I i
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Mathematical Topics in
Fluid Mechanics
Volume 1
Incompressible Models
Pierre-Louis Lions
University Paris-Dauphine
and
Ecole Polytechnique
CLARENDON PRESS • OXFORD
1996

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© Pierre-Louis Lions, 1996
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PREFACE
Our goal in this book is to present various mathematical results on fluid
mechanics models such as, for instance, Navier-Stokes equations both in the
incompressible case and in the compressible case. Most of these results are
new even if some have been announced in various places. For each of these
recent results, we shall give complete proofs and we try to present as much
as possible self-contained proofs that do not assumefrom the reader really
technical prerequisites other than a basic training in (nonlinear) partial
differential equations. The book is divided into two volumes, the first one
being essentially devoted to incompressible models while the second one is
concerned with compressible models and asymptotic problems.
Before we briefly describe the topics covered here, we wish to mention
that this book does not pretend to be a complete survey of the existing
mathematical results even if we recall quite a few works on fluid mechanics
equations. We selected what we consider to be the most significant results
and in making such a biased selection we certainly omitted many relevant
contributions to the field. We tried .to compensate for these omissions by a
rather extensive bibliography (even if some of the references included there
are not quoted in the text). Let us also warn the reader that this book is
concerned only with newtonian fluids and that many important subjects,
such as the numerical approximation of the models we study, turbulence
models, qualitative properties of solutions (bifurcation theories, attractors,
inertial manifolds), reactive flows and combustion models, magnetohydro-
dynamics (MHD), multi-phase flows, and free boundary problems, are not
even touched on here. Also, as we shall see, many basic open questions
are left unanswered and we shall recall a large number of open problems.
More than two centuries after the introduction by L. Euler (and later by
Navier) of the fluid mechanics equations, much remains to be understood
mathematically even if considerable progress has been (slowly) made. We
only hope that these notes will be a small contribution to the formidable
task of the mathematical understanding of fluid mechanics models.
Let us now go a little bit into the contents of this book. We begin
in chapter 1 by recalling the fundamental equations modelling newtonian
fluids together with the basic approximated and simplified models—more
are to be found in the following chapters. Much more could be said on the
derivation of these models and we strongly advise the reader to consult such

viii
Preface
classical references as G.K. Batchelor [25], and L. Landau and E. Lifschitz
[281]. Many references may be found in the bibliography here.
The rest of the book is divided into three parts. The first one (which
concludes volume 1) is concerned with incompressible models and is
divided into three chapters (2, 3, 4). We begin in chapter 2 with a study
of the so-called density-dependent Navier-Stokes equations. We present in
section 2.1 the most general existence results for such problems, namely
global existence results of weak solutions in arbitrary dimensions with
possibly vanishing density (i.e. a vacuum is allowed in some regions). These
results are due to R-J. DiPerna and the author—they were announced in
[126], P.L. Lions [307]—and extend previous results due to various
authors like S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov [17], S.N.
Antontsev and A.V. Kazhikov [16], J. Simon [436],[437]. Complete proofs
are indicated in sections -2.3 and 2.4, while in section 2.2 we discuss
regularity questions, stationary problems and mention various open questions like
uniqueness, although we devote section 2.5 to partial "uniqueness results"
showing that global weak solutions are equal to a strong one if it exists.
The following chapter (3) deals with the classical Navier-Stokes
equations for homogeneous, incompressible fluids. In section 3.1 we deduce
from the analysis performed in chapter 2 the celebrated results of J. Leray
[283],[284],[285] concerning the global existence of weak solutions, and
we recall various classical facts on Navier-Stokes equations that can be
found in many existing books (see for instance P. Constantin and C. Foias
[102], and the bibliography). Sections 3.2 and 3.3 are devoted to some
recent "regularity" results in three dimensions consisting of variants or
extensions of results shown by C. Guillope, C. Foias, R. Temam [203],
L. Tartar [470], R Constantin [98], R. Coifman, P.L. Lions, Y. Meyer
and S. Semmes [95]. These results aim to make precise the regularity of
the global weak solutions. Finally, in section 3.4, we consider briefly the
so-called Rayleigh-Benard equations and we indicate existence results of
global weak solutions.
Chapter 4 is the last chapter of this part on incompressible models. The
first four sections are devoted to the classical Euler equations: we first
recall (4.1) the state of the art on Euler equations. Let us mention at this
stage that more details can be found in A. Madja [316] and J.Y. Chemin
[90]. We next (4.2) make a few remarks on the two-dimensional case,
comparing the multiple notions of weak solutions and showing the existence
and uniqueness of weak solutions for almost all initial conditions in L2. We
then briefly discuss in section 4.3 the fundamental open question of a priori
estimates in three dimensions, and we give some details about the simple
examples introduced in R.J. DiPerna and P.L. Lions [126] showing the lack
of "intermediate" a priori estimates. Finally, we introduce in section 4.4 the
notion of "ultra weak" solutions, which we call dissipative solutions, whose

Preface
ix
only merits are their global existence and the fact that they coincide with
classical solutions as long as they exist. Despite their weakness, they will
turn out to be an extremely useful tool for recovering Euler equations from
some compressible models in part IIL The last two sections are devoted to
other incompressible models, namely density-dependent Euler equations in
section 4.5, and the models obtained via the hydrostatic approximation in
section 4.6.
We then present in Appendices A-E various "technical" results used in
chapters 2-4.
The second volume will consist of parts II and III. The second part deals
with compressible models. The first three chapters deal with the so-called
(somewhat inappropriately) compressible isentropic Navier-Stokes
equations and detail the results (and their proofs) announced by P.L. Lions
[303],[304], [305]. The first chapter (5) deals with the compactness
properties of sequences of solutions: we first explain (in section 5.1) difficulties
encountered in such models which are due to the possible propagation of
oscillations in densities (related to acoustic modes) and we state the
available compactness results. Those results are shown in section 5.2, while the
slightly more difficult case of Dirichlet boundary conditions is treated in
section 5.3.
Chapter 6 deals with stationary (or time-discretized) problems
associated with the compressible isentropic Navier-Stokes equations: we state
our existence and regularity results in section 6.1 (in two and three space
dimensions), we prove these results in section 6.2 and we treat the
isothermal case in two dimensions in section 6.3. All these three sections are
concerned with time-discretized problems. The final section of this chapter
(6.4) is concerned with real stationary problems.
Next, chapter 7 is concerned with global existence results for the Cauchy
problem. Our main results are stated in section 7.1 and two "different"
proofs are respectively given in sections 7.2 and 7.3 which rely crucially on
the results and methods introduced in chapter 5. Finally, we consider in
section 7.4 the case of other (and more realistic from the applications point
of view) boundary conditions.
Chapter 8 collects additional results and information on compressible
models and related systems of equations. Section 8.1 is concerned with
the exact compressible model and we present, in particular, some rather
preliminary existence results. We come back in section 8.2 to isentropic
and isothermal models and we present some global existence results of
a different nature. Next, we investigate in sections 8.3 and 8.4 a
shallow water model and we present some results essentially taken from P.L.
Lions and P. Orenga [309]: section 8.3 is devoted to the global existence of
weak solutions while section 8.4 is concerned with the global existence of

X
Preface
smooth solutions. Then we discuss in section 8.5 the compressible isentropic
Euler equations (i.e. the inviscid case) in one space dimension, reviewing the
results due to RJ. DiPerna [124], G.Q. Chen [92], P.L. Lions, B. Perthame
and E. Tadmor [311], and P.L. Lions, B. Perthame and P.E. Souganidis
[310]. We also show in that section some new bounds (valid for all weak
solutions and not only entropy solutions). Finally, we discuss in section 8.6
a tentative low Mach number model, directly inspired by A. Majda [315].
The final part of the book consists of a single chapter devoted to
asymptotic limits. Sections 9.1, 9.2, and 9.3 are devoted to incompressible limits
(small Mach number, density nearly constant) from the solutions of
compressible, isentropic, Navier-Stokes equations built in chapter 7. In
particular, we consider in section 9.1 the problem of recovering the global weak
solutions (a la Leray) of the incompressible Navier-Stokes equations, and
in section 9.2 we strengthen the convergence results in two space
dimensions. Finally, in section 9.3, we obtain the solutions of the incompressible
Euler equations in the smooth regime (globally in time in two space
dimensions, and on the maximal time interval in three space dimensions).
This analysis relies upon the notion of dissipative solutions of Euler
equations introduced in section 4.4. Next, section 9.4 is devoted to the rigorous
derivation of the linearized system (around a constant flow) and thus of a
simple acoustics limit. Finally, section 9.5 is concerned with some
asymptotic problems (of homogenization type) for the compressible, isentropic
Navier-Stokes equations.
Paris
December 1995
P.L.L.

CONTENTS
1. Presentation of the models 1
1.1. Fundamental equations for newtonian fluids 1
1.2. Approximated and simplified models 9
Part I: Incompressible Models
2. Density-dependent Navier-Stokes equations 19
2.1. Existence results 19
2.2. Regularity results and open problems 31
2.3. A priori estimates and compactness results 35
2.4. Existence proofs 64
2.5. Uniqueness: weak = strong 75
3. Navier-Stokes equations 79
3.1. A brief review of known results 79
3.2. Refined regularity of weak solutions via Hardy spaces 92
3.3. Second derivative estimates 98
3.4. Temperature and the Rayleigh-Benard equations 110
4. Euler equations and other incompressible models 124
4.1. A brief review of known results 125
4.2. Remarks on Euler equations in two dimensions 136
4.3. Estimates in three dimensions? 150
4.4. Dissipative solutions 153
4.5. Density-dependent Euler equations 158
4.6. Hydrostatic approximations 160
Appendix A. Truncation of divergence-free vector fields in
Sobolev spaces 165
Appendix B. Two facts on P^2(IR2) 173
Appendix C. Compactness in time with values in weak
topologies 177

xii
Contents
Appendix D. Weak Ll estimates for solutions of the heat
equation 178
Appendix E. A short proof of the existence and uniqueness of
renormaiized solutions for parabolic equations 183
Bibliography of Volumes 1 and 2 196
Index
233

Contents
xiii
INTENDED CONTENTS OF VOLUME 2
Part II: Compressible Models
5. Compactness results for compressible isentropic Navier-Stokes
equations
5.1. Propagation of oscillations and compactness results
5.2. Proofs of compactness results
5.3. The case of Dirichlet boundary conditions
6. Stationary problems
6.1. Existence and regularity results for discrete time
problems
6.2. Proofs
6.3. The isothermal case in two dimensions
6.4. Stationary problems
7. Existence results
7.1. Global existence of weak solutions
7.2. Existence via regularization
7.3. Existence via time discretization
7.4. Other boundary conditions
8. Related questions
8.1. Remarks on the full compressible model
8.2. Remarks on the isentropic and isothermal models
8.3. Shallow water models: global existence of weak
solutions
8.4. Shallow water models: regularity of solutions
8.5. One dimensional compressible isentropic Euler
equations
8.6. On a low Mach number model
Part III: Asymptotic Limits
9. Asymptotic limits
9.1. The incompressible limit and the convergence to Leray's
solutions

xiv
Contents
9.2. Further results in two dimensions
9.3. The incompressible limit and the convergence to Euler
equations
9.4. Acoustics and the linearized system
9.5. A homogenization problem
Index

1
PRESENTATION OF THE MODELS
1.1 Fundamental equations for newtonian fluids
We recall here the standard derivation of the classical fluid dynamics
equations in eulerian form. For the evolution of a fluid (or a gas) in N spatial
dimensions"(TV >"i),* the description involves (N + 2) fields, namely the
mass density, the velocity field and the energy. We shall derive the
corresponding (N+ 2) evolution equations in the case of a fluid filling the whole
space, and we postpone the discussion of boundary conditions. Finally, let
us mention that more details on the considerations which follow can be
found in G.K. Batchelor [25], L. Landau and E. Lifschitz [281], and P.
Germain [170] (see also the bibliography of this book).
We begin with the evolution equation for the (mass) density p(= p(x, £)).
Applying the principle of conservation of mass, we simply observe that for
any volume O—say, a smooth bounded open set of IR^—the variation of
mass inside C?, i.e. fc ^f rfx, must be equal to the flux of mass on dO. Since
particles of fluids are moving along the integral curves of X = tz(X,t), this
flux is clearly equal to — fdopu • ndS, where we denote by n the unit
outward normal to dO. Therefore,
J JLfa = - [ pu-ndS, (1.1)
JO & JdO
and, if we apply Stokes' formula, we deduce from (1.1)
/o{|+divW}=0.
Since O is arbitrary, we obtain finally
^+div(pu) = 0. (1.2)

2
Presentation of the models
Another way (clearly equivalent) to derive (1.2) is to say that the transport
of mass on a time interval (t,t+h) and the conservation of mass yield
p(X(t+h),t+h)J(h) = p(x,t)
where X(t) = x, X(s) = u(X(s), s) and J(h) is the jacobian of the
transformation (x »-+ X(t+h,x)). Standard considerations on ordinary differential
equations give: J(h) = 1 + hdivu(x,t) + o(h); therefore, we obtain
p(x,t) + h{^ +u-Vp+ (divu)p} (x,t) + o(h) = p(x,t),
and we recover (1.2).
We now turn to the evolution of the velocity field u = (tzi(x,£),...,
Utf(x,t)) or equivalently of the momentum pu. If we follow the first
argument above and if we apply the principle of conservation of momentum, we
find
/ —(fm)dx = - / pu(u-n)dS+ [ pfdx+ [ F ndS. (1.3)
JO & JdO JO JdO
The two last terms of the right-hand side of (1.3) represent the forces
acting on the fluid, namely volume forces corresponding to external forces
(gravity, Coriolis, electromagnetic forces), and surface forces which, roughly
speaking, are due to the fact that we are dealing with a continuum and
which can be (somewhat improperly) thought of as contact forces with, for
instance, the fluid particles lying outside O. Clearly, F is a tensor and if we
set a = --F, the tensor a is called the stress tensor. Classically, a fluid in
motion is submitted to two kinds of stresses corresponding to compression
effects and viscous effects, and one writes .
j = -pl + r (1.4)
where p, a scalar function, is the pressure and r is the viscous stress tensor.
We denote by 1 the identity matrix (tensor), i.e. 1 = (6%j)ij.
Therefore, using (1.3) and (1.4) and the Stokes formula, we deduce finally
— (pu) + div(pu ® u) - div (r) + Vp = pf (1.5)
at
or (in a given orthonormal basis)
d
— (pui) + djipUiUj-Tij+pSij) = p/i, for 1 < i < N. (1.6)

Fundamental equations for newtonian fluids
3
Here and below, we write equivalently dj = ^- and systematically use the
convention of implicit summation over repeated indices.
Let us observe also that if we expand the derivatives of pUi and pUiUj
in (1.6) and we use (1-2), we can also write (at least, if all functions are
smooth)
P-£+P(v V)ti - div(r) + Vp = pf. (1.7)
Another way (again equivalent) to derive equation (1.7) is to use
Newton's equations, noticing that the acceleration of a fluid particle at (x, t) is
& {u(X(t),t)}|t_, = (fj + (u - V)ti) (*,*). We then obtain
/ p\-z: + (u'V)u\dx = I pfdx+ J F-ndS
JO ^ "t J JO JdO
and (1.7) follows easily.
We have now to describe the scalar unknown p and the unknown tensor
r. Postponing the discussion of p, we recall a few facts concerning r. First
of all, the conservation of angular momentum—which we shall not describe
here—leads to the following fact: r is a symmetric tensor. Next, a classical
fluid is a continuum where the constitutive law for r is of the following
form
r = r{Du,p,T) (1.8)
where the temperature T will be discussed later on. Then, if we postulate
that r is a linear function of Du, is invariant under a change of reference
frame (translation and rotation) and that the fluid is isotropic, we deduce
from elementary algebraic manipulations that necessarily
r = \divul + 2pd (1.9)
where d is the so-called deformation tensor
d = \{Du + tDu) (1.10)
and A,/x are the so-called Lame viscosity coefficients. Observe that in
general A and p. are functions of p and T in view of (1.8). The three
assumptions mentioned above that lead to (1.9) correspond to the so-called
newtonian fluids, the only case we shall consider in this book. In other
words, we shall always assume here (1.9) and very often we shall consider
A and p as fixed constants (i.e. independent of p and T). Furthermore,
the kinetic theory of gases (monatomic gases) indicates that the Stokes
relationship should hold, namely
A - * (!,!)

4
Presentation of the models
This is the three-dimensional situation. For an "iV-dimensional gas" we
would obtain A = —7^, while of course we still have A = — %ji for the
evolution of a "3-dimensional gas" depending only on one or two coordinates.
For most fluids and gases, experiments indicate that A+ ^ is very small
and this is why it is often set to 0 for common fluids, an assumption we
shall not need in the rest of this book. In practice, and this is also crucial
for the mathematical analysis of these models, we have
2
I* > 0, A + — /m > 0. • (1.12)
Since r = 0ifA = /i = 0, this case is called the inviscid (non-viscous) case,
while if fi > 0, A + /x > 0, we have the viscous case.
We finally derive the last equation which corresponds to the
conservation of energy, which expresses the first law of thermodynamics. Before
we discuss the equation in more detail, let us make a physical comment:
the derivation of the energy equation relies upon the assumption that, in
a fluid in motion, the fluctuations around thermodynamic equilibria axe
sufficiently weak so that the classical thermodynamics results hold at
every point and at all times. In particular, the thermodynamical state of the
fluid is determined by the same state variables (thermodynamic pressure
p, internal energy per unit mass e, thermodynamic temperature T, density
p) as in classical thermodynamics, and these variables are determined by
the same state equations.
Next, we observe that the total energy E is given by the sum of the
kinetic energy p\u\2/2 and of the internal energy pe. Then the conservation
of energy reads
iJAt£+-)<~-L<i£+')-<s\ (1,3)
+ / pf-udx + I u-(a-n)dS + Q.
Jo Jdo '
The two last integral terms of the right-hand side of (1.13) correspond to
the work done by the forces and, assuming there are no sources of heat
(we are dealing with non-reacting fluids), Q is simply the amount of heat
received (or lost) across the boundary dO characterized by a heat flux. In
other words
Q = - J q-ndS
Jdo
for some vector q to be determined. Using Stokes' formula, we deduce from
(1.13) and (1.4)
|W^!+e)}+div{uK^+e)+p]}j (ii4)
= div {r • u) — div (q) + pf • u. J

Fundamental equations for newtonian fluids
5
In order to close the system, it remains to describe p, e and q. If we choose
as independent (thermodynamic) variables p and T (the thermodynamic
temperature), then p and e are functions of p and T, i.e. obey some given
state equations of the following form
p = p{p,T), e = e(p,T). (1.15)
We shall come back to this point later on. There remains to describe the
heat conduction, or in other words to examine the dependence of q upon
T (and p). If we suppose that the fluid is isotropic, we are led to
q = -k(p,T,\VT\)VT (1.16)
for some scalar function A; which, in most cases, is taken to be simply a
function of p, T or even a constant called the thermal conduction coefficient.
We wish now to recall some basic facts from thermodynamics that will
be useful later on. First of all, there exists a state variable called entropy
(or entropy per unit mass), denoted by s, satisfying
ds = I{de + pd(i)}, (1.17)
and the basic assumption on thermodynamic equilibria yields the so-called
Gibbs equation
ds I (de d /1\ I /B| .
where ^ denotes the total time derivative, i.e. along fluid particle paths,
namely
Then the entropy of a given volume O is given by Ja ps dx. Applying the
second law of thermodynamics, we find
/ —{ps)dx > — / psu-ndS— I —-ndS
Jo 0* J do J do *
or
- (ps) + div (up*) > -div(|). (1.20)
On the other hand, we deduce from (1.18) and (1.2)
ds 1 / de ,. \ ,
"Tt =f("*+pd,vu)- (121)

6 Presentation of the models
In addition, multiplying (1.7) by u, we have
Pl(!i2")+p(u*V)Ji2'~div(TU) + U'Vp
= pf • u — r • Du = p/ • u — r • d
or, in view of (1.2),
|(p^)+div{u(p!^)}^div(ru)+u.Vp = pf-u-T-d. (1.22)
Comparing (1.14) and (1.22), we find
— (pe) + div (upe) + p div u = -div g + r • d, (1.23)
hence in view of (1.2)
p—+pdivu = — divg + r-d,
and inserting this relation into (1.21), we find, using (1.2) once more
f) 11
— (ps) + div (ups) = -ydivg + -r-d. (1.24)
In particular, we find, comparing (1.20) and (1.24)
r-d-^g-Vr > 0, (1.25)
an inequality which must thus be satisfied for all (p,u,T). In particular,
choosing u = 0, we deduce
-q • VT > 0,
an inequality which, combined with (1.16), yields k > 0, a fact which is
consistent with experiments. Similarly, if we choose T to be constant, we
deduce
r-d > 0.
For a newtonian fluid, we deduce from (1.9) for all u
2/i|d|2 + A(divu)2 > 0
and this is easily seen to be equivalent to (1.12), namely the "natural"
constraint on \x and A.

Fundamental equations for newtonian fluids
7
At this stage there only remains to discuss the "thermodynamical
constraints" on the state equations (1.15); therefore, we choose p and T as
independent variables for the thermodynamic relations. Then we deduce
from (1.17)
dr t dT' dp T\dp p2r v ;
and, in particular, s can be deduced from the expressions (1.15). But (1.26)
also implies the following compatibility equation: ^ (^ ^f?) = ^ (^ {|| —
£}),Le.
7{'-'#}-& «*>
This relationship "constrains possible laws for p and e.
Additional constraints can be derived from the second law of
thermodynamics. Choosing now (e,r) or (s,r), where r = i, as independent
variables, one deduces from the second law the following equivalent
statements
s(e,r) is concave in (e,r) (1.28)
or
e(s,r) is convex in (s,r). (1.29)
These statements are obviously equivalent since (1.28) is equivalent to the
convexity of the set {(s, e, r) / s < s(e, r)} while (1.29) is equivalent to the
convexity of the set {(s, e, r) / e(s, r) < e}, and those two sets are identical
since (147) yields ff =' ± > 0.
Let us mention two consequences of (1.28)-(1.29). First of all, p or r
being fixed, ff = T and thus 0 = f£ Hence, (1.29) yields
e(p,T) is increasing in T. (1.30)
Similarly, 5 being fixed, f^ = -p, |^ = £ and thus (1.28), (1.29) imply
p(p,T) is increasing in p for 5 fixed (s = s(p1T))
p(p,T)T~l is increasing in p for e fixed (e = e(p,T))
:.}
In conclusion, the restrictions on the state equations (1.15) are given by
(1.27) and (1.29), while (1.26) indicates how to derive s once (1.15) is given.
We wish now to conclude by giving two examples and we begin with the
most common model, namely the case of an ideal gas. An ideal gas is a fluid
which obeys the following laws: Mariotte's law, namely p = p/(T), and
Joule's law, namely e = e(T) for some scalar functions / and e. Observe
then that (1.27) implies that f(T) is linear and we deduce
p = RpT, e = e(T) (1.32)

8
Presentation of the models
where, because of (1.31), R > 0, e is increasing in T. The constant R
is called the ideal gas constant. Writing Cv = e'(T) (> 0), Cp = J£ =
R + e'(T) where /i = e+^ is the so-called enthalpy, we see that Cp > Cv
and we set 7 = ^.
Common fluids and gases in "normal" conditions (i.e. not with high
densities or at high temperatures) are often well described by the ideal
gas model, which assumes furthermore that Cp and Cv are constants, i.e.
e = CVT with Cv > 0 and 7 = 1+^ > 1. We then deduce from (1.26) that,
in this case, s = R log (^-—7—) = Cv log t^t. It is possible to deduce
from the kinetic theory of gases that 7 = ^ffi (for a monatomic gas) or
7 = I if N = 3. The most interesting region for physical applications is
As a mathematical exercise, it is worth looking at the case when we use
only Joule's law, namely e = e(T). Then, because of (1.27), we deduce
e = e(T), p = Pl(p)T
(1.33)
and because of (1.30) and (1.31), e(T) and Pi(p) have to be increasing (or
at least non-decreasing). In fact, these monotonicity conditions on e and p
are, in this case, equivalent to (1.28) since, up to some irrelevant constant,
s = h(e) = P(r) where fc'(e) = ^ (c = e(T)), ^(r) = pi(J).
Let us conclude by summarizing the model we derived for newtonian
fluids
dp
dt
+ div(ptz) = 0
d(puj)
dt
d(pe)
dt
+ div (puui) — dj{fx{djUi + diUj)}
- di(Xdiv u) + dip = p/i, for 1 < i < N,
+ div (pue) + pdiv u - dj(kdjT)
= ^{diUj+djUif + XidWu)2
>
(C)
where A, ji, p, e are functions of p and T; k is a non-negative function of p, T
and possibly |VT|; A,/x satisfy (1.12); and (1.27), (1.29) hold. The most
common case being the one when A,/z,A; are constants, we have p = J?pT
(R > 0), e = CVT (Cv > 0) (the ideal gas) and we set 7 = 1 + ^. Finally,
5, defined (up to a constant) by (1.26), satisfies
d(ps)
dt
+ div (ups)
= +kdW (*VT) + I(fi^ + aiUi)2 + A(divU)2). J
(1.34)

Approximated and simplified models
9
When /x > 0, A + ^ > 0, the system of equations (C) is called the
"compressible Navier-Stokes equations11.
1.2 Approximated and simplified models
In this section we wish to discuss successively incompressible models, low
Mach number expansions, ideal Quids and some simplified mathematical
models. Many more approximations or simplifications are possible
(shallow waters, quasi-geostrophic approximations, etc.), some of which will be
studied in the rest of this book, in which case we will say a few words on
their derivation.
We thus begin with incompressible models or, more accurately, models
for incompressible fluids, which are very important for applications since
many common fluids (liquids) are incompressible or only very slightly
compressible. Mathematically, the incompressibility means
divu = 0. (1.35)
Indeed, if at fixed time t > 0, we consider the volume of an open set O
filled with the fluid, then, at time t + h (h > 0), the corresponding fluid
particles will fill the set
Oh = {x(t+h,y) / y e O}
where i(r) = u(x(t),t), x(t,y) = y. The volume of Oh is given by
J{Ky) dy
where J(h,y) is the jacobian of (y »-» x(t+h,y)). Finally, it is well known
that we have
J(t) = divu(x(r), r) J(r).
Therefore, the volume of Oh is equal to the volume of O (for all O, h > 0)
if and only if (1.35) holds.
Next, we may follow the same derivation of the equations as in the
preceding section and we recover the conservation of mass (1.2) and the integral
form (1.3) of the conservation of momentum. Notice that since div u = 0,
if p is constant initially it remains so, in which case we say the fluid is
homogeneous (p = pg (0,oo)). Also, the stress tensor a has to preserve
the incompressibility and we deduce from the hypothesis of a newtonian
flow
a = 2\xd-p\ (1.36)
l

10
Presentation of the models
where \i > 0. The hydrostatic pressure, p, is in fact a Lagrange multiplier
associated to the incompressibility constraint (notice that CijdiUj = 0 for
all u such that divtz = 0 if and only if a^ = p6ij for some p). Notice
finally that divd = \dj(diuj + djui) = \&u% s*nce divu = 0, and we
obtain finally
ft
(pui) + div (puui) - djifxidiUj + djUi)) + d^ = p/if
fori <i<N, divu = 0,
(1.37)
or, if/z is constant,
— (ptti) + div (puui) - fiAui + d^ = pft,
fori <i<N, divu = 0.
(1.38)
In conclusion, we look for the (p, tz,p) solution of the following system of
equations
|£+div(pu) = 0
d(pUj)
at
+ div (puui) - dj(n(diUj + djUi)) + dip = pfi
fori <i<N, divu = 0.
(1.39)
If/i > 0 (and is constant), i.e. if viscous effects are present, this system of
equations is the so-called non-iomogeneous incompressible Navier-Stokes
equations (quite often, incompressible is omitted), and a particularly
relevant case consists in choosing p = p > 0, in which case (1.39) reduces
to
~P -ST + P^iv (UUi) ~ y-^ui + d%P = P/i»
dt
(1.40)
for 1 <t < N, divu = 0.
This system is called the homogeneous, incompressible Navier-Stokes
equations, often simply called the Navier-Stokes equations(!). Replacing /z by
v ~ p> V by ^, we see it is enough to study the case when ~p = 1.
It remains to write down an equation for the internal energy e (= e(p, T)):
following the procedure described in the preceding section and taking into
account the incompressibility condition (1.35), we find
^- + div (upe) - div (kVT) = | (diUj + djUi)2
(1.41)
where k > 0 may depend on p, T. The most common case corresponds to
k > 0, k constant (i.e. independent of T) and, if we are in the homogeneous

Approximated and simplified models
11
situation, e = e(p, T) is only a function of T, for instance e = CVT, for
some Cv > 0.
We next discuss low Mach number expansions. We just want to present
here some formal asymptotics, some of which allow us to go from the
compressible models introduced in the preceding section to the incompressible
models we stated above. First of all, let us recall the precise definition
of the Mach number: M = }u[ . Therefore, letting M go to 0 means
that, keeping p,T of a typical size 1, we consider u of order e where e > 0
is a small parameter (going to zero). We also rescale the time variable,
considering finally
Pe_(*i t) = p(x, -J, ue(x, t) = -u\x2-), 1
(1.42)
where (p, u, T) solve (C) and A, fx, k depend upon the scaling parameter e
in a way to be determined.
We begin with the "traditional" derivation of the homogeneous,
incompressible Navier-Stokes equations. We then take n(e) = en, \(e) = e\,
k(e) = ek, f(e) = e2f, where \,n,k may be functions of (p,T), and we
deduce from (C)
-£ + div(ucpe) = 0
d{peue)
dt
d{peee)
dt
+ div (pcue <8> tzc) - div (2fidc + X div ue)
+ v(J?p«) = */
+ div (peuce£) - div (A:VTC) + pe div ue
= e2{%(diucj + dju^i)2 + A(divUc)2}. J
(1.43)
The equation corresponding to the conservation of momentum indicates
that pc should behave like P(t) + e2p + o(e2) where P(t) is independent
of x. One way to ensure this fact consists in assuming that pe and Te are
initially constant (up to terms of order e) in order to avoid initial layers,
and that pc is initially constant up to terms of order e2 (this is clearly the
case if pe and Tc are also constant up to terms of order a2). Then, at least
formally, if we set
pc = p + epx + 0{e2), Te=T + eTi + 0{e2), uc=u + 0{e)

