8t Therefore, in order to conclude, we just need to check that pn and pnun are bounded in L2 (0, T; LP) for some p > 2N/(N + 2). This is clear for pn which is bounded in view of (5.19) in L°O(0,T; L'') and r > N/2 > 2N/(N + 2) (N > 2). Since un is bounded in L2 (0, T; H'), we also deduce from (5.19) that pnun is bounded in L2 (0, T; LP) where
2N2 ifN>3, p 2. The equality (5.32) follows from (5.33) choosing Qa = b -+t for b > 0, letting 6 go to 0+ while remarking that,(3a (p) - ,3, (p) p = VT -+P - 2 p/ vw-+p converges as 5 goes to 0+ (in Lz i for instance); and the proof of part 1) of Theorem to a 5.1 is complete. Remark 5.15 We observe for future purposes that if Q E C'(0, oo), 3'(t) = o(t ) as t goes to 0+ and ,(3'(t)(1 + t)-a is bounded on (1, oo) where a = (q - 2)/2, then (5.33) holds with .3(p), u replaced by ,3(p11), un (we agree that 3'(p)p = 0 if p = 0). In particular 84(pn)/8t is bounded in L2q/(q-1) (0,T; Lq/(q-1)) +
Ll (0, T; L1) ((C L'(0,T; Ll)) and thus (5.30) holds with gn = ,6(pn). Since ,Q(pn) is bounded in L2 (0, T; L2), we can use Lemma 5.1 as in the preceding proof to deduce that 8(pn)un converges (in the sense of distributions) to ,Qu if ,3(pn) converges weakly in L2 to some ,0.
5.3 Proofs of compactness results in the whole space case This section is devoted to the proof of part (2) of Theorem 5.1 and more precisely of (5.20). But before we begin this rather delicate proof, we wish to present first a formal argument that yields the compactness of pn and which is based upon the energy equation. The reason why we present this argument, which can be justified modulo various technical bounds that we do not know how to obtain, is that we believe it is a more direct one than the actual proof while it contains some of its essential features. Furthermore, we think this argument could lead to some interesting analysis of numerical schemes for the system of equations (5.1)-(5.2).
Formal argument for the strong convergence of pn The main assumptions we need for this formal argument are that p, u (the weak limits) have to satisfy (5.9) (or a variant of it written below), (div u)- E
L1(0, T; L°O(f )), and (pn, un) satisfies (5.9) (with (p, u, f) replaced by (pn, un; f,)) or even the following weaker version Iunl2 2
d
d
R
pn
+
a 'Y
< J p"un. fndx. n
pIDun12 + l;(div un)2 dx
1(Pn)ry dx + n
(5.34)
Compactness results
16
Without loss of generality, we may then assume that (e)" converges weakly in Lq/7 x (0, T)) to some p : since (t - t1) is convex, we deduce that p > p7. Using part (1) of Theorem 5.1, (5.34) yields 2
dt
in
p
+'Y a l p dx +
-
I
Jn µI DuI2 + t;(div u)2
dx
0. Combining (5.35) and (5.36), we obtain
d
jPe dx + (-y - 1) j(div u)pe dx < 0.
(5.37)
In particular, fn pe (t) dx < exp [('Y - 1) fo I I (div u) I I Loo ds] fn pe (0) dx. Therefore, if po converges to po in L'7 (a), pe (0) = 0 and we deduce that pe = 0. This shows in fact that p" converges in Ll' (SZ x (0, T)) to p and the main assertion of
part (2) of Theorem 5.1 is shown. 0
Remark 5.16 At this stage, it is worth looking at the conditions which are needed in order to justify (5.9) or (5.36). Typically, if we assume that p (or p7) E L2 (n x (0, T)), we have in addition to (5.1) the following equation
+ div (pu (9 u) = T E L2(0, T; H-1(1))
(5.38)
(considering for instance the case of Dirichlet boundary conditions, the other cases can be treated in a similar way). And it is then possible to justify (5.36).
Proof in the case when St = RN, N > 3 and when pn is bounded in L-f+1(RN x (0,T)) n L°°(O,T; L°(RN)) with s> N We begin with this case in order to explain some of the main ideas developed in this proof, leaving out all the technical difficulties associated with boundary
conditions (or N = 2) or the fact that "good" bounds on p" are not always ava.i I able.
Proofs of compactness results in the whole space case
17
Step 1: Formal proof. Furthermore, we wish to begin with a formal presentation of the argument that we shall develop and justify below. We use two main ingredients that both follow from (5.1)-(5.2) : indeed, we first observe that we have 19P Log
p
+ div (u p Log p) + div u p = 0
(5.39)
and taking the divergence of (5.2) (-0)-'div
a
(pu) + (-A)-laij(Puiuj) + [(µ + )div u - app'] (-A)-'div (Pf)
=
(5.40)
From (5.40), we extract div u and obtain
(µ + )div u = ap7 +
- (-0)
(=A)-ldiv
(pf) 8j(Puiu.i) -
5j(-0)-'div
(5.41)
(pu).
Then, inserting this relationship into the right-hand side of (5.39), we deduce, using (5.1)
(P + 0 + 5
aP &g + div (up Log p)] {p(_A)-'div (pu)} + div
_p(_p)-'div
+
(Pu)(-0)-'div
(pu) +
(Pf ) p(-0)-laij(Puiuj)
hence finally (µ
+
e)
aP
P
&g
+ div (up Log p) + apl+7 =
_ p(_A)-'div (pf) + +p {(-0)-'aij (Puiuj)5i- u
[p(-0)-'div (pu)]
+ div
[pu(-0)-'div (pu)]
V(-0)-ldiv (pu)}.
(5.42)
We are going to use systematically these (and related) identities. First of all, we can write (5.42) with (p, u, f) replaced (p", un, fn) and we pass to the limit in (5.1), (5.2) and (5.42). Part(1) of Theorem 5.1 and a systematic use of Lemma 5.1 then yield (we shall come back to these points later on): (5.1) holds and (5.2) at the limit is replaced by a
(pu)+div (pu(9 u) - pAu - Vdiv u +Vp= pf,
(5.43)
where P is the weak limit of a(pn)-. In addition, we deduce from (5.42) (p + l;)
+ div (us) + apl+7 =
- p(_A)-'div (Pf) +
(9
at + p {(-A)-1a=j(Puiuj) - u
[p(-z)-'div (pu)]
+ div
[pu(-0)-'div (pu)]
V(-z)-'div (pu)}
(5.44)
Compactness results
18
where p1+7, s denote respectively the weak limits of (pn)1+7, ppLog pn (extracting subsequences we can always assume they converge weakly). The first three terms on the right-hand side of (5.44) are obtained easily using Lemma 5.1 as we shall see below. The fact that we can pass to the limit in the term pn 81j(pnui un V(-A)-ldiv (pnun)} and recover the expression p{(-0)-1 8ij (puiuj) (pu)} is also aconsequence of Lemma
81j(puluj)u - V(-0)-ldiv (pu)} as the commutator of uj and aj(-0)-1div acting upon pu, i.e. [uj, Rij] (put) where Rij = (-A)-1 81j is a "nice" Calderon-Sygmund 5.1 and of the following crucial observation: one can write
singular integral operator which is nothing else than the composition of two Riesz a1(-0)-1/2 and aj(-A)-1/2 ). Next, we have a (limited) transforms (namely
"Sobolev" regularity of u", u (in L2(O,T; H1)) and this is enough, as we shall explain below, to make the operator [uj, Rj] a smoothing operator which allows us to pass to the limit. Next, we deduce from (5.1) and (5.43) exactly as we deduced (formally) (5.42) from (5.1) and (5.2)
(µ + )
+ div (u s) -p(-0)-ldiv
_
+ pi [p(-0)-ldiv (pu)]
(pf)
+N {(-0)-1a1j(puiuj) - u
+p
+ div [pu(-0)-1div(pu)]
V(-0)-ldiv (pu)}
(5.45)
where s = p Log p. Comparing (5.44) and (5.45), we obtain 19
(s
- s) + div
[u(s - s)] +
a
[p1+7
- p 7] =
0
(5.46)
where 7 = p/a is the weak limit of (pn)7). The conclusion follows from rather standard functional analysis and .onvexity
considerations since we know that s < s a.e. and p1+7 > pp7 a.e. In particular, integrating (5.46) in x, we deduce d
dtINs-sdx < 0 R
while fRN s -
s dxlt=o = 0 since we assumed that po converges strongly to po
and fRN s - s dx < 0 for all t > 0. Hence, s = s a.e. and, since t Log t is strictly convex, we deduce finally the strong convergence of pn towards p which is the crucial information required to complete the proof of Theorem 5.1. Of course, the above argument is just a sketch of the actual proof we may now begin.
Step 2 : Preliminaries. We justify here some of the considerations developed in step 1 above.
Proofs of compactness results in the whole space case
19
First of all, if u E L2(O,T;H1 c) and p E Li c satisfy (5.1) then using the fundamental regularization lemma stated in Part 1, chapter 2, we see that we have
/3(p) + div [u3(p)] + (diva) [p3'(p)
- Q(p)] = 0
(5.47)
for any C' function /3 from [0, oo) into R such that
3C > 0,
dt > 0,
(5.48)
Ii3'(t)I < C(1 + t°1)
with a = (q - 2)/2 (if p E Li ; if q = 2, (5.48) reduces to assuming that
/3'
is bounded) . Next, approximating /3(p) = p Log p by /36 (p) = p Log (p + 6), we write (5.47) with 0 replaced by Q6 for b > 0. Then, we observe that we have p Q6 (p)
-,06(p) = p2 (p + b) -' -'pin LQ (or L2) as b - 0+.
In this manner, we justify (5.39) passing to the limit as b goes to 0+. In particular (5.39) holds both for (p", u") (replacing of course (p, u) by (p", u")) and for the
weak limit (p, u) provided we show that (5.1) holds-a fact that we establish below.
Next, we justify (5.42) (with (p, u) replaced by (p", u")). The equations (5.40)-(5.41) are immediate (in view of the properties of -A on RN, N > 3). Then, in order to justify (5.42), we have to make sure that each of the terms of the right-hand side of (5.41) cafr be multiplied by p (or p") and to justify the manipulations on the term p J(-A-1div (pu)). Multiplying pry by p is meaningful since we assumed that p" is bounded in D+1. Similarly, the assumptions made upon f (or f") namely (5.18) ensure that we can multiply (-0)-'div (pf ) (-0)-' aij(puiu3) and we observe that, by by p. Next, we consider the term assumption, puiu. E L°° (0, T; L' (RN)) n L' (0, T; L'(RN )) where 1
1
N-2
r
s
N
(p"ui u,' is bounded in that space) since uiuj E
L'(O,T;LN/(N-2)(RN)) by
Sobolev inequalities. In particular, the operator (-z )''a{j (=RJR;) is bounded in all L' spaces (for 1 < r < co), so we see that
(-A)-'ai3(puiui)
E L' (O, T;Lr(RN))(nL'(O,T;Lq(RN))
where 1 < p < oo, 11q = 1/pr + 1 - 1/p. Hence, we can multiply this term by p since p E L°O (0, T; L3 (RN)) and
s+r = s+N N 2
2
N-2 N =1.
N The only fact left in order to justify (5.42) is the multiplication of p by
(-j)-'div (pu) and writing
Compactness results
20
pa [(-0)-'div (pu)] {p(-A)-ldiv (pu)}
_
_ _
19
- L (-L)-'div (pu)
{p(-A)-'div (pu)} + div {p(-A)-ldiv (pu)} + div
(pu)(-0)-ldiv (pu) (pu(-o)-'div
(pu)) - pu
V(-A)-ldiv
(pu).
This is not difficult to justify so we only sketch the argument. Indeed, smoothing Ic E Co (RN x R), p by convolution in (x, t) (pE = p * ice, 'E _ (1/EN+1)(,/e),
Supp rc C B1, e E (0,1]) so that (5.1) becomes W + div (pu)E = 0 we can perform the above manipulations replacing p by pE since pE is smooth. Then we let a go to 0 and we recover the equallity between the first and the last terms. In doing so, we essentially only need to check that p . [(-A)-ldiv (pu)] , p(-A)-1
div (pu), pu(-0)ldiv (pu) and pu - V(-0)-ldiv (pu) make sense (in Ll ,,): in fact, the last of these four terms is very much similar to the term studied above namely p(-A)-181j(puiu1) and the two preceding ones are even simpler. Finally, per(-A)-ldiv (pu) makes sense because of (5.41) and because p div u makes sense since div u E L2 (RN x (0,T)) and p E Lf+' (RN x (0, T)) (-y + 1 > 2) . We may now begin the passage to the limit as n goes to +oo in (5.1), (5.2) and (5.42). Without loss of generality, we may assume that pn Log pn, pn)7, pn = a(pn)7, (pn)7+1 converge weakly respectively to s, pti, p = a 7, p'r+l. Observe that pn Log pn, (pnyt (and thus pn) are bounded in some L" space for some p > 1 and the weak limits are taken in these spaces while (pn) T+1 is bounded in L' (RN x (0, T)) and the weak limit to_+ is in the sense of measures (or in the sense of distributions). In particular p1+7 is a bounded non-negative measure on RN x [01T). We have already shown in the preceding section that (5.1) holds at the limit
while (5.2) leads to (5.43) (i.e. (5.2) with p = ap7 replaced by p = a 7). In order to conclude this step, we explain how one can pass to the limit in pn)un,pn(-i)-'div
the expressions
pn(-A)-ldiv (pnfn),
pnun(_O)-'div (pnun). For(pnfn), all these terms, we are going to use Lemma 5.1. pn(Log
First of all, pn(Log pn)un goes to s u for the same reasons as given in the proof of part (1) of Theorem 5.1 (see section 5.2 above). Next, for the remaining three terms, we use Lemma 5.1 with g' = pn, pn, pnun, hn = (-A)-1div (pn fn), (-A)-'div (pnun), (-A)-ldiv (pnun), respectively (recall that we already know that pnun converges to pu). The three terms are treated similarly and we just
detail how to use Lemma 5.1 for the last one (which is the "worst" of the three). Indeed, pnun is bounded in L°° (0, T; L1 n Lq(RN)) with 1/q = 1/2s + 1/2 (write pnun = pn pnun) and in L2(0,T; L'' (RN)) with 1/r = (N - 2)/2N + 1/s < 1/2. In addition, (5.2) immediately yields the fact that (5.30) holds. And
Proofs of compactness results in the whole space case
21
(-A)-ldiv (p"u") is thus bounded in L°O(0,T;W1l (RN))nL2(0,T;W1o (RN)) so that (5.31) holds with q1 = 2, q2 E [2, Nr/(N - r)). And we conclude easily. Step 3 : Passing to the limit in the commutator. Since (5.45) (and thus (5.46)) is deduced from (5.1) and (5:43) exactly as we deduced (5.42) from (5.1) and (5.2), we only have to show that pn { [u7 , Rjj](pnui)
} w, p {[uj, R=j](pu=)} in D'
(5.49)
in order to conclude that (5.46) holds. This is precisely what we prove here. Since Dun is bounded in L2 (RN x (0, T) ) and p"u" is bounded in L2(0,T; LP(RN))(nL°°(0,T; L1 (RN))) where 1/p = (N2)/2N + 1/s < 1/2, we may use the general results of Bajsanski and R. Coifman [25], and R. Coifman and Y. Meyer [109] on commutators of the form [uu , R,=j] to deduce that [uu,Rj ](p"ui) is bounded in L1(O,T;W1,4(RN)) where 1/q = 1/2 + 1/p. Next, we wish to deduce (5.49) from this regularity and by using Lemma
5.1 twice. Indeed, we first use Lemma 5.1 with gn = pnui and hn = u, (more precisely with Ri j (p"ui) since we already know that p"ui u, -and thus R,j(p"ui u )-converge weakly to pu8uj-and thus to R=j(pusuj) respectivelyby the proof of part (1) of Theorem 5.1). Then, pnui (or R= j (pnui)) is bounded in L2(O,T; LP(RN)) with p > 2, (5.30) obviously holds in view of (5.2) and uj satisfies (5.31) with q, = 2, q2 = 2 since Dud is bounded in L2(RN x (0,T)). converges Therefore, applying Lemma 5.1, we see that Un = [u7 , Rij] weakly to U = [uj, Rjj] (pui) (for instance in L1(O,T; L°2 (RN)) for q2 > 1
satisfying 1/q2 < 1 + 1/s - 2/N while, as shown above, Un is bounded in L' (01 T; W1,-?
(RN)).
Hence, we may apply Lemma 5.1 with g" = pn, p1 = oo, p2 = s, hn = Un,g1 = 1, q2 = s/(s - 1) since (5.30) clearly holds in view of (5.1) and (5.31) follows from the bound on Un together with the inequality 1 2 1 Nq 1 -N=1-N+S q2q 1
i.e. s > N. This completes the proof of (5.49) and thus of (5.44)-(5.46).
Step 4 : Pointwise convergence. We start from (5.46) and we wish to prove that r = s - s , where s = pLog p, vanishes identically on RN x (0, T). From this fact, we shall then deduce the pointwise convergence of p" to p. We begin by observing that, since p" I Log p" I can be bounded by C [(p" )1+b+
(p")1-a] for any S E (0, 1), s, s and thus r belong to LOO (0, T; LP (RN)) for all 1 < p < N (for instance). In addition, using once more the Sobolev inequalities, L2N/(N-2) (RN)) we know that u E L2(0, T;
We next claim that r > 0 a.e. and that p1+7
-p7>0
(in the sense of measures). This is a simple consequence of the convexity of the functions
Compactness results
22
(t i- t Log t), (t '-+ t(1+7)/7) and (t H t1+7) on [0, oo). Indeed, we obviously have
(pn)l+7 =
{(pn)7}(1+7)/7
therefore pl+7 > p7 (1+7)/7 and pl+7 > (p)1+7 7/(1+7)
1/(1+7)
and p < { (p1+7) reg } Therefore, p7 < {(pi+ir),eg} where µreg denotes the regular part of the measure it with respect to Lebesque measure. Hence, Formally, the sign of r is clear but the (weak) singu7P < (p1+7) Ceg < larity of Log at 0 requires some justification. One possible argument is to write the following convexity inequality pl+7,
pn Log pn > V Log cp + (Log cp + 1) (pn -W) = (Log cp)pn + pn -W a.e.
for all cp E Lt bounded away from 0. Letting n go to +oo, we deduce
> (Log cp) p + p - cp and we conclude choosing cp = p + 6 for 5 > 0 and letting S go to 0+. Next, in order to prove that r vanishes, we first deduce from (5.46)
+ div (ur) < 0 in RN x (0, T).
(5.50)
And we simply need to integrate this inequality over RN x [0, t] and to use the sign of r. Once more, we need to justify all this. First of all, we remark that a (pn Log pn) ,
are bounded in L°° (0, T ; W -1,1(RN)) for example. And using
the result shown in Appendix C of Part 1, we deduce that pn and pnLog pn converge weakly respectively to p, s weakly in LP(RN) for 1 < p < s uniformly in [0, T]. In particular, p, r E C[0, T]; LP - w) where LP - w means LP endowed with the weak topology. Furthermore, since po converges to po in L' (RN) and thus in LP (RN) for 1 < p < s, we deduce that p(O) = po and r (O) = 0 a.e. in RN. Finally, in order to justify the integration on RN of (5.50), we only have to show, using cut-off functions of the form c,(./R) where cp E Co (RN), 0 < co 1, co = 1 on B1, Supp cp C B2, R E (1, +oo), that we have
1T1
luIr
R I Vcp ()I dx - 0 as R --> +oo.
(5.51)
N L2N/(N-2) (RN))
, r E Loo (0, T; LP(RN)) for 1 0. From the strict convexity of On, one deduces immediately that pn converges to p in measure on each BR x (0, T) (for all R E (0, oo)). Therefore, pn converges top in Lpl (0, T; LP (BR)) n Lq(BR x (0, T)) for all 15 pl < oo, 15
p2<s, 1 0 small enough (so that all terms below make sense (namely 0 < 0 < 2v
min(q - -y,
- 1))
Cu + )(div un)(Pn)e = a(p')p'+A + (pn)A(_0)-ldiv (Pnfn)
_
=
[(pn)0(_Q)-1d1V (pnun)]
(pn)ry+A
-
(Pn)A(_0)-l0.
+
(Pn)e
(Pnui uj ) (-0)-1div (Pnun)
+ (pn)A(_0)-1diV (Pnfn) - (Pn)A(_0)-laii(Pnui uj )
n A (-0)-div (pnun )] - div [u n (p) n A (-0) - ata [(P) 1
+(p')eu'
V(-A)-'div
I div (pnun)]
(p'u') + (1 - 0)(div
u")(Pn)A(-0)-'div
(Pnun)
or finally
(p + )(div 'un)(P")e = a(pn)4A +
(Pn)e(-0)-'div (Pnfn)
_ [(pn)9(_0)-1div (p'u')] [u'(P")e(-A)-ld1V (pnun)] - div + (Pn)e
[un V(-0)-ldiv
+(1 - 0) (div
(pnun)
(5.53)
-
un)(P")e(-o)-1div
(-A)-1ai.(Pnui
uj )]
(p'un).
We next pass to the limit in (5.52)-(5.53). To this end, we denote by 7,7;79and Q the respective weak limits (in I," (RN x (0, T)) for p = q/0, p = q/ (^I+())
Proofs of compactness results in the whole space case
25
and p = 2q/(q+26) respectively-recall that by assumption q > 2 and q > -y-of (pn)9, (pn)7+e, (div un)(pn)e. Exactly as in steps 2 and 3, we obtain
(pe) + div (upe) = (1 - A) Q,
(µ + OQ = ap'l+e
+ p(-p)-'div (P.f) _'
(5.54)
[(_y'div (Pu)] -div [u pe(0)-1div (pu)] +7[u V(-0)-idiv (pu) - (-0)-18ij(Puiuj)] + (1 - 6)Q(-A)-'div(pu). 8t
(5.55)
In addition, (5.43) still holds, from which we deduce in view of (5.54)
(p + e)div upe = ap" pe +
-
pe(-0)-idiv
(Pf) - Pe(-j)-'
5i(_Pe(-A)-'div
(-0)-'div
(pu)) +
= a p pe +
pe(-L)-'div
(Pf)
(Pu:uj) (pu)
[pe(-,&)-'div (pu)]
-rat
- div [u pe(-0)-idiv (pu)] + Pe [u
- (-0)-laij(P uiuj)] + (1 -
V(-0)-'div e)Q(-0)-idiv
(Pu)
(pu).
Comparing this equality with (5.55), we deduce
(p + )Q - ap7+9 = (µ + )divu pe - app pe
a.e.
(5.56)
Next, we observe that we have (by convexity, see also step 4 above)
(p7+91 e/(7+e) >
(pe/
(pir+e)1"8)
>- p7
a.e.
hence we deduce from (5.56)
Q > (divu) pe
a.e.
(5.57)
In particular, this inequality combined with (5.54) yields A
pe + div (u pe) > (1 - 6) (divu) pe.
(5.58)
We then wish to conclude about the pointwise convergence of pn as in step 4 above by proving that (p9)1/e _ p. We observe that, on one hand, (pe)1/9 < p
Compactness results
26
and (p)1/el,.o = PI, = Po while, on the other hand, we may deduce at least formally from (5.58)
{iIe} at 49
+div
{U1/e}
> 0.
(5.59)
This looks like a rather innocent manipulation but it turns out that a proper justification requires a bound on p in L2,t: in other words, it is precisely at this point (and at this point only) that we need to assume q > 2. Indeed, we may regularize (as usual, by Lemma 2.3, Chapter 2, Part I) (5.58) and find for all 6 E C'([O, oo)) with say /3' E Co ([0, oo) ) a at
{/3()} +div {u/3()] ? (1 - e)(divu) 01(-p-6)7 +(div u) [i3G) - 701(Pe)]
_ -A (div u) 6'(7)7 + (div u) 6(P9). {coM(pe)}1/e
where cpM = MV(./M), M > 1, cp E
We then choose Q =
C0 ([0, oo)), V(x) = x on [0,1], Supp cp C [0, 21, and we obtain 1/e 1/e a {co(p)} + div u {coM(P8)} Cat
> -(divu) {caM (pe))}
_ (divu) {caMp8}
1 a -1 W(
1/e-1
[caM(pe)
{WM(pe)}1/e
)P8 + (divu)
- cP'M(P9)7]
1(Pe>M)
-Co Idiv ul MO' 1 (7>M) where Co = sup { I'(x) I co(x) - cp'(x)xl /x E [0, oo) } . We then deduce easily (5.59) provided we show that Idly ul Ml/e 1(Pe>M) converges to 0 in L1(RN x (0, T)) as M goes to +oo. First, we observe that pel/e < p hence Idiv ul M11e 1 (Pe>M)M'/e). Next, divu E L2(RN x (0, T)) and p E L2 (R" x (0, T)). Therefore, we have 11/e-1
f 0
T
dt
JRN
dx Idivul M1/e 1(P>Ml/e)
as M -4 +00 At this stage, we have shown that p" converges to p in L}(BR x (0, T)) for all p E [1, q) and in LP1(0,T; LP2 (BR)) for all pi E [1, oo), P2 E [1, r) and for all R E (0, oo). The argument made in step 5 is still valid here and we complete the proof of Theorem 5.1 in the case when 11 = RN and N > 3. 0 < Ildivu 1(P>M1/e)IIL2(RNx(o,T))IIP1(P>M1/e)IIL2(RNx(o,T))
0
Proofs of compactness results in the whole space case
27
Remark 5.17 We wish to observe that we used in the proof of Theorem 5.1 bounds on pn in LOO (O, T; L'' (RN)) for some r > N/2 and in L4 (RN x (0, T) )
for some q > y. We assumed that q > 2 and we used the L2 bound in the last argument above: in fact, when y < 2, inspecting the above argument we see that we only need a bound on pn in L2'°O(RN x (0,T)). In fact, we only need to know
that the limit p belongs to L2i0O(RN x (0, T)). Indeed, if this is the case, we remark that we have
fTdtj C
dx Idiv ul Mlle 1(p>Miie)
II(divu) 1(p>M1/e)IIL2(RNx(O,T))Mi/e meas (p> M1/e)1/2
< C I I (div u)1(p>M1,e) I IL2(RN x
(O,T)) -'
0
as M -- +oo.
Remark 5.18 Let us now prove the statement announced in Remark 5.13. First of all, let us observe that, even if we no longer assume the strong convergence of po to po, the property (5.29) holds. In fact, the proof made above immediately yields, replacing (pn)e, pe by /3(p'1), 8 respectively, the fact that (5.29) holds for any function /3 E Cl([O, oo)) such that Q'(t) goes to 0 fast enough as t goes to +00 (say, 3' E Co ([0, oo)) or If3'(t)I t1-6 is bounded for some 8 as in the preceding proof). Next, if 0 is an arbitrary continuous function on [0, oo) such that Q(t) 0-7 and ,Q(t) tq/2 go to 0 as t goes to +oo, we approximate it by a sequence of functions Qk E C1([0, oo)) such that f3k E Co ([0, oo)), Qk converges to 83 uniformly on each [0, R] for all R E (0, oo) and Qk (t) V-7, /3k (t) tq/2 go to 0 as t goes to +oo uniformly in k. Without loss of generality, we may assume that f3k (pn) converges weakly (in w - L°° *) to 13k. Then, (5.29) holds for each 13k. We deduce that (5.29) holds for 3 by letting k go to +oo. Indeed we have IIi3k(Pn)(divun) - 3(pn)(divun)II Ll(RNx(0,T) < I Idiv un1IL2(RN ),(O,T)) I I Qk(Pn) - I3(Pn)II L2(RN x (O,T))
IIQk(Pn)(Pn)7 - 8(Pn)(Pn)7II L1(RNx(O,T)) IIP"IILQ(RNx(o,T))II /3k(Pn) -,Q(Pf)I'Lq/(q_7)(RNx(o,T))
and for each e E (0, 1), there exists a constant CE > 0 independent of n such that Iak(Pn) - /3(P' )I
c Min [(Pn)q/2e
+ SUP {100)
(Pn)q-71
- Q(t) I /0 < t < CE }
.
0
Remark 5.19 (Connections with compensated compactness). We wish to explain the part of Remark 5.13 and assertion (5.29) that is related to the div-curl
Compactness results
28
lemma of the compensated-compactness theory (F. Murat [401], [402], L. Tartar [530], [531]). Indeed, on the one hand, we have some information (and LP bounds, hence W-1,p compactness for appropriate p) on div t,x (00(P)un) (see equation (5.33)) for convenient 3. On the other hand, we can project equation (5.2) on the space of gradients decomposing (orthogonally) "arbitrary" vector fields v as Pv + Qv where curl (Qv) = 0, div (Pv) = 0. We then find
at
Q(pu) + Q (div (pu ® u))
- (p + e)Vdiv u + V ap = Q(pf);
(5.60)
notice indeed that Du = V (div u) - curl curl u. Therefore, we get some informaun -a(pn )-y -,Gn) where Oon = Qdiv (pnun tion (i.e. bounds) on curlt,y ((p'+C) div Q(Pnun) un). Using, at least formally (one needs to work out the appropriate functional setting, which is a bit delicate here but nevertheless can be done), the div-curl lemma, we deduce that /3(pn) [(p+ )divan -a(pn)ry - Y ')n+unQ (pnun)] weakly passes to the limit. f3(pn)(,)n This is precisely the quantity given in (5.29) with the extra term unQ(pnun)). Next, we observe that 7pn = -(-0)-1 div [div (pnun 0 un)] _
-(-0)-10ij(pnui ujn) and that Q(pnun) = -V(-0)-'div (pnun). Hence, the additional term is nothing but V(-0)-'div (pnun)
/3(P") {un
_
uj )}
which is the crucial term we had to analyse in detail in the above proofs; it is the term that involves the commutators [ui , Rij]. It turns out that here the div-curl lemma sheds some light on the argument but unfortunately on the easy part of
the argument. 0 We wish to conclude this section with a few facts. First of all, the general structure hidden behind the product &n) {(p + e)div un - a(pn)" } is studied briefly in Appendix B, where we also show the following fact. Let P be a general
pseudo-differential operator of order -1 (like for instance (-0)-1/2, (_A)div,
(-0)-'curl, (-A)-'D). Then the following convergence holds in D'
O(pn) {P(iuu' + Vdiv un
-
a0(pn)-Y)}
n , {P(-pOu + Odiv u - aV
pry }
for the same class of non-linearities 3 as before. In particular, choosing P = (-A)-lcurl, we find
/3(pn) curl un n /3 curl u in V. Similarly, choosing P (v) _ (-0)-' ak vi and P = Rki(-0)-'div (Rki = RkRi) we deduce for all 1 < i, k < N 8(pn) {p C3kun + eRki div un - aRki(pn)' } n Q {pakui + Rkidiv u - aRki Pry} %
Proofs of compactness results in the whole space case
29
and 3(PP) {Rkt { (µ
+ )div un - a(pn)" } }
n Q {R, i {(µ + l )div u - a py}}
.
Therefore, we also have for all 1 < i, k < N
fl(pn) {8kui - Rki div un} W. 8 {aku - Rkidiv u}. All these properties obviously contain information about the possible behaviour of sequences of solutions. However it is not clear how one can use them and this is the reason why we did not incorporate them in Theorem 5.1 (nor in its proof). Our final observation concerns the decay of "oscillations" in the context of Theorem 5.1 when we no longer assume that po converges strongly to po. In this case, as we have seen above, oscillations on pn may persist for all t > 0 but we want to show that in some sense their strength decays as t increases. Indeed, we claim that we always have
at{P-(Pe)pie}+div{u[p-(pe)pie]} ry + 1, we have the following precise identities
tIe
8 cat{P-(Pe)
}+div{u[p-(p9)
p7+e Cpe) n
pie
a
+µ+
1
6-1
_ p7(pe)lie j=OinV', for all 0 < 8 < 1,
{ p Log p - p Log p J + div {u p Log p - p Log p] } +
a u+t; [pT+71-pp =0 in D'.
The relationship between our "claim on the strength of oscillations" and the preceding identities becomes clear once we recall that p - (pe)lie, pLog p p Log p are non-negative and vanish if and only if pn converges (in Lt)
to p.
We briefly sketch the proof of these identities. First of all, it is enough to consider the case of (p - (pe)lie) since the other case can be deduced from it.
Indeed, we just need to observe that lie (p - (pe)'/e) converges, as 8 -- 1, to p Log p - p Log p. Next, our proof relies upon a truncation of pn and we shall simply use pn A R (R E (0, oo)) to simplify notation. In fact, this is not absolutely correct since (t - tAR) is not C' and we need in fact a further layer of approximation (smoothing t A R or directly working with an increasing, concave
truncation function): we ignore this irrelevant technical detail here. Then, the proof of Theorem 5.1 yields the following identity (for all 0 < e < 1). 5j (p A R)e + div (u(p -A R)8) = (div u) {(1
- 8)(p A R)e + RelP>R}
Compactness results
30
-µ+c(1-6) [PPAR)e_iIPAR)eJ a +µ + {Rep'1lP>R - P7 Re1p>R} in V. C
In fact, the equality also holds with 1P>R (depending on the type of smoothing we use for t A R). Next, if q > -y + 1, we let R go to +co and easily deduce
(1 - A) [p7+e - p7 pe] in V.
(pe) + div(u pe) = (1 - 6)(div u) pe + +C
5j In the general case, we use the fact that p7(p A R)e > p7(p A R)e, p'Y1P>R > 7 lP>R. These inequalities are very particular cases of general inequalities that we show in the next section. Therefore, we have
5j (pe) + div (u pe) > (1 - 6)(div u) pe in D'. At this point, we may follow the proof of Theorem 5.1 and recover the desired identities. In fact, with a bit more work (a similar proof will be given in chapter
6), one can show that the above equalities hold if q > -y (and q > 2). Let us also finally mention that the term µ+C [p7+1 - 7p] creates some damping of oscillations measured by (p Log p - p Log p) (for instance). It is worth noting that,
roughly speaking, the damping increases as a goes to +oo which corresponds to the incompressible limit (low Mach number limit) and as p + l; goes to 0 which corresponds to the inviscid limit (to the Euler equations), which are two asymptotic regimes where "some compactness" is to be expected.
5.4 Proofs of compactness results in the other cases We are going to conclude in this section the proof of Theorem 5.1 first in the periodic case, next in the case of Dirichlet boundary conditions and finally in the particular case when Sl = R2.
Proof in the periodic case. The proof is essentially the same as in the case when 1 = RN and N > 3 (see section 5.3) except for one modification concerning the inversion of -A with periodic boundary conditions. More precisely, for each
periodic function g on RN such that fn g dx = 0 we denote w = (-A)-'g the unique periodic solution of
-Ow = g in RN, w periodic,
fo
wdx=0.
(5.61)
This operator makes sense not only for periodic functions g E LoC (RN) , 1 < p < oo) but by duality for any distribution g = E «j =1 act gcx where m > 1, ga E is periodic. In addition, whenever it makes sense, (-A)-1 commutes with derivatives.
Proofs of compactness results in the other cases
31
With this convention, we deduce from (5.2) and (5.43) the following relationships
(p + ) div un = a [(pn)7 -
fi2(Pn)7 dx]
+ (-0)-idiv (pn fn)
(pntf )] - (-0)-iai,,(pnui uj) - -5ia-0)-'div [( I
and
-f
A) i div (pf) ( , a + ) div u = a p 7 a p7 dx] + (-0 (5 . 63) (-A) iaa7(puiuj) ) i div (pu)] . [( at The proof then follows step by step the argument made in the preceding section and we do not repeat it here. We simply have to observe that (pn)e (fn(pn)7 dx) converges (weakly) to pe(fn p7dx) (use for instance Lemma 5.1, noticing that fn(pn)7 dx is independent of x and thus is smooth in x!). Proof in the case of Dirichlet boundary conditions. In the case of Dirichlet
-
boundary conditions, the proof given in the preceding section has to be modified
in two places. First of all, one has to localize the argument which yields the following limit
{(p + l;)div un - a(pn)1'} (pn)e {(p + ) div u - a
71 pe
in D'(S2 x [0,T]).
(5.64)
Next, the conclusion about i) the pointwise convergence and then ii) the full convergence in C([O,T]; LP(11)) for p < r has to be carried out taking care of boundary difficulties. First of all, we deal with the localization of the proof of (5.64). This is rather
straightforward (but somewhat tedious). Indeed, we can still write
i div (p'u') + 8ij (pnut uj) - (p + )0 div un + 0(a(pn)7) = div (pnfn) in D'
(5.65)
and
div (pu) + ai,(puiuj) - (p+t;)Adivu+ A(ap7 = div (p f) in D'
(5.66)
Therefore, letting cp be an arbitrary cut-off function namely cp E C00001), 0 < cp < 1, Supp cp D K for an arbitrary fixed compact set K C Il we deduce at div (WPnun) + ai; (WPnui uj)
- (p + e)0 (V div un) + A(aW(Pn)7)
= div (cp pn fn) + Fn
a div (Wpu) + aij (vuiui) - (p + ) A (w div u) + A(cpap7
Compactness results
32
=div(cppf)+F where we have
Fn =
div un at (Pnun VV) + (atjW)Pnui uj + 2aicp aj (Pnu uj) - (µ + C)Acp fn
-2(µ + )VW Vdivun + Acpa(pn)7 + 2aVcp . V (e)"
- P"
. Vcp
and
F=
a
(Pu V W) + (aij cP)P'uiuj + 2a,cp aj (Puiuj) - (µ + C)OV div u VW
or equivalently Fn = ascp 8j (Pnunju7) + (aij w)Pnun uj
- (µ + e)dcp div u'
- (2µ + C)Ocp V div un + p un Vcp + AV a (pn)7
(5.67)
+aV cp V (e)-' and
F = a=cp aj(Pujuj) + (a0jcP)Pujuj - (µ + )Ocp div u
7.
(5.68)
We may then follow without further changes the argument developed in the (pn)e(-A)-1Fn preceding section and establish (5.64) once we observe that converges weakly (say in D') to pe (-A)-1 F for small enough 6 > 0. Once more, we only need to apply Lemma 5.1 remarking that Fn is bounded in LP(0, T; W ',P)
for p > 1 close enough to 1 (depending only on q, r and ry) and choosing 9 satisfying in particular q > A p/(p - 1). Let us only mention that the operator (-0)-1 can be taken as the inverse of -A on the whole space since all functions considered are supported on Supp cp or on fl with Dirichlet boundary conditions.
We then deduce from (5.64) the following inequalities as in the preceding section
a
1/9
(pe)
+ div
1/9
u (pel
while (-6)11E) < p and (pe)1/eIt=o
=
> 0 in D'(St x (0, T))
(5.69)
plt=o = po in 11; and in order to be able
to conclude that p = (pe)1/e, we need to integrate (5.69) over ft Formally, this is clear since u vanishes on aQ. To make it rigorous, we recall that since u E L2(0, T; Ho (11)) then u/d E L2(SZx (0, T)) where d(x) = dist(x, aQ). We then
denote by (E = ((-) where ( E C' Q0, oo)), ((t) - 0 if 0 < t < 1/2, 0 < C < 1
Proofs of compactness results in the other cases
33
on [0, oo), C(t) - 1 if t > 1 and e E (0, 1) is fixed. Multiplying (5.69) by (,(d), we see that we only need to show that we have
f
T
r
dt
Jn
dx p I u V (E (d)
0
as a - 0+.
(5.70)
This is the case since
pu VCE(d) I = p
Jul
C'
(d)
(IVdl =1 a.e.)
< t>0 IC'(t)I 1(.o IC'(t)I ), and (5.70) is proven since p E L2(fZ x (0, T)).
At this stage, we have shown that pt3 converges to p in LP(CZ x (0, T)) fl LP1 (0, T;172 (e)) for all 1 < p < q, 1 < pl < oo, 1 < P2 < r. The convergence of p? in C ([0, T]; LP (11)) for all 1 < p < r or equivalently in C ([0,T] ; Ll (fl)) is shown exactly as in the preceding section keeping in mind the above justification of the integration over Q. The only new ingredient is the proof of the fact that p E C ([0,T] ; L' (a)) since the argument in the preceding section uses a mollification and thus is not clear near the boundary. This difficulty is solved by combining the above truncation argument together with the regularization argument. More precisely, we obtain as in the preceding section for c E (0, 1), pE E C ([0,T]; L1(cl )) where 1ZE is, say, defined by {x E fZ /d(x) > e/2}, smooth in x for all t E 10, T], satisfying
ap,
div (up,) = rE in Q, x (0,T), PCIt=o = p° at +
where II p° - pE I I LI (n!) -' 0
, I I rE I I LI (n!
(5.71)
x (o,T)) ` 0 as e - 0+. It is rather
straightforward to show that PC - PIIc((o,T1;1,1(fl)) __1'O as E - 0+. We then deduce from (5.1) and (5.71) d
T
n
jIPP6IC6(d)dx
< .fn
IrEIdx+J
IVCE(d)I Jul Ip - PEI dx
hence
(I (t)
T
sup oT fn , 1
I P -PEI dx
_O o
I)
Jo
T
dt f
e
dx Jul (p + PE) d
Compactness results
34
0
as
This allows us to conclude as in the preceding cases and thus to complete the proof of Theorem 5.1.
Proof of the convergence assertions (5.21)-(5.22). We begin with the convergence of pnu". In order to prove it, we use once more a mollifier rcc = rc(E) where r. E Co 00 (RN), 'c > 0, fRN rc dx = 1, Supprc C B, and we let gE = g * cE for an arbitrary function g: notice that, in the case of Dirichlet boundary conditions, if g is defined on Si x (0, T), gE is defined on iE x (0, T) when SzE = {x E Si / dist (x, 8S2) > e}. We first observe that we have for all
N/2 N+2 , pp l < N z and thus IIuE - un I I L2 (o,T;LP/ (P-1)) converges to 0 as e goes to 0+, uniformly in n. In addition, (5.20) implies that sup, .1<E I IPn(. + e) PnIILP converges to 0 as e goes
-
to 0+, uniformly in n. Therefore, in conclusion, (p"u")E - pnun converges to 0 in L2(0,T; L1) as a goes to 0+, uniformly in n. Next, (p"u")E is obviously smooth in x , uniformly in n and in t E [0, T]. Therefore, remarking that, because of (1.2), ac (p"u")E is clearly bounded in L2(0,T; H') for any m > 0, we deduce that (p"u")E converges to (pu)E as n goes to +oo in L1(0, x (0, T)) (say) for each e > 0. Then, using the bound on pnun in LOO (0, T; L27/(7+1)(SZ)), we deduce from those facts that pnun converges to pu in
L' (f x (0, T)) and the first part of (5.21) follows easily. The second part of (5.21) is an immediate consequence of the first part together with the bound on un in LPN/(N-2) (K) ) L2 (K x (0, T))-in fact the convergence is even true in LP(0, T;
for all 1 < p < 2 by the same argument. The L2 convergence in (5.21) of un is easily deduced from the fact that p"Iu" uI2 converges in L' to 0 (and the
-
strong convergence of p" uniform in t) since
p"Iu" -uI2 =
p"IunI2
-2
(p"un) . U + pnIul2
converges weakly to 0.
In particular, (5.21) yields the a.e.-convergence of p"uEuj (or of a subse-
quence) to puiuj as n goes to +oo (V 1
_
q/(q - 1), b > 1, c > s/(s - 1). This is how the above proofs yield the following
Corollary 5.1 Theorem 5.1 still holds if we replace (5.18) by (5.72).
General pressure laws
5.5
In this section, we shall investigate a variant (or extension ) of the preceding results and proofs. We are concerned here with the case of a general barotropie
fluid, that is the equations are still (5.1)-(5.2) but in (5.2) the term apt is replaced by p = p(p) where p is a general "pressure laid', i.e. p is assumed to be a continuous non-decreasing function on [0, oo) vanishing at 0 (this is an irrelevant normalization of p). We already explained in chapter 1 of part I how the monotonicity of p follows from thermodynamics and we shall see here that our mathematical analysis relies in an essential way on that property. In other words, we study the system of equations consisting of (5.1) and
+ div (pu (9 u) - p0u - l;V div u + Vp(p) = pf
(5.73)
and we consider the same three possibilities for boundary conditions (whole space, periodic case, Dirichlet boundary conditions). Let us only briefly mention the analogue of (5.8) in that general situation or in other words the local energy identity
at
pI2I2
+ div
pu 22
+l;(divu)2 +
µA 22 - l; div (u div u) + plDu12 (5.74)
(q(p)) + div (u[q +p]) = pu f
where q is defined (up to a constant) by dt(q(t)/t) = p(t)/t2 for t > 0 . From this identity, we can deduce various bounds as we did in section 5.1. We do not want to give too many details here but let us just make a few remarks in that direction. There are two cases worth considering: first of all, if p(t) is such that fo ds < oo then we can choose q(p) = p fo' tt dt and then only the behaviour of p at infinity matters in order to obtain a priori bounds. For instance,
if lim,-+oo p(s) Is" > 0 when y > 1, then the same bounds as in section 5.1
General pressure laws
37
I2
Next, if p(t) t-' goes to a positive constant as t goes to 0+, we can choose q(p) = pfi tt dt and q behaves like p Log p as p goes to 0+. In the case of Dirichlet boundary conditions or in the periodic case, the change of sign in q does not affect the a priori bounds since q is bounded from below and we easily reach a priori estimates for plui2, plLog pI in LO°(0,T; L'(11)), Du E L2(St x (0, T)). In the case when 1= RN, we have to be slightly more careful. We can recover some
bounds using the existence of some p E L1(RN) such that Vq'(p) E L°O(RN): take for example p = e-IzI, assuming for instance that p E C1([0, oo)). We then write
a +div {u[P__++P_PQ'(/3)]} at PI22+{q(P)-q(p)-gV)(P-P)} ediv(u div u) + pl Du12 + t; (div u)2 = pu f - pu Vq'(p) µD 12
-
-
hence, if f E L'(0, T; LOO (RN)), we have for some a > 0
[f1+ 2
dt
{q(P) - q() - q'(P)(P - )} dx] + a < IIp IUI2
JR
IDuI2 dx
IILI(RN) IIPIILI(RN) IIf - Vq'(P)II LO(RN)
Recalling that do fRN p dx = 0, we deduce bounds for Du in L2 (RN x (0, T)) and
for pIuI2 and {q(p) - q(p) - q'(p)(p - p)} in L°°(0,T; Ll(RN)). Notice indeed that this last quantity is non-negative because q is convex since q'(t)t-q(t) = p(t) and thus q"(t)t = p(t) > 0 (in the sense of distributions).
We now consider a sequence of solutions (pn, u') of (5.1), (5.73) and we make the same assumptions on this sequence as in section (5.2) except that we need to modify the assumptions on pn. We assume that (pn)n>1 is bounded in C ([O,T];L'(S2)), (p(pn))n>1 is bounded in L°°(O,T; L1(SZ)) and we replace (5.19) by (pn)n>1 is bounded in LQ(S1 x (0,T)) nL°O(0,T;Lr(S2))
in the periodic case or if it = RN, N > 3, (pn)n>1 is bounded in L9 (K x (0, T)) n L' (0, T; Lr (11)) in the case of Dirichlet boundary conditions or if S1= R2,
(5.75)
for some q > 2, r > N/2, and we also assume that we have ((pn)sp(pn))n>1 is bounded in L'(K x (0,T))
(5.76)
for some s > 0, where K is an arbitrary compact set included in S1.
Theorem 5.2 Theorem 5.1 still holds and, in addition in part (2), p(pn) converges to p(p) in L'(K x (0,T)) for any compact set K in SZ.
Compactness results
38
Proof. Most of the proof of Theorem 5.2 is the same as that of Theorem 5.1. In particular, we obtain the analogue of (5.29), namely
,8(P") I
div u" - p(p' )} n j {(µ + C) divu - fi} in D'(1 x (o, T)) (5.77) 0(t)t_9
go to 0 for any continuous function 3 on [0, oo) such that ,Q(t)t- /2 and as t goes to +oo. The rest of the proof is exactly the same as in the proof of Theorem 5.1 using Q(p) = pe for 6 small enough (9 > 0) and the following crucial lemma that we
apply with pi(t) = te, p2 = p. Lemma 5.2 Let P1,P2 E C([O, o0)) be non-decreasing functions. We assume that pi(p"), p2 (p") and pi (p") p2(pn) are relatively weakly compact in L'(K x (0, T)) for any compact set K C St. Then, we have Pi (P) P2 (P) ? Pi (P)
P2 (P)
a.e.