12
Presentation of the models
we deduce from (1.43)
p(p,T) = P(t)>0 is independent of x (1.44)
*(?.T)Pi + H(P>T)Ti = Pi(T) is independent of x (1.45)
|? + div (up) = 0 (1-46)
a^ + div (pu ® t») - div (2/*d + A div u) + Vtt = p/ (1.47)
dt
d(pe)
+ div (pue) - div(fcVT) + P(t) div u = 0 (1.48)
where e = e(p,T), p = p{p,T), k = k(p,T), tt = lim£^0+ j7 (pc-P-ePi).
From now on, we assume, to simplify the calculations, the ideal gas laws:
p = RpT, e = CVT. Then we deduce from (1.48) that pe = (Cv/R)P(t) =
—^ P(t) is independent of t; this, in fact, requires some boundary
conditions like, for instance, if we consider the evolution in the whole space IRn, u
and VT vanishing at infinity. Hence, P(t) — Pq > 0. We then deduce from
(1.48) and (1.46)
- (log T) + u • V(log T) = -(— + u-VJ(logp)
= divtx = ^—div(A;VT).
7-H)
Hence, if T is constant initially, T is equal to that constant for all (x, J).
Therefore, p and T are positive constants and divtx = 0. In conclusion,
(1.46)-(1.48) reduce to
p — 4- ~pu • Viz — JlAu + V7T = p/, div u = 0
where /Z = ^u(p, T), p > 0, T = ^, and we recover the homogeneous,
incompressible Navier-Stokes equations.
We next present some variants of the above formal derivation. First
of all, we consider the case when p.(e) = £/z, \{e) = sA, k(e) = ek€ and
ke —> k > 0 as e —► 0+, f(e) = s2/. We still assume the ideal gas laws
(p = RpT, e = CVT) and we now assume that p is initially constant up to
terms of order e2 but we do not assume that p or T are initially constant
(or equivalent to constants). With the same notation as above, we still
recover (1.46)-(1.48) with k replaced by k. Exactly as before, we deduce
that P = Pq > 0 and
-yPodivu = div(ifcVr).
(1.49)

Approximated and simplified models 13
In particular, if fc = 0, i.e. k(e) = o(e), divtz = 0 and (1.46)-(1.48) become
dp
dt
d(pu)
_ +div (up) = 0, divtx = 0,
at
dt
+ div (ptz <g> u) - div (2/Zd) 4- Vtf = p/,
i.e. the non-homogeneous, incompressible Navier-Stokes equations. Notice
that JZ = /z(p, Qj and that, if /z is independent of p and T, div(27Zrf) =
/zAtx. _
Next, if k > 0, we obtain the following new system of equations
-^+div(tzp) = 0, 7divu = div (fcvf-H
-^ + div(ptz ® tz) - div (2/Zd + A div u) + Vtf = p/.
We would like to point out that we are not aware of any formal
derivation of incompressible models including a temperature equation (or energy
equation) like (1.41). One possible physical explanation for this apparent
lack of consistency between compressible and incompressible models is the
following assertion: compressible models are valid for gases and the
incompressible limit yields particular incompressible models, while the general
incompressible models aim to describe liquids where compression effects
are neglected.
We next briefly make a few remarks on perfect Quids which correspond
to the particular case when A = \i = k = 0. The compressible models (also
called compressible Euler equations in this case) take the following form
^ + div(ptz) = 0
^+div (ptz®tz) + Vp(p,T) = pf
I("(!T + e))+divW',Jl!++'>e+p}} " »fu\
(1.51)
and in the ideal gas case p(p, T) = RpT, e = CVT, 7 = !+<£-. Furthermore,
we might expect that the entropy equation (1.34) simply becomes
dps j. / x 9s _
-r— 4- div (ups) = 0 or p — 4- pu • Vs = 0.
ot ot
However, this is the case only on regions where p,u,T are smooth. In fact,
it is well known that shocks (discontinuities) appear in finite time and that

14
Presentation of the models
the entropy equation does not hold "on shocks". Instead, only the following
inequality, which is deduced from (1.34), remains true:
-^+div(ptzs) > 0. (1.52)
at
As long as shocks do not form, the entropy equation is valid and if
initially s is constant, it remains constant: s = so- In the ideal gas case,
this leads to p = (ite5o/Co)p7 = cop7 (cq > 0) and we obtain the so-called
isentropic gas dynamics system
^+div (pu) = 0
-^ + div (pu <g> u) + coVp7 = 0.
at
(1.53)
This system has clearly a somewhat limited physical validity in view of
what we have just recalled but has recently received a lot of (mathematical)
attention. For a general pressure law, s = so means p — p(p) (= p(p,T)
with T determined by s = sq) where p is increasing in p (see (1.31)).
For a perfect fluid the incompressible models take the following form: in
the homogeneous case, we obtain the classical Euler equations
du
— +div(u®u) + Vp = 0, divu = 0 (1.54)
at
(the constant density p = p is scaled out by considering p/p instead of p).
In the inhomogeneous case, we obtain the density-dependent Euler
equations
^ + div(pu) = 0, divu = o)
d( ) ( (]L55)
-^p + div (pu ® u) + Vp = 0. J
We now turn to some models obtained by further simplifications or
different (singular) asymptotic limits. We first consider the compressible system
(C) with ^, A, k constants (/z, A > 0, k > 0) and we neglect the heating due
to viscous dissipation, an approximation which is reasonable except for
hypersonic gases. We then obtain for the temperature (or energy) equation
-^ + div (pue) + div up - kAT = 0. (1.56)
We also assume Joule's law: e = e(T), p = p(p)T (for instance the ideal
gas law e = CVT, p = RpT) where p is increasing. Next, we take k = 0 (we

Approximated and simplified models 15
do not claim this is a relevant physical assumption!) obtaining an equation
which is easily seen to be equivalent to the following entropy equation
ds
p-zr+pu-Vs = 0. (1.57)
at
In particular, if s is constant initially, we now expect to deduce s = So
(we do not expect shocks since A,/i > 0), and we obtain the compressible
isentropic Navier-Stokes equations
?R+div(pu) = 0
^f/ + div (pu <g> u) - fxAu - (A+/x)V divtz + Vp(p) = pf
at
) (1-58)
where p(p) = p(pT) (and s(p,T) = So) is increasing in p because of (1.31).
In particular, in the ideal gas case, p(p) = cop7 with cq = Res°tCv.
Let us make at this stage a general remark, which is related to isentropic
or barotropic pressure laws. Precise state equations like p = p(p, T) are not
an easy matter nor are they fully understood. In particular, many semi-
empirical laws exist (depending on the physical phenomena to be studied).
For instance, one can find the following pressure law for water in R. Courant
and K.O. Friedrichs [108] (p. 8): p = A(p/poY1 - B with the proposed
values 7 = 7, A = 3001, B = 3000 (atm.) and p0 is the density at 0°C.
We finally describe some (mathematical) formal asymptotics. First of all,
we take p = p(p)T, e = CVT and we let Cv go to +oo (!). The temperature
equation then yields
p— + pu-VT = 0
or
fYT1
p — + pu ■ VT - kAT = 0
at
if £r—► k > 0. A particular solution is clearly T = To > 0. We then obtain
the compressible, isothermal Navier-Stokes equations
Jl+div(pu) = 0
—^+div(pu®u) - fiAu-(\+p.)Vdivu + T0Vp(p) = 0
dt
) (1-59)
and p(p) = Rp in the ideal gas case (p(p) is increasing, as usual).
The other limit consists in letting k go to +oo. We deduce T = T(t)
(independent of x), and in the ideal gas case, the function T is determined
by the conservation of energy
*{/»¥*»}-/
pf -U

16
Presentation of the models
where M = f p (independent of £), or equivalently by the entropy
RJtjplog^~y~~ = Jf{2ldiUj+djUil2+X{dWu)2}'
We now conclude this section (and the chapter) with a brief discussion
of initial and boundary conditions. Our main goal in this book is to study
the Cauchy problems for the models derived and described above. In other
words, we study the systems of equations for t > 0 prescribing the values
of p, pu1 pe at £ = 0. Of course, for the models introduced in this section in
which one does not write an energy equation or one takes p = ~p constant,
we only prescribe p and pu, or u at t = 0.
The question of boundary conditions is much more delicate and would
require a detailed discussion. Our ambition in this book is somewhat
limited since we shall consider problems set in a domain f2, with standard
(mathematical) boundary conditions on dQ for (possibly) p,u,T. We shall
essentially always restrict ourselves to three cases:
(i) Q is a smooth, bounded, connected, open set in IR^, and we impose
Dirichlet-type boundary conditions on dQ on p and u while we impose on
T Dirichlet, Neumann or mixed boundary conditions. A typical example
would be the case of a homogeneous Dirichlet condition, i.e. u • n = 0
on dCt or u = 0 on dQ if viscous terms are included in the model, in
which case no conditions on p on dQ are imposed. If we incorporate a
temperature equation, a rare event unfortunately in this book, we can
impose for instance homogeneous Neumann conditions: |^ = Oon dQ.
Here and above, n denotes the unit outward normal to dQ.
(ii) Q = llili(0>£i) wi*h ^i > 0 (V 1 < i < iV) and all functions are
periodic in each x* of period Li, for all 1 < i < N.
(iii) Q = IR^ and (p, tx, T) "vanish at infinity" or are "constant at
infinity".

PART I
INCOMPRESSIBLE MODELS

2
DENSITY-DEPENDENT
NAVIER-STOKES EQUATIONS
This chapter is devoted to the so-called density-dependent Navier-Stokes
equations or inhomogeneous, incompressible Navier-Stokes equations,
namely
^ + div(pu) = 0, p > 0, infix (0, +oo) (2.1)
at
-^+div(ptx®tz)-div(2^d) + Vp = p/,
divtz = 0, infl x (0,+oo)
i
where d = \ (diUj + djUi), f is given on Q x (0, +oo) and fi = fi(p) is a
continuous, positive function on (0,+oo). Of course, (2.1), (2.2) are to be
complemented with boundary and initial conditions.
The above model was derived in chapter 1 but we wish to add to that
derivation the fact that it can be also seen, due to the possible dependence
of fi on p, as a model for the evolution of a multi-phase flow consisting of
several immiscible, incompressible fluids with constant densities and
various viscosity coefficients.
2.1 Existence results
Let us first describe the boundary and initial conditions for (2.1)-(2.2)
which we shall consider here. We study only three model cases (recall that
N > 2):
(i) (Dirichlet case) fi is a smooth, bounded, connected open subset of
R"and
u = 0 on dQ. (2.3)

20 Density-dependent Navier-Stokes equations
(ii) (Periodic case) ft = U^Li(^Li) (£* > 0, V 1 < t < N) and (p,u) are
periodic so that we can consider that (2.1)-(2.2) hold on JRN x (0, +oo)
with (p, u) periodic in each X* of period L*, for all i € {1,..., TV}.
(iii) (IR^ case) ft = IRN and we want p to be bounded (for example)
while u satisfies, in a sense to be made precise,
u -> Uoo as |x| -» +oo, for all * > 0 (2.4)
where u^ is fixed in JRN.
We now discuss initial conditions. In view of (2.1)-(2.2), we need to
impose conditions on p and pu at t = 0. Observe that we cannot directly
impose initial conditions on u in case p vanishes on some pant of ft, i.e. if
there is some vacuum. We then consider,.if ft is bounded,
po > 0 a.e. in ft, p0 € £°°(ft), ]
m0 € L2(Q)N, m0 = 0 a.e. on {p0 = 0}, > (2.5)
|mo|2/Po € Lx(ft) J
where we agree that |mo|2/po = 0 a.e. on {po = 0}. We also want to impose
p|t=o = Po on ft, pu|t=o = mo on ft. (2.6)
Of course, in the periodic case, we may extend po and mo periodically on
WLN.
If ft = IR^ (case (iii) above), (2.5) is replaced (if u<x> $ 0) by
po > 0 a.e. in IRN, p0 € L°°(RN),
mo - pqUoo € L2(1RN), mo = 0 a.e. on {po = 0},
|mo-potXoo|2/Po 6 Ll(JRN).
For technical reasons, we need to assume, in the case when ft = JRN, in
addition to (2.7), one of the following three conditions
(1/Po) l(po<s0) e LlQ&N), for some S0 > 0, (2.8)
(2.7)
or
or
JV
(p-po)+€L^JRN), for some £ € (0, oo), p € (—, oo), (2.9)
ifJV = 2, / pg<x>2(p-1)(log<x>)rdx < oo]
for some p € (1, +oo], with r > 2p — 1
ifiV>3, po € L%'°°(IRN)
(2.10)
where we define <x>= (1 + |x|2)1/2.

Existence results
21
Finally, we assume that the force / satisfies
/ 6 L2(Q x (0,T))N, for all T € (0,oo). (2.11)
Since we shall always make these assumptions in the rest of the chapter—
unless explicitly mentioned—we shall not recall them.
Our main existence result will state the global existence of weak
solutions that could be called "solutions a la Leray" by analogy with the
classical global existence results for the homogeneous, incompressible Navier-
Stokes equations obtained by J. Leray [283],[284],[285], and we wish now
to define precisely what we mean by weak solutions. We look for
solutions satisfying, for all T e (0,oo), R € (0,oo): p € L°°(Q x (0,oo));
u 6 L2(0,T;Hi(Q))N (Dirichlet case (i)), u € L2(0,T;Hfa)N
(periodic case (ii)), u 6 L2^,T\Ex\BR))^zxA u e L2(0,T;L^(IRN)" if
N > 3 (whole space case (iii)) ; p\u\2 e L^^TjL1^)) if Q is bounded,
p|u- u°°\2 6 ^^(CoojL1^)) if ft = JRN; Vu e L2(Q x (0,T));
p 6 C([0,oo);/?(«)) if fl is bounded, p e C([0, oo); L*>(BR)) i£ Cl = JRN
for all 1 < p < oo. Finally, if (2.8) or (2.9) hold, we require that u €
L2(JRN x (0,T)) for all T € (0,oo).
Here and everywhere below, BR = {y € TRN / \y\ < R}, Hl(0) =
{/ € L2(0) I dj 6 L2(0)}, flj(n) is the subspace of Hl(Q) consisting
of functions whose trace on dQ vanishes, H*ev = {/ € H1{Br) for all R <
oo, / is periodic in Xi of period Li for all 1 < i < N}. Furthermore, (2.1)
holds in the sense of distributions (for example) in Q x (0, oo) (case (i)) or in
fftN x (0, oo) (cases (ii) and (iii)) and the initial condition on p (contained in
(2.6)) is meaningful since we require p to be continuous in time (with values
in LP or £foc). Of course, we require that divu vanishes as a distribution
(on Q x (0,co) or JRN x (0, oo) in the periodic case or if Q = 1RN). In the
periodic case (ii), p is periodic in Xi of period L{ for all 1 < i < Ny for
all t € (0,oc). It only remains to explain the meaning of (2.2) and of the
initial condition on pu (contained in (2.6)).
We shall use a weak formulation based upon a class of smooth test
functions, namely the class $ of <f> € C°°(TRN x [0, oo))N, <j> has compact support
in Q x [0, oo) in the Dirichlet case or if Q = 1RN, <p is periodic in Xi of period
Li for all 1 < i < N in the periodic case, div0 = 0 on RN x (0,oo). We
want u to satisfy, for all <j> € $,
-m0- 0(x, 0) dx + // \-pu • — - pUiUj drfj
Jn JJqx(o,oo) *> en > (212)
+ -vidiUj + djUi){di<t>j + dj<t>i) -pf • 4>\ dxdt = 0. J
Let us also emphasize the fact that, in view of classical density results—
see, for instance, R. Temam [472], R. Dautray and J.L. Lions [115]—we

22 Density-dependent Navier-Stokes equations
could equivalently take test functions 0, say in the Dirichlet case, such
that div<£ = 0, <j> e L2(01T;Hi(Q))N and ff € L2(Q x (0,T))N for all
T € (0, oo), <f> vanishes on Q for t large, V<£ € L2(0, T; L*(fl)) (for instance)
where p>2ifiV = 2,p = iVifiV>3. This last integrability requirement
follows from the fact that, by Sobolev embeddings, we have, for all T €
(0, oo), if N > 3 (for example)
p\u\2 e L-CO.TjLHn)), p\u\2 € L^TjL^n))
hence /m^ € Z,2(0,T;Z^(fl)) (V 1 < ij < N).
At least formally, we expect solutions of (2.1)-(2.2) to satisfy the energy
identities:
— / p\u\2 dx+ [ fM(diUj + djUi)2 dx = / 2pf • u
at Jq Jq Jq
if Q is bounded, or
— / p\u-Uoo\2dx+ / nidiUj + djUi)2dx = / 2p/-(u-uQO)
if Q = R^. These identities are obtained upon multiplying (2.2) by u (or
by u — Uqo), observing that, because of (2.1), (Jj (pu) + div (pu® u)) • u =
((^+/mV)u).u = (p^+pu.V)l^ = {^(pH!) + div(^l^)}
and integrating by parts over CI.
As it is often the case when dealing with global weak solutions of
nonlinear partial differential equations, the global weak solutions we obtain
satisfy the following energy inequalities
— f p\u\2dx+ [ n(diUi + djUi)2 dx < 2 f pf -u inl^O.oo), (2.13)
dt Jci Jn Jn
I p\u\2dx + I j p.(diUj + djUi)2dxds
Jn JoJn
< [ -^Ldx + 2 J j pf -udxds a.e. t € (0,oo),
Jn Po Jo Jn
if fi is bounded, and if Q = TRN
— / p\u—Uoo\2dx+ / fi(diUj + djUi)2 dx
dt JfrN Jj^N
< 2 / pf • (u-Uoo) in 2>'(0, oo),
JnN
(2.14)
(2.15)

Existence results
23
/ p\u-Uoo\2dx + I I n{diUj + djUi)2dxds
J jr." Jo JnN
<[ l^-^^l2 dx + 2 f( pf-iu-u^dxds
JjR* PQ JO JQ
(2.16)
a.e. t € (0,oo).
We may now state our main existence result.
Theorem 2.1. There exists a global weak solution (p, u) of (2.1)-(2.2)
satisfying the boundary and initial conditions described above and the energy
inequalities stated above. Furthermore, we have for allQ <a < 0 <oo
meas{x e WLN / a < p(x,t) <J3} (e [0, +oo]) .
is independent of t > 0.
i
In particular, ifft = 1RN andpo-p00 e IfiJR*) forsomel <p < oo, p°° e
[0,oo), p-p°° e C((0, ooJjL^R*)) and ||p-P°°||lp(r^) is independent of
t>0. Also, ifpo = p on Q for some p € [0, oo), then p = p on Ct x [0, oo).
Remarks 2.1. 1) Global existence results for non-homogeneous,
incompressible Navier-Stokes equations were first obtained by A.V. Kazhikov
[254]—see also A.V. Kazhikov and S.H. Smagulov [260], S.N. Antontsev
and A.V. Kazhikov [16], S.N. Antontsev, A.V. Kazhikov and V.N. Mon-
akhov [17]—in the case when /x is independent of p, po is bounded away
from 0. These results were extended by various authors and in particular by
J. Simon [435],[436],[437] allowing po to vanish but still with a constant /x.
The methods of proofs do not adapt to the situation treated here, namely /i
depending on p. Furthermore, the continuity in t of p (with values in Lp) or
properties like (2.17) were not known with the above generality. Particular
cases of results similar to Theorem 2.1 were obtained by R.J. DiPerna and
the author relying on general results on "transport equations" shown in
[128] and were announced by R.J. DiPerna and P.L. Lions in [126]. This
is the approach we detail (and extend) here.
2) Many questions are left open; we shall come back to these in the
next section. Let us only mention a rather technical matter: the weak
formulation (2.12) yields the existence of a pressure field p (defined up to a
"distribution in t only") which is essentially a distribution (one can analyse
its singularity in terms of negative Sobolev spaces, or sums of negative
Sobolev spaces)—see J. Simon [437] for more details; however, we do not
know if p (appropriately normalized) e L\oc(ft x (0, oo)).
3) The final statement of Theorem 2.1 is a trivial consequence of (2.17).
Obviously, if p = p> 0, txisa weak solution of the homogeneous,
incompressible Navier-Stokes equations (1.40) (the classical Navier-Stokes
equations!). We shall come back to this particular case in chapter 4.

24 Density-dependent Navier-Stokes equations
4) Let us also explain how (2.17) yields the statements on p—p°°\ indeed,
(2.14) implies that the distribution function of p — p°° is independent of
t > 0, therefore \\p—P°°\\lp(rn) is independent of £. Since we already know,
by definition of weak solutions, that p € C([0,oo);I/^)C(IRiV)), we deduce
the fact stated in Theorem 2.1, namely p e C([0,oo); 1^(11*)), from the
following classical observation from measure theory. Let un € LP(JRN) be
such th&t un —>u in £,{^(11^), and un and u are equimeasurable, i.e. have
n
the same distribution function. Then un —>u in 1^(11^). Working with
— n
^/|cjn|signo;n instead of un if p = 1, we see that it is enough to consider the
case when 1 < p < oo. Then, obviously un converges weakly in 1^(11^) to
u) while [|u;n||LP = |M|z,p- The strong convergence in Z^IR^) follows.
5) At the end of this chapter (in section 2.4), we will briefly explain how
the proof of Theorem 2.1 yields even more general results where we are
allowed to take p° e Lp(ft) if ft is bounded, p° e Lfoc(JRN) if ft = IR*
and p > y if TV > 3 while p > 1 if N = 2. In this case, we need to
assume that (t i—► p,(t)) is bounded and bounded away from 0 on [0, oo)
(0 < /i < ^(t) < Jl< oo for all * € [0, oo)).
6) We will show in section 2.3 that, for any weak solution (p, u) as in
Theorem 2.1, the following property holds: if p € Z^IR^ x (0, oo)) solves
(2.1), p|t=0 = po and p\u\2 € Z,1^" x (0,T)) for all T e (0,oo) if ft = IR",
then p = p. In other words, tx being fixed, p is the unique solution of (2.1)
with the initial condition given by po.
7) We wish to point out that the galilean invariance of the fluid mechanics
models allows us to restrict our attention, without loss of generality, to the
case when Uoo = 0. Indeed, if ft = IR^ and (p,u) solves (2.1)-(2.2),
then (p(x + V(t),t), {u(z + V(t),t) - v(t))) also solves (2.1)-(2.2) where
v e L^OtTiTR.") (VTe (0,oo))andy(0 = /or;(5)d5- In particular, if we
choose v(t) = Uqo, we only have to work with (p(x+u<x>*, t), u(x+Uoo*, t) —
Uoo) and thus we only need to study the case Uoq = 0.
8) Let us finally mention another consequence of (2.17) in the whole
space case (ft = JRN). We claim that if (2.8) or (2.9) hold then u €
L2(JRN x (0,T)) for all T € (0,oo). Indeed, if (2.9) holds, (2.17) implies
that (2.9) holds with p(t) replacing po. We then deduce from the Sobolev
and Holder inequalities, for all t > 0, defining a = (p — p)T, that
£HIl*(II") ^ llallL'(R") IMIzJ(R.V) l|Vti||J^ij
where § + ^j*"2 = 2=±. Observe that 9 e (0,1). Our claim follows
easily since Vtx e L2QRN x (0, T)) for all T e (0, oo).
If (2.8) holds, we observe that (2.17) also implies the equality
||(l/p(0)l(p(t)<6o)llLi(iRN) - IKVpo)

Existence results
25
Then, u = ul{p<6o)+ul{p>so) and ul{p>6o) e L^TjL^IR")) since
M l(p>h) ^ 7£ \/£M-In ad^011*we have
MU) = (VpN)^)1(p<^o) e L-co.rjtfCR")),
for allT€ (0,oo).
By Sobolev embeddings, we also have, if N > 3: u 6 L2(Q,T\L2NttN~V
(IRN)) (V T 6 (0,oo)) and thus \u\l{p<6o) € L'OR* x (0,T)) (V T 6
(0,oo)), and even ^^(O^jL^IR^)). If iV = 2, we obtain the L2 bound
using the results shown in Appendix B. □
Finally, we would like to make a general comment on the initial
condition imposed on pu and on the weak formulation (2.12). Indeed, we
did not require any time continuity (with values in an arbitrarily negative
Sobolev space) upon pu and, in fact, there is no reason why, in the sense
of distributions, pu should converge to ttiq at t goes to 0+. The only
information we can obtain is that, roughly speaking, pu converges to mo up
to a "gradient-like" distribution. A convincing argument which shows that
we cannot expect more is to consider the case when po> p> 0. Then, by
Theorem 2.1, p > p on Q x [0,oo). If {pu)(t) -» mo in Lloc(Q) as t —> 0+,
then, since p e C([0, (»);£{*), u(t) -> u0 = 2J> in L^Q) as t -> 0+. In
particular, we must have
div(uo) = 0 in V(Q). (2.18)
We did not impose such a condition on uq since, in general, uq does not
exist when po vanishes.
This is clearly a delicate point that we wish to clarify—in particular
because we shall encounter similar difficulties in the study of
incompressible limits (chapter 9). In order to keep ideas clear and avoid unpleasant
technicalities, we only state and prove the following result in the case when
fl = EVV and u°° = 0 even if similar results can be obtained in the case
of Dirichlet boundary conditions—the periodic case or the case Q = IR^,
u°° ^ 0 are easily adapted from the case we treat. In order to avoid
(too) negative Sobolev spaces, we introduce some notation: we denote by
Ri = (-A)~1/2 ^ (1 < i < N) the usual Riesz transform and we denote
respectively by i?A, R and R- the operators (-A)"1/2rot , (-A)~1/2V
and (-A)~1/2div. We shall also need the following elementary variant of
Hodge-de Rham decomposition:
Lemma 2.1. Let N > 2, p e /^(IR^) such that p > p > 0 a.e. on IRiV
for some p € (0, oo). Then there exist two bounded operators Ps.Qs on

26 Density-dependent Navier-Stokes equations
L2(TRN)N (whose norms depend only on p and ||p||l~(ir")J such that for
all me L2(TRN)N, (p,q) = (Ppm,Qpm) is the unique solution in L2(JRN)
of
m = p + q1 i*(-p)=0, RAq = Q. (2.19)
Furthermore, if pn € L°°(TRN), p<pn<? a.e. on TRN for some 0 < p <
p < oo and pn converges a.e. to p, then (PPnmny QPnmn) converges weakly
in L2(SRN) to {Ppm1 Qpm) whenever mn converges weakly to m.
Remarks 2.2. 1) Notice that we always have J^s A (Ppm) • (Qpm) dx =
0 and thus P* = j;PP{p')i Qp = pQp(p-)- I*1 other words, the above
decomposition simply corresponds to a change of metric induced by ^ in
the usual decompositions.
2) If ntn converges strongly in L2 to m, then PPrtmni QPnpn also converge
strongly. Indeed, we have in view of the weak convergence stated above
(recall that pnfn^pfi 7- fn~* - / if fn-*f weakly in L2)
n Hn n p n
/ \PPnmn\2dx=[ {^-PP„(pnPp„mn)}.mndx
~> / (-PpCpPpmjI-mdx = / \Ppm\2dx. D
n ./iR" l P > MN
Proof of Lemma 2.1. We first recall the usual "div-curl" decomposition,
namely the Hodge decomposition
m=pi+qi, Rpi=0, RAqi = 0. (2.20)
Recall that p\ = Pim, q\ = Q\m define bounded operators on L2. Next,
(2.19) is equivalent to finding a unique v € L2(TRN) such that
p = pi + Rv, R-(-p\=0
or equivalently to finding a unique v € L2QRN) such that
R-(Rv+Pl) = 0 in BN. (2.21)
P
The unknowns p and 9 are then given by p — p\ + Rv, q = qi — i?v.
The equation (2.21) is an unusual way of writing the elliptic equation
div(i(W + p!)) =0 in Ew

Existence results 27
where V = (-A)"1/2v. The only advantage of (2.21) is that N = 2 does
not play any role there. In particular, the solution of (2.21) is the unique
minimum over L2(BlN) of
Jjf
1 in..,2 . Pi
n \Rv\* + Z±-Rvdx.
ir" 2p p
The bounds follow trivially since we have
/ -\Rv\2dx = - f ^-Rvdx
Jm." P Jirn P
hence
< (£)-1/2l|Pilli»(ii») < (P)-1'2 \H\lhr")-
In order to prove that the weak limit of PPnmn is Ppm, we use a standard
"hilbertian strategy". We first observe that PPn is bounded on L2(JRN)
uniformly in n in view of the preceding bounds. Next, for any ty 6 L2(JRN),
p ^ —► V> in L2(JRN) (by Lebesgue's theorem) and thus it is enough to check
that
/ (Ppnmn) 'P^-* f (Pfim) • 1>dx.
JnN Pn n Jtrn
But, we have, in view of Remarks 2.2 (1),
/ T(pp~mn) • P^ = / mn- — ppn(f"J>)<&
Jm.N Pn JIRN Pn
and it is enough to show that PPn<p converges (strongly) in L2(TRN)N to
Pp<p for all v? € L2(1RN)N. In fact, it is enough to show that PPn(p converges
weakly in L2(TRN)N to Pp<p since we have then, using again Remarks 2.2
(1),
/ ±\PPM2dx= f ±(PPn<p)-<pdx = f M-(f)ii
JJRN Pn JHN Pn Jm.N XPn'
- / {P,<p)'(£)dx = / -\PM2dx
n JnN \pJ Jm.N P
while -^ (PPn<p) converges weakly in L2(JRN)N to -j*{PP<p)-
Without loss of generality, we may assume that PPn<p,QPn<p
(extracting subsequences if necessary) converge weakly in L2(TRN) to some p,q €