(5.78)
Remark 5.20 The assumption on pi (p"), P2 (P") is equivalent to requiring that we have {Ip1(P")I + IP2(P")I + IP1(P")P2(Pn)I } 1(P..>M)
--- 0 in Li(K x (O,T))
as M - +oo,
uniformly in n > 1, for any compact set K C Q. Remark 5.21 In fact, we do not need the full strength of (5.78); we only need it when pi (t) = to and P2 = p. But, it turns out that there is no gain in generality (for p) since we claim (as is quite standard) that requesting that (5.78) holds for such a choice (and for an arbitrary weakly convergent sequence p") is in fact equivalent to requesting that p is non-decreasing! In order to prove this claim, in view of the proof below of Lemma 5.2, we observe that if we choose a sequence (pn) n> i such that p" oscillates between two values a and b (a # b > 0 are
fixed) with probability 1/2 (pn = a if t E (2k/n, (2k + 1)/n), = b otherwise, for 0 < k < [(n + 1)/2] -1 when St = (0,1), N = 1 for example) then (5.78) reduces to
2 aep(a) + beP(b)
-
2
(2 ae + 2
-
be)
(2p(a) +
Zp(b))
or equivalently, (ae be) (p(a) p(b)) > 0. Since a, b > 0 are arbitrary, this means that p has4 to be non-decreasing on [0, 00).
Proof of Lemma 5.2. We first make a few preliminary reductions. Without loss of generality, we may assume that pi (0) = P2 (0) = 0 (subtracting a constant from either pi or P2 does not affect (5.78)) and thus P1, P2 > 0. Next, we observe that it is enough to prove (5.78) when pi and P2 are bounded. Indeed, if this is the case, (5.78) holds for pi A K, P2 A K for all K E (0, oo). Then, we have for
i = 1,2 and for all ME (0,00)
Other boundary value problems
0 : P, (P,) - pi(Pn) A K -< pi(P")1(pn>M) if K >
39
sup lpi(t)) = pi(M)
0 0
st p
+
2
a -Y - 1 {pry
'Y(POO)ry-1p + ('Y t r
< fn PO
2
fo
a
n
t r +CoJ dsJ dx plu-ui+p+pry, + -y o
a
1
(t)
ds J dxIDuf2
+v
uo -
- 1)(p')') dx
{Po -'Y(POO)ry-1Po + ('Y
(5.87) 1)(PO°)ry} dx
x
for some v > 0 which depends only on p and t;, some constant Co > 0 and some compact set K C Sl which depend only on U.
Compactness results
42
Bounds then follow easily from (5.87) upon noticing that, by convexity of the function (t i- t-1), { pry - ry(p°O) ry-1 p + (ry -1) (p°O) ry } = j7 (p) is non-negative and that we only need to bound fK p + pry dx by C(K) fK j7 (p) dx. If this is the case,
we then deduce from (5.87) the following a priori bounds valid for all t > 0:
f P(t) Iu(t) -
ul2
n
t
dx
+ j7(P(t)) dx + fo
CleC,t f PoIuo n
r
Jn
dxIDuj2 (5.88)
- u12 + j7(Po) dx
for constants Cl > 1, C2 > 0 which are independent of t, p, u, po, mo = Pouo The above claim on j7 (p) is straightforward since we have
JK7H J pry dx +
-
2meas(K)(7-1)
fx
pdx
r pry dx + 1meas(K)-i7-1) f pd2 J K K 2
7
ry(P°O)7-1
K
pdx
-C
for some positive constant C which depends only on 'y, meas(K) and p°O.
In the tube case, one can argue in a similar fashion introducing p(x), u(x) satisfying the following requirements: p, u = (u1, 0) are functions of x1 only, P', ui E Co (R), (p, ul) _ (p+, ul) for x1 large enough, (p, ul) = (p-, ul) for
-xi large enough and p > min(p+, p) > 0 on R. Then, we perform similar
computations to the ones above writing now j7 (p) = p7 - .y p -1 p + (-y - 1) per' and
a
1 1
j7(p)
+ div
{u1(p_r1p)}
= -'IPU .
Vp(P)7-2
= -'yPu1P
(xl)(P)7-2.
In this way, the analogue of (5.85) is given by the following identity
a 1.7-Y(P) +div u at Plu 2u12 + -Y-
ary1(P7-,r-1P)+p
2
ry
- µ0u- (u - u) -l;Vdivu (u - u) -(Pu V)u - u VP7 -
.yPu1P'(xl)(P)7-2
(5.89)
from which we deduce _ 2 d Pfu 2ul + 'Y a 1j-t(p) dx + n liVu V(u - u) dt n dx = nf -pul ui + (ul)'Pry -
f
f
div (u - u) dx
ryPulP'(P)7-2
(5.90)
Other boundary value problems
43
and this identity leads, exactly as before, to the bound (5.88) where of course j7 is defined as above. We see that, in both cases, we obtain some a priori estimates on Du (or equivalently on D(u-u)) in L2(O,T; L2(0)) and on jy(p), plu-uI2 in L°O(0,T; L1(0)) for all T E (0, oo). If N > 3, this implies in particular that we obtain an estimate L2N/(N-2)(S2)) and on p in L°(0,T; L-1 (BRnO)) for all R, T E on u-u in L2(O,T; (0, oo) and thus on u in L2 (0, T; L2N/(N-2) (f n BR)) for all R, T E (0, oo). If N = 2, we claim that we can obtain an estimate on u in L2 (0, T; LP(BR n 1k))
for all R, T E (0, oo), 1 < p < oo. Indeed, denoting p = p°O, in the exterior case, we deduce from the bound on jry(p) in L°° (0, T; Ll(S2)) that there exists a constant C > 0 such that, for all t E [0, T], meas {x E S2 /p(x, t) < 1 p} _< C; notice that j.y is bounded away from 0 on [0, 2 Therefore, for R large enough (R > Ro): fBRnn dx p(t) > Z (Inf p) {meas(BR n 0) - C} , for all t E [0, T].
Let us observe that R0, C depend only on T and on the bound on j.,. Next, we observe that we have for all R E (0, oo)
J Rnn Iv12 dx < CJB
for all v E H'(BR n fl) (5.91)
Rnn lDvI2 + hIvI2 dx
and for all h E L7(BR n n) (recall that -y > 1) such that fBRnn h dx > v > 0. Furthermore, the constant C apppearing in (5.91) depends only on v, R and on bounds upon h in L1'(BR n 12).
The proof of (5.91) is easily done by contradiction: if fBRnn Ivnl2 dx = 1 and fBRnn I Dvn 12 + hn I vn 12 dx n 0 where fBRnn his dx > v > 0, h7L is bounded in L"(BR n 11), then vn converges in Lp(BR n 12) for all 1 < p < 00 to meas(BR n SZ)-1/2 while we can always assume that his converges weakly in Llf (BR n fl) to some h satisfying fBRnn h dx > v > 0. We then reach a contradiction since on the one hand, fBRnn his Ivnl2dx w+ 0 and on the other hand, fBRnn his I vnl2dx (fBRnn h dx) meas (BRn2)-1, and (5.91) is shown.
n
We may then conclude, using (5.91) with h = p(t), v = u(t) for all t E [0, T], R > Ro, that when N = 2, u is bounded in L2 (0, T; L2(1 n BR)) and thus in L2 (0,T; Lp(S2 n BR)) for all R, T E (0,00), 1 < p < 00. Let us finally observe that all the estimates on p follow from the LOO (0, T; L1 (S2 n BR)) bound on j.r (p) = p1' - -tjr-1 p + (-y - 1)j5 '-recall that in the exterior case p - p°° and that in both cases Mina p > 0. We shall need later on for the proof of compactness results an equivalent (and simpler to use) formulation of that bound. The formulation is shown in the next lemma.
Lemma 5.3 j.y(p) E L1(12) if and only if (p p)1(1 p- ,al >6) E
-
E L2(11) and (p -
Ll (11), for any 6 E (0, Mina p).
Proof. Obviously, on the set fl p - pI < 6} both p and p are bounded by 0 and bounded from above (Mina p < p < Maxa p, p > Minis p - 6 > 0, p < Maxis p + 6). Since -y > 1, we thus deduce that j.y(p) is equivalent to Ip the set {I p pl < S}.
-
-
pl2
on
Compactness results
44
Next, on the set {ip - Al > b}, we just have to observe that we have for some v E (0, 1), C E (1, oo)
i IP - All lMinn p} < meas{Ipn - pI > 2 Mini p'} + 1(Min p)62 f pn l un ul2 dx < 00
-
-
In addition to the above "natural" bounds and assumptions, we need some further bounds on p" which, of course, we shall need to establish in the course
Other boundary value problems
45
of proving existence results at least for some range of exponents 'y. We assume that for all R, T E (0, oo) and for all compact sets K C fl p" is bounded in L4 (K x (0, T)) n L°O (0, T; L''(SZ n BR))
(5.93)
for some q > 2, q > ry, r > N/2. Finally, we need some (technical) condition in the exterior case: p" is bounded in L2(0,T;
L2rr/(N-2))
if N > 5, for all T E (0, oo).
(5.94)
Notice that we already assumed that p' is bounded in L°° (0, T; L2 (Q)) ; therefore (5.94) automatically holds when N > 5 if 'y > 2N/(N + 2).
Theorem 5.3 (1) Part (1) of Theorem 5.1 also holds here. (2) If, in addition to the above assumptions, we assume that po converges in Ll (St n BR) (for all R E (0, oo)) to po then we have for all R, T E (0, oo)
p" w pin C ([0,T]; LP (11 n BR)) n L' (K x (0,T)) for all l < p < r, 1 < s < q,
p"u"
(5.95)
pu in Lp (0, T; L°t(1l n BR))
for all 1 < p < oo,
< cc
0 (p")e {(µ + e) div u" - a(p")I} pe {(µ + e) div u" - a p'F}
in D'(Q x (0, oo))
at(pe) + div (upe) > (1 - 6)(divu)pe in D'(SZ x (0,oo)).
(5.98) (5.99)
In addition, since (pe)1/e) E L2 (K x (0,T)) for any compact set K C St and for all T E (0, oo), we may follow the proof of Theorem 5.1 to deduce
Compactness results
46
0 {()1/9} +div {(()1/e} > 0
in D'(SZ x (0,oo))
(5.100)
while we have (5.1), (pe)1/e _< p a.e. in SZ x (0, oo) and (pe)1/elt=o = plt=o in SZ.
Subtracting (5.99) from (5.1) and setting f = p - (pe)1/e, we obtain
8 +div (u f) < 0
in D'(SZ x (0, oo)),
f > 0 a.e., f It=o = 0 in Q. (5.101)
We then claim that f E LOO (0, T; Lz (Q)), flu - ul2 E Loo (01 T; Ll (Q)) for all
T E (0, oo). The second fact is obvious since 0 < f < p and pl u - ul2
E
LOO (0, T; Ll (Q)). In order to prove the first claim, we only have to show that (Pe)1/e - P E LO°(0, T; L2 (SZ)). But, (p" )e (p)e = (p + (p" - p))e (p)e is bounded in L°° (0, T; L2 /e (11)) in view of the results of Appendix A. There-
-
-
fore, (pe) - (p)e E L°°(0, T; L2/e(SZ)). Next, we write (pe)1/e _ p = [(p)e + 1/e (pe and using once more the results of Appendix A, (p)eJ we deduce that (pe)1/6 - p E LO°(0,T; L2-'(Q)) for all T E (0, oo). Let us
-
remark that, if N >-5, (5.94) implies in the same way that f E L2 (0, T; L2N/(N+2)A),
Next we wish to deduce from (5.100) that f - 0. Once this is proven, the proof of Theorem 5.1 applies and yields the L' (for s < q) convergence of p" in (5.95) and (5.96)-(5.97). Formally, the fact that f vanishes follows from (5.100) simply by integrating the inequality "by parts" over Q. In order to justify this integration by parts, we use a cut-off function. In fact, we really need two cut-off functions: one to truncate the integration at infinity and the second to truncate the integration near BSZ. We skip the second one since this is the same argument as in the case of Dirichlet boundary conditions (even if it is really needed on each BR n SZ for R fixed in (0, oo)). Therefore, we concentrate on the more delicate
truncation at infinity. In order to do so, we introduce cp E Co (Rk), 0 < cp < 1, Supp cp C B2, cp - 1 on B1 and we set cpR = cp(R) for R > 1. In the exterior case we take k = N while in the tube case we take k = 1. Multiplying (5.100) by cpR(x - ut), we deduce
x Rut jfccR(x-ut)dx=jf(u-u).VP( dt R
-
P
U j 0k co
x Rut
t 83 uk dx,
for t>0. }
j,k
(5.102)
Next, if T > 0 is fixed, we see that Vcp (x R t) # 0 implies that R < lx - utl < 2R and thus R - l l ul l L,,,,T < l xl < 2R + I l ul l Therefore, for R large enough, (5.100) reduces to L,,.T.
Other boundary value problems
dt
a Rut
fWR(x-ut)dx= J
n R
/'
Jn
47
dx
n
(5.103)
flu - ul1(R_c<JzI 0; we look for p > 0, u solutions of
a p + div(pu) = h
in 52
(6.1)
(6.2) a pu + div(pu 0 u) - pAu - eVdiv u + V(ap") = pf + g in 52 where y > 1, p > 0, p + > 0 and f, g, h are given function on 11, h > 0 in 52, h00.
In sections 6.2-6.7 below, we shall be concerned with our "usual" boundary conditions, namely i) Dirichlet boundary conditions in which case 52 is a bounded,
open, smooth set in RN (N > 2) and we require u to vanish on 852, ii) the whole space case in which Il = RN and (in some weak sense) p, u vanish at infinity, iii) the periodic case in which case (6.1)-(6.2) hold on RN and p, u are periodic in each x_ (1 < i < N) with a fixed period Tt > 0-we then agree that
--
= The above problem is naturally obtained from (5.1)-(5.2) through timefN1(0,T).
-
discretization replacing all time derivatives ap/at, apu/at respectively by of (pn pn-1), (pn,un_pn-l,un-1) and in all the other terms p, u respectively by pn, un.
of
In this way, we obtain (6.1), (6.2) with a = ot, h = of pn-1, f = fn, g = 1 of p n-1Un-1
In section 6.2 below, we state our main existence and regularity results while
we prove the key a priori bounds in section 6.3. These bounds are obviously crucial and involve two kinds of estimates: first of all, LP bounds on p where N P = N-2 (^f - 1) ) are obtained and next, if 'Y > 1 when N = 2 or ifY '> 3 when N = 3, further bounds and regularity estimates are shown in section
Stationary problems
50
6.3. Section 6.4 is devoted to compactness issues on sequences of solutions: of
course, the results are of a similar nature to those in the preceding chapter but the proofs are much simpler. Finally, in section 6.5 we prove the existence (and regularity) results stated in section 6.2, thus completing our study of timediscretized problems when p = apy, ry > 1, a > 0. Section 6.6 is devoted to the rather particular isothermal case, that is the case when ^y = 1 which we can study in two dimensions (N = 2). It will involve an adaptation of the concentrationcompactness theory (see P.-L. Lions [347], [348]). Next, in section 6.7, we study real stationary problems, namely
div(pu) = 0, div(pu 0 u) - µ0u - V div u + V (ap'') = pf + g in 1
(6.3)
with the same boundary conditions as before. Existence and regularity results will be obtained using the results and methods developed in the preceding sections and by letting a go to 0+ in (6.1)-(6.2) once h is conveniently chosen as we explain now. First of all, we wish to point out that solutions of (6.3) are non-unique (and have to be non-unique!). Indeed, we can always take p = 0 and solve for u (u = 0 if g = 0) at least in the case of Dirichlet boundary conditions. Furthermore, if f is a gradient and g - 0 (resp. f = 0 and g is a gradient), then we can take u = 0 and we simply have to solve aV pr' = p f
aVp'' = g If f = VO (resp. g = V O), then
p=
in 1l , p
> 0 in SZ
(6.4)
in 1 , p > 0 in Q.
(resp. 6.5)
(_lcyi
is a solution of (6.4) for any C E R (resp. p = (± + C)11_1 for any C E R such that a + C > 0). In other words, the system (6.3) is underdetermined. From a physical viewpoint, this is to be expected since we have to prescribe the total mass of the gas, in other words we may (and have to) prescribe
in
p dx = M, p>0 in
11
(6.6)
where M > 0. The case M = 0, i.e. p = 0 is easily settled since it leads to the elliptic equation: -µ0u - CV div u = g in &I-notice that in the periodic case, we need fn g dx = 0 and that if SZ = R2 we need g to decay fast enough and fR2
g dx = 0.
We shall precisely show in section 6.7 the analogues of the results shown in the preceding section for the system (6.3) and (6.6). The constant (6.6) will be enforced by letting a go to 0+ in (6.1) while choosing h such that « fn h dx = M: notice indeed that if p solves (6.1), then fn p dx = a fn h dx.
Existence and regularity results for time-discretized problems
51
In section 6.7, we also consider a different normalization of solutions (p, u) than (6.6), namely
in
p'dx
= K, p>Oinll
and we show that, if p is chosen large enough, then there exists for any ry > 0 and for any K > 0 at least a bounded solution (p, u). Furthermore, by a topological degree argument, it is possible to assert that these solutions belong to a continuum containing p - 0, u - 0 (for instance when g - 0). This will allow us to deduce that there exists a maximal M E ]0, +oo] such that there exists a bounded solution (pM, um) of (6.3) and (6.6) if 0 < M< M and, if (or FPM Lp (,,)) goes to +oo as M goes to M. We do not M < oo, pM know whether M can be finite in some situations. Let us finally mention that the reason why we can enforce a constraint like (6.7) comes from the fact that we can approximate the equation div(pu) = 0 by
div(pu) + a p' = ah in SZ
(p > 0 in 1k)
,
for a > 0,
and if fn h dx = K, than we have automatically fn p' dx = K. Section 6.8 is devoted to problems set in unbounded domains but where we no longer assume that p vanishes at infinity, like for instance exterior problems. In section 6.9, we investigate the regularity of solutions of the above stationary problems. We show there that if infess p > 0 and p E LO° then p and Du are Holder continuous assuming that all the data are smooth and we briefly discuss situations where these conditions are met. We also present a class of examples that show that the assumption on a bound from below on p is crucial: indeed, solutions (p, u) such that p E LO' (Q) may not belong to C x C' (or to Wo e x w2 ) 10C 1
even if the data are COO.
In section 6.10, we consider related problems such as the case of a general pressure law. Finally section 6.11 is devoted to general compressible models with an addi-
tional unknown scalar function which corresponds to the temperature. We show there how we can extend at least some of our results to "full compressible models".
6.2 Existence and regularity results for time-discretized problems We shall make the following assumptions on the data f, g, h : h > 0, h : 0 and h E L~ (S2), h E L1 (f)
if Q = RN
,
(6.8)
g E L2N/(N+2) (SZ) if N = 3 ;
gEL9(f)forsomep>1 if N=2andS1R2; if SZ =R 2 , g E L9 (11 2)
jgu dx
C
r
(f
for some p > 1 and 3C > 0, 1/2
hJuI2 + IDuI2 d2
for all u E Dl,' (R 2),
(6.9)
Stationary problems
52
where Di/2 (R2) is the space of functions u E Li , (1[22) such that Du E L2 (1122 )see Appendix B of part I for more details on that space. In particular, in view of the results shown in this appendix, the above conditions hold on 1R2 if IgI < Cvfh- + gi where gi > 0, g1+6 1x16 E L'(R2) for some 6 > 0. Next, we shall
assume that f satisfies
f EL4(SZ)with q>
(N + 2Ny - 2N
if N>4 or if N=3,y>2;
q> 3) if N=3, y max(21 y
y-1
if N = 21 St
1E22;
q=
22
y
(6.10)
if St = R2.
In the particular case when 11 = R2 , we will use also the additonal assumption hlxl6 E L' (R') for some b > 0 , f E L° (1[22) with q > max(21y)
(6.11)
Y
Our first result concerns the existence of a solution of (6.1)-(6.2). First of all, we have to define what we mean by solution. We look for a solution (p, u) of (6.1)-(6.2) in the sense of distributions, such that p > 0, p E L7 (a), p E Li (St) if SZ = RN; p, it are periodic in the periodic case; u E H' in the periodic case; L2N/(N-2)(RN) if St = RN with N > 3; u E H11.c(R2) in the Vu E L2(RN), u E case when S2 = R2; u E Ho (SZ) in the case of Dirichlet boundary conditions; and pIul2, hlul2 E L'(fl). Our main existence result is then the
Theorem 6.1 We assume (6.8)-(6.10) or (6.8)-(6.11) and y > N/2, if N > 2 , y > 5/3 if N = 3. Then, there exists a solution (p, u) of (6.1)-(6.2) satisfying pEL2-,
(f) if N=3, y>3orif N=2, 1154R2;
p E Lloc (1[22) if 1 = R2 ; pE
L2-f (122) + L4 (R2) for any q E (2y, +oo] , pl xI b E Ll (1[22)
(6.12)
if SZ = R2 and (6.11) holds ; (71) (St) if N=3, y4, pEL
and if N > 4, or if N > 3, 5/3 N/(N+1)), that diva-a/(µ+l;) p'r belongs to Lq' (SZ) + L2(11) where q* = Nq/(N - q) = (N/(N - 2)) ((7 - 1)/7): since Vu E L2(SZ), this consequence is in fact contained in (6.12). Next, we with to point out that, when N > 3 or N = 3, -f:5 3, then q:5 2N/(N + 2). Finally, let us observe that (6.14) implies easily that we have
I h 122 + pI 22 + 7 1(ap - hp"-1) + pIDuI2 + 6(divu)2dx Js2 7
2. This integral (in) equality is indeed easily deduced from (6.1) except in the case when fl = R2 where we have to worry about the integration over R2. The only (real) difficulty being in the justification of the integration of div IU [p 12 + p r] } over R2, we solve it by using a cut-off function (as usual) cpR(.) = cp where R > 1, co - 1 on B1, Supp cp C B2i 0 < cp < 1 and we write, denoting < x >= (1 + IX12)112, P2
div
2
u p11 + 7
1 pry
co, dx
N/2: if it were the case, our existence proof would apply.
Remark 6.3 The bound (6.12), shown in the next section, can be understood as follows. First of all, when N > 4 or N = 3, -y < 3, we shall show that p7 "behaves" like pIuI2 or equivalently that p7-1 "behaves" like Iu12. By Sobolev's
embeddings, we obtain (6.12). Next, if N = 2 or N = 3, y > 3, N/(N - 2)(^y 1) > and in these cases (6.2) will imply that p7 "behaves" like Du E L2. However, we shall see in the regularity results which follow that the bounds in LN/(N-2)(7-1) are still valid in those two cases. We do not know whether the exponent N/(N - 2)(-y - 1) is optimal.
Remark 6.4 The proof of Theorem 6.1 shows in fact that we can estimate all the norms of p, u, pIuI2, hI uL2, curl u, div u - a/(µ + C) p7 occurring in the definition of solutions or in (6.12)-(6.13) in terms of bounds on the data corresponding to (6.8)-(6.11).
Existence and regularity results for time-discretized problems
55
We now turn to regularity results. In these results, we assume that f E L° (0), g E LP (0), h E Lr (ft) where p, q, r do not necessarily correspond to the exponents given in (6.8)-(6.11) and we postulate the existence of a solution as in Theorem 6.1. Combining the two types of results, namely existence and regularity, is straightforward and we do not detail it here. We begin with the two-dimensional case.
Theorem 6.2 (N = 2). Let f E L° (SZ) with q > 2y/(2-y - 1), g E
LP (SZ)
with p > 1, h E Lr (1) with r > 'Y. We assume there exists a solution (p, u) of (6.1)-(6.2) satisfying (6.12). Then, we have
PE L"' (K); DUE L'(K) if s 2;s2=
-
1
(6.16)
2 if p < 2, s1 < oo if p = 2, si = +oo if
ifq 'y, f E LQ n L2 with q > 2 then there exists a solution of (6.1)-(6.2) satisfying (6.12), (6.18) and (6.19).
Remark 6.5 When N = 3, ry = 3, the assumptions made upon f, g, h in the case of Dirichlet boundary conditions or in the periodic case reduce to g E LP with p > 5, h E L' with r > -y and f E L' if a < q:5 2, f E LQ if q > 2.
Remark 6.6 In the case when n =1R2, it is possible to show global regularity results on R2: we then need to assume a sufficient decay of p (which is in fact ensured by a similar decay of h). We shall not consider here such extensions or variants for the sake of simplicity.
Remark 6.7 Of course, (6.16)-(6.19) really mean that we obtain a priori bounds in terms of the data f, g, h, except, however, in the case N = 3, 'y = 3 which is a critical case.
Remark 6.8 In order to understand the above regularity results, it is worth taking an example: let us take the case when f and h are smooth, then Du E L' (St) where 1 = 1 - .1 and we obtain the same regularity as for the elliptic equation: - pAu - CV div u = g. However, we do not quite obtain the same regularity for the second derivatives of u, since instead of obtaining D2u E LP (11),
we prove that D curl u E L' (11) and D(div u - µ+E p') E LP (11); and we shall build examples in section 6.9 where D2u (and thus Dp") and where p is not continuous! The very particular example of gradient forces already shows that there are clearly some limits to the regularity we can expect.
Example 6.1 We take g - 0, h - ap and f = Dqi. Then, it is easy to check that necessarily u - 0 and thus aVpry = pO0. Obviously,
(7 -1) +
Pay
,-1
im+A
if-f>1' p=c°
if ry=1
is a solution for all A E R. Let us take the example of 'y = 3, N = 3: then, as we shall show in the next sections, it is possible to extend the existence theorem to the case when f E L312 , hence 0 is not in general in LP for p > 3 and thus p does not belong (in general) to LQ for q > 6: notice that Theorem 5.1 yields precisely the fact that p E L6!
It is worth looking at all solutions of aV p" = pVg. To this end, we first consider the case when -y = 1 and assume that V0 E L oc, p E Llo, for some a, /3 > 1, a + 1 < 1. Then we deduce that Vp E Llama and thus VLog(b + p) _
V¢ for any 6 > 0. Since P0¢ converges to V1(>0) in Li (if Q < oo), we deduce that, unless p - 0, Log (b + p) converges as 6 goes to 0+ to some 10
element of W, hence Logp E W10 and VLogp = a \70. Finally, we deduce that p=c on each connected component for some A E R2. Next, we consider
A priori estimates
57 p
ry-1
the case when ry > 1 and deduce as before that fo a ds converges as 6 goes to 0+ to some element of Wlo which has to be 1 p' -1. Therefore, we find: -1 _ ti-i Op
= a7
1(p>0)V q.
If ,8 > N, then this implies that p"-1 is Holder continuous. Therefore, {x/p(x) > 0} is an open set and we deduce that on each connected component of this open set
p= (7_1+)+J for some A E R.
Remark 6.9 As we shall see from the proofs made below in the next sections, it is possible to improve slightly the assumptions on f for the existence thorem (Theorem 6.1) combining in fact the proofs of Theorems 6.1 and 6.2. Indeed,
in theqcase when y > 3, we shall see that it is possible to assume only that
-
f EL
(Il) where q > 2-y/(2-y 1) when N = 2 (and 11 34 R2, the whole space case SZ = R2 requires as usual some modifications we do not wish to detail here),
q = 6-y/(5-y - 3) when N = 3 and q = (2N(-y - 1)) /((N + 2)7 - 3N + 2) when N > 4. o
Remark 6.10 All the regularity results, including assertion (6.13), are only local in the case of a bounded domain i and Dirichlet boundary conditions. It is an important open question to determine whether they can be obtained globally in Q. This issue is very sensitive on the type of boundary conditions we impose since, as we shall see in the next sections, it can be solved positively in the case when we impose the following natural but somewhat more complicated boundary
conditions: u n = curl u = 0 on 8c if N = 2, u . n = 0, curl u x n = 0 on all if N = 3. Other types of boundary conditions allowing for regularity results up to 0 the boundary will be discussed at the end of the next section.
6.3 A priori estimates We split the a priori estimates into various categories. Of course, all these estimates are purely formal at this stage and will have to be justified later on.
Step 1: Energy-like estimates. We multiply (6.2) by u taking into account (6.1) and we find h
122
-µAI 22 +pIDuI2 -e div(udivu)
pu f Using (6.1), we see that we have
u Op = div U __L p7 'Y-1
- --!-(h ry-1 - cep)
(6.20)
Stationary problems
58
This equality conbined with (6.16) and (6.1) yields 2
hl
2 +ap
IZ
2
2
aryl
+
2
(ap7 - hp7-1) +div u
pi
2 + 7 1p 'Y
'Y
- µ0I 2 + plDul2 - .div(u div u) + (div u)2 = pu f + u . g
.
(6.21)
Hence, integrating over fl, we deduce
l22 +
Jh--+a-----n I
+ e (div u)2 dx =
a2 (a p,r - hp1) 7
+ pI Dul2 (6.22)
Jn pu f + u g dx .
In particular, we obtain fn hIuI2 + pI u12 + (DuI2 + pr' dx < C 1 + fn Pl ul If I + Jul191 dz
(6.23)
where, here and below, C denotes various positive constants which depend only on bounds upon the data f, g, h. Let us begin with the case N > 3. In the case of Dirichlet boundary conditions or if St = RN, then we can bound fn Iullgldx by IIuIIL2N/(N_2) and thus, in view of Sobolev's inequality by II9II LZN/(N-2) IIDuI IL2 In the case of periodic boundary conditions, a similar bound holds with II91I L2N/(N-2) I IullH1 II9IIL2N/(N-2)
and we observe that since h E L", ry > N/2
,
h # 0, the quantity (fn hIuI2+
I Dul2dx)1/2 is an equivalent norm on H1. Therefore, we find
PIu12+e dx+IlullH1 2, we bound (fn pl ul If I dx) by I IPI IL-Y IIuI I LsN/(N-2) I if I ILa if q = 22)--2N N-y (notice that 9 + ,1-y + 2N2 = 1). If 'y > 2, f = RN and f e LQ
(N-
2N2
+7=1 7-2N, we replace IIPIIL, by lIPlLL. where r + and 1 < r < dy. We then need to observe that integrating (6.1) over RN yields fRN p dx = « fR, h dx. These considerations show that, in the case when ry > 2, we obtain bounds on p in L-Y (f1L1 if Q = RN), u E H1 (or L2N/(N-2) with Du E L2 if S2 = RN) and in particular plul2, hIuI2 are bounded in V. with q >
N2
If N/2 < -y < 2 (i.e. N = 3 and 2 < ry < 2 or if N = 4, -y = 2), we bound fn pl ul l f I in the following manner: we detail it only in the case when
N=3, ! es(2 - ry), an inequality which is obvious if ry > 2 (and thus s = 1) and which holds if 1 < y < 2 since in this case s = a and e = yry l ''r-1 hence -y(1- s) > es(2 - ry) is equivalent to (ry -1)(2 - r) > (r -1)(2 -'y) or to r < In this case (St = R2 and (6.11) holds), we have shown a priori estimates for p in L' n Lt(R2) ; pIuI2, hIu12 in L1(1R2) ; u in D1'2(]R2) and p < x >a in L'(R2) for some small a E (0, 1). In fact, in view of (6.11), we also obtain an estimate for p < x >a in L1(1R2) for the same 6 as in (6.11): indeed, we argue as above and multiply (6.1) by < x >a using the L2 bound on f it we just proved. 0
Step 2: Proof of (6.12)-(6.13) in the case when 11 = 1RN(N > 3) and in the periodic case. We begin with the proof of (6.12) and then turn to the proof of (6.13), each time in the case when 11 = IRN, N > 3. Then, we shall briefly explain how to modify the proofs in the periodic case.
The idea of the proof of (6.12) is extremely simple: we observe that (6.2) yields
a V pry = F1 + div F2 - div(p u ®u) (6.27) where F1 = g - apu + pf is bounded in Indeed, p is bounded in L2N/(N+2).
L' n L with ry > N/2 and thus in LN12 while it is bounded in L2N/(N+2) . Next, F. = pVut + div it ei ((el, ..., eN) denotes the canonical orthonormal basis of RN) and thus F2 is bounded in L' . Similarly, f E L° where q + ,11 < 1 - ZN = 2N and thus p f is bounded in L2N/(N+2). As is well known, F1 can be written as div F3 where F3 is bounded in L' : for instance, solve 0q5 = F1 in RN, V E L2 (RN), ' E L2N/(N+2) (RN) and take F3 = V O. Therefore, we may rewrite (6.26) as
a V p' = div G - div(p u (9 iu) where G is bounded in L2 (RN), or equivalently in terms of Riesz transforms
ap' =
Ri R3(P uiu3
- Gi3).
Next, if N = 3, ry > 3, we deduce from (6.27) IIP''lIL2(R3) 1, we can choose (uniquely) v, w in Next, we observe that we have (pu 0)u = (v 0)u + (w 0)u. In addition, Lr,,
div w = div pu = h - a p is bounded in
L" (RN ), therefore Dw is bounded LNry1(N_ry)(RN) if y < N, LO(RN) for all
in L1'(RN) and thus w is bounded in Q < oo if y = N, L°° (RN) if y > N. In particular, w is bounded in LN(RN)
in all cases and thus (w 0)u is bounded in L2N/(N+2)(RN). Finally, for all 1 3.
where
63
This concludes the proof of (6.13) when
The proof in the periodic case is almost exactly the same except that in (6.23) we just have to replace a p'Y by a (plf fn p'r dx), a modification that is easily handled since we already know that ho p7 is bounded. 0
-
Step 3: Proof of (6.12)-(6.13) in the case of Dirichlet boundary conditions (N > 2). We begin with the proof of (6.12). We may argue as in step 2 above to deduce
a V p'f = div G - div (p u ® u)
(6.31)
where G is bounded in L2(11). Notice indeed that in the case when N = 2, we already know that p is bounded in L7 and u is bounded in L' for all q < oo in view of the estimates shown in step 1 above. When N = 2, or if N = 3, -y > 3, (6.12) follows easily since we have
e -in p1dx
Lz
< C IIVP'IIH-1 < C IIGIILZ + C IIp IUI2 IIL,
whereq=6ifN=3, q>2 if N=2.
< C (1 + IIPIILq )
When N > 4 or N = 3, 2 < ry < 3, we take r = N 2 1 (recall that 1 < r < N 2 and r < 2). We then conclude as before once we observe that the
following inequality holds for any 1 < r < oo:
f
II(PIILr(n) < C IIGIILr(n) +
dx InI11r
n
if V = divG.
(6.32)
Indeed, by standard density arguments of {div e /q5 E Co (11) }, we deduce sup
LcouI dx /IIII Lr/(r-1)(0) < 1 in tb dx = o
< C IIGIILr(tl)
: 1, then III - f-n 0 dxllLr/(r-1)(n) < 2'
Next, if 101 = 1 and 11011
where s = T 21-the inequality is obvious for r = oo, r = 2, r = 1 and follows by interpolation-and thus
II(PIILr()) =SUP f cp n
dx
/II
< C IIGI ILr(n) + sup
< C IIGIILr(n) +
IILr/(r-1)(n)
fn
q. We agree that 7-1" = Lq if q > 1 in order to simplify notation. Since d {(p + t;) div u - apry } vanishes on an, we may now easily complete the proof of (6.13).
Step 4: Proof of (6.12) in the case when Il = R2. First of all, we begin with the proof of (6.12) following the argument given in step 2. In particular, we may still write (6.27) where F2 E L2 (R2) and F1 = g - a pu + p f. Notice that pu is bounded in L' fl L2-,/('+1) (11 2) and that p f is bounded in L' fl L2''I ('Y+1) (R2) if (6.10) holds while pf is bounded in L1 fl L''(R2) with r = 9 + 1-y, < 1 if (6.11) holds. Next, we deduce from (6.9) that on the one hand g E LP (R3) for some p > 1 and on the other hand there exists v E D1"2(R2) such that v E L2(R2) and g = -Av + hv. Hence, g = div G1 +G2 where G1 E L2 and G2 E L1 nL2 In conclusion, we have
a V p" = div G + F - div(p u ®u)
(6.33)
where G is bounded in L2(1R2) and F is bounded in L' fl L''(R2) for some r > 1 (which we can always take to be in (1,2)). Next we observe that p u ® u is bounded in L1(R2) fl L q,,, (R2) for all q < 'y since u is bounded in D1'2 (R2). Furthermore, if (6.11) holds, we have already seen that & 16 is bounded in L' (R2). Hence, for e E (0, 2('y - 1)) and for any measurable set A fR2
lp
uiu,12
1A dx L}E
f
< (R2 lA
P2+,
< x >3 2+1 dx
(fR2
' (<x>°
dx
A priori estimates
65
where < x >= (1 + IxI2)1'2 and _a is arbitrary. We may then choose a = e and deduce from the bound on u in D11'2 (R2) that ei, (1R2 p < x >8 dx 1A P''dx 1A I Pu=ui l2 dx < C
2(1-e)
(j2
fR2
where e + ie = enough we find
e))-1
2+2E and 6 = (2(1-
f
I P uuI2 dx < C
A
e . In conclusion, choosing a small
(j1Ap2dx)°
(6.34)
as e goes to 0+. In particular, a/7 < 1 for e small enough. where A goes to 2 Next, we deduce from (6.28) that ,y
p = 71 +7r2 +
-a1 RjR3(puiuj)
r
where 7r1 is bounded in L2(R2) and 7r2 is bounded in L"(R2) for p E (2) 2) if r < 2, p E (2, +oo] if r > 2. Hence, we have, choosing A = {x E R2 /p" >- 2 I1r2I },
pY lA < 2 Iir1I +
IR%R.i(Pu;.uj)I
a and decomposing RR3(puiuj) into RjR3(1Apu=u1) + RjRR(lAcpusu3)
l2p2y1Adx 2. Reiterating the above argument, we find in a finite number
of steps that curl u, (p + C) div u - a p7 E W 1'q for some q > 2 and thus (p + C)divu - ap7 E L. Then, letting m go to +oo in (6.37), we deduce that p E L°O and thus div u, curl u E L°°, hence Du E BMO, and (6.23) is shown in the case when s = oo. (6.24) then follows in view of (6.35) since pf E Lq, g E L", p(u V)u E L' for all r < oo and hu E L°°. The case when s < oo is treated by the same bootstrap argument that we do not wish to repeat. However, let us explain that the bootstrap argument will make the exponents grow indefinitely until they reach or exceed values due to p f , grh. For instance, if h E L'' (St), then the above argument shows that p E LQ7 where r = and thus Du, p7 E L'3 where s3 = Tryl r ; similarly, D(div u - µ+C p7 ), D curl u E L 13 where s = 3 + 337 = 33 771 = r if r < 00 (and t3 < oo if r = +oo). Similarly, if g E LP, D(div u - +t p7), D curl u E L`1 where t1 = p if p < oo, t1 < oo if p = +00. And Du, p7 E L'1 where s1
2. Finally, if f E L9, we find that L"2 and 82 = t2 - , = D(div u - µ+f p% D(curl u) E Lt' while p", Du E ryry l 2 z to ry and t2 < 2 is equivalent , t2 = q 1 + s if t2 < 2, i.e. S2 =
q 2, we deduce from (6.35)
- ap'% - µp1 {cp curl u}
= cP [Pf + g - hu - P(u . 0)u] - [(µ + t;) div u - ap'y ] pcp - µ curl u01 co. (6.38)
First of all, we see that the term cp[p f + g hu] can be written (cpp) f + cog h(Vu) and thus is naturally localized. Next, the term cop(uV)u can be written (gyp) [(mu) ti0] u choosing co = V53 where E Co (fl). Therefore, the only difference with respect to (6.30) mainly lies in the terms [(µ + t)div u - app] Dcp and curl u V l cp which obviously belong to L4 (1) and thus are more regular than cpp(u V)u which belongs to L' (Q) where r = q + 97 if q > 2 and r < if
q=2.
Stationary problems
70
The proof made in the periodic case will thus clearly extend provided we can
localize (6.36) and (6.37), and also the regularity argument made in the case f E L9 with q < 2 (and g, h are bounded for instance). First, we write (6.36) with i > 1 and we deduce + (m - 1) + P+ arrtcp i` pm V
div (r
hp
= MW
+ p'"
P.
+
(1+!?!) co
V
[ap - (A + )div u]
u V V.
Hence, we deduce at least formally rn
IIcP''II m+y o}, I ldiv v all L, < e IIDvl I L, , Ilcurl v - b1 1L2 < e II DvII L, where a, b E L' are fixed, v3 + civi + c2v2 = 0 at x3 = 0 where c1, c2 are Lipschitz , IIc2IILIQ < e, and e > 0 can be made as small as we wish (by and
-
IIci11L,,,
localization). Since A(v3 + civi + c2v2) = Ova + T = [V (div v) - curl curl v] 3 + < C(1 + T where IITIIH_1 < C(11vlIL2 + e llvvllL, ), we deduce that C IIVVIIL,) and thus Ilaivi+82v21IL2 , 11a2v1+a1v2II L, , I1a3VIIIL2 , II83v2IIL, 0. Indeed, if I = 0, the infimum is achieved at some u E Hi (St) such that u # 0 , u n = 0 on 8SZ and divu = curl u = 0 in 0. In particular, u = V where 0 E H2 (St) and AO = 0 in St , 94 = 0 on of Hence, 0 is constant and thus we reach a contradiction. In conclusion, we have shown the following inequality
in
(p + )(div u)2 + ,(curl u12dx > v
I Dull + Iu12dx
(6.47)
for some v > 0, independent of u E HI (11) satisfying (6.42). The combination of (6.46) and (6.47) allows us to repeat the arguments developed in step 1. In the case of (6.42)-(6.45), the argument is somewhat simpler since (6.46) is now replaced by
r h 122
Jn
+ ap l u2 + /'
+J
an
7
ry 1(apy
p(Au, u)dS =
- hp'r-1) + plDul2 + e(div u)2 dx Jn
(6.48)
pu f + u g dx.
Recalling that, by assumption, (Au, u) > 0, p > 0, we may then conclude easily at least when > - N . Indeed, in that case, we have for some 6 E (0, N )
(div u)2 > -6p(div u)2 > -N6plDul2 a.e. in St.
Stationary problems
76
If
= - N, we need to do a bit more work. Following the proof made below
(for the boundary condition (6.44)), we can show the following inequality for all u E H1(SZ)N such that u n = 0 on 8SZ
fn
1L(DuI2
- *(divu)2dx >
(6.49)
v fn (Du(2 + (u(2dx
for some v > 0 independent of u. Indeed, the argument given below shows that we only need to prove that if u E H' (fl) , u n = 0 on 8S1 and (Du (2 - (div U)2 = 0 N facts imply or equivalently Du N div u I then u 0. But, obviously, these
that u = ax+b for some a E R, b E RN and thus 0 =J div u dx = Na meas(SZ), therefore a = 0 and b = 0 since {n(x)/ x E 8S2} = S '. We next consider the case when (6.42) and (6.44) hold. First of all, writing Du = div(2d) - Vdiv u, we obtain n
h 122 + c p (22
+
7 1(ap'r 'Y
+ ( - µ)(div u)2 +
- hp7-1) + 2µ(d(2
Jan
2µ(Au, u)dS = J pu f + u g dx. n
(6.50)
We shall only detail the case when e = N-2),,, since the case when > N 2 µ is N in fact (clearly) simpler. We just have to show that there exists v > 0 (depending upon h, SZ and µ) such that the following inequality holds for all u E H1 (11) satisfying (6.42) /'
Jn
(22
+ 2µd12 - N (div u)2dx > v in (Du(2 + (u(2dx.
h
This will be achieved in two steps. First of all, we remark that (d(2 -
(d -
,
(6.51)
n
divu
I12
and that (dJ2 = 2 (Du(2 +.I 8iuj O ui, hence
n (d12dx = 2 f IDu12 + (divu)2dx + Since u.n = 0 on aQ
,
N
(div u)2 =
88ui dS. J s1 2 u3ni
u3ni ajui = -(Ku, u) > -C1u12 where K denotes the
curvature tensor. Hence, we have
f (d2 -
(div u)2dx > 2 I (Du2dx +
2
fn
(divu)2dx -
rn (Du(2dx
- C fn (u(2dx
an
1/2
1/2
2
C
fn (Du 12 + Iu12dx
4 J (Du(2dx - C [iui2cix ffn
n
where C denotes various constants independent of u. This inequality shows that (6.51) holds if h 1 since (d12 - N (div u 112 > 0 a.e.
A priori estimates
77
Next, we prove that (6.51) holds if h E L7 and h # 0. It is clearly enough to show that
I = Min
in
hlul2 + 1d12 - 1(div u)2dx /u E Hl(SZ),
0 on 811,
in
>0.
IuI2dx=1
In order to prove this claim, we argue by contradiction and assume that I - 0. Obviously, this infimum is achieved for some u E H' (1) such that u n = 0 on all , hu - d - N div uI - 0 a.e. in 52. In particular, 2
__
2
alai
1
N-1
j>_2
8z 8xi
u
2
N1
1
j>2
8x?9
u1 in D ' (52).
In other words, ui (and in fact U2, ..., UN) solves a uniformly elliptic linear secondorder equation with constant coefficients, namely 2
ax?j +
2
7- u1 = 0 1 1: ax3 i>2 -7
in
)
D'(cI), u1 E H1(S2).
In addition, ui vanishes on a set of positive measure since h # 0. This implies that ui - 0. We argue similarly for U2, ..., UN and reach a contradiction with the constraint f0 JuI2dx = 1. The contradiction proves our claim. We now turn to the second point we wish to explain, namely the regularity (in terms of LP spaces) of D(div u - 'Ua p7) and D curl u. First of all, we observe
that in all three cases, we may write the boundary conditions as (6.42) and
curl u x n = Lu x n on
(6.52)
acZ
where Lu is a linear multiplication operator with smooth coefficients. In particular, in two dimensions (6.52) reduces to curl u = Lu on 852. With these boundary conditions, the regularity issue is the following: assuming that u E W1,q(52) , p7 E Lq(SZ) where q > 2 (for instance!), we wish to extract some regularity on div u - µ+t p7 and curl u from the following equation
-v ((µ + e)div u - ap7) + µ Curl curl u = cp E Lr (52).
(6.53)
If N = 2, Curl = a and r > 97 + Q if q = 2 , r = -L + . ifq > 2, while, if N=3 Curl =curl andl1 ifq +13 ifq=3, r = 1q7 +1q N r- q71 +?q r 3ry 1
if q > 3. More precisely, we want to prove that curl u, div u - a p7 E W i,r (SZ) When N = 2, this is rather straightforward since curl u = L u E W 1-1/q,q (act)
on 8f and there exists 1i E W1,q(c2) such that curlu = V on 80. Therefore, we
Stationary problems
78
have µ Curl(curl u - ip) = cp + V ((µ + )div u - ap7) in 11, curl u - ip = 0 on 8SZ and thus taking the curl of this equality, we find
-µ 0 (curl u - i) = curl cp in 11, curl u - ip = 0 on all. Hence, curl u - lp E Wo ,r and curlu E W1,'' since q > r. Then, (6.53) shows that div u - µ+E p 7 E W1,r.
When N > 3, a more general argument is needed. Let us first explain the case when 11 is a half-space, say SZ = {x E RN /xN > 0}. Then, we have UN = 8u: {xN = 0} for 1 < i < N - 1, where 1pi is the trace on 8S2 of aXN - 0i, = 0 on 0 a function in W1,q(SZ). Introducing u E W2"q(11) such that ui = on {XN = 0} for 1 < i < N - 1 and uN - 0, and replacing u by u in (6.53), we see it is enough to consider the case whenz - 0. Then, if we decompose u = v + w, div v = 0, curl w = 0 in SZ, vN = 0 on {xN = 0}, we deduce from (6.53)
F = -µ0v + Vir E L'(St)
= 0 on {xN = 0} , div v = 0 in Q.
, VN = a2N
Next, we remark that we may extend VN and FN in an odd way and vi Ft in an even way to {x,, < 0} for 1 < i < N - 1, obtaining
, 7r
and
-µ 1v + V7r E L'(RN), div v = 0 in RN. Therefore, D2v E L''(RN) and thus D curlu = D curl v E L''(SZ) and we conclude.