28
Density-dependent Navier-Stokes equations
L2(]RN). Therefore, we have obviously: <p = p + q, RAq = Q. Next,
— (PPn(f) converges weakly to ^p and we thus obtain R • (-p) = 0. The
uniqueness of the decomposition (2.19) completes the proof of Lemma 2.1.
D
We may now go back to the issue of the "time-continuity" of pu in
Theorem 2.1. We shall use the following notation: / € C([0,T];Lj;(IRiV))
if / e L°°(0,T; L2(TRN)) and if / is continuous in t with values in L2(JRN)
endowed with the weak topology (picking up a metric for this topology on
a large ball of L2(JRN) containing the values of f(t) for t e [0,T]). Recall,
as explained above, that in the following theorem we consider the case
n = IR^ and Uoo = 0.
Theorem 2.2. Let (p,u) be a weak solution of (2.1)-(2.2) as in
Theorem 2.1. Then, for all T e (0,oo); R A (pu) e C([0,T];L^) with {R A
(pu)}(0) = RAm0 and £{RA(pu)} e L2(0,T;H-l(ELN)) + X where
X = L^Tj^-^OR")) with 1 < p < oo, q = -^ if N > 3,
1 < p < oo, q = -^y if N = 2 and p0 is bounded away from 0, and
X = &(0,T;W^'qQRN)) with 1 < p < oo, 1 < q < •£- if N = 2; we
can also take X = L°°(0,T; W"1^1^)) for any e > 0 if N = 2. In
addition, if we assume that po is bounded way from 0, then we have
1) puiy/pu,u e C([0,T\;Ll)N for all T e (0,oo).
2) pu (respectively y/pu,u) converge weakly in L2(1RN), as t goes to
0+, toPPomo (respectively^ (PPomo)j ^ (^p0mo)J- Furthermore,
if divu0 = 0 in Vf(JRN) where uq = *aa, pu (respectively y/pu,u)
converge strongly in L2(JRN), as t goes to 0+, to mo (respectively
y/pou0,u0).
3) The energy inequality (2.11) holds for all t > 0. □
We are going to prove Theorem 2.2, admitting temporarily Theorem 2.1
whose proof is given in the following sections.
Proof of Theorem 2.2. First of all, taking <j> = R A rp (V> € Cg°(IRiV x
(0, oo)) in the weak formulation (2.12), we deduce easily that the following
holds in the sense of distributions
^{RA(pu)}+dj{RA(puju)}-dj{RA(p('Vuj+dju))} = RA(pf). (2.22)
Recalling that Rk is bounded on LP(IRN) for all 1 < p < oo, 1 < k < N,
we remark that for all T € (0, oo)
RA(pf) e L2(IRN x (0,T)), RAiniVuj + dju)) € L2(JRN x (0,T))

Existence results
29
since p,/z 6 L°°(TRN x (0,T)), / 6 L2(1RN x(0,TY), Vu e L2(TRNx(0,T)).
Next, we observe that pu,u e Z^(0,T;£^1R )) (v ^) and by Sobolev
embeddings u e I2(0, T; L^ (IR*)) if JV > 3, hence pujU 6 Ll(0, T; Z,*^
(IR*)) for all T e (0,oo). Therefore, if N > 3, ptx^u e Z^TjL^IR"))
(V T) for 1 < p < oo, q = ^5. If N = 2, by the definition of a
weak solution, u e L2(0,T;F11OC(1R2)), hence, by Sobolev embeddings,
u e L2(0,T;Lfoc(R2)) for all 2 < r < 00 and pUjU e L^TjL^IR")) n
^(0,^1,^(11^)) for 1 < p < 00, 1 < q < ^, and for all T e (0,oo).
Finally, if N = 2 and po is bounded away from 0, say /?o > £ > 0 a.e. on
IR^, then (2.17) yields p > p a.e. on 1R2 x (0,00). Therefore, in this case,
u e L°°(0,T;L2(IR2)) n tfJo^H^TR2)) (V T) and, by the Gagliardo^
Nirenberg inequalities, u e 1^(0, T;£*(R2)) for 1 < p < 00, g = 2^y
(V T). Collecting all these bounds, we deduce from (2.21) the regularity of
^ {R A (pu)} stated in Theorem 2.2.
This yields of course the continuity in t of R A (pu) with values, say, in
W^ll(WiN) but, by definition, R A (pu) is bounded in L2(TRN) on each
interval (0,T) (for all T 6 (0,oo)). Therefore, i* A (pu) 6 C([0,T];L^)
(VT) and RA(pu)\t=0 = m0 because of (2.12).
We now turn to the proofs of claims l)-3) in Theorem 2.2 in the case
when Po > P a.e. in JRN for some p > 0. As we just saw, this implies
p>p>0 a.e. on JRN x (0,oo) and thus u e L°°(0,T-L2(1EIN)) (VT).
We next prove that pu 6 C([Q,T\\L2W) (V T). With the notation of
Lemma 2.1, setting m = pn, we see that RA(Pim) = JRAm e C([0,T];L^)
(V T) while, by definition, R • (Pim) = 0. By classical properties of such
div-curl decompositions (ellipticity!), we deduce that P\m 6 C([0,T]; L^))
(V T). Next, we claim that we have
m = Pp(Pim). (2.23)
If it is the case, we deduce the fact that m € C([0,T];L^) from Lemma
2.1. Now, (2.23) follows from the obvious properties
R A (P\m — rn) = R- Q\m = 0 by definition of Pi, Qi ,
R-(±m) = R-u = 0 since divu = 0.
The rest of claim 1) is an easy consequence of, on the one hand, the fact
we have just proved, namely pu € C([0,T];L^)) (V T) and, on the other
hand, the properties of p, namely p e C([0, oo)\ Lp(Br)) (V i? , V 1 <
p < 00). Indeed observe that if tn —> £ > 0, , * , -^7-7—* A % , ^ in
£p(£*) (V R, V 1 < p < 00) and remain bounded in L^IR") uniformly
in n, and 1) follows by observing that gnfn —*gf weakly in L2, if fn —>/
n
weakly in L2, <?n —► # in Lj1^ and #n is bounded in L°° uniformly in n.

30
Density-dependent Navier-Stokes equations
Claim 3) follows from 1): indeed, for all t > 0, we deduce from (2.14)
the existence of tn e (0, t), tn —► t such that
n
Of pH2dx)(tn)+ / / nidiUj+djUifdxds
< / \l!^Ldx+fn f pf-udxds.
Jjrn Pq Jo Jjr*
We only have to let n go to +oo observing that
M([ n P\u\2dx\ (tn) > (J n p\u\2 dx\ (t)
since y/pu e C([0,T\;Ll) (VT).
It remains to prove claim 2). First of all, in view of the properties of p,
it is clearly enough to show that pu converges weakly in L2(1RN)1 as t goes
to 0+, to Pp0(mo) while, if div (^) = 0 in &(RN) (or R • (a*) = 0),
yfpu converges strongly in L2(1R ), as t goes to 0+, to 7^8=. Since m =
pu e C7([0,1];!^), we only have to show that m(0) = Ppomo hi order to
prove the weak convergence. But, in view of what we have already shown
#A(m0-m(0)) = 0, R.(^l) = R.U(0) = 0
V PQ J
since u e C([0,1];L^). Hence, m(0) = Ppo(m0).
Finally, if div (^) = 0, we see that m(0) = mo. Hence, y/pu converges
weakly in L2(IR^), as t goes to 0+, to -~?8=. But, we also have, since (2.4)
holds for all t > 0 (claim 3) proved above),
IIv^IIl2(IR")(0 ^
7710
VPo
L*(R")
and the strong convergence in L2(JR ) is proven. □
We conclude this section by mentioning that the first statement in
Theorem 2.2 can be expressed, in fact, in terms of Pi(ptz), i.e. of the projection,
say in /^(IR^), on the subspace of divergence-free vector fields—we used
this fact in the proof of Theorem 2.2. In fact, the orthogonal decomposition
(PiiQi) is also possible in the case of Dirichlet boundary conditions (and
in this case Pi is the projection onto the subspace of divergence-free vector
fields u with u • n = 0 on dQ) or in the periodic case. If Q = HjV or in the
periodic case, we obtain immediately

Regularity results and open problems
31
Theorem 2,3. Let (p,tx) be a weak solution of (2.1)-(2.2) as in Theorem
2.1. Then, for all T € (0,oo), Px{fm) e C([0,T];L^(n)) and £ {Pi(fm)} e
L2(0,T;H-l(n)) + X where X = Lp(0,T; W-l>*(JRN)) with 1 < p < oo,
g = «j^5 if JV >3, l<p<oo, g = ^£j if iV = 2 and p0 is bounded away
from 0, and X = L*>(0,T; W^QR*)) with 1< p < oo, 1 < q < ^ if
N = 2. If N = 2, we can also take X = L°°(0,T; W"1^1^)) for any
e > 0.
In the case of Dirichlet boundary conditions, one must replace W~l*q(Q)
by V'l^(Q) = V£'9(Sl)' where F01,9(fi) te the closure for the Wl« norm of
the space of functions <p in C7o°(n) such that divy? = 0.
2.2 Regularity results and open problems
We begin this section by discussing some open problems on (2.1)-(2.2).
First of all, the uniqueness of weak solutions is completely open in all
dimensions. Of course, we expect this to be the case if N > 3 since,
in view of Theorem 2.1 (and Remark 2.1 (3)), a particular case of our
weak solution is p = ~p € (0, oo) and, in that case, u is a weak solution
"a la Leray" of the homogeneous, incompressible Navier-Stokes equations
(the classical Navier-Stokes equations), and, for this particular case, the
uniqueness of weak solutions is still an open question. But, even in two
dimensions, the uniqueness of weak solutions is not known for (2.1)-(2.2).
We show in section 2.5 some partial "uniqueness" results indicating that
any weak solution is equal to a strong one if the latter exists.
Of course, the uniqueness of solutions is closely related to the regularity
of solutions: "smooth enough" solutions are indeed unique—this is not
difficult to check and results in that direction can be found in [17] for
example. For the same reason as for the uniqueness, we cannot expect full
regularity results (like Pq,uq € C°° yield C°° solutions for all t > 0) to
be known since they would imply regularity results for the homogeneous
Navier-Stokes equations. However, as we shall see and recall in chapter
4, regularity in the preceding sense holds for the homogeneous Navier-
Stokes equations if N = 2 and if N = 3 various regularity results are
available. If N = 3, we do not have any further regularity information on u
different from what we stated in section 2.1 (or easy consequences of what
we stated). In particular, as we already mentioned in Remark 2.1 (2), very
little is known on the pressure field.
But, even when N = 2, regularity does not seem to be available at least
when fx depends on p. However, in the very particular case when N = 2, \x is
independent of p, i.e. ^ is a positive constant and po is bounded away from

32
Density-dependent Navier-Stokes equations
0, it was shown by A.V. Kazhikov [254] (see also S.N. Antontsev and A.V.
Kazhikov [16], S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov [17])
that one can obtain more regularity and therefore also uniqueness results.
We do not want to treat this case in great detail and we refer the interested
reader to [17] for complete proofs. But we wish to explain the main idea
of the proof of one regularity result, from which further regularity can be
deduced by classical arguments (one way is to differentiate the equation
and apply again the argument we are going to present).
More precisely, we wish to explain why u e £2(0,T; H2) n C([0,T]\ Hl),
fj* e £2(0,T;L2) (V T) if p0 > P > 0 a.e. for some p e (0, oo), u0 = ** €
Hl and divtzo = 0, N = 2, fi € (0,oo). We simply want to obtain, at
least formally, a priori estimates on u and ^ corresponding to the claimed
regularity, and we shall do so only in the case when ft = IR2, u^ = 0
to simplify the presentation. First of all, as we mentioned several times
before, p > p > 0 a.e. on IR2 x (0, oo). Next, using (2.1), we write (2.2) as
p—+p(u- V)u-nAu + Vp = p/,
divu = 0, in IR2 x (0, oo).
du
(2.24)
We then multiply (2.24) by ^, integrate (by parts) over IR and find for
all t > 0
£Llf \2 + ^i{\LlVul2dX^ - M^2)
du
dt
£,2(1R2)
II M |Vu| ||l»(r») + Hpd|Il-(r») II/IIl2(ir2)
du
dt
L2(1R2)
Hence, using the Cauchy-Schwarz inequality repeatedly, we deduce
du||2 d
L2
dt
Hn>) + Jt HVt*'(iR2) ^ c{NliWVull£<(iR2)
+
2
L2(H2)
}
where C denotes various constants independent of tx, t. Next, we recall that
we already have a bound (deduced from (2.14)) on u e £°°(0,T; L2), Vu €
£2(0,T; L2) and that the following inequality holds for all v e Hl(lR2) (a
particular case of the Gagliardo-Nirenberg inequalities)
\\vhw) < CNliw II^Hlw)-
Therefore, J^/R2 \u\4dxdt < C||tx|li«(0j^;i(2OR2))IIVtx||22(Q^^2(lR2)) < C.
Using (2.24) once more, we finally deduce that we have for all £ > 0
du
dt
d
l»(R») + A |Vu,^Ra)
<^(l + \\Vu\\lHlR2))+e\\D2u\\
(2.25)
2
L2(1R2)

Regularity results and open problems
33
where C0 > 0, /QT CQ{t) dt < C (for all T € (0, oo)).
Next, we observe that we have for all t> 0 in view of (2.24):
||-/zAu+Vp||L2(R2) < C\\f\\LHm +C
du
dt
L2(1R2)
+ c||H|Vu|||L2(]R2).
Since divu = 0, we can use classical regularity results on (linear) Stokes
equations—see for example R. Temam [472]—to deduce
MIh2(ir2) < c{lMlz,2(iR2) + H/H^ia*)
du
dt
L2(1R2)
+ IIN|Vu|||L2(1R2)}.
Exactly as above, this yields for all e' > 0
Hhh^) < -c^ + c
du
dt
L2(1R2)
+ £'Nltf'(lR2)
(2.26)
where Cx > 0, J* C2{t)dt < C (for all T e (0,oo)). Hence, choosing
e' = 1/2,
NlW2) ^ <*(*) +cf
dt
L2(1R2)
(2.27)
where C2 > 0, J* C2(t)dt < C (for all T e (0,oo)). Inserting (2.27) in
(2.25) and choosing e — ^, we deduce finally for all t > 0
du
dt
L2(]R2) dt
jt HVulli,^ < C8(t)(l + ||Vti||ia(Iia))
where C3 > 0, /QT C3(t) dt < C (for all T € (0, oo)).
The desired a priori estimates on §& in L2(JR2 x (0,T)), u in L°°(0,T;
H^TR2)) and thus u in L2(0,T;H2(1R2)) (V T) follow using Gronwall's
inequality. D
Remark 2.3. Further regularity on u(p,p) can be deduced from the
regularity we just obtained. One way is to differentiate and apply similar
arguments. Another way is to observe that, since u e L2(Q,T;H2), u
satisfies: |u(xi,t) - u(x2,t)\ < C{t)\xi— X2I |log{min jxi —a:2|» 5}| for some
C € L2(0,T). This implies that, for each T, pis Holder continuous in (x, t)
on [0,T] and this implies that D2u is Holder continuous in (x,t). O
We would like to mention another interesting open question: suppose
that po = li> for a smooth domain D (c CI), i.e. a patch of a homogeneous

34 Density-dependent Navier-Stokes equations
incompressible fluid "surrounded" by the vacuum (or a bubble of vacuum
embedded in the fluid). Then, Theorem 2.1 yields at least one global weak
solution and (2.17) implies that, for all t > 0, p(t) = Ip^ for some set
such that vo\(D(t)) = vol(D). In this case, (2.1)-(2.2) can be reformulated
as a somewhat complicated free boundary problem. It is also very natural
to ask whether the regularity of D is preserved by the time evolution.
Finally, we conclude this section with a few remarks on stationary
problems associated with (2.1)-(2.2), namely
p > 0, div (pu) = 0, div u = 0,
p(u • V)u-txAu+Vp = pf in ft , u e H£(Q)N, p e L°°(ft)
looking, for example, at the case of Dirichlet boundary conditions, and
/i € (0, oo) independent of p. Choosing for instance / e L2(Q)N, we claim
that in general (2.28) has a "huge number of solutions". First of all, some
"trivial" solutions are obtained by setting p = A € [0, oo) and solving the
stationary, homogeneous, incompressible Navier-Stokes equations:
A(it- V)u-p,Au+Vp = Xf in ft,
ueH£(Q)N, divu = 0 in ft,
and we know (see for example R. Temam [472]) that, for each A > 0, there
exists a solution (at least one) u € H2(Q) (p e H2(Q)) at least if N = 2 or
3. In addition, uniqueness holds for instance if N = 2 and A||/||£2 is small
enough (/x and ft being fixed).
In fact, there are many more (stationary) solutions of (2.28) than the
preceding ones. Indeed, take for example N = 2, ft = I?i, /i = X2p(r),
h = -*ig(r) where r = [x\ + x^)1'2, g e L2{BX) (i.e. J^1 g2(s) ds < oo).
Then, we look for solutions of (2.28) having the following forms : p =
p(r) > 0, u\ = X2^(r), v>2 = -x\ip(r). Obviously, div it = div(pu) = 0. It
is easy to check that p(u • V)tx = V(rpip2) and -Aui = -**((r^') 4- tp)' =
-X2W + ?V>')> *Atx2 = *iW + J^)- Therefore, if p > 0, p € L°°
is given, solving (2.28) amounts to solving — ^" - ^tp' = pp, i.e. ^ is
determined by
tf(l)=0, ^(r) = ~ I' s3p(s)g(s)ds, (2.30)
f* Jo
and thus for each p, we obtain one stationary solution (smooth if p and g
are smooth)!
A similar example can be built in the periodic case: take /1 = g(x2 + 4f)
where g is odd, periodic of period L2, p = p(x2 + -^) where p > 0, p is even,
(2.28)
(2.29)

A priori estimates and compactness results
35
periodic of period L2/2 (g € L°°) and solve —\iuft = pg on H, u periodic
of period L2. Then, p, u = (tx(x2),0) solve (2.28). This indicates that the
right way to formulate the stationary problem (2.28) might be to constrain
(2.28) with an additional requirement on the distribution function of p, a
direction that needs to be investigated in more detail.
Let us finally mention that the regularity analysis of (2.28) follows closely
the known results on the steady-state homogeneous Navier-Stokes equar
tions: in particular, if N < 4, u e H2(Q) and u e W2*(Q) if / e &(Q)
for any 2 < p < 00. In fact, since the case N = 4 does not seem to be
well known, we shall come back to this point in chapter 4. However, even if
/ € C°°(r2), we cannot expect more regularity on u in view of the preceding
examples: this is due to the fact that p may not even be continuous.
2.3 A priori estimates and compactness results
Let us first explain the organization of this section. We shall work mainly
in the periodic case and after each proof we shall explain how to modify
the preceding proofs in the Dirichlet case or in the case when Q = JRN.
Next, we begin with a priori (formal) estimates and then we state and
prove some general compactness results on sequences of solutions. These
compactness results will play a fundamental role in the existence proofs
since they will allow us to deduce the existence of the global weak solutions
upon passing to the limit in conveniently approximated problems and using
the compactness results shown in this section.
We thus begin with a priori estimates. We first remark that (2.1) and
the incompressibility condition (divtx = 0) immediately imply that the
distribution function of p(t)—considered as a function of x—is independent
of t. In other words, (2.17) holds. This is in fact nothing but the celebrated
Liouville's theorem. A direct formal proof consists in observing that if
0 € C1([0,co);lR), 0(p) satisfies
^M + div{u(3{p)) = (l+ti.v)/3(p) = ^(p){^+ti-Vp} = 0.
Therefore, integrating over Q (periodic case or Dirichlet case), we find,
using the boundary conditions, that
or equivalently that (Jn /3(p) dx) is independent of t.
In particular, choosing 0 = gn € C^fO.oo^IR), 1 > gn > 0 such that
gn(t) = 0 if ti [a,0] (where 0 < a < 0 < oo are fixed) and gn(t) = 1 if

36 Density-dependent Navier-Stokes equations
t € [a + £ , (3 - i] (take n > ^^), we deduce (2.17) from the preceding
fact upon letting n go to +oo. In particular (2.17) yields the following L°°
a priori estimate
0 < p(x,t) < UpoIIl- a.e. (2.31)
(in fact ||p(*)||z,~ = Hpo||l~ for all t > 0!).
The other a priori estimate that we can obtain simply follows from the
energy identity: indeed, we expect, at least formally, in view of (2.1), that
(2.2) implies, multiplying by u and integrating by parts, that
— / p —-dx+ / ti(diUj+djUi)djUidx = / pf-udx
or
o 31 / P\u\2 dx+n vidiUj + djUi)2 dx = I pf -udx. (2.32)
* at Jn I Jn yn
Next, the right-hand side of (2.32) is bounded, in view of (2.31), by
Qf P\f\2<**) Qf P|u|2&) < iipoiiie ii/iil» HvWIl*,
and, because of (2.31), p. = p.(p(x,t)) > /£ = inf {/x(A) / 0 < A <
IIPoIIl*} > 0, and thus
i / fiidiUj+djUifdx > = J (diUj+djUi)2dx
= J [ \Vu\2 + 2diUjdjUidx\.
In addition, we find, integrating by parts,
/ diUjdjUidx = / (diUi)2dx = 0.
Jci Jn
In conclusion, we obtain for all T € (0, oo)
fi / p\u\2dx\t)+p.f f\Wu\2dxds
< lipoid J* H/lJLa HvWl»* + 1/
^<*r, Vt€[0,r|.
A>
(2.33)

A priori estimates and compactness results 37
In the case of Dirichlet boundary conditions, using the Cauchy-Schwarz
inequality, we thus deduce
l|Vu||La(nx(o,D) ^ C (2-34)
sup ||pM2||Li(n) < C (2.35)
0<t<T
where C denotes various constants which depend only on T, Q. and bounds
on UpoIIl*, H/llL^nxfo.T)), IIPo|uo|2||Li(fi) = II^^ILi^)"
In the case of Dirichlet boundary conditions, we then deduce from
Poincare's inequality
H|z,2(0,7V/1 (n)) ^ c- (2.36)
In the periodic case, we claim that (2.35) also holds. In order to see this,
we introduce
</> = 4 fdx where 4 = j^rr I
and we deduce from (2.35)
( I pdx\\ <u> |2 < 2 J p\u\2dx + 2 I p\u- <u> \2dx
< C + 2||pol|Loc||t4-<tl>||ia
< C + C||Vu|j22
for all t € (0,T). Then, we can assume without loss of generality that
po ^ 0 (otherwise the problem is trivial: p = 0), in which case we deduce
from the argument made above on p that we have for all t > 0
( / pdx\{t) = / p0dx = Mq > 0.
i
Hence, we have
rT
\<u> \2dt < C (2.37)
which, combined with (2.34), yields (2.36).
We conclude this brief discussion of a priori estimates by explaining the
modifications of the preceding arguments needed to treat the whole space
case (Q = 1R^). First of all, the derivation of (2.17) is simply identical.
Next, instead of multiplying by u, we now multiply by u — u^ and find in
a similar way
\±\ plu-tioop + yi/ \*u\2dx < ||po||i2 ||/||l» IIVp(«-«.
I at jftN — Jj^n
oo)\\L*>

38 Density-dependent Navier-Stokes equations
and we still obtain (2.34), while (2.35) is now replaced by-
sup ||p|u-Uoo|2Hli(1Rn) < C. (2.38)
0<t<T
In particular, if p0 > p a.e. on IRN for some £ > 0, then, by (2.37), we
deduce p(x,t) > p > 0 a.e. on TRN x (0,oo). Therefore, (2.38) yields
\\u-Uoo\\mo,T}HHm.N)) ^ C» (2-39)
and we deduce from Sobolev embeddings if N > 3
»"-"-Il.(^*i»(r-)) * C (2-40)
At least formally, we in fact deduce (2.40) from (2.34) if N > 3 in all cases
even without assuming that po is bounded away from 0. In particular, if
N > 3, we obtain
\\u\\mO,T;HHBR)) < <?, (2.41)
for all R € (0, oo) (C depends now on R).
Finally, we claim that, even if N = 2, (2.41) holds. In order to prove
this claim, we first observe that, assuming again that po ^ 0, we can find
for all T e (0, oo) fixed, some Ro € (0, oo) such that
mQ = inf / p(xyt)dx > 0. (2.42)
In fact, as we shall see, rno and Ro depend only on po and on "a modulus
of continuity in t of p in Lloc". Indeed, arguing by contradiction, if such
an Rq does not exist, we find that for each n > 1, there exists tn € [0,T]
such that p(tn) = 0 a.e. on Bn. Extracting a subsequence if necessary, we
may assume that tn-+t e [0,T]. Then, since p € C([0,T]; If^c), we see
that p(J) = 0 a.e. on 1RN. Then, by general uniqueness results shown later
on in this section, this implies that p = 0 on IR x [0, oo) and we reach a
contradiction.
This proof, however, does not yield uniform bounds, i.e. bounds
independent of the solutions, and we re-prove (2.42) below by a different and
more efficient argument. But, before we do so, we wish to explain how
(2.42) yields (2.41). Indeed, we just have to copy the argument that led to
(2.37) in order to obtain
i.
T
| <u>R \2dt < C, for -R>-Ro, (2.43)
o

A priori estimates and compactness results 39
where <u>r- jB udx, and (2.43) combined with (2.34) yields (2.41).
We finally give another proof of (2.42) that yields uniform bounds. Since
the difficulty encountered here will be encountered many times in this book,
we state and prove a general lemma which is more general than what we
really need here.
Lemma 2.2. Let T e (0,oo), pn 6 C([0,7];Ll(BR)) for all R e (0,oo).
We assume that pn > 0 a.e. and that p# = pn(0) satisfies, for some
Ro e (0, oo) and for some v > 0 independent ofn,
i.
p^dx > v > 0. (2.44)
Br0
We also assume that pn satisfies
^ +div(mn) = 0 in £>'(]R2 x (0,oo)) (2.45)
ot
where mn = m" + m^, and we have, for some C > 0 independent ofn,
HmillLi(H2x(o,T)) < C, ||m5||L2(IR2x(o,r)) < C. (2.46)
Then there exist R > Ro and no > 1 such that for n > no we have
Before we prove this result, let us explain how we use this lemma in the
above context: assume that (pn, un) is a sequence of solutions of (2.1)-(2.2)
with the bounds already shown, which we assume to be uniform in n. As
we have seen in Remark 2.1 (7), it is enough to treat the case when u^ = 0,
hence we have for all T 6 (0, oo)
sup f pn\un\2dx < C, ||pn|U~(R2x(0,oo)) ^ a
We claim that if pg = pn(0) satisfies (2.44), then (2.47) holds. Indeed, we
just have to check that (2.48) holds with mj = mn = pnun, and this is
obvious since
f^\mn\2dx < (^P>n|2ds)||pn||L~(]R>).
Proof of Lemma 2.2. Without loss of generality, extracting subsequences
if necessary, we may assume that |m"| and (mjl2 converge weakly in the

40 Density-dependent Navier-Stokes equations
sense of measures to some bounded, non-negative measures on IR2 x [0, T]
denoted respectively by /xi and /X2- Next, we choose tp € Cq°(IR2) such
that tp = 1 on JBi, 0 < tp < 1 on IR2, </?(x) = 0 if |x| > 2 and we define
d = suPlR2 |VV|, C2 = (/Ra \VV\2dx)l/2.
We then consider a > 0 such that Cxa + C2a1/2T1/2 < »//2. Then, we
observe that there exists R > max (Rq, 1) such that for i = 1,2
/ l(fl-i<|x|<2«+i) d(*i < a/2.
Jjr2x[o,T)
Then, for n large, we also have for i = 1,2
'0 JJR2
dt dx l(R<M<2R) tf < a
JO JJR2
1/2
where /# = |m?|, & = |m£|2.
We next multiply (2.45) by <pr{x) — <p(%) and integrate over IR2 x [0,t]
(for all t € [0,T]) to find in view of (2.44)
(7 l|x|<2*p"<&)(*)
> i/- /" <ft/ dxlii<|x|<2A(W| + |m5|)|Vv?ii|
Jo Jtr2
>^--—a-C2J dt[l2 1R<\x\<2r\™2\2 dx
>V-Cla- C2Tl<2alt2 > V-
- ~ 2
in view of the choices of a and R. The proof of the lemma is then complete.
□
We finally briefly explain some bounds in the case when Q, = IRiV. Let
us first observe that (2.17) obviously implies that (2.8) and (2.9) hold
uniformly in t > 0. Next, if JV = 3 and (2.10) holds, p{t) € L^'°°(1RN) for
all* > 0 and \W)\\L^,X(^N) = \\Po\\L^,X(JRNy Furthermore, as explained
in Remark 2.1 (8), if (2.8) or (2.9) hold, we obtain a priori estimates on u
in L2{TRN x (0,T)) and thus in L2(0,T; fl^flR")) for all T € (0,oo). We
now consider the case when (2.10) holds and JV = 2: recall that we wish
to show that this bound propagates (in t). In order to prove this claim, we
wish to multiply (2.1) by <p(x) =<x>2(p-1) (log <x>)r. Of course, we
need to justify the integration by parts to be performed but this point can

1/m
mr-l
m-1
A priori estimates and compactness results 41
be checked easily in view of the bounds that follow. We then obtain, since
pP also satisfies (2.1), for some m € [2, oo) to be determined later on,
— / (Ptpdx
dt Jjtf
< I f?\u\\Vtp\dx < C f f?\u\ <x>~l(pdx
JlR2 JJR2
• ( / (?"£* (p^ <x>";::r (log <x>);^r dx 1
< CXm\\p\\v£ ( f ^ pP^^r <x>$3 (log <x>)^)dx)
where a is chosen in (^+l,oo) and Xm = (/Ra &jgj(log<x>)~a dx) m.
In view of Appendix B, Xm is bounded in L2(0,T) (V T € (0,oo)), and
our claim follows if we choose m large enough so that m > 2p and thus
<p-z£r <£>^r (log <x>)^rT is bounded on 1R2.
We now turn to the fundamental compactness results that we need in the
existence proofs presented in the next section. We consider, for the reasons
explained above, the periodic case and we suppose that two sequences
pn,un are given satisfying: pn € C([0,T];I1(Sfl)) (V R € (0,oo)), pn > 0
is periodic in n of period Li (V 1 < i < N), un € L2(0,T;H^r)N where
T € (0,oo) is fixed. We define pg = pn(0) and we assume
0 < pn < C a.e. on fi x (0,T) (2.48)
divun = 0 a.e. on Q x (0, T), ||un||L2(0lr;/n(fi)) ^ c (2-49)
^•+div(pnun) = 0 in V'(TRN x (0,T)) (2.50)
p£-+Po inl1^), un —tx weaklyinI2(0,T;#ier), (2.51)
n n F
for some po which thus satisfies 0 < po < C a.e., and where C denotes
various positive constants independent of n. Notice that, because of the
bound (2.48), the convergence of pg to po also holds in LP(Q) for all 1 <
p < oo and that, because of (2.49), divu = 0 a.e. on Q x (0,T).
Theorem 2.4. 1) With the above assumptions, pn converges in C([0,T];
LP(Q)) for all 1 < p < oo to tie unique periodic solution p, bounded on
ftx(0,T), o/
^ + div (ptz) = 0 in ^(H" x (0,T)), (2 52)
p € C([0, T]; Z,1 (ft)), p(0) = po a.e. in Q.