It only remains to treat the case of a general domain. First of all, we decompose u as before: u = v + w, v, w E W1,q (SZ) div v = 0 , curl w = 0 in St
, v n = 0 on a Q. Then, (6.48) becomes -µ0v + Vir = F E L'(SZ), curl v = curl u,
v n = 0 on 852,
(6.54)
div v = 0 in Sl
and in particular, curl v x n = ip x n on 8SZ where 0 E W'(). Obviously, we only have to prove that v E W2"''(f ). This follows in fact from general regularity results on elliptic systems but it is possible to give an elementary direct proof. By standard localization and partition of unity arguments, it is enough to prove the regularity in an arbitrarily small neighbourhood of a given point of 8SZ which, by translation, we can take to be the origin, assuming that v (and ir, F) are supported in this neighbourhood. Furthermore, by a rotation, we may assume that, locally near 0, St = {xN > O(x')} where 0 is smooth, q5(0) = 0, VO(0) = 0 and we denote x' = (xl, ..., SN_1). We then change variables y' = x', YN = XN - l¢(x') and set v(y) = v(x), *(y) = ir(x), (y) = F(x) and .
we find
-p0v + Vfr = F + G in {xN > 0} , ON= h on {xN > 0}
(6.55)
A priori estimates
79
div v = D in {XN > 0} av=
aXN
= Ei on {xN = O}
where IGI < E [ID2vI + IDfrI] + G'
,
fo r
(6.56)
--
11 , (u'),,>, and we always assume at least that p" > 0 is bounded in L'nLq(11) for some q > 1, that un is bounded in H1(S2) N (except in the case when 12 = RN where we assume L2N/(N-2)(RN) if N > 3) that Dun is bounded in L2(RN) and u" is bounded in and that pn Iun I2 is bounded in L1(1l). Let us recall that, in the periodic case, we denote S2 = FIN 1(0, T1) and pn, un (and all other data) are defined on ][8N and periodic of period Ti > 0 in each xi (1 < i < N). Finally, if SZ is a bounded, smooth, open, connected set in RN, we always assume that un satisfies (6.42): we do not require un to vanish on 8S2 and we thus incorporate in our analysis the boundary conditions (6.43), (6.44) or (6.45) as well as the case of Dirichlet boundary conditions where un vanishes on 011. Without loss of generality, we may assume, extracting subsequences if necessary, that Pn converges weakly (in
L' n Lq) to some p > 0 E L' n Lq and that u" converges weakly in H'(1),
a.e. and strongly in LP(fl) for 2 < p < N2 N (except in the case when S2 = RN where the strong LP convergence holds on all balls of RN) to some u E H'(1) (with the same modifications as above for un in the case when S2 = RN) . Let us observe that plul2 E L1(fl): indeed, by the Egorov theorem (for example), we deduce that for all e > 0, there exists EE such that meas(EE) < c (if Q = RN, lungs meas(EE n B11E) < E) such that converges uniformly to lull on E. Hence, we have
C>
pnlunI2dx >
pnlunl2dx
JEe E and our claim follows upon letting c go to 0+. Jn
Plul2dx
as n goes to + oo,
Compactness
81
Next, we assume that we have
anpn + div(p"u") - EnApn = hn where en > 0
,
an > 0
,
an w a
,
fn
0
,
(6.60)
in 12
hn E L1(1)
,
hn > 0 and
hn n h in L'(12). Finally, we assume that the following assertion holds: div un - b(pn )^1
converges a.e. in 12
(6.61)
where b>0, 7>0. Theorem 6.4 If q > 2 and q > 7, then pn converges to p strongly in LP(Q) (resp. Li'C(RN) if 12 = RN) for all 1 < p < q as n goes to +oo.
Proof of Theorem 6.4. We follow the strategy of proof introduced in chapter 5 with a small technical modification due to hn and the possibility of zones of vacuum (where pn or p vanishes). We thus consider (E+pn)e where e > 0, e > 0: O has to be taken small enough in the argument below and will be determined later on. We claim that we have ean(e + pn)e + div {un(e + pn)e} - EnA(E + pn)e pn)e-1 + (1 - O)div > O [hn + E div un + anE] (E + un(E + pn)e
in Q. (6.62)
In order to justify this computation which is straightforward if pn (and 4.un) is smooth, we use our "standard" regularization proof which only requires e to be small enough (e _< 1 and O _< q/2 even if q were not assumed to be larger than 2). We simply observe that when cp is smooth and positive then e)We-2IV I2 -Ave = -OAV Ve-1 + e(1> -eocp ye-1. We next denote by ip the weak limit of a sequence cpn. With this notation, we see that divan - b(pn)7 converges in L''(SI) (Lr,, (RN) if 0 = RN) to divu bp 'for all r < 2, r < q/7. Using this information, we may pass to the limit in (6.62) (extracting subsequences if necessary) and deduce easily
-
p)e-1 Oa (E + p)e + div u (E + p)e > 9 [h + aE] (E + + 9 E div u(c + p)e-1 + (1 - O) div u (E + p)e
+ (1 - e)b
{p7(E +
(6.63)
p)e - 7 (E + p)e}
provided we choose O < q(1 - T ). We thus choose O > 0 in such a way that
O < min (1,q(1-
*)).
We then compute the inequality satisfied by ((E + least formally
1/e
and we find at
Stationary problems
82 p)e)1/e
a ((E +
+ div u ((E + p)e)1/e
s-1
J
> [h + aE] (E +
p)e-1
((E
S
p)e-1
+ E div ZL(E +
+
+ P)e)
ee) b {p(+p)e -
1
1
1
(E+P)eJ ((E+p)A)g (6.64)
However, the justification of this inequality is a bit delicate. Let us first observe that, denoting /R(t) = Rf (R) for R > 1 where 0 E C°°([0, oo)), 3' E Co ([0, oo)), 1 > /3' > 0, 3(t) = t if t E [0,11 , we have C OR ((E
+ p)A)
1/e
> (h + aE)(E +
1/e
+ div 1U/3R ((e
+ p)e) p)e-1 PR ((E + p)e) PR ((E
+Edivu(E+p)A-1 PR ((6
+ P)e) + p)8) PR ((6 + p)e)
+le - (div u) (e + p)e/3R ((E + p)A) PR ((E + p)el
1
-1
(E+p)A}/3R((E+P)e),QR((E+p)e
+(divu)/3R ((E + p)A)
1/9
-1
J
We then wish to recover (6.64) upon letting R go to +oo. First of all, we remark that we have (diva) {PR(( + p)9
< Cldiv uI
1e
-
+ (E
((6+p)e)
1 /A
P)e PR (E + P)e PR E + pe) 1
1
((E+p)e>R) < Cldiv ul(e + p) 1 (e+p)e>R)
and thus these terms go to 0 in L since divu E L2, p E L oC . Here and below C denotes various positive constants independent of f and R, and we have made use of the following inequality (E + p)9 _< (E + p)9 due to the concavity of (t H t9) on [0, oo) (recall that 0 < e < 1). Next, we estimate 16 div u(E +
p)9-1 PR ((E
+P ) 9PR
p)e-1 (E + p)e < C E Idiv u(E + < C E9 Idivul (E + p)1-9-
((E + p)9)
-l
This inequality shows that the left-hand side which obviously converges a.e. to E div u(E + p)9-1 (E + also converges in L as R goes to +oo and furthermore we have p)91/9-1
Compactness
E divu(E+p)e-1 (E+p)e in Lloc
1
83
< CEO Idiv ul (E +
P)1-0 __ 0
(6.65)
as E-->0+.
Then, we observe that we have
(h + a)(E + p)1f((E
+ p)e) OR (( + Pr)
g-1
1
> (h + aE)(E + p)e-1 (E + P)e
1((E+p)e 0
and
7 (E+P)eJ
QR
{p- + p)9 - 3
(E+p)ei-11
(E+P)e}
Here, Here, we have used the fact that (pY(f + p)e - p7 E + p)
0-
> 0 a.e. in view of 1/e
Lemma 5.2 of chapter 5. Then, the above inequality satisfied by /3R ((E + p) e)
shows that these quantities are bounded in
[h+ae]
while they converge a.e. to
and VI(f + p)e - pry (E+ Al I E-+-P-)
(E+p)e-1
The inequality (6.64) then follows from the above bounds and Fatou's lemma. Furthermore, we have shown that (h + aE) (E +
p)e-1
(E + p)9
are bounded in
L1
1 ,
p7(E + p)e - p1 (E + P?
loC uniformly in
(6.66)
E.
We next claim that (6.64) implies a
+ div {opal/e} > h+ 1 ee
b
pas-1. (6.67)
{v+e
Indeed, using (6.65) and (6.66) (and Fatou's lemma once more), we see that (6.67) follows from (6.64) upon letting E go to 0+ once we observe that we have
h (E + p)e-1 (E + p)es-1
>h
a.e.
Indeed, h > 0 a.e. and (e + t)e-1 = ry [(E + t)e] where -y(s) = s1-1/e is convex p)e1-1/e on (0, oo). Therefore, we have: (E + p)e-1 > -y ((E +-7M = (E + a.e.
Stationary problems
84
On the other hand, passing to the limit in (6.61), we obtain
a p + div {up} = h . Recalling that pe
(6.68)
1/e < p a.e. and subtracting (6.68) from (6.67) we deduce
1-6
b
E)
{ p7+e
l
_ P Pe } Pe -1 < div(ur)
where 0 < r < p a.e. In particular, r E L1 n L9(St) , rJu12 E L'(1). Integrating this inequality over SZ (and justifying it as in sections 5.3 and 5.4 of chapter 5), we obtain
(p+e_) fn
pe*1
dx < 0
while pry+e - pry pe > 0 a.e. Therefore, we have (p.r+e _ pry
Since 0+8 > Pry) ~ If Pry+e = Pry
pe )
pe
* -1
= 0 a.e. in St.
, p-'+" > (Pe) 2_ a.e., we deduce that -Y
Pry+e = (PI)
a.e. on {Pe > 0}
Hence, Pn converges strongly to p in Lry+e ({pe > 0). Finally, on the set {pe = 0}, clearly (pn)e converges strongly to 0 in L1(f n BR) (VR < oo). Therefore, we deduce that (pn)e converges to pe in L1() n BR) (VR < oo). Theorem 6.4 then follows easily. 0
6.5 Existence proofs We present in this section the proof of Theorem 6.1 and more precisely we prove the existence of solutions satisfying the bounds stated in Theorem 6.1 and obtained formally in the preceding section. We shall then conclude this section with the proof of the existence assertion in Theorem 6.3 in the case when N = 3, 'y = 3, thus completing the proof of Theorem 6.3. Step 1: Preliminaries. In order to establish the existence of solutions, we shall need several layers of approximations. First of all, we approximate (6.1)-(6.2) by
ap + div(pu) - EDp = h in S2 + 2pu.Du+a p 2 + Zdiv(pu®u) - pAu - V div u + aV Pry = P.f + g in c
h2
(6.69)
where f > 0. In the case of periodic boundary conditions, (6.69) really means that the equation holds on RN (and that p is periodic). In the case of Dirichlet
Existence proofs
85
boundary conditions, where 11 is a smooth bounded domain, we complement (6.69) with Neumann boundary conditions (for instance) on p
ap = 0 On
(6.70)
on 8SZ
where n denotes the unit outward normal to c9SZ. However, (6.69) is not the only approximation we shall need. In the case when N > 4, we shall have to approximate (6.69) further by
ap + div(pus) - EDp = h 2u
in SZ
(6.71)
+ 2 pub Du + a p 2 + z div(pu6 (9 u)
- ILDu - V div u + aVp1' = pf + g in ci
,
with the same conventions as above-in particular, we add the boundary condition (6.70) in the case of Dirichlet boundary conditions. In the equations (6.71) above, b E (0,1] and for each function cp we denote, in the periodic case, cPs = cp * rc6 where rc E Co (RN), 0 < r., < 1, Supp rc C B1i rc is even and res = 1 rc(6 ). In the case of Dirichlet boundary conditions, we denote u6 = (4u) * X6 and (Vp")6 = (6(rc6 * Vp"r) _ Cs(Vres * p'r) where C6 E CO '(0), Supp (6 C {x E 11 /dist(x, ac) > b} , Cs = 1 on {x E 0 /dist(x, 49S2) > 2S} and 0 < Cs
0 depends only on 11. But before we explain how to solve (6.71) and then (6.69) and how to deduce Theorem 6.1, we wish to make some preliminary reduction. We claim that it is enough to prove the existence of a solution of (6.1)-(6.2) satisfying the bounds stated in Theorem 6.1 and obtained formally in section 6.3 in the case when f, g, h are smooth. Indeed, the existence then follows in the general case by approximating f, g, h by fn, gl, h" respectively in their respective integrability classes: in this way, we obtain a solution (pn, un) of (6.1)-(6.2) with f, g, h replaced by fn, gn, hn and (pn, un) satisfies, uniformly in n, -the bounds stated in Theorem 6.1. In particular, extracting a subsequence if necessary, we may assume without loss of generality that (6.62) holds. Furthermore, the restrictions on y in Theorem 6.1 ensure in particular that pn is bounded in Ll fl L4 (SZ) where q > 2, q > y. Therefore, we may apply Theorem 6.4 and we can pass to the limit proving in this way the existence of solutions in the general case.
Step 2: Existence for (6.69) (N = 2, 3 ; fl # RN). Therefore, we assume from now on that f, g, h are smooth (with compact support if fl = RN). We begin with the case when N = 2 or 3 and Q 34 RN. We first list a priori bounds satisfied by solutions (p, u) of (6.69) that we assume to satisfy: p, u E W2,q(fl)(dq < oo)in the periodic case, this really means that p, u E Wio, (RN), and that p, u are periodic. First of all, we perform the same "energy" computations as in step 1 of section 6.3 and we find
(ei p)1uj2 + (pu). 0I - µ0 + µI Du,2 - div(u div u) + '(div u)2 + div (aui_1 p'r) + -y' p"r-1(ap - e0p - h) = pu f + u - g hIuI2 +
z
Stationary problems
86
hence, integrating over 1, we deduce fh!.-
+
+ap1
1(p1- hp1)
+ p1 Du12 + e(div u)2dx
0 in 11 (1-t)hu+ div u + aV p1 = p(tf) + tg in Sl. Furthermore, the proof made above immediately yields uniform bounds in t r. [0,1] for such fixed points in W2,q for all 1 < q < oo. This is enough to apply standard fixed points results (such as Leray-Schauder topological degree results) and to deduce the existence of a solution (p, u) of (6.69)-(6.70).
Existence proofs
87
Step 3: Existence when N = 2,3; y > 2 and n # RN). We now pass to the limit when N = 2, 3; -y > 2 and 11 ,E RN-in fact, the proof below may be extended to the case when y > 1, N = 2 and S1 34 RN. We wish to let a go to 0+ in (6.69) recalling that pE > 0, uE is bounded in H1, pE is bounded in L'y (and PEIuEI2 , hluE12 are bounded in L') uniformly in e E (0,1]. Furthermore, the bound (6.72) holds. Before passing to the limit, we are going to show that pE is bounded in Lq
whereq=2yifN=2orN=3, y>3andq=3(y-1)ifN=3, 2 0 in L'' for all 1 < p' < p). Indeed, we first observe that (6.72) yields for all 6 > 0 (recall that -y > 2) lie
IVP I2 1(Pt>_6)
IIL2
0 E Lq in Lq' for 1 < q' < q
Existence proofs
89
while uE converges weakly in Hl to some u and also converges strongly in L" for 1 < p < rr a Next, we pass to the limit in (6.69) writing 2 2
+21pu. Vu+
apu+ 1div2
2
(pu®u) = apu+div(pu®u) -
eOpu 2
Opu = -
div(EDp 0 u) + 1 e(Vp 0)u] converges to 0 2 a (in the sense of distributions for instance) in view of (6.75). Let us observe in addition that we can also show that Du, converges to Du in LP for all 1 p < 2 since div uE - µ+f (p6)'f and curl uE are relatively compact in Ll for all 1 r < q/y. We have thus built in this way a solution of (6.1)-(6.2) satisfying the properties listed in Theorem 6.1 including (6.13) at least in the case when N = 2,3 and y > 2. We have already explained in section 6.2 how to obtain when N = 2 or N = 3, -y > 3 a local energy identity (which, by the way, shows that Du, converges in L2 to Du). Let us observe that when N = 3, 2 < -y < 3, the local energy inequality (6.14) follows from the above construction. Indeed, we have
and observing that [-
shown above that (pc, uE) satisfies 2
h IuI2 + 2apIuI2 + div u [P-j-- + a71 p I
,y
+ 7a
1
[app
- hp7-1] -
y
2
AAI 2 +14IDuI2
-
ediv (u div u) + e(divu)2 -
+
Eary
(6.77)
Eary
ry-1 div(e-1 Vp) 0-2I Vpl2 = pu. f + U. g
and thus hIuI2
In
2
+ 2 apjuj2 + pI DuI2 + e(div u)2 +
- pu f - u.gdx = 0.
VpI2 (6.78)
We then pass to the limit in (6.77) as e goes to 0+ and we recover (6.14) observing that earyp7-2IVp 2 > 0 and that Ep7-1V p converges to 0 in L1 (say).
Step 4: Existence when N > 4, 7 > N - 1 and fl 54 RN. We want now to adapt the preceding proofs to the case when N > 4, -y > N - 1 and 11 34 RN. In this case, however, we first have to solve (6.71) and this is done exactly as in step 2 above once we observe that ub and (Vp")b are smooth and that the fundamental "energy" bounds still hold in this case (and in fact hold uniformly in 6 E (0, 11 and c E (0, 11). Let us detail this important point. We deduce from (6.71)
hu + pub Du + EDpu - pAu - V div u + a (Vp7)6 2
hence
= p f +g in St
Stationary problems
90
(22
hIuI2 + pub v + IeApIuI2 _ µ1 122 + µIDuI2 - l; div(u div u) + t;(divu)2 + a u (Vp7)6 = pu f + u g in )
,
therefore we also have
(o_.) - µ0 2 122
2hIuI2 + 2apIuI2 + div
2
+ µIDuI2 - ediv(u divu) + l;(divu)2 - a [u6 Vp7 - u (Vp7)b] ary -e div (p'V p) + (ap7 - hp7-1) + +div
(o1)
Earyp7-2IVpI2
+
= pu f + u g
ry-1
in 11.
(6.79)
Next, we observe that we have in the case of Dirichlet boundary conditions j'1
ub V pry
- u (Vp7)bdx = fJ
xn
dx dy [C5(y)u(y) - Vp"(x)ica(x - y)
-u(y) (s(y)V p7(x) rc6(y - 2:)] = 0 since rc6 is even. In the periodic case, this integral also vanishes for obvious reasons. Therefore, integrating (6.79) over n, we deduce easily (as in section 4.3)
IIIIHI + e IIP"2IIHI < C,
IIPIIL-Y < C
(6.80)
where, here and below, C denotes various positive constants independent of e and 6 E (0,1]; and, as in step 3 above, we also obtain
f IIVPIILZ < cc 1/7
(6.81)
.
Using the bound (6.80) as we did in step 2 above, it is then easy to construct solutions (u,,6 , pE,6) of (6.71) (or of (6.70) in the case of Dirichlet boundary
conditions) in W2' for all 1 < q < oo such that (6.80) (and (6.79)) hold. We may now pass to the limit as 6 goes to 0+, for e fixed in (0,11. Extracting subsequences if necessary, we may assume that uE,b
,
PE,b , (pE,6)7"2 converge
weakly in H1, a.e. and strongly in L" for 2 < p < N 2, respectively to some pE a.e. and in L4 for 1 < q < N try, LN/(N-2)7). In partticvular, (uE,6)6 = ((6 u6,6) * r.6 converges a.e. and and pE E in LP to uE for all 2 < p < Nt . In this way, we recover a solution (u, PE) of (6.69)-(6.70)) satisfying in addition (6.80) and (6.81). More precisely, we have in the sense of distributions (while (6.70) holds in a straightforward weak sense) both (6.69) and
uE, PE, PE/2 E H1 (so that pC,b converges to
hu + pu- Du+ 2div(Vp0 u) -2-VP - Vu - µ0u - tV dE u + aVp'r = pf + g apu + div(pu ® u) - 2div(Vp (D u) +
2(Vp
V)u
(6.82)
- µ0u-lVdivu+aVp7 = pf +g. The passage to the limit is straightforward except for the terms (p u6.V )u, Vp.Vu and (Vo7)b which require a few explanations: first of all, we observe that PE,6 is
Existence proofs
91
bounded in L9 where q = N 2 If > N while (uE,6)6 is bounded in H1 since, in the case of Dirichlet boundary conditions, D(uE,6)6 = D(6 uE,6 + (6DuE,6 and J D(61 luc,6I < 6 1(26>d>6) IuE,6I < C "d61, denoting d = dist(x, 8ft). Therefore, pE,6(uE,6)6 is bounded in LP for some p > 2 and thus converges to pEuE
in P. In addition, from the equation satisfied by pE,6i we deduce that V PE,6 is bounded in LP for some p > 2 ( p = a - N + 9 = 2 - N + N-2) Furthermore, EOPE,6 = apE,6 - h + (u6,,6) VPE,6 + pE,6 div(uE,6)6 is bounded in L' where = 1- 2 + q and thus V p,,6 converges strongly to VpE as S goes to 0+ in L2 (and in L" for all 2 < p' < p). We also deduce, by the way, that VpE E L' and Ape E L''. Before letting e go to 0+, we wish to make a few observations: first of all, we always have, by an easy passage to the limit, 2hIuI2
1
+ 2apIuI2+pIDuI2+ S(divu)2+ a"l (apy - hp ') 4ea
+
IVp7/2I2-
pu f
(6.83)
0.
Next, if N = 4 or N = 5, we claim that we also have if ry > 2 a7
(h + ape) IuE,2 + 2
+ div [u,
PEI 2
7-1 (apE - hpE -i)
+-
1 pE
1
+ pIDu6I2 +((divu.)2 - E + E a'Y pr,-2 I Vp I2
a
7-1
- Y1 2I2 - ( div(u6 div
6.84)
(pE-1VpE)
div g
PEuE ' f + uE
uE)
in 11.
The inequality is easily deduced from (6.79) using the preceding bounds and convergences for pE,6, V pE,6 and uE,6: for instance, uE,6IuE,6I2 converges to uE IuE I2
in La for a < 3 N 2 while p,,6 converges to pE in Lb for b < N2 and 3(N-2)
< 2N
3(N-2
+
= 1.
Similarly, pE 61VPE,6
= pe,62
Y-T
[v(p2)J
+
and
PE 12 while V (P'/2) b2) converges weakly in L2 pe'.2 converges strongly in L2 (say) to to V(pE/2) (in fact, it converges strongly). Two terms require some explanation, namely piDuE,6I2 + ((div UE,6)2 (since we only assume that p > 0 and p + ( > 0)
and uE,6 (Vp5)
6
- (uE,6)6 V pE6. The latter converges to 0 in the sense of
distributions since we have for all cp E Co (S2) fixed and for S small enough Jin
Rb6
st xcl
(Vp',5)5
-(
7
dx
dxdy u6,6(x) VPE,6(y) rc6(x - y) [W(X)
- P(y)I
Stationary problems
92
IIlX0
dxdy uE,b(x)pE,6(y) Vy {rcb(x
- y) [W(x) - V(y)J}
and we conclude easily since uE,b converges to uE in LP for p < NN2, e,,6 converges to pE for q < N2 and since we can write Dy {rc6(x
- y) I WW - W(Y)II
8
_ where I roI < C x
-
x
y b
VV(X) - (x
- y)
N rc (x_Y)v V + r,6
+, J -vrc() V ,(x)
10rc
+1
ptDuEI2 +l;(divuE)2. This is straightforward since we know that fnplDvl2 + t;(div u)2dx is a positive quadratic form on Ho and thus is convex. Hence, we have for all cp E C0 li minf
J
pI D(cpuE,s)f 2 + (div uE,6)2 dx > fn J2 D(W%,)12 + (div(cpu6))2 dx
from which we deduce easily using the strong convergence in L2 of uE,b lim inf
Jn
1W2I DuE,6 I2 + Ecp2(div uE,b)2dx
=
Jn
W dm
fco2(izIDu2+(divu)2)
,
and we conclude the proof of (6.84). We may now let E go to 0+ and we follow the proof made in step 3. We prove exactly in the same way that pE and EDp, are bounded respectively in L9 and
L" where q = N 2 (-y - 1) and .1 = 4 + zN ; notice that all the computations LN/(N-2)-r. made in step 3 are easily justified since pE E Let us also observe that the convergence (6.75) is ensured by the inequality (6.81). The rest of the proof
is then exactly the same, and we obtain a solution of (6.1)-(6.2) satisfying all the properties listed in Theorem 6.1. Let us conclude this part of the proof by showing that (6.14) (the energy inequality) holds if N = 4 or 5 and -y > N . This fact follows from (6.84) upon letting a go to 0+. We only have to explain how one can pass to the limit in the
Existence proofs
93
terms pEt I uE I2 or uE pE and why c pE -' VpE converges to 0 in L' (say). The last point is easy once we observe that E
E P7-1 IVPEI = /2-11I
1
E
PC
IVPEI
PE/2
pE and OPE is bounded in L2 in view of (6.80) while p7/2 is bounded and a > 2 since 7 > . We could also observe that in La where a = NNZ E pE -1 VpE _ V (, pE) converges to 0 in the sense of distributions. Next, in order to pass to the limit in pEu( (u6 12 and uEPE , we recall that we already obtained (up to the extraction of subsequences) that uE converges to u in Lb for 1 < b < while pE and pE converge respectively to p and p7 in Lc for 1 < c < N2 of - 1) and in Ld for 1 < d < N 2 ryry 1. We then conclude easily since I
z
3
N-2 2N
N-2
1
N y-1
+
N-2 N-2 2N +
ry
N -y-1
N.
2N -21 (3ry-1)
6 At this stage it only remains to prove Theorem 6.1 in the cases when 1 < both ry 0 and p given by
p = ap'r + Epm
(6.86)
whereEE(0,1], m=2ifN=2or3, m>N-1 ifN>4. Then it is not difficult to check that the preceding proofs yield a solution (PE, uE) of (6.85) (periodic in the periodic case, such that uE E Ho (SZ) in the case of Dirichlet boundary conditions) satisfying: PE E LN/(N-2)(m-1) if N > 3, pE E
L2m if N = 2, uE E H', D (div uE - 111 p, and D curl uE E L'(K) where r = 1- N + N-2 $m-1) and K = 12 in the periodic case while K is an arbitrary compact set included in fl in the case of Dirichlet boundary conditions, and we have
Stationary problems
94
n
2(h+ap6)IuEI2 +
a ,y
y1(apE -hpE-1) m 1(ap - hpm-1)
+
+m
t; (div uE )2 dx
3 and thus q' > q, m > r. We then deduce easily from this inequality that pE is bounded in L4 and that pE" is bounded in L'. We now wish to obtain as in section 6.3 (steps 2-3) bounds on D(div uE 7+-,-p,) and on D(curl tion 6.3 and write
Therefore, we follow the argument introduced in sec-
-V [(p + e)div uE - pE] + curl (µ curl uE) = H1 - div(p6u6 0 uE)
where H1 is bounded in 7-(' where s < + if N = 2, s = 1 - N +
7 1 if
N
N > 3 and where we agree that W = L3 if s > 1; in fact, H, is even bounded in
La for some a > 1. Next, we claim that div(p6u6(&u6) is also bounded in 7-(s. This was shown (at least formally, that is assuming pE, uE have smooth) in section 6.3 and we give a short proof of it. Decomposing pEuE into pEuE = wE + zE where
div wE = 0, curl zE = 0 (and wE n = zE n = 0 on On in the case of Dirichlet boundary conditions, while wE and zE are periodic in the periodic case), we write div(p6u6 (&
uE) = div(w6u6) +
Then, div(w6u6) is bounded in fl3 in view of the results of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes [110] while zE is bounded in W 1,q and thus div(z6u6) is even more regular. We then deduce as we did in section 6.3 (steps
Existence proofs
95
D(curl u,) are bounded in %,'(K) where K = SZ 2-3) that D(div uE - µ+E in the periodic case and K is an arbitrary relatively compact smooth open set such that K C 11 in the case of Dirichiet boundary conditions. In particular, we are now in a position to apply the compactness results shown
in section 6.4. Let us only observe that (6.56) holds (extracting a subsequence if necessary) because of the bound we just proved and since epE' converges to 0 in L' for all 1 < r' < r as e goes to 0+. Therefore, extracting subsequences if for all necessary, we may assume that pE converges to some p > 0 E L9 in 1 < q' < q , uE converges weakly in Hi and strongly in L" for all 2 < p < NN-2 to some u as e goes to 0+; and we easily recover a solution of (6.1)-(6.2) which satisfies all the properties listed in Theorem 6.1. In the case when 11 # RN, the only information left to check is the energy inequality (6.14) when N = 3, 3 < 7 < 2, and we easily deduce it from (6.88) upon letting a go to 0+ provided we show that euepE' converges to 0 in L'. Indeed, u, is bounded in L6 , epE' converges to 0 in La for all 1 < a < 3 yry 1 L9'
and 6+3y-1) =firy-1 6 where < x >= (1 + 1x12)1/2 exactly as we did in section 6.3 (step 1). Furthermore, following the proofs made in section 6.3 (steps 2-4), we obtain bounds on pn in Lq (Bn) except in the case when N = 2 where we obtain bounds in L2,t(BR) for each fixed R E [1, oo) and in L27 + L7(Bn) for any q E (2, +oo] if (6.11) holds. The fact that these bounds are uniform in n is due to the fact that in (6.32) the measure of 12 = Bn appears with an exponent equal to 1 - 1 < 0.
In addition, we also obtain bounds in f-lt (Bn-1) for D(div un -+ pn) and D(curl un) if N > 3 and in fl'' (BR) for each fixed R E [1, oo) if N = 2. We may then pass to the limit as n goes to +oo using the compactness result shown in section 6.4. The details of this passage to the limit are very similar (and in fact somewhat simpler) than those made in the preceding step (step 5) and we thus leave them to the reader. At this point, we have completed the proof of Theorem 6.1. O The proofs of Theorems 6.1-3 are now almost complete: it only remains to prove the claim made in Theorem 6.3 about the case N = 3, y = 3. More precisely, we wish to prove there exists a solution of (6.1)-(6.2) satisfying (6.12)-
(6.22) and (6.23) when N = 3, -y = 3, f E Lq with q > 2 , g E LP with p > 6/5, h E L'' (n L' if SZ = RN) with r > 3. One possible argument consists in approximating the problem by considering 'yn > 3 and letting -fn go to 3. In view of the results already shown and their proofs, we obtain solutions (pn, un) corresponding to the exponent N satisfying the following bounds uniform in n: Pn > 0, pn is bounded in L''" (Q) (nL' if n = RN ); un is bounded in Hi (SZ) if SZ # RN; Dun is bounded in L2 and un is bounded in L6 if ft = RN; D(div un µ+£ pnn) and D curl un are bounded in L615(w) where w = 0 in the periodic case, w = RN if I = RN and w is an arbitrary smooth open set such that Co C S2 in the case of Dirichlet boundary conditions. In addition, we have
f
t
h lul l
+pn 2 + alyn (apn" 'yn - 1
hpnn-i)
1
(6.91)
+ pl Dunl2 + t; (div un)2 dx = fPflUfl.f+Ufl.9dX}
a7'n
'(h + apn)lunl2 -{"''n div fun Pn
2
Iu2 l
+
l (aPn7n - h
-fn
a - 1Pnn
7n
11
P7nn-i)
- A lug l
2
(6.92)
+ µl Dun l2
- S div(un div un)2 + (div un)2 = Pnun f + un g
in SZ.
The fact that these identities hold (instead of the mere inequalities proved above) has been shown in section 6.2.
The isothermal case in two dimensions
97
We may then pass to the limit and recover a solution (p, u) as n goes to +oo and yn goes to 3+ : the proof is very similar to proofs we have already made several times before. In order to prove the regularity statements contained in Theorem 6.3, we remark that, in view of the proof made in section 6.3 (step 8), we just have to show that un converges to u in L6. Notice that we only know a priori that un (and pn) converges to u (and p respectively) in LP for p < 6 (L oc
ifSZ=RN). One possible proof consists in using the result shown in Appendix C since IunI2 is bounded in L3, while undivun , pnunlun and unpnn are bounded in a-In (apnn hpn^-1) converges L3/2 and pnun . f + un g - z (h+ap)Iun12 - 7n-1
-
weakly in L' to pu f +u g - (h + ap) IuI2 - a (ap3 - hp2), and (6.92) allows us to apply Appendix C.
z
2
It is also possible to use the energy identities satisfied by un and by u to deduce the strong convergence of Du in L2 and we shall only sketch the argument: indeed, passing to the limit in (6.91), we deduce that we have
pIDunl2+e(div un)2 dx = n 12
lim
inpu- f
22 -ap
122
- 32 (ap3-hp2) dx.
On the other hand, it was shown in section 6.2 that we also have
fiIDuI2+(divu)2dx = fPtL.f+u.g_hi!- -apt
2
-
(ap3-hp2)dx.
Comparing these two equalities easily yields the strong convergence in L2 of Dun and thus the convergence in Ls of un.
6.6 The isothermal case in two dimensions We consider in this section the case when y = 1 which corresponds "physically" to an isothermal situation. In other words, we consider solutions of
ap + div(pu) = h , p > 0
(6.93)
apu+div(pu®u)-piu-CVdivu+aVp = pf+g
(6.94)
where a > 0; a > 0; h, f and g are given and h > 0, h
0; p > 0, p + > 0.
The only situation we can solve is the two-dimensional case and therefore we assume throughout this section that N = 2. In order to simplify the presentation, we begin with the cases when the equations (6.93) and (6.94) are set in a bounded domain, that is we restrict ourselves to the periodic case (in which case, the equations hold in R2 and all functions are required to be periodic as usual) or to the case of Dirichlet boundary conditions (in which case, the equations hold in a bounded open smooth domain 1 in R2 and we request that u vanishes on 81). Later on we shall treat the whole space (R2) case. Let us first make precise what we mean by solutions of (6.93)-(6.94): we look for (p, u) solving (6.93)-(6.94) in the sense of distributions and such
Stationary problems
98
that p E L', pLogp E L' and u E H1 (Ho in the case of Dirichlet boundary conditions, H1 ,(R2) in the periodic case). Let us observe that these conditions imply that pIuI2 E L1 since we have
pIuI2 < A p Log p + A exp
(2i)
for all A > 0,
using the straightforward inequaliy valid for all A > 0, a, b > 0
ab < A a Log a + A exp
b -1
(6.95)
.
Then, we see that pIuI2 E L' since edI"I2 < oo for some c > 0 if u E H1-in fact, the inequality is true for all c but without bounds-see J. Moser [400] for more details on such facts.
Let us also mention that if p E V (p > 0) then p Log p E L' if and only if p Log p 1(p>K) E L' for any K < oo since we have obviously for some positive constant C depending on K
I
plLogpl 1(P 0 such that fn eco I"1' dx is bounded if v E H1-or Ho in the case of Dirichlet boundary conditions-and IIvIIHI < 1).
Theorem 6.5 Under the assumptions (6.96)-(6.98), there exists a solution (p, u) of (6.93)-(6.94) satisfying in addition fo ap - h dx = 0, h 1(p=o) = 0 a.e. and if exp cl f I2) E L1 for all c > 0
inn
(h+ap)I uF +a(apLogp-hLogp)+pIDul2 2 +C(divu)2-pu f -
(6.99)
0.
The isothermal case in two dimensions
99
Remark 6.12 Let us make precise the meaning of the integral contained in the inequality (6.99). First of all, -h Log p > 0 if p < 1 hence this term makes sense since
h Logp 1(p>1) < h Logh 1(p>1) + e p 1(p>1) E V. In particular, (6.99) yields the following bound
h ILog PI E V. a9i where 90,
axe
i=1
2
Ugo + E
a
Of course, if g E L1 and glLoglgl
1 gi 1112
Finally, pu f E L1 since pIuI2 and
E L2 ,4.b e n
91, g2
2
9i =1
=1
and fn u . g dx = fn u - go - E
(6.100)
au aau t
az dx (an obviously meaningful integral).
E L1 then u g E V. pi f 12
E L1-the latter being true because
of the condition (6.97) on f . 0
Proof of Theorem 6.5. The proof is divided into several steps. First, we approximate (6.93)-(6.94) replacing aVp by aVp1' for y > 1 (and letting y go to 1+). Next, we collect a priori bounds and prove that p , plLog pl, p1 ul2 , hlul2 are bounded in L1 and that u is bounded in H1. Then, we pass to the limit as y goes to 1+ : this is the heart of the matter and in this case the only difficulty lies with the term pu ® u. The strategy of proof will be to use the concentrationcompactness method of P.-L. Lions [348] and to prove that pu ® u passes to the limit except for a "defect measure" which is purely atomic. We then show that such a defect measure cannot be present in the equation (6.94). The final step of our proof consists in deriving (6.99) from the corresponding energy inequality satisfied by approximated solutions.
Step 1: Approximation. As explained above, we consider an approximated system of equations, namely
aPn + div(pnun) = hn, P > 0, aPnun + div(pnun 0 un) - p1un - V div un + aV pn = pnln + gn where y = 1 + n for instance-we simply write y instead of yn in order to simplify notation. The data hn, fn, gn are convenient approximations of h, f, g which belong to L°° and are defined as follows: fn = f 1(ifk no), and no is large enough so that h # 0, we deduce from (6.95) and (6.97) a priori bounds uniform in n on un in H1 , pn and pn Log Pn in L1 provided we show that we have for all 6E (0, 1) 1)
(Pn-
1 1
J
a
dx > a9
h
Log
C
1
where C denotes here and below various positive constants which are independent
of n (but may depend upon 9). Indeed, we just need to observe that the following list of inequalities holds:
fn
(pn
- 1) (apn - hn)
> 9a
n Pn
/////
1(Pn >
max(a
Pn-1 -1 -1
-g',1))
1-
-Y
> Oa f Pn Log Pn 1(Pn>max(_ i2
fPnLoPn 1
1(1 -C
hn
fn ' -1
all - e)
and finally
-1
r
1(hn>a(1-A)) dx ry-1
hn Loga lhn
6
> -C fnh(ILoghI +1) 1
7-1
hn,
101
7-1
(a(1 - 6 Rn-1
1(hn>_a(1-A)) dx
dx > -C,
-1) (apn - hn)
fj
1)
dx > 0 ,
- 1) (aPn - hn) 1(h apn - fn aPn ILogpnI 1(Pn -C.
In conclusion, we have shown that un is bounded in H1, and that pn and pn Log pn are bounded in L1. Let us also observe that these bounds allow us to deduce from (6.101) hn
C(-y -1) > Jn pn -pn-1 d2 > fn en J
>pn (L)'
1 (Ln) ry
-,
- 'Y-1 Pn d2
dx.
Since hn < hn Rn ' , we see that On is bounded in V.
Step 3: Convergence and defect measures. Without loss of generality, extracting subsequences if necessary, we may assume that un, pn converge weakly respectively in H1, Ll to some u E Hl , p E L' such that p E L1 and p Log p E L' (and p, u are periodic in the periodic case while u E Ho in the case of Dirichlet boundary conditions). In addition, we may assume that un converges a.e. to u and in LP for all 1 < p < oo.
We want to pass to the limit in the equations satisfied by pn and un and thus we have to analyse the non-linear terms pnun, Pn, Pnfn and Pnun ®un Only the latter creates serious difficulties which we shall address later. We begin with pnun and pn fn: we claim that PnUn and pn fn converge weakly in Ll to p u
and p f respectively; indeed, we have by Egorov's theorem for all e E (0,1), a measurable set E such that meas (E) < e , u, f are bounded on E' and un, fn converge uniformly on EC respectively to u, f . Hence, we may write Pnun - Pu = (Pnun - Pu)1 E + l E pn (un - u) + (Pn - P) 1.-U Pnfn - Pf = (Pnfn - Pf)1E + lEcPn(fn - f) + (Pn - P) 1Edf
and we see that the last terms converge weakly in L1 to 0. The second terms converges in L' to 0 since we have
Stationary problems
102
fEC Pnlun - uI+pnlfn - fldx
< EP[Iun-ul+Ifn-fll
0
Finally, we claim that the first terms can be made arbitrarily small in L1, uniformly in n, by taking a small enough: indeed, pulE and pf 1E converge to 0 in Ll as a goes to 0+ and we have, in view of (6.95), for all A E (0,1)
e" + e I"dx
CA+A
f(PnIUnl+PnIfnI) 1E dx
E
r
< C (A +A e1/2
< C (,\
1/2
2 eI[..I
dx
n
e1/2
+
C(A))
and we conclude taking A > 0 small and then e > 0 small enough. Next, we claim that pn converges to p weakly (in the sense of distributions or in the sense of measures). Indeed, without loss of generality, we may assume that pn converges weakly in the sense of measures to a bounded non-negative measure ir and we have by the convexity of the function ([0, oo) t H V)
(Pn)' >- 1 +'y(Pn - 1) a.e. Hence, ij(pn)1dx > p. On the other hand, we have seen at the end of step 2 that we have
1
fPfll(>K)dx < Jn pn dx - if Pn1(pn 0 converge weakly in L' to p, and let un, vn converge weakly in H1 to u, v respectively. We assume that pn log pn is bounded in L1. Extracting a subsequence if necessary, we may assume that Pnunvn converge weakly in the sense of measures (on f) to a bounded measure v. Then, there exists an at most countable set I (possibly empty), distinct points (xi)iE1 C Il, and constants (vi)iEI C R - {0} such that
v = puv+Evibyi
,
E IV,1112
< 00.
(6.103)
Step 4: Elimination of the Dirac masses. In view of the facts shown in step 3 and of Lemma 6.1, we deduce the existence of an at most countable set I (possibly empty), distinct points xi E l (i E I) and non-negative symmetric 2 x 2 matrices vi (vi # 0) such that we have
ap + div(pu) = h ,
-u-e0 div u+aVp = pf +g.
c pu+div iEI
Let us recall that p, pu, pu 0 u, h, pf E L', g E H-1, and Du E L2. We are going to deduce from (6.104) that vi = 0 or in other words that I = 0. Let us argue by contradiction and consider some point xio E SZ for some io E I. The matrix vio is non-negative and symmetric and thus if vio # 0 we know that (vi. )11 or (via )22 is strictly positive. Let us consider for instance the case
Stationary problems
104
when (vi. )11 is strictly positive, the other case being treated similarly. Making a translation, we can assume without loss of generality that xio = 0. Let b > 0 be such that [-b, b] 2 C 11 and let W E Co ((-S, b)) be such that: 0 < w < 1, and ]. We then multiply the equation satisfied by u1, contained in V = 1 on 2 the system (6.104), by cp(x2)xl cc (E) for C E (0,1) and we obtain easily
J (Pul + ap) V (El) W(X2) dx + E(vi)11 W
(()i) cp((xi)2) + (Vio)11
i
< f F cp(x) xico (e1 1 dx + +,/
G co(x2)
- E(Vi)11
[w (C1 /
(xi-) W1
+
J
-
(E)1
cp'(x2)x1
W, (e1 / J
W
(e1) dm2
dx - J (pU2 + ap) E W/ (E)ca(x2)dx
V ((xi)2)
iEI
where F E L1(fl), G E L2(SI), and m2 is a bounded measure. Since pui +ap > 0, cp > 0 and (vi) 11 > 0, we deduce from this inequality (Vi.)11 < Ce + Cf + J(pu2 + ap) 1(Ixil e-p'6 a.e. Therefore, we obtain for all e, S E (0,1)
I
Jhe-P/6dx < Ce+
C
+e-E"6
log e
I
and choosing S = e2, we deduce letting a go to 0+ the equality fnh1(p=o)dx=0. It only remains to show (6.99) which we wish to deduce from (6.101) or equivalently from
in
-
2(hn+c pn)IunI2 +a ryI [a(Pn-Pn) hn(Pn-1-1)] + plDunl2 + (div un)2 - Pnunfn - un gn dx = 0.
(6.106)
We already know that we have ll nm fn pI Dun 12 + (div un)2 dx > j pIDul2 + e(div u)2 dx and
fun.gn-*ju.gdx
fpnIUnI2dX4jpIUI2dX.
,
In addition, by Fatou's lemma, we immediately obtain hnlunl2
lim n
n
dx > [hItLIdX c1
(in fact, it is possible to show that hnlunl2 converges in L' to hIuI2 as n goes to +oo)
We also claim that fa pnun fn dx converges to fn pu f dx; indeed, we prove exactly as we did for pnun that pnun fn 1(Ifnl <M) converges weakly in L' to pu f 1(I fkM) for all M E (0, oo). Then, our claim is shown provided we prove
that sup f I Pnun . Al 1(If..I>M) dx - 0 n
as M - +oo .
Indeed, we have for all e > 0 using (6.95) with A = 62
In
IPnun . fnI 1(Ifnl>_M) dx
_M) dx
M)
dx
Stationary problems
106
IfnI2
+2 f0 exp
-1
E2
1(Ifnl>M) dX 1/2
< Cc + C
r e2lfl2/e2 dX
meas(I f I ? M)
n
< CE + C(e)e-M j elfI dx st
and our claim is shown. At this stage, we see that (6.99) follows from (6.106) provided we show the following inequality linm ry 1 1
(6.107)
and « with hn and h re-
Notice that we have changed notation, replacing spectively.
We begin by proving (6.107) when h E LOO in which case hn = h for n large enough. Then, we observe that the function ([0, oo) E) t - ,x,11 [t"-t-h(t7-1-1)]) is convex and thus we have for any function cp E L°O (11) such that infess cp > 0
y11 [p;n'-Pn-h(Pn-1-1)] >
1
cpry-1
-
1
- hurry-2
-Y
'Y
(Pn - (p)
a.e.
}
We easily deduce from this inequality that the following holds lnm
>
1
7-1 f Pn-Pn-hn(Pn-1-1)dX Jn
V log W - h log V + 1 + log cp -
(p-cc) dx.
Next, for e E (0,1), M E (1, oo), we choose cp = min (max(p, e), M) and we see that the right-hand side is equal to
f(Pve) log(p v e) - h log(p V c) dx + J [M log M - p log p] 1(p>M) dx n
+
1(p<E)
(1+loge_ h
+ n 1(p>M)
(p-e) dx
(1+logM_)(P_M)dx
denoting a V b = max(a, b). Since p (1 + I log pi) E L1, it is easy to check that the
second and the fourth integral go to 0 as M goes to +oo and thus, letting M goes to +oo, we are left with
f(P V ) log(pve)-hlogpvedx+n J 1<E1+loge-
(p-e)dx.
107
The isothermal case in two dimensions
Then, we can bound the second integral by 1(p<E) [E(1 +I logEI) + h dx] ,
an expression that clearly converges to 0 as e goes to 0+ since h 1(p=o) = 0 a.e. Also, (p V e) log(p V e) converges in L1 to p log p while -h log(p V E) increases to -h log p as e decreases to 0+. Therefore, the first integral converges to fn p log p - h log p dx as c goes to 0+ and (6.107) follows in the case when h is bounded. Next, in the general case, we observe that we have for all M > 1 and for n large enough
fn 'Y >
1 1(Pn-11) (Pn - hn) dz
f
n 1(h<M)
-1 1
(P7n-1-1)(Pn-hn)d2
1
(Pn-1-1)(Pn-hn) + in 1(h>M) 1(1M) ,
1 1(hn-1-1)hn dx
1(Pn-1-1)(Pn -hn) dx - f 1(h>M) (log hn)hn dx
In
(Pry,,-1-1)(Pn-hn)dX
> in
1(h>M)
logh hdx
Rn-1.
Using the above proof on the set SZ fl {h < M} (on which h is bounded!), we deduce lim n
1
St'Y-
1
(pn-1-1) (pn-hn)d2x
fn 1(h<M) [P log p - h log p] dx - fo 1(h>M) log h h dx = fn 1(h<M) P log P - i(h<M) h log p 1(p? 1) -h 109P 1(h<M) 1(pM) h log h dx .
We thus conclude easily letting M go to +oo since p log p, h log h, h log p 1(p>1)
belong to L' and [-h log p 1(h<M) 1(p 0 in view of the estimates obtained by J. Moser [400]). We then conclude letting M go to +oo, then c go to 0+ and finally S go to 0+.
We may now begin the proof of the lemma in the case when un - vn, u - 0. Also, we only treat the case when un E Ho (fI), the general case following by a simple extension argument for instance. First of all, extracting subsequences if necessary, we may assume that J Dun 12 andpn log+ pn converge weakly in the sense of measures to some bounded (on S1) non-negative measures µ, in respectively, where we denote log+ t = log(max(t, 1)) = max(log t, 0) (V t > 0). Next, we claim that we have for all Sc E C(?!), cp > 0
J
cp3 dv < 8 (fcc2 dµ
dm
(6.108)
.
Admitting temporarily this claim, we conclude easily. Indeed, we deduce easily from (6.108) that we have for all cp E C(SZ), cp > 0 nL
J cp3 dv
cp > 0,
cp(xi) = 1 (for i fixed in I) and letting S go to 0
Vi < $1 µ({2i}) m({2i}) . Hence, we have for all i E I 1/2
1 8x
0,e>0
ab < e a log+ a + min(b, E)
e(b-,)+/e
and we deduce for all e < 0
f
n
(P Pn (
n)2 dx < e fn SP pn log+ Pn d2
(6.109)
+ f- go min((goun)2, e) e((SPun)2-E)+/E dx .
Next, we observe that we have
f I0(cpan)12 dx = fn
21 Dun12
+ 2goVgo Vv,, un dx + IOcp12 un dx
J
(P2 dFi
since un converges a.e. and in L2 (for instance) to 0.
Therefore, if we choose e > co where co = 8 f V2 dµ, we deduce from the above convergence and J. Moser's sharp inequality [400] that e(Wun)2/e is bounded in La for some a > 1. Since un converges to 0 a.e., we deduce easily
that
1 go min((
n)2, E)
e((Wun)'-a)+/E
dx _4 0 . n
Hence, passing to the limit as n goes to +oo in (6.109) yields
f lp3 dv < e
r Vdm,
and (6.108) follows upon letting c goes to co. The proof of the lemma is complete.
0
Stationary problems
110
Another possible proof of Lemma 6.1 (still based on the results of P.-L. Lions [348]) is to apply the following lemma that we state, in order to simplify the presentation, in the case when un E Ha (SZ) converges weakly to 0 in Ho l. Let us recall that, in view of the preliminary reductions made in the beginning of the above proof of Lemma 6.1, this particular case is sufficient to obtain Lemma 6.1 in its full generality.
Lemma 6.2 Let SE be a bounded open set in R2, let un E H0 (1) converge weakly to 0 in Ho'. Then, there exists a set I C Il which is at most countable and possibly empty such that we have (u^)2 converges to 1 in L"(f) for all c > 0, p E [1, oo), (i) if I = 0, then a= ("^)2 converges (ii) if I is finite, i.e. I = {x1, ... , x,,,,} C SZ, then, for all b > 0, e < to 1 in LP (St - Um, B(xi, b)) for all e > 0, p E [1, oo), (iii) if I is infinite and I = (xi)i>I C SZ (where all the points xi are distinct),
then there exists, for each k > 1, ek > 0, ek goes to 0 as k goes to +oo such that, for all b > 0, eck (u")2 converges to 1 in LP (c - Ui==1 B(xi, S)) for all pE [1,00).
This lemma immediately implies that v is purely atomic since we have in view of (6.95)
P.(un), < ePn
logPn+ee 1 k
U
- U B (xi, 6)
< C Ek
i=1
and thus, in all cases, we deduce that v is supported on I. Proof of Lemma 6.2. Without loss of generality, we may assume that J Vun 12 converges weakly in the sense of measures to some bounded non-negative measure
p on SZ. We denote by I the set of atoms of j. If I = 0, we use an observation shown in P.-L. Lions [348] to conclude. If I is finite, we take, for each S > 0 fixed, < E COO (RN), 0 < 1, ((X) = 1 if Jxi > b, ((x) = 0 if Ixi < b/2. Then, we consider un = 11m 1 C (x - xi)un and we observe that we have IDunI2 (fj; ` 1((x - xi)) u = µ weakly in the sense of bounded measures on S2.
n
Since µ has no atoms, we deduce as above that e+(u )2 converges to 1 in L1'(Sl) for all 1 < p < oo and (ii) is shown.
Finally, if I is infinite, we consider, for each k > 1 fixed and for each b > 0 fixed, un = (n1 ((x-xi)) un; and IVun12 converges weakly to µk = ((x-xi)) µ Obviously, itk < µ, the measure of the largest atom of µk is less than or equal to maxi>k+1 µ({xi}) = bk, and bk goes to 0+ as k goes to 11k
i=1
The isothermal case in two dimensions
111
+oo. We conclude easily using the fact, shown in P.-L. Lions [348], that ei Iunl2 is bounded in L1(11) if e > s . 0 We conclude this section by deducing from Theorem 6.5 an analogous result in the case when !Q = R2. In all the rest of this section, we therefore assume that 2 = R2 and we request that the following conditions hold
hEL1, hloghEL1, hlxl8EL1 for some 6E(0,1], h#0, h>0 VK>0
,
3 9 E (0, 2acY2)
,
(6.110) (6.111)
1(IfI>K) exp(If 12/9) E L'(R 2)
3C>0, VuECr (R2),
feudx
< C (L2 IVul 2+ hlul +2 1B1 I(I2 dx
(6.112)
1/2
The latter condition holds as soon as g = g1 +g2 where 9,(y)-1 is bounded on R2 and 92 satisfies for some m E (1, 2] JR2
1921' <x>2(m-1) [log(<x> +1)]0 dx < oo;
denoting <x>=
with
6 > 2m - 1
(1+Ix12)1/2. Indeed, (6.112) then follows from the results shown
in part I [355], Appendix B. Of course, we may even assume that g2 may be split into several pieces each of which satisfies the previous assumption for some m. Let us also observe that we may discard 1B1 Iu12 in (6.112) without loss of generality: indeed, let Ro E (1, oo) be such that h # 0 on B&, then (fBRO IVU12+hIuI2 dx)1/2 is an equivalent norm to the usual H1 norm on H1(Bp, ). This fact is easily shown since fBR hIuI2 dx is a continuous quadratic form on H1(B,) in view of (6.110). We may now state our main result
Theorem 6.6 Under the conditions (6.110)-(6.112), there exists a solution (p, u) of (6.93)-(6.94). 0
Proof. We approximate (6.93)-(6.94) by the same system of equations set in a ball BR (for R large enough so that h # 0 on BR) with Dirichlet boundary conditions. Using Theorem 6.5 we obtain a solution (PR, UR) satisfying the conditions listed in Theorem 6.5. In view of steps 3 and 4 of the proof of Theorem
6.5, we just have to obtain a priori estimates on PR, PR log PR, hIuRI2 in L' and DUR in L2. Notice that such bounds imply, as seen above, bounds on uR in D1,2(R2). Since we are going to deduce these bounds from (6.99), we also approximate f by fR = f 1(If1e_isi6) dx +
r PRIuRI2 + 4 J BR 2 P I
PR(-logPR)1(pR<e-1=I6) dx
JBR'BR
BR
K)dz
1/2
+CJ +a
JBR
PR 1(pR<e-i=i6)dX
2PRIURI2+4hIuRI2+2IDuRI2+ bPR1og+PRdx
+C
1+f
PR I x I6 dx R
for some v > 0 independent of R, where C denotes various positive constants independent of R, PR and uR, and where b E (20 , 1). In the above string of inequalities, we have used conditions (6.110)-(6.112) and the inequality (6.95). In particular, we obtain JBRPRIuRI2
+ hIuRI2 + IDuRI2 + PR I log PRI
-h log PR 1(pR 0, the coercivity follows from the inequalities shown in section 6.3, upon observing that if u E Ho (11) N and (d - N div u) = 0 a.e. in 11, then u - 0 in SZ: indeed, as shown in section 6.3, each component of u, i.e. U1, U2, ... , UN, solves a linear second-order elliptic equation in 1. Similarly, when e > N 2 µ, if u E H1(SZ)N and div u = d = 0 a.e. in fl, then, as is classical in continuum mechanics, one can check easily that u = Qx + uo for some uo E RN and for some N x N antisymmetrical matrix Q. Therefore, u - 0 and we conclude.
Proof of Theorem 6.7. Step 1: A priori bounds. We are going to obtain the solution (p, u) that we seek by approximating (6.3) by the system (6.1)-(6.2)-with Dirichlet boundary conditions-with h = am meas (ci)-1 (in fact any h > 0 such that fn h dx = M could be chosen). We may now apply Theorems 6.1-6.3 and we obtain a solution (pa, ua) of (6.1)-(6.2) satisfying all the properties listed in Theorems 6.1-6.3 and
M Jc2ameas(SZ)
IuaI2 2
+apa IuaI2 2
a-ya I 7 _ pry-1 11'r 7-1 +-y - 1 meal 1l LPa
+µlDuo, l2+C(divua)2dx < J Paua f +ua gdx,
(6.119)
}
n
L pa dx = M
(6.120)
(since a fn pa dx = fn h dx = aM). We are going to obtain a priori estimates on (pa, ua), uniform in a E (0, 1]. First of all, we deduce from (6.119) and Sobolev embeddings VIItiaIIHo 0, where C denotes various positive constants independent of a and Co denotes various positive constants independent of a and of 11 f 11 Loo (n)
Stationary problems
116
and where L2N/(N+2) is to be replaced, here and in all that follows, when N = 2 by L" for an arbitrary p E (1, ry). The above inequality allows us to deduce IIuaIIH1
< C +CO IIfIIL°°
IIPaIIL2N/(N+2)
.
(6.121)
Next, we argue as in step 3 of section 6.3 and deduce from (6.2) IIPa)ILr s C+CIIPaIIL2N/(N+2) +COIIPaIua12IILr +C
IIP«IIL
C + C II Pa II L2N/(N+2) + COIIPaJIL-rr IIUaII i + CIIPaIIL7
if N > 4 or N = 3, y < 3, and r = 2 if N = 2 or while N = 3, 'y > 3. Let us observe indeed that -L + N 2 = r if r = N 2 where r =
yry 1
N-""i
3.
z-f+
Combining this inequality with (6.121), we deduce IIPatILr* 0, then we find for M small enough three solutions given by u = 0, near Yi (i = 1,2) and p3 = [(4 +.1)+ ti-1J1/(i'-1) Pi = [( + Ai)+ near 21 and Z2 Where A,, A2, A > 0 are small and such that fn Pi dx = fn P2 dx = '27
fnp2dx=M.0
We have seen in Example 6.3 that (6.3) has in general no solution in the periodic case. The result which follows (and the methods of proof) shows that the only real obstruction to existence is the lack of coercivity (or control) on "constant flows", i.e. u = constant. We treat in fact two cases, namely the case when f - 0, fn g dx = 0, and the case when we have a dissipative force and more precisely we consider the example of an electromagnetic force with a non-trivial magnetic field. We thus consider in the periodic case the system div (pu) = 0
,
(6.124)
div(pu ® u) -pOu-eV div u+aV'p'' = pf +g+ (u x B) x B
where p, u, f, g, B are periodic and the meaning of (u x B) x B is
-
(uiB2 when N = 2 and the usual exterior products when N = 3. We make the following assumption on (B, f, g) : B, f, g E L°° and we have u2Bi)(B2)
either B is not parallel to u a.e., for all u E RN, u 54 0 ; 1 or B = b(x)u for some b E L°°, periodic, b # 0, u E RN, r
i orB-O, f -OandJgdx=0. u#0, f -0and In
n
I
(6.125)
Stationary problems
118
We then have Theorem 6.8 Under assumption (6.125), the same result for the system (6.124) as Theorem 6.7 for the system (6.3) holds.
Proof. We only have to show that (6.121) (or some appropriate modification) holds under condition (6.125). The rest of the proof is essentially the same. In fact, the only modification is really the existence of approximated solutions that is not granted by the results obtained above. However, showing the existence of solutions of the following approximated system of equations
div (pu) + ap = aMmeas (c)-1
,
.
pf +g+ (u x B) x B
apu+div(pu(&
(6.126)
is a simple adaptation of what we did above in the preceding sections (in the case when B = 0), and we thus skip this part of the proof which presents no new difficulty.
Therefore, it just remains to prove (6.121). The only new term in the energy identity is
j[(uxB)xBJ.udx = -JnIuxB12dx where u x B = u1B2 - u2B1 when N = 2, and we obtain easily the following inequality
j
vJn IVuI2dx+
r
Iu x BI2dx < C
(i+j pf
(6.127)
In the case when B is not parallel to a non-trivial constant vector, one shows easily that (fn vIDuI2 + Iu x BI2 dx)112 is a norm on (Hp.,)N which is equivalent to the usual norm since fn Iu x B 12 dx does not vanish when u is constant (and
u 0- 0). This allows us to copy the argument made in the case of Dirichlet boundary conditions.
In the case when B - 0, then, by assumption, fo g dx = 0 and f = 0 where ' is periodic; therefore, fn gu dx = fn g (u f-n u) dx < CII VUIi L2 (n) and we
-
obtain a bound on Du in L2 (12). In addition, integrating over St the momentum equation in (6.126), we find that fn p u dx = 0. Then, we have
M
Lu
Lu
[uJu] dx
< CI DuIL2 IIPIIL,
< C IIPIIL,
Therefore, we have obtained the following modification of (6.121)
IIuIIH1 < C (1 + IIPIIL,)
(6.128)
where C may depend upon M. Next, we follow the argument in the proof of Theorem 6.7 observing that div [pu 0 u] = div [pu ®
(u - 1 u) I +
(j(o )/
div pu
Stationary problems
=div
pu® u-
f
n
u
+
119
If u n
a
M meal(1)
-ap
We thus deduce the following bound IIPIIi7r
Jin
IDvI2 + I DwI2 + IvI2 dx
Indeed, if this inequality were not true, we would find sequences un, vn, wn such
that Dun and B x un converge in L2 to 0 while fn I vn I2 dx = 1; hence, un converges in H1 to some constant u and thus vn converges to u - (u, u) I uI -2 u whose L2 norm is 1, while B x ii = 0 a.e. and a must be proportionnal to U. The contradiction proves our claim. We then go back to (6.127) and write
J
=C 1+
f
=C 1+ J
n
Therefore, v is bounded in H1 and Dw is bounded in L2; in particular Du is bounded in L2 and (u f-u is bounded in H1. Furthermore, multiplying the
-
momentum equation by u/IuI and integrating over 11, we find that fn pw dx = 0
and thus
M
fwH
JP[WJWJdX
From these bounds, we deduce once again (6.128) and we may follow the argument above.
Remark 6.17 Let us observe that we have shown in fact the existence of a solution which satisfies fn pu dx = 0 if B - f = 0 and fn g dx = 0, or (fn pu dx, 1) =
OifB-b(x)u,uERN,u54 0,bEL°Operiodic, b00, f -0and
The rest of this section is devoted to the study of the same stationary problem (6.3) with a different parametrization from the prescription of the total mass. Namely, we shall use the constraint (6.7) in place of (6.6) for some convenient p. Typically, p will be chosen large enough but it need not be so. In fact, we shall
Stationary problems
120
use this idea in two types of situations: first of all, when y = 3 , N = 3 in which case we choose p E (1, 2); and when y > 0 is arbitrary (for all N > 2) in which case we choose p large enough (p > po(y, N) to be determined later on) and we shall consider "modified Dirichlet boundary conditions". Before we state precisely such existence results, we wish to make a general comment and draw some conclusions. First of all, we are going to prove the existence of a continuum C of solutions containing the trivial solution p = 0,
u = uo and such that for all M E [0, oo) there exists an element (p, u) in C satisfying fn pP dx = M, where uo denotes the solution of
-pAuo - V div uo = g
in St
,
uo = 0 on 00
(6.129)
(or different boundary conditions as we shall see later on). By continuum we mean a closed connected set in an appropriate function
space (typically p E Lq for some q > p, u E H1 for instance). In fact, the proofs below also show that, in Theorem 6.7 (or Theorem 6.8), we also have the existence of a continuum C of solutions, containing the trivial solution (0, uo),
in LP x(W1,q)N(for all p 2, one can check that, in general, there exists a critical mass beyond which no bounded solution exists. Indeed, let us take g 0, f - VD where is Lipschitz (or even C°°) on 1, (D < 0, max.- ( _ D(xo) = 0 for some xo E S1 and (P behaves like -jx - xo1m near xo (m = 1 or 2 for instance). As we have seen several times above, in this case for any solution (p, u) we have u - 0 and aVp"T = pV , pry (and thus p) E W1,°°(S2) if p E L. It is then easy to check that we have pti-1 = (.1)(A -') or p = b(\ - -1>)-1/(1-7) where A > 0 (and b Next, as A goes
to 0_, fn pdx converges to Mo = fn a'f dx < oo if lmf < N. If m = 1, our claim is shown provided y < 1 - N and, if m = 2, we need N > 3 and y < 1 - N . In fact, it is possible to show that no solution other than the ones above (with A < 0) exist with u - 0 and p E V: indeed, we have then pP E W1,1 and a E+P V p 1 = EP V15, hence, letting e go to 0+, we deduce easily
that
Opt'-1 = -l(ogo) 0' a.e. in fl. In particular, pry-1 is Lipschitz, thus
Stationary problems
121
pry- 1 + 4P is constant on Q. Obviously, this constant denoted by A is non-negative and we conclude.
P > 0 a.e. on St and
We may now state our main results.
Theorem 6.9 Let 'y = 3, N = 3 and p E (1, 2). Then, there exists a continuum C of solutions of (6.3) in LP x W1,P for all 1 0, containing (0, uo) and such that, for all M E [0, oo), there exists (p, u) E C satisfying (6.7).
Remark 6.19 Of course, one can obtain a similar result in the periodic case with the same assumptions as those made in Theorem 6.8. The second result we now wish to state is of a slightly different nature. First of all, we consider in this case arbitrary 'y > 0 in arbitrary dimensions (N >_ 2) but we modify the boundary conditions imposing (6.42) (the normal velocity vanishes) and either (6.43), or (6.44) with the conditions > N 2 µ and (6.118) -A.. Those conditions are imposed in order to or (6.45) with the condition enforce "enough coercivity" as explained in section 6.3 and in Remark 6.15. We shall not recall below these assumptions and conditions.
Theorem 6.10 Let N = 2 or N = 3, y > 0. Under the above conditions and if p is large enough (depending only upon N and -y), there exists a continuum of solutions in Lq x W 1,q (V q < oo), containing (0, uo) and such that, for all M E [0, oo), there exists (p, u) E C satisfying (6.7). Remark 6.20 As we shall see, we have in -fact for (p, u) E C p E L°° ,
curl u E W 1'q (V q< oo)
,
divu-
µ
a
p l E W 11q (d q< oo) .
This follows from the existence proof but it can also be deduced from the regularity analysis (results and proofs) performed in the preceding sections.
Remark 6.21 We are not able to prove a similar result in the case of "pure" Dirichlet boundary conditions for lack of compactness of p in LP "up to 892".
Remark 6.22 The analogue of Remark 6.18 also holds here.
Remark 6.23 Analogous results to Theorem 6.10 can be shown when N >_ 4. However, the form of constraints to be used has to be modified slightly (or at least it seems so!). The modifications being too technical, we prefer to skip such extensions to higher dimensional situations which, anyway, are not physically meaningful.
Remark 6.24 In Theorem 6.10, the trivial solution uo obviously corresponds to the equation -pAuo - V div uo = g in S2, with the same boundary conditions as those mentioned before Theorem 6.10. In particular, uo is not the same in Theorems 6.9 and 6.10.
Stationary problems
122
The proofs of Theorem 6.9 and 6.10 follow the same line of arguments with, however, important technical differences. Therefore, we wish to begin by explaining the common strategy of proof. The main (and simple) idea is to approximate (6.3) by
div (pu) + aPP = a
M , meas(g) apPu+div (pu(gu) - pAu-t;0 div u+aVpr' = pf +g inSl,
(6.130)
in the case of Theorem 6.9, and in the case of Theorem 6.10 by
div (pu) = 0 div(pu (9 u)
in 12,
J in
pP dx = M,
(6.131)
div u+V {app'+apP} = g f +g in St
where a E (0,1) and p > 3 is large enough. Of course, p is required to be nonnegative in SZ and u to satisfy the same boundary conditions as in Theorem 6.9 or 6.10. Then, essentially copying the existence proofs for the system (6.1)-(6.2), we shall obtain the existence of solutions of (6.130) (or (6.131)). In fact, we shall obtain a continuum Ca of solutions (p, u, M) in L2 x H1 x [0, oo) or Lq X W l,q x [0, oo) (p < q < oo) such that, for all M > 0, there exists (p, u) satisfying (p, u, M) E Ca. The existence of such a continuum is a rather direct application of the existence proofs and of topological degree arguments (together with a simple compactness observation described below). More precisely, we prove that
Ca is unbounded in L2 x H1 x [0, oo) (resp. Lq x W1,'? x [0, oo)), closed and connected for the weak topology (resp. in Lq X W i,q x [0, oo) and the strong topology) and Ca fl {(p, u, M) / 0 < M < R} is bounded in L2 x H1 x R (resp. Lq X W i,q x R) for all R E (0, oo). We then set
C = { (P, u, M) / 3 an
Mn n M ,
n 0, 3 (pn, un, Mn) E Cc,,,, (Pn, Un) n (PA t) }
the convergence being the weak L2 x H1 convergence in the first case, strong Lq x W' ,q convergence in the second case. We shall then prove, using compactness
results, that (p, u) solves (6.3), pn converges in L" (in particular) strongly to p
and thus fn pP dx = M. Then, as shown in Appendix D, C is a continuum, contains (0, uo, 0) and we shall prove that C n {(p, u, M) / 0 < M < R} is bounded for all R E (0, co) and C n {(p, u, M) / M = R} is non-empty for all R E [0, oo). The conclusion follows easily setting C = {(p, u) / 3 M E (p, u, M) E C} and observing that C = { (p, u, fn pP dx) / (p, u) E C} . We may now begin the proofs of Theorems 6.9 and 6.10.
[0, oo)
,
Proof of Theorem 6.9. We recall that throughout this proof -y = 3 and N = 3.
Stationary problems
123
Step 1: Bounds for solutions of (6.130). We begin with formal a priori estimates for solutions of (6.130). First of all, integrating over SZ the equation satisfied by p, we immediately obtain fo pP dx = M. Next, exactly as in section 6.3, we obtain the following energy identity (writM ing h = meal O ) 2
Jcthi!_ + app 12 + a'y (P7 - hp-1) + µI DuI2 +t;(divu)2-pu f-g-udx =0
(6.132)
from which we deduce as in the proof of Theorem 6.7 (see (6.121) and the following inequalities) IIUIIHI
s IIPIIL 3 implies that ai > 1). The usual (by now) regularity analysis then implies, if ai > 5, i.e. p > 6, that apb + app is bounded in L2(11). In addition, the argument detailed in section 6.3 shows that (p + l;) div u - (a p'" + app) and curl u are bounded in W1,al (1) and thus in LQ1(Q) where /3i = 33a1 can be made arbitrarily close to 3 by choosing p large enough. Then, the lemma stated and shown immediately after this proof implies that app' + app and thus div u, curl u and Du are bounded in Lpt (11). Hence, p(u 0)u is bounded in Lag (11)where-L=1+ pl + pl - 3 = p + « -1 and a2 can be made arbitrarily close to 3 by taking p large enough. Iterating the above 32 argument, we obtain a bound on a p7 + app, and Du in L02 where 32 = 3-a2
«
Stationary problems
127
and thus a bound on p(u V )u in La(S1) provided we choose p large enough so that 82 > 3 and thus u is bounded in LOO, and p + 102 = 1 < Then, exactly Ct as above, we deduce that ae + app is bounded in L°° and thus p is bounded in LOO, and that div u and curl u are bounded in L°O and therefore u is bounded in W1,11(52) for all 1 < q < oo. The rest of our claim then follows easily. We may then complete the proof of Theorem 6.10 by sending a to 0+, using the result of Appendix D. Once more, the compactness requested by the condition (D.5) is a straightforward consequence of the above bounds on p and u and of the compactness analysis performed in section 6.4. O 3.
Lemma 6.3 Let U E W1,9(52) for all 1 < q < oo satisfy u n = 0 on 852; and let p E L'(0) > 0 satisfy div (pu) = 0 in 52. Let cp be a continuous function on [0, oo) and let us assume that div u - V(p) E L' for some r E [1, +oo]. Then we have (6.139)
IIco(P)IILr s Ildivu-go(p)IILr.
Remark 6.25 This lemma and the estimate (6.139) is very much related to the estimate (6.32) shown in section 6.3 above. It is in fact possible to combine the argument made there and the above setting to reduce the regularity assumption on p and assume that p E L", U E W 1,q where 1 < p, q < -oo and 1 + 1 < 1. We do not give the details of this technical extension since we do notpneeA it. 0
Proof of Lemma 6.3. We begin by showing that we have for any continuous function 0 on [0, oo) :
fu.Vb(P)dx = 0 =
fdiv(u)b(p)dx.
(6.140)
Indeed, we observe that we have for any function 3, div [u/3(p)] = (div u) [,8(p)
- Q'(P)P]
in
S2 .
Therefore, (6.140) holds if we can choose 8 in such a way that Q'(t)t - /3(t) _ fi(t) on [0, oo). Then, we remark that we may assume without loss of generality
that ti(0) _ &'(0) = 0 and that b is smooth (once (6.140) is shown for such smooth functions i, the general case follows by density). Thus, we may choose ,3(t) = t fo s ° ds and (6.140) follows. At this stage it only remains to use (6.140) with = Icpl''-2cp (at least if
r < oo, the case r = +oo follows upon using (6.139) with r < oo and letting r go to +oo). Indeed, we find f I w(P) I r dx =
f{() - div u} I'(p)
0
in (6.141) and p+ = p°O = p°° > 0 in (6.142). We entirely leave out the case when u°° 34 0 since this case already raises the delicate issue about problems set
129
Exterior problems and related questions
in bounded domains with non-homogeneous Dirichlet boundary conditions for u
(and thus Dirichlet boundary conditions for p on some part of the boundarythe so-called "inflow" part). This issue is investigated in the next section. Also, in case iii), we do not consider the case when p+ # p_° in (6.142) since we can construct non-existence situations in this case as we now show. Let (p, u) be a solution in that case; we assume (for instance) that p E L°O(11), Du E L2(11), u E L2(fl) fl L°°(0) (notice that we can use Poincare's inequality since w is
bounded), g - 0 and f - 0 (or f - V with 4P smooth, vanishing at xi = ±oo fast enough). Multiplying the momentum equation by u, and integrating over (-R, +R) x w, we find easily (a.e. R (=- (0, oo))
µa2 (R, x') - u(R, x') - div u(R, x') ui (R, x')
dx' w
+u(R,x') a P(R,x')1u(R,x')12 +a72 1 p(R,x')7
+
i
(-R, x') u(-R, x') + t; div u(-R, x') ul(--R, x')
- ui(-R, x') 11p(-R,x')ju(-R,X1)j2+a -Y
R
+J
r
IR
dxi
Jw
'
1
p(-R, x')7
2
dx' µ1Du12 + (div u)2 = 0.
We next observe we can find Rn going to +oo such that fw dx'{IDu(Rn, x')12+ 1Du(-Rn,x')12+1u(Rn,x')12+1u(-Rn,x')12} goes to 0 as n goes to +00. Letting
R = Rn and n go to +oo in the preceding equality yields the fact that u - 0 and thus Vp7 = 0, i.e. p is constant and our claim is shown. We thus begin with our analysis of the case when u°O = 0 and pO° > 0. In order to keep ideas clear, we consider the case when fl = RN and then explain how to modify the results or the arguments in the exterior case (case ii)) or in the tube case (case iii)). We thus wish to solve div (pu) = 0 in RN,
p > 0 in RN
pf +g inRN,
(6.143)
with the following conditions at infinity (to be interpreted mathematically appropriately)
p(x) - p°° > 0,
--+
as
1x1 -- +oo.
(6.144)
The data f, g are always supposed to satisfy f, g E Li fl L°° (RN). Less restrictive conditions are possible but we skip such easy (and technical) extensions.
Furthermore, we take N > 3 for rather clear reasons (due to the decay of Green's function for second-order elliptic operators): let us mention in passing that if N = 1 and p, u are "constant at infinity", integrating the momentum equation immediately yields fR p f + g dx = 0. Thus, if f > 0 and fR g dx > 0, there are no solutions, even if g =- 0, p - 0 if f > 0 in R and thus u is constant.
Stationary problems
130
Next, if N > 3, we shall always look for solutions (p, u) satisfying at least: L2N/(N-2)(RN), p E p°O + LP(RN) + Lq(RN) (when 1 < Du E L2(RN), u E p, q < oo are to be determined). These conditions are the precise mathematical (weak) translation of the boundary condition at infinity (6.144). We begin our analysis with the case -y > 3 and afterwards we shall look into more general exponents. Theorem 6.11 Let N > 3 and -y > max (3, 2) . Then, there exists a solution (p, u) of (6.3) such that Vu E L2(RN), U E L2NI(N-2)(RN), p - poo E L3(RN) n L°° (RN) if N = 3, (p - p°°) E L2 (RN) n Lq (RN) with q = N 2 (-y -1) if N > 4.
Remark 6.28 Of course, when N = 3 and -y > 3, we may apply the regularity results shown in the previous sections and we deduce that p E L°° (RN), curl u and div u - ' E W1,q (RN) l- {c2 E W1l' (RN) SuPyERN fy+Bl I,I q + IDcolq dx < oo}) for all 1 < q < oo and Du E BMO(RN). Remark 6.29 The behaviour at infinity of the solutions (p, u) is not clear. It is easily seen however that the best (that is the fastest) possible decay at infinity C.Indeed, if we take -r, lp(x) - p°°I < 777:7 is: Iu(x)I < Ix 2, IDu(x)I < first p = p°°, f = 0 and div g = 0, we thus find a solution u of
-µ 1u - eV div u = g in RN,
L2N/(N-2)(RN),
uE
Du E L2(RN)
and, in general, u decays at most like 1/IxIN-2 while Du decays at most like
1/IxIN-1 in general if g E Co (RN). Next, if u = g = 0 and f = V LNI(N-1)(RN))
W E L1 n LOO(RN), 4 E
(i.e.
then we may choose p to be defined
by
p7-1
= (P°O)ry-1 +
7
- 1 ID
a-y
and we see easily that the decay stated above is in general optimal. In view of these examples, it is natural to conjecture that u E L7I 0o(RN), Du E LN/(N-1),oo(RN) and p - p°° E LN/(N-1),oo(RN). O Proof of Theorem 6.11. Once more, the proof is divided into several steps. The first one consists in obtaining formal a priori estimates. Next, we build conveniently approximated problems (through a "double-layer" approximation) and we justify the a priori estimates derived in step 1. Finally, we conclude the proof in a third step passing to the limit.
Step 1: Formal a priori estimates. We first recall the "usual" local energy identity with a small modification, namely
div u
P
122
+a
1 Y
(pry-(P°°)ry-1P) 11 _µ0I 22 +pIDul2
(divu)2 = pu- f
inR'.
( 6.145 )
Indeed, we have incorporated the extra term div {u(-a(p°O)'1-'p)} which clearly vanishes since (6.3) holds.
Exterior problems and related questions
131
Next, we use the boundary conditions at infinity (6.144) and we deduce at least formally JRNII()= JRN
Using the Sobolev inequalities, and the assumptions on f and g, we deduce easily the following bound Hull L,
(RN) + II Dull L2(RN) < C (l + IIPIILO0+LQ(RN)) .
(6.146)
Here and below, C denotes various positive constants independent of u, p. Next, taking the divergence of (6.143) (the momentum equation) and using (6.144), we deduce easily that we have (at least formally) ap'y = a(p°O )"y + (µ+1;) div u + RjRj (pu=u j )
div (pf) - (-0)-1 divg
-(-O)-1
in RN
(6.147)
(recall that R, denotes the Riesz transform 88(-0)-1/2). In particular, we deduce IIP'IILo+Lq/ry(RN) < C 1 + IIDuIIL2(RN) + II PII L°O+L9(RN) IIUI12
(RN)
+ IIPIILo+L9(RN)
}
Notice indeed that when N = 3, q = 2, N Z = 6 and thus Z7 + .1 < 2 since N 7-1 < 2 and N-2 + 1 = N-2 C1 + 1 = z > 3, and when N > 4,
-
N-2 ry N q N Combining this inequality with (6.146), we finally obtain y
ry-1 J
q
IIPIII,o+L9(RN) 3, this inequality shows that we have a priori bounds on p in L2N/(N-2)(RN) L°° + Lq(RN), on u in and on Du in L2(RN). In addition, we may now go back to (6.147) in order to deduce when N = 3 that p'7 - (p°O )'y
is bounded in (L3/2,o° n L°°) + L2 + L3 + Lr wherer 1 =q1 + 31 < ?; indeed, 3 p f and g belong to L' n Lq for some a > 3, puiu j belongs to L3 + Lr and
-
div u E L2. Therefore, p1' (p0O) f is bounded in L312'°° +L 3 . Notice also that by applying the bounds obtained in the previous sections, we also obtain a bound on p in L°°. In conclusion, p't - (p°°)'y is bounded in (L 3/2,o0 + L3(R3)) n L°O(R3).
-
Therefore, (p - p°°)1(jp-p°O j2pm) are bounded in L' (BR) while p is bounded in LOO + Llf (BR) uniformly in R > 0. We next need to obtain further a priori bounds (essentially on p) in order to be able to pass to the limit as R goes to +oo (and then as a goes to 0+). In order to do so, we use systematically the observation made in Appendix E and deduce bounds on p'Y - f p' in various sums of LP spaces from the equation (6.149) and the bounds already shown. Before we do so, we first wish to estimate carefully f p" . Obviously, we have f pry > (f p) ry = (pOD) lf. Next, we remark that we also have (in view of the bounds shown above)
-
-
fB p'' dx < R
(LR
P-'-' dx P°° +
< (L, p" dx
('r-1) /'r
CR-N
poo +
CR-N
Stationary problems
134
_ M + 1 and in M > 1. Next, when N > 4, we obtain in a similar way using Appendix E that p is
bounded in L' + Lq where q =
(7 - 1) and that p - p°° is bounded in
L2 n Lq. Let us finally observe that the arguments made in section 6.3 show that D(µ + ) div u - apry is bounded in L'(BM) or lf(BM) (if r < 1) for all
r < oo if N=3, and r = .1+1-N-L ifN>4andthus(p+l;)divu-apry is compact in LP(BM) for all p < oo if N = 3, p < N 2 ry-i if N > 4, for each fixed M E (1, oo), uniform in R > M + 1.
Step 3: Passages to the limit. We first let R go to +oo, denoting by PR, UR the solution of (6.149). Without loss of generality, extracting subsequences if necessary, we may assume that UR converges weakly in H1 to some u and UR converges in L oC to u for 1 < p < NN2 while PR converges weakly in Ll to some
p>0(q=27if N=3,q= N 2(7-1) ifN>4),andwehave pIul2ELi(RN), P - p°° E L2 n D. In addition, p E L°° (RN) , Du E LP (RN) for all 2 < p < 00 if N = 3. Finally, we may assume that div UR - µ+C (PR)" converges a.e. and in L PC for all 1 < p < q/7 in view of the bounds shown in the preceding step. Next, we wish to apply the results and methods of section 6.4 in order to conclude that PR is converging to p in Li for all 1 < p < q. However, we cannot just apply Theorem 6.4 because of the behaviour at infinity and we need to look into the proof. In fact, following the proof of Theorem 6.4, we only need to check
that
LN
u VW,,r dx - 0
as n --> +oo
where r = p - (_p8)'19 (recall that pe is the weak limit of pR, 9 E (0, 1) is chosen small enough) and Wn = Sp (n), cp E Co (RN), cp(x) = 1 if Ixi < 1. Obviously, PO - (pop)e E L2 n Lq/e since the set {pR < p°O/2 or pR > 2p°°} has a finite
measure bounded uniformly in R and the function (x - xe) is Lipschitz on [p°O/2, 2p°0]. Then, we deduce that (pe)'/e - p°° E L2 by a similar argument (it also belongs to L°° + Lq and thus to L' in view of Lemma 6.4). Therefore, r E L2 n Lq(RN) and we may apply (or follow) the rest of the proof of Theorem 6.4.
We may now pass to the limit in the equations (6.149) thus satisfied by (p, u). In addition, we deduce from (6.150)-(6.151) and Fatou's lemma the following:
f
N
2
#, IuI2 + 2 pIuI2 + µiDuI2 + (div u)2
+
a7
7
1
(pry-i
- (p0O)ry-i) (p-p00) dx < fit
(6.154) N
pu f +
dx .
We next need to pass to the limit as a goes to 0+, and we first have to obtain a priori bounds on (p, u) independent of a. In order to do so, we use the argument
Stationary problems
136
developed in step 1 above, the only difference being the terms involving a in (6.149).
Obviously, the inequality (6.146) is still valid since the terms involving a in (6.154) are non-negative. Then, going back to (6.154) and using (6.146), we obtain
a
r
1/2
N
IuI2 + pIuI2 dx
4 while G L2N/(N-2) (RN) (uniformly in a E (0,1]). Finally, if N = 3, the is bounded in analysis performed in section 6.3 shows that p is bounded in LOO (R3 ).
We may now pass to the limit letting a go to 0+ and writing p = pa, u = ua. Extracting subsequences if necessary, we may assume without loss of generality that, as a goes to 0+, pa converges weakly in Li C to some p E L' + Lq (RN ) such that p - p°O E L' (RN) n (LOO + L3 (RN)) (s = oo if N = 3, s = q if N > 4), L2N/(N-2)(RN) and strongly in L O0 for 1 < p < NN2 to ua converges weakly in L2N/(N-2)(RN) satisfying Du E L2(RN). Finally, as in step 2 abovesome u E and in the preceding sections-we may assume that div ua - +C (pa)d' converges a.e. and strongly in L O0 for 1 < p < q. We conclude using Theorem 6.4 as we did
Exterior problems and related questions
137
above when we passed to the limit in R: indeed, r = p - (pe )110 E L3 n Lm(R3) if N = 3, E L2 n (L°° + Lq(RN)) if N > 4. Then, if N > 4, fRN Iu - OVnIrdx n f(n p7 r a.e.
We thus find that f Ben pry r dx < CJ n1/2. Next, we recall that L3 (R3 ). Therefore, we deduce for all n > 1
f
n<jxj 2. We assume that y > max (3, 2) . Then, there exists
n a solution (p, u) of (6.3) in 11 such that u E Ho (St), p - p°O E L2(SZ) n L°O(SZb) for all b > 0 with SZb = R x w6i w6 = {x E w / dist (x, 8w) > a} when
N=2or3andp-pOOEL2(11)nLq(SZ)with qN 2(y-1)ifN>4.
We next explain how we may obtain another type of existence result for the same range of exponents y as in the case of a bounded domain. This will require the force term to be small (up to a gradient). More precisely, we write
f = VV + f, f, f E L1(RN) n L°O(RN)
(6.157)
and we shall assume that f is small. We only consider here the case when fl = RN, the other cases, namely the exterior domain case, or the tube-like domain case, being treated similarly.
Theorem 6.14 Let N > 3 and let y > 3 if N = 3, y > 2 if N >_ 4. If f is small enough in L1(RN) n L°°(RN), then there exists a solution (p, u) of (6.3) L2N/(N-2)(RN), satisfying u E Du E L2(RN), p - poo E Lr(RN) n Lq(RN) with
r=3,q=min (3('y-1),2y)if y>2,N=3and r=2,q= N 2(y-1) if N > 4, and p - p°° E L3 (RN) + L3(ti-1) (RN) if
< -y < 2, N = 3. 3
Remark 6.31 As can be seen from the proof below, one only needs f to be small enough in L2N/(N+2)(RN) n La(RN) wherea 1 = 2N N+2 - q'i
Remark 6.32 If y = 3, N = 3, exactly as in the preceding sections, there exists a solution (p, u) such that p e LO°(R3), Du E LP(R3) for 2 < p < oo.
Exterior problems and related questions
139
Sketch of proof of Theorem 6.14. We only present a sketch of the proof of Theorem 6.14 since it follows essentially the scheme of the proof of Theorem 6.11 with a few major modifications concerning in particular the a priori bounds. Observing that we have (at least formally) Du12 JRN pl
+ (dlv u)2 dx < fR N pu f + u g dx ,
we deduce the following variant of (6.146), namely II uIIL2N1(N-2(RN) + II DuIIL2(RN) 3 (and thus in particular 3 _< N _< 5); indeed, the case -y > 3 is contained in Theorem 6.11 and if y = 3, we just take co small enough and deduce from (6.159) a bound on p in L°° + Lq(RN) and the rest of the proof is the same as in Theorem 6.11. Next, when -y > 3, we deduce from (6.159) that we have for co small enough IIPIILo+L9(RN) < Cl(EO)
or
IIPIILOO+L9(RN) > C2(E0)
(6.160)
where Ci(eo) < C2(Eo) are respectively the smallest and the largest positive roots of [xy = C(1 + x + Eox3)]. Let us remark that Ci(eo) converges to the positive root of [xl' = C(1 + x)] while C2 (eo) (Cc 2)1/(3-y) converges to 1 and in particular C2(eo) converges to +oo as co goes to 0+. The strategy of the proof is to show by a continuity argument that a solution satisfying the first (upper) bound exists for co small enough. In order to do so,
we consider a solution (p, u) of (6.149) with f replaced by ft = VV + ti where t E [0,1]. From the arguments and methods introduced in section (6.7) we may even assume that we have a continuum C of solutions (p, u, t) in L4 X Ho x [0,1]. Note there is here a minor technicality due to the fact that the continuum is included in Lq x Ho x [0,1] but the topological properties (closed and connected) hold for the strong topology of L" x W01 'r x R for p < q, r < 2. This difficulty is circumvented by observing that the alternative (6.160) is easily shown to be valid with Lq replaced by LP where p < q is close to q. This is why we shall ignore this problem in this sketch. Next, we claim that the alternative (6.160) holds for any such -(p, u) and for all R > 1 (large enough) and a E (0,1]. This claim really follows from the proof of Theorem 6.11 and the Remark 6.29. We just need to observe that the term
LR pVV udx = LR aV(p-p°°) dx
Stationary problems
140
can be easily bounded by a fB,, (p7-1 - (p°° )7-1) (p - p°°) dx. Indeed, without loss of generality, we may assume that V is bounded and V E LN/(N-1) (RN). Hence, we have o
f
BR
0 small enough.
Not only does this show that (6.160) holds but also it shows that, when t = 0, then IIPIIL-+L4 < C1. But this implies that for all (p, u, t) in the continuum C we must have II PI I L°° +LQ < Ci; indeed, (6.160) imply that the sets {(p, u, t) E C / IIPIILo+LQ < C' 2C2 }
and {(p, u, t) E C / IIPIIL-+L9 > C'
C2 }
make a disjoint open covering of the continuum C, and since the first one is non-empty, the second one has to be empty. Then, (6.160) implies our claim. At this point, we have shown the a priori bound we needed in order to be able to reproduce the proof of Theorem 6.11. 0 We now conclude this section with a few additional results that we state only
in the case when 0 = RN, the extensions and adaptations to the cases of an exterior domain or of a tube-like domain being straightforward. Once more we look at the situation when u "goes to 0" as IxI goes to +oo while p "goes to a constant". But, instead of prescribing the "value of p" at infinity, namely the constant p°O above, we are going to use different parametrizations in a slightly similar way to what we did for stationary problems on bounded domains in section 6.7. More precisely, we introduce a positive function on RN denoted by w such that: w E L1 n L°°, f'RN w dx = 1, inf essBR w > 0 for all R E (0, oo), I f I < Cw a.e. on RN for some C > 0. Such a function clearly exists: take for e-IXI2 instance w = (If I + e-Ix12)(fRN If I + dx)-1. The type of parametrization we shall use is described by the following constraint
Exterior problems and related questions
fR N
141
pw dx = M
(6.161)
where M > 0. Our main result is then
Theorem 6.15 Let N > 3, let -y > 3 if N = 3, ry > 2 if N > 4. Then, for each M > 0, there exists a solution (p, u) of (6.3) satisfying (6.161) such that L2N/(N-2)(RN), Du E L2(RN), p E L°O(RN)+Lq(RN) with q= N 2 (y-1) 4L E
ifN>4or3
E
Lr (RN) n Lq (RN) for some p°O>0with r=3if N=3,ry>2,r=2if N=4,
p - p°O E L3(-'-1)(RN) + L3(RN) for some pO° > 0 if N = 3, 3 < -t < 2.
Corollary .6.1 Let N _> 3, let ry > 3 ifN = 3, y > Z ifN > 4. Then, there exists p E (0, +oo] such that, for each p°O E [0, p), there exists a solution
(p,u) of (6.3) such that u E Lr(RN) n Lq(RN) (or L3(RN) +
L2N/(N-2) (RN),
L30y-1)(RN)).
Du E L2(RN), and p - p°O E
Remark 6.33 It is an interesting open question to decide whether we can take p = +oo in the preceding corollary. Of course, Theorems 6.11 and 6.14 provide examples of situations where indeed we know that p = +oo. Remark 6.34 It is possible to show that Corollary 6.1 still holds for a < 7 < 3 if N = 3. The proof of this claim follows closely the proofs of Theorem 6.15 and Corollary 6.1 above: one builds continua of (approximated) solutions satisfying fR, ppw dx = M for some large p. This constraint allows us to obtain a priori bounds on u in , Du in L2, p and u in L , Du in L o° for all 1 < q < L2N/(N-2)
oo, and p - p°O in L3 + L3(,y-1) for some p°O > 0. As seen from the proofs below, we only have to check that f R, pn w dx converges to fR3 pew dx as n goes to +oo
if pn satisfies the preceding bounds uniformly and if pn converges to p in LP10 for all 1 < p < oo. This is the case since fBR pnw dx converges to fBR p'yw dx as n goes to +oo for all R E (0, oo) while we have (1-9)'y
0-f/P
pn w dx
R)
(L,>RPnwdx)
where B + 1 - 6 = 1 P 'Y
r
J3
e7/P
w dx
,
_ IIwIIL3/2f1L3(y-1)/(3ti-4)(lxl>R)
and the right-hand side converges to 0 as R goes to +oo uniformly in N > 1. 0
Sketch of proof of Theorem 6.15 and Corollary 6.1. We first explain how we can obtain a priori bounds. We only have to modify a little bit the proof made in step 1 of Theorem 6.11. Indeed, we now estimate the integral
f
RN
pu f dx < CII DuII L2(IItN) IIPwII L2N/(N+2)(RN),
Stationary problems
142
and we deduce as in step 1 of the proof of Theorem 6.11 the following bound II DuII L2 +
< C (1 + II PW II L2N/(N+2))
IIuIIL2N/(N-2)
Next, we use (6.147) which holds for some p°O > 0 which is also an unknown and we write (6.162) e = (p°")" + F1 + F2
where F1 E L2, F2 E L" with p = N 2 2171 (or p = 2 if N = 3, 'y > 3) and IIF1IIL2 IIPWIIL2N/(N+Z))
In particular, we have II
PII
(1 + (P°O)" + II PW II L2N/(N+2) + IIPIIL°°+L9 11pW112 L2N/(N+2) ) II PII L-+L9
and 9
1IIPWIIL98
IIPWIIL2N/(N+2) < IIPWIIL1
with 0+ 1 q-9
N+2 2N
< 2), presents some similarities with exterior problems. We consider the stationary problem (6.3) in Q with the following boundary condi-
tions: u = uo on BSZ, p = po on {x E 9) / uo n(x) < 0} where n denotes the unit outward normal to acz, uo and po are given functions on &Q. The existence of stationary solutions in such situations is essentially an open problem.
Stationary problems
144
6.9 Regularity of solutions We begin with an example that shows that, even though all the data are smooth, we cannot expect the density p to be continuous or in W1,P for any p > 1 and the velocity field to be C' or in W2,P for any p > 1. As we shall see later on, this is related to the possible presence of vacuum since we shall prove that, in the absence of vacuum, the density p is (Holder) continuous and the velocity field is at least of class C'. Example 6.4 Let SZ be the unit ball centred at 0 (= B,) in R2 (for instance). We are going to build a solution (p, u) such that p - 1 Bi z and u E CO0(B1/2 UBi/2), u V C'(B,). Indeed, we choose p(x) = 1 if Ixi > 2= 0 if Ixi < .1 and we first take an arbitrary divergence-free vector field u E C°O (B1) or even Co (Bi) such that x Vii(x) = 0 on 8B112. Then, we set u(x) = u(x) if 2 < Ixi < 1. Next, we take = 0 in order to simplify the presentation even though the construction is easily modified to all cases where + p > 0, it > 0. Then, we choose u1 E C°°(B1/2)
such that ui(x) = u(x) if lxi = 2 and (2 V)ui = -µ j if Ixi = 2, and we set u(x) = ui(x) if Ixl < 1. Notice that u is Lipschitz on B1, but u ' Ci(B1), u ¢ W 1.1c (Bi) and p ¢ W11-.1 ,1(B1). Next, we define g to be an extension to B, of
-pLu1 such that g E Co (B1). Finally, we define f to be an extension to B, of -µou+ (u V)u - g E C°°(Bi/2) such that f E C°°(B1) and thus f E CO' (B1). Since the flux pu FXT vanishes as Ixl - 2 and as Ixi - a -, we see that we have
div (pu) = 0
in
Bi
while by definition of u, p, f, g we have both on B1/2 and on BI1/2
pf +g Finally, we conclude by observing that this equation holds on B1 since the jumps on 8Bi/2 cancel
(.v) ui(x) + a[1- 0]
-µ 1(1XX1 O u(x) - (1XX1
X
=0
1
by construction. 0 Example 6.5 We want to show here that the above construction can be adapted to the full compressible model, namely
div(pu) = 0 (pu
V)T - kLT + (y-1)(divu)pT = 1(auj +a;uti)2 - (divu)2.
(6.165)
Indeed, we choose once more p, u on B1 and u on B112 as in Example 6.4. Next, we wish to build u on B1/2. Writing x = (r, 0) where (r, 0) are the polar
Regularity of solutions
145
coordinates, we choose cp E C°° ([0, 2]) such that V vanishes for r small, w (2) 0 and cp' = 1 for r close to 2. We are looking for ul defined on B112 by
_
ui(r,6) = r T (2,0) co(r) Next, we wish to solve the temperature equation with the appropriate boundary conditions, say, for example, T = 1 on 8B1. We thus look for a solution T of
-kiT + (pu . 0)T = 2 82uj +ajui)2 - (divu)2 in Bi
,
T=1 on 8B1 i
where u is defined by ui in B1/2 and by u on Bi12 so that u is Lipchitz on B1. In
_ O(r) E Co ({ < lxi < 1})2 extended by 0 to B1-and look for T = T (r), then u = * T (2) W(r) for r < a fact, if we further choose u =
(-82") awhere
and (azuj + ajut)2 - (div u)2 is a radial function while (pu 0)T _= 0. In this case,2we simply solve
-kLT = [2 (auj + 8jui)2] + T (2)2 a(r)1(r 0.