42
Density-dependent Navier-Stokes equations
2) We assume in addition that pn\un\2 is bounded in L^^TjL1^))
and that we have for some q e (1, oo), m > 1
<—(pn,txn), <p>\ < C||^||L,(ofT5Wm>f(n)) (2.53)
for all(p€ L*(0,T; Wm^(Q)) periodic such that div<p = 0 on JRN x (0, T).
Then, for all 1 < i < N, y/^u^ converges to yfpiii in #(0,T;Lr(ft)) for
2<p<oo, l<r< ^^, and u? converges to u{ in Z*(0,T;L*^r(ft))
for 1 < 6 < 2 on the set {p > 0} (ifN = 2, ^^ is replaced by an arbitrary
r in [1,00)/
Remarks 2.4. 1) Part 1 is essentially contained in R.J. DiPerna and P.L.
Lions [128] and we re-prove it for the reader's convenience.
2) It is possible to weaken the bounds on pn and un. For instance, if we
keep (2.49), it is enough to assume instead of (2.48) that pn is bounded,
uniformly in t € [0,T], in LP(Q) where p > -$~~—one can even treat the
case when p = -M^ if -^ > 3. In fact, if we consider renormalized solutions
instead of solutions in the sense of distributions, the above result holds
with p =■ 1! This is shown in R.J. DiPerna and P.L. Lions [128].
3) The same result holds with some obvious adaptations in the case of
Dirichlet boundary conditions replacing L2(0,T;#£er) by L2(0,T;^(fi)),
and assuming that (2.50) holds in ft x (0,T) and that <p e C£°(ft x (0,T))N
with divv? = 0 in Q x (0,T) in (2.53). □
Proof of part 1 of Theorem 2.4. The proof is divided into several
steps. Without loss of generality, we may assume, extracting a subsequence
if necessary, that pn converges weakly to some p in LP(Q x (0,T)) for all
1 < p < 00 where p satisfies (2.52), p is periodic. In addition, since pnun
is bounded in L2(0,T;L*(ft)) with 1 < q < ^ (q < 00 if N = 2), we
deduce easily from (2.50) that pn converges to p in C([0,T], W^m^(Q))
for 1 < p < 00, m > 0; see for instance J.L. Lions [293] for very general
compactness results of that sort. If we equip 1^(0) (1 < p < 00) with the
weak topology and an associated distance over a large ball containing all
values pn(t) (n > 1, t e [0,T]), we also deduce easily that pn converges to
p in Cr([0,T];L^(fJ)), and, in particular, p(0) = po a.e. in CI.
Then, we first prove (step 1) that p uniquely solves (2.52). Next, we give
a general regularization procedure for solutions of transport equations like
(2.50) (step 2). In step 3, we complete the proof of part 1.
Step 1. In order to check that p solves (2.52), that is
^ +div(/m) = 0 in Z^IR" x (0,T)),
ut

A priori estimates and compactness results 43
we have only to show that pnun converges to pu in 2^(11^ x (0, T)). This is
in fact rather straightforward since pn converges to p in L2(0, T; H~1(Br))
for all R < oo, while uncp converges weakly to ucp in L2(0,T;Zfo(5/0) f°r
all tp € C^{^N x (0,T)) supported, say, in BR x (0,T). Hence,
npnun tpdtdx =<pn1un<p> -> <p1u(p>= / / pmpdtdx,
_i" n Jo ./ir"
and our claim is shown.
We next explain why p uniquely solves (2.52). More generally, if g €
L°°(TRN x (0,T)), g periodic, g e C([0,T\;L^(Q)) (1 < p < oo) satisfies:
g(0) = 0 a.e. on JRN,
^ + dW(ug) = 0 in &QR.N x (0,T)),
then g = 0. Indeed, we deduce from the regularization property proved
below (step 2) that \g\ also solves the same equation. Then, we simply
integrate the equation in x using the periodicity to find
Jtfn\9\dx = 0 in V(0,T)
and /n \g\ dx = m(t) 6 C([0,TJ) satisfies m(0) = 0. Therefore, m = 0 and
Step 2. A general regularization for solutions of transport
equations. This regularization is based upon the following classical lemma that
we re-prove for the reader's convenience. We denote by u/c = pr<*;(j) a
smoothing sequence, i.e. u> e Cq?(1Rn), /eiv udx = 1, Support(u;) C Si,
u/>0 and £€ (0,1].
Lemma 2.3. Let v € Wl<a{m.N), g € L0{m.N) with 1 < a,j3 < oo,
i + i < 1. Tien, we have
||div (vp) * a;e - div (v($ * cjc))\\lh1rn) \
< C\\v\\wi,a{JRN) \\g\\L0(RN) J
for some C > 0 independent of e, v and <j and 7 is determined by^ = £ 4- 4.
In addition, div (v#) * u/c — div {v((j * u;c)} converges to 0 in L')'(IRiV) as £
goes to 0 if 7 < 00.
Proof of Lemma 2.3. Once (2.54) is proven, the rest of Lemma 2.3
is clear using the density of Cj?(RN) in W^flR^) (if a < 00) or in
L0(JRN) (if 0 < 00). Next, in order to prove (2.54), we define Ct =

44
Density-dependent Navier-Stokes equations
div (vg) * uc — div (v(g * u;c)), which is nothing but a commutator, and we
write C£ = r€ — (divv)(g * ve) where
f 1 /x '—v\ 1
rff = / -(v(y)-v(x))-Vc4/f—-—) -$9{y)dy.
Obviously, we have
\(divv)(g*u£)\ < VN\Dv\\g*ue\.
On the other hand, we have, using Holder's inequality,
|r.| < c\4 {-\v{y)-v{x)\V\ *•[/ \g\*\
where 1 < s, £ < co, ^ +j = 1,1 <t < J3, 1 < s <a and C denotes various
positive constants independent of e, v and g.
Next, we write
1 I3 U Z"1
-(v(y)-v(x))\ = -/ Vv(x + X(y-x))-(y-x)
£ ' \€ Jo
< I \Vv(x + \(y-x))\
Jo
dX
Therefore
7 \-(v(y)-v(x))\ads < dX \Vv(x+Xew)\a \w\s dw
JB(x,cy£ ' Jo JBi
< I dX I \Vv(x+Xew)\3dw = |Vv|3 *xe
Jo JBi
where ^(z) = fjdXjfa lBjk.(z) = jvMtjff)""1 " l) **.«"" (and
/m" Xe = meas(Bi)). Thus we obtain, defining Xe = (meas(S5))~1 \bc
\Ce\ < C{\Dv\\g*u>t\ + (\Dv\U%)l''{\g\**Xe)V< a.e. onIR*, (2.55)
and we conclude easily since we have, by classical properties of convolutions,
for all e > 0
\\9*»<\Il* < Ml*, \\(\Dv\s *xe)1/s\\La < \\Dv\\L. WxAi
M'**)1"^ <\\9\\l*,

A priori estimates and compactness results 45
and UxJIn = meas(Bi).
In fact, the manipulations leading to (2.55) need to be justified and one
way is to argue by density on (2.55). □
In particular, we deduce from Lemma 2.3 the following fact: if v €
L2(0,T;Hl(TRN)), g e L°°(JRN x (0,T)), v and g are periodic, divv = 0
a.e. and
§ + div(v<7) = 0 on V'(1RN x(0,T)), (2.56)
ot
then, for any 0 e C(IR;IR), (3(g) also solves (2.56). Indeed, in view of
Lemma 2.3, we have
|fe + v.V* = re in JRNx(0,T),
where ge = g * w«, rc—>0 in L2(Br x (0,T)) (V R < oo). Hence, if
0 € Cl(]R;]R), /%e) satisfies
at
+ div(v/?(<fe)) = (^ + v • V) 0k,) = /3'GfeK in TRN x (0, T).
Then, our claim follows upon letting e go to 0, at least when /? is C1. If
f3 is merely continuous, we simply approximate it (uniformly on [—||p||l*i
H^llx,*]) by C1 functions and pass to the limit in the sense of distributions.
Let us point out that we already used this fact (with (3(t) = \t\) in step
1 above. Let us also remark that this regularization allows us to show that
g 6 C{[0,T]\Lp(Q)) for all 1 < p < oo: indeed, we observe that, for all
jtJjge-gv\pdx < ^\re-rv\>>dx^ Qf l^-^dx) " .
Hence,
sup Wge-gvU" < \\g(0)*we - g(0)*uJv\\LP + / \\re-rv\\LP(n)dt.
[o,ti Jo
Therefore, ge converges to g in C([0,T];Ip(ft)).
Step 3. We have only to show that pn converges to p, say, in C([0,T];
L2(Q,)) (to deduce the convergence in C([0,T];L*(ft)), for all 1 < p <
oo). We already know from step 2 that p e C([0,T];L2(ft)) and from the
argument given before step 1 that pn converges to p in C([0,T];L5,(fi)).
Therefore, we have only to show that pn(tn) converges in L2(Q) to p(t) if

46 Density-dependent Navier-Stokes equations
tn (e [0,T]) converges to t1 while we already know that pn(tn) converges
weakly in L2(Q) to p(t).
Hence, the proof of part 1 is complete if we show that we have for all
t>0:
f(pn(t))2dx = [(ptfdx-* f p2dx = [ p(t)2dx. (2.57)
In fact, the convergence is obvious in view of (2.48) and thus we have only
to check the fact that pn(t) (resp. p(t)) has the same L2 norm as pfi (resp.
Po)- Next, in view of step 2, (pn)2 (resp. p2) also solves (2.50) (resp.
(2.52)) and the claimed conservations simply follow upon integrating these
equations and using the periodicity of all the functions considered. □
Remark 2.5. As mentioned in Remark 2.4 (3), the proof of part 1 is easily
modified if we replace the periodicity requirement on pn,txn by Dirichlet
boundary conditions, namely un = u = 0 on dCl1 or in other words un,tx e
L2(0, T; Hq(Q)). Of course, in that case, all equations are set in Q x (0, T).
The only argument which needs some explanation is the "integration over
Q of div(pu)" where p e L°°(Q x (0,T)), u e L2(0,T; H&(fl)). This is done
by observing that, by classical Hardy-type inequalities, ^ € L2(Q x (0,T))
where d = dist (x, dQ). Then, we consider, for e small enough, <p€ € Cq?(Q)
such that
0 < (f€ < 1 in Cty (p€(x) = 1 if d(x) >e,)
<pe(x) = 0 if d(x) < |, |V^| < - in ft
for some C > 0 independent of £. Then we have
| <div(pu),<pe>\ = / pu-V(pe\
m
and (/(d<£) l#dx)^0mL2(0,T)ase->0+. □
Proof of part 2 of Theorem 2.4. We first prove that we have
/ dt fdxpn\un\2 -> / dt( dxp\u\2. (2.58)
Indeed the condition (2.53) shows that ^ {Pi(pnun)} is bounded in
L"(0,T;W-m*(n))N while, by assumption, pnun and thus Pi(pnun) are

A priori estimates and compactness results
47
bounded in L°°(0,T;X2(fi))N. Hence, by classical compactness theorems
(see for instance J.L. Lions [293], R. Temam [472]), Px{pnun) is compact
in L2(0,T\H~l(Sl))N. In particular, since pnun converges weakly to pu
(step 1 of the proof of part 1 in Z,oo(0,T;X2(fi))iV for the weak-* topology,
Pi(pnun) converges to Pi(pu) in L2(0,T; J-T-1^))^. Hence, we have
f dt fdxpn\un\2 = f dt(pnun,un)L2(Q)
Jo Jn Jo
= / dt{Pi(fun),un)vin) = / dt <P1(pnun)1un >H-ixHi
Jo Jo
-> / dt<Pi((m)1u>H-ixHi = / dt (Pi(pu),u)L2ia)
n Jo Jo
= / dt (pti,tx)La(n) = / dt / dx p\u\2
Jo Jo Jq
where we use the fact that Pi(txn) = tzn, P\(u) = u since divtxn = divtx = 0
in V'(1R x (0,T)). In the case of Dirichlet boundary conditions, the
passage to the limit for pn|un|2 is shown in exactly the same way, replacing
Once (2.58) is shown, we observe that y/pPun converges weakly in L2(Q x
(0, T)) to y/pu. Indeed, in view of step 2 of the proof of part 1, y/p™ also
solves (2.50) and converges to y/p in C([0,T]; I^(fi)) (1 < p < oo) because
of part 1. Then, applying step 1 of the proof of part 1, we deduce our
claim, namely the weak convergence of y/p"un to y/pu in L2(Q x (0,T)).
This weak convergence, combined with (2.58), yields the strong
convergence in L2(Q x (0,T)) of yffFun to y/pu. The convergence of y/p"un
stated in part 2 of Theorem 2.4 then follows from the bounds we assumed
on pn,txn and y/fFun. The final statement of part 2 concerning the
convergence of un to u on {p > 0} is shown if we show that un converges in
measure to u on {p > 0}. But we deduce from the fact just shown that,
extracting subsequences if necessary, y/p"un converges a.e. on £2 x (0, T) to
y/pu. In addition, because of part 1), we may assume that y/p" converges
a.e. on £2 x (0,T) to y/p. Hence, on the set {p > 0}, un converges a.e. to
u and the proof of Theorem 2.4 is complete. □
We conclude this section by explaining how the preceding result, valid
in the periodic case and in the case of Dirichlet boundary conditions, can
be adapted to the case CI = 1RN. We begin by stating conditions and
assumptions on /pn,txn; of course, we no longer require pn and un to be
periodic and only request un to be bounded in L2(Q,T\H1{Br))n for all
R € (0,oo). We still assume (2.48) and (2.50) while (2.49) and (2.51) are
now replaced by
divun = 0 a.e. on IRN x (0,T) (2.59)

48 Density-dependent Navier-Stokes equations
pt^po in Ll{BR),
n
un -* u weakly in L2(0, T; H1 (BR)), for all R € (0, oo)
(2.60)
l(p«>tf)
F? (resp. F2n)
(resp.
= F? + F?,
is bounded in V
L(0,r;L]
Ll{0,T;L°°{n.N)))
\
(Hw)) I
J
(2.61)
for all 6 > 0. Since we shall deal mostly with situations where pn|wn|2 is
bounded in Iroo(0Jjr;£1(IR/')), we only wish to observe at this stage that
such a bound obviously implies that un l(p^>s) is bounded in L°°(0,T;
L2{JRN)) and thus (2.61) holds.
Theorem 2.5. 1) Under the above conditions, pn converges in C([0,T];
Lp(Br)) (for all 1 < p < oo, R € (0, oo)) to the unique bounded solution p
of
dp
Qt +div(pu) = 0 in V(1RN x (0,T)),
p\t=o = Po a.e. in JRN
(2.62)
such that
u
1+ x
!<,„>*) € LHO.TjL^IR^+XHO.T.L00^)). (2.63)
2) We assume in addition that pn\un\2 is bounded in L°°(0,T; L^IR*)),
Vwn is bounded in L2(JRN x (0,T)), if N > 3 that un is bounded in
L2(0,T;L^(JRn)) and that (2.53) holds with Q = JRN and for all <p e
L*(0,T; Wm*(lRN)) such that div(p = 0 a.e. on IR*. I^hermore, we
assume that either un is bounded in L2(RN x (0,T)) or N > 3, pg €
LV'OOQR"), p0 € L^'°°(IRN) or N = 2 and
sup sup / (pn)p <x>2(p~1) (log <x>)rdx < oo ,
0<t<T n ./]R2 ^ (2.64)
for some p € (1, oo] with r > 2p — 1.
Then, for aiJ 1 < i < N, y/(Fu? converges to y/pm in LP(0,r; £*(£*))
for 2 < p < oo, 1 < r < ^f4> 0 < i2 < oo, and u" converges to tx» in
Le(0,T;L^(BR)) on the set {p > 0} for 1 < 9 < 2, 0 < R < oo.
Remark 2.6. 1) Similar extensions to those described in Remark 2.4 (2)
are possible for the preceding result.

A priori estimates and compactness results 49
2) Part 1 of Theorem 2.5 allows us, in fact, to extend slightly some of
the uniqueness results obtained by R.J. DiPerna and P.L. Lions in [128].
D
Proof of part 1 of Theorem 2.5. The proof is divided in three steps.
Step 1. Truncations and consequences. We introduce p£ = (pn — 6)+
for 6 e (0,1]. Obviously, (2.61) yields for alU > 0
l(on>0) is bounded in I
<x> {Ps>{)) > (2.65)
L\0,T]L™(1Rn)) + L\0,T;L\IRn)))
where we define <x>= (1 + jx)2)1/2.
We are going to show below (in step 3) that p? converges in LP^Br)
uniformly in t € [0,T] to some p6 > 0 (e L°°(IR^ x (0,T))) for all 1 <
p < oo, T e (0,oo). This will be done using, in particular, some general
uniqueness results established in step 2 below that also show the uniqueness
statement contained in part 1 of Theorem 2.5.
We wish to show now why such a convergence of p£ yields the convergence
of pn in C([0, T]; L*>(BR)) (V 1 < p < oo, V R e (0, oo)) to some p which,
obviously, is bounded on IR^ x (0,T) and solves (2.62). Then, we show
why (2.63) holds in the limit.
First, we observe that we have for n,rn > 1
\pn-pm\ < \(pn-6)+-(pm-6)+\ + 26.
This is enough to ensure that (pn)n is a Cauchy sequence in C([0,T];
Lp(Br)) (Vl<p<co, ViJe (0,oo)) and thus converges to some p.
Obviously, p = lim$|o+ T 7>s (anc^ one can in ^act deduce a posteriori from
the uniqueness statement and its proof the fact that ~p6 = (p—6)+).
Next, we show that (2.63) holds. To this end, we observe that in view of
(2.61)
un
^ ^Xs(pn)\ < Mn(t) + Fn (2.66)
<x> I
where Mn > 0, Mn is bounded in L^O, T), Fn > 0 is bounded in Ll(JRN x
(0,T)) and** € C([0,oo), [0,oo)) satisfies: 0 < Xs < 1 on [0,oo), xsh) = 1
if t > 5, Xs(t) = 0 if t < 6/2. Obviously, for all R 6 (0,oo), xs(pn)
converges in C([0,X]; Lp(Br)) (V 1 < p < oo) and is uniformly bounded on
JRN x (0, T) while un converges to u, for example, weakly in L2(Br x (0, T)).
Therefore, ^Xs(pn) converges weakly in L2(BR x (0,T)) to ^x*(p)-
On the other hand, we may assume without loss of generality (extracting
subsequences if necessary) that Mn converges weakly in the sense of
measures to a non-negative bounded measure M on [0, T) while Fn converges

50 Density-dependent Navier-Stokes equations
weakly in the sense of measures to a non-negative bounded measure F on
R.Nx[0,T\.
Then, we deduce from (2.66) and all these convergences
U Xs(p)\ <M + F. (2.67)
<x>
Let us then denote by M and F, respectively, the absolutely continuous
parts (with respect to the Lebesgue measure) of M and F. Since -^ Xs(p)
€ Ll(BR x (0,T)) (V R 6 (0,00)), we deduce finally
<x>
and (2.63) is shown. □
HP>6) <
Xs(p)
<x>
< M + F
Step 2. Uniqueness. We consider here /i,/2 bounded solutions in
C([0,n;L1(BR))(Vil€(0,oo))of
4r + div(v/)=0 in P'OR* x (0,T)) (2.68)
ot
such that /i(0) = /2(0) a.e. on IR* and
T^Z 1(IAI>*) € ^(O.Tj^OR^)) + L1(0>T;L«(lRiV)), (2.69)
for all i = 1,2, 6 > 0. Then, if divv = 0 a.e. on 1R.N x (0,T)—we
could as well assume divv € L1(0,r;Loo(]RAr)) as in [128]—and v €
^(O.T; W&JOR*)), h = /2 a.e. on JRN x (0,T).
Let us first remark that / = /i — /2 satisfies the same properties as f\
and J2 with, of course, /(0) = 0 a.e. on IR^. Indeed, we just need to
observe
1(l/l>«) ^ 1(l/il>*/2)+1(|/2|>tf/2)-
Next, we use Lemma 2.3 and step 2 of the proof of Theorem 2.4 to deduce
that, for all 6 > 0, (|/| - S)+ satisfies exactly the same properties as /. In
other words, we can assume without loss of generality that / > 0 a.e. on
HJvx(0,T)and
-!4- l(/>0) € iHo.Ti^QR")) + L1(0,r;I°°(lRivr). (2.70)
Indeed, observe that / = 0 follows from (|/| - <5)+ = 0 for all <5 > 0.

A priori estimates and compactness results 51
We deduce from (2.70) the following fact
J^L i(/>0) < M{t) + G(x, t) a.e. in 1RN x (0, T), 1 (g n)
g 6 L1QRAfx(o,:r)), g > o, m 6 i^o.r). M ^ °- J
Next, we consider <p 6 Cq°(IR), even, 0 < <p < 1, y?(x) = 1 if |x| < 1,
(p(x) = 0 if |x| > 2, <p nonincreasing on [0,oo), and we multiply (2.68)
by <pn(ecw <x>) where v>n(*) = ¥>(*), n > 1, C(«) = fiM(s)ds.
Integrating by parts over IR^ x (Q,t), we find for alii € (0,T]
t Aft*8"5?)* + jf t ** M^))
•{^}-{«W<«>-~}/-«>-
In view of (2.71) and the properties of <p, we deduce
/ f(t)v(ecit)^^)dx < C f [ dsdxGli<J>>ne-c(.)y
Jn" \ n J JQ JfrN
This is enough to show that / = 0 upon letting n go to +00 since the
right-hand side goes to 0 as n goes to +00.
Step 3. Convergence of p£. Without loss of generality, we may assume
that p£ and (p£)2 converge respectively to some pj, p26 weakly in L°°(JRN x
(0, T)) - *. Furthermore, exactly as in the proof of Theorem 2.4, we know
that pj, pf satisfy the equation (2.62), belong to C([0,T]; Lp(IRN)-w) and
satisfy p?|t=0 = (po - £)+, P2|t=o = (po - £)+2 = (pJ)2lt=o a.e. in IR*.
In addition, by Lemma 2.2 and step 2 of the proof of Theorem 2.4, we see
that Cps)2 also satisfies (2.62).
Therefore, if we show that ^ l(?T>a), M. 1-^ € £1(0, T; ^(IR*))
+L1(0,T;L1(IRN)) for all a > 0, we deduce from step 2 above that pf =
(p?)2. Hence, p£ converges to pj in L2(Br x (0,T)) and thus in Lp(Br x
(0,T)) for all R e (0,oc), 1 < p < 00. Since the bounds on ^ l(pj>a)
and on Jg. L-y> , are proven in exactly the same way, we just show them
for J|L l(pj>a)- We first remark that (2.61) implies that we have
^ !(*■>«> < Mn(t) + Fn(x,t) a.e. in RN x (0,T)
where JIT.P > 0, Mn is bounded in Lx{0,T), Fn is bounded in Lx(JR.N x
(0,T)). In particular, we deduce for some C independent of n (and 8)
-r-r Kpn6\ < C{Mn(t) +Fn(x,t)} a.e. in 1RN x (0,T).

52
Density-dependent Navier-Stokes equations
Then, step 1 of the proof of Theorem 2.4 shows that unpg converges to up~s
weakly, say, in L2(Br x (0,T)) for all R € (0, oo), and we deduce as in step
1 above
I til
<x>
pi < C{M + F} a.e. in TRN x (0,T)
(2.72)
where M € Ll(0,T), F € V-QR* x (0,T)) are respectively the absolutely
continuous parts (with respect to Lebesgue measure) of the weak limits (in
the sense of measures) of Mn,Fn. In particular, (2.72) yields the desired
fc**: ^ l(^>a) e LHOWQR*)) + Ll(0,T;L1(JRN)) for all a > 0.
At this stage, we have shown the convergence of p™ to ps in 1P{Br x
(0, T)) for all 1 < p < oo, R 6 (0, oo). In order to show the convergence in
C{{0)T)\Lp{Br)) for all R € (0,oo), 1 < p < oo, it is enough to consider
the case when p = 2 for instance. Then, we fix R in (0, oo) and we choose
<p e C£°(IRN), (p > 0, <p = 1 on £*. We claim that we have, for all n > 1
and for all t€ [0,T],
/ \vpn6{t)?dx = f \tp{fi-6)+\2dx
Jm.N Jm.N
+ /7 dxds{(p?)2un.V<p2},
f \<pps(t)\2dx = f \<p(p0-6)+\2dx
+ // dxds{{p!)2U'Vip2}.
Jo MN '
(2.73)
(2.74)
Indeed, using step 2 of the proof of Theorem 2.4 once more, (p£)2 and
(p?)2 satisfy respectively the same equations as pj and p$ and belong to
C([0,T\;IS>(Br>)-w) for all 1 < p < oo, R 6 (0,oo). Then, (2.73),
(2.74) follow upon multiplication by <p2, and show, by the way, that p$ G
C([Q,T)\L*>(BR>)-w) for all #' 6 (0,oo), 1 < p < oo. In addition, we
check as in the proof of Theorem 2.4 that p£ converges to p$ in C([0,T];
L2(Br')—w) for all #(0, oo) and thus, in particular, <ppj converges to (pps
in C([0, T]; L2(IRN)-t<;). This fact, combined with (2.73) and (2.74), shows
that <pp£ converges to ipps in C([0,T];L2(IRiV)) provided we show
f( dxds{{pns)2un-Vv2}
JoMN
-> dxds{(p~t)2u-V(p2}, uniformly on [0,r].
n JoJvi" J
(2.75)

A priori estimates and compactness results 53
This is easy since (p£)2 converges to (p6)2 in L2(BR< x (0,r)) (V Rf e
(0, oo)) and un converges to u weakly in L2(BR> x (0,T)) (V R' e (0, oo).
Therefore, <pp% converges to <pps in C([0,T\;L2(1R.N)) and thus p£
converges to p? in C([0,TJ; L2(Br)) for all # 6 (0, oo), and the proof of part
1 is complete. □
Proof of part 2 of Theorem 2.5. Let us first observe that step 1
implies easily that y/p"un converges weakly to y/pu in L2(JRN x (0,X))
(for instance). Thus, exactly as in the proof of Theorem 2.4, we have only
to show that we have for all R € (0, oo)
/ dtf dxpn\un\2 -> / d*/ Gbp|u|2. (2.76)
7o VBr n ^0 J Br
The proof of (2.76) is divided into four steps. First, we show that P(pnun)
converges to P(pu) in L2(Br x (0,X)) (V R e (0, oo)) where we denote by
P the orthogonal projection in L2(TRN)N onto divergence-free vector fields
(P = Pi with the notation of section 2.1). Then (step 2), we show that
(2.76) holds in the case when un is bounded in L2(JRN x (0,T)). Step 3
is devoted to the proof of (2.76) in the case when N>3,pe L^'°°(JRN),
and we treat the case N = 2 with the condition (2.64) in step 4.
Step 1. Compactness of P(pnun) in L2(BR x (0,T)) (V R € (O.ooM.
Since P = Id +V(-A)_1div is bounded on each Sobolev space Wm'q(TRr)
(m > 0, 1 < q < oo), we deduce from (2.53) that ^P(pnun) is bounded
in L*(0,T;W-m*'(1RN)). On the other hand, by assumption, P(pnun) is
bounded in L°°(0,T;i:2(IRN)). Hence, Appendix C implies that P(pnun)
converges to P(pu) in C([0,T};L2(JRN -w). We then wish to show that,
for each fixed R 6 (0, oo),
(dtf dx\P(pnun)\2 -> / dtf dx \P(fm)\2 dx. (2.77)
JO J BR n JO J Br
We then write ip = 1#R.
Then, we take as in Lemma 2.3 a regularizing kernel cj£ (e G (0,1]) and
we observe that {<pP(pnun)} * cje converges to {<pP(pu)} * cj£ in C([0,T];
L2(RN)). Therefore, we have for all e € (0,1]
f dt f dx {<pP(pnun)} ■ ({P(pnun)} * ujc)
JO JlRN
-+ f dt [ dx{<pP(pu)}'({P(pu)}*uje),
n Jo Jtrn
since P(pnun) converges weakly in L2(RN x (0,T)) to P(pu).

54 Density-dependent Navier-Stokes equations
In view of this convergence, it only remains to show that the following
integral can be made arbitrarily small for e small enough uniformly in n
{ dt fdx <pP(pnun)}{P(pnun)} - P(pnun)*uc)
•/0 J JR.
= f dt f dxP(<pP((m)){(pnun)-(pnun)*cje}.
Jo MN
First of all, we remark that P(pnun) is bounded in L°°(0,T;£2(]RN)) and
thus P((pP(pnun)) is also bounded in L°°(0,T;L2(1R")). Furthermore,
from the definition of P, we deduce easily the following bound
\P(<pP(pnun)\ < JjL a.e. |x|>i*+l, t€(0,T).
These two facts imply that it is enough to show
(2.78)
limsup \\pnun-(pnun)*ue\\LH0tT.L2(BM)) = 0,
for all Af€(0,oo).
To this end, we drop the superscript n and write
(pu) *ue—pu = (p*ue)u — pu
+ Vu(x+X(y-x))-(y-x)p(y)we(x-y)dydX.
JoJjrn
Next, we remark that we have
nVu(x+X(y-x)) • (y-x)p(y)ue(x-y)dydX
^\\p\\l~(jrn)£ / \Vu(x+X(y-x))\u;e(x-y)dydX
Jo Jm.N
and if we take the L2 norm on 3RN of the last integral, we can estimate its
square, using the Cauchy-Schwarz inequality, by
I d\ [ dyl [ \Vu(x+X(y-x)\2cje(y-x)dx\
Jo JjR" Ue" J
= f dX f we{z)dz f dx\Vu{x+Xz)\2 = [ |Vu(x)|2 dx.
Jo Jm." Jtrn Jm.N

A priori estimates and compactness results 55
On the other hand, we have by Sobolev embeddings for some p = p(N) €
(l,oo)
\\\(pn*Uc)-pn\\un\\\L>(BM) < C\\f*U>€-fr\\L*{Q9TlL>{B„)y
This is enough to prove (2.78) and to conclude since the compactness shown
in part 1 yields
lira sup ||pn*o;e-/9n||L2(o,r;Lp(BA/)) = 0.
e~*° n>l
Step 2. The case when un is bounded in L2. We complete here the
proof of part 2 in the case when we assume that un is bounded in L2(TRN x
(0,T)). By definition of P, we know there exists nn e L^O,^1'2^"))
(P1,2(IR2) if N = 2—see Appendices A and B for more details on these
spaces) such that
pnun = P(pnun) + Vtt11. (2.79)
Since un is divergence free, we deduce for all S > 0
un = —-^{P(pnun) + 6un+ V7rn} (2.80)
(L2(0,T]Vl'2(R2)) if JV = 2) J
div("lTp7 {V7r? + P(/9ntxn)}) = 0 in P7 (2.82)
div(^^{V7r? + (5txn}) =0 in I/, (2.83)
and we write with obvious notation similar decompositions for tx, p that
involve 7r,7r$ and its.
We next remark that we have
Jr» I >>"+* i J** p + l
since we deduce easily from (2.83) (using the density of C^(TRN) into
X^QR*) or ^(IR2) if JV = 2; see Appendix A)
Jw
L^T7{|V*?|2 + iu"-V*?}dl = 0-

56
Density-dependent Navier-Stokes equations
In particular, we obtain
lim sup
6-0 n>1
^^ " v^"{^(p(Pn«n)+V,r?)}
L*(TRN x(0,T))
and similarly
Urn
5—0
v> " v^ { T7£ (p(P») + V7r*)}
I»(R"x(0,T))
= 0.
= 0
(2.84)
(2.85)
Therefore, in order to complete the proof, we need only to show that for
each 6 > 0
V^bb) {*<P"«B)+Vir?> - ^(^) {P(pu) + Vir,}
in L2(BRx(0,T)), Vi*€(0,oo).
But we know from part 1 that pn and thus y/p~", 7^~pj converge respectively
t0 P. VP. pi* in C([0,T1;LP(Bii)) (V 1 < p < 00, V R € (0,oo)). In
addition, we also know from step 1 above that P(pnun) converges to P(pu)
in L2(Br x (0,T)), V R e (0,oo). Therefore, it only remains to show that
Vtt£ converges to Vits in L2(BR x (0,T)).
As seen from the following result, the above convergence is, in fact, a
consequence of (2.82) and the convergences we just recalled.
Lemma 2.4. Let hn be bounded in L2(JRN)N, let (a£-)i<i,i<iv be
bounded in L°°(1RN). We assume
N
3 v > 0, V n > 1, a.e. in x 6 IRN, V £ 6 JR.", ]T ag-&& > i/|£l2,
(2.86)
a^^aij in Ll(BR) for ail Re(0,oo), (2.87)
hn^h in Ll{BR) for all Re(0,oo). (2.88)
We consider the unique solution fn 6 rli2(]Rw) (if AT = 2, /n 6 P1-2(H2)
witi/Bi/ndx = o;of
(2.89)
and we denote by f the solution of (2.89) with a£-,/i" replaced by ay.ft..
Then, /n converges in ^{Br) to f for ail fi € (0,00).

A priori estimates and compactness results
57
Proof of Lemma 2.4. First, we observe that (by the density of Cq^IR^)
in Vl'2{JRN), or C£°(IR2) in P1,2(JR2)^-see the argument of Appendix A)
fn is bounded in Vlt2(lRN) (resp. 2>1,2(IR2)) and converges weakly in
Vh2{JR.N) (P1,2(IR2) if N = 2) to / since hn converges to h weakly in
L2(JRN)N. In particular we deduce, from the Rellich-Kondrakov theorem,
that fn converges to / in L2(BR) for all R € (0, oo). Then, for R 6 (0, oo)
fixed, we consider <p 6 C^CB.N) satisfying: <p > 0 on IR^, ^ = lon Br;
and we multiply (2.89) by ipfn (or <pf). We then obtain
> (2-90)
(2.91)
*,J—J.
*»J — * *iJ — * *
•ij—i
= — / y* a,,- ~- -r-^- fdx - I V^ auk -— (<pf) dx.
Jwl» fa ' dx, dxi J Jaffa ' dxi* I
We then claim that the right-hand side converges, as n goes to +00,
to the right-hand side of (2.91). Indeed, a% is, uniformly in n, bounded
and converges in Lp(Supp<p) for 1 < p < 00 while fn and hn converge in
L2(Supp<p) therefore a£ J^-/n, a^h* converge in L2(Supp<p) to,
respectively, dij J^- /, a^hj. In addition, §£■ and gf- (<p/n) converge weakly in
L2(JRN) to, respectively, J^ and jf^ (<pf), and this shows our claim.
Hence, the right-hand side of (2.90) converges to the right-hand side of
(2.91). But we also have
"/ \v{r-f)\2dx
Jl&N •~'1 ^ &xi 9xi) \ dxj dxj J
Jb." M dx> dxi J*K Mi dx* dx=
£-*. lj \ dx{ dxj dxj dxi)
-f
JlRN
i,J=l

58
Density-dependent Navier-Stokes equations
and the lemma is shown if we prove that this upper bound goes to 0 as n
goes to +00. This is the case since we just proved that the first term goes
to Jjr" £ij=i av §£~'§i:dx' while the second term converges obviously to
the same quantity in view of the properties of a£. Finally since a£ ^, or
aij §h converges in L2(JRN) to dij J-£, dij j^- respectively (by Lebesgue's
lemma), the last integral converges to 2 J^s Ylij^i aij ~Sxl ^ <&> ^d we
conclude. D
Step 3. The case when N > 3, pn,p€l^'°°(IR/v). Let us first observe
that the proof of part 1 of Theorem 2.5 and the proof of step 2 of Theorem
2.4 immediately yield the fact that meas {pn > A}, meas {p > A} are
independent oft e [0,T] for all A > 0 and thus pn,p € LTO(0,T;L£>~(IRN)).
We are going to use the results of Appendix A and more precisely Theorem
2. To this end, we introduce the solutions u^.ur^ for a.e. t € (0,T),
R € (0, oo), e e (0,1] of, respectively
- AunRy£ + \ p»ul£ + Vp£,ff = -Aun + \ pnun in V'{BR),
uR>e € Hl0{BR), divu£)£ = 0 a.e. in BR;
- AuRtC + - puR>c + VpR,e = -Au + -pu in V(Br),
} (2.92)
(2.93)
ur,€ 6 Hi(BR), divtxfl,c = 0 a.e. in BR.
We may then apply Theorem 2 in Appendix A with /n = 7 pn, the
assumptions required in Theorem 2 being satisfied in our case in particular because
of part 1 of Theorem 2.5 proven above, and we obtain, for all e € (0,1],
«5,.
V
( pn\uR>e\2dx + e f \VuR<e\2dx
JBR JBr
< I pnuR<cun dx + e I Vun • VuR>c dx,
J Br J Br
a.e.i € (0,T),
ur,c->u inL2(0,T]Vl'2(TRN)) as R -> +00,
sup||un - urJ\L2{BmX(0iT)) — 0 as R -► +00,
for allM€ (0,oo).
(2.94)
(2.95)
(2.96)
Finally, we also have
u
n
R>e-uR,e weakly in L2{Q,T;Hl{BR)),
for all R 6 (0,00), e € (0,1].
(2.97)

A priori estimates and compactness results 59
Indeed, we deduce H1(Br) bounds on u^ from (2.94)—recall that pn|un|2
is bounded in ^(O^Z^IR*)) by assumption. Then, (2.97) follows from
the uniqueness of the equation (2.93) passing to the limit in (2.92) (recall
that, by part 1, pn converges to p in C([01T];Lp(BR)) for all 1 < p < oo).
We then write, for all M G (0,oo) fixed
\\y/^Un - y/pu\\LHBMx(0,T)) < C SUp ||un - UnR%e\\L2{BM x(0,T))
n>l
jv/X(0,T))j
where C denotes various positive constants independent of n, R, e. Using
(2.95) and (2.96), we deduce
\\y/^Un - y/pu\\LHBMx(Q,T)) V gg
< \\VF^£-y/pu\\L2(BMX
(0,T)) +WC(R) J
where we denote by u€ (R) various positive constants that depend only on
e and R, and such that ve(R) —> 0 as R —> +oo, for each e € (0,T].
Next, we remark that (2.94) yields
npn\u\e\2dxdt < I J pnun -unR%edxdt + Ce (2.99)
Jr ' JO J Br
while we have
npu • uri€ dxdt , / / p|utf,ff|2 dx eft
Jr JO J Br
p\u\2 dxdt
II
J0 JlR
fO JJRN
as R goes to +oo, for all e G (0,1].
(2.100)
Indeed, p e Loo(0,T;Lt»co(]R7V)) and by the (sharp in Lorentz spaces)
Sobolev embeddings (2.95) implies that UR$e converges to u in L2(0,T;
L^(1RN)).
We then claim that we have for all Re (0, oo), e e (0,1]
npnun -unRedxdt -> / / (wuR,cdxdt. (2.101)
Jr ' n Jo J Br
Indeed, since u^c and uriC are divergence-free vector fields and vanish
outside Br, each integral can be rewritten as an integral over 1RN (or Br)
of P(pnun) • tx£ , P(pu) • u#|C respectively. Then, in view of step 1 above,

60
Density-dependent Navier-Stokes equations
P(pnun) converges to P{pu) in L2(Br x (0, T)) while uRe converges weakly
to uriC in L2(Br x (0,T)) because of (2.97) as n goes to +oo. This proves
the claim (2.101).
Combining (2.99), (2.100) and (2.101), we obtain
\\y/FuR,e - VpurAlhbkxW) < Ce + uc(R)
+ dt dxpnun • uRe + pu • uR<c - 2<v/p?rVpuRe • u
JO JbR
and
lim \\y/p~*uR>e - y/pu\\mBRx(o,T)) < Ce + ue(R) (2.102)
n
since uR e converges weakly to ur)€ while -y/p" converges strongly to yfp.
_ Adding, up (2.98) and (2.102), we finally deduce
Hm \\y/p~*un - y/pu\\L2{BMX(0tT)) < Ce + ue(R),
n
and we conclude upon letting first R go to +00 and then € go to 0.
Step 4. The case when N = 2, (2.64) holds. We first remark that,
because of part 1 of Theorem 2.5, (2.64) yields
sup
o<t<rJiR2
/ pP <x>2(p~l) (log <x>)rdx < 00.
Jjr2
(2.103)
We are going to show in that case that yf^un converges to yfpu in
L2{B? x (0,T)). To this end, we use the notation of Appendix A. We
observe first that fR(un) is bounded in L2(0,T;^1,2(IR2)) uniformly in
n,R, fR(u) is bounded in L2(0,T;P1,2(IR2)) and that fR(un) converges
weakly in L^O.TjP1'2^2)) (or L2(0,T;H^(BR))) to fR(u) as n goes to
+00 (V R € (0,00)). In particular, Theorem 1 of Appendix A yields for all
Ro 6 (0, oo)
rT
sup / / pn\un\2 - pnun • fR{un) dx
dt
'BrqJo
I f P\u\2-
pu ■ Tr(u) dx dt
0
as R —► +oo.
Next, we claim that we have
rT
(2.104)
sup(7 dx f dt pn\un\2 - pn\un\ \fR(un)\) )
n \J\x\>R0 JO J
+ I dx J dt p\u\2 - p\u\ \fR{u)\ -+ 0
J\x\>Ro Jo
as R —*■ +oo.
(2.105)

A priori estimates and compactness results 61
Indeed, we choose s € (2p—l,r) and we use Holder's inequality to obtain
rT
, \W
'\x\>Ro
Up
J dx I dt f\v
J\x\>Ro JO
< (I dxp <%>2{^V> (log <x>y)
■tXL^to<*>r*Y ^
< (log y/l+R20y^ (f dx fp <x>2^ (log <x>)r) P
' Wv\\L*(0,T;V*'Hm.2)) WWWlH0,T;V1'2(R2))
where / = pn,p, v = un,u, w = tzn, fj*(tzn), u, Tr{u).
The limit (2.105) is shown using the bounds (2.64) and (2.103) on pn,p
and the bounds recalled above on TR(un), Tr(u)._
Finally, we notice that we have, since Tk(txn), Tr(u) are divergence-free
and vanish outside Br
II pnun-fR{un)dtdx = / / P(pnun)-fR(un)dtdx
Jo Jjr? Jo Jm?
rp rp
= // P(pnun).fR(un)dtdx -► / / P(pu)-fR(u)dtdx
JO JBR n JO J Br
= [[ P(pu)-fR(u)dtdx = / / fm-fR(u)dtdx,
Jo Jjr2 Jo Jjr2
for all Re (0,co).
The convergence is a consequence of the strong convergence in L2(Br x
(0,T)) of P(pnun) towards P(pu) as shown in step 1 above, and of the
weak convergence in L2(Br x (0,T)) of TR(un) to Tr{u) recalled above.
Combining the preceding convergence with (2.104) and (2.105), we
deduce easily
npn\un\2dxdt -* I [ p\u\2dxdt,
. _l2 n Jo JlR2
and we conclude the proof of Theorem 2.5. □
Remarks 2.7. 1) It is plausible that the additional bounds assumed upon
un on pn in part 2 of Theorem 2.5 are superfluous. They are used to allow

62 Density-dependent Navier-Stokes equations
us to pass to the limit in pn|txn|2 and if this passage to the limit were
true (without these extra conditions), more general existence results than
Theorem 2.1 in the case when fi = IRN would be possible. We indicate in
the next two remarks different arguments that can be used to pass to the
limit but which, unfortunately, do not give better results than the methods
introduced above.
2) We begin with a method taken (and adapted) from [16] (see also
the references therein). This argument requires un to be bounded in
L2(0,T\Hl(RN))N, N = 2 or 3 and some conditions on p explained
below. We just sketch the argument, letting (pn,un) be a sequence of
(weak) solutions of (2.1)-(2.2) satisfying, uniformly in n, the a priori
estimates described at the beginning of this section. The method of proof
consists in integrating (2.2) in time between t and t + h and multiplying
by un(t+h) — un(t): this yields, using the a priori estimates,
/ (pn{t+h)un(t+h) - pn(t)un(t)) - (un(t+h) - un(t)) dx
J1RN
< C7/i1/2(||tx-(t+/i)||^x(IR^) + \\un(t)\\milRN))
at+h /• \ 1/2
ds dxy/p^\un\4)
■ (||Vt*n(t+A)||L.(R*> + Wun(t)\\LH1RN)).
Next, ifiV = 2oriV = 3
't+h
ft+n f
j ds dx >/pr\un\\un\3
Jt MN
2 fWL'n|u"'S1/U>"|64
ds KH&V) * chl/4-
Hence, we find for all h € (0,1)
f dt f {pn(t+h)un(t+h)-pn{t)un(t))'(un(t+h)-un(t))dx < Chl/4.
Jo Jm.N
It is in fact possible to extend this argument to N > 4 multiplying by
un(t+h) * ue - un{t) * ue (instead of un{t + h) - un{t)). This leads to a
bound like (CTi1/2 + -§r h + Ce) = C^A W) if £ = fciA W) u^g the fact
that ||un - un * uc\\L2imNx(0yT)) < Ce.

A priori estimates and compactness results
63
Next, we deduce from the preceding inequality
f dt f Uf^it+h^it+h) - y/pn(t)un(t)\2dx
JO JlRN
rp
< Ch1/4 + f dt I dx un{t) • un(t+h) \y/^)- y/pn(t+h)\2
Jo JjRN
< C7i1/4 + f dt f \un(t)\\un(t+h)\\pn(t) -pn(t+h)\
JO JlR"
< u(h)
where lj(K) —► 0 as h —> 0+ and u(h) does not depend on n. The last
inequality requires some assumptions on pn like for instance p£ = P + fo
with p > 0, /0n e L*(JRN) for some q 6 (f, oo), /0n -^ /0 in £*(IR"). This
condition as explained above yields L2(0,T\Hl) bounds and one deduces
easily from part 1 of Theorem 2.5 (and its proof) that pn = p+/n where fn
converges in C([0,T];Lq(JRN)). This is enough to yield the above bound.
The above "time-continuity in L2" of yffFun allows us to obtain
compactness in L2(Br x (0,T)) of y/fFun, using the compactness of pn (see
above and part 1 of Theorem 2.5) and the fact that un is bounded in
L^T^IR")).
3) Another method of proof consists of using some particular projections:
we introduce PRl the projection from L2(JRN)N onto {v e £2(EtN)N, v =
0 a.e. on BCR, divv = 0 in P^IR^)}; notice that necessarily v • n = 0 on
9Br by trace theorems. Then, if we consider pn,txn as in the preceding
remark, it is not difficult to check that for all Rq € (1, oo), there exists Rn €
(#o,/?o+l), such that £ {PRn(pnun)} is bounded in £*(0,T; W~1«(BrJ))
for some q > 1. This is enough to ensure that PRn(pnun) is relatively
compact in L2(BRn x (0,T)) (for instance) and converges to Pr(pu) if Rn
(or a subsequence) converges to some R e [i?o,-Ro + l]. On the one hand,
we have then
Pnun = PRn{pnun) + ^izn in BRn)
pn\un\2 = PRn(pnun)un + div(un7rn) in BRn
for some 7rn e £°°(0, T; Hl(BRn)) (which we can normalize by JB 7rn dx =
0), and, if we let n go to +oo, we obtain
p \u\2 = Pr(pu) • u + div (p) in Br
where Jtx|2 is the weak limit of |txn|2 and p. is the weak limit of un7rn.
Notice that p e L2{0,T\L^A{BR)) (in fact, it is bounded in that space

64 Density-dependent Navier-Stokes equations
uniformly in Ro) if un is bounded in L2(TRN x (0,T)) or if pn is bounded
in L^'^CJR1*) (assume for instance N > 3). On the other hand, we also
have
p\u\2 = Pr(pu) -tz + div(/x) in BR
with the same bounds on p,. In particular, we deduce, upon letting R go
to +00,
divftZ) = p(H2-|u|2) 6 L^^TjL1^)) > 0
where /x - /x € L^OjTjL^'^IR^)). It is then easy to conclude that
/Z = 0 and thus v^u*1 converges to pa in L2(B^ x (0,T)) (V R e (0, oo)).
Let us finally observe that this method of proof—which requires many
rather technical justifications that we leave to the reader—requires either
un to be bounded in L2(JRN x (0,T)) or pn to be bounded in L°°(0,T;
L^'00^")), i.e. p£ to be bounded in L*'00^"). D
2.4 Existence proofs
In this section, we give complete proofs of the existence part of Theorem 2.1.
We split the argument into three steps. In the first one, we solve an
approximated problem and thus construct approximated solutions. Next, in step
2, we use the a priori estimates and the compactness results obtained in
the preceding section to pass to the limit and build solutions of (2.1)-(2.2),
and this will prove Theorem 2.1 in the cases when Q is bounded, namely
the periodic case or the case of Dirichlet boundary conditions. Finally, in
a third step, we treat (and deduce) the case when Q = IR^.
Step 1. Construction of approximated solutions. Our goal here is
to construct solutions of the following approximated system
-^ + div(t^p) =0 in V (2.106)
at
-J^ + div (pu€ ® u) - div (2M) + Vp = pf£ 1
mV , divtx= Oinl/. J
If we consider the periodic case, then (2.106)-(2.107) hold in Z^IR^ x
(0, co)) and all unknowns are assumed to be periodic of period Ti > 0
in Xi, for each i G {1,... ,iV}. Let us recall that we define in this case
fi = rii=ri(0,T;), while, if we treat the case of Dirichlet conditions, (2.106)-
(2.107) hold in V'(Q x (0, oo)) and we require u to vanish on dQ x (0, oo).

Existence proofs
65
We now have to explain the real meaning of (2.106), (2.107) or, in other
words, the precise definition of u£ and p,e which are regularizations of u and
p,(p) respectively, depending upon a parameter s e (0,1]. In the periodic
case, we simply take ue = tz*u;c, fic = M*(p) *uc> fe = /*^o where u£ is a
regularizing kernel as in the preceding section and /x£ is defined below. In
fact, for technical reasons, we take fe = (f*ve)(e(t) where (c 6 C°°([0,T]),
Ce(t) = 1 if * > 25, 0 < C*(0 < 1 if * G [0,T], Cff = 0 if t < e. Observe that
we still have dxvu€ = 0.
In the case of Dirichlet boundary conditions, we set /2(p) = /x£(p) m ^>
= 1 in ftc and define p,e = /xc(p) *^cln- The definition of ue in that case is
a bit more delicate since we wish to smoothe tx, while keeping the Dirichlet
conditions and the divergence-free property. One possible (explicit) way
is the following: if tx € L2(0,T;H$(Q)) (for example) where T 6 (0,oo)
is fixed, we set ue to be the truncation in ftc of u (extended by 0 to ft)
as defined in Appendix A and we define ue by ue * o;c/2- Clearly, uei
which vanishes near dft, is smooth in x (recall from Appendix A that
u£ € L2(0,T;H$(Q))) and satisfies divtxff = 0 in IR^. Finally, we set
fe = Cr(/ l(d>2c)) * ue where d = dist (x, dft).
We would like also to make a simplification on fi(-): since, anyway, all
values of solutions p remain uniformly bounded (typically in an interval
[0, HpoIIl*])* we can assume without loss of generality that p,{t) is constant
for t > 0 large and in particular that (t »—► P>(t)) is bounded on [0, +oo).
Then, /xc is a C°°([0, oo[) function, bounded away from 0, which is constant
for t > 0 large and such that sup[0oo) \p,€ — p,\ < e.
We now discuss the initial conditions associated to (2.106)-(2.107),
namely
p|t=o = Poi pu\t=o = m§ on fi (2.108)
where p§ = (p0)e + e, m§ is defined below using mg = (mop^ ' )e(po )*.
In the periodic case, for / = po,Po imoPo > we define f£ = / * u€. In
the case of Dirichlet conditions, (po)e = Po * ^cln> (Po )* = Po * ^eln
and (m0Po 2)c = (m0Po 2 l(d>2*)) * ^ where p0 = Po on ft, = 1 on
ftc and d = dist (x, 5ft). Obviously, p§ 6 C°°(ft), mg e C£°(ft). Let us
immediately remark that we have for some Co > 0 independent of e
e < Po < C0 (2.109)
p£o^Po in ^ (ft) (l<p<oo), 1
mg-^mo inL2(ft), m^(p§)-1/2^m0p^1/2 inL2(ft).J
The last convergence in (2.110) is easily deduced from the following facts:
(Po/2)e < (Po)l/2 in ft, (Jfc)±Jfr in L2(ft).

66 Density-dependent Navier-Stokes equations
We finally build mg. First of all, we decompose, as in section 2.2, m% in
the following way
mo = Po^o + V^o, %,ql € C°°(ft),
divtZo = 0 in ft, TZo-n = 0 ondft
(denoting by n the unit outward normal to dft). Let us observe that q$ is
determined, up to an additive constant, by the equation
div(-^(V?S-m§)) =0 infi, ^ = 0 on 3ft, (2.112)
and we finally set
me0 = Po"o + V^, where u% € C^°(fi),
H«o-^ollLa(n) < e, divuo = 0 in^-
(2.113)
We then deduce from (2.109) and (2.110)
ml^mo inL2(Q), meQ(p£Q)-l/2-^mQpo1/2 inL2{Q). (2.114)
Observe that we have tuq =ml + Po(uq-Uq), m§(po)~1/2 = "^o(Po)~1/2 +
(pS)1/2K-^).
In fact, as we explained in section 2.1, (2.108) is not really meaningful
since (2.107) shows that pu is determined "up to a gradient" and thus the
initial condition, contained in (2.108), on pu\t=o really means an equality
modulo a gradient. Since p§ satisfies (2.109), and divtxg = 0 in ft, we
may—see also section 2.1—impose
p|t=o = Po in ft, u\t=o = Uq in ft. (2.115)
We then state and prove the following existence result.
Theorem 2.6. With the above notation and assumptions, there exists a
solution (p,u) of (2.106)-(2.107) and (2.115) such that p,u € C°°(TRN x
[0, oo)), p, u periodic in the periodic case; p, u € C°°(ft x [0, oo)), u = 0 on
dftx [0,oo).
Remarks 2.8. 1) The regularization procedure we are using is directly
inspired by J. Leray's original work on (homogeneous) incompressible Navier-
Stokes equations ([284]).
2) Obviously, we have e < p < Co on ft x [0,oo) (since divu£ = 0 in
ft x (0,oo), and u£ is periodic or vanishes on 9ft).

Existence proofs
67
3) It is in fact possible to prove the uniqueness of (p, tx), using for instance
the type of arguments developed in section 2.5. □
Proof of Theorem 2.6. We are going to show the existence of a solution
by a fixed point argument. In fact, this fixed point argument will yield
a solution (p,tx) with the following regularity: p G C(Q x [0,oo)), u €
L2(0,T;H2(Q)) nC([0,T\;Hl(Q)), ff 6 L2(Q x (0,T)) for all T G (0,oo)
in the case of Dirichlet boundary conditions and a similar regularity in
the periodic case. To limit the length of the proof, we only treat the case
of Dirichlet boundary conditions: the proof in the periodic case follows
the same line of arguments and is in fact much simpler. Finally, we fix
T € (0, oo) and work on [0,T].
We now define the mapping whose fixed point will yield a solution. Let
C be the convex set in C(H x [0,T]) x L2(0,T;H&(Q)) defined by
C = {(p,u) g C(Q x [0,21) x L2(0,T;Hl(Q) /
£ < P < Co in ft x [0,T] , divtT = 0 a.e. on Q x (0,T),
INlL2(o,T;/fl(ft)) < -Ro} where Ro > 0 is to be determined.
We define a map F from C into itself as follows: F(p, u) = (p, u) as defined
below. First of all, we solve
-^ + div(uffp) = 0 infix (0,T), p|t=o = Po in ft, (2.116)
where u£ is constructed from u as ue was from u above. Observe in
particular that u, € L2{Q,T]Ck(H)) for all k > 0, divu€ = 0 in Q x (0,T),
uc vanishes near 3ft (a.e. £ € (0,T)). The solution of (2.116) by classical
(and elementary) considerations on (divergence-free) transport equations
is given by a simple integration along "particle paths", i.e. solutions of the
following ordinary differential equation
j y
— = uc(X,s), X(s\x,t) =x, X €fi, t € [0,T\. (2.117)
In view of the properties of u€, there exists a unique solution X of (2.117),
continuous in (s,t) € [0,T]2, smooth_in x € Q such that d%X € C([0,T] x
Q x [0,T]) for all a and X(s;x,t) € ft for all (s,t) € [0,T]2, x € ft. Then,
we have
PM) = Po(X(0;x,t)), Vx€ft, V*€[0,r]. (2.118)
Obviously, £ < p < C0 in ft x [0,T],_p € C([0,T];C*(ft)) for all A: > 0 and
in view of (2.116) |f € £2(0,T;C*(ft)) for all jfc > 0. Furthermore, p and

68
Density-dependent Navier-Stokes equations
|2 are bounded in these spaces uniformly in (p, u) G C. In particular, the
set of p built in this way is clearly compact in C(Q x [0, T\).
We now build tx: first of all, we set \xt = /x(p)c with the same construction
as above and we wish to solve the following problem
P "ST + P«c • Vu - div (2ficd) + Vp = pfc in Qx (0,T),
eft
divu = 0 in Clx(Q,T), w|t=o = «o in fi.
u € La(0>T;H9(ft))nC([0>T];Ho1(n));
> (2.119)
0u > (2.120)
vp,^ € L2(nx(o,r)). '
This is nothing but an inhomogeneous (linear) Stokes equation with rather
smooth coefficients: the regularity of p, txc has been discussed above, \ic €
C°°(Tl x [0,T]), /ff € CS°(ft x (0,T]) and txg 6 C£°(ft), div tig = 0 in ft.
We postpone the discussion of this problem and admit temporarily that
there exists a unique solution tx of (2.119)^(2.120) (depending continuously
on data). This fact is established in Proposition 2.1 below. Then, when
(p, tx) G C, tx is bounded in L2(0,T; #2(ft)) while §* is bounded in L2(H x
(0,T)). Therefore, tx is compact in L2(0,T;^(H)). This shows that the
mapping F is compact on C.
Hence, if we wish to use the Schauder theorem in order to conclude the
existence of a fixed point, we have only to choose Rq in such a way that
IMlL^o.Ts/f^n)) — ^o- To this end, we multiply (2.119) by tx, integrate by
parts using (2.17) and obtain easily (all manipulations are justified by the
regularity of p and tx) for all t € [0, T]
I p—-(xyt)dx + fi / \Vu\2{x,s)dxds
< Co\\f\\LHQx{0,T))(j J \u\2(x1s)dxdsj
\l/2
hence,
SUp ||ti(t)||La(n) + ||ti||La(0fT;if0»(n)) ^ Ci (2.121)
t€[0,T]
where C\ depends only on /I, jf, C but not on i?o,p,u. We then choose
Rq > C\.
In order to conclude, we still have to show that a fixed point (p,tx)
is in fact smooth. This is easily done by a bootstrap argument that we
only sketch. First of all, we observe that txff 6 C([0,T];Cfc(fl)), p,/xff 6
C°'1/2([0,T];C*(ft)) for all k > 0, and using ^-theory (see V.A. Solon-
nikov [444],[445] for instance), or direct proofs similar to the proof of

Existence proofs
69
Proposition 2.1, we deduce from (2.119)-(2.120) that u e £P(0,T;
W2,p(^))> If 6 ^p(° * (°>r)) for all 1 < p < oo. With this
regularity on tx, we can bootstrap and gain more time regularity on uc then p and
thus more regularity (in (x,t)) on u.
Before stating Proposition 2.1 which fills the only gap left in the above
proof, we first observe that (2.119) may be written as
c^r +bi-Vu-aAm + P- = gi infix (0,T), 1 < i < N,
at oxi
div u = 0 in ft x (0, T), ti|t=o = u° in ft,
(2.122)
where g e L2(Q x (0,T)), c 6 L°°(ft x (0,T)), a e L~(0,r; W1'*^)),
-|f e-Ll(Q,T;L°°(n)), b e L2(0,r;L°°(ft)), c > <5, a > 6 a.e. on ft x (0,T)
for some 6 > 0, u° e #o(^)-
Proposition 2.1. There exists a unique solution u of (2.120)-(2.122).
Proof of Proposition 2.1. We only prove that the a priori estimates
contained in (2.120) hold. The proof will show the uniqueness of solutions,
and the existence follows in a straightforward way from a priori estimates
by standard arguments that we leave to the reader.
Next, in order to prove a priori estimates, we multiply (2.122) by ^,
sum over i and integrate (by parts) over ft to find for almost all t € (0,T)
. fldui* , If d |T_
7J*I dx+2LaTt^u
< f \9\\^\ + \b\\Vu\
Jn I ot I
2<*r
at
+ |Vo| |Vu|
3u
a*
dx.
Hence, using the Cauchy-Schwarz inequality, we find
6 f\du<2
< cfl + ||6||i-(n) + ||Va||i-(n) +
UMdx+12Jt(ia^2dx)
da
dt
L-(n)/ Jn
)/.|V»|'
dx.
Since 6 e Z2(0,r;L°°(ft)), Va e L2(0,r;L°°(ft)), ff e L^CTjL00^)),
we deduce from Gronwall's inequality an a priori estimate on u in L°°(0,T;
Hq(Q)) and on ^ in L2(ftx(0,T)) depending only on the datac,a,6,p,tx0.
In particular, we may then write (2.122) as
-aAu + Vp-h in ft, divu = 0 in ft, u€i/o(ft)
(2.123)
for almost alH e (0,T). In addition, h e L2(ft x (0,T)) and its norm is
bounded in terms of the data. From here on, whenever we say bounded, it

70 Density-dependent Navier-Stokes equations
means that the bound depends only on the data. From the previous bound
on u, we deduce in particular that
Vp = h + div (aVtx) - Va • Vu
and thus is bounded in L2(0,T; H l{Sl)). Therefore, if we normalize p by
imposing
/ pdx = 0, a.e. te (0,T),
Jn
we deduce that p is bounded in L2(Q x (0,T)) (see [472] for instance).
Then, we write (2.123) as a usual Stokes problem, namely
-Au + Vp=h infi" ~divti = 0 in Q, ueHi(Q), (2.124)
where h = J - 2*2 is bounded in L2(fi x (0,T)), p = f, and we
conclude that u is bounded in L2(0,T;#2(fi)) by classical regularity results
on Stokes equation (see [472] for example).
This completes the proof of Proposition 2.1 and of Theorem 2.6. □
Step 2. Passage to the limit. First of all, we collect a priori estimates
and follow the arguments of section 2.3. Since divu£ = 0, we immediately
obtain for all (3 6 C(1R, 1R)
/ (3(pe) dx = I P(p£Q) dx for all * 6 (0, oo). (2.125)
Jn Jn
Here and below, we denote by (tic,pc) the solution built in step 1 (observe
that u£ is the regularization of u6, namely (vf)€ *^c/2 in the case of Dirichlet
boundary conditions).
Next, exactly as in section 2.3, we obtain the analogue of (2.34)-(2.36),
namely
\\ue\\L2{Q)T;HHCi)) < C, (2.126)
sup \\P*\u*\2\\Lim < C (2.127)
0<*<T
where C denotes various positive constants independent of e.
Because of (2.110), we may then apply part 1 of Theorem 2.4 to
deduce that p€ converges, up to the extraction of subsequences, to some p in
C([0,T];L*(Q)) (Vl<p<co, VTG (0,oo)) which is bounded, satisfies
(2.125) with pe and p§ replaced respectively by p and po, and thus satisfies
(2.17). Furthermore, p satisfies (2.1) (and is periodic in the periodic case)
where u is a weak limit in L2(0,T; fT^fi)) (VTG (0,oo)) oiu*. Of course,
u is periodic in the periodic case and u 6 L2(0,T;Hq(Q)) in the case of
Dirichlet boundary conditions.