(6.166)
Let us point out at this stage that we simply assume that y > 0 and that we have shown in the preceding sections existence results of solutions in two and three dimensions having the regularity assumed above at least for some range of exponents y. It remains to decide when (6.166) holds, and we postpone the
Stationary problems
146
discussion of (6.166) until the end of this section where we present some situations
where solutions are known to satisfy (6.166). Of course, Example (6.4) above shows that p may not be continuous if (6.166) does not hold.
Theorem 6.16 Under the above conditions, p and Du are bounded and (uniformly) continuous. Furthermore, if we denote 8 = IA+f (inf ess #f (IDull Loo -1 , k = [8] (the integer part of 0) and a = 8 - k E [0,1), then we have
pECb'a,
UECb+l,aifa
pEC6-1,Q
0;
(6.167)
u
-µ+
curl u E Cb'Rfor all (3 E (0, 1) if a = 0
Remark 6.36 In the case when a = 0, the proof below shows that p satisfies IDk-lp(x)
- Dk-lp(y)I
f2 a.e. on RN. Indeed, we first observe that we have in D'
u V(f2-fi) +V(f2-fi) )+dx
(P- l ) / P M1/P
JRN
for some positive constant M (= IB1I R4N). Therefore, we deduce
v-
f
/'
1
II(divu)+IIL°O
1/p
N(f2-fl-S <x>)P. dx
) _
0,
Xx(0) = x E RN
(6.171)
,
and this solution is unique. In addition, one deduces easily from (6.170) the following bound:
IXx(t)_X (t)I < exp {(log Ix-yf )e-c1t}
if0 < t < To,
for all X, y E RN such that I x - y I < 2 , where To is defined by log log 1
Ix
(6.172)
- y I e-c1 To =
Regularity of solutions
149
Next, we introduce the following value function defined on RN by
{J00 .f (x) = inf
[co(xr)+b['ye(t)_1]e0(t)J exp [_jtf.ye0(8)ds]
(6.173)
e meas. from [0, oo) into [ao,13o]}
where ao = inf ess f (= log (inf ess p)) and Qo = 7 log IkaIILLet us immediately point out that, if we know that u were Lipchitz, then-see for instance P.-L. Lions [346]-f would be uniformly continuous and would be the unique (viscosity) solution of the associated Bellman or Hamilton-JacobiBellman equation, namely
uvf+
sup a0 C1 and we observe that T < To if we choose K large enough and Ix - yI < -11. Hence, we deduce for all x, y E RN
with Ix-yI 0 which we may always assume to be in (0,1) (in fact, 6 = K Let us also observe that f is clearly bounded since cp is bounded and 0 is bounded by definition. 0
Step 3: f = f and thus p is continuous. We are first going to show that (6.174) holds in the sense of distributions. One possible way to prove this claim
consists in regularizing u by convolution uE = u * tc6 (for e E (0,1)) and approximating f by fe defined in the same way as f replacing u by uE. Of course, we denote as usual s;E = - ri. () with n E Co' (RN), x > 0, Supp rc C B1, fRN , dx = 1. We thus introduce the solution XE = Xx'E of
exp
Xe = -uE(Xe) for
t > 0,
X6(0) = x E RN
and the value function fe defined, for all x E RN, as follows
ff(x) = inf
-Jo t'Yeh'9ds
0 meas. from [0, oo) into [ao, Ao]}
.
Then, exactly as in [346], fe is bounded, uniformly continuous and the unique viscosity solution of 'ue
V1,6
+
sup a0:50 0, e E (0,
a-11
for all x E RN
.
Regularity of solutions
IXe-XI < C,
IX£-XI Ilog IXE-X II +E
151
loge
IXE-XI(s) < 2
if
for all s E [0, t]. This inequality easily yields for all x E RN, E E (0, a-1] if t E [0, Tel
I XE -X I < me
where me is the solution of mE = Cl [mc log
me(0) = 0
,
1 me
+ e log 1
me(t) E (o, a]
for t E [0, T e]
for t E [0,TE]
,
mc(TE) = 2
A simple argument allows us to check that TE goes to +oo while me goes to 0 uniformly on [0, T] for all T E (0, oo) as e goes to 0+ We then deduce the uniform convergence on RN of P to 1. Indeed, we have for all x E RN, by a similar argument to that in step 2 above T
Ife(x) -.f(x)I : CJ 2, and we denote E(x) = C (M) for all x E RN, for all C E (0,1]. We then write for all e E (0, 1) and for all x E RN
2 y F(y)
u(x) = CN +CN
(x --y) dy
x-y
]RN
JN Ix-ylr'
and we denote b uE (x) the first integral and u2 (x) theeecond integral in t1ie
right-hand side. Next, we have by symmetry for all x E R and for all e E (0, 4 )
N (F(Y)-F(x)) (x-y) dx
IuE(x)I = LIRN Ix
_< Cl log
yl eI-a
1(1x-yl 1, using the thus div u - bprr, curl u E C"' (RN) for all 0 estimate shown in [346], we see that f , p and p7 are even Lipschitz continuous. Therefore, D2u E BMO since curl u E Cb'Q and divu E Wb'°O. Next, we differentiate (6.168) with respect to Xk for each k E {1, ... , N} and we find
(u.V)8kf +c8kf +(8ku- V)f = 8kw
in D'
where c = 6ye'r1. Notice that c E W 1,°°(RN) and that c > v = II DuII
(6.179) (inf ess p)"7 _
L'0.
Then, exactly as in steps 1-3 above, we show that ek f is the unique bounded solution of (6.179) and coincides with the following continuous explicit expression
akf(x) =
rt exp (- / c(Xx(s))ds dt
1000
(6.180)
o
where
k is the solution of N
E jaju(Xx)
for
t>0
(6.181)
j=1
satisfying (0) = ek (the k-th vector of the canonical orthonormal basis). In addition, 8k f E BUC(RN) and satisfies (6.176) for some b > 0. A simple way to convince ourselves that (6.180) does provide a natural candidate for the solution of (6.179) is to use a method due to N.V. Krylov [3131. We define an extended function on RN x RN, f, (x, ) = V f (x) and we deduce from (6.179) that f, satisfies the following equation N
eiaiu =1
.
f = C - DAP,
(6.182)
Regularity of solutions
155
and, we should of course expect f, to be given by 00
rt
f i (x, i) = fo V (X x (t)) - C(t) exp -
J0
c(X X) ds dt
for all x, i E RN
where C solves (6.181) with (0) = rl. Of course, (6.180) corresponds to the particular case where 71 = ek.
We then argue as in steps 4-5 using Lemma 6.5 twice to obtain that D2u E BUC(RN) and then that V E Cb'a for some a > 0. In particular, Dcp is Lipschitz continuous. In order to conclude, we adapt the argument in [346] and write for all x, y E RN and for all T E (0, oo)
IDf(x)
- Df(y)I 0, 0 < y < 1, then the second choice yields q(p) = a 11,Y (1 - p-0-10); of course, since we assume p to be Cl on [0, oo), we shall no longer consider this slightly more singular case. In view of the regularity we assumed on p, the condition fo p 1' ds < oo is simply equivalent to p'(0) = 0. In everything that follows, we assume that p'(0) = 0. This assumption plays in fact no role in most of the results discussed below, except when 11 = RN. In this case, if p'(0) > 0, then we may choose q (p) = p' (0) p log p + p fo s s p ° ds
and the lack of positivity of q or equivalently of p log p can be handled by
somewhat more technical arguments similar to those introduced in our study of the isothermal case (in two dimensions) in section 6.6. We next discuss the existence and regularity of solutions of the time-discretized problems (6.184). The results contained in section 6.2, namely Theorems 6.1-6.3, are still valid for the problem (6.184) provided we assume that h E L°° n L' (SZ) and that p satisfies for some y > 1
lira inf p'(t) t-++oo
t-l-'-1)
> 0.
(6.188)
In addition, whenever in Theorems 6.1-6.3, we obtain some Lq integrability upon p then we have in addition in this case that pi/1' E Lq. Let us mention that the assumption upon h made above is only a simplifying assumption and that h E L°° may be replaced by various (depending on which of the analogues of Theorems 6.1-6.3 we consider) integrability requirements that depend on p in a more technical way. The proof of this claim follows closely the proofs of Theorems 6.1-6.3 made in sections 6.2-6.5. We only explain two points concerning a priori estimates and one concerning the crucial compactness properties of (sequences of) solutions. About
a priori bounds, one obtains obviously as usual the global conservation of mass which yields an L1 bound: fo p dx = a fn h dx; and, one obtains bounds on Du in L,2 on p in L", on q(p) in L1 and on (p+h)Iu12 in L' by the following (formal)
Stationary problems
160
energy identity easily derived from (6.184) upon multiplying the momentum equation by u and using (6.186)
f
DuI2
(P+h)l u12 + µI n 2
+ e(div
u)2
+ aq(P) dxx
(6.189) P
J12
The second modification concerns the bootstrap argument used to obtain "regularity" (that is the improved integrability) of p and Du. Using Lemma 6.3 (section 6.7) with cp = , together with the method introduced in section 6.3 to obtain similar estimates without assuming p bounded, it is then easy to adapt the arguments introduced in section 6.3. The final adaptation needed in the proof that we wish to mention concerns the compactness results obtained in section 6.4 and more precisely Theorem 6.4. Of course, the setting remains the same replacing (pn)7 by p(pn), assuming in addition that p(pn) is bounded in Lgl7. Then, we claim that Theorem 6.4 holds if a > 0 with essentially the same proof. The only modification concerns (6.67) which is now replaced by 9
a(pe)s"e+div {u)h1'0} > h+ 16 b {p(P)Pe - p(P) Pe}
(6.190)
Next, we use Lemma 5.3 (section 5.5, chapter 5) and we deduce
a(p8)119 + div {u()"°} > h.
(6.191)
Therefore, we deduce with the notation of the proof of Theorem 6.4
div {ur} + ar > 0 ,
where
r = p - (pe)", > 0
and we conclude easily that r = 0 a.e. and thus pn converges strongly to p. Let us emphasize the fact that this proof relies upon the strict positivity of a and thus we shall have to come back to this compactness issue when studying stationary problems (6.185).
We now turn to stationary problems (6.185) and we immediately mention that the analogues of Theorems 6.7-6.10 are still valid assuming (6.188) and that p is strictly increasing on [0, oo) (exactly as above, p E Lq is now replaced p E Lq and p(p) E Lq/7). The only new point in the proofs is the compactness analysis which corresponds to Theorem 6.4 in the case when a = 0. In this case, we wish to deduce, following the proof of Theorem 6.4, the strong convergence of pn to p from the following identity {P(P)PO - p(P) Pe
}
=0
a.e. on
{x,> 0}
.
Let us observe that (pn)e converges to 0 in L1({pe = 0}) and thus the preceding equality is clearly equivalent to
Related problems
p(P)PB
- p(p) pe = 0
161
a.e.
Let us then recall that Lemma 5.3 (section 5.5, chapter 5) implies that this quantity is non-negative as soon as p is non-increasing: the argument below will in fact yield a "different" proof of this fact. We claim now that, when p is strictly increasing, it vanishes if and only if pn converges strongly to p. Indeed, denoting we have (Pn)e = pi, pe = Pl, pl(t) = p(P)PB
- p(P)PB
= pi(Pi)P1 - p1(Pl)Pl
= [pl(P1) -pl(Pi)][pi - P11
or in other words this quantity is nothing but the weak limit of IIn = (pi (pi) i) which is clearly non-negative if pl is non decreasing and strictly P1(Pl)) (P1 positive for pi # pi if pi is strictly increasing. Therefore, the above expression vanishes if and only if IIn converges strongly to 0 (in L1) and our claim follows
-
easily.
Remark 6.39 The strict monotonicity of p is crucial in order to assess that bounded sequences of solutions of the problem (6.185) are compact (in some LP space). Indeed, if p is constant on an interval [a, b] (C [0, oo)) with a < b, then
we may choose f - g = u - 0 and p to be any (smooth or not) function taking only values in [a, b]
!
We now turn to stationary problems (6.185) set in unbounded domains and more precisely to the settings studied in section 6.8. Under the condition (6.188) and assuming of course that p is strictly increasing, then the results obtained in section 6.8 can be readily adapted to (6.185). The main modifications concern the integrability properties of p and p(p). Indeed, if we take, as an example, the case L2N/(N-2)(RN), of Theorem 6.11, then one obtains similar a priori bounds: u E Du E L2 (RN), p E L°° + Lq (RN) and p(p) E L°° + Lq/-' (RN) where q = if N = 3, q = N 2 (-y -1) if N > 4. If N = 3, then one shows also that p E L°O (R3 ) and, following the proof of Theorem 6.11 in section 6.8, we obtain a bound on p(p) - p(p°°) in (L3/2'°° + L3 (R3)) n LOO (R3) = L3(R3) n L°° (R3) . Then, if p'(pOD) > 0, we deduce a bound on p - p°O in L3(R3) n L°°(R3). If p' (p°°) = 0, the situation is slightly different. For instance, when we assume that p satisfies in a neighborhood of p°O : Ip(t) - p(p°°) I > Sit - p°O l"` for some m > 1, 6 > 0, then we deduce a bound on p - p°O in L3m(R3) n L°O Similarly, when N > 4, we obtain a bound on p(p) - p(p°O) in LN/(N-1),°° + L2 (RN) if ry > N -1 and in Lg17 + L2 (RN) if ^y < N -1. Therefore, if p' (p°O) > 0 then p - p°° E L2 n Lq(RN) while if p satisfies the above condition near p°O then P - P°° E (L' + L2m(RN)) n (Lq + Loo(RN)) Let us conclude this brief study of problems with general pressure laws by recalling that the regularity of bounded solutions can be studied exactly as we did in section 6.9 in the pure power case. The necessary modifications of the arguments introduced in section 6.9 are explained in Remark 6.36 (section 6.9). Let us observe that the analysis requires p' to be bounded from below on compact sets (R3).
Stationary problems
162
of (0, oo), a condition that can be viewed as one form of the strict monotonicity required for compactness results of solutions of stationary problems.
The second topic we briefly address in this section concerns the case of "Stokes" equations which correspond to neglecting the term p(u.V)u (or div (pu(& u)). In order to restrict the length of this presentation, we only consider the case
of stationary problems in a bounded open smooth domain S2 of RN (N > 2) namely
div(pu) = 0,
pf +g
p> 0,
with the normalization (for instance) fn p dx = M E [0, oo). It is easy to check that the arguments introduced in the preceding section yield the existence of a solution p E L2"Y(12), u E H' (S2) as soon as y > N+2 with either Dirichlet bound-
ary conditions (respectively one of the three variants (6.42)-(6.43), or (6.42)(6.44), or (6.42)-(6.45)). In addition, any such solution satisfies p E Ll (St), u E WWo` (12), (µ + ) div u - ap'r E Wl (S2), and curl u E WI q(S2) for all 1 < q < oo (respectively p E L°° (S2), u E W l,q (S2), (µ+e) div u - ap'' E W.(12), and curlu E Wl,q(S2) for all 1 < q < oo).
6.11 General compressible models We only wish to discuss in this section stationary problems for the full compressible Navier-Stokes equations (with a temperature equation, or a possibly nonconstant entropy) with Dirichlet-type boundary conditions in a bounded smooth open domain fl C RN (N >_ 2). We shall not address here time-discretized problems, which in fact may be written in various forms depending upon which unknowns are being used, like density, velocity and temperature or entropy or total energy, and of course everything we do below can be adapted to appropriate periodic cases as we did in section 6.7 (Theorem 6.8). We thus consider the following system of equations div (pu) = 0
in S2,
p> 0 in 2
,
(6.192)
div(pueu)-µ0u-eVdivu+V(pT) = pf+g in St , u-n = 0 on 812, (6.193) div (puT) + (-y-1)(divu)pT - div (kVT) = 6[2pldl2 + (e-p)(divu)2] in n (6.194) The vector fields f, g are given say in L°°(S2)N, -y is a constant in [1, oo), µ and
are positive constants and µ > 0, k is a non-negative constant or possibly a non-negative function of T and finally b will be either (-y-1) or 0. We denote, as usual, d = (Du + Dut), the deformation tensor, and of course T stands 2 for the temperature which will always be a non-negative function. Notice that this notation allows for the non-physical constant y = 1 and that 8 = (^y -1), -y > 1 correspond to the correct equations from a physical viewpoint while S = 0 corresponds to the classical (at least for non-hypersonic gases) assumption which consists in neglecting the heating due to viscous friction. Of course, the quantity
163
General compressible models
2pIdI2 + (l
- p) (div u)2 should always be non-negative and this is equivalent
to requiring that p and 2p + N(C - p) > 0. We have already assumed that p > 0 and we assume-in order to simplify the presentation and avoid further technicalities-that we have 2p + N(e- p) > 0, i.e. C > N2 p. We now have to present the boundary conditions we add to the system (6.192)-(6.195) and more precisely to the equations (6.193)-(6.194) for the velocity field and the temperature. For the velocity field u, we either impose Dirichlet
boundary conditions (u = 0 on aft) or one of the three variants used several times before, namely (6.43) or (6.44) or (6.45). Boundary conditions for T are slightly more delicate and we only mention one meaningful possibility here in order to illustrate our methods. We impose the following boundary condition
8 + A(T -To) = 0
on
8St
(6.195)
where A is a non-negative constant and To is a given non-negative function on
all say in L' (all) bounded away from 0. Let us recall that we denote by n (= n(x)) the unit outward normal to all at x, and of course, if k = 0, then we do not impose any boundary condition on T. Before we start discussing some results about the above system of equations (and boundary conditions), we wish to recall from chapter 1 (volume 1) some classical identities involving either the total energy or the entropy. In fact, at least formally-or in other words if p, u, T are smooth and T > 0-the equations we shall obtain either for the total energy or for the entropy are equivalent to (6.194), provided of course that p and u solve (6.192) and (6.193). First of all, multiplying (6.193) by u and writing [-p0u-e0 div u = -2p div ddiv u] we obtain div
pu
+
'Y
1T
- 2p div (du) - (C -p) V (u div u)
ILH-1-
-div(kVT)+ 1= pu f +
(6.196)
8
7-1
in
fl .
Next, we deduce from (6.194) and from (6.192)
log T+(-y-1)(divu)p - div
k TT =
kI T2I2 +
and thus we find
div(pus)-div
TT = kI TT (1c)
I2
+T [2pIdI2+(t;-p)(divu)2]
where we denote the entropy by s = log T - (-r -1) log p = log (per} .
(6.197)
164
Stationary problems
We are going to investigate several cases in this section. We begin with the case when b = 0 , k = 0. In this case (6.197) reduces to
div (pus) = 0 . Then, in view of (6.192), we may simply take s - so E R (the entropy is constant!). Then, we find, in view of the definition of s: T = apps-1 where a = es0, and we are back to the situation studied in the previous sections, namely pT is replaced by apl. Next, we consider the case when k = 0 , b = (7-1) > 0. We then claim that in general there are no solutions; indeed, integrating (6.197) over n, we deduce easily that u - 0. Therefore, the whole system (6.192)-(6.194) reduces to V(pT) = p f + g, and we reach a contradiction as soon as g is not a gradient vector-field (curl g # 0) and f - 0. Even if g = 0, there are no solutions or at least no non-trivial solutions (p # 0); indeed, if, for example, f is given by (0, x1), 0 E SZ and N = 2, then any solution would satisfy
0 = curl (pf) = -a(pxi) i.e. p =
for some a E V. Then, p E L' implies that a
a(x2)
a.e.
0 and thus p - 0. Another
case we wish to consider is the case when b = -f - 1. First of all, if we assume in addition that k = 0, then (6.193) reduces to
div (puT) = 0, and we may choose T to be a positive constant, i.e. T > 0 and we are led to the isothermal case, i.e. the case -y = 1 with the notation of the preceding sections. This problem was studied in section 6.7 if N = 2 or 3 where we obtained, with the boundary conditions (6.43) or (6.44) or (6.45), the existence of a continuum of solutions (p, u) with fn pP dx = M E [0, oo), p large enough (and p E L°°, U E W 1,4 (f) for all 1 < q < oo). In particular, see for more detail Theorem 6.10 in section 6.7, we obtain for some interval [0, Mo) and Mo E (0, oo], the existence of solutions (p, u) with the above regularity and fn p dx = M. Next, still in the case when -y = 1, we assume that k > 0 and is a constant even though we might consider as well quite general positive functions of T. First of all, if a = 0 (respectively if To is a constant), then we may choose T to be any positive constant (respectively T = To > 0) and exactly as before we are back to the isothermal case. Next, if A > 0 and To is not necessarily a constant, then T solves the following elliptic equation
inn , a +A(T-To)=0 on all,
(6.198)
and we deduce from the maximum principle that we have 0 < inff ess To < T < sup ess To an
a.e. in Sl
(6.199)
and multiplying (6.198) we easily deduce that T is bounded in H1(f) (independently of (p, u) satisfying of course (6.192) and (6.193)). Then, arguing as in
General compressible models
165
section 6.7 (Theorem 6.10), it is not difficult to show the existence of a continuum of solution (p, u, T) in LQ(S1) X W1,q(11) X W2,q(11) (for any q E [2, oo)) such curl u E W 1,q (!a) and that: p E L°°(S2), Du E BMO, divu - µ1 pT E W
T E W2,q(cl) for all q E (1, oo), T satisfies (6.199) and for each M E [0, oo) there exists a solution (p, u, T) such that fn pP dx = M where p is chosen large enough (see section 6.7 for more details). In particular, we deduce the existence of some Mo E (0, +oo] such that, for all M E [0, Mo), there exists a solution (p, u, T) with the above regularity satisfying fn p dx = M. We need to make precise that, as in Theorem 6.10, we assume that N = 2 or N = 3 and that we use one of the three boundary conditions (6.43)-(6.45) which allow us to obtain the "regularity" of
div u - ,+z pT and curl u "up to the boundary". The two last situations we study below where 'y > 1, k > 0 and 6 = -y - 1 or 5 = 0 are more delicate and also more meaningful from a physical viewpoint. For reasons similar to those recalled above, throughout the rest of this section we shall make all the same assumptions as the ones we just recalled above and more specifically we assume that N = 2 or N = 3 and we impose the boundary condition (6.44). And exactly as above or as in section 6.7 (Theorem 6.10), we shall study (6.192)-(6.194) with the boundary condition (6.195) and the following normalization:
in
pP dx = M
(6.200)
where M is arbitrary in (0, oo) and p will be chosen large enough later. Finally, we assume that A > 0 since otherwise there does not exist in general a solution. Our main result is the following.
Theorem 6.18 If N = 3 and 6 = ry - 1, we assume in addition that k depends upon T and satisfies (1+T)O1 > k(T) > v(1+T)°C
for all T > 0 with a > 1 , for some v E (0,1) .
(6.201)
Then, for any p large enough, there exists a continuum C of solutions (p, u, T) of (6.191)-(6.194) in LP(Q) x W1"p(Sl) x W2,P(S2) such that: p E L°°(f ), u E W 1,q (f1) for all 1 < q < oo, Du E BMO,
div u - pT and curl u E W 1,q (11)
for all 1 < q < oo, T E W2() for all 1 < q < oo and inf-ff T > 0. Furthermore, for any M E [0, oo), there exists a solution (p, u, T) E C such that fn pp dx = M. In addition, there exists Mo E (0, +oo] such that there exists for all M E [0, Mo) a solution (p, u, T) E C such that fn p dx = M.
Remark 6.40 In the above result, we have taken, when N = 2 or when N = 3 and 5 = 0, k to be a positive constant. In fact, the proof below yields the same result if k depends upon T and (6.201) holds (for example) for some a > 0. We do not know if this assumption suffices when N = 3 and 6 = 'Y -1.
Stationary problems
166
Remark 6.41 As in all the results of this type obtained in the previous sections, we do not know whether the critical constant (mass) Mo is finite or infinite. This is of course a fundamental open question.
Remark 6.42 We have already shown in section 6.9, Example 6.4, that in gen-
eral solutions (p, u, T) with the regularity stated in the above result are no longer regular if there is a vacuum (p ¢ C, u ¢ Cl). On the other hand, if there is no vacuum, i.e. p is bounded away from 0 (assumption (6.166)), then the analogue of Theorem 6.16 holds with almost the same proof; we skip the straightforward modifications and adaptations. The exponent 0 is now given by
0 = +£ (inf ess p) (mina T). In addition, T E C,b +2,« if a # 0, T E Cb+" for IL all ,8 E (0,1) if a = 0.
Remark 6.43 If we replace (6.44) by the boundary condition (6.43) (respec-
tively (6.45)), then we have to replace the right-hand side of (6.194) by 6[p(curl u)2 + ( +µ)(div u)2] (respectively S[j I DuI2 + (div u)2]). With these modifications, Theorem 6.18 still holds in these cases. In addition, if we impose (6.45) and if we keep (6.194) then Theorem 6.18 is still valid provided we assume
that A -kin > 0 on 811 where x is the curvature tensor. 0
Sketch of proof of Theorem 6.18. Since most of the proof is similar to the proof of Theorem 6.10, we only explain how to obtain a priori bounds. In addition, the case when 6 = -y - 1 being more delicate than the case 6 = 0, we detail the a priori estimates in the first case and then explain the modifications required to treat the latter case. Furthermore, we begin with the two-dimensional case and then analyse the three-dimensional one.
Step 1: A priori bounds when S = -y - 1, N = 2. We first integrate over 11 the energy identity (6.196) and the entropy identity (6.197). Using the boundary condition (6.196) and the normalization of p, we then deduce JTdS < C (1 + IItIILP')
n
I TT12 2
fn
+
IDT12
fn
,
dx < C,
(6.202)
(6.203)
where C denotes various positive constants independent of p, u, T. We are going to deduce from these two inequalities bounds on T in La (f') for all 1 < q < oo. Indeed, we first observe that (6.203) implies that V log(1+T) is bounded in L2(1l) and thus, by an easy functional analysis observation, log(1+ T) - fan log(1+T)dS is bounded in HI (Q). By classical Sobolev embeddings in two dimensions, we deduce for any q fixed in [1, oo)
C>
jexp q(1+T)
- fan log 1+T dS dx = I(1+T)q -C°
where Co = fan log(1+T)dS. Then, we remark that we have in view of (6.202)
General compressible models
0< Co = fan log(1+T)dS < log
f
167
n(1+T)dS < log{C(1 + IIUIILA')}
Therefore, we deduce IITIIL4 < C (1 +
(6.204)
IIuIILp')
In particular, we may choose q = 1 and we deduce from (6.203) for all e E
(0,1),e>0 2
IDuldx< In
2E
Tdx J IDTI dx+2fn
2 E
+ CeIIDuIILI
since we may always assume without loss of generality that p > 2 (and thus p' < 2). We then deduce from this inequality a bound on u in Wl"l(ft) and from (6.204) a bound on T in Lq(fZ) for all q E [1, oo). Going back to (6.203), we deduce for all r E [1, 2) IDTI2 n
IDul''dx < C
+T
dx < C.
At this stage, we have shown that T and u are bounded in Lq (1) for all
1 < q < oo and that u is bounded in W'() for all 1 _< r < 2. We then use (6.193) and elliptic regularity to deduce that u is bounded in Wl,'' for all
1 3 is easier (as can be seen from the proof below). First (6.293) by u, we deduce easily that we have 31
IIuIIH1 < C(1 + IIPTIIL2) < (1 + IITIIL2p,(p-2))
,
(6.208)
.
(6.209)
and, (6.194) yields
II - div(kVT) + (pu . V)TIILI < C (1 +
IITIIL2pi(p-2))2
We next claim that we have IITIIL3c1+ag>,co < C
II
- div(kVT) + (pu V)TIILa + 1
.
(6.210)
We have made in chapter 3 (volume 1) several arguments of a similar type and
this is why we only sketch the proof of this claim. Multiplying (-div(kVT) + (pu V)T) by T A R for any R > 0, we find denoting M = II - div(kVT) + (Pu V)TIIL1
A R) (T -To) dS < C R M 1 kIVT121(T 1
in
V (T A R)1+a/2I2 < CRM + fan k(T A R) (T -To)1(T 1 using the bound on T in L3a IIT A RIIi +3a < CII(T A R)l+a/2IIH1
< C (uv[2' A R)1+a/2JIIL2 + II(T A R)1+a/2IIL1) < C(1+M)112 R1/2 + R(1-5a/2)
< C(1+M)112 R1/2
and thus meas {R > T > R/21:5 C(1+M)3 R-3(1+a). This inequality also holds for R < 1 since T is defined on St which is bounded. Then, we deduce for all
A>0 00
meas {T > Al _
meas
{2(1)X > T > 2n A }
n=o 00
< C(1+M)3 A-3(1+a)
2-3(1+a)n
n=O
< C(1
+M)3-3(1+a)
.
This inequality implies our claim (6.210). We then combine (6.209) and (6.210) and we obtain IITIIL3(1+a)..
C (1 + I I T I I i(pi P-2) )
We may choose p > 4 and thus, if a E (3 , 31 , 3a < 2 < p-2 6. In order to do so, we argue by contradiction and we assume that we have a sequence of solutions Tn satisfying the boundary condition (6.195) and
T,,>0 in11,
in1Z,
(6.211)
where c,,, is bounded in L2p/(p+2) (like div (pu)) and bn is bounded in L6p/(p+6) and div bn = 0 in 11 (like pu)-in fact, we could replace 2p/(p+2) by any exponent
greater than 2 and s by any exponent greater than 3-and we assume that +p
IITn1IL2p/(p-2) goes to +oo as n goes to +oo while IITnhLLI(an) is bounded. We
then set Tn
In = IITnfiL2p/(p-2)
and observe that Tn solves the same equation as Tn namely (6.211). In addition,
cnTn is bounded in L'. We may then apply the same argument as in step 2 (taking a = 0) and we obtain for all R > 1
fn
I°TnI21(Tn 0) in three situations : i) the periodic case, ii) the case of the whole space and iii) the case of (homogeneous) Dirichlet boundary conditions. Other situations of interest will be studied later on in this chapter. The precise notion of solution we use is detailed in section 5.1 together with the notation and the conditions on the data (force f , initial conditions po, mo) that we shall use.
Let us also recall from section 5.1 the natural a priori estimates satisfied by solutions, bounds which follow from the energy identity (and possibly Sobolev embeddings). We always assume that po, mo satisfy
poEL1nL7(fl),
po>0a.e.in 11, po00
mo E L2-'/(-'+1)(S2) , mo = 0 a.e. on {po = 0} IMo12/po E L1(0) (defined to be 0 on {po = 0}).
(7.1)
We may now recall these "natural" a priori bounds that we write simply as "p, u(...) E X" where X is some functions space. What this really means is the fact that we have a bound in X that depends only on bounds on the data (po, mo, f) in the spaces they are assumed to belong to. First of all, in the case of Dirichlet boundary conditions, we assume that f satisfies
f E L1(0, T; L2-rl ('r-1) (c)) + L2/(1+a) (0, T; Lr(c))
(7.2)
where a = (2 - -y)+, r +-!(1- 2) + 2 + 1 g« = 1 and q= i2N2 if N > 3, q is arbitrary in [2, oo) if N = 2. Then, we have u E L2(O,T; Ho(c)
,
p E L°°(0,T; L'(&1))
,
PIu12 E L°°(0,T; L1(st))
.
(7.3)
Next, in the periodic case, we assume that f r= L1(0, T; L27/(,'-1) (SZ)). Then, (7.3) holds with Ho (SZ) replaced by H1 (0) provided ^y >- N+2 if N > 3.
A priori bounds
173
Remark 7.1 It is possible to combine (in a rather technical way) Remarks 5.1 and 5.4 and obtain a slightly more general condition on f than f E L1(0, T; L2-r/(7-1)(SZ)), which still ensures (7.3). But the condition we can obtain is very technical and less general than (7.2) and thus we do not know if (7.3) holds in the periodic case under condition (7.2) only. 0 We now consider the case when fl = RN and N > 3. Then, under assumption (7.2)-we can of course in this case replace L27/(7-1) by L27/(-r-1) + L°O-we have L2N/(N-2) (RN)), Du E L2 (RN x (0, T)) , u E L2 (0, T; (7.4) p1u12 E L°O(O,T; L1(RN)) , p E L°°(O,T; L1 n L-'(RN)) .
Finally, if SZ = RN and N = 2, we assume that f E L1(O,T; L2-r/(-f-1)(SZ)) + Li (0, T; L' (f?)). Then, we have u E L2 (0, T;L2 (BR)) for all R E (0, oo), Du E L2 (R2 x (0,T) ), 2)). pluI2 E L°°(O,T; L1(R2)) , p E L°°(O,T; L1 n 1
(7.5)
Throughout this chapter, unless explicitly mentioned, we always assume that the various conditions detailed above on the data (which lead to (7.3) or (7.4) or (7.5)) hold, and we shall not recall them. We may now turn to the crucial a priori estimates we want to derive in this section.
Theorem 7.1 Let (p, u) be a solution of (5.1)-(5.2). We assume that f E Li (0, T; L3/2) if N = 3 and -y > 6 and f E L1(0, T; L2) if N = 2 and -y > 2. We assume in addition that p E Lp(K x (0, T)) with j5 = max(p, 2) and p = y+ N 'y -1, K = SZ except in the case of Dirichlet boundary conditions where K is an arbitrary compact set of 11. Then, p is bounded in LP(K x (0, T)) in terms of bounds on the data only.
Remark 7.2 A somewhat technical extension of the proof allows us to prove the above result assuming only that p E L'10i We then have to show that p E LP (if p > 2) and prove the bound in LP by essentially the same proof as the one given below, but using first some truncations of the power functions introduced below in a way which is similar to what we did in the regularity arguments introduced in section 6.3 (chapter 6).
Remark 7.3 It is possible to provide some "explanation" for the exponent p occuring in the above result. Indeed, on one hand p E Lr(Lt) ("energy estimate") and on the other hand, if we simply expect p'y and pu ® u to have the same integrability, we deduce that pry-1 should belong to Lt (LN/(N-2)) since
u E L2 (Lx /(N-2)) by Sobolev embeddings (again the "energy estimate"). These
two bounds imply by interpolation that p E Li,t with p = 'y + N y - 1; notice by the way that if N = 2, we cannot quite say that u E Lt (Lz°) but nevertheless we can reach the exponent p. The heuristic argument leads to the following "conclusion" : the derivation of improved bounds on p requires an improvement
Existence results for Cauchy problems
174
of the "energy estimates", and this is obviously a fundamental question. Let us mention however that we cannot be too optimistic about bounds on p as can be seen from the following remark. Remark 7.4 A recent example by V.A. Weigant [550] indicates that, even with rather smooth data, we cannot expect bounds on p in L' for p large, at least if -y < 2 when N = 2. Notice that the exponent p in Theorem 7.1 is greater than
yif y> N,i.e.y>1if N=2. D Proof of Theorem 7.1. Of course, there is nothing to prove if y < 2 since p E L'(0, T; L1 n L'w). Therefore, in all that follows, we assume that y >
Step 1: Proof of Theorem 7.1 in the case when 11 = RN, N > 3 or in the periodic case. We present the proof in the case when St = RN, N > 3, and indicate briefly afterwards the modification to be incorporated in that proof
in order to treat the periodic case. We let 9 = p - -y and we recall from the proof of Theorem 5.1 (chapter 5, section 5.3, see in particular identity (5.53)) the following identity
ap7+e = at
[(pe)(-A)-1
div(pu)] +
+ (µ + t;)(divu) pe + (9-1)(divu) +pe[R=R,(putu3)
- u2RRR(pui)] -
div[u(pe)(-0)-1
div (pu)]
pe(-0)-1
(7.6)
div (pu)
pe(-0)-1
div (pf).
Then, we recall that plr and pIuI2 are bounded in L°"(O,T; L1) while Du is bounded in L2(RN x (0, T)) and u is bounded in L2(0,T; L2NI(N-2) (R N)). In particular, pu is bounded in L°O(0, T; L27/(7+1))nL2(0, T; L") withr1 = N-2 2N + 1 and thus, pe(-A) div(pu) E L°°(O,T; L1 n L3) where + 2ry1 77 _ If N + 1 27 < 1. Integrating (7.7) over RN x (0, T), we then deduce
-
-
1T1 PA d x
dt
C 1 + fdx fRN dx [Idiv u(1 + I (-O1 div (pu) I) + PBIRtiRj(Puiuj) - ujR=Rj(pu;)I +
peI(-A)-1 div (p.f)I,
(7.7)
We next observe that the three first terms in the integral of the right-hand side of (7.7) are bounded in L1 (RN x (0, T)), except when N = 3, y > 6 where the first term is bounded by CII PII ii
P
: indeed, Idiv uI I(-0)-1 div (pu)I pe is bounded
C L1. since 2+,-1.-N+e = 1. This argument is incorrect when r > N and this is possible only when N = 3 and y > 6 (recall that N > 3 in this step). In that case (of marginal interest) we argue as follows: we write III div ul
pe(-A)-1
div (Pu)IIL1,t
IIDuIIL=,, IIPIILP II pull LiP/(P-se)(L=P/(SP-BB))
A priori bounds
< C IIPIILz t g < CIIPIILzjt
175
IIPUIIL(1oy-6)/(y+3)(L3x (1oy-6)/(13-(+3))
2(2-y-3)/(57-3)
(7+3)/(5-y 3)
6))
IIPUIIL$°(L?.y/(7+1))
< CIIPIIL=,i IIPIIL /(L=)3) IIuIIL2(Le)(5ry-3) IIVPUIIL-(L=)/(5ry-3) C IIPIILz,,
+ p-29 = 1, 1 + e + 5p-66 1 = 1, 6p = 3 107-6 < 6p 5p-69 137+3 2 p 2p /(N-2)) 3. Similarly, p°R%R;(pu2u,) and peuiR4-RR(puj) are bounded in L' (Lx Lt°(Li) Lt°(Lile) C Lt'z since (again) N 2 + It = NN-2 + N2 = - 1.
since we have 2i +
- 3-
p
Finally, we consider p0(-0)-1 div (pf ). If N > 4 or if N = 3 and 2 < 7 < 6, pe(-A)-1 div (pf), in view of (7.2), is bounded in Lto(L7/e n Lz) +L?(-A)-1/2{Lz'La}] c L'(L2nL')+L2(Len
Lz)wherea1-72N -TNa X
c
'v
^1
+-
'
271 2=
1
77
=-L+1--L yy l . Therefore, C is bounded in
(-A)-1/2{Lr(L-()
LeOO(Lx/e)
.
Lt/(3-y)(Lz)]}
(Lxy/(y-1)) +
[Li
C Lt (Lx) +L /(3-y) (Lx )
t
where 1=e+-1+
2=1-2y 2, we assume that f E L' (0, T; L2). Therefore, C is bounded in
Li°(L'y/e) (_A)-1/2 {L°°(Lz) Lt (Lz)} C L2,t 0+1
since
=1.
The two remaining terms, namely A and B, can be handled as follows. First
of all, pu = f f u can be estimated as follows IIPU
IIL,P(LyP/(P+1)) < C
IIPIIL
,,
Then, we have IIAIIL=,t < C IIPIIL=,t
since 1 +
e
+
p P1
-
2
IIPUIIL,P(L=P/(P+1))
= 2 + 20+1 = 1 and 2 + 2 + 2p = 1. Hence, (7.9) yields IIAIIL=
< C IIPIIL
(1/2)
(7.10)
Next, we use the fact that u is bounded in L2(O,T; H1) and thus in L2(O,T; BMO). Then, by the Coifman-Rochberg-Weiss commutator theorem [1111, we have for almost all t E [0, T) IIRiR;(Puiuj) - uiRiR;(PU;)IIL2P/(P+1) < CIIUIIBMO
IIPuLL2P/(P+1)
Therefore, we have I)RjR.i(Puiu.i) - ujRiRR(Puj)IIL2p/(P+1) t 0 on SZ.
We may then follow the argument in section 5.4 and we obtain easily
t
(`p
div(pu)) + 8ij (cppuiuj) - (µ+)O(co div u) + aA(pcoy) = div(cpp f) + F
where F is given by F = (ei, cp) puiuj + 28icp8, (puiu3)
-
u)
.
+aOcppy-2(µ+C)VW V divu+2aVco Opy-pf VV.
(7.12)
The above equality then yields the following "local" version of (7.6), namely acppy+e =
[p°(-0)-1(p div(pu))]
at + div[upe (-O)-1(cp div(pu))J
-
(7) .13
+ pe{R=R3(wpuiu3) - u V(-0)-1(Wdiv(Pu))} + (0-1)(divu)pe (-0)-1(cpdiv(pu)). Then, if N > 3, the proof made in step 1 immediately yields an L1,t bound on copy+a and thus on fK pP d2 since all the terms, except the new one, namely
Existence results for Cauchy problems
178
pe(-0)-1F, are handled in exactly the same way. We thus only have to obtain an Lit bound on pe(-A)-1F. One then checks easily that, of all the terms
entering the definition (7.12) of F, only the term 2aVcp Op1' requires some analysis. Indeed, all the others are in fact "smoother" than the terms handled in step 1. If -y < N, we estimate pe (-A) (Vv - V pry) in Lr (La) using the bound on p in Lr(L t) where a is arbitrary satisfying 1
0
a
-y
+
N-1 N
=
2
1
N
y
+1-
1
1+
N
1
1
N
y
and in particular we may take a = 1. Notice that this argument also applies in the two-dimensional case. If -y > N, we first write p9(-A)-1(Ocp Op'') _ p°(-A)-1(div (p70cp) p1Ocp). Then, we estimate the two terms as follows
-
II(-0)-1div(p"V
with
)IIL:/,(L=) < CIIP1IV l"ILP1
div(p1' 4')IIL'/-1 (Lb) :5 CIIP''IAWHIL=/e
with
1 _-y a
p
1
-y
b
p
1
N' 2
N*
We then deduce from (7.13)
f
T cppP dx dt
C1+
(fTf
dt
(7.14)
providedy + 1a 2. The first of these two and the term cp"'Wo(-0)-1(div terms is easily bounded provided we choose m and cp in such a way that we have
Ipcp"`I < C.
(7.15)
We then argue as in step 2 and we find, provided (7.15) holds, the following estimate
1T1
+m pP dx
< C 1 + IIIIL/t IIJ IIL+ IIwmpe(-o)-1(div
(P''VV) - P"I(P)IIL=,t
.
Hence, if we choose m such that mp/B > 1 + m and 2p > 1 + m, then the preceding inequality yields T
f0
dt
J
dx cpl+m pP dx
< C 1 + II
mpe(-0)-1(div
(p7vcp)
II(-o)-1(div
C 1 + IIcomPolILs/o
C 1 + IIVmPOII L=/t IIP''IIL=t
< li
1 + IIcmPeIILP/e z,t
-
(p''V) -
1 + m. In particular, we may take m = 1, then (7.15) automatically holds and we conclude since p > 1
andp=2-y-1 > 28=2(ry-1). 0
Existence results for Cauchy problems
180
Remark 7.5 The proof above (step 3) is entirely local and only requires (p, u) to be a solution satisfying the following local bounds: plul2 and p1 are bounded in L°O(O,T; L C(1)), and u is bounded in L2(0,T; Hoc(S2)). Then, we obtain a bound on pin L7 (O, T; LP (1)) . 10C Remark 7.6 In the case of Dirichlet boundary conditions and when N > 3 and 2 < 'y < N, the proof made in step 3 shows that we may in fact choose cp to be strictly positive on SZ and equal to dist (x, 01) in a neighbourhood of act. Thus, we obtain in this case the following estimate T
dt
J
f
n
dx pp dist (x, & l) dx _ 2 i.e. -y > 2 if N = 2, ry > s if N = 3. Therefore, improving the requirements on p in the compactness results of chapter 5 or improving the bounds on p in Theorem 7.1 would lead to improved conditions on -y in two or three dimensions.
However, the first restriction on 7, namely -y > 2, is absolutely essential to our analysis. Let us give one example of the manner this condition is used in our proofs (see also chapter 5 for similar arguments) p E LO°(0,T; L") and u E L2 (0, T; L2N/(N-2)) (Sobolev embeddings) imply that pu ® u E L' (0, T; LP)
for some p > 1 if and only if 7y > E.. Improving this restriction is a fundamental
open question: for instance, can one prove an analogue of Theorem 7.2 when -y > 1 and N = 2 or 3?
Remark 7.9
7 if N = 3, y > 6 if N = 4, the proof of
Theorem 7.2 yields the following (version of a) local energy inequality in the sense of distributions say in the case = 0 (in order to simplify notation) 8 (.1plU12 (PIuI2+
at
a
py
+ div u p
ry-1
- µA
2
2 -v
ary 2
-1 pu.f ^f
p'
(7.18)
where v is roughly speaking-we do not wish to detail here the precise meaning of what follows-a bounded non-negative measure in t with values in LN/(N-2) (or in LP for any p > oo if N = 2) supported on the set l p = 01-the vacuum. Let us mention by the way that we do not know if p remains strictly positive when po is strictly positive. In particular, we can check that v = 0 if the set {p = 0} has zero measure, and, exactly as in Remark 7.7, the equality in (7.18)-which is to be expected at least formally-is another interesting open problem.
Remark 7.10 Theorem 7.2 is the only known global existence result for general initial conditions. Many references are given in the bibliography which prove the existence of smooth (or relatively smooth) solutions-with uniqueness results in some cases-in special regimes. For instance, locally in time smooth and
182
Existence results for Cauchy problems
unique solutions do exist-see also the following remark for further details on that aspect-and global in time solutions close to an equilibrium (i.e. p close to a strictly positive constant and u close to 0) exist. For the latter type of results, probably the most general result is the recent work by D. Hoff [2511 which shows that if po is sufficiently close (in the L°° norm), then a global solution exists with
(p, u) staying close in L°° x Li to (p, 0). 0
Remark 7.11 We wish to mention here the recent examples by V.A. Weigant [550 presenting the formation of singularities in finite time of "smooth" solutions
(p, u) of (5.1)-(5.2). More specifically, if -y < 1 + N (and thus if 1 < y < 2
when N = 2), one can find f E LQ(l x (0, T)) for some q > N such that the maximal, local in time, unique solution (p, u) of (5.1)-(5.2) with Dirichlet boundary conditions (f is the unit ball) such that u E W 2',t ,4, p E LO ° (Wz ,4 ),
inf p > 0 for all t in the existence interval, 7 E Lr(Li), blows up in finite time. In other words, the maximal existence interval [0, To) is finite (T0 < oo) and (sup p)(t) goes to +oo as t goes to To. These examples show various facts of considerable interest: i) first of all, the special regimes briefly described in Remark 7.10 above which allow the global existence of smooth solutions do not capture some really non-linear and delicate phenomena for this system of nonlinear partial differential equations, and ii) we have to be careful with the type of regularity we may (or can) expect for p and u. 0 7.3 Existence proofs via regularization Both this section and the next section are devoted to the proofs of Theorem 7.2. In fact, we present two different proofs, one in each section. Of course, both involve approximating and "regularizing" the system of equations (5.1)-(5.2) but the type of approximation we propose in each section differs substantially. In section 7.4, we make a natural time discretization of (5.1)-(5.2), which allows us to use the results of chapter 6 and we then check that we can pass to the limit and recover a solution of our original system. In this section, we build solutions with the properties listed in Theorem 6.2 by a convenient approximation (and regularization or simplification) of (5.1)-(5.2). In fact, the approach here is much more in the spirit of what we did on time discretized problems in chapter 6. However, unlike what we did in chapter 6, we present various regularization procedures which keep some of the basic properties of (6.1)-(6.2) like non-linear transport for the density p, compactness analysis as in chapter 5, bounds on p as in section 7.1 above. These procedures can be seen as a collection of "tricks" rather different from those introduced in chapter 6 for stationary problems. Of course, not only do we not claim that this approximation, namely the one developed below, is the only one (or even the simpler one!) that yields Theorem 7.2, but we want to make clear that several other approaches are possible, some of which use some ingredients of the proofs below. We decided to present a particular one even though it is a "multi-layered approximation" for several reasons: first of all, as stated above, it has no similarity with the
Existence proofs via regularization
183
particular approximation strategy introduced in chapter 6; next, we believe that some of the tricks mentioned below might be useful in other contexts; and finally, it also makes contact with some problems discussed in chapter 9. As indicated above, our approximation involves several layers and we thus split our presentation into several steps (one per layer at least). This is why most of the proof of Theorem 7.2 presented in this section will be made "backwards": if equation(s) (B) approximates equation(s) (A), we first explain how solutions of (A) can be obtained from solutions of (B) before explaining how to obtain the latter-possibly via solutions of (C). Finally, we should warn the reader that some of the details (in the various steps) are quite tedious, so we have tried to present the main ideas. Anyway, there is always the possibility of having a glance at the approximation tricks introduced in this section and immediately jumping to the next section! Finally, we begin our first proof of Theorem 7.2 by considering the periodic case and then we treat the other cases, namely the case of Dirichlet boundary conditions and the whole space case.