Existence proofs
71
In particular, this convergence implies, in view of the construction of /zc,
that we have
K^tip) in C([0,T];L*(ft)) 1
(Vl<p<oo, VTg (0,oo)), ase-O,/ K ' )
p£f€->pf in L2(ftx(0,T)) (VTG (0,oo)), as£->0. (2.129)
In addition, from the results shown in Appendix A, ue is also bounded
in tfiOtTiH1^)) and ue converges weakly in L2(0,T;Hl(fl)) to u—this
is obvious in the periodic case.
These bounds imply that (2.53) holds with q = 2, m = max(^r -1,1):
indeed, \itd? is bounded in L2(Q x (0,T)) while pctxc ® txc is bounded in
L2(0,T;ZP(n)) with pe [1,2) if N = 2, p = jfe if JV > 3, and thus is
bounded in L2(0,T; H~3(Q)) with 5 > 0 if N = 2, s = f - 1 if JV > 3.
We then deduce from part 2 of Theorem 2.4 that yfcfu* converges to
y/prn in Z*(0,T;Lr(ft)) for 2 < p < oo, 1 < r < ^^ and thus pffuff
converges to pm in Lp(0, T; £r(ft)) for the same (p, r).
These convergences allow us to recover (2.2) from (2.107) upon letting e
go to 0. In fact, we recover (2.12) (the weak formulation of (2.2)) provided
we show in the case of Dirichlet boundary conditions that
/ p%ul-<t>dx —► / rriQ'(j>dx, as e —> 0,
Jq Jq
for all 0 G C§°(Q)n such that div0 = 0. This is clear in view of (2.113)-
(2.114) since we have
/ PoV>Q-<t>dx= / mQ'<t>dx —► / niQ-cfrdx, as e —> 0,
Jq Jn Jq
for all <f> G L2(fi)N with div<£ = 0 in P'(fl), 0 • tx = 0 on dQ.
The only fact left in order to complete the proof of Theorem 2.1 is the
energy inequalities (2.13)-(2.14). This is in fact relatively easy since (pc,txc)
also satisfies some energy identities obtained as in section 2.3 by
multiplying (2.107) by u£ and integrating over £2, using (2.106) and the boundary-
conditions. We find then for all £ > 0
-£ f pe\u£\2dx + f pcidiul+djutfdx = 2 [ p€fe-u€dx. (2.130)
dt Jq Jq Jq
We have seen above that yfp*u€ converges in L2(Q x (0,T)) (in particular)
to yfpu, /ic converges in C([0,T];LP(£2)) (V 1 < p < oo) and is uniformly
bounded on ft x (0,cc), while f€ converges to / in L2(Q x (0,T)), for

72 Density-dependent Navier-Stokes equations
all T e (0,oo). This is enough to imply (2.13) provided we show for all
<p€C£°(0,oo), <p>0
(2.131)
lim / dt f dx tp(t) licip'Xdiu'j + djU?)2
c Jo Jn
> I dt dx(p(t) fi(p)(diUj + djUi)2. J
Jo Jn )
In order to show (2.131), we observe that we have
0< / dt J dxtpWticidiiu'j-u^ + djiul-Ui))2
Jo Jn
= / dt I dx <p(t) /xe(SiuJ + djul)2
Jo Jn
+ I dt I dx <p(t) ti£(diUj + djUi)2 dx
Jo Jn
- dt dx <p(t) fxe(diUj + djUi)(diUj + dju\).
Jo Jn
Since <p(t)(diUj + djUi)2 € Ll(Cl x (0,oo)) and fie is uniformly bounded
and converges in measure on Q x Supp(<p) to /x(p), we deduce easily that
<PHe(diV>j + djUi)2 and <p1/2 p,£(diUj + djUi) converge, respectively, to <pfi(p)
(diUj + djUi)2 in L1^ x (0,oo)) and to y>1/2/x(p)(ditij + 8^) in L2(Q x
(0,oo)). In addition, ^^(diUj +djU?) converges weakly in L2(Q x (0, oc))
to <pl/2(diUj + djUi). Therefore, the two last integrals converge, as e goes
to 0, to /0°° dt /n dx <p(t) /j,(p) p>(p)(diUj + djUi)2. This implies (2.131).
Next, in order to prove (2.14), we first integrate (2.130) between 0 and t
to find
/ pe\u£\2dx(t) + I ds J /MeidivZ + djulfdx
Jn Jo Jn
= 2 J ds J pef£ -uedx+ J pl\u%\2 dx
Jo Jn Jn
for all t > 0. Then, we observe that we have
J piWdx = [ ±\m'0-V4\2dx
J ci Po Po Po
Jn Po Po
(2.132)

Existence proofs
73
Since Uq = 0 on dCt (in the case of Dirichlet boundary conditions) and
div Uq = 0, we finally obtain
/ p$K|2 dx + f JM- to = / l2|L (far. (2.133)
•/n </n Po -/n Po
Since j^tn converges in L2(Q) to -%, we deduce (2.14) from (2.132)
l*V Po
exactly as before. a
Remark 2.9. In fact, it is often possible to sharpen a little the energy
inequality (2.14), replacing I212L. by p0|uo|2 for some u0 to be determined
satisfying div uq = 0. However, we cannot do it in full generality and we
have to make someassumptions on po-
The first case we can treat is when infessn Po > 0- Then, exactly as in
section 2.1, we can check that Uq converges in L2(Q) to t*o = -Pp0(mo); in
the case of Dirichlet boundary conditions, V% = mo — uo is determined by
the elliptic equation
*,(a=»)-o mn, v..i^l (2m)
(Vgo—™>o) • n = 0 on dfi, J
and it is clear that (2.14) holds (in fact for all t > 0 since u € C([0,T];L^)
for all T e (0,oo), see section 2.1) with 1~£L replaced by pol^ol2-
The second case allows po to vanish. For instance, we assume that fi is
connected, po = 0 a.e. on Q — a;, po > 6 > 0 a.e. in u; where 57 € fi, u
is smooth, and we only consider the case of Dirichlet boundary conditions.
First of all, we observe that |V«g| = ^-^ (Po)1/2 is bounded in L2(fi).
Next, we can normalize q$ in such a way that jdCi <j§ d5 = 0. Therefore,
if we extract subsequences if necessary, q$ converges weakly in H1 (Q) to
qo satisfying fd~ qodS = 0, V<?o = 0 on Q — u and thus qo = 0 on Q - u.
Hence, qo € H}(u). In addition, tzg is bounded in Lfoc(u) and we may
assume that tz§ converges weakly in L2(K) (V K compact C a;) to some
uo € Lfoc(u) such that po|^o|2 € £ *(<*;), and we have
[fWfdx+fKS^dx < f^-dx- l^-dx (2.135)
7a; 7a; P0 JQ P0 Ju; P0
mo = Pouo + Vgo in a;, div ( ) =0 in a;. (2.136)
v Po '
We next claim that there exists a unique qo € /Zq (uj) which satisfies (2.136)
(assuming that m0 € L2(u;), ^ 6 L2(u;)), and we have J^ ^^^x =

74 Density-dependent Navier-Stokes equations
h V(ft'm<? dx. If this claim were established, we would deduce that (2.135)
is in fact an equality and thus Po|tx<)|2, ? converge in Ll(Q) to po|uo|2,
' ^°l , respectively, where we extend these functions to ft by 0 outside u.
This is enough to conclude that (2.14) holds with po|uo|2 replacing l£2L.
In order to show the above claim, we have only to show that for any
solution qo € Hq (u) of
d.v / V^mjN _ Q
V On J
in u>
Po
we have Ju ^fjzL dx = fu v<?°Qm<? dx. Then, we multiply the equation by
?oC(f) where d = dist (x,dw), e > 0, C € C°°([0,oo]), £(<) = 0 if t < 1/2,
C(0 = 1 if t > 1, 0 < C(<) < 1 on [0, oo), and we obtain
f (Vgo-mp) • Vg0 ^^ A Vgo-m0 Vd ,/dN)
7W po W 7a, po e W
go<&: = 0.
It only remains to show that the second integral goes to 0 as £ goes to 0.
Indeed, we have
1/
VqQ — mn Vd ,/d
Po
<
va ,/a\
— C(-)qodx\
In fact, the above analysis can be extended to the case when p > ad(x)7,
p < /?d(x)7 a.e. in u for some a,/? > 0, 7 > 0. This is possible in view of
the following variant of Hardy's inequality
This inequality follows easily from the following computation: we have for
all feC%° (0,oo)
hence /~ j£, <te < ^/~ <££ dx. D
Remark 2.10. We observe here without proof that the existence and
compactness results can be extended to the case when po (> 0) is assumed

Uniqueness: weak =■ strong
75
to be in L*(ft) where p > 1 if N = 2 and p = f if W > 3. We still assume
that lEjL € L^fi) (m0 = 0 a.e. on {p0 = 0}). Then weak solutions are
defined exactly as in section 2.1 except that p € C([0,oo);Lp(fi)), and one
can adapt the preceding proofs to show that Theorem 2.1 holds in that
case D
Step 3. Existence in the case when Q = IR^. We use the existence
results we just proved with Q = Br and Dirichlet boundary conditions on
8Br. We then obtain approximated solutions (pr,ur) and we let R go
to +oo. More precisely, we denote by (pr,ur) a global weak solution of
(2.1H2.2) in Br where R € (0,oo) with the boundary condition (2.6),
restricting of course po> mo to Br. Recall that, as explained in Remark 2.1
(7), we may assume without loss of generality that Uoo = 0.
Next, we observe that all the estimates shown in section 2.3 in the case
when ft = JRN hold uniformly in R large. We may then apply
Theorem 2.5 to deduce the relative compactness of pR, y/pRUR and prur in
C{\Q,T\,L>{BM)) (K P < oo), IS{Q,T-Lr{BM)) (2 < p < oo, 1 < r <
Tfj*i) respectively (for all T € (0,oo), M € (0,oo)). Finally, this
compactness allows us to prove the existence of solutions in 1R letting R go
to +oo and using the same arguments as in step 2 above. □
2.5 Uniqueness: weak = strong
In this section, we show that any global weak solution coincides with a more
regular solution as long as such a "strong" solution exists. More precisely,
we prove that a weak solution is equal to a strong solution whenever the
latter exists. It is not difficult to check that smooth solutions exist for a
certain time interval—at least if po does not vanish—and the result that
we are going to present then implies that any weak solution is equal to the
smooth one on this time interval.
In order to simplify the presentation, we only treat the periodic case
and the case of Dirichlet boundary conditions even if similar results can
be obtained in the case when Q = IR^ by convenient adaptations of the
arguments below. We then consider a global weak solution u of (2.1)-
(2.2) and (2.6) as built in Theorem 2.1. We assume (for instance) that
/ € £2(0,T; L°°(n))jtnd fix T 6 (0, oo). We next assume that there exists
a solution p,tl G C(Q x [0,T]) (resp., in the periodic case,
periodic) of (2.1)-(2.2) in ft (resp. in 1RN) with Vu € I2(0,r;I°°(n)),
Vp € L2(0,r;L°°(n)), H € L2(0,T,L°°(Q))) and with u = 0 on dQ x
(0,T). Furthermore, we assume that jj. is locally Lipschitz on [0, oo) and

76
Density-dependent Navier-Stokes equations
that p, u satisfy
p\t=o = Po in ft, pu|t=o = m0 in ft.
(2.137)
Notice that this equality implies in fact that thq = pou(0) with divxx(O) = 0
in £2. Let us notice, that, of course, (2.2) holds with some pressure field p
that belongs to Ll(0, T; L°°(ft)) + £2(0,T; W^°°(Q)).
Theorem 2.7. Assume in addition that p ^ 0. Then we Lave tx = u a.e.
inftx(0,T).
Proof of Theorem 2.7. We first recall that we have for (almost) all
*€(0,T)
- / p\u\2dx+- / / nip^diUj+djUifdxds
* Jn * Jo Jn
^ / / pf -udxds + - I
Jo Jci 2 Jn
|m0p
Po
(2.14)
dx.
Next, we remark that, in view of the regularity of tx, we deduce from the
weak formulation (2.12) of (2.2) the following equality
/ pu-udx + - / / tx(p)(diUj+djUi)(diUj+djUi)dxds I
Jn 2 JQ Jn I
= / mo -u(Q)dx+ / / pf -udxds + / / pn • { — +u-VxZJdxds
(2.138)
a.e. * 6 (0,T). Then we write
du - ^
p — + pn - VxT - div (2/x(p)d) + Vp
= p/ + (p-/9) (~ +xZ - Viz) + p{u-u). VxZ - div(2(|i(p)-fi(p))d). J
(2.139)
If we first multiply (2.139) by tx and integrate over 1*2 x (0, t), we find
/ / r 757 +PU' VxZftzdrrds
2 / / MidiUj+djU^idiUj+djUi) dxds
I p/ • u + (p—p) f — + xZ • VxZj • u + p(u—u) • Vxl • u
+ - / / (/x(p)-/x(/^)(diUj^
zJoJn )
•fi
JoJn
} (2.140)

Uniqueness: weak = strong
77
Combining (2.138) and (2.140), and using (2.137), we obtain for almost all
t€(0,T)
/ pu-udx + / / /j,(p)(diUj+djUi)(diUj+djUi)dxds
Jn Jo Jn I
= f\sf+ffpt.v+lsf.u
Jn Po Jo Jn
+ (P"~p)("^~ +u ' ^0 ' U + P(u"^xx) • Vtx- uda:
ds
} (2.141)
+ - / / (p(p) - vipMdiUj+djUiXdiUj+djU^dxds.
1 JoJn '
Finally, we multiply (2.139) by u and integrate over Q x (0,t) to find
- / p\u\2dz + - / / v(p)(diUj+djUi)2dxds
* Jn l Jo J a
* Jn Po Jo Jn
'du
(2.142)
4- (p—p) ( "q7 + ^ • ^) * ^ + p(u—u) • Vu • H
+ 9^(p) -Mp))(^j+^i^)(^iSj+^tri)dxd5.
Then, if we add up (2.14) and (2.142) and substract (2.104), we obtain
- / p\u-u\2dx+- I I pWidiiuj-urf+djiui-Uitfdxds
< f •{u-u)(p—~p)dxds
Jo Jo.
1 JoJn
+ / / ^~^H"^7 +^' ^*V ' &~u) ~ p(u~u) • V(u-u)dxds.
Hence, we deduce from the assumptions made upon u that we have for
almost all t € (0,T) and for all € > 0
/ p\u-u\2dx + / / |V(u-u)|2cbds
,/n t JoJq \ (2.143)
< / / C{s)p\u-u\2 + e\u-u\2 + C£(s)\p-p\2dxds,
JoJci )

78
Density-dependent Navier-Stokes equations
where C,Ct denote various non-negative measurable functions in L1(0,T).
Next, we wish to estimate \\p - p||L2(n)- We write
T^(P"P) + div{tx(p-p)} = {u-u)-Vp
and deduce easily (see section 2.3 for related arguments) for all t € [0,T]
/ p\u-u\2 + \p-p\2 dx + I I \V{u-u)\2dxds
Jn Jo Jn
< f ds C(s) J p\u-u\2 + \p-p\2 + e f ds J \u-u\2 dx
Jo Jn Jo Jn-
(2.144)
Next, we observe that there exists e > 0 such that we have for all v e
H 1(to) and for all p € L°°(Q) such that /n pdx = /n p0 dx > 0, ||p|U*(n) <
\\Po\\l~(si)
£ f \v\2 dx <\ [ \Vv\2 dx + l f p\v\2 dx.
Jn 2 Jn 2 Jn
Indeed, if this were not the case, we would find vnipn satisfying
/ \Vvn\2dx+ f pn\vn\2dx - 0,
Jn Jn n
-f\vn\2dx=l, p0>0,
Jn
pn —>pw - L°°(Q) — *, / pdx = I podx. \
n Jn Jn '
Hence vn converges to 1 in H1^), and pn\vn\2 ^ P w — Zr!(fi). The con-
n
tradiction proves our claim.
Inserting the above inequality in (2.144), we then conclude that u = u,
p = p a.e. in Q x (0,T), by applying GronwalTs inequality. □
Remark 2.11. Modifying a little the above proof (using Sobolev's
inequality), one can extend the preceding result to the case when ^, /,
Vp e L2(0,T;LP(fi)), Vu e L2(0,T;L°°(ft)) where p = N if N > 3, p > 2
if N = 2. D

3
NAVIER-STOKES EQUATIONS
This chapter is devoted to the classical Navier-Stokes equations in the
homogeneous, incompressible case. The system, described in section 1.2,
can be deduced from (2.1)-(2.2) by setting p = p where pis a positive
constant and by introducing the kinematic viscosity v = /i(p)/p and a
reduced pressure field p/~p. We then obtain
du
-rr+u- Vu-vAu + Vp = /, divu = 0 in ftx(0,T) (3.1)
at
where T > 0 is fixed and / is given on CI x (0,T). Of course, (3.1) is
complemented with boundary conditions (the same as in chapter 2) and an
initial condition
u\t=o = uo in Q. (3.2)
Without loss of generality—otherwise we simply subtract a gradient term
from u—we may always assume that we have
divuo =0 in fi. (3.3)
3.1 A brief review of known results
We begin with the celebrated results due to J. Leray [283] (see also [472],
[293] and the bibliography for more references on the subject) concerning
the global existence of weak solutions. In order to simplify the
presentation and notation, we denote by Hl {Hs, W771^) the usual Sobolev space
Hl(JRN) in the case when Q = JRN, or H^T = {u e Hj^QR*), u periodic}
in the periodic case and by H~l the dual space {H~s, W~m'p'). In the
results which follow, we assume
no e I2(ft), / 6 L2(0,T;H~l). (3.4)

80
Navier-Stokes equations
In the case of Dirichlet boundary conditions, we assume in addition
u0-n = 0 on dQ. (3.5)
Recall that (3.5) is meaningful since (3.3) holds and u0 G L2(Q) (hence
uq - n € H~l/2(dQ)). Again this is not a restrictive assumption since we
can always decompose (uniquely and continuously) any Uq € LP(Q) into
a gradient term (in 1^(0)) and a divergence-free vector field in Z^(ft)
satisfying (3.5) (for all 1 < p < oo).
In the case of Dirichlet boundary conditions, we need to introduce some
functional spaces for rather delicate reasons to which we shall come back
in detail later on. We set for 1 < p < oo,
V°>P(Q) = {u 6 Lp(ft), divu = 0 in fi, u • n = 0 on dQ}
Vl>p(Q) = {u e W^P(Q) , divu = 0 in ft},
and we recall that f>(Q) = {<p e Co°(ft), div<p = 0 in Q,} is dense in
V°'p(ft), Vl>p(Q) respectively for the Lp, Wl* norms. Finally, we denote
by V~l*p the dual space of Vx* where £ + ^r = l(l<p< oo).
Next, we recall the weak formulation of (3.1) as given in chapter 2 for a
more general system without checking that all terms written below make
sense, since this point will be a straightforward consequence of the
regularity we assume for weak solutions: we request that we have for all
(p e C°°(Q x [0,T]) such that div<p = 0 and with compact support in
nx[o,T)
rT
j dtdx I vVu • Vy? — UiUjdi<pj — u • — \
to Jn
rT
= / <f,<p>H-*xHidt+ I uo-<P, divu = OinP/(nx (0,T)).
Jo ° Jn
(3-6)
In the periodic case (or in the case fi = IR ) we replace Hq by Hl and <p
is then assumed to satisfy: (p G C°°(JRN x [0,T]), divtp = 0, tp is periodic
in x for all * e [0,T].
This formulation implies that (3.1) holds in the sense of distributions
for some pressure field which is a distribution. Observe also that the term
UiUjdj<pj can be written as —UjdjUiipi (as soon as Vu € L2(£l x (0,T)),
u 6 L2(Q x (0,T)) for example).
In the case of Dirichlet boundary conditions, (3.6) is also equivalent to
a more abstract formulation involving the spaces VliP(Q) (and V~liP).
As we shall see, the weak solutions satisfy: u e Jj2(0,T;71?2(n)), u 6
L°°(0,T;L2(ft)), \u\2 € L2(0,T;L^(ft)), and thus (3.6) implies that

A brief review of known results
81
f* € L2(0,T; V1'*^) (in fact §j* should be written v! since it is
considered as a (time) derivative of a function with values in some Banach space),
and (3.6) is then equivalent to
u + vAu + B(u, u) = /
where f,Au € L2(0,T; V"1'2), £(ti,ti) € L2(0,T; V1'^) are defined
</,v>=</,t; >fr-ixjsrs, v*> 6 V1,
<w4tx,t;>= I Vu-Vvdx, VveV1,2
<£(u,tx),t;>= (u-V)u-vdx = - / UiUjdiVjdx, VvG^1'^.
Observe finally that, since ft is bounded, V1,7V c V1,2 and we deduce that
V"1*2 C V~l'T&* (identifying L2 with its dual as usual).
We now state some global existence results of weak solutions: the first
two results concern the case when Q = IR (or the periodic case) in two
dimensions (N = 2) and in dimensions N > 3 respectively, while the next
two results are devoted to Dirichlet boundary conditions with N = 2 or
N > 3 respectively.
Theorem 3.1. (N = 2, Q = IR2 or the periodic case). There exists a
unique weak solution u of (3.1 )-(3.2) with the following properties: u €
L2(0,r;fr1)nC([0,7l;L2), f € ^(O.Tjtf-1).
Furthermore, there exists a unique p € L2(Br x (0,T)) (for all R e
(0,co)j such that Vp € I2(0,T;.H'"1), /Qpdx = 0 a.e. t € (0,T) where
Q = Bi for example if Q = TR2 or Q = Q in the periodic case, and such
that (3.1) holds in the sense of distributions.
We have for all t € [0,TJ
- / \u(x,t)\2dx + v / / \Vu\2dxds
2 Jn Jo Jq v
= 2 / \uo\2dx+l <f(s),u(s)>H-ixHids. I
(3.7)
Theorem 3.2. (TV > 3, ft = IR^ or the periodic case). There exists
a weak solution u of (3.1)-(3.2) and a pressure field p such that (3.1)
holds in the sense of distributions, and the following properties hold: u 6
I2(0,T;fr1)nC([0,ri;Li)nC([0,ri;L'(Bjl)) (VI < s < 2, \f R 6 (0,oo)),
§* € ^(O.Tjtf-1) + (L3(0,T; W1'*^) nI*(0,T;i7)) /or 1 < 5 < oo,

82 Navier-Stokes equations
1 < g < 2 and r = j^, p € L2(BR x (0,T)) + La(0,T;L^) for
1 < 5 < oo, R € (0,oo), Vp € L2{Q,T\H~l) + L<(0,T;i7) for 1 < g < 2
and r = ^^_2; and we have
i / |tz(x,i)|2dx + i/ / / \Vu\2dxds
2 7n ./o in .
^« \uo\2(ix + <f,u>H-1xH*ds, forallt>0.\
(3.8)
|Q/Kx,t)|2dx)+i/^|Vtx|2dx <<f,u>H-*xH* inV{Q,T).
(3.9)
Rzrthermore, if iV = 3, there exists a solution satisfying in addition
!(i|u|*)+div(u{i|u|>+p})
-uA^-+u\Vu\2 < u/ in V'.
(3.10)
Theorem 3.3. (N = 2, DirichJet boundary conditions). There exists a
unique weak solution u of (3.1)-(3.2) which satisfies: u € L2(0,T; Hq(CI))
(orequivalentlyu € L2(0,T; V1-2);, u 6 C([0, T]; X2), §* 6 L2(0, T; V"1-2),
and (3.7) holds.
Theorem 3.4. (N > 3, Diricblet boundary conditions). There exists
a weaJc solution u of (3.1)-(3.2) satisfying (3.8),(3.9) and such that u €
L2(0,T;H£(CI)), u 6 C([0,T];L2,) n C([0,T];L*(fl)) for alJ 1 < s < 2,
§* € L2(0,T;Vr-1'2) + (LJ(0,T;Vr-1'*7)nL«(0,T;Lr)) fori < s < oo,
l<g<2andr=77^.
Remarks 3.1. 1) We do not claim any originality in the results presented
above except the slightly unusual presentation and, maybe, some regularity
(or partial regularity) information on ^ and p. The results presented in
Theorems 3.1 and 3.2 are essentially contained in J. Leray [283],[284],[285]
and further references can be found in R. Temam [472], J.L. Lions [293]
and the bibliography for instance.
2) Let us remark that u{^ \u\2+p} makes sense in (3.10) (since we know
that, for instance, \u\2 e L3/2(0,T;L9/5),p e L3/2(0,T;L9/5)+L2(0,T;L2)
while u e L3(0,T; L18'5) n L°°(0, T; L2) (by Sobolev embeddings) and f +
yg = j| < 1. The meaning of uf in (3.10) has also to be clarified: since
/ 6 Z/^Tjtf-1), u e L2{Q,T\Hl), uf is the distribution defined by
<uf,(p> = <f,uip> observing that mp € Hl for smooth test functions <p.