Step 1: Periodic case; preliminary reductions. We first claim that it is enough to prove the existence of solutions satisfying (7.17) in the case when f is smooth in (x, t) and periodic, po is smooth, periodic and positive on SZ = 11N 1 [0, Tti] (T1,... , TN are the periods), and mo is smooth and periodic on st. Then, for general data (f, po, mo) (as in Theorem 7.2), we build a sequence of data (f", po, mo) such that f" converges to f in L1(0, T; L27/(7-1)( ))N, po converges to po in L" (St) and mo = po v" where v" is smooth and periodic, in L2(f)N. v" converges to VJW Then, if we have shown the existence in the case of smooth data, we obtain a solution (p", u") of (5.1)-(5.2) satisfying (7.17) with (f, po, mo) replaced by (f ", p0 n, M0 n) Next, we deduce easily from (7.17) (and from (5.1)) as in section 5.1 of chapter 5 that p" is bounded in L°° (O, T; L'Y(f) ), p" lu" I2 is bounded in L°O(O,T;L1(11)) and u" is bounded in L2(O,T;H'(11)). Furthermore, p" E LP (11 x (0, T)) with p = y + N -y - 1 > 2 by assumption (see Remark 7.8
above) and we may apply Theorem 7.1 to obtain a bound (uniform in n > 1) on p" in LP(1 x (0, T)). We may then apply Theorem 5.1 (section 5.2 of chapter 5) to deduce, extracting a weakly convergent subsequence of (p", u") in L'' (11 x (0, T)) x L2 (0, T; Hi (St)) for instance, if necessary, that p" converges strongly to p in C([0,T]; L''(Q)) for all 1 < r < y while p" u" converges strongly
to pu in L-(0, T; L"(11)) for all 1 < s < oo, 1 < r < 2, and (p, u) is a solution of (5.1)-(5.2) satisfying the properties listed in Theorem 7.2 including (7.17) thus proving our claim. We also wish to observe that, when N = 2 and -y > 2 , it is enough to prove the existence of a solution (p, u) of (5.1)-(5.2) such that p E LQ (S2 x (0,T)) for all 1 < q < p = 2-y -1. Indeed, we then claim that, in fact, p e LP(S) x (0, T)). In order to prove this claim, we inspect carefully the proof of Theorem 7.1 (step 2) replacing p by q arbitrary in ('y, p) and 9 = -y -1 by 9 = q -p. We may then copy
Existence results for Cauchy problems
184
the proof made in step 2 (Theorem 7.1) and find a bound on p in L9 (SZ x (0, T) ) independent of q E (-y, p). This obviously implies that p E L"(fZ x (0, T)) and we conclude. In addition, we only have to consider the case when -y > 2 in two dimensions. Indeed, once we have obtained a solution (p, u) for -y > 2 with p E L21-1(SZ x
(0, T)) satisfying (7.17), we may deduce a solution (p, u) for ry = z with p E We only have to use L2(SZ x (0,T)) satisfying (7.17) upon letting ry go to Theorem 7.1 (and its proof) which yields a uniform bound on p in L2' I ' (IZ x 2 (0, T)). Then, observing that 2 , we may apply Theorem 5.1 as we did above and recover a solution in the case when -y = Let us finally explain one aspect of the proofs that we skipped until now, namely the possibility of passing to the limit in (7.17). This is indeed straightforward since (7.17) is equivalent to 2+.
2.
fo dt ffi dx WW
2PIu!2
+,y a 1P7+ f
ds[µIDu(x,s)12+t;(divu)2(x,s)-pu f]
(f(t)dt) f
f22
Jtno po
+ rya 1 PO d2
for all cp E Co (0, T), cp > 0. We may then pass to the limit in this inequality as soon as Du converges weakly in L',t, p converges in Lz,t and Ct(Lz) for some
p > 2 (in particular this is the case if it does so for all p < 'y and y > 2 !), po converges in L2, po converges in L7 and pu is bounded in the dual spaces of those to which f belongs, which is always the case in view of the assumptions made upon f and the bounds proven on (p, u)!
Step 2: Periodic case; smoothing pat + pu V. As seen in the preceding step, we may assume from now on that all the data are smooth (and that when N = 2 and when -y > 2 , we only have to build a solution (p, u) with p E L4 (SZ x (0, T) )
for all 1 _< q < 2-y - 1). In order to build a solution (p, u) of (5.1)-(5.2) as in Theorem 7.2 (except when N = 2 as just recalled) including the inequality (7.17), we are going to approximate the system of equations (5.1)-(5.2) by (recall that all functions are periodic)
+ div (pu) = 0
,
p > 0 in RN x (0, T)
div ((pu)E ® u) - AAu - V div u + aV p" at (p£u) +
(7.19)
=pEf inRNx(0,T),
where e E (0,1) and cp, denotes, for an arbitrary periodic function cp, cpE = W * nE
where rc£ _ - ,c(), K E S(RN), ic > 0 and in RN and fRN rk dx = 1-for
Existence proofs via regularization
(27r)-N/2 a-1x 2/2
instance !c = have obviously
and is =
(29re2)-N/2
185
exp -(IxI2/262). Since we
+div(pu)E = 0, we may write equivalently (at least formally) the second equation in (7.19) as 8u
pf inRN x (0,T).
peat +(pu)£
Therefore, the approximation we use consists in smoothing in the momentum equation the "total derivative" (P At + Pu V). Let us assume temporarily that we have already built a solution (p, u) of (7.19) satisfying plt=o = po, ult=o = mo/(po)e and the same integrability requirements as in Theorem 7.2-in fact, we shall build in step 3 below a solution (p, u) with p E LP((SZ x (0, T)) and p = (1 + N)-y if N > 3, p E [1, 2-y) if N = 2-and the following analogue of (7.17) for the system (7.19):
f
n
2 pelul2 +
7
3, pcu is bounded in LOO (0, T; n L2 (0, T; Lr) with 1r = N-2 2N
1 3, 1 < p < 2-y - 1 if N = 2. Obviously, pk also converges-for the same topologies to p-since a goes to 0. Since, at = div [(pkuk)£], we may still use Lemma 5.1 (section 5.2 of chapter 5) in order to deduce as in the proof of part (1) of Theorem 5.1 (section 5.2 also) that pkuk and pkuk both converge weakly in L2(0, T; L'') to pu where r = 7 + 2N if N < 3, 1 < r < -y if N = 2. In addition, since pkuk is bounded in L°°(0,T;L2"r/(7+1)), we also deduce that pkuk converges weakly to pu in LO°(0,T; L27/(7+1))-weak*. In order to conclude, we have to check that (pkuk)£ ® uk converges (say in D') to pu O u and that part (2) of Theorem 5.1 also remains true. And in fact we only have to check that assertion (5.19) in part (2) of Theorem 5.1 still holds and even more precisely that we can adapt the proof of part (2) of Theorem 5.1 made in sections 5.3-5.4 of chapter 5. Both facts involve the same difficulty: replace (pkuk)£OUC by pkuk®uk or [RiRj((pkuj)Eui)-uiR2Rj(pkuj)] (for some small enough 0 > 0) by (pk)e[R4 Rj(p9 uj ui) uiRR,(pk uj)]. If these substitutions are possible, then we conclude easily adapting the proof of Theorem 5.1. It turns out that these substitutions are straightforward once we observe that we have (dropping the index k in order to simplify notation) for all 0 < 0 < 1
-
(PU) E (x, t) - (PE U) (x,
t)1 = JRN P(y, t) [u(y, t) - u(x, t)] r.E (x -y) dy 1-e
< [(P119)E]0 [JRN
I U(y, t) -U(x,
t)1i/(1-e)
r e (x-y)dy
Therefore, if we take p E [2, N N2) (p < oo if N = 2) and define r by T = we obtain for almost all t E (0, T) choosing 0 = 3 7
-}- P1
187
Existence proofs via regularization
II ((Pu)E - (Pcu))(t)IILr
r/P r [IRN
fn
Iu(y, t) -u(x, t)I P/roc (x-y)dy
dx Iu(y,t)-u(xt)IP/rKE(x-y)dyjr)
1/P
f dxl f N
< II(P1/r)EIIL
Jft dx
pP+--'
n
1(p>,) - b
T
J
dxpn in
dxp 1(p 0. Next, u6 is bounded in L2 (0, T; H') while we have (skipping S) au
l;
1
div((pu), ®u) + µ 1 Au + PC
-div
it div PE
+ ((PU)eV!) PE
+aV
P
a
a W, + f -
1
div u
PC
PE
- Vu + l V
u-µ O (!).v) u PE
CV
1
S
6 (PI )Eu
- aV p
+f
PE
divu
P£
(p")eu,
p Pe
V div u -
PE
and thus at is bounded in L2(0,T; H-1) + LOO(O,T; W-1,1). This implies that ub converges in L1(1 x (0, T)) to u and thus in L'' (0, T; L2 (11)) (Y r < oo) or in L2(0,T; LP (11)) (d z
< 2 2)
Step 4: Periodic case; truncation of the pressure. In this step, we propose our final approximation (or final layer of approximation). We are going to construct solutions of (7.24) (with the regularity mentioned in step 3) by taking limits of solutions of the following system of equations
ap+div(pu)+Sp"=O, p>0 U1;
PC
au
at
l
-µ0u-eVdivu+aV(pAR)1 (
(-7 30 )
= PEf in RN x (0, T) ;
all functions are periodic in x; (p, u) satisfy the following initial conditions: PI t=o = Po, ult=o = mo/(Po)e; and R > 1 is a truncation parameter. Let us recall that a A b denotes min(a, b). We are going to explain in the next step why there exists a smooth solution (p, u) of (7.30): p E C([0,T]; W1,q) and u E W2,1,q = {cp E Lq (O, T; W2,q) with
E Lq (St x (0, T)) } , for all 1 < q < oo. In fact, if we replace p A R by a smooth truncation [i.e. (R(P) = R((p/R) where C(0) = 0, S' E Co (R), ('(t) > 0, ('(t) = 1 if It) < 1, and C'(t) = 0 if Itl > 2], we can even check there exists a smooth solution (p, u) E C°° (S2 x [0, T]) .
Next, we explain how one can recover of solution of (7.24) satisfying the
properties listed in step 3 by passing to the limit as R goes to +oo in the system (7.30). In order to do so, we have to explain (as in the previous steps) how to pass to the limit and that we have the following a priori estimates: p is bounded in LO° (0, T; Lq), u is bounded in Lq (0, T; W 1'q) and is bounded in Lq(0,T; W- 1,q) for all 1 < q < oo. We begin as usual with the bounds (that may of course depend upon S but are independent of R). First of all, the energy identity in the case of (7.30) takes the following form as is easily checked by elementary computations:
Existence results for Cauchy problems
192
a
p+ div IuI
+S(pp)EIuI2-µs
= pfu f
[(PU)e_+aU7R(P)]
+a5o' (P) Pp
2
(div u)2
(7.31)
in R x (0, T)
dt and ryR (P) = /3R (P)p, hence /3R (P) _ X 11 (p A R)'1-1P + R11-1(P - R)+, QR(p) = 711 (P A R)7-1 and 7R(P) = fll (p A R)1-1p. where 13R (P)
= P fop
As usual we deduce from the equation satisfied by p and (7.31) that u is
bounded in L2(0, T; H1), p and OR (p) are bounded in L°° (0, T; L1) and thus pE is bounded in L°° (0, T; Ck (S2)) for all k >_ 0, pp is bounded in L1(S2 x (0,T)), p' Iu I2 is bounded in LO° (O, T; L1), (pp) E I uI2 and f3(p)p" are bounded in L1(f Z x (0,T)).
We next claim that (7.28) holds uniformly in R > 1 and in S as long as S is chosen as in the preceding step. Indeed, we only need to observe that the argument made in step 3 still applies replacing pP+'r -11(p> 1) by pp (p A R)'' (p> 1) (= y-f 1 pP 1O(p) Therefore, u is bounded in LOO (0, T; L2) and, in particular, (pu)e is bounded in LP (0, T; Ck) (for all k > 0) in view of the bound on pP mentioned above-recall that p > 1 + oy > 2. We are now going to prove that p is bounded in LOO (0, T; P), u is bounded in
L'(0, T; W l,) and au is bounded in L' (O, T; W-14) for all s E (1, oo). Indeed, we have for any q E (1, oo)
a at P4 + div (up') +
g5pp+q-1
= (q-1) div u pq .
Therefore, we obtain, denoting s = (p + q -1)/(p - 1), IIP(P-1)
II PII
1 + -y, we deduce from (7.32) and (7.33) the bounds we claimed above
provided we can bound the last term of the right-hand side of (7.33) for all s E (1, oo). We then check this by a bootstrap argument: in view of the bounds
Existence proofs via regularization
193
shown above, we can take s = 2, r = oo and the above argument yields a bound on pin L°° (0, T; LP- 1) and thus in L°O (0, T; L'Y). In particular, if 'y > 2, then pu is bounded in L°° (0, T; Li ). We may then choose r = s and deduce from (7.33) IIUIIL.(O,T;wi')
0). 0
Step 5: Periodic case; solution of (7.30). We only explain how to obtain a priori estimates on a solution (p, u) of (7.30). We have already obtained in the preceding section estimates on p in C([0,T]; Lq(SZ)), u in Lq(0,T;Wi,q(SZ)) and on i in Lq (0, T; W -1 ,q (S2)) for all 1 < q < oo. We show here that it is possible to obtain bounds on p in C([0,T];W1,q(Sl)), at in C([0,T];Lq(SZ)), U in Lq (0, T; W 2,q (SZ)) and on au in Lq (SZ x (0, T)) for all 1 < q < 00. With such bounds, it is relatively easy to build a (unique) solution and we leave it to the reader. Let us also recall (from step 4 above) that we may even obtain C°° solutions if we replace p A R by a smooth truncation. In order to prove the bounds we just stated, we adapt and extend a method of proof due to V.A. Weigant and A.V. Kazhikhov [552]; see also A.V. Kazhikhov [294]. We first prove that p is bounded in L°°()). Indeed, we write
Existence results for Cauchy problems
194
at
(log p) + u
/(log p) + div u + bpp-1 = 0
(7.35)
and
div u =
µ
-
+ (P A R)ry
-
+
(-0)-1 div (b(pP)Eu)
Jt
(-A)-i
(p A R)ry +
div (P£f )
- RiRj ((Pui)Euj) - a
(-L)-1 div
(PEu)
All the terms, but the last one, of the right-hand side of the preceding inequality are clearly bounded since pE f , (p")Eu are bounded in LOO (SZ x (0, T)) and (pui)E is bounded in L°° (0, T; Ck (Ii)) for all k > 0 while u is bounded in L°° (0, T; Ca (SZ))
for all a E (0,1) by Sobolev embeddings. Similarly, -D = bounded in L' (n x (0, T)). Therefore, we deduce from (7.35)
a
-5j (log
p+P)
+bpp-
1
div (peu) is
= XF
where W is bounded in L°° (SZ x (0, T)) since u V4D = uiRiRj is bounded in L°O(11 x (0, T)). Since 5P-4 > 0, this equation implies, using the maximum principle, a bound from above on log p + This yields the desired bound on p in L' (f2 x (0, T)). Let us observe, in passing, that the above equation also yields a bound from below on p. Next, we differentiate the equation satisfied by p and we obtain easily for all q c- (1, oo) using the bound on p in L°O(SZ x (0, T))
at I VPI9 + div (uI VPI4) < C [I DuI I VPIq + ID2uI IOPIQ-1]
.
This inequality implies that we have on (0, T) dt IIVPIILq(n) N IIDuIILOO(nx(O,t)) < C 10g(1 + oma t IIV(P A R)''IIL9(n))
therefore
Existence proofs via regularization
195
IIDuIIL-(nX(o,t)) < C 1og(1 + o0
a (p u ee) + div (pEE u ® uE) 1
+ aV (p')7 + pue = P.f ,
µ0u'_ V div uE
(7.39)
Existence results for Cauchy problems
196
where f is an extension of f to fl x (0, T) by 0 and is then extended periodically to RN x (0, T). The proof made above (steps 1-5) immediately adapts to the system
(7.39) and thus yields a solution (ps,uC) of (7.39) satisfying all the properties listed in Theorem 7.2 in the periodic case replacing obviously (7.17) by
f
1(Pe)7
1 PE Iuf I2 +
a
dx +
'Y
< fc] 2
Imo Po
I2
a
+
Jo
po dx +
'Y
ds rt
J_
tt
dx
ds
Jo
n
[IDuf2 + l;(div u,)2 + dx pu"
EPIteI2
a.e. t E (0, T)
(7.40) Of course, p£ I t=o = Po, peui I t=o = mo and (p"o, rno) are periodic extensions of
(po, mo) by 0. Exactly as in step 1 above, it is enough to consider the case when f is bounded on RN x (0, T). This inequality (7.40) yields, as usual (by now), bounds on (p`)l and p1Iu2I2 in L°°(0,T; L' (fl)), on uE in L2(0,T; H1(SZ)) and on EpjU I2 in Ll(St x (0,T)) uniformly in e E (0,1). In addition, the proof of Theorem 7.1 (see in particular Remark 7.5) in section 7.1 yields a bound on pC in LP (K x (0, T)) for any compact set K C SZ where p = 'y + N'y - 1. Extracting subsequences if necessary, we may assume that p£ converges weakly in L''(0, T; L'r (fl)) for all r E (1, oo) as a goes to 0+ to some p (> 0) E LO°(0,T; D (fl)) n LP(K x (0, T)) for any compact set K C S1, and we may assume that u` converges weakly in L2 (0, T; H1(!)) as E goes to 0 for some u. In addition, since T J ue is bounded in L2(11 x (0, T)), VfP- u£ converges to 0 in L2 () x (0, T)) and thus lp- u =- 0 in fl x (0, T). Therefore, u = 0 a.e. in (St - St) x (0, T). Since U E L2 (0, T; H1 (fl)) and n is smooth, this implies that u E L2 (0, T; H01).
Finally, in order to conclude, we simply observe that Theorem 5.1 (in chapter 5, section 5.2) can be applied in our context inside the domain fl x (0, T). Indeed,
the proof of part (1) of this result is still valid here while the proof of part (2) also applies as one can easily check observing that we only need the limiting velocity namely u to belong to L2(0,T; Ho (SZ)).
Step 7: St = RN. Let us explain briefly one possible way to deduce the existence
of solutions in the whole space case from the results shown in the preceding steps. First of all, as explained in step 1 above, we only need to consider the case when f E L' fl L°° (RN x (0, T)). We then consider the solution (pR, uR) of (5.1)-(5.2) set in the ball BR(X (0, T)) with Dirichlet boundary conditions on BBR. We choose R > Ro so that fBR Po dx > 0. The existence of such a solution (satisfying the properties listed in Theorem 7.2) is ensured by step 6 above. In particular, we have for almost all t E (0, T) JBR
PRIURI7 + a 1(PR)dx +
ds J J ft1 < R
BR
dx pI DuRI2 + Z; (div uR)2
Existence proofs via time discretization
197
The conservation of mass (equation (5.1)) and this inequality yield, as usual, bounds uniform in R on pR in L°O(0,T; L' n L1' (BR)), on ARI URI2 in LOO (0, T; Li (BR)) and on DuR in L2 (BR x (0, T)). If N = 2, we deduce, as in Remark 5.1, a bound on uR in L2(O,T;Lq(BM)) for all 1 < q < oo, M E (0,00) and in L2(0, T; BMO)-considering UR as a function on R2 by extending it to R2 by 0. Next, a careful but straightforward examination of the proof of Theorem 7.1
shows that pR is bounded in LP(BM x (0, T)) for any M E (0, oo), with p = -y + N 7 - 1, uniformly in R > 1 + M. It only remains to apply the compactness analysis of chapter 5 (Theorem 5.1 in particular) in order to recover a solution of (5.1)-(5.2) in the whole space. 0 7.4
Existence proofs via time discretization
This section is devoted to the presentation of another proof of Theorem 7.2. This proof is based upon a simple time discretization of equations (5.1)-(5.2). Let M > 1; we denote At = T-r and define by induction for k > 1 (pk, Uk) to be the solution of Ot pk + div pkuk
(pkuk) = Qt pk-1
+ div(pkuk ® Uk) - µiuk - Vdiv uk + a0(pk),'
(7.41)
(7.42)
1
= pk fk + 1 Mk-1 and mk = pkuk, po = po, m° _= m0, f k is a convenient discretization of f to be detailed below. More precisely, since the uniqueness of solutions of such problems is not known, we shall in fact simply take one solution whose existence follows from the results of chapter 6, and Theorem 7.2 will follow upon letting At go to
0 (or M go to +oo).
Step 1: Preliminaries. Exactly as in the proof of Theorem 7.2 given in the previous section, we shall only consider the periodic case since the other case can be "deduced" from it, even though the proof presented below can be adapted to
the other cases directly. Similarly, we may assume that f is smooth. Then, we solve (7.42)-(7.43) requesting all unknown functions and all data to be periodic in each x2 of period Ti > 0 (V 1 < i < N). And we may simply take f k to be f (x, kOt); if we were not assuming f to be smooth, we would have to take f k to be (of 4k-1)ot f (x, s) ds for instance. In addition, we may assume that po is smooth and positive on RN and that uo is smooth (or simply L°°). Finally, when N = 2, we have explained in the preceding section why it is enough to study the case when -y > 2 and to prove the existence of a solution (p, u) satisfying p E L2 ,t for all q E [1, 2-y -1). We may now use the results of chapter 6, namely Theorems 6.1-6.3 (section
6.2), which ensure the existence of (pk, uk) (for 1 < k < M) periodic, solving
Existence results for Cauchy problems
198
(7.41)-(7.42) and satisfying, for 1 _< k _< M, uk E H1, Pk E Lq with q =
1 fN>4orN=3and3,
uk E W l,a for q E [1, oo) if N = 2 or N = 3 and y > 3, and the following energy inequality ffZ 21
(pk-lIuk12+pkIukI2) +
Lt('y-1) ary Otmk-1.uk +
< j Pkuk.fk +
(Pk)ry + uI Duk12 + e(divuk)2 dx 1
At
pk-1(pk)7-1 dx. (7.43)
Of course, integrating over the period the equation (7.41), we obtain for 1 < k
_ 0 for all a, b > 0 since (t H V ) is convex on [0, oo), and summing (7.45) from k = 1 to k = Q, we deduce easily the following bounds sup IIP`IILv + IIPeIutI2IIL1 < C
O N if N > 3, -y > 2 if
N = 2.
Step 3: Passage to the limit when -y > N. We assume in this part of the proof that -y > N. We first observe that (7.41) and (7.42) may be written as
at + div (pfi) = 0
at
(Vu)
+ div (pfi (9 fi) - µ0u - V div u + aV (p)ry = p f .
(7.54) (7.55)
Next, we remark that (7.47) yields in particular the following information
p - p( - At) , p - p converge to 0 as At goes to 0+ , in Lq (0, T; Lry) for all 1 < q < oo.
(7,56)
Indeed, we first observe that the following elementary inequality holds (since -y > 2) for all a, b > 0
201
Existence proofs via time discretization
(-y-1)a' + P - rya'1-1b > via-bI"
for some v > 0.
,
Therefore, (7.47) yields M
1113-13(' - Ot)II L7(Ot,T;L1) =
E At lip, - P,-IIIiy < COt k=1
and (7.56) follows easily recalling that 13 and 13 are bounded in LOO (0, T; Lt).
We also claim that (7.47) yields
pu - (3i) ( - At) , 1311-1311 converge to 0 as At goes to 0+ , 2^y in Lq(O,T; L'') for all 1 < q < oo, 1 N, which is precisely the case here. We use the notation of this proof and denote by 7 the weak limit of 0 for any W.
Existence results for Cauchy problems
202
First of all, as shown above, we have on RN x (0, T) (with periodicity)
+div(pu) = 0, a(Pu)
1
+div(pu(9 u)-j u-l;Vdivu+aV pT = pf.
Next, we observe that we have for 1 < k < M (with an easy justification) k-1
k
(log pk + 1)
P
+ div (ukpk log pk) + (div uk)pk = 0 I
Lit
and since (1 + log pk) (Pk o f -1)
At
pk log pk - pk-1 log pk -1, we deduce
(pk log Pk _ pk-1 log pk-1) + div (ukpk log pk) + (div uk)Pk < 0
on RN x (0, T) . (7.59)
This information combined with the convergences established above and (7.50)-(7.53) allow us to copy the proof of Theorem 5.1 and yield the desired strong convergence of p, provided we show that we can pass to the limit in p[R%Rj(,Mifij) - 4iR{Rj((p4Lj)( - At))]. This is the case since we just have to remark that JP[uiR%Rj(Pfij) - i2 R%Rj((Puj)( - At))] I . > 7 + NZ, recall that -y > N) in view of (7.57). In conclusion, we have shown the existence of a solution (p, u) of (5.1)-(5.2) satisfying the properties listed in Theorem 7.2 except that we have only shown that p E L-Y+ 1(cl x (0, T)) instead of L4(1 x (0, T)) with q = -y + 1 -y - 1 (and q < 2-y -1 suffices for N = 2).
Step 4: p E L4(f x (0, T)). We explain in this step how it is possible to recover the integrability of p a posteriori without assuming it as we did in Theorem 7.1. The possibility of doing so was mentioned in Remark 7.2, where we discuss an even more general possibility than what is needed here. We thus follow the proof of Theorem 7.1 (steps 1-2 in particular) but instead of writing (7.7), we replace pe by an appropriate test function /3(p) to be determined below satisfying as least /3 E C1 ([O, oo)), 8 > 0 on [0, oo), 0 and /3' are bounded on [0, oo). We obtain the following identity
ap'P(P) = a fn Pry
i3(P)
+'
[3(p)(-0)-1
div (pu)]
+ div [u/3(p)(-0)-1 div(pu)] + (p+1;)(divu),0(p) div (pu) + (0'(P)P-l3(P))(div +Q(P)[IZ Rj(Puiuj) uiRjRj(Puj)] 0(p)(-0) u)(-0)-1
-
-
div (P.f) (7.60)
Existence proofs via time discretization
203
Next, if N > 4 or if N = 3 and 3 < ry < 6 (recall that we assume that -y > N), we may check easily that p)` ,Q (p) is bounded in L'(1 x (0, T)) provided fl satisfies Cte-1 fort > 0. 0 0. Indeed, let us recall that we have at(log P-p)
p-cp) +
µ+a pry
(7.62)
= RiR? (Puiuj) - uiRi=R7 (Puj )
where cp =
µ+C(-0)
div(pu) E L°O(11 x (0, T)).
Then, we observe that, if N < q < p, for almost all t E (0, T) II RjR.7(Puiuj) - u=RiR7(Puj)II LZO < CII [RiRj,ui](Pua)jjw=,9
< C IIuIIw=,p IIPuIILo < C IIuII ,,i,p IIPIIL< C IIuIIW=., exp(2IIWIILoo) [IIPeILO eXP(-IIcPIILo
a
itJ
1
p(s)s-2 ds >_ p(t)
Jt
s-2 ds
= p(t)
t -1
.
207
General pressure laws
Finally, we may follow and adapt the proofs of Theorem 7.2. We obtain in this way the existence of a solution (p, u) of (7.63) satisfying all the properties listed in Theorem 7.2, and in addition p(p) (1 + pe) E L1 (RN x (0, T)), with (7.17) replaced by
f
N
pu2 + q(p)dx(t) + 1lmol2
fo
ds f dx{ pIDuI2 + (div u)2 }
rt < LN2PO + q(po) dx +J ds / dx pu f, a.e. t E (0, T). o
(7.66)
ff
Of course, we need to assume that the initial conditions (mo, po) satisfy in addition to (7.1): q(po) E Ll (RN). We now turn to the other case, i.e. when fo p(s)s-2 ds = +oo. Let us recall that we choose in this case q(t) = t fl p(s)s-2 ds. We then assume that, at least formally, we have for any function p >- 0 in L' fl L°O(RN) : 2
at
PI 2 + [q(P) -- q(p) - q'(p)(P-p)]
- p0 122 +div u [pL2-+(p+q-q1(7i)p)] 2 + pIDuI2 + l;(div u)2
-
div (u div u)
(7.67)
= pu (f - 0(q'(p)))
which yields d
d
f +
22
N
I
pI
+ [q(P) - q(p) - q'(p)(P-p)] dx iIDuI2+t;
N
(7.68)
(div u)2dx = JNpu (f -V (q'(p))) dx. IPo s
E L1(RN) and {q(po) Therefore, if we assume that po E Ll n L1(RN), q (-p) - q' (p) (po - p) } E Li (RN), then we deduce a priori bounds on p, pI uI2 and {q(p) - q(P) -q'(p)(po - p)} in L°O(0,T; L1(RN)), on Du in L2(RN x (0,T)) and L2N/(N-2) (RN)) if N >_ 3, provided we assume that p satisfies on u E L2(0,T; (for example)
V(q'(p)) E LOO(RN) Let us give a few typical examples of q and p.
(7.69)
Example 7.1 p(p) = p + app', a > 0, 7 > 1. Then, we have: q(p) = p log p + 7 -r (p'r - p). We may for instance choose p = exp (- < x >a) where 0 < a < 1, <x>= (1+IxI2)112, and (7.69) clearly holds since we have IVq'(p)I =
p I
+a7#7-2VpI < CID log pI < C.
;
Next, we compute q(p)--q(p)-q'(P)(p-P) = [p log(P/p)+7i-'p]+a[PI+(7-1)pry-
Ip -'p] and we deduce easily that this quantity belongs to L', assuming that
Existence results for Cauchy problems
208
p E L1, if and only if p E Li' and p log (p/ p) + p - p or p log p + p < x >01E L'.
We then claim that this latter requirement is also equivalent to p log p and p < x >a E L'. Indeed, we have
f Ip log p+<x>a Idx > I
p(log p+<x>*)dx
J (P?pl/2)
RN
>;
p<x>adx
J
and
p <x>a dx < LNh12 <x>a dx < oo . O jp&) is bounded in L°° (0, T; L') and we conclude since we also have
JNP 2N if N _> 3, otherwise we drop the L2(0, T; W l,r (Sl)) part in the above definition. Then, the boundary condition on p really means that p(i.n)- on an x (0, T). In addition, we shall impose a solution (p, u) to satisfy p E L' (OSt x (0, T) ; lu n1dS (& dt).
Since our proofs in chapter 5 or in section 7.1 are essentially local, the only new argument needed concerns the energy bounds, i.e. the bounds on p in LOO (0, T; L'f (1l)), on fpu in L°°(0,T; L2 (Q)) and on u in L2(0, T; H, (11)). In
order to obtain these bounds, we multiply (5.2) by (u n) and obtain, at least formally, with some straightforward computations
a
p
+
2
- µ0 )u 2ul
a p7 ry-l
+ µI
+ div u p D(u-u)I2
2
+a
ry
ry-
l p7
+ a(div u)py (7.73)
-l; div[(u-u) div(u-u)] + l;[div(u-u)}2 5t
=pf-
) (u-u)+((p .V)E.(u-u))+µdiv((u-i)vu)
+l; div((u-u) div u) -µ D(u-u) Du - div(u-u) div u Therefore, integrating by parts over SZ, we deduce
.
Other boundary-value problems d
f f f
pIu-uI2 +
dt n
+ =
n
2
pI
a ry
an
dSa
211
ry- 1 (a.u)+ P 'Y
D(u-u)I2 + l;(div(u-ii))2 dx
r rc)+ f P f - aU--) (u-2c) - 1(u
(7.74)
n dS a ry -1
(u-U) - a(div-u)p1' - pD(u - u) - Du - e div(u-u) div u dx. +
The first term of the right-hand side is bounded because of (7.72), the last two terms are easily estimated by the Cauchy-Schwarz inequality, and we argue for the second term as in section 5.1 (chapter 5) or as in section 7.1 since we made
the same assumption on y as on f. Finally, we estimate the two remaining terms easily using the bound on Du in L1(O,T; LOO (11)) and writing
fo
in
(div u) p1' dx < C II DiuI L- (n) fn p'r dx ,
(pu.V)u(u-u) dx < IIDuIIL-(n) [fn p iu-ul2dx + IIPIILI IIuIIL21/(-y-1)
.
We then deduce easily from Gronwall's lemma the desired bounds. Notice that we also obtain a bound on p in LY(8it x (0, T) ; dt). Having thus derived the "energy bounds" it is now easy to adapt the proofs of Theorem 7.2, and we obtain the existence of a solution (p, u) satisfying the same properties as in the case of homogeneous Dirichlet conditions and, in addition, p E L'(8ft x (0,T) ; ®dt). We now turn to problems set in unbounded domains. Let us immediately mention that we shall set homogeneous Dirichlet boundary conditions, even though it is rather straightforward to combine the arguments developed below with those introduced above and thus to treat as well non-homogeneous Dirichlet conditions on the boundary of the unbounded domains we are going to consider. We shall consider two types of situations: i) the exterior case when it = Oc, O is a bounded, smooth, open domain in RN-we agree that we can allow 0 to be empty in which case fZ = RN and no conditions are imposed on 8O!, and ii) the tube-like case when fl = R x w and w is a bounded, smooth, open domain in ISBN-1. We have already mentioned and studied these situations in chapter 5, section 5.6. In both cases, we wish to solve (5.1)-(5.2) and look for a solution (p, u) of (5.1)-(5.2) in fl x (0, T) satisfying the initial conditions (5.6), p > 0 and the following boundary conditions. In the exterior case, we impose
u=0
on 80 x (0, T)
(p,u)(x,t) --+ (p"0, u') as IxI -- +oo, for all t E (0,T) where p°° > 0, u°O E RN
(7.75) (7.76)
Existence results for Cauchy problems
212
In the tube case, we request that (p, u) satisfies
u=0
on (R x ow) x (0, T)
(P, U) (Xi, x', t) - (P+, a+) as xi - +oo , (P, u) (xi, x', t) -- (P , u-) as xi -- +oo, f o r all x' E w, t E (0, T) ,
(7.77)
(7.78)
where p+, p- > 0, u+ = (ui , 0), u- = (ui , 0), ui , ui E R. Both (7.75) and (7.77) are to be understood in the sense of traces of functions which are in H1 (essentially, see the definition of the precise space below), while (7.76) and (7.78) will be understood in a weak "integral" sense.
More precisely, we look for a solution (p, u) of the above boundary value problems, satisfying with the notation of section 5.6 (chapter 5): i) p - p E - p =- #1 in the exterior case, and p E Q[0, T]; LP (11 n BR)) n L°°(0,T; L2-1 C([O,T];L7(QnBR)-w) for all R > 1, 1 < p < -y, ii) u-i E L2(0,T; H1(IlnBR)) for all R > 1, u- E L2 (0, T; H1(1)) in the tube case or in the exterior case when if N > 3 in the exterior pO° > 0 and N > 3, and u-u E L2(0,T;L2N/(N-2)(S2))
case, iii) plu-112 E L°° (0, T; L1(St)) and iv) p(u-u) E C([0, TJ ; L2 + L' (11) - w)
where r=3 if-y>2,r=-
if'y 3 and p°O > 0. Indeed,
1(p)p-/2)(U-U) is obviously bounded in L2 (S2 x (0,T)) while u - u is bounded in L2(0,T; (fl)) and sup esstE(o,T) meas{x / p < pO°/2} < 00. Let us also mention that these bounds were proven in section 5.6 in the case L2N/(N-2)
when f - 0 only. It is not difficult to check that the proofs can be adapted provided we assume that f E L1(O,T; L1 n (it is possible in fact to extend slightly this condition in a manner somewhat similar to condition (7.2)). We conclude by observing that Theorem 7.1 can then be easily adapted and yields a bound on p in LP(K x (0, T)) for any compact set K C 1 where p = 'y+ N'y-1. Similarly, following the methods of proofs of Theorem 7.2, we obtain the existence of a solution (p, u) of the above problems such that p E LP(K x (0, T)) for any compact set K C 11, under the same restrictions upon ^f as in Theorem 7.2. Of course, the crucial compactness result namely Theorem 5.1 used repeatedly in the proofs of Theorem 7.2 is to be replaced by Theorem 5.3 (section 5.6, chapter 5). L27/('f_1))
RELATED PROBLEMS 8.1 Pure transport of entropy In this chapter, we shall consider various related problems, many of which (but
not all of them) can be studied with the methods introduced in the preceding chapters. We begin in this section with the study of models where entropy is purely transported (along particle paths). From a physical viewpoint, see for instance chapter 1, this amounts to assuming that the thermal conduction coefficient can be taken to be 0 and that one can neglect the heating due to viscous dissipation-an approximation which is often made except for hypersonic gases. Then, the entropy s solves the following equation P
as
or, if (p, u) solve (5.1), equivalently (at least formally)
a
(ps) + div (pu s) = 0 .
In the ideal gas case, the pressure is then given by p = RpT = Rprle'lc' where R > 0, C > 0, y > 1 are given. Of course, replacing s by s/C,,, we may assume without loss of generality that C = 1 and take p = Rp''e' (and R = -y - 1). Therefore, we look for a solution (p, u, s) of (8.2),
at + div(pu) = 0, p > 0 8pu
4.div(pu®u)- Au-POdivu+77,n
=f
and
p = Rp7 e' .
(8.4)
As usual, we assume that p > 0, p + > 0, and we consider as in the preceding chapters the case of Dirichlet boundary conditions (u = 0 on 811 x (0, T)) where the equations hold in 1 x (0, T) and 11 is a bounded smooth open domain in RN
Related problems
214
(N > 2), the case of the whole space where the equations hold in RN x (0, T) and (p, u) vanishes at infinity and the periodic case where the equations hold on RN x (0, T) and all unknowns (and data) are assumed to be periodic in each xi (for 1 < i < N) of period T= > 0. Let us immediately warn the reader that, unless explicitly mentioned, we shall always consider these three cases for all the models and equations studied in this chapter. We also prescribe initial conditions Plt=o = Po
,
Putt=0 = mo
,
in 1
Pslt=o = So
(8.5)
where po, m0 and So satisfy po
- OEL'nL'f(SZ), 1PO 4
(8.6)
E L1(SZ) , mo E L27/(ti+1) (SZ) and So E L°O (St) and I So 15 Cl Po a.e. in SZ 2
for some C1 > 0, and mo = 0 a.e. on {po = 0}, as is defined to be 0 on {po=0} and p000. I
Obviously, we may expect from (8.2) a LOO bound on s and more precisely (8.7)
IISIIL-(cx(0,T)) < C1.
It may be worth remarking (once and for all) that s is not really well defined on the set where p vanishes since the equation (8.1) degenerates completely on this set. On the other hand, this does not affect the pressure p given by (8.4) and we may ignore completely this difficulty, agreeing for instance that s - 0 on {(x, t) / p(x, t) = 0}. In fact, if s" E L' (P x (0, T)) solves (8.2), then replacing the values of s by 0 on {p = 0} and denoting by s the resulting function, one sees that s is still a solution of (8.2). Next, we observe that we have at least formally
8
pJU12
at
2
+ -y-1 R P
+div u
2
p_u + 2
Rry
y-1
p-f e'
(8.8)
+µ0l22 +µfDuI2-Cdiv(udivu)+t;(divu)2 = pu f. Indeed, we have clearly
at (p'ie') + div (uppe') = (-y-1)(div u)p7e'
.
Then, because of (8.7), we deduce exactly as in section 5.1 (chapter 5) the same a priori bounds on (p, u) under the same conditions on f . Similarly, one can
check that the proof of Theorem 7.1 (chapter 7, section 7.1) can be adapted mutatis mutandis and thus the same result holds. All these observations lead to the following existence result.
Pure transport of entropy
215
Theorem 8.1 Under the same conditions on f and -y as in Theorem 7.1, and if (po, mo, So) satisfies (8.6), there exists a solution (p, u, s) of (8.2)-(8.3)-(8.4) with s E L°O(fZ x (0, T)) , ps E C([0, T]; LP (11)) for all 1 < p < oo, satisfying the 1, initial conditions (8.5) and such that p E LP(1 x (0, T)) with p = y + except in the case of Dirichlet boundary conditions where p E LP(K x (0, T)) for any compact set K C ft. In addition, (p, u, s) satisfies the following energy inequality for almost all t E (0, T) y -N
J
+Rpresdx+
0, sE = 0 if pE = 0 (sE E L°D(f x (0, T)) and is bounded in LOO (fl x (0, T)) uniformly in e E (0,1) ), we obtain in the sense of distributions as£ PE
at
+ PEu .
We + rESE = T£
and thus (by one more regularization justification that we skip) p£
3(S£) + P£uV 3(`SE) + r,P'(` e).£ = #'(`SE)r£
or
5i (PEQ(sE)) + div(pEuf(SE)) + rE f Q'\S£)SE
-i
(SE) = Q/(SE)r£
We then recover (8.10) upon letting e go to 0+: indeed, sE converges a.e. on J p > 0} to s and sE is bounded in L°° (Sl x (0, T)). Therefore, f3(9E) converges in L9 (((l n BR) x (0, T)) n J p > 0}) to 3(s) for all 1 0 (small enough), (p
i3(Sn))B
div un - µR
- (P3(S))B (divu
- µR
(P)
weakly, denoting by 7 the weak limit of co (recall that we are using the same notation as in chapter 5). In view of the strong convergence of p' shown above, we deduce that we have
(p/3(s))° div u" -n(p/3(s))e div u weakly (in 12' say). We may then adapt the proof of part (2) of Theorem 5.1 to deduce the strong convergence of pn/3(sn) to p/3(s) (in C([0, T]; LP (Q)) n L'(K1 x (0, T))
for all 1 < p < r, 1 < s < q) for any non-negative continuous function /3 and
Pure transport of entropy
217
thus for any continuous function /3. Let us briefly sketch the argument: denoting f n = p",8(sn), we have :
5,(fn)e+div(un(fn)e) _ (1-0)(divun)(fn)0; thus we deduce letting n go to +oo
of +div(uf) = 0,
+div(ufe) = (1-0)divu(fe)
and we obtain
a(7-(fe)1/e)+div(u(f--(f9)"°)) = 0. Hence, fo f - (fe)'1e dx(t) = 0 for all t >_ 0 and our claim is shown. Taking /3 = 0, we obtain the strong convergence of pn, and the proof of the compactness claims is complete. 0 We conclude this section with the case of a general state equation for p (and s). For the same reasons as those mentioned at the beginning of this section, we solve (8.2) and (8.3) but replace (8.4) by (8.11)
P = AP, s)
where p is a given continuous function of p E [0, oo) and of s E R. As is natural from a physical point of view-it is indeed a consequence of the second law of thermodynamics, see section 1.1, chapter 1, part I, volume 1-we assume that
p is non-decreasing with respect to p for each s fixed and, in order to avoid ambiguous definitions on the vacuum, we assume that p(0, s) = 0 for all s E R. Finally, we assume that p satisfies for some y > 1 lim inf ,sl 0
for all R E (0, 00),
(8.12)
and in order to avoid the technical difficulties associated with the behaviour of p near p = 0, we assume in the case when 11 = RN that p satisfies 1
fo
p(t, s)t-2 dt < +oo ,
for all s E R.
(8.13)
Let us also mention in passing that it is also possible to analyse more general situations than (8.13) using the ideas and methods developed in section 7.5 (chapter 7), and we denote q(p, s) = p fo p(t, s)t-2 dt. Of course, in (8.6), we replace po E L7 by q(po, so) E Ll.
Theorem 8.2 Under the above conditions and the same assumptions as in Theorem 8.1, there exists a solution (p, u, s) of (8.2)-(8.3)-(8.11) satisfying the initial conditions (8.5) such that s E LOO (SZ x (0, T)), ps E C([0, T]; LP(f )) for all
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218
10,=0ifp=0,'82= Pp/32(s)ifp>0,=0ifp=0. Therefore, we may rewrite (8.19) as
a (p,3) + div(pu,3) = 0 , while we have of course (take Q
e
(pat) +
div(pu(132))
=0
(8.20)
1)
+div (pu) = 0.
(8.21)
Then, by the same argument as the one used to derive (8.10), we deduce from (8.20) and (8.21) the following equation [p('82
at
- (Q)2)] + div [au(#2 - (Q)2)] = 0
(8.22)
.
In addition, we have in view of the assumptions made upon the initial conditions
pO(s)It=o = po/(so), p/32(s)It=o = pof2(so) . Therefore, we have p(/32 (4)2)It_0 = p/32 (p8)21t=o = 0 in Sl. Since, we
-
-
obviously have 02 > (4)2 a.e., we deduce from vthis initial condition and (8.22) (which we integrate over Sl) the fact that p(j32 (4)2) = 0 a.e. and our claim is
-
shown.
In particular, we have T
dxp"IQ(sn)41 < C
dt
fT dt
o
=C
dt o
Jn
r dxpfI$(Sn)_al2 n
dx p"($(s"))2 - 2pn/3(sn)Q + pn(Q)2
n 0.
In particular, we may choose p(t) - t and (8.16) is shown. Next, in order to prove (8.17), we observe that we have for all S E (0,1) and for all R E (1, oo)
f 0
T
/'
dt J dx Ip(p', Sn) _p(p", s)I [1(p^ R, Isl < Ro Iq
where Ro = supn> 1 I I s" 11 L:°t . In view of the assumptions made upon p, we deduce that the above integral converges to 0 uniformly in n as S goes to 0+ and R goes to +oo. Next, we conclude, remarking that we have T
J0
dt
f
t2
dx
p(pn,s"')_p(pn,sn)11(b 0.
v
>0
(8.28)
(in other words, we assume that po < Ceo for some C > 0), then we deduce from the maximum principle
T74 > ap
a.e.,
(8.29)
an inequality from which we deduce a bound on p in L°° (0, T; L7). Next, we observe that we have
8
E (y-1)
(8.30)
[IDu+DutI2 + ( -µ)(divu)2J
hence
vIDuI2-y(divu)p > -Cp2
5 and 49
(inf essp) + C(inf essp)2 > 0.
Therefore, if we assume that we have
inf esspo = R inf ess (poTo) = f3 > 0
(8.31)
we deduce a lower bound on p and thus on T using (8.29) inf essp >_
inf essT >[(1+i3ctr'] R
(7-1)17
(8.32)
We may also deduce from (7.30) the following equation
a (pl/7) + div u 1/7
2 - 1 [LIDu+Dutl212
div u
2
1/7
1
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> ry - lvp (1--1r)IDuI2. y Since p is bounded in L°° (0, T; L1), we deduce, integrating this inequality in x and t, a bound on IDuI2 p-(1-(1/7)) in L'( SZ x (0, T)). Furthermore, we may write IDuI2''/(27-1)
=
(()) IDuI2
p
from which we deduce a bound on Du in L2(O,T; L27/(27-1)(S2)). Let us mention that this is the "best" bound we can obtain on Du and in particular we are unable to obtain a bound on Du in L2 ((fl x (0, T)). This bound does not seem to be "strong" enough to implement the strategy of proof of Theorem 7.1 (section 7.1, chapter 7) in order to prove some Li,t bound on p for some q > 1 (or some Li,t bound on p for some q > y). Indeed, the LOO(Lt)
bound on p and the above bound on Du does not yield, even when N = 2, a 1 better bound on pu ®u than Lz,t; indeed, we have, if N = 2, 1-r + 2 (2-y2-1 - 2 = 1 ! The second obstruction we encounter in proving an existence theorem for the problem we are studying here is the compactness analysis of solutions: indeed, even if we postulate Ll bounds on (p, p) (or simply Li,t x Lz,t bounds for some q > -y, r > 1), the proof of Theorem 5.1 (chapter 5) can be adapted to yield the following information
div u - p] Q(p, p) =
div u - p Q(p, p)
but we are unable to conclude from this information any compactness of p or p. 11
8.2 A semi-stationary model We consider in this section the following system of equations i9p
+div(pu)= O , p> 0
(8.33) 0
with the same boundary conditions as before (see section 8.1), where a > 0, µ > 0, µ + t; > 0, ry > 0. Let us observe that we have not included force terms (right-hand sides for the second equation) in order to avoid unnecessary (and straightforward) technicalities. There are various motivations for the study of the model (8.33). First of all, we have shown in chapter 5 (section 5.2, Remark 5.8) how solutions of this system of equations allow us to build solutions of our initial system (namely (5.1)-(5.2)) which exhibit persistent oscillations. The second motivation is the model derived in W.E. [165],[166] for the dynamics of vortices in Ginzburg-Landau theories in superconductivity, which is precisely of the above form. As it stands, the above model is slightly ambiguous in two cases: i) when
St = R2 and ii) in the periodic case. In the periodic case, u is defined by the
A semi-stationary model
225
second equation up to a constant and we thus need to add one more constraint like for example
In
dx u(x, t) = 0,
for all t > 0.
(8.34)
In the case when fZ = R2, requiring that u vanishes at infinity needs some explanation (while it is an obvious requirement if 11 = R" and N > 3). The simplest way to argue is to write the explicit integral relationship between u and p7 (assumed below to be in L' (R2) for all t > 0) namely
u
I
X
for all t > 0.
in R2 ,
* P7
(8.35)
Of course, we need to complement the above system of equations with an initial condition on p namely (8.36)
P!t=o = Po ? 0
where po E L' (0), po E L7 (f) if y > 1 and po I log po I E L1(11) if y = 1. Before stating our main results, we wish to make a few observations. First of
all, if 0 = RN (N > 2), we see that we have a div u =
µ
+ p7
,
curl u = 0 in RN
,
for all t > 0
(8.37)
C
while we have in the periodic case
div u =
µ+ p7 _3:p1 dx
curl u = 0 in RN
,
(8.38)
for all t > 0 .
We may now state our main results.
Theorem 8.4 (The periodic case). Let y > 0. Then, there exists a solution (p, u) of the above problem satisfying: p E C([0, T]; L1), p E C([0, T]; L7) n L2'(1 x
(0,T)) if y > 1, p E L1+7,°°(f x (0,T)) if y < 1, pl log pl E L'(0, T; L1) if y = 1, u E L2(0,T; H1), and u E LOO (0,T; W1+1/7) if y < 1, for all T E (0, oo). Furthermore, we have for any ,Q E Co ([0, oo)) and for all T E (0, oo)
8) +
di v(u,Q (p)) + ( div u)[Q' (p)P-P(p)] = 0 ,
II P(t) II too t-11''
is bounded on (0,T)
(8.39) (8.40)
If po E LP (Q) for some p E (1, oo) (resp. p = oo)
then p E Q0, T]; LP) n LP+7(1 x (0,T))
(8.41)
(resp. E L°° (f x (0, T )))y
If in less po > 0, then inf oes s) p > 0 for all T E (0, cc), fl x
(8.42)
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If y > 1 or if y < 1 and inf essn po > 0, then p E C([O,T];W1"p) (resp. L°O(O,T;W1"°°)) whenever po E W 1,P for some p > 2 (resp. p = oo).
(8.43)
Finally if y > 1 (resp. if -y < 1 and inf ess po > 0) solutions p in L°O (0, T; L°° n W 1,P) (resp. which are in addition bounded from below) are unique if p = N when N > 3, p > 2 when N = 2.
Theorem 8.5 (SZ = RN). Let -y > 1. Then, the preceding result holds except for (8.42) and provided we replace u E L2 (0, T; H1) by u E L°O(0, T; Lwt-4-1'°°),
Du E L2 (RN x (0, T)) and Du E L' (0, T; L1/7)if y < 1.
Theorem 8.6 (Dirichlet boundary conditions). Let y > 0. Then, there exists a solution (p, u) of the above problem satisfying: p E C([0, T]; L1), p E C([O, T]; L'1) nL21(SZ x (0, T)) if -y > 1, p7 E C([O, T]; L1) nL2(SZ x (0, T)) and p E L1+1r (0, T; Li + ' (11)) if -y > 1, pI log PI E LOO (0, T; L') ify = 1, u E L2 (0, T; Ho ), u E L°° (0, T; W 1,1/7) if -y < 1, curl u and div u - 14+t p- E LOO (0, T; LOO ), for all
T E (0, oo); and (8.39) holds. Furthermore, we have for any compact set K C SZ and for all T E (0, oo): IIP(t)IIL-(x)
is bounded on (6,T), for allb > 0,
(8.44)
If po E L oC(SZ) for some p E (1, oo) (resp. p = oo)
then p E C([O,T];LP(K)) and p E LP+-'(K x (0,T))
(8.45)
(resp. LOO (K x (0,T))).
Remark 8.8 Let us make a few remarks on the above statements: i) First of all, in the case when SZ = RN, we have restricted ourselves to the case when y > 1 in order to avoid the technicalities associated with y < 1 although this case can be treated by a convenient adaptation of the considerations introduced in section 7.5 (chapter 7). ii) In the case when SZ = RN, it is possible to treat other situations with different "conditions at infinity" adapting the arguments of section 7.6 (chapter 7). Then, whenever the condition (inf essRN po > 0) makes sense, the property (8.42) also holds.
iii) The fact that p E Ly can actually be deduced from the bound on ti/'tIIPIIL and on II PII L-(L=) Indeed, we have, for all R > 0, denoting Co = sup[o,T](t1/^'IIPIILo),
meas {(x, t) / p(x, t) > R}
1, we just have to replace by p log p. viii) Next, we observe that all bounds on p are easily translated into bounds on u using elliptic regularity. ix) Finally, we have to clarify the meaning of pu in (8.33) at least when y < 1. Then, we observe that
p E Lr(L')nL'(L2) where 1
R) < A + a
J po dx + T sup -} e)(sup n
(0,T)
f2
J p dx
,
(0,T) n
and the bound in L1+",°° is shown. The proof is obviously similar (and in fact simpler) when Q = RN. The bound mentioned in (8.41) also follows from (8.39) choosing /3(t) = tP: indeed, we find dt
fP1'dx+(p_1)fpP(1_j',fY)dx = 0 n
t
and we conclude integrating with respect to t. Again, the proof is similar (and simpler) when ) = RN. The proof of (8.40) is also straightforward once we remark that we have
+ u.V p + +
P1+-r = +
p7
p on RN x (0,1) .
(8.49)
Indeed, applying the maximum principle and writing simply sup p for sup ess p, we deduce
4 (sup P) + + (sup p)'+' < Co (sup p) µ
on (0,1)
where Co = 14+C sup(o,l) (f p1f). This differential inequality then easily yields the bound (8.40). Once more, the proof is simpler when f = RN (Co = 0).
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The bound (8.42) also follows from (8.49) and the maximum principle: indeed, we deduce
(inf p) +
N
µ
+ (inf p)1+7 > 0
on RN x (0, oo)
hence, inf p > (ryat(inf po)*y + µ + 1;)-1/-Y (µ + 6)-1/7 inf po on (0, oo).
We now turn to the proof of (8.43): differentiating (8.33), we obtain easily, denoting by C various positive constants independent of u and p, d
jIVPI7'dx
< C L IDul IoPIP dx + C fo ID2uI < CII DuII
Loe fn I VPIP dx + CIIPIIL-
< C(1 + IIDuIIL-)
f
IoPIP-1
f
p dx P-1 P
IVPIPdx
1/p
ID2UIPdx
IoPIP dx
where we used the L°O bound on p obtained in (8.41) and elliptic regularity noting that we have -(a-y)pry-1Vp and IIP''-1VPIILP < IIPII7O'IIVPIILP
if -Y > 1,
< (inf ess
if y < 1.
We conclude easily recalling the classical inequality (whose proof is a simple adaptation of Appendix F for instance) II DuII L- < CIIPIIL- log 1 + IIVeIILP IIP''IILwhich is bounded (as above) by (C 1og(1 + II VPII LP))
Step 2: Existence and compactness. Various proofs of the existence part of the above results are possible: the simplest probably consists in solving first the system (8.33) when po is smooth. This can be done by various arguments including simple fixed point methods based upon the uniqueness proven below. The existence results in the case of Dirichlet boundary conditions or when S = RN can then be deduced as in the preceding chapter (7). For a general initial condition, we then just have to regularize the initial conditions po and pass to the limit. The compactness of p then plays the usual (by now) crucial role. The proof of the compactness of p is similar (and much simpler) than the one in chapter 5 and we only need to clarify one point, namely the continuity at t = 0
A semi-stationary model
231
of p and (pe)i/e (with the notation of chapter 5) with values in Li (or even in Li') of p: indeed, the L°O bound on p for t > 0 and the transport equation satisfied by p makes the rest of the argument considerably simpler. About the continuity in t, we first observe that the L°° bound on p for t > 0 together with (8.39) immediately imply the fact that p E C((0, oo); LP) for 1 < p < oo and /3(p) E C([0, oo); LP) for 1 < p < oo for any ,3 E C'(R) such that /3 and f3't are bounded on R (for instance). Next, in order to check the continuity at t = 0 of p (7)'10 py), various arguments can be made. First of all, we may use (8.48) and 1
deduce for all T E (0, oo)
J (p(t)-R)+dx < r (po-R)+dx --+ 0
o supo 0, hence pe E C([0, oo); Life) and thus p E C([0, oo); L1). The same argument shows that p E C([0, oo); Li'): indeed, p E C([0, oo);
Li' - w) n C(0, oo; Li') while, for all t > 0, IIPIILI S IIPoIIL-, in view of (8.47). Finally, (P9)l/e E C([0, oo); Li) since pe E C(0, oo; LP), (PB)1/0 AR E C([0, 00); LP)
for all 1
0. In addition, g, h are bounded in LOD(0,T; L1/7) if y < 1, while g, h are bounded in L°° (0, T; W -E") if y > 1 for any e > 0 and for all T E (0, oo)-notice indeed that p7 is bounded in L°° (0, T; L1) and thus Du is bounded in LOO (0, T; W -E,1) for any E > 0 and for all T E (0, oo). Since g and h are harmonic, we then deduce, in all cases, that g, h E L°O (0, T ; LOO) . This turns out to be sufficient to prove (8.44), (8.45) and the Li +7'°O bound on po. We begin with the proof of (8.44). We then write for any cp E Cow(fl) with
0 0, the existence of t2 E (0, S2) such that p(ti+t2) is bounded in L o' where p2 = pi + y. Reiterating this argument, we find, for some fixed no > 1 that depends only on y, some pno > q, where q has been determined above. We conclude choosing Si = 62 = ... = bno = no6 . The proof of (8.45) is similar (and considerably simpler). First of all, it is enough to consider the case when p > max(1, y). Next, we write in a similar way to the argument above with the same notation (vpp) + div (ucppp) +
(p-1)cphp1 + (u'V g )pp
hence
sup (in cppP dx + J (O,T)
1 and N > 3 (the case N = 2 is easily adapted), u E L2 (0, T; L2N/(N-2) (S2)), therefore we conclude if p < N+2 y while if p > N y we 22
obtain a bound on pin L P ' ' ' (D x [0, T]) fl LOO (0, T; L I (0)) where pi = N22 y N/(N-2) (f )). We may now use L2N/(N-2) (0, and thus u is bounded in T; W1, Sobolev's embeddings and complete the proof reiterating the argument. If y < 1, the argument is essentially the same except for the numerical values entering the preceding bootstrap method: for instance, we start in this case using the bound on u in LO° (0, T; W ','I" (SZ)) and thus in L°° (0, T; LQl (S1)) where a = y - N if y > rr , qi is arbitrary in (1, oo) if y = N , qi = +oo if y < N We conclude with a brief sketch of the proof of the bound on p in Li+7,' (K x (0, T)) for any compact set K C Q. First of all, we prove by a bootstrap argument that p E L-Y+e(K x (0, T)) for any compact set K C ) and for any 0 E (0,1). In order to do so, we write (p'v) + div (up°co)
= (1-0) + p7+e' + (1-0)hpecp + µ
cppe
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235
where cp E Co (SZ), co = 1 on K and 0 < cp < 1 on SZ. Therefore, we have for each
compact set K C Sl and for any 9 E (0,1)
JTJ K
T
< C 1 +J dtJ
IuIPe.
SuPPV
0
Since u is bounded in LOO (0, T; W1,117), we conclude immediately if -y < -L. If 7y > N , using the fact that pe is obviously bounded in LO (0, T; L110), we deduce Once we have choosing 0 = 1 + N a bound on p in L1+11N(0, T; L a bound on p in L"(O,T; L P(SZ)) for some p, we use this bound to deduce that pe is bounded in LP/0 (0, T; Lple (SZ)) and u is bounded in LPh7 (0, T; W , /7 (1)) n
L°°(0,T;W1,1/7(SZ)). This allows us to take a larger 0 and reiterate the above argument. In this way, we obtain a bound on p in L7+e (0, T; L'ly.,e (SZ)) for all 0 E (0,1). Finally, we write for all R E (0, oo)
((P-R)+cP) + div(u(p-R)+cP) + +
p7R
1(p>_R)W
= -hR 1(p>R)co + Therefore, we have for all T E (0, oo)
1T1 dx p7R 1(p>R)cp < C 1 + f /
dt JUPP dx Iulp
.
ip
The right-hand side is obviously bounded since p is bounded in L7+0 (Supp cp x (0, T)) while u is bounded in L°O (0, T; W1, 117 (0))nL(7+B)/7 (0, T;
W"('Y+e)/7 (SZ))
for all 0 E (0,1). In particular, choosing 0 close enough to 1, we deduce that u is bounded in L''(Suppcp x (0, T)) where rr11 = ry + 9. This completes the proof of our claim. 0
Remark 8.14 We wish to conclude this section with a general remark on this semistationary (or quasi-semistationary) model when we replace ap7 by a general (barotropic) pressure p = p(p) E C([0, oo)). In the periodic case, (for instance, it is easy to check that the above arguments yield the existence and uniqueness of a smooth solution as soon as lim inft-..+o° p(t)t > 0, p E C' (0, oo) and either p E C' ([0, oo)) or po is bounded from below (in that case the solution is bounded from below and is unique among such solutions). On the other hand, the existence of a global weak solution seems to require (at least with our analysis) the assumption, which is natural from a physical viewpoint, that p is non-decreasing. This observation obviously raises the issue of the necessity of this assumption for the above simple model and also for the models studied in the previous chapters.
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8.3 A Stokes-like model In this section, we study the following model which corresponds, roughly speaking, to a Stokes-like approximation to the momentum equation of the system of compressible isentropic Navier-Stokes equations. More precisely, we look for a solution (p, u) of ,jT
+div(pu) = 0, p0
p 8u - µ0u
at
- 6V div u + aVp" = 0
where p > 0, a > 0, µ > 0, µ+e > 0 and -y > 1. We consider as usual the periodic case, the case when SZ = RN and the case of Dirichlet boundary conditions even though more general boundary value problems can be handled as well using the methods and techniques introduced in the previous chapters. We might as well treat more general pressure laws (and in particular the case when 0 < ry < 1) or force terms like p f in the right-hand side of the momentum equation in (8.51) (the "u-equation"). This model is, in some sense, intermediate between the semistationary model studied in the previous section and the "full" system (5.1)-(5.2) studied in chapters 5 and 7. It was studied systematically in the periodic case, when 7 = 1 and the flow is potential by V.A. Weigant and A.V. Kazhikhov [552] (see also A.V. Kazhikhov [294]). Also, we shall see that the methods used to study (8.51)-some of which are direct adaptations of what we did in the preceding chapters-allow us to study and solve some shallow water models discussed in the next section. As usual, we complement the system (8.51) with the initial conditions PI t=o = Po,
UIt=o = uo
(8.52)
where po E L" (11), uO E L2(SZ), po and uo are periodic in the periodic case, po E L1 (RN) if SZ = RN. In addition, if y = 1, we assume that poI log poI E L1(SZ).
We may now state our main results.
Theorem 8.7 (N > 3). We assume that N > 3 and that 'y > NN
Then, there exists a solution (p, u) of (8.51) satisfying (8.52) and u E L'(0, T; Hi (SZ)) n C([0, oo); L2(n) - w), p E C([0, oo); L1(SZ)) n C([0, oo); L1(S2) - w), p E Lq(K x
(0, T)) with q = N -y for all T E (0, oo), where K = SZ if SZ = RN or in the periodic case and K is an arbitrary compact set included in SZ in the case of Dirichlet boundary conditions.
Theorem 8.8 (N = 2). We assume that N = 2 and that y > 1 except in the case when SZ = R2 where we assume that y > 1. Then, there exists a
-
solution (p, u) of (8.51) such that u E L2(0,T; Hi (Q)) fl C([0, oo); L2 (Q) w), u E C([0, oo); L2(K)) if -y = 1, p E C([0, oo); L'())), p E C([0, oo); L, (Q) - w) if ^y > 1 and p log p E L°° (0, oo; L1(SZ)) if 'y = 1, p E Lq (K x (0, T)) where q = 2 if 'y = 1 and q < 2-y if 'y > 1, for all T E (0, oo) and where K = 11 except in
A Stokes-like model
237
the case of Dirichlet boundary conditions where K is an arbitrary compact set included in St. In addition, (p, u) satisfies Ifpo E Lq(11),Duo E
L(q+-f)/2-t (n) then
p E L°O(0,T;Lq(K)) nLq+'1(K x (0,T)) , Du E Ll+q/'f (K x (OT)) if q < oo , , D u E Lp ( K x ( O , T)) for all P-'00 if q = oo,
(853)
}
u E LOO (O, T; L'' (K)) with r < oo ifq = 3y,
r=+ooif q>3y,
for all T E (0, oo) and for any q E [3y, +oo]. Furthermore, (p, u) satisfies in the periodic case or when 1 = R2 for any q E (2, oo) and for all T E (0, oo) If po E Wl-q(SZ) and D2uo E Lq/2(fl) then
(8.54)
p E C([0, oo); W 1 iq (SZ)) and u E W2,1,q (SZ x (0, T) ),
and solutions (p, u) (satisfying (8.52)) such that p E L2(0, T; W l"q(f )) and u E L1(0, T; W1,00 (SZ)) are unique on SZ x (0, T).
Remark 8.15 In Theorem 8.8, one can obtain (and deduce from (8.54)) further regularity results for p, u and their higher derivatives. We do not wish to detail those straightforward extensions. Remark 8.16 Once more, we do not know if the bounds on p, u, Du hold up to the boundary in the case of Dirichlet boundary conditions and, as a consequence, we do not know whether solutions are smooth and unique in that case. Remark 8.17 The case y = 1 (in Theorem 8.8) was treated for periodic boundary conditions in the potential case by V.A. Weigant and A.V. Kazhikhov [552], and in the general case by F.J. Chatelon and P. Orenga [99] where the following boundary conditions are considered
u n = curl u = 0
on
(8.55)
8SZ
(where n denotes as usual the unit outward normal to 8SZ). These boundary conditions allow us to obtain regularity results (and uniqueness results) on St, i.e. up to the boundary. We shall in fact somewhat simplify and extend these proofs and present various new arguments (even in these particular cases). Let us finally mention that our analysis of Dirichlet boundary conditions (N = 2) is taken from P.-L. Lions and P. Orenga [357], where, in fact, related equations for shallow water models, are treated; we shall detail these in the next section.
Remark 8.18 In (8.54), we assumed that D2uo E Lq/2(1). The role of this assumption is to ensure that the solution u of
8u-
µ
Du- Vdivu=0 inn x 0( oo) u lt=o
= uo
inn
(and periodic boundary conditions or SZ = R2) satisfies u E W2"1,q(ft x (0, T)), i.e. D2u E Lq (St x (0, T)) for all T E (0, oo). The precise condition on uo which is needed is too technical for the (slight) gain of generality to be given here.
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Remark 8.19 If N = 3 and ^y = 1, it is also possible (and easy) to prove the existence of a solution (p, u) with the regularity mentioned in Theorem 8.7, i.e. p and p log p E L°° (0, oo; L 1), u E L°° (0, oo; L2) n L2 (0, T; H1) (V T > 0) and p E L(111+2)11 . The a priori bounds are shown exactly as in the proofs below and the existence can be easily deduced from various approximations and a straightforward passage to the limit. The restriction upon N is needed in order to make sense of pu-notice that u E LtN+2)12(LP) where p > N2 2 if and only if N < 3. This point seems to have been overlooked in [294] where the existence is claimed for all N. O Remark 8.20 In the periodic case and if -y = 1 and N = 2, the uniqueness holds under less restrictive conditions on the solution (p, u). Indeed, in this case, the argument introduced by V.A. Weigant and A.V. Kazhikhov [552]-and described in Remark 8.11 in the preceding section (section 8.2)-shows that the uniqueness holds assuming only that p E L' . We now briefly sketch the proof of Theorems 8.7-8.
Step 1: General a priori bounds. We begin with the proof of the a priori bounds which are available in all dimensions. First of all, we observe that if y > 1 we have 2 JJu 2 a -1 ups
p2+
-
1
p' + div
-µ0I 2
(8.56)
div( diva) + pIDul2 + (divu)2 = 0
while if y = 1 we have 2
2
+ ap log p + a div {up(log p + 1) } - µA 12 at p 12
(8.57)
- t; div (u div u) + pIDuI2 + e(div u)2 = 0.
We deduce from these (formal) identities a priori bounds on u in C([0, oo); L2(Q)) n L2(0,T; HI (Q)) (V T E (0, oo)), on p in C([0, oo); L" (Q)) if i > 1, and on p and p log p in C([0, oo); Li (1k)); recall that, as usual, dt fn p dx = 0. Next, we prove Ly,t bounds on p where q = NN2-y. The proof follows in fact the proof of Theorem 7.1 (section 7.1, chapter 7). This is why we only briefly
sketch the proof in the periodic case in order to show the modifications to be made in the proof of Theorem 7.1. We thus obtain in the periodic case the following identity, denoting 0 = q - -y =122 ,y, ap7+e
= a h, prype +
+ 49 [pe(-A)-'divu] + div [upe (- A) - 1 div u] +(0-1)(divu)pe(-L)-1(divu) - peuiRiRjuj .
(8.58)
Integrating with respect to x and t, we deduce easily the bound we claimed on p once we observe that we have
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239
II (div u) pe (-A)-' (div u) II Li,t
< IIdivu11L=,tIIP°(-A)-1(divu)IILz,< < IldivuIIL2.,t < IldlvUIIL= t
0 fixed is straightforward: first of all, the energy identity yields bounds on u in Lt ° (L2) n Lt (Hz) and on pin Lt ° (Lx) n Lz
if y > 1 and on p(1 + I log pl) in
Lr(L'), pp(1 + ( logpl) in Li.t if 'y = 1. These bounds imply (by the regularity theory of parabolic equations) a bound on u in and on LZ+(p-1)/7(W, 1,1+(p-1)/7) au in and thus, if p is large enough, on u in LI in particular. We next observe that (8.58) still holds provided we replace ap7+e by ap7+e + This allows us by a simple bootstrap argument to deduce bounds on p in Lt° (Lq ), on u in Lt (WW,q) and on et in Lt (Wz 1,q) for all 1 < q < oo. We may then write Lt+(p-1)ry(Wz'1+(n-1)/7)
b9pP+e-1
div u -
= div
a
µ
p-- (p+t;)A (div u - + p7
a up7 µ+1;
8a- pp+7 + a( - 1) (divu)p7 µ+ µ+" y
and we deduce, if po, uo are smooth, bounds on div u - µ+{ p7 (and on curl u) in Lt (Wy,q), on Ft (divu- +{p7) (and on 8 curlu) in Li (W. 1,q) for all 1 < q < 00 A and thus in particular LI bounds on div u - µ+E p7 (and on curl u). We finally obtain LOO estimates on p by writing
AC
p7+1
- (P7_divu)P (tt +
and applying the maximum principle. Estimates on higher derivatives are then obtained by differentiating the equation for p and arguing as in the proof of Theorem 7.1 (section 7.3, chapter 7); see also the proof in step 3 below. Let us conclude this part of the proof by mentioning that in the case when -y = 1 (and N = 2) the existence proof may be simplified substantially by using a Galerkin approximation on the equation for u or a simple regularization and we detail this point in the next step.
Step 3: Another existence proof when N = 2 and -y = 1. We consider here the periodic case and we simply mention that our arguments apply as well to the whole space case (S = R2) with a few obvious adaptations. We next discuss
A Stokes-like model
241
briefly the case of Dirichlet boundary conditions. As we mentioned above, various approximations are possible and we give one example, namely
j +div(Pu,) =0, P>0 paatu
-p
=0
with the initial conditions (8.52) where we replace po and uo by (po)e and (uo)e respectively. Here and below, we denote c = cp * ,cE where K. _ -K (E) , Ic E Co (R2), Supp is C Bi, 0 < rc on R2, K is even on R2 and fR2 rc dx = 1. Exactly as in step 1 above, we obtain the following energy identity
d fr p 12
7- f
2
n
+ ap log pdx +
Jn
pIDuI2 + l;(div u)2 dx = 0
(8.61)
from which we deduce a bound on p(1 + I log pI) in Lt ° (Li) and on u in Lt (Hx) . This bound suffices to deduce the existence (and uniqueness) of smooth solutions
of (8.60). It is then straightforward to pass to the limit and recover a solution (p, u) of (8.51) satisfying (8.52). We next claim that p,, is bounded in L2(SZ x (0, T)) and thus p is bounded in L2(SZ x (0, T)) for all T E (0, oo). Indeed, we find easily for T E (0, oo) fixed apE =
(PE(-0)-1divu)
+pdiv((pue)E(-i)-idivu) -p(Pue)f
V(-A)-idivu -
hence we have using the bounds on u in L2 (0, T; Hi) and on p(1 + I log pI) in L°O(0,T; Li) 11pC112
T
C 1 + IIPEIIL2(nx(O,T)) + f f puE V(-0)-1divu,dx o
C
(l
n
T
2
+ IIPEIIL2(nx(O,T)) + fo dt(1 +
uE PI
1 + IIUEIIH
vE I
I
1 + IIUEIIHI
Jdx
< C (l + IIPCIIL2(nx(O,T)) since i+
tlc
is bounded in Hi uniformly in t and thus, in view of J. Moser's inequality [400], there exists a > 0 such that fn exp[cx i+ HI i+ IUC HI ]dx is bounded uniformly in t, and therefore HI
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242
rp
Jn
Ivel
Iuel
P
dx
0, we also deduce that inf essn x (o,T) p > 0 for all T E (0, oo).
It only remains to show the existence of a solution satisfying the a priori bounds obtained above. There are many ways to do so; the simplest is probably to admit temporarily the existence of smooth solutions when the initial data are smooth, a fact shown below. Regularizing the initial conditions and passing to
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the limit, our claim is easily shown observing that the above bounds are then uniform.
Step 5: Another proof when 7 = 1, N = 2. We wish to present now a different proof of the Lq bounds in the case when -y = 1, N = 2. This proof is somewhat simpler and allows us to show (8.54) for all q > 1 but the bounds we obtain require the initial conditions po and u0 to satisfy: ua belongs to a compact set of H1 while p and p log p belong to a compact set of L' (St). Then, by the argument shown at the end of step 3, we already know that u belongs to a compact set of C([O, T]; L2)nL2(0,T; H1) while p belongs to a compact set of C([O, T]; L1)nLx,t and p log p belongs to a compact set of C([0, T]; L1) for all T E (0, oo) fixed. Next, we deduce from (8.63) if q > 1 IISIIL1+9(I x(0,T)) < C 1 +
IIPUIIL9+1(W
1.9+1)
and thus, in view of (8.66), we have II PII L°O(O,T;LQ(n))
+ IIPIIL1+9(nx(O,T)) :5 C 1 + (IPUIILi+Q(W; 1.1+
At this stage, we need to recall that
IIPUIIw-1.1+9 < IIPIIL1+9IIUIIL2 and thus
we conclude immediately when IIu!ILr(L2) is small enough. In general, this is where we use the compactness of u in C([O, T]; L2); indeed, we then deduce that, for each e > 0, we may find ui, u2 such that II U1 II Lr (L2) E, u2 E L' and u = ui + u2. The above inequality then yields IIPIIL1+9(nx(0,T)) :5 C + CEIIPIIL1+9(fx(O,T)) + CIIPU2IIL1+9(W 1.1+9)
C+
CEIIPIILI+9(c x(O,T)) + C(E)II PII Ll+Q(Lr)
C + CEIIPIILI+9(nx(O,T)) + C(e)IIPIILI+Q(nx(O,T)) IIPIIL (L=)
C+CEIIPIIL1+9(nx(O,T)) +C(E)IIPIIL1+9(cx(O,T))
where 9 E (0,1) is determined by 10 + 1 - 0 =
11, 1
I
= 1+q + 2. The desired
bound on p then follows easily choosing c small enough. Proving the existence of a solution satisfying those bounds can be done using for instance the approximation (8.59) with p > 2. Indeed, one can then show that the preceding bounds are uniform in 6, while, for each 6 > 0, similar arguments
to those made above easily yield a priori bounds like (8.54) (in fact, one then obtains a bound on p in LP+q-1(St x (0, T)) and these bounds allow us to obtain smooth solutions exactly as we do in the next step. The only fact that requires some detailed explanation is the derivation of Lit bounds on p uniformly in 6-the energy bounds are obviously valid here and yield an additional bound on 8pp(1 + I log pI) in Ly,t. We then write the analogue of (8.58), namely ap2
= a (in P P + (i +C) (div u)p + a (p(-0)-1div u)
A Stokes-like model
+div(up(-A)-ldivu)
247
- pu=R=R?uj +bpp(-A)-1(divu)
and we conclude easily since (-O)-idivu is bounded in Lt°(Hi) while bpp is bounded in Lt (Hz 1) since we have if bpp < 1
bpp (1 + I log b pp I) < 1
,
< pbpp (1 + I log PI)
if bpp > 1.
Step 6: Higher regularity and uniqueness. We consider here the periodic case since the arguments adapt trivially to the case when Sl = RN. We begin with the proof of (8.54). We observe that we have
f
Vplgdx < C
dt n
I
f I Dul Iopl'dx + f n
< C IIDuIIL-(n)IIVPIIL9(n) + IID2ulILQ(n)IIVPIIL4(n)
hence for all t E (0, T) (where T is fixed in (0, oo)) using parabolic regularity theory
It
JPtIIVPIIL9(n) 0
f d f Iul -u2l2dx + V ID(ul -u2)I2dx 0 fixed, (8.69) is solved exactly as the original problem (in the periodic case) with exactly the same results. Then, it is straightforward to check that all the estimates shown above hold uniformly in e and we recover a solution satisfying the above bounds upon letting e go to 0+.0
8.4 On some shallow water models In this section we briefly discuss some models for shallow water. These models always involve a conservation equation for the height denoted by p > 0 a
ap + div (pu) = 0
in fZ x (0, oo) , p > 0
(8.70)
together with an equation for the velocity u of the following form aPu
+ div (pu ® u) + D + bVp2 = 0
in St x (0, oo)
,
(8.71)
oEw_
where b > 0 is given. Various models (or approximations) are possible for D like for instance
D = -v pLu
(8.72)
V = -v 0(pu)
(8.73)
D = -v div (pVu) .
(8.74)
or
or
Finally, at low Reynolds number, it is possible to replace (8.71) by
D+bVp2 = 0.
(8.75)
For more details on these various approximated models, we refer the interested reader to, for example, P. Orenga [4261 and C. Bernardi and O. Pironneau [64]. Here and below, n is a domain in R2 and we consider the following two possibilities, namely the periodic case and the case of Dirichlet boundary conditions. In the first case, namely when we impose the equation (8.71), we prescribe initial conditions for p and pu while in the second case ((8.71) is replaced by (8.75)) we only prescribe initial conditions for p.
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In the first case (i.e. (8.70)-(8.71)), the Cauchy problem is completely open for the models involving (8.73) or (8.74). Therefore, we consider (8.72) and we first observe that (8.71)-(8.72) (in view of (8.70)) can be replaced (at least if
p>0) by
j+(u.V)u-ziiu+aVp
=0
in 1 x (0,oo)
(8.76)
where a = 2b. We may then impose the following initial conditions Pl t=o = Po >- 0
in 12
in 0.
ult=o = uo
,
(8.77)
We restrict ourselves to the most natural situation of Dirichlet boundary conditions (u = 0 on 8S2). In order to state precisely our result, we introduce the best (i.e. minimal) constant Co for which the following inequality holds
f
u'
a8Uu=usdx
I IuI2d
IDuI2dx
Co
f
2
(8.78)
for all u E Ho (SZ)2-notice indeed that u E L4(11). Then, we claim that, if the following condition holds 2
Jn
I uo I2dx +J Po log PO < 2C2
n
o
+fn
PO log
j
po dx dx,
(8.79)
n
all the results stated and shown in the preceding section are valid here, and in fact, all the proofs and arguments presented and developed in that section immediately adapt to the setting studied here (notice in addition that this corresponds to -y = 1). The main modification concerns the energy bounds and this is precisely where we use the condition (8.79). These results are in fact taken from P.-L. Lions and P. Orenga [357]. Finally, the energy bounds are obtained as follows: first of all, we observe that fo p dx = fn po dx and thus fo p log (p/ fn po dx) dx > 0 (by the convexity of the function (t H t log t) on [0, oo)). Next, we multiply (8.76) by u and deduce (at least formally) using (8.78) dt
+v
2
< Co
f
n
n
IDul2 dx
1/2
Iul2dx
f IDuI2dx. n
We claim that the condition (8.79) remains true for all t > 0: indeed, if it holds up to time t, then Co (fn IuI2dx)1/2 is smaller than v and thus dt (2 fn I ul2dx + fo p log p dx) < 0. Therefore, (8.79) holds for all t > 0 and we deduce a priori
On some shallow water models
253
bounds on u in C([0, oo); L2(fZ)), p and pl logpl in C([0, oo); L1(fZ)) and on Du in L2(f) x (0, oo)), thus completing the proof of the energy a priori bounds. We now turn to the second case, namely the case when (8.71) is replaced by (8.75). If we use the condition (8.72), then (8.75) combined with (8.72) yields the following equality (at least if p > 0)
-vAu + aV p = 0
(8.80)
where a = 2b, and we remark that the system (8.70), (8.80) has already been studied (and solved) in section 8.3 above. Finally, if we use (8.73) (or (8.74)), the situation is rather different. In the case of (8.79) with periodic boundary conditions, we observe that we have
(p2 - -}n p2 dz
div (pu) =
(and curl (pu) = 0)
.
v Therefore, this system ((8.70), (8.73) and (8.75)) reduces to the following integrodifferential equation e+_P2=
in
vip2dx
and it is very easy to check that, if plt=o = po E L2(SZ) > 0, then there exists a unique solution p> 0 E C([0, oo); L2(f1)) fl C1([0, oo); L1(cl)) fl L3(St x (0,T)) (for all T E (0, oo)) such that p is bounded if po is bounded and p is always bounded on 1 x (8, oo) for each 8 > 0. On the other hand, if we use (8.73) with Dirichlet boundary conditions, we claim that such a model is not satisfactory since p may become negative: indeed, if plt=o = Po is smooth, then we have (at least formally)
f
i§F (X, 0)
2
Y)
2
V :=1
Po(y)
dy
8xaayi G(x,
(where G is the Green's function for the Laplace operator (-0) with homogeneous Dirichlet boundary conditions) and this expression may be negative on the set where po vanishes. In order to convince ourselves that this is indeed the case, it suffices to take (for example) fZ = {(x1i x2) E R2/X2 > 0}.
Ei
Then, 1 ax8- y; G(x, y) = 2a 1 away: log y = (Y1, Y2)). A simple computation then solves 2
E 8xza2Gays i=1
1
(x1
F._ y1
where
(y1, -y2) (and
- yl)2 _ (x2Ix + y2)2
Ix - yl4
- M,
hence, if we take po to be supported in a small enough neighbourhood of (0,1) and we pick x outside this neighbourhood (so that po vanishes at x), we have
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b r ((xi-yi)2 - (x2-+2)2)
8t
7rU ,ln
Ix - W
dy
which can be made negative if we choose x1 smaller than (1 + x2)2.
Finally, let us point out that we believe that the model involving (8.74) is also inconsistent but we have not succeeded in building convincing examples.
8.5
Compactness properties for compressible models with temperature
We discuss in this section compactness properties of solutions of the "full" compressible Navier-Stokes equations (namely with a temperature equation). More precisely, let (pn, un, Tn) be a sequence of solutions of the following system of equations
j + div(pu) = 0 in fl x (O, T)
,
p> 0
in St x (O, T)
at (pu) + div(pu ®u) - pAu
- V div u + Vp = pf
Ip = 2p
2
12
+e
+ div ju
(8.82)
in SZ x (0, T)
2
- div (kVT)
p- 2 +pe+p
-p) div(u
(8.81)
(8.83)
in f x (0,T)
where p = p(p, T), e = e(p, T); d = .1 (Du + Dut); f (= f7) is given (exterior forces) on SZ x (0, T), and we assume (to simplify) that f n is bounded in L°O (SZ x
(0, T)), T E (0, oo) is fixed; k = k(T) > 0 for T > 0 and k E C([O, oo)) (for example); p and C are given and p > 0, + (N -1) p > 0; 1 is a bounded, smooth open domain in RN and N > 2. We choose to study the compactness properties of (pn, un, Tn) in the case of Dirichlet boundary conditions even though our arguments apply to the other usual cases (like the periodic case, the case of the whole space, the case of an exterior domain) and are essentially local in nature (some conclusions are global but their global character is in fact derived from local properties). We thus assume that un satisfies (V n > 1)
u=0
on an x (0, T)
(8.84)
and in order to fix ideas we impose Neumann boundary conditions on Tn namely
aT =0 on 8S2 x (0, T) (8.85) an where n denotes, as usual, the unit outward normal to ail. Of course, this is nothing but a simple example and it will be clear from the proofs below that many other possibilities exist for which the arguments we introduced are easily adapted. Let us also point out that the boundary conditions (8.84) and (8.85) are, as is well known, in fact integrated in either the requirement that un E L2(0,T; Ho (SZ))
Compactness properties for compressible models with temperature
255
(for (8.84)), or the fact that (8.83) (with (8.85)) holds in a weak form where we simply multiply (8.83) by an arbitrary function cp E CO° (1 x [0, T ]) N and write (for example) [2PIuI2 +Pe] co(x,0)dx Jn in f T - J n dx dt (2 pI ul2 + pe) - (2 plUI2 + pe + p) (u
([2pIul2 + pe] co) (x,T)dx
.
V w)
o
+ y d (u (9 Dcp) + (e- u) div u div cp + kVT V (p
=
fdxfdtpu.fco.
Let us next recall (see for example, chapter 1 in volume 1 [355]) that (8.83) is equivalent (at least formally) to
4 (pe) + div (upe) - div (kVT) + (divu)p = 2,IdI2 + (C-p)(div u)2 in 11 x (0, T).
(8.86)
We also recall (see also for more details [355]) that the laws defining the pressure p and the internal energy e in terms of the density p and the temperature T must obey the classical principles of thermodynamics. More precisely, p and e have to
be such that there exists a function (called the entropy) s(p, T) such that s, as a function of e and r = p is concave in (e, r), satisfies as
1 ae
8T
TOT
'
8s 8p
1
8e
T 8p
p pz
(8.87) .
The existence of such a function s has various consequences: in particular, e(p, T)
is increasing with respect to T, p(p, T) is increasing with respect to p for s (= s(p, T)) fixed and fixed, and we have
p(p,T)T-1
is increasing with respect to p for e (= e(p, T))
p - T L = ae .
(8.88)
P
Relevant examples of e, p, s are given by: i) (ideal gas) e = CoT, p = RpT, (T11 -1)) = Co log (p T r) where R, Co > 0 are given constants and s = R logRl/Co _ -y = 1 + ; ii) (Mariotte's law) p = RpT, e = e(T) with e' > 0 on [0, oo), fT s= -tie/ (t) dt - R log p ; and iii) (Joule's law) e = e(T) with e' > 0 on [0, oo), p = q(p)T with q' > 0 on [0, oo), s = if e' (t) dt f1 q(o) do,. We shall assume throughout this tsection that (for example) e, p E C1([0,oo)2), e(p,0) = 0 for p > 0, s E C1((0,oo)2), p(0,T) = 0 for T > 0, ps+(p,T) E C([0, oo)2), (p,T) > 0 on [0,00)2 and that p satisfies
-
T
p(p, T) is non-decreasing with respect to p, for all T > 0.
(8.89)
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The latter condition is natural from a physical viewpoint at least if we postulate Joule's law since p(p, T )T -1 must be non-decreasing with respect to p for e = e(T) fixed, that is for T fixed. Let us finally recall that, at least formally, the following identity (the entropy identity) holds
aat (ps) + div(pus) - div
k
VT
_T (8.90) µ = k T2 + T + T (div u)2 in SZ x (0, T) . Let us now turn to our assumptions on (p'n, un,T'). We assume that the se2
L
Idle
quence (pn,un,T")n>1 is bounded in L°° (Q x (0, T)) and (thus) that (un, Tn)n>, is bounded in L2(0,T; H1(1)) (Ho in fact for un, taking into account the homogeneous Dirichlet boundary conditions (8.84)). As mentioned above, (pn, un,Tn) solves, for each n > 1, (8.81)-(8.83) and (8.86) (even though (8.86) can be deduced from (8.81)-(8.83) as we shall see below).
The existence of the entropy s is not needed for our analysis but, if we were to assume it exists, we would then assume that (8.90) holds requesting 2 that pn(sn)- E L°°(0,T; L1(f )), DTn E L1(SZ x (0,T)) and log (Tn A 1) E L2(0,T; Hl) (and thus meas {(x, T) /Tn = 0} = 0, log Tn E L2(0,T; H1)). When this is the case, we immediately deduce from (8.90) integrating it over 12 x (0, t) for any t E [0, T] rt
I pn(sn)-(2, t) dx -} J 0 Jn
0}), for all 1 < p < coo. In addition, if pouo converges in L' (11) to pouo, un converges to u in L2(0,T;H1(SZ)) and in L"(SZ x (0,T)) for all 1 _< p < oo,
and so does Tn provided e does not depend upon p and L2 (SZ) to
polo converges in
poTo.
Remark 8.21 The only serious restriction to the applicability of the preceding
result is in fact the L' bound we assumed on (p, u, T). Indeed, and this is in fact the main obstruction to the construction of solutions in general, very few bounds on (p, u, T) are available (and come from mass, energy and entropy conservation or identities): more specifically, if we assume the existence of an entropy (as explained above), we obtain bounds on p in C([0,T]; L1); pIuI2 and z pe in C([0, T]; L1); ps in C([0, T]; L1), DT and T I DTI2 in Li,t (provided we assume that s+ < C(1 + e) on [0, 00)2). We shall come back to this delicate and crucial point in the next section. Next, if we are willing to postulate bounds that do not seem to be within our reach-and we are in order to examine the stability and compactness features of compressible Navier-Stokes equations-L' bounds can be substantially relaxed. For instance, in the ideal gas case (e = C0T , p = RpT), if we assume Lt (Hy)
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bounds on (un, Tn) (bounds which can also be relaxed), then a bound in Lz,t on pn is enough to ensure the convergence of pn to p in C([O,T]; L'(11)) n LP(1 x (0, T)) (V 1 < p < 2), using a bound on ps in C([O, T]; L1) which thus requires a bound on p°s° in Ll.
Remark 8.22 The remarks made in chapter 5, section 5.2 on the propagation of oscillations (namely Remarks 5.8 and 5.9) can be easily adapted. In particular, the example built in Remark 5.8 (based upon homogenization) in the isentropic
case carries over to the setting studied here, at least when we assume that e does not depend on p (i.e. only depends on T). Indeed, it suffices to replace the system (5.24) by
j + divy(pu) = 0 in RN x (0, oo) -pAyu - Vy div u + Vy p(p, To) = 0 in RN x (0, oo),
[pu dy = [mo dy
for all t > 0
JQ
Q
(8.91)
which, as shown in section 8.2 above, admits unique smooth solutions. We then solve the following auxiliary equation -divy(k(To)Vy9) + (divy u) p(p,To) = 2µI Dyu+DyuT I2 + (e-p)(divy u)2
(8.92)
in RN x (0, oo)
(with fn 9 dy = 0). All unknowns (p, u, 9) are assumed to be periodic in each xi (1 _< i < N) of period 1, Q = (0, 1)N and To > 0 is given. We may then construct solutions of the above system which have the following asymptotic expansion (more precisely what we write below are the leading terms of their asymptotic expansions) pn
p(nx, t)
,
un ti 1 u(nx, t) n
,
Tn -z,- To + T2- 9(nx, t)
.
Proof of Theorem 8.9. Step 1: Convergence of pn and consequences. Exactly as in the proof of Theorem 5.2 (sections 5.3-4, chapter 5)-see also the Appendix B-we obtain the following information
div un -
µ
+ pn Q(Pn)
n
(divu_ +p Q(P) µ
(8.93)
weakly in L2 (1 x (0, T)) .
Then, in order to conclude the proof of the convergence of pn, we only need to show the following inequality PP
0 > ppe
(8.94)
for any 9 E (0, 1) (in fact, for any 9 E (0, oo)). Indeed, once this inequality is proven, the rest of the argument is the same as in the proof of Theorem 5.2.
259
Compactness properties for compressible models with temperature
Next, in order to prove (8.94), we use the condition (8.89) and the bound on Tn in L2(0, T; H1(1k)). Indeed, we first claim that we have for any continuous functions ,3, -y on [0, oo)
f3(P)'Y(T) = Q(P) 7(T) .
Since T n and pn are bounded in LOO (St x (0, T) ), it is enough to prove this claim
for 3 and y bounded with continuous and bounded derivatives. Then, -y(Tn) is bounded in L'(0, T; H1 (0)) while at O(pn) (= -div(un8(pn))-(divun)[ai(pn)pn is bounded in L°O(0,T;W-1'1)) +L2(St x (0, T)). We may then apply Lemma 5.1 (section 5.2, chapter 5) to complete the proof of our claim. The proof of (8.94) is then straightforward; indeed we write p(P,T)P =
=
I
00
J 0 1(P>_a)Pe a (A, T)dA =
f
°°
f
(A,T) dA
1(P>a)P8
P
19P
1(P>a)P0
and similarly
pPe =
f
p(A,T)da
°°
a
1(P>APe ap(A,T)dA.
We may then conclude since (8.89) precisely means that aP (A, T) > 0 for all A > 0, T > 0 and pe 1(P>1) < Pe 1(P>,\) in view of Lemma 5.2 (section 5.5, chapter 5). Having thus proven the convergence of pn to p, we deduce as in the proof of Theorem 5.1 the convergence of pnun (and pTTn) to pu (and pT respectively) in LP (11 x (0, T)), and thus of un (respectively T') to u (respectively T) in LP ((Q x
(0, T)) n {p > 0}) for all 1 < p < oo. In particular, extracting subsequences if necessary, we may assume that un (respectively Tn) converges a.e. to u (resp. T) on the set {p > 0}. Since p(0, T) = 0 for all T > 0, we deduce that p(pn, Tn) = pn,
plunl2un, pne(pn,T') = peen, pnunen converge respectively to p = p(p,T), plul2u, pe(p,T) = pe, pue in LP(SZ x (0,T)) for all 1 < p < oo.
Step 2: Convergence of (un, Tn) in L2(0,T; H1). We begin with the convergence of un in L2 (0, T; H0) . The idea of the proof is very simple: one writes, at least formally, the following identity rT
J
o 2pnlunl2dx(T)+ I J
fT
/'
r
dsJ dxuIDu"I2+t(divu")2 in
ds J dx pnun f n + (div un)pn i
+I 1
co dx
.
2
Letting n go to +oo, we deduce that (fo ds fo dx pnun fn + (div un )pn + fo co dx) converges to fo ds fo dx pu f + (div u)p + fn 2 co dx as n goes to a
Related problems
260
+oo. And, since (p, u) solves (8.81)-(8.82), the latter quantity is also equal, at least formally, to the following quantity (f0 ds fn dx µI Du 12 + t; (div u)2 + f 2 plul2dx(T)). This is enough to ensure the strong convergence of u" jr L2 (0, T; Ho) to u. Therefore, we only have to justify the above identities obtained by multiplying
(8.82) by u. In order to do so, we shall mollify (8.82) with respect to t and we (E ), with n > 0, , is even on IR, fR rc(t)dt = 1, Supp rc C introduce nE = [-1, +11 and e E (0,1]. Extending all functions for t E [-1, 0) and t E (T, T + 1] on R by 0 for instance, w e obtain, denoting co (x, t) = f cp(x, s) r., (t s)ds for
-
any function cp,
f f T
dt
0
dx Gb (Pu)f ' u - P(u ®u)E Du
n
+ ptDu, Du + & div uE div u - Vip,div u
=
1T1 dx b(Pf)F u + 1T1
[ mor(t) -
i(Pu)(T)rc(t-T)]
for any t' E Co (0, T). The last term of the right-hand side obviously vanishes for a small enough, namely if e < inf Supp and E < T - sup Supp iP, and from now on we assume that this condition holds (so we can safely ignore all boundary terms) and we wish to let s go to 0. We obviously have T fdtf dx - p(u ® u)E Du + pi)Du, Du lim
+ O div uE div u - V)p,div u - 1P (pf )Eu j0T
dt
in
dx-p(u ®u) Du + µlDul2 +
(div u)2 -p div u - dip f u
.
Let us also observe that we have 2
1
dx - p(u (9 u) Du = -
in
dx pu V 12
.