A brief review of known results
83
3) We shall see below—see also sections 3.2 and 3.3—further regularity
properties of weak solutions. Let us recall however that uniqueness of weak
solutions is an outstanding open problem (even for more regular data / and
uq). It is clearly related to regularity issues: in particular, if we postulate
more regularity on the weak solutions, the uniqueness follows. It is possible
to show (this result is due to J. Serrin [428]) that if there exists a more
regular solution then the weak solution coincides with this one. These
results of the type "weak = strong" are very much in the same spirit as
the one shown in section 2.5. However, the optimal—possibly the full—
regularity of global weak solutions is not known: for instance, if N = 3,
Q = H3, / = 0, uq is smooth, results due to J. Leray [283],[284],[285], L.
Caffarelli, R.V. Kohn and L. Nirenberg [77] show that weak solutions are
smooth except for "small sets" (zero one-dimensional HausdorfF measure
in [77]) containing the possible singularities. The solution is also known to
be smooth for t small and for t large, and for all t if uq and / are small in
appropriate spaces. Of course, if N = 2, much more is known: for instance,
if u0 6 Hl(fl) (or tf^IR2)), / 6 L2(Q x (0,T)) then u 6 L2(0,T; H2(Q))n
C([0,T\]H&(O)), §£ 6 L2(Q x (0,T)) and p e I2(0,T; Hl(fl)), and if / is
smooth, u is smooth for t > 0.
Another topic which has been extensively studied concerns the space of
initial conditions tzo (take / = 0, Q = JR3 for example) in which there
exists a (unique) solution for t small or a global small solution. The most
general result in that direction is probably the result of M. Cannone and
Y. Meyer [80] which states that if N = 3, Q = Ut3, u0 € X3, / = 0, u0 is
small in B& ' then there exists a global solution u € C([0, oo); L3), and
the solution is then automatically C°° for t > 0 as we shall see below.
Let us finally mention that some marginal improvements of the regularity
of Ut and p shall be given in section 3.2 below, particularly in the case of
Dirichlet boundary conditions.
4) We wish now to explain some of the difficulties encountered in the
case of Dirichlet boundary conditions. First of all, let us remark that the
information contained in Theorems 2.3-2.4 on |jj* does not say much if we
insist upon looking at ^ as a distribution. Indeed, if we can check that
V~l*(fi) is W-l*(n)/{Vq/q € £p(fl)}, then % e L2(0,T; V"1'2^))
(say if N = 2) does not imply $ € L2(Q,T\H-l(Sl)). In fact, even if
N = 2, we do not know if §j± € L2(0,T;^-1(fi)). Of course, there is a
distribution tt such that §j± - Vtt e £2(0,T; H'l(Q)), and it is possible to
choose 7r harmonic (in x): indeed, we can write
du
— = -AC/ + Vtt, div*7 = 0 in ft, U = 0 on dQ. (3.11)
The regularity stated in Theorems 3.3 and 3.4 yields:

84
Navier-Stokes equations
U € I2(0,r;i7o1(^))+(^(0,r;W01,lfe(fi))nL«(0,r;W2'r(fi)))
(3.12)
for 1 < s < oo, 1 < q < 2, and r = Nq£_2- ^Tom th*s we deduce that
g_V7reL2(0,T;tf-^^
and we finally observe that (3.11) implies An = 0. In other words, up to
a gradient harmonic distribution, |^ has the regularity we expect. From
(3.1) (which holds for a certain distribution p), we deduce that Vp has also
the same properties as ^ ("natural regularity" up to the gradient of an
harmonic function).
This question does not arise when Q = IR or in the periodic case since
if 7r is harmonic and V7r e H~l (and periodic in the periodic case) then
Vp = 0. A related issue is the possibility of projecting (3.1) on divergence-
free vector fields. Let us recall that we denote by P the projection (in L2
say) onto divergence-free vector fields, i.e.
p = Id + V(-A)-Xdiv.
Both in the periodic case and in the case when Q = IR , P commutes with
translations and thus with derivatives so it is bounded (and a projection)
from any H3 into H3 (s e IR). In fact, since P is bounded in LP (1 < p <
oo), P is also bounded in any H3iP (s € IR, 1 < p < oo). Then, we can
write in those two cases
or equivalently
^ + P((u • V)n), - uLui = P(f)i (3.13)
^ + 4~ (P("jUi)) ~ "Aui = P{f)i. (3.14)
Cft UXj
On the other hand, when we consider the case of Dirichlet boundary
conditions, it does not seem to be possible to write (3.1) in such a concise
form. The natural replacement for P is the orthogonal projection (in L2)
onto divergence-free vector fields v € L2(Q)N such that v • n = 0 on dQ—
recall that if v e L2(n)N, divv e L2(Q) then v • n e H'l^2(dQ). Then
P is also bounded in LP(Q) for 1 < p < oo but it no longer commutes
with derivatives. In fact, it is easy to check that P is bounded in Wl>p(Q)
for 1 < p < oo but P does not leave WqiP(Q) invariant (we only obtain
P(u)-n = 0 on dQ and not Pu = 0 on dQ). In particular, we cannot deduce
(by duality) any information on P in W"1'P(Q) including a definition.
The fact that P(u) does not make sense if u € H~l(Q) (say) can be
seen from the fact that P(—Au) does not make sense if u € Hq(Q) even if

A brief review of known results
85
divtx = 0 in Q. Indeed, if u € H2(Sl) n H^(Q), then P(-Au) € L2(ft) and
we claim that P(—Au) is not bounded in H~l if tx is bounded in H^(Q)
even if divtx = 0 in Q. To prove this claim, we argue by contradiction and
thus assume that P(—Au) extends by continuity to a continuous map from
H$(Q) into H-l(Q). Then, let u e Jf2^) n-ffj(fi) be such that divtx = 0,
Au-n £ 0 on dQ (Au € L2(Q), div (Au) = 0 in Q so that Atx-n makes sense
in H~l/2(dQ))] examples of such a u are not difficult to build. We next
choose un e Cq?(Q) such that un converges to u in H£(Q) and divun = 0
in Q. Clearly, P(-Aun) = ~Aun converges in H~l(Q) to -Au while, by
assumption, it should also converge to P(-Atx), and we reach the desired
contradiction since Au - n ^ 0 on dQ and thus P(—Au) ^ — Au.
This argument shows that P(-Au) defined on H2(Q) n K1,2(fl) cannot
be extended by continuity to Vl>2(Q). Of course, even if it does not seem
a natural thing to try, we might attempt to define P(—Au) on Vl>2(Q) in
a, different manner using the orthonormal basis of V0f2(Q) composed of the
eigenfunctions Wi (i > 1) of the Stokes operator, namely
-Awi + V7Ti = Xi w{j Xi € 1R, w{ € Vll2(Q)
where Xi (> 0) are the eigenvalues. The set {wi / i > 1} is an orthogonal
basis of Vl'2(Q) and we have
OO m OO
u = ^UiWi, / \u\2dx = ^\ui\2 for alU € V°'2(Q),
i=i Ja i=i
/ \Vu\2dx = YXi\ui\2 foralUeK1'2^),
where Ui = fQuWi dx.
Then, a possible attempt to define P(—Au) is to consider the limit (if
it exists) of P(—A(^s=i ^i^i)) as N goes to +oo or in other words the
limit of £i=i ^i^i^i since P(—Aw^ = Ai^. This is not possible: indeed,
arguing by contradiction, if X)i=:1 KuiWi converges as N goes to +oo in
some space, say H~l(Q) (we could also assume that it stays bounded in
H~l (Q) and with a little more work we would reach a similar contradiction)
for all u € Vl'2(Q), then YliLi ^iu%^% converges to T(u) where T is a linear
mapping continuous from Vl*2(Q) into H~l(Q). We claim that T(u) =
P(-Atx) if u 6 #2(fi) n Vl'2(Q): if this claim is proven, we conclude easily
in view of the fact shown above. Then, if u € H2(Q)nVl*2(Q), Au e L2(Q)

86
Navier-Stokes equations
and thus
n N r N r
y^XiUiWi = S^Wi uXiWidx = y^t^j / (-Au)wjdx
tA tA Ja iA J<>
* r
in L2(Q) as JV goes to +00.
The specific difficulties encountered in the case of Dirichlet boundary
conditions are intimately related to the simple observation already
mentioned above that there exist non-trivial (non-constant) harmonic functions
in L2\ Indeed, there exists he L2, his harmonic in ft: hence, V/i e H ~l(Q)
and div (V/i) = 0. In the periodic case we immediately see that h is
constant and thus V/i = 0. If Q = IR^, T € ff"1, curlT = 0, divT = 0 in
V(1RN) then we also obtain T = 0.
5) The energy inequality (3.8) shows that u(t) converges to uq in L2(Q)
as t goes to 0+. □
We now sketch the
Proof of Theorems 3.1—3.4. First of all, the existence of weak solutions
is a particular case of Theorem 2.1 taking po = 1 and thus, by (2.17),
p = 1. Notice that Theorem 2.1 also yields the energy inequalities (3.9)
and (3.8) for almost all t > 0. The fact that (3.8) in fact holds for all
t > 0 is then a simple consequence of the continuity in time (with values
in Llj) of u. Let us remark that the fact that u € C([0,T];Z^) is a
consequence of Theorem 2.3 since P(u) = u. Also the continuity in t with
values in L\oc for p < 2 (or IP in the periodic case or in the case of Dirichlet
boundary conditions) follows upon decomposing u into u\ + U2 where u\
solves: % - ^Atxi + Vpi = /, divtxi = 0, m € C([0,T];L2), txi|t=o = u0;
and where tx2 solves: ^ — 1/AU2+VP2 = — (tx-V)tx, div 1x2 = 0, tX2|t=o = 0>
tx2 € W2'1'^^ x (0,T)) (see V.A. Solonnikov [444],[445] and section
3.3 for such estimates), therefore u2 € C([0,T]; W*^'##(Q)). It only
remains to explain the additional regularity information on ^, Vp (step
1), the uniqueness statements if N = 2 (step 2), and the local energy
inequality (3.10) (step 3).
There is however one more point to clarify: what we claimed above
about the applications of Theorem 2.1 is not entirely correct since we
need to assume that / e L2(Q x (0,T)) in order to apply Theorem 2.1
while the results above only require that / e L2(0,T; Jf-1(fi)). The
reason why we neglect this technical point is the following: when p is
constant, say p = 1, then all the a priori estimates and passages to the

A brief review of known results
87
limit are valid if we only assume that / € £2(0,X;i7 1(f2)) and thus
the proofs already given easily adapt to that case. Another way to
argue is to approximate / in L2{Q,T\H"l{9)) by fn e L2(Cl x (0,T)). We
then apply Theorems 2.1 and 2.3 and obtain weak solutions un, which
as we shall prove below satisfy the properties listed in Theorems 3.1-3.4.
Finally, we recover the desired results passing to the limit as n goes to
+oo.
Step 1. Regularity information on |jp Vp. In the periodic case or if
Q = JRN, we simply use (3.13) or (3.14) (which are easily deduced from the
definition of weak solutions). If N = 2, we recall that u e L4(f2x(0,X)) and
thus P{ujUi) (Vz,j) e L2(ftx(0,T)), hence §£ € L2(0,T; fT"1). Of course,
this yields the regularity statements made upon p and the continuity of u
in time with values in L2. This also allows us to justify (3.7). If N > 3, we
remark that u 6 L°°(0,T;L2) n L2(0,T;L^) (by Sobolev embeddings)
while Vu 6 L2(Q x (0,T)). Therefore, u-Vue L«(0,T;Lr) for 1 < q < 2,
r = t$^- while u • VtX; (V i) e £*(0,T; W~l> *&*) for 1 < s < oo since
u • Vi*i = div(uui). The regularity for ^ then follows from (3.13) and
(3.14).
The regularity of p stated in Theorem 3.2 is deduced from (3.1) in the
following way: we take the divergence of (3.1) and we find
-Ap = diiujdjUi) - div/ = diUj djUi -div/
= dijiuiUj) - div f in IR" '
and in the periodic case p is periodic—p can be normalized by requesting
that fQpdx = 0 where Q is the periodic cube or Q = j?i if fi = IR^. The
regularity of p then follows from elliptic regularity and from the bounds
on UiUj, (u • V)ui (1 < i, j < N) obtained above. Of course, we could
also obtain the regularity of Vp from equation (3.1) in view of the bounds
shown above on (u • V)tz and ^.
In the case of Dirichlet boundary conditions, the argument for ^ is the
same except that, for reasons detailed above, we have to replace W~l'p by
Step 2. Uniqueness if N = 2. We only need to observe that if u,v €
L°°(0,T;L2)nL2(0,T;Hl) and thus u,t/ € L4(Q x (0,T)) and divv = 0
a.e. in Cl x (0,T), we have for all t € [0,T]

88 Navier-Stokes equations
dx ds [(u • V)u - (v • V)v] ■ (tx-v)
7n Jo
= / dx / ds [(u—v) ■ Vtx + (v • V)(u-v)] ■ (u-v)
Jn Jo
= [dx [ ds[(u-v)-Vu]-(u-v) > - / ||Vtx||L2 ||tx-t;||24d5
> -Co / ||Vti||L2||ti-t;||La||V(ti-t;)||L2d5
</o
for some C0 > 0 independent of u, v. Then, if u,v are solutions of (3.1)-
(3.2) as in Theorem 3.1 or 3.3, we deduce easily from the above inequality
(and the regularity of u, v, §j*, ff) for all t > 0
^llti-vllLW + ^jT1 II V(u-u)||£2d*
< C0 / ||Vti||L2||ti-t;||L2||V(ti-t;)||L2d5
«/o
and the uniqueness follows from Gronwall's inequality.
Step 3. The local energy inequality (3.10). In order to show (3.10),
we go back to the construction of weak solutions performed in section 2.4
in our special case, namely p = 1. In other words, we consider ue G
C°°(TRN x [0,T]) (vanishing at infinity if fi = IR^, periodic in the periodic
case) as a solution of
-^ + (^ * we) ■ Vue - uAue + Vj>£ = U in fft* x [0,T], 1 i
divu£ =0 in IR^ x [0,T] J
ue|t=o = ti0*we in IR^ (3.17)
where fs € C°°(TRN x (0,T)) converges to / in L^O.TjiJ-1), /e
vanishes near t = 0 (V x), fe is periodic in the periodic case and fe €
CS°(IRN x (0,T)) if ft = IR^. Let us emphasize that this is essentially
the original approximation of (3.1) introduced in J. Leray [283]. We also
know that uc converges weakly—extracting subsequences if necessary in
L2{0,T\Hl)—to a weak solution u satisfying the conditions listed in
Theorem 3.2. In addition, us converges to u in 2/(0, T; 2/(I?r)) for 2 < s < oo,
q < j77^4 and for all R € (0,oo). In particular, |ue|2 converges to \u\2 in
2/(0, T; !*(£*)) for 1< r < oo, 1 < g < ^, R € (0,oo).

A brief review of known results
89
If N = 3, we deduce that |uc|2 converges to \u\2 in L$(0,T',Lq(BR)) for
q < | while ue converges to u in L3(0, T; £*(£*)) for g < f (V il € (0, co)).
Hence, (uc*uc)\uc\2 converges to u\u\2 in L1{Brx (0,T)) for all .R 6 (0, co).
In addition, the bounds obtained in step 1 on ^ and p are easily shown
to hold for (3.16) and are uniform in £ € (0,1). In particular, pe converges
weakly to p in L2(0,T\Lq{BR)) for q < § while uc converges to u in
L2(0,T;L<*(BR))foTq<6.
Next, we multiply (3.16) by uc and we obtain on TRN x [0,T]
|(i|tz£|2)+div((u£*u;c){i|u£|2+pc})-^^ + HVU£|2 = ««/..
Without loss of generality, we may assume that |Vtxc|2 converges weakly
(in the sense of measures) to a bounded non-negative measure D on IR^ x
[0,T]. By standard functional analysis considerations, we deduce that D >
j/|Vtx|2. This fact, together with the convergences established above, allows
us to pass to the limit as e goes to 0 in the above equality to recover the
inequality (3.10), thus concluding our proof. □
We conclude this section with an observation on the regularity of
solutions of Navier-Stokes equations if N > 3: we postulate the existence
of weak solutions u of (3.1)-(3.2) with the properties listed in the above
existence results and such that u € C([0,T];LiV(n)). The result that
follows shows that if / is smooth then u is smooth for t > 0. More precisely,
we have the following classical result
Theorem 3.5. Let N > 3, let f e L2(Ct x (0,T))nLr(flx (0,T)) for some
r € [2,iV). Let u be a weak solution of (3.1)-(3.2) as given by Theorem
3.2 or 3.4. We assume that u e C([0,T]; LN(Q)). Then, for each e > 0,
u 6 L^6,T]W2^(n))y p 6 L*(e,T;Wl«(n)) and % e L*(£,T;L*(Q)) for
2<q<r.
Remarks 3.2. 1) If uo is smooth enough (in the appropriate Besov space),
the argument below shows that we can take e = 0 in the preceding result.
2) If / is smooth, then it is straightforward to deduce from the above
result the fact that u is smooth for t > 0.
3) The proof presented below can be adapted to the case when u is
assumed to satisfy u e £a(0,T; L0{Q)) where 2 < a < oo, (3 = ^ (or even
more generally Lai(0,T;L^(fi)) + L^(0,T;L^(Q)) + C([0,T];LN(Q))
where 2 < ai,c*2 < oo, fa = ^z^, i = 1,2). The above result remains
valid.
4) It is possible to extend slightly the above result by requiring u to satisfy
the following property: for all s > 0, u = U\ +112 where ||ui ||z,~(o,r;L".~ (n))
< e, u2 6 L°°(Q x (0,T)). □

90
Navier-Stokes equations
Proof of Theorem 3.5. We first observe that for all e > 0 there exist
(1x1,1x2) such that
u = m +tx2, l|tti||Loo(0|T;LN(n)) < ** w2 € L°°(ft x (0,T)), (3.18)
and we may even assume 112 to be smooth on Q x [0, T) (periodic in the
periodic case or in Cq°(Q, x (0,X)) in the other cases).
We next wish to make a few remarks on the following linear equations
— + u • Vv - vAv + Vp = g,
divv = 0 inftx(0,T), t;|t=0 = 0 in ft J
(3.19)
with the same boundary conditions for v as for u. We first claim that if g €
Lr(fix (0,T))C\L2(Q x (0,T)) for some r € [2, N) then there exists a unique
solutionv € L*{0,T;W2>*(n)), § 6 L«(0,r;L«(n)), Vp€ L«(0,T;L«(fl))
of (3.19) for all q € [2,r]. The existence (and uniqueness) follows from the
a priori estimates we explain now. First of all, we have (multiplying by v)
a priori estimates in L2(0, T; H1) n L°°(0, T; L2). Next, we remark that we
have by Sobolev embeddings and because of (3.18)
H-Vv||L,(nX(o,T)) < e||Vv|| , J^-fnj\ ^ c4D2v\\Li(nx(o,T)),
ll«2-Vv||L,(nX(o,T)) < CV||Vv||L,(n)
^ ^rllvIll,«(nx(0,T)) H-^ VHL«(nx(0,T))
1 — 0
^ ^'«llvllL2(nx(0IT)) HVHL'(nx(0,T)) 11"^ VWl"(CIx(0,T))
< CellvIII»(nx(0IT)){ll^llLf(nx(0IT)) + IP^IU'^x^.T))}
1-7
where, above and below, C denotes various positive constants independent
of v, e,q in [2,r], Ce denotes various positive constants independent of v, q
in [2, r], 7 = f 6 (0, j], f + ^ = J, 5 = ^ (if * < ^, s = +oo if
9 > ^, s arbitrary in (2,oo) if q = ^).
Then, we use Lq(Q x (0,T)) estimates for linear (Stokes) equations due
to V.A. Solonnikov [444],[445] (which are in fact valid in all dimensions)
and we deduce
\\v\\Li(0,T;W2<i(n)) +
dv
Hi
L«(nx(0,T))
+ l|Vp||L«(nx(0,T))
< Ce||i?2t;||L,(nx(o,T)) +ccWv\\lHnxiotT))
\dv\
\Li(Clx{
{II Hi IL*(nx(o,T)) + llZ)2vl^'(nx(°.T)) J + c\\9h«(nx(o,T))

A brief review of known results
91
Since we already have a priori estimates on v in L?(Q x (0, T)), the desired
a priori estimates are shown.
The next step consists in showing that there exists a unique weak solution
(€ L2{0,T\Hl) n Z°°(0,T;Z2)) of (3.19) or in other words that if q = 0
then v = 0. To this end, we first observe that u • Vv 6 £2(0,T; £">$+*)
and thus §jf € L2(0,T;H~l) (or I2(0,T; V"1-2) in the case of Dirichlet
boundary conditions). Therefore, v € C([0, T];L2) and we multiply (3.19)
by v to obtain for all t € [0, T]
I \\v{t)\2dx + u f f \Vv\2dxds
J a 2 Jo Jn
= - / / (u'Vv)-vdxds = - / / u-V^-dxds = 0,
Jo Jn Jo Jn 2
the last computations being easy to justify since u € C([0, T];LN), u • Vv 6
Z-^O.TjZ,^1), v € L2(0,T;L^) (argue by density on v for instance).
We may now complete the proof of Theorem 3.5 by observing that v\ =
tu {pi = tp) solves (3.19) with g = tf+u € Lri{Qx (0,T))nL2(Q x (0,T))
with rx = min(p,4) hence V! € £*(0,T; W2«(n)), Vpi 6 L«(fl x (0,T)) for
2 < 9 < ri- If p < 4, we conclude, while if p > 4 (hence iV > 5), we observe
that v2 = t2u solves (3.19) with g = t2f + 2vt € Lr»(ft x (0,T)) n L2(12 x
(0,T)) while r2 = min (p, i^i) if iV > 7, r2 = p if iV = 5 or 6 (since
«i 6l4"(flx (0,T)) by the regularity just established). If p < 4^,
we conclude. If p > 4^| (hence N > 11), we consider vz = <3tx and
reiterating the preceding argument, we prove Theorem 3.5. □
The same type of technique can be used to prove the regularity of weak
solutions of stationary Navier-Stokes equations if N = 4 (such results are
classical if N = 2 or 3). More precisely, we consider stationary weak
solutions of Navier-Stokes equations, in the case of Dirichlet boundary
conditions to fix ideas, namely solutions of
- t/Au + (u • V)u + Vp = / in ft, 1
ix ? (3-20)
u€H£(Q), divtx = 0 in a J
Then, if / € H~l(Q), there exists at least one solution u of (3.20) (see
for instance R. Temam [472]). We claim that if N = 4, any such solution
belongs to W2^(Q) if / € L*(H) and q € [t$£j, JV), and in addition Vp €
Lq(Q). In particular, if this claim is shown, then / € L°°(Cl) implies
u € W2><>(n) for q<N and thus u € Ca(Q) for all a 6 (0,1). By standard
regularity results, we then deduce u € W2<q(Q) for all q < oo, and if
/ € Ca(ft) (a € (0,1)) then u € C72>a(fi).

92
Navier-Stokes equations
In order to prove the above claim, we argue as in the proof of Theorem
3.5 and we remark that, by Sobolev embeddings, u G L4(Q) (recall that
N = 4) hence for all e > 0
u = ui + tx2, u2 e £°°(ft), ||tii||L4(n) ^ £-
Next, for any p 6 Lr(fi) fl L7**5'^), there exists a unique solution t; of
- v&v + (u • V)v + Vp = g in fl,
veHi(Q), divv = 0 infl
and v 6 W2'«(ft) for g e_[^r], where r € [^,oo).
The proof of this claim relies upon the above decomposition of u and
follows the same lines as the corresponding argument in the proof of Theorem
3.5. In particular, taking g = /, r = g, u has to be this unique solution
and thus has the claimed regularity.
3.2 Refined regularity of weak solutions via Hardy spaces
In this section, we review some results due to R. Coifman, RL. Lions, Y.
Meyer and S. Semmes [95] which concern some (marginal) improvements of
the known regularity of weak solutions. They rely upon multi-dimensional
Hardy spaces and they are valid in the periodic case or in the case when
Q = JRN. We shall discuss corresponding results in the case of Dirichlet
boundary conditions in the next section. In order to simplify the
presentation, we only consider the case CI = IR^ since the adaptations to the
periodic case are straightforward.
We first recall the definition and some of the main properties of Hardy
spaces introduced by E. Stein and G. Weiss [455] (for more facts on these
spaces see C. FefFerman and E. Stein [149], R. Coifman and G. Weiss [96]).
The Hardy space, denoted by W1(IRiV) to avoid confusion with Sobolev
spaces, is a closed subspace of Ll(JR ) defined by
nl(lRN) = lfeL\TR.N)/8up\ht*f\eLl(TRN)\ (3.22)
where ht = $r Mt)> h e Cq°0RN), h>0, Supp/i C 5(0,1); in fact, it can
be shown that this space is independent of the choice of h. Also, 7il can
be characterized in terms of Riesz transforms Rj as
Hl{TRN) = {feLl(JRN)/Vl<j<N1RjfeLl(JRN)}. (3.23)
(3.21)

Refined regularity of weak solutions via Hardy spaces 93
In addition, we have
Rj is bounded from Hl into ft\ (3.24)
provided we equip ft1 with a norm taken to be, for instance, || sup(>0 \ht *
/|||Li(IRiV). Hl is a separable Banach space whose dual is BMO(lRN)
and which is the dual of VM0(RN)—the "completion of C^(1RN) for the
BMO norm" (supn f0 |6—j-Q b\ where the supremum is taken over all cubes
ofIR").
In [95] it was shown that Hardy spaces can be used to analyse the
regularity of the various nonlinear quantities identified by the compensated
compactness theory due to L. Tartar [468], [469] and F. Murat [349], [350],
[351]. In particular, it was shown that EB e HlQRN) ifE,Be L2(JRN),
curl E = div B = 0 in ©'(IR/*) and we have for some C > 0 independent of
E B
\\E - B\\nl{JRn < C\\E\\L2{1RN)\\B\\LH1RN). (3.25)
If u € H^TR")", divu = 0 a.e. in ET, we can use this result to deduce
IKu-vkii^jr*) < c||u||L2(IRiv)||Vui||L2(]RAr),,l
for all 1 < i < N, J ]
\\diUj djUkWnnjR*) < C||Vu|||2(IR^),,l
for all 1 < i, k < N. J
We then consider a weak solution u of the Navier-Stokes equations (3.1)-
(3.2) (in the case ft = JRN) and we recall that we always assume uq 6
L2(JRN), f € L^O.Tjfl-^IR")) (at least) while u satisfies the conditions
listed in Theorems 3.1-3.2. Let us also observe that we can assume without
loss of generality that the force term / satisfies
div/ = 0 in V'(JRN x(0,T)). (3.28)
Indeed, we can always decompose / = f\ + Vir where div/i = 0, f\, Vir 6
L2(0,T;H~1(1RN)), incorporate t to the pressure p and replace / by f\.
Notice that for "most" functional spaces X = LP(0,T;Hs'q(m.N)) (s € Ht,
q € (1, oo), p G (1, oo]) if / € X then /x € X.
With this normalization of /, we have
Theorem 3.6. The following properties hold
dijP € ^(O.TjW1^")) (1 < i,j < N), }
v J ) (3.29)
pS^^T;^'1^)) if N>3,\
peL^TiCo^)) if N = 2, J

94 Navier-Stokes equations
(u-V)ti, Vp € L^O.TjW^lR*))*,
uel^O.TiCo^)) if N = 3 and
/eL^O.TjL^JR3)),
If Duq is a bounded measure on 1R ,
and if Df is a bounded measure on TRN x [0,T
then Du e L<»(0,T;Ll(TRN)).
Remarks 3.3. 1) The case N = 2 of (3.29) for the regularity of p is due
to L. Tartar [470]. The regularity shown in (3.31) was first obtained by
L. Tartar (unpublished), and C. Foias, C. Guillope and R. Temam [152]
by different methods. Finally, (3.32) is a slight improvement of a result
originally proven by R Constantin [99], where it was shown that curlu e
Loo(0,T;L1(IRN))ifiV = 3.
2) If N > 4, (3.31) may be replaced by u G L^TjL^'^lR")) with
a similar proof.
3) Since p is defined up to a constant, (3.29) really means that we
normalize p in such a way that p goes to 0 as |x| goes to +oo (in L7rrjrl(IRiV)
sense if N > 3).
4) If we go back to the above modification of / (/ = fi + V7r; div fi = 0;
/i, Vtt e L2{0JT;H"l(TRN)))1 let us observe that (3.32) holds if Dfi is a
bounded measure on JRN x [0,T).
5) Theorem 3.6 is stated in [95] but the proofs of (3.31)-(3.32) are only
sketched there. This is why we give below a complete proof of this result.
D
Proof of Theorem 3.6
Step 1. Proof of (3.29)-(3.30). Recall that p satisfies (3.15) and
that / satisfies (3.28). Then, using (3.26),(3.27) and the known
regularity of u, we deduce immediately that (u • V)u € L2(0,T;'H1(1RN))N,
Ap 6 L^TiJpQR?)) and Ap = dih{ in EtN where hi 6 L2(0,T;
rtQR"))" (1 < i < N). Since dijP = JUR^-Ap), diP = -RiRkhk
(1 < i,j < N), we deduce from (3.24) that diiP G L^O.TjT^flR*)),
Vp € L2(0,T;rtQR*))" (1 < t, j < N). Hence, (3.30) is proven while the
rest of (3.29) follows from the regularity of D2p and Sobolev embeddings.
Step 2. Proof of (3.31). If N = 3, / 6 L^O.TjZJ'^lR3)), we claim
that we have for each i € {1,..., N}
~i-*/Aui € L^O.TjL*'1^3))- (3.33)
(3.30)
(3.31)
)
(3.32)

Refined regularity of weak solutions via Hardy spaces 95
Indeed, in view of (3.29), we have only to show that (u • V)tXi G Ll{Q,T\
Z,*'1 (1R3)), and this follows from Sobolev embeddings since they imply that
u 6 L2(0,T;L6'2(IR3))3 while we have Vtx* € Z2(0,r;L2(IR3))3. Next, we
remark that the solution U{ of
-rj--vAui=0 in!R3x(0,r), tii|*=o = u? inH3
ot
is given by Ui(i) = uj * ((4tti/«)~3/2C""^"). Hence, ||ui(0llL~(R») <
||t*?||L3(R») (27rt/^~3/4 and we deduce easily that «< € i7(0,r;Co(IR3)) for
all 1 < p < 4/3. Therefore, (3.31) follows from
Lemma 3.1. Let N > 3, let g € £*((), TjL^'^lR*)) and Jet v be the
solution of
~-uAv = g inJRNx(0,T), v\t=0 = 0 in!RN. (3.34)
at
Then, for almost all t € (0,T), v(*) € C0(IR3) and v e ^(O^CoiJR3)).
Proof of Lemma 3.1. By density, it is enough to show that we can
estimate |M|li(0|t;L-(II»)) in terms of MIl»(o/t;l*'1(ii"))" Using the den~
sity of functions piecewise constant in i, we see that it is enough to show
such an estimate when g = l(o,t0)
h where *0 € (0,T), h € L^MflR").
Then, rescaling (£,x) (i.e. considering ^ (-#-, 7-)), it is enough to consider
the case when *o = 1 provided we obtain an estimate of v in £*((), 00;
L°°(]R )). In addition, replacing h by /i", the Schwarz spherical
decreasing rearrangement, we increase, for all t > 0, IMOIIl^OR") (see, for
instance, C. Bandle [20], A. Alvino, P.L. Lions and G. TVombetti [8],[9])
and thus, without loss of generality, we may assume that v and h are
non-negative, spherically symmetric and nonincreasing with respect to |x|.
Then, /0°° |Kt)lli-(R") dt = /0°%(0,0 dt.
On the other hand, V(x) = J*0°° v(x, t) dt solves
-uAV = h in B.N or equivalently V = — N__2 * ht
and V 6 C0(TRN) since /i € L^MQR"), j^t 6 L*^>00(1RN). Hence,
V(0) = \\V\\LX{1RN) < C||/i||L^,I(RiV) (withC=^)
and the proof of the lemma is complete. □

96 Navier-Stokes equations
Step 3. Proof of (3.32). We essentially follow an argument introduced
by P. Constantin [98]. Formally, we differentiate (3.1) and we obtain for
alll<z,fc<7VonIRiVx(0,r)
^7 (dkUi) + (u • V)(dkUi) - uA(dkUi) = dkfi - dkdip - dku5 djUi.
eft
In view of the assumptions made upon /, (3.29) and the fact that u €
L2(0,T;if 1(IRiV)), the right-hand side of the above equation is a bounded
measure on JRN x [0,T). Still arguing formally, we deduce
— \dkUi\ + (u-V)\dkUi\-vA\dkUi\ < m
at
where m is a bounded non-negative measure on JRN x [0, T), and integrating
over JRN x [0,£] we obtain a uniform (in t) bound on f^s \dkUi(x, t)\ dx.
It only remains to justify the above argument for any weak solution. To
this end, we consider, for h € (0,1], i,fc € {1,...,JV} fixed, Vfl(x1t) =
£ (ui(x+hek,t) - Ui(x,t)). We have obviously on IR^ x (0,T)
^ + u . Vv* - i/At/fc = ^(fi(- + hek)-fi-diP(-+hek) + diP))
+ t (uj(-+hek)-Uj) • djUi(-+hek). J
(3.35)
Exactly as above, we deduce that the right-hand side, denoted by m^, is
bounded in L^IR^ x (0,T)) uniformly in h € (0,1]. Since u e £2(0,T;
Hl(JRN)) and vh € L°°(0,T;L2(1RN)) D L2(0,T; £*(]&")), we deduce
from Lemma 2.3 (section 2.3) that we have
— (vh *u£)+u- V(vh *u£) - vA(vh *ue) = mh*uj£ + rch
where r£h —£0 in L^O.TjL^B^)) n I1(0,T;I^i(lRJV)) for each h > 0
fixed. Then, we write, recalling the classical convexity inequality (—A|/| <
(-A/)sign/inP'),
— \vh*<jje\ + (u-V)\vh*<jjE\-vA\vh*ujE\ < \rrih * we\ + \r%\,
and we recover, letting e go to 0+,
d\vh\
m + (u ■ V)\vh\ - vA\vh\ < \mh\ in IR* x (0,T),
K|| _ =wl \Vh\ ex2(o,r;fr1(iRJV)nx00(o,r;X2(]RiV))
} (3.36)

Refined regularity of weak solutions via Hardy spaces 97
where w°h = £ \ui(- + hek) - tij| e Ll(JRN) n L^IR") and is bounded in
L1(1RN) uniformly in he (0, lj.
Next, we multiply (3.36) by tp(^) where tp € Cg^IR*), <p = 1 on Bu
(p(x) = 0 if |x| > 2, 0 < <p < 1 on IRN and we find, integrating over
IR^ x [0,*] for all t € [0,X], denoting by C various non-negative constants
independent of t, n and /i,
/ \vh{x%t)\H>(-)dx<C+±ff \vh\(-AJ^))dxds
Jtun \n/ nz JoJjrn \ \nJ/
+ - I I \u\\vh\\V(p(-)\dxds.
n Jo Jtrn ' Vn/'
Hence, we have
sup / |v/i|<te <C-\—j / / \vh\dxds
[0,T] J(\x\<n) n JO J(n<\x\<2n)
Q
+ — II«IIl»(0,T;£,2) \\Vh\\L*>(0,T;L*)
it
or
sup / \vh\dx < (<? + —)+-^ f f \vh\dxds. (3.37)
[0,T]J(\x\<n) V Tl / n J0 J(n<\x\<2n)
In particular, we have
sup / |v/i|cfa
[0,T] J(\x\<n)
< c+%V+^([T[ \vh\Uxds\l/2nN'*.
n nz \J0 7(„<|x|<2n) /
If N < 4, we deduce that sup|0Tj J^n |v/»|dx < C. If TV > 5, we deduce
that
sup / \vh\dx < C + Cityn*?1
[0,T] J(\x\<n)
and we insert this bound in the right-hand side of (3.37) to obtain
sup /
[O.TJ J(\x\<n
Mdx < c + ^ + £+C(h)n^,
If N < 7, we obtain the same bound as before. If N > 8, we go back to
(3.37).