We thus need to compute the limit of (fo dt fn dx ac (pu)E u) and we write this expression as (fo ds fo dt fn dx 0(t) p(x, s) u(x, s) rcE(t - s) u(x, t)). Next, we observe that we have by elementary computations which are easily justified 0=
f
dx
d 2
f
dx'(t) j dt in J IT
ds p(x, s) u(x, s) rcE(t-s) u(x, t}
0
ods p(x, s) u(x, s) rc(t-s) u(x, t)
+ a f i(t) p(x, s) u(x, s) rc'E(t-s) u(x, t) ds 0
Compactness properties for compressible models with temperature T
+
- 5 (x, t) ds
(ti(t) p(x, s) - ip(s) p(x, t)) u(x, s) rc, (t s)
2J
f
+2
261
T
t1(s) u(x, S) a(a ) (x, t) rce(t
0
T
- 2 JO
i(s) u(x, s) u(x, t) Ke(t-s)
- s) ds
at
(x, t) ds.
Therefore, we deduce ds dt
J0
o
_-
fT +
Jn
dx iP(t) p(x, s) u(x, s) rcE(t-s) u(x, t)
r dx dx ti'(t) Jn IT IT r
2 J0 Jo
ds dt
T 2
Jo
ds p(x, s) u(x, s) K (t-s) u(x, t)
dx [ i(s)p(x, t) - b(t) p(x, s)] u(x, s)r.,(t-s)
fnn
j (x, t) - 2Ty(s) u(x, s) u(x, t) Ke(t -s) div (pu)(x, t) . The first term of the right-hand side converges, as c goes to 0+, to jT
dt
-2
f
dx ik'(t) Plul2
n
while the last term goes to
+a f
T
dt J n dx Ti(t) pu OIu12
.
We are left with the following expression 1
2
TT
[b(s)P(x, t) - Ty(t)P(x, s)] u(x, s) Ke(t-s) I I ds dtJ dx (x, t) n Tds - -1 Jo TJ dt n dx u(x, s) u(x, t) at [((s)P(x, t) -(t)P(x,
dd
_ -2
TT
J
ds dt
o
in
dx u(x, s) u(x, t)
((s) t (x, t)
p(x, s))
Ke(t-s) + [ b(S)P(x, t) - ')(t)P(x, s)} KE(t-s)
TT =2
ds dt
dx
t) V
s) u(x, t)) Ke(t-s)
J0 0 Jn +u(x, s) u(x, 00' (t) P(x, S) Ke(t-s) + Tux, s) u(x, t) 1 01 (S + A(t-s))dA ((t-s) KE(t-s)) p(x, t)
s))Ke(t-s)]
Related problems
262 1
-OW
J0
V (u(x, s)u(x, t))((t-s) rc'' (t-s))
(pu)(x, s + A(t-s)dA
which goes to 0 as e goes to 0+, since (t - s) r.' (t - s) _
-s)
t E 88 rc' (tt
and f R trc'(t)dt = -1. In conclusion, we have shown that we have (in the sense of distributions) 2
J
pIuI2dx+J 4JDu12+l;(divu)2dx =
fpi.u +p divudx on (0,T)sswhich
was precisely the formal identity we had to justify.
In the case when e does not depend upon p, the argument for T' is almost the same once we observe that F" = [-(divul)pn + 2µIdn12 + (e-y)(divu")2] converges in L1(1 x (0,T)) to F = [-(divu)p + 2pldI2 + (e-µ) (div u)2]. Hence, we only need to write (8.86) as
in n x (0, T)
i [pnf (Bn)] + div [p"unf (Bn)] - O9n = F'
where on = K(Tn), Q o K = e, K(7) = f' k(s) ds for all A > 0. In view of the aforementioned convergence of Fn, (fo dt fn dx Fn9n) converges, as n goes to +oo, to (fo dt fn dx F) . In addition, the convergence of Tn on {p > 0} shows that we have 9 = K(T) on {p > 0} and thus 49
(p/(g)) + div (pu,3(9)) - A9 = F.
Finally, justifying the computations exactly as we did above, we have
in
p"ry(6")dx(T) + P
Jn
f /0
py()dx(T) +
T
f
dt
f
pT
dxlV8' I2 = J 0
SZ
r
dt I dxIV2 = R
fT
dt
inn
dxF"8"
r dt I dx F6
.
o
Hence, V9n converges in L2 (St x (0, T)) to 09 while (for example) we already know that J 9 ' converges in L2 (SZ x (0, T)) to 9 (and fn p dx = fo po dx > 0 ' for all t E [0, TI). Therefore, on converges in L (0, T; H1())) (since IIV (L2 + lk/ IIL2 is an equivalent norm on H1) and thus in LP() x (0, T)) (V 1 0 and q is a continuous, non-decreasing function on [0, oo) such that q(0) = 0
converges to a positive limit as t goes to +oo for some a > 1 and fo 1.JIds < oo-this last assumption is only made to simplify the presentaq(t)t_a
tion and is not really needed for our analysis. We begin with the case when 6 > 0 and later on explain the modifications to be made when S = 0. Next, we choose e(p, T) = COT + 6 fo q(s)s-2 ds. One then checks easily that there exists an entropy s = s(p, T) given by s = Co log T - fa q(s) s-2 ds, and, by a straightforward computation, that s is concave as a function of e and rr = n Also, we shall need k to really depend upon T since we assume that k E C([0, oo)), k > 0 on [0, oo) and k(t)t-b converges to a positive limit as t goes to +oo for some b > 1. In order to fix ideas, we shall only work with k(t) = ko+k1tb
where ko > 0, k1 > 0 and b > 1. Finally, we restrict our attention to N = 2 or
N=3.
We are going to show in this section that, if we choose a and b large enough, then it is possible to construct global solutions for such compressible systems. However in order to do so, it is necessary to explain carefully the meaning of solutions. We shall look for (p, u, T) "satisfying" (8.81)-(8.83) with periodic boundary conditions (as usual we could consider as well all the other boundary conditions studied and used in the text above) and some form of the entropy identity (8.90). The density p is required to satisfy p E C([0,T]; Ll(SZ))f1C([O,T]; L2(S )w) and plt=o - po in 12 where po > 0 E L1 fl La(f2), po # 0. The temperature T satisfies: T > 0, T E L2 (0, T; H1 (11)), log T E L2 (0, T; HI (11)), and in particular meas { (x, t) E SZ x (0, T) / T (x, t) = 0} = 0, Tb/2 E L2 (0, T; Hl (SZ) . The velocity u satisfies: pu E C((0, T]; L27/(7+1) (SZ) - w), pint=o = mo with mo - 0 a.e. on {po = 0}, u E LP(O, T; Wl,q(SZ)) for some p, q < 2 which depend upon N and b
and that can be made as close to 2 as we wish by taking b large enough-this point will be detailed later on. In addition, pIu12, pT and p log T E Ll (0, T; L1) and 2pJu12, CopT, Cop log T - p fo q,s ds = ps(p,T) converge (in the sense of distributions) as t goes to 0+ respectively to co = Im012/po, eO, so (which are thus assumed to belong to L'(S2)), and we assume that eo = so = 0 a.e. on {po = 0} and that eo - S fo ° q(s)s-2 ds = Co exp(c0 [so + fo P0 q(s)s- 2dsD on {po > 0}. Finally, we require u to satisfy IDu12 T-1 E L1 (S) x (0, T)). We next need to make precise the meaning (and the formulation we shall use)
Related problems
264
of equations (8.81)-(8.83) with (8.90). In fact there are many possible variants and we just give one possibility. Equations (8.81)-(8.82) hold in the sense of distributions provided we check that p E L' (l x (0, T)) or that paT E L1(cl x (0, T)), all the other terms being clearly defined in L1(f x (0, T)) at least. Let us recall that we assume that f E L°° (11 x (0, T)) (for instance) in order to simplify the presentation. The equation (8.83) will be understood as follows: there exists a distribution E such that 2
a
5 p
lu12
2
+e
+div
= pu f
Lulr
p 2 + pe+p
in
Sl x (0, T)
+div(E)
.
}
(8.95)
In fact, this says little more than the natural identity for the global total energy, and additional information is in fact deduced from the entropy identity. We now make precise (8.90) (i.e. the entropy identity): we request that there exists a bounded non-negative measure m on fl x (0, T) such that we have div (pus)
at (ps) + = m + koLL + kol0LI2 + k1EB + b
+
[2µI
dl2 + µ(div u)2] ,
where L E L2 (0, T; H1(11)),
IvBl2
B
L < log T
,
(8.96)
B > bTb ,
E L2 (0, T; H1 (Q)) and we agree that
B
_
4Ivv12. This formulation makes sense provided we check that plugs E Ll (fix (0, T)) or that plug I log TI and palul E L1(Sl x (0, T)). In conclusion, we have introduced a rather complicated formulation of the full system of compressible Navier-Stokes
equations that we need to motivate and we shall do so below. In addition, this formulation is meaningful provided we check that paT, pluI3, palul, palulT and pul log TI E L' (11 x (0, T)), a fact whose discussion we postpone since we first wish to discuss the above formulation, and explain the role of the auxiliary unknowns, E, B, L. We need to introduce them in order to take into account possible losses of compactness (or the lack of L'(Hy) estimates on u) we cannot overrule because of the lack of a priori bounds. If we were able to prove L° bounds-as assumed in the preceding section-then indeed we could simply set E = 21u - d + m = 0, L = log T and B = iTb, and this would be in fact a consequence of the compactness results (and proofs) developed in the preceding section. However, the lack of such a priori bounds forces us to introduce the weak limits of log T, namely L, and 6T6, namely B. We shall prove below some compactness results of T on the set J p > 0} and thus we have as a by-product of the arguments presented below in fact L = log T on the set {p > 0}. Let us also mention that, in view of the definition of O, L, B (as weak limits), standard functional analysis considerations yield the following additional inequality valid a.e. on 11 x (0, T):
Global existence results for some compressible models with temperature
265
(8.97)
Finally, we want to explain that if (p, u, T) satisfies the above formulation and, for instance, belongs to LO° (SZ x (0, T)) and if there is no vacuum, i.e. meas J p = 0} = 0, then (p, u, T) is in fact a "standard" solution of the original system of equations (8.81)-(8.83) and satisfies (8.86), (8.90) (in the sense of distributions). Indeed, by formal computations which can be justified as we did in the proof of Theorem 8.9 for similar formal computations, we deduce from (8.96) (taking L = log T, B = 1Tb a.e.) that there exists a bounded non-negative measure fn on SZ x (0, T) such that jt (CopT) + div(CopuT) + (div u)p - div(kVT) = 2pJd12 + (t; -p)(div u)2 + rn while we have
(1&12) +div (u (2pJu12))
div(2pm.d) -
div(udiv u) = pu f .
5
Therefore, we deduce in particular
d (f
pIuI2+CopTdX = fpu.fdx+dni(c2t).
We may then compare with (8.95) (integrated over St) and we deduce that m 0. In other words, (8.86) holds from which we deduce that (8.83) holds (and also that (8.90) holds). Let us also mention that it is plausible that the above
argument can still be made without assuming that meas{p = 0} = 0 by an appropriate modification of (8.96), but we do not want to pursue this argument here since, anyway,. it requires bounds on (p, u, T) which we do not know how to obtain. The only merit (if any!) of the above argument is to show that the auxiliary unknowns we introduced are merely reflections of losses of compactness and do not really affect the physical equations, since they can be recovered if the solutions we build are bounded. We finally turn to the discussion of the integrability of the various quantities mentioned above (plul3 , paT, pa lul, pa IuI T and pl ul I log TI). In fact, we claim that these quantities are integrable for a and b large enough. We shall obtain below some a priori bounds on pin L°O(0,T; L°(Q)), on pT and pi u1 2 in L°°(0,T;L1(11)), on log T, T and Tb/2 in L2(0,T;H1(11)) and finally on DT in L 1(Q x (0, T)). Therefore, if N = 2, T is bounded in L' (0, T; Y (n)) for all r < oo while, if N = 3, T is bounded in Lb (0, T; Lr (S2)) with r = 3b. Therefore, Du ( DuT ) 1/2 T 1/2 is bounded in LP (0, T; LQ (n)) with p1 = 21 + 21b ) -1 = + 2r1 and we remark that p and q go to 2 as b goes to +oo. As we shall see f elow this will yield (at least for a, b large enough) bounds on u in LP (O, T; W 1"4 (Q)). In order to make sense of the above quantities, we shall need some further a priori estimates. We shall prove below that pe pa (T +5) is bounded in L1(O x 12
266
Related problems
(0, T)) for some 9 > 0 which goes to N a - 1 as b goes to +oo. Admitting temporarily all these bounds we may now check that pJuJ3, paT, pau, palutT and pluI log T are bounded in L1 (S2 x (0, T)) and in fact in L' (0 x (0, T)) for some t > 1. Obviously, it suffices to show this claim in the case when "b = +00", i.e. when T E LOO (Q x (0, T)) and u E L2(0, T; H1(SZ)). Since pa+0 is bounded in L' (SZ x (0, T)), the claim is immediate for the quantity paT and the one for pa lulT follows from the one on pa Jul. Next, writing pJuI h log TJ = (fp Jul)\ I log TI, we deduce that it is bounded in Lt0(L,,) Lr(Lza) Lt (Hi) C Lt (Ly) for some
t > 1 provided a > 1 if N = 2 and a > z if N = 3. We now turn to
pluI3
when N = 3 (the case N = 2 being obviously simpler since u (and p) enjoy better bounds): first of all, plul3 is bounded in L213(Lx) with -11 = 1 + a writing plu13 = p. Ju13 while it is bounded in Lt (L6/7) writing pIuJ3 = (pIu12) Jul. We only have to deduce from these bounds that piu13 is bounded in L' (LA) for some ,Q > 1: indeed, we find A = 2 a + 6) = s + 2a < 1 if a > 3. When N = 2, the condition is that a > 1. We conclude with the most delicate term, namely pa Jul (and thus paluIT): this is where we need the additional bound on p. Once more, we consider only the case when N = 3 since the case when N = 2 is easier. Then, we remark that pa Ju( is bounded in Lt (LO) with « = a+e + a, Q = a+T + 1 while a-1/2 + .1. Next, we observe a-112 1 it is also bounded in L7 t (La) x with If1 = a+9 2' b = a+9 that, as a goes to +oo, as+9 and as+e2 go to 5 therefore goes to -1, 10 --1 goes to 30 while ,11 goes to b and b goes to is . And we conclude since 5 10 + -11 55= 1 111 4 23 while 5 50 + 5 50 - 6 < We may now state our main result. 1
Theorem 8.10 With the above notation and conditions, there exists a global solution (p, u, T) of (8.91)-(8.93) and (8.90).
Remark 8.23 In the case when b = 0, the same result holds (with the same proof) provided we degrade the notion of solution (even more!). The difficulty lies with the fact that we no longer know that pus and in fact p'u E L oC: indeed, we have some information on pep and, unless T is known to be bounded from below, we cannot deduce (at least we are not able to deduce) that pau E L' 10C, Then, we only obtain a distribution E' such that (8.96) holds with pus replaced by V.
Sketch of proof of Theorem 8.10. We shall only discuss the proof of the a priori estimates and the compactness analysis, the only missing element being the actual construction of a solution which can be deduced from a series of approximations very similar to what we did in the previous chapter and sections. The details, which are quite lengthy and tedious, will be omitted here. We thus begin with the a priori bounds. First of all, the energy identity immediately yields a bound on pT, plul2 and 4na in L'(0, T, L1(Sl)). Next, the entropy equation (8.90) implies
Global existence results for some compressible models with temperature
sup
0 1 of, say, smooth solutions satisfying the previous bounds uniformly in n and such that (pn, u", T") converges weakly to some (p, u, T) in [L°O (0, T; L'(11)) (weak-*) nLa+e(1 x (0, T))] x LP(0, T; W l,q(11)) x L2 (0, T; HI (SI)). In addition, we assume
that the initial conditions satisfy the bounds introduced above uniformly in n and that po = pnlc=o converges in L1(Q) to po (# 0). Then, we argue as in the proof of Theorem 8.9. Using the bound on T11 in L2 (0, T; H1(St) ), we deduce that we have /3(p) 'Y(T) = Q(p) -y(T)
(8.98)
from any /, y E Cb(R) for instance. Using the bounds on (pri)g+9(Tn + S), we deduce from (8.98) for any 9' E (0, 9) q(p)(T+S) = q(P) (T+S)
,
q(P)Pe'(T+S) = q(P) Pe' (T+S)
and since we always have q(P) peg > peg q(P)
we obtain finally P", P > Pe, P
This inequality allows us to prove that pn converges to p in C([0, T]; LP (f)) for all 1 1.
Remark 8.24 We wish to observe that we can apply the same type of methods in order to analyse stationary problems (see chapter 6 for related results) namely div (pu) = 0 in 0,
p>0
in SZ,
in p dx = M
div (pu (9 u) - pEu - l;V(divu) + V(q(p)T) = pf in
SZ
u= 0
,
on 8SZ
IL
'
(divu)q(p)T - klT
fQAnl
= 2pldl2 + (6-p)(div u)2 in fl an + AT = ATo
on 811
where M > 0 is given, k > 0, Co > 0, A > 0 and To is (for instance) a positive constant. We only discuss (in order to restrict the length of this remark) the case of boundary conditions contained in (8.99) where fl is a bounded, smooth, simply connected domain of R2. We assume for example that f E L°O (SZ), that q(p) is a smooth (Lipschitz for example) increasing function on [0, oo) such that q(0) = 0 and q(t)t'° > 0 for some a > 1. Then, we claim that there exists a solution (p, u, T) of (8.99) such that T, T E Lq (SZ) for all q E [1, oo) ; T and log T E H'(SZ) ; q(p) E LP(St) and u E WW''(SZ) for all 1 < p < 2 ; s 'DT E L'(11) and the following properties hold
div u p 12u2
+ CopT + p
- div (kVT)
= pu f + 211 div(u d) + (l; - p) div (u div u)
div (pus) - div
=
(8.100) in SZ
,
VT
T T Idj2+ Tp(divu)2+T2 IVTl2
and p E LIO (St), U E WWo (Q), T E W oq (SZ), div u
(8.101)
inSZ,JI
-
IL
(Q) +E q(p)T E W11, 0C
and curl u E Waq (Q) for all 1 < q < oo. Let us also mention that the latter
Related problems
270
regularity statements are valid up to the boundary if we modify the Dirichlet boundary conditions as we did several times in chapter 6. We only prove the a priori bounds. First of all, integrating (8.101) on 11, we deduce
TI2
fn
+
IVT12
dx +
Ja T
dS < C
(8.102)
where C denotes various positive constants independent of (p, u, T). We first deduce from this bound a bound on log(T A 1) in H1(1) and thus on T in Lq (SZ) for all 1 < q < oo. Indeed, we have 0 < 18n log
Tnl
dS < log
/
T ^ 1 dS < C
./a
and the bound on log(T A 1) in HI (0) follows since V log(T A 1) = TT 1(T 2, this simply means that we impose a local energy inequality
at
(PluI2+1P7)+div{u{PIu2+ a71 p 1 j 2 - ) -
0 by an appropriate scaling. The above observation is immediate once we remark that X is nothing but the fundamental solution of (8.110) (Xlp=o = 0 , ap Ip=o = So). In particular, we have P=
f
X(P, e-u) d
,
Pu =
f
eX(P, c-u) de .
(8.115)
and
a
2Pu2
f2x(P,e_u)de.
(8.116)
Finally, one can check (see P.-L. Lions, B. Perthame and E. Tadmor [359]) that 77 is convex in (p, pu) if and only if g is convex. We may now define precisely an entropy solution (p, u) of (8.106)-(8.107): (8.111) is required to hold for any 17 given by (8.113) with g convex on R. This makes sense if we require (p, u) to be bounded on Rx x [0, oo). Another natural case consists in assuming that (p, u) has a finite energy, i.e. p E LO° (0, oo; L,' ,), pl ul2 E L°°(0, cc; L'10c): in that case, we need to restrict the growth of g assuming
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that g is subquadratic. We also need to precise the entropy flux. A straightforward computation then shows that -P (P, U) = fR g()[9e + (1-9)u] X(P> e-u) d
(8.117)
where 9 = y-1 2 We next describe the so-called kinetic formulation of the notion of entropy solution, a formulation introduced in [359] which allows us to write in a single equation both the system of equations (8.106)-(8.107) and the preceding set of entropy inequalities. Indeed, it is shown in [359] that (p, u) E LOO (R x (0, 00)) is an entropy solution of (8.106)-(8.107) if and only if x = x(p, a-u) satisfies 2
19
at + ax [(9 + (1-e)u)x] for xE]R, eER, t>0
ae2
(8.118)
where m is a bounded non-negative measure on R2,, x (0, oo). In order to be more specific, let us observe that if x is given by x(p, 1; - u) for some functions (p, u) of (x, t), then p and pu can be recovered from x using the formula (8.115). The meaning of (8.118) is relatively clear: indeed, for each t; E R, x(p, c-u) is an entropy and thus we expect to have
a [(0 + (1-9)u)X] = 0 in view of (8.117), at least when p and u are smooth. The right-hand side in (8.118) therefore accounts for the possible losses of smoothness of (p, u). The fact
that the two first moments of x do satisfy the natural, associated conservation laws then yields the second derivative with respect to and finally, the sign of m simply reflects the entropy inequalities. Let us observe in addition that, clearly enough, if p and pu are of class C' in an open set 0 of R x (0, oo) then m - 0 for (x, t) E 0, l; E R. Finally, the terminology "kinetic formulation" comes from the strong similarity of the above formulation with kinetic models. In fact, it goes beyond a simple similarity since , which is nothing but an extra "hidden" variable, plays the same role as a velocity (one can also think of as a fluctuation variable). In addition, when -y = 3, then 9 = 1 and the transport operator in the left-hand side of (8.118) simply reduces to (a +l; ax ) . In that case, it is possible to approximate (8.118) by a "Boltzmann-like" kinetic model, namely replacing x, (- as )respectively by f ,
(X-f) where f =f(x,e,t) >0andx=X(P, u-C), Then, letting a go to 0+, we may recover (formally but also rigorously) (8.118) and, roughly speaking, the collision term E (x - f) converges to a distribution (with the precise form of - a for some m > 0) which is "supported" on shocks. We refer the interested reader to P.-L. Lions, B. Perthame and E. Tadmor [359, 360] for more details. We may now state our main existence result.
On compressible Euler equations
275
Theorem 8.11 Let (po,mo) E L°°(R2) be such that po > 0, mo/po E L°°(R). Then, there exists an entropy solution (p, u) E Loo (R x (0, oo)) of (8.106)-(8.107) such that PIt=o = Po, Pul t=o = mo.
Remark 8.25 Obviously, u is not uniquely defined on the vacuum set l p = 0}. More generally, the uniqueness of entropy solutions is an important open problem.
Remark 8.26 Aspects of the proof of the preceding result were given in a remarkable paper by R. DiPerna [150] when ^f = 2k+1 with k > 1, the proof being completed by G.Q. Chen [101] with an extension to the case when 1 < 'y 3 The case when ry > 3 is treated by P.-L. Lions, B. Perthame and E. Tadmor [359] while a general proof (which is also not too complicated) is presented by P.-L. Lions, B. Perthame and P.E. Souganidis [358] for 1 < ry < 3. 0
As usual, the above existence result is shown in [358] by approximating
(8.106)-(8.107), adding "viscosity terms" namely (-e a-.P), (-e a), with e > 0, respectively in the right-hand sides of (8.106)-(8.107) and by passing to the limit using some compactness properties that we state, as usual, for "exact" solutions and not for "approximated" solutions. More precisely, the following result is shown in [354] for ry > 3 and in [358] for 1 1 be a sequence of entropy solutions of (8.106)(8.107) that we assume to be bounded in LO°(Rx (0, oo)). Without loss ofgenerality, we may assume that (pn, un) converges weakly in L°° (Rx (0, oo))w-* to some (p, u) E LOO (R x (0, oo)). Then, pn and pnun converge in L"((-R, R) x (0, T))
top and pu respectively for all 1 _< p < oo, R E (0, oo) and T E (0, oo). Therefore, un converges to u in LP({(-R, R) x (0, T)} fl {p > 0}) for all 1 < p < oo, R, T E (0, oo) and (p, u) is an entropy solution of (8.106)-(8.107).
Remark 8.27 We wish to emphasize that, contrarily to what we saw for compressible Navier-Stokes equations in chapter 5, no assumption is made about the behaviour of (pn, pnun) at t = 0. In particular, oscillations may be present initially and are immediately wiped out for positive time. In other words, the inviscid case (i.e. Euler equations) enjoy better compactness properties than the viscous case (i.e. Navier-Stokes equations), a fact which may look slightly surprising in view of the better regularity expected for solutions in the viscous case. A tentative "physical" explanation of this phenomenon is the following: as the initial oscillations grow (in frequency), shocks develop in shorter times and truncate these oscillations thus restoring compactness. In other words, the shocks, which are a consequence of the non-linearity of the system, do create the above compactness!
Remark 8.28 In the very particular case when 'y = 3, a much more precise result is given in [359] which states some partial Sobolev regularity for entropy solutions. These bounds, in turn, immediately imply compactness. The proof (see [359] for more details) is an immediate consequence of the kinetic formulation.
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It is a very interesting open question to decide whether such a direct proof is possible for general 'y's (proving directly from the kinetic formulation either partial regularity results or even the mere compactness of solutions as stated in Theorem 8.12). The actual proof of Theorem 8.12 relies upon the compensatedcompactness approach initiated by L. Tartar [531] and extended by R. DiPerna [150], and uses the kinetic formulation in a straightforward (although somewhat technical) manner in order to eliminate the possible losses of compactness which are measured in terms of Young measures a la Tartar [530].
Remark 8.29 As mentioned above, the existence proof in [358] uses a viscous approximation where, however, we introduce very specific second-order derivatives of p and u (of p and pu in fact). This raises a very natural question which is still open: can we pass to the limit from solutions of the compressible NavierStokes equations to (entropy) solutions of the Euler equations? In other words, if we only add the natural viscous term (-E e ) in the right-hand side of (8.107) and solve the resulting system, can we let a go to 0+ and recover entropy solutions of (8.106)-(8.107). 0
We now turn to a slightly different topic which concerns bounds on solutions. If the initial conditions are bounded, then LO° bounds on an entropy solutions
(p, u) follow easily from the kinetic formulation (see [359] for more details). s Furthermore, it is shown in [359] that if po uo 2 + Po = po + po E L'(R), then p(3'y-1)/2 +pIuI3(x, t) dt. the kinetic formulation yields a bound on sup essXER fo We wish to describe now another bound which follows from a general observation that seems somewhat related to the compensated compactness theory and more specifically the so-called div-curl lemma (see F. Murat [401,402], L. Tartar [531] and also R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes [110]). We first state
and prove this general observation and then apply it to the Euler equations. We assume that u, v, w are given functions in L10C (St) where 11 is a bounded open set in R2 and that u, v, w satisfy
u>0, w>0, uw>V2 (9U
49V
a-+aw
=0, a7X+ay
a.e.
=0 inD'(S2).
(8.119) (8.120)
Then, we have
Theorem 8.13 uw
- v2 E L'10j C(SZ) and, for any compact sets K, k included in
12 such that K C K, we have JK
uw - v2 dx dy < C(IIuII
L1(K1) +
IIVIIL1(K') +
IIwhILl(K')/
for some positive constant C which is independent of u, v, w.
(8.121)
On compressible Euler equations
277
Remark 8.30 The preceding result admits many variants. We may for instance
replace in (8.119) the assumption u > 0, w > 0 a.e. by u > -f, w > -g where f, g E Li ". We may also relax in a similar way the condition uw > v2 a.e. and we may also allow for appropriate right-hand sides in (8.120). Another variant consists in assuming that u, v, w E Ll (11), 11 is smooth and simply connected and 'n= (') Proof of Theorem 8.13. Let cp E C000(0) be such that cp = 1 in a neighbourhood of K and 0 < co < 1 and Supp cp C K. In order to simplify the presentation, we also assume that Supp cp is simply connected (otherwise, the argument can be adapted somehow but becomes quite technical). We first deduce from (8.120) .n=0.
the existence of cx,,3 E W 1,1(w), where w is a simply connected domain such that Supp cp C w, such that U
as
ay,
as
v
aQ
a/3
ay, w
v
ax,
MW.
ax
In particular, we have on w az+ap
= 0.
Thus, there exists 7r E W2,1 (W) such that a = ev , l3 = -
.
In other words, we
have
u=
a27r
w=
aye'
a27r
a29r
axe'
ax ay
V
and (8.119) is then equivalent to the convexity of ir. Next, we use a regularization kernel r.E = 1 rc(E) where rc E Co (R2), ,c > 0, fez rc dz = 1 and Supp rc C B1. Then, for s small enough we may write, denoting tpE = ip * rcE for any function t/,,
1g2 cp(uewe-v) dxdy
_
R2
W det(D2ire) dx dy 1
a2
alre a,r,
ax ay
1R2[OXOY
_ 0. In particular, if initially the energy is finite, this implies that curl u is bounded in L°O(0, oo; L'') (at least). We may thus expect a uniform (in t) "regularity" (and thus compactness) of the divergence-free part of u. It is also tempting to expect that the compactness result Theorem 8.12 holds, when N = 2, for the density p and for the potential part of u. There is little rigorous evidence for this hope. We can simply invoke a naive passage to the limit as it, go to 0+ in the identity which measures the dissipation of the "strength of oscillations", obtained in chapter 5 for Navier-Stokes uations. This formal passage to the limit forces quantities like pl+7 - P pry (or pe+ti Pe pry to vanish at the limit, an indication of the "automatic compactness" of pn). If the compactness of pn were established, one can, at least formally, argue convincingly that u'', or equivalently the potential part of u'', is compact. Indeed, we write un = Virn + w'n where wn and irn. are bounded in L°° (0, T ; W"'q) for
-
all 1 < q < oo, and as " is bounded in L°°(0, T; Li) for all 1 < q < oo, and n n applying the div-curl lemma ([402]) with and (U% n j , we obtain air
pat + p Ju12 - puw = p hence p[Ju12
-
IU12]
+ pl'a12 - puw
= 0, and the compactness of u on (p > 0) then follows.
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280
Consistent with the above speculations are two simple classes of examples that we briefly describe now. First of all, in the radial case, that is when p and it are radial and u = 07r, we are essentially in the one-dimensional case and the compactness of p and u = 0ir holds. Next, we consider the case when p, ul and u2 do not depend on x2 so ui is the potential part and u2 the divergence-free part. Then, (p, ui) is a solution of the one-dimensional system of equations and thus the compactness of p and ul is ensured by Theorem 8.12, at least if we assume that (p, u1) is an entropy solution (otherwise we may have to solve the interesting and natural open question which consists in asking whether Theorem 8.12 is still true when we allow for only one entropy, namely the energy). Then, u2 = w solves (writing x for x 1 and denoting u = u l)
(P'w)+ax(puw) = 0.
(8.125)
The above quantity (8.124) then indicates that we should expect to obtain a solution of (8.125) with some smoothness, at least under some conditions on the initial condition pwlt=o = no. This is indeed the case as we now show. We assume, for instance, that wo = no / po and d /po are bounded and we consider, in order to simplify the presentation, the periodic case (the proof being easily adapted to more general situations). Let is, = E r(E) and rc(z) = (21r) We then denote pe = p * ice , ue = (Pul)e /Pe . Observe that, if po 0 0, then pe > 0, pe E Wt '°O (Cz°), (pu)E and uE E W1'°° (Cr) and
+ a (Peue) = 0. Next, we set wo,e = (wo)e and we observe that I
1 < C po,E for some positive
constant C independent of e. We may then solve the following linear transport equation
8+ a=0 uE
for x E R, t > 0;
w1t=0 = Wo,e
and we find a smooth solution which obviously, satisfies a
au
at
ax
aw
a
0,
hence, we have for any continuous function a dt
f dx pe
/pE
= 0.
In particular, and thus ai are bounded in L' . It is then straightforward to pass to the limit and obtain a solution w of (8.125) in Wi,t °. Furthermore,
On a low Mach number model
281
one can build, in this manner, such a solution which satisfies for any convex continuous function Q and for all t > 0
fdxPfl(r)/P)
< f dx Po Q
do
/Po
8.8 On a low Mach number model In chapter 1 (volume 1) [355], we derived formally, letting the Mach number go to 0+, the following model
apu
at +div (pu) = 0, p > 0
(8.126)
div u = co A(1/p)
(8.127)
+ div (pu ®u) - div Nn(p)d) + 0ir = 0
(8.128)
for some scalar unknown r, where co > 0 is a fixed constant and µ is a continuous
positive function on (0, oo). Finally, we denote, as usual, d1? = 1(a + e ) for 1 < i, j < N, and we consider either the case when the equations are set in RN, or the periodic case when again the equations are set in RN and we require the unknowns to be periodic. In fact, we shall concentrate below on the case of the whole space requiring u to vanish and p to be constant at infinity in an appropriate sense. The periodic case is then a straightforward adaptation. However, we do not discuss here the case of Dirichlet boundary conditions in order to avoid some rather lengthy considerations on boundary conditions. The above system of equations and more precisely its extension to combustion models, i.e. models for reactive flows, has been introduced by A. Majda [363]
and studied in particular by P. Embid [169] as fas as the local-in-time wellposedness is concerned. We want to make some mathematical observations on the existence (and regularity, uniqueness) of global solutions in the case when N = 2. Before we discuss further the results we are going to obtain below, we wish to point out that a special and important class of solutions of the above system consists in taking p - p, a positive constant. Then, the preceding system reduces to the classical (homogeneous) incompressible Navier-Stokes equations
(see for instance chapter 3, volume 1 [355]). Therefore, our goal is to obtain global existence results for finite energy solutions which contain as a special case the known results on incompressible Navier-Stokes equations. More precisely, we impose the following initial conditions PI t=o = Po,
ult-o = uo
(8.129)
where uo E L2 (RN ), po E LOO (RN) and 0 < a < po < ,Q a.e. on RN for some
a < ,3 E (0, oo) and we assume in addition that po - p E H1(RN) for some positive constant p and that div uo = coo (-L). For the reasons explained above, we would like to obtain the existence of global solutions without size restrictions
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282
on uo or more precisely on its divergence-free part. Unfortunately, we are unable
to do so when N > 3 and we only succeed in doing so when N = 2 and po is close top in some precise sense detailed below. From now on, we thus assume that N = 2, and we begin by proving various a priori estimates which are central for the existence of global solutions. These a priori estimates, as will be clear from the arguments below, are obtained by using a variety of equivalent formulations of the system of equations (8.126)-(8.128), equivalent formulations that we first wish to describe. First of all, we may write (8.126), using (8.127), as i9p
+ (u'V)P + cop0
=0
P
or equivalently (at least formally)
log p + coo 1
(log p) +
=0.
(P)
(8.130)
Next, we may write in view of (8.127)
u=v+co01,
divv=0,
P
(8.131)
where v has to "vanish at infinity". Then, we write
Pu = pv + co pV P = pv -coo log p . Therefore, modifying the pressure field (* = r - co
log p), we find
(pv) + div (pu (9 u) - div (2p(p)d) + Vfr
= 0.
(8.132)
Also, if we go back to (8.130), we deduce 9+v
Op+coI0 2 P
2
(8.133) P
or
O
P
eP0e-P = 0
and finally, setting W(p) = -(p2 + 2p + 2)e-P + (p2 + 2p + 2)e-p,
at (AP(P)) +
coLe-P = 0 .
(8.134)
We first show some a priori estimates on p. First of all, we remark that (8.130)
yields by the maximum principle
On a low Mach number model
283
on R2 x (0,00).
0 < a < p(x, t) :5,8
(8.135)
Next, we multiply (8.133) by W(p) and we obtain by integrating by parts
d fR
z
Ico(P)I2 dx + co
f
Rs
P2e-2pIVPI2 dx = 0.
And, in view of (8.135), we deduce a bound on p - p in Lt (Hz) n LrcL2). We next wish to obtain a bound on p - p in Lt (Hi) n Lt ° (HH ), on 7t in L2 't and on v in Li (Hz)nLt°(L2). We begin by multiplying (8.132) by (-Ap) and we deduce using (8.135), where C denotes various positive constants independent of u and p, 2 dt
f
IopI2 dx + Q2
f
IopI2 dx _< C 2
f
R2
{IVpI2 + IvI IVPI} IkpI dx.
Hence, we deduce, recalling that fR2 ILpI2 dx = fR2 ID2pI2 dx, for some v > 0
f IVPI2dx+v
dt
0.
We then need to estimate carefully the last integral of the right-hand side of (8.137). In order to do so, we observe that, for all c > 0, there exists a Lipschitz (or C°°) function µE such that
max I µ(t) - µE(t) I < e.
tE[a,fll
Then, we write
R(P) dij o
(i))
8jvi dx
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284
=j 2
FO(P)
s
P
J f2
a1v= (9i vi
3
2
a'jp8 vi dx
dx +
P
µE (P) p2 8 vs dx, ai 2P
and we observe that we have
f
2
µE(P)
a
acv;, dx
P2P
fR2
a= (µ2 a9 p) ajvi dx P
L(P2
C
Hence, we obtain, observing that the first integral vanishes since div v = 0 1
fR2 p(p) d1, V
acv=dx
p
< C (1
+ tmf
Qf I
Ft'E(t)I)
I vpI2I Dvl dx + 2
f
I D2Pi IDvI dx.
We may then go back to (8.137), choosing a small enough, and we deduce for some v > 0
T fplvI2dx+v f max (1, fR2 No l2 dx). We then claim that if we fix R, a, ,0 and T E (0, oo) then we can deduce a priori bounds on v and Vp in LOO (0, T; L2 (R2)) and on Dv and D2p in L2(0, T; L2 (R2)) provided fR2 IDpo I2 dx is small enough (depending on R, a,,3 and T). These bounds obviously imply a priori bounds on v, Dp and thus u in L4(R2 x (0, T)), hence, in view of (8.132) and (8.135), on at in L2(R2 x (0, T)). Indeed, we may argue as follows: we first postulate that II VPII L4 (JR2 x (o,T)) o [St
- sli(t)] is equivalent to xP'
for x small and to x4 for x large. In particular, Lq(1) is reflexive for p > 1.
We shall show in Appendix E below that LP = L" + Lq (with equivalent norms) if 1 < q < p < oo. Finally, we conclude this list of elementary properties with the following fact.
Let F be a continuous functions on R such that F(0) = 0, F is differentiable at 0 and F(t)I tI -e -- a # 0 as Itl -- +oo. Then if q > A, F(f) E Lp/e(SZ)
if f E Lp(SZ).
(A.6)
Indeed, F(f) is equivalent to F'(0) f for f small, therefore for 6 small enough IF(f)I 1(IF(f)I vi 6,
2 It > I DuI + CN 2 > viN-2
iEI
(C.1)
6xi
iE1
where CN,2 is a "universal constant" > 0 (it is the best Sobolev constant, see to u if and only [348] for more details). Of course, un converges strongly in L if I is empty or in other words if the atomic part of the bounded nonnegative measure v vanishes.
The next result gives a sufficient condition ensuring that property. More precisely, we assume that we have N-1
IDunI2 = fn +
Dy fn when fn is bounded in L7° IaI=1
if jai = k and fn is weakly compact in L'.
}
C.1 If (C.2) holds, then un converges strongly to u in L41-4
(C.2)
293
A remark on the limiting case of Sobolev inequalities
Proof We deduce easily from (C.1) that we have N-1
f+E Do fa
fEL1,faEL'
where
(C.3)
IaI=1
if IaI=kE{1,...,N} If 134 0, we deduce from (C.1) and (C.3) that there exists µo > 0 such that N-1
Flooxo < f+ E Da f* where g E V.
(C.4)
IaI=1
Then, we multiply (C.4) by cp(E) where W E C000(RN) , Supp V C B(xo,1), p(xo) _
1, e > 0, and we find for some C > 0 independent of e N-1
0 0 , µ+6 > 0 N-1
µI Dun I2 + t; (div un)2 = fn +
in L WN-T if
I a) = k
Dz f a where f," is bounded IaI-1
(C.5)
and fn is weakly compact in L1.
Theorem C.2 If (C.5) holds, then un converges strongly to u in L IM, The other extension we wish to mention is the following : let (un),,,1 converge
weakly (in D') to u, we assume that un is bounded in Wm,P(RN) where m >
A remark on the limiting case of Sobolev inequalities
294
1 R > 0.
(D.7)
Next we claim that we have for all t > 0
3xEE, (z,t)EC.
(D.8)
296
Continua and limits
Indeed, we use (D.4) and (D.3) to obtain a subsequence xnk which converges to
some x E E. By construction, (x, t) E C. In particular, (D.8) implies that C is unbounded in E x R. Our last claim is that C is a continuum, that is a connected set in view of the properties shown above. The classical fact is easily proved by contradiction: assume thus that there exists a continuous mapping cp from C into {0,1 } such that cp-1{0}, cp-1{1} are non-empty. Without loss of generality (replacing cp by 1 - cp), we may assume that p(xo, 0) = 0. By classical extension theorems (Dugundji's theorem for instance, or setting on Cc, cp(x, t) = inf(y,s)EC{cp(y, s) + a(s) [d(x, y) + It - sl] } for some convenient continuous a _> 1), we may extend cp to a continuous mapping from E x R into R (or even [0,1]). Since cp-1{1} 9k 0, there exists (x1, t1) E C such that cp(x1, t1) = 1. Hence, there exist a subsequence x , tnk nk, (xnk, tnk) E Cnk such that Xnk t and cp(xnk, tnk) 1. In particular, for k large enough, cp(xnk, tnk) > 1/2.
k
k
k
We now use the fact that Cnk is connected (and the equality cp(xo, 0) = 0) to deduce the existence of ynk E E , snk E [0, tnk] such that cp(ynk, snk) = 1/2. We then use (D.3) to deduce the existence of a subsequence (yn', sn') such that s E [0, T] and by construction (y, s) E C. Since cp is yn' -& y , sn' continuous, we find cp(y, s) = 1/2. The contradiction proves our claim.
APPENDIX E ON SUMS OF L SPACES We wish to extend to sums of LP spaces the observation made in the course of proving Theorem 6.1 in section 6.3, namely for all 1 < p < oo (and for all
0ER+LP) IIV)
-I
0111,P
infII & > cER
>- 2
cIILP
P
IIipf 1IILP
(E.1)
First of all, we recall that for 1 < p, q < oo, LP + Lq and LP n Lq are Banach spaces endowed with the norms (respectively) inf { I 11P1 I I LP + 111'21 I /V = th + 02} and max (I I 'I ILP , 11011L J. Next, we recall that (LP + Lq )' = LP' n LQ' if 1 < p, q < oo (if p or q = +oo and p # q, it is still correct) where we denote as usual p' = P , q' = q-1 In fact, what we mean by this equality also contains the fact that the dual norm of LP + Lq is equivalent to the norm of LP' n L4' (say on RN and thus on any subdomain with the same constants, extending all functions to RN by 0). Indeed, the "equality" between the two vector spaces is obvious and the embedding of LP' n into (LP + Lq )' is one to one and continuous since we have clearly for all f E LP' n L4 and for all cP E LP + L9, Lq'
i.e. cP = (PI + (P2, (P1 E LP, Lq
f f((P1 +(P2) 1, meas{I f I > 1} < oo while if
p = 1, there exists Ro > 0 such that meas { If I > Ro} < oo. Indeed, if p > 1, we write (at least formally, and justify easily)
meas{I fI > 1} < f If 11of 1>1)
< Max
f
fV /II(PII LPf1Lq
On sums of LP spaces
298
(max[meas{IfI > 1}1/P
,
meas{IfI > 1}1/q])
while if p = 1, we obtain similarly meas{IfI > Ro}
Ro}, meas{IfI >_ Ro}1/qJ
In order to simplify notation, we take Ro = 1 when p = 1 (replace if necessary f by f /Ro). Then, we have IIf 1(1111)IILq- = Max if f 1(111>-1)w /II(PIILq < 1
=
Jf 1(111,1) (Po with (Po = f
1/(q-1)
1/(q-1)
1 (III>>1) II f 1(11151) I ILgI
and we deduce as above that f 1(111>1) E L° since IwoLP
-
f
IfI
P
1/P
/(q-1) 1(1112_1)
((q'' P)-1/(q-1)
IIf
lun>1)I
Lq
(the case p = 1 is handled as above).
-1/(q-1) IIf 1(IrI>1)IILgf
, and q' p
-q1
1
< 1 if p > 1
On sums of LP spaces
299
These "classical" facts being recalled, we may now turn to our main observation, namely the following inequality max
[fi
f =0 1f J
< Co inf II CER
,
II!IILP, < 1
1
,
- CI I Lp+Lq
(E.2)
(Lp+Lq
COI 11b - f'/ I
Mo max
[ffi / ff = 0
,
IIfIILP,
>
IIfIILq, < 1
,
where 1 < p < q < oo, all functions are defined on a set St in Rn with meas (St) < oo (just to fix ideas, in fact the inequality holds for all 1P E R + LP + Lq ) and the positive constants Co, Mo are independent of ip and of f2.
Let us first observe that the first inequality of this chain is a direct conse-
quence of the facts recalled above since f f = f f (ii - c) for all c E R if f f = 0. Next, the second inequality is obvious. Therefore, we only have to check the last inequality. Replacing & by 0 f, we may assume without loss of generality that f i = 0. Next, we use once more the facts recalled above and write for some positive constant C1 independent of 1i
-
IIIILP+Lq
< C1 max
[fi
/IIfIILp,
0. The proof given below adapts easily to such straight-
forward extensions. Finally, we wish to emphasize the fact that the results given below are not proven under minimal regularity conditions on the coefficients a=j, bs, c (or on the initial condition). These conditions should be considered as one simple set of assumptions which make possible the results presented below. First of all, we recall the classical LP estimates: if 1 < p < oo and Gi - 0 (V 1 < i < N) then we have (F.5) C IIFIIL=,t IIUIIW2,t,P W2,p(RN))
at E LP (l N x (0, T))) and C denotes various positive constants independent of u (and of the data). Next, we claim that we have for any p E (1, oo) where Wi't ,p = {cp E LP (0, T;
IIUIILP(O,T;W1,P(RN)) +
II utII
,
LP(o,T,W-1LP(RN))
< C{IIFI ILz t + IIGIIL=,t }.
(F.6)
This inequality may be proved by the following simple argument: let uo, uk (1 < k < N) be the solutions in of Ww;t'P
N
au«
au«
at -
a=j ax
= G« in RN x (0,T)
ax
u« I t-o
= 0 on RN
,
0 < cx < N
where Go = F. Then, U = uo + E u j satisfies j=1
au
N
at -
_
a2 u a=j
axjam j
N
-
au
bz
ate=
'9G=
= F+ i=1
+EE axi k=1%j=1 ij1
+cU
Sac,
N
a2uk
aXk axjaXj
IE £=1
b1
au any
+cU
in RN x (0, T).
Then, we introduce the solution v E Wz,t ,p of the following equation
67-Eij 8v N -1 _ a fj =
Ek=1 EN
82v
x
N - %=1 s
,j=1 Xk
bs
8
+cv
EN bs aU + c U in R N X (0,T) i=1
with vlt_0 = 0 on RN. Obviously, u = U - v and thus using (F.5) systematically we deduce N IIuIILP(O,T;W1,P(RN))
6t-1/2)ILogtl is bounded by C(1 + ILogSI) if t E [0, T]
b E (01 1).
,
Finally, we estimate the first integral as follows using (F.8): ft
ds
+v...
line -12
Lemma 2.3
lines -10, -8, -6, -5, -3
P6
line -12 line 4 line 8
line -2 line 3 (2.88) (2.90)
p.55 p.56 p.57
W-m,,q'
line 9
line 1
p.54
Lq,
line -4
Lq' (0, T; W-m'q'
(RN)) P(cPP(Pnun)) P(cop(Pnun)) IVu(x + A(y - x)) I L1(0, T; L2 (BM)) in the first norm L2 (BR)
=-f fBR
line -3, -2, -1
f ... cp dx
p.58 p.59 p.61 p.61
lines 3, 6
f ...cp dx
line -6
L1'2(Rn))
line 3
1 -AP = -2det(D2(p)
line -6
=
line 12
v goes to 0
line -7
Du = Vdivu - vlcurl u
line 14
(4.21)
line -20
... on c9Sl} in the case ...
-2(-0)-1...
(4.33)
e-C(uo,n)n
line -5
P((L2 n Cl'a)2)
line - 7 line 15 line 8
line 15 line 17
line -13 line -7 line -2 line 6 lines 4,5
line -13 last line line 9
lines -7, -6 p.171
ev EL 2 (01 T; H-1)
last line line 18 last line
line -13 p.149 p.150 p.151
339
line -5
radially decreasing fBl udxl Ja(0) xu-
- (-0)-
1
...
(-o)-lvvlc, = Vu C([0, oo); 17(R2))
_ -fB,w(x, t)dx --L -U2(0)
-
u1(0))
df0..