98 Navier-Stokes equations
In conclusion, we have shown
\dx < C. (3.38)
sup / \vh\
[0,T] JlRN
We deduce (3.32) from (3.38) letting h go to 0+: indeed, since u 6 L2(0, T;
Hl(1RN))N, vh converges to dfeix; in L2(RN x (0,T)) as h goes to 0+ and
(3.32) follows. □
3.3 Second derivative estimates
In this section we want to present various a priori estimates on weak
solutions of Navier-Stokes equations and their second derivatives in x. This will
yield similar estimates upon $%. We first consider the case when ft = IRN
(and the periodic case) and next we consider the case of Dirichlet boundary
conditions which presents specific difficulties already mentioned in section
3.1.
We thus begin with Q = IR^ and we mention without further detail that
all the results we state and prove below are also valid mutatis mutandis
in the periodic case. Also, as explained in the preceding section, we
always normalize the force term in such a way that (3.28) (unless explicitly
mentioned) holds. Then we observe that in view of the results shown in
the preceding section, we have for any weak solution as built in Theorems
2.1-2.2
(tx-V)tx, Vp eLa(O,T;L0(JRN),
L^T^OR")), Ll(o,T;LT^>l(lRN)} J
(3.39)
whenever l<a<2, ^ = 1 — 2^. In particular, we can choose a = (3 =
fl±f,i.e. | if iV = 3.
Next, any weak solution ix is the sum of ixi and 1x2 where 1x1,1x2 are
respectively solutions of
—■ - uAui = 0 in IRN x (0,T), ui|t=0 = t*o in IRjV (3.40)
-^ - i/Aix2 = g in IR" x (0,T), ix2|t=o = 0 in IR* (3.41)
where g = / — (ix • V)ix — Vp. Of course, U\ is smooth for t > 0 and its
global regularity on JRN x [0,T] depends on the properties of uq, and, in
view of (3.39), g 6 La(09T;L^(TRN)) if / 6 La(O,T]L0(RN)). Notice
that the reduction to (3.28) leaves invariant this property together with

Second derivative estimates 99
theL^Tjtt^R")) or the L^TjL^'^JR*)) regularity. Therefore,
by classical Lp (Lq) estimates on heat equations (see also Appendix D),
^,D2xu2 g La(O,T;L0(JRN)) if / € La(O,T,L0(RN)) (3.42)
with l<a<2, 4 = 1- ^*. In conclusion, we find
^,D2xu € La(0,T;L'3(m.N)) if / € L*(O,T;L0(1RN)),
* l , 2~a
1< a < 2, -5 = 1 - -rr~
f3 Na
(3.43)
if uo belongs to an appropriate (Besov) space and in particular to
WS'0(JRN) where s - 2^1 (we still assume that u0 € L2(JRN)), while
if u0 € L0(TRN) (n£2(lR*)), we obtain for all e > 0
^,D2xu € L"(e,T; £*(*")) if / € ^(O.TjL^lR")),
l<a<2, -*=l--TT- •
0 Na
} (3.44)
Of course, these results really mean that we have a priori estimates in
these norms for weak solutions in addition to such regularity information.
The borderline cases are more delicate: in particular, we do not know
if (3.42) holds mth'La(O,T',L0(JRN)) replaced by ^(O.TjW1^))—
this does not seem to be known for the heat equation! As we shall see
later on, we can prove that §», D\u e ^(O.TjL1^)) for all p e
[1,2). A similar difficulty occurs with L1 (0,T;L^'1^)): however,
in that case, using the results of Appendix D, we can deduce
conclusions similar to (3.43)-(3.44) replacing §*, d2xu € Ll{0,T\LT^<l(JR.N))
by fjf, D\u € ^''(E") a.e. t € (0,T) (we already know they belong
to IS(1RN) for p < j^j) and f*, D2xu 6 Ll>°°(Q>,T;L^>l{lRN)) where
a € L1,oo(0,T;X) means meas
{NOII* > ^} < X for a11 A > °» for
some
Ce [0,oo).
At this stage, we obtain the conclusions (3.43)-(3.44) and if we insist
upon having a = /? we find ot = /? = $^f. However, as shown by P.
Constantin [98], one can obtain a better integrability on D\u by a different
argument: in [98] it was shown that D\u 6 2^(51^ x (0,T)) if p < 4/3,
N = 3 (under appropriate conditions on /,tio)- We shall show below that
Dlu € L£>°°(1Rn x (0,T)) (for all N > 2) by a variant of the argument
in [89]. Before we even state a precise result, we would like to explain

100
Navier-Stokes equations
formally the origin of this exponent |. Differentiating (3.1) (and taking
/ = 0 to simplify) we obtain
£7 (9kUi) + (u - V) dkUi - v&(dkUi) = -(dkUjdjUi + dikp)
and we remark that the right-hand side belongs to Ll(TRN x (0,T)) since
u e L2(0,T;Hl(TRN))N by the definition of a weak solution and by (3.29)—
we could, as in [98], avoid the use of (3.29) by taking the curl of (3.1). The
maximal regularity we can deduce from this fact and the preceding equation
is Dl(dkUi) € Ll(JRN x (0,T)). In fact, even if the convection term (u-V)
were not creating additional difficulties, this is not correct since the L1
maximal regularity is not true for the heat equation, but let us ignore this
borderline problem for the sake of the argument. Then we recall that dkui €
L2{1RN x (0,T)) and thus, by interpolation, Dx{dkm) 6 27(51" x (0,T))
where ± = \ (\) + \ 1 = | and we recover the claim D2xu 6 L4/3(IRN x
(0,T)). As we shall see below, this formal (and false!) argument can be
almost justified and the only price to be paid is the replacement of L$ by
L^00! Let us also observe by the way that "D^u € L1" is a good guess
since it also yields "D2xu 6 L1 (0, T; L^A(TRN)Y (Sobolev's embeddings)
which is the borderline case of (3.43) (and (3.39)) taking a = 0. Also,
^, Dxu 6 LlQRN x (0,T)) implies that D2u 6 L2{0,T;Ll{^N)) which
is the other borderline case of (3.43) taking a = 2. Notice however that
these two (equally false!) deductions do not use the L2 integrability of
Du which seems to be the key to the improvement of the 2/(111" x (0,T))
integrability of D\u from p = $^y to P = I (or almost). Notice finally
that only in the case N = 2 (the nice case), ^±| coincides with |.
We now state precisely this integrability result; as in the preceding
section and above, u is any weak solution as built in Theorems 3.1-3.2 and /
has been normalized to have zero divergence.
Theorem 3.7. We assume that Duo,Df are bounded measures
respectively on TRN and JRN x [0,T]. Then, we have
D\u e L$>°°(1RN x(0,T)) and
1 f fT N
R>0 -« J]R"J0 .^
Remarks 3.4. 1) Taking the curl instead of an arbitrary first derivative
of (3.2) in the proof below, we obtain (3.45) with |jD2tx|2 1(\du\<R) replaced
by IVcurlupl^ur^i^) assuming only that curliio and curl/ (before or
after normalization) axe bounded measures.
(3.45)

Second derivative estimates 101
2) As we shall see below, the proof of Theorem 3.7 also shows that
Av 6 Mb(lRN), v € L2(JRN) =*> Dv € L*-°°(1RN) (3.46)
^-lAV6^6(]RiVx[0>ri),f|t-o€Af»(lRiV)> 1 (347)
v € tfflR" x (0,T)) =» Dv € L^QR" x (0,T)). J
Here and below Mb denotes the space of bounded measures (with a norm
A
denoted by || • ||m). Even though we do not know if in Theorem 3.7 L*'00
can be replaced by L4/3, we can check that such an extension is not true
for (3.46) or (3.47). Indeed, we claim that Dv e L4'3(1RN) is not true
in general since, if-it were,-we would have estimates of Dv in £4/3 in
terms of \\Av\\m + \\v\\L2 or ||f - jAv||m + ||v(0)||m + \\v\\L,, and, if we
choose for (3.46), N = 4, ve(x) = min (^y, £)C(z) where C € C£°(1RN),
C(x) = 1 if |x| < 1, e 6 (0,1), we check easily that ||Av£||m is bounded while
Hv.11^ = (\S3\ I log £\ + C)1'2 and ||Vve\\L</3 = (|S3| 24/3| log e\ + C)Z/\
and we reach a contradiction. In the case of (3.47), we choose vc = (27r(t+
£))-N'2 exp -^ for * € [0,T], x € 1RN, AT = 2, T € (0,oo), e 6 (0,1).
Obviously, ^ - \Ave = 0, ||ve(0)|U = 1 and
II««IIl>(R>x(o.D) = (j[ (47r)-1(«+e)-2(<+^)^) = c(log (l+|))1/2
On the other hand, we have
H^Wxcw,, = c(^(t+,)-*)3/4= c(iog(i + f))3/4
and we reach a contradiction. □
Proof of Theorem 3.7
Step 1. We prove
rT \
(3.48)
sup - / / IVd^l2 l/.a.
R>0 *i JjrnJo
for all 1 < z, fc < N.
Let us first explain formally the proof of this estimate (directly related to
the idea of renormalized solutions for elliptic and parabolic equations—see
for instance R.J. DiPerna and P.L. Lions [128], P.L. Lions and F. Murat

102 Navier-Stokes equations
[308] and Appendix E). First of all, we differentiate (3.1) and we find for
all 1 < i, k < N
^ (dfciti) + u ■ V(dkUi) - uA(dkUi) = dkfi - dip - dkUj djUi. (3.49)
at
In view of Theorem 3.6, of the properties of weak solutions and of the
assumption made upon /, we see that the right-hand side is a bounded
measure on EtN x (0,T). Then, we multiply (3.49) by TR(dkUi) where
TR{z) = z if \z\ < R, = R if z > R, = -R if z < -R, and we find,
integrating by parts over 1RN,
if SR(dkUi)dx+ f (u-V)SR(dkUi)dx
at JfcN jfrN
+ u f \VdkUi\2l(\dkVi\<R)dx < CR
Jm.N
since \TR(z)\ < R for all z € 1R, where SR(z) = ^ if \z\ < R, R\z\ - &-
\i\z\> R. Hence,
f f \VdkUi\2l{\dhUi\<R)dxdt < CR+ f SR(dkUi(0))dx < CR,
JO JjR" JjRN
since 0 < SR(z) < R\z\ for all zeJR.
In order to justify this computation, we argue as in the proof of Theorem
3.6 and we find, defining Vh(x, t) = £ (ui(x+hek, t) - i*i(x, t)) (i, k fixed in
(l,...,iV},/l6(0,l)),
-!—+u* Vvh - v&vh = gh bounded in Ll(TRN x (0,T)),
at
vh\t=0 = v°h bounded in Ll(]RN).
We then multiply by Tr{vh) justifying the computations exactly as in the
proof of Theorem 3.6 and we obtain finally
f f \VTR{vh)\2dxdt = f [ \Vvh\2li]Vhl<R)dxdt < CR
Jo Jtr" Jo Jjrn
(3.50)
for some C > 0 independent of R 6 (0,oo) and h e (0,1). The only new
point to check is the fact that we have for each h, R
n
Jo Jw
T t
\u\\SR(vh)\ \{n<\x\<2n)dxdt -> 0 as n-»+oo,
0 JJRS

Second derivative estimates 103
and this is immediate since u,Vh €l2(IR"x(0,T)).
Once (3.50) is established, we conclude easily upon letting h go to 0.
Indeed, we then deduce that TR{dkUi) 6 ^(O.T;^1^)) and
f f \VTR(dkUi)\2dxdt < CR,
Jo Jm.N
for all Re(0,oo), l<i,k<N.
Since, as shown above, D2u € £joc(]R.w' x (0,T)) for some q > 1
(observe indeed that since / € Mb(JRN x (0, T)) and / € L2(0, T; H~l(JRN)),
f € £*((), r;L4/3(IRN)) for all p < 4/3), we conclude since VTR(dkUi) =
VdkUil{ldkU.\<R) a.e. on JRN x (0,T).
Step 2. We prove that D\u € LS'^ilR" x (0,T)). For each (i,k) €
{1,..., iV}2 fixed, we set v = dkUi and we recall that v 6 L2((JRN x (0, T))
while, by step 1,
sup — / dx I dt |Vv|2 1(|V|<.R) < cx>.
r>o -K Jut* yo
We then decompose Vv as follows for each R 6 (0, co)
|Vv| = |VV|l(N<jR) + |Vv|l(M>fl), 1
||Vv l(|v|<fl)||L2(Rwx(0,T)) ^ C^V2- J
Next, we wish to estimate |Vv| l(|w|>fl). In order to do so, we write
/ / \Vv\l{M>R)dxdt
= z2 / l^vl Un2i<\v\<R2i+*)<tedt
< ^ C(R2j)V2 meas{(ar, t) € IR^x (0, T) /
R2j<\v(x,t)\<R2*l}1/2
< Cl J2(R2j)2meas{(x,t) € JRNx(0,T)/
x 1/2/ oo v 1/2
R2j < \v(x, t)\ <R2*x }J I ^(R*)'1)
<CR-X'2(j J l{R<M)\v\2dxdt\ <CR~1/2.

104 Navier-Stokes equations
This estimate, combined with (3.51), yields
|Vv| = di + d2, ||di||L2(iRNX(o,T)) < CRl/2,
II^IIlhiR^xCo.T)) < CR~~ •
This is enough to complete the proof of (3.45) and of Theorem 3.7. Indeed,
we have for all A > 0
meas {|Vv\ > A} < meas Id\ > — \ + meas {^2 > - }
" C(^ + E1^) = ^"^ ifWechOOSeil = A2/3- D
The estimate (3.45) and the argument used in step 2 above can be used
to derive some regularity information on ^~ and D\u in
for all p e [1,2) which correspond, roughly speaking, to the regularity of
(u ■ V)u and Vp contained in (3.39), namely (u • V)u, Vp € L2(0,T;
ft^Hl )). More precisely, we have
Theorem 3.8. We assume that Duo,Df are bounded measures on TRN,
JRN x [0,T) respectively and that f e L^O.T;!1^*)) for ailp € [1,2).
Then, we have
Dlu,^ e Lp(0,r;L1^)) foraii p€[l,2). (3.52)
Remark 3.5. If we do not normalize / (to have zero divergence), we need
to assume that V(-A)-1div/ is a bounded measure on HN x [0,T) in
addition to the assumptions made in the above theorem. □
Proof of Theorem 3.8. We use the estimate shown in step 1 of the
proof of Theorem 3.7 and the fact (shown in Theorem 3.6) that Du 6
L^TjL^R*)) while, of course, Du e L2(TRN x (0,T)). Therefore,
Du e Lq (0, T; L^t (IRN)) for 2 < q < oo. Next, we write with the notation
of the proof of step 2 of Theorem 3.6
/ \Vv\dx = Y] / |Vv|l(2i<iw|<2i+i)flk;+ / \Vv\dx
Jtr» J^Jk» J(\v[<i)
OO
3=0

Second derivative estimates 105
where
\ xl2 \
>
/;(*) = (J N\Vv\2li2i<\v\<2i+i)dx) ,
Lit) = (J | Vv\21(M<1) <fcc) ,
Qj(t) = meas {x € BN / 2j < \v{x,t)\ < 2j+1} 1
forall;>0, a.e. *€ (0,T). J
In view of (3.45), we have for some C > 0 independent of J,*
ll/;IUw) < C*l2, ||/||L»(o.T) < C. (3.53)
On the other hand, we recalled above that v € £*((), TjL^IR^)) for
2 < q < oo. Hence, choosing q < oo (g > 2), we deduce
oo
^q^Vt e L'-^O.T). (3.54)
i=o
Using (3.53) and (3.54), we may then write
r / °° x 1/2 / oo n 1/2
/ ivV|dx < /+(E^?2"iA E^2iA
and / € L2(0,T), (E£ o *i 2''^)1/2 € L'fo-^O.T), while (E~o/?
2^^t)1/2€jL2(0,T) since we obviously have, because of (3.53),
rT oo
-J?*r
di
VI II
oo
C^2J'2-J'^-
oo
j=0
= C(l-2-^r)
-l
We have thus shown that Vv, and thus D2u, € £2 *T" (0, T; L1 (EtN)). Since
9 is arbitrary in [2,oo), we have proven Theorem 3.8 for D\u. The claim
for ^ then follows from equation (3.1) and (3.30). □
Remark 3.6. The method introduced above can be used (and modified
slightly) to build global weak solutions for two-dimensional Navier-Stokes
equations (N = 2, ft = IR^ for example or the periodic case) when the
initial data uq satisfies: uq € L2,00(IR2), ujq = curluo € ,M&(IR2). Indeed, one

106 Navier-Stokes equations
only needs to obtain appropriate a priori estimates and the existence
follows upon regularizing uq and passing to the limit. These a priori estimates
can be obtained as follows: first of all, u = curl u should solve
-57+ u- Vu;-W\u; = 0 in R2 x (0,oo), u\tssQ = u0 inR2 (3.55)
ut
(we take / = 0 to simplify the presentation). Indeed, if N = 2, curl ((tx •__
V)tx) = (tx • V)curltx since divtx = 0. Next, using symmetrization results
due to A. Alvino, P.L. Lions and G. Trombetti [8],[9], we deduce that for
each t > 0, u(t) is "dominated" by ZJ(t) = Uq * e~ *'»* (iirvt)"1 where Uq is
the (Schwarz) spherically symmetric decreasing rearrangement of uo and
domination means in particular that all Lp and Orlicz norms of u(t) are
less than W(t). Therefore, u € L^°°(0,oo;L2(IR2)) D L^oojL^lR2))
and ||w(*)IIL^tiR2) ^ i^t ll^o||Mb- In particular, we deduce that u G
L~(0,cx);L2'~(IR2)). In addition, (3.55) yields as in the proof of
Theorem 3.7
S f dt [ dx|Vo;|2l(M<ii) < -llmollm, for all R > 0. (3.56)
K Jo Jr3 v
Next, we estimate, in a manner similar to the proof of Theorem 3.7, for all
A > 0, R > 0
meas {\Du\ > A, |u| > R}
oo
= ^meas {\Du>\ > A, 2jR < \w\ < 2j+1R}
j=o
oo
1/2 r. , ^0iDTl/2
< £meas{|0w|>A, \u\ < 2i+1R} ' meas{M > 2jR}:
3=0
<cy (iZ2^1)1/2 (my1 < cr-^x-1
i=o
in view of (3.56) and the fact that u € L2'°°(IR2 x (0, oo)). Therefore, we
deduce for all A > 0
meas {| .Do; | > A}
= meas {\Du\ > A, \u\ < R} + meas {\Du\ > A, \u\ > R}
< C{^ + j^} = CA-4/3 choosing R = A1/2.
Hence, Vw € L*'°°(IR2 x (0, oo)) and thus D2u € L3.°°(IR2 x (0, oo)). Even
if more estimates can be derived for u, Du,D2u, |j-f,p, the estimates listed

Second derivative estimates
107
above are already enough to pass to the limit in regularized problems and
build a solution. □
We now conclude this section by considering in more detail the case of
Dirichlet boundary conditions. First of all, we take a € (1,2) and set
i = 1 - j£. If we assume that / € La(O,T}L0(Q))—and we always
normalize it to satisfy div/ =_0 in X?'(ft x (0,T))—then by Irregularity
results for linear Stokes equations (see V.A. Solonnikov [444],[445] for
instance) and extensions due to Y. Giga and H. Sohr [183] about L\{L%)
regularity, we deduce immediately
§^«,Vp € L«(e,T;i/(Q))
(3.57)
for all e > 0 and we can take e = 0 if u0 € W2^-^'0(Q). We next
decompose p as follows: p = Po +Pi where po.Pi satisfy
- Ap0 = 0 in ft, podx = 0 a.e. t € (0,T), 1
— Api = diUj djUi in ft, pi = 0 on 9ft. J
(3.58)
Observe that since (u • V)u 6 L^O.TjL^ft)) (and diUjdjUi = div((u •
V)u)), Vpi € La(O,T;L0(Sl)) and thus Vpo € La(e,T;L^(ft)). From now
on, in order to simplify the presentation we take e = 0. Therefore, we
deduce from (3.58) and using the arguments developed in the preceding
section
Vpi € la(0,r;^(fi)), Vp0 € L«(0,T;^(ft)),
(3.59)
I?2ft6l1(0IT;W1(n)),
Vp^L^O.TjL^'^ft^nL^O.TjW^ft)),
pi€L1(0,r;L^'1(«)),
(u • V)ti € L2(0,T; W1^)) n L^O.^LT^OR.")), J
(if iV = 2, we replace L^'^ft) by C(ft))
po € La(0,T;tf£c(ft)) for all k > 0.
(3.60)
(3.61)
Here and everywhere below, whenever we write LP(X\oc(Q)) for some
function space X wejnean LP(X(K)) for any relatively compact subdomain K
of ft such that K C ft.

108 Navier-Stokes equations
Next, we claim that if Df, Duq are bounded measures respectively on
ft x [0,T), ft, then we have
Du € Loo(0>T;Lie(ft))> (3.62)
D2u € Lp(0,T;Llc(Cl)) for all p < 2, (3.63)
< 00,
(3.64)
sup - / dt / dxY] \VdkUi\2 l(\dkUi\<R)
R>0 & Jo JK i^Z1
for all compact sets K C ft,
D2u € ^~(0,T;4f(n)). (3.65)
Let us recall before explaining the proof of all these estimates that (3.62)
is the analogue of the estimate (3.32) (Theorem 3.6), while (3.64)-(3.65)
correspond to Theorem 3.7. Finally, (3.63) is the analogue of (3.52); we do
not know if a similar estimate holds for ^. If we go back to the proofs
made in the case when Q = IR , we see that we considered in each case
Vh = X (^i(x + /iefc,i) - Ui(x,t)) for h G (0,/io) (where x € Cth0 = {x e
ft, dist(x,dft) > /i0}, t € [0,T]) which satisfies
2£ + («.VH-"Atfc=mfct } (36g)
mh is bounded in L1 (0, T; Lj,,. (ft))
and as in the proof of Theorem 3.6
d\vh\
dt
+ (u'V)\vh\-uA\vh\ < \mk\. (3.67)
Then, if we fix a compact set K C ft, we choose tp € Cq* (ft) and ho in
such a way that 0 < (p < 1 in ft, f = 1 on K, Supp <p C fth0- We may next
multiply (3.67) by (p and we find
I \vh\<pdx(t) < C+ J I \u\\vh\\V<p\dxdt
Jq Jo Jn
< C + C f [ \u\2 + \Vu\2dxdt < C
Jo Jq
and (3.62) follows.

Second derivative estimates 109
Similarly, (3.64) follows upon multiplying (3.66) by <pTr(vh)- We then
obtain as in the proof of Theorem 3.7
2l(\vK\<R)dxdt
T
f [ <p\Vvh\
Jo JQ
<CR + \[ [ VvhTR{vh)-V<pdxdt\+ [ [ \u\\Vy\\SR(vh,)\dxdt
\Jo Jn I Jo Jn
< CR+\[ J SR(vh)A<pdxdt\ +R J J \u\ \Vtp\ \vh\ dxdt
\Jo Jn I Jo Jn
< CR.
Once (3.64) is established, (3.65) follows exactly as in the proof of Theorem
3.7.
Finally, we claim that if / € L1(0,r;L£:T,1(fl))l then
D2u, |i € L^T;^'1^)), (3.68)
u 6 X^O.TjCiocCfi)) if iV = 3. (3.69)
Indeed, we observe that we have, in view of (3.60),
In particular, we deduce
— - i/AJ (u(p) = gip- 2uVu • V<p - uAcp in IR* x (0, T)
and g<p - 2uVu • V<p - uA<p e L^O.TjL^'^lR^)). Therefore, (3.68)
follows from the result shown in Appendix D while (3.69) is a consequence
of Lemma 3.1.
Let us summarize the regularity information we have obtained with the
Theorem 3.9. Let f € L2(0,T;tf-1(ft)) n La(Q,T;L0(Q)) n Ll(Q,T;
L^'l(1R.N)) for 1 < a < 2 (with % = 1 - ^§) satisfy: div/ = 0
in V'(Q x (0,T)), Df is a bounded measure on Q x [0,T). Let uQ €
W-2(i-£),/J(ft)r\L2(9) for 1< a < 2 satisfy: divu0 = 0 in V'(Q), u0 -n = 0
on dQ, Duq is a bounded measure on Q. Then, there exists a weak solution
u (or (u,p)) as in Theorems 3.3-3.4 satisfying in addition (3.57) (with
s = 0), (3.59)-(3.61), (3.62)-(3.65), (3.68)-(3.69) and the local energy
inequality (3.10) if N = 3.

110
Navier-Stokes equations
The only new information is the local inequality (3.10) if N = 3 which
was first obtained by L. Caffarelli, R. Kohn and L. Nirenberg [77]. Let us
emphasize that contrary to all the other information listed in Theorem 3.9,
we do not know if any weak solution satisfies (3.10). But we can build at
least one that satisfies (3.10). In fact, the existence procedure taken from
chapter 2 and recalled in section 3.1 (see (3.16)-(3.17) in the case £2 = JRN)
yields such a solution. Indeed, one can check without any difficulty that all
the estimates derived above hold uniformly: in particular, Vpc is bounded
in L*(Q x (0,T)) (take a = § and thus (5 = |) and this is enough to derive
(3.10). Indeed, if we compare with the proof of Theorem 3.2 (step 3), we
have only to explain how to pass to the limit in u£p£. The preceding bound
shows that—normalizing pe such that f^Pedx = 0 a.e. t e (0,T)—pc
converges weakly to some p in L* (0,T;L15/7(£2)) while we already know
that u£ converges to u (a weak solution) in L5(0,T;Lg(£2)) for q < |» and
we conclude since j£ + jg = | < 1.
3.4 Temperature and the Rayleigh-Benard equations
In this section, we study the Navier-Stokes equations ((3.1)-(3.2))
complemented with an equation for the internal energy e (or equivalently for
the temperature T), namely (1-41) written under the assumption that p is
constant, namely
■^ + div (tie) - div (JfeV0) = ^ (dmj + djUi)2 in Q x (0, T). (3.70)
In order to simplify the presentation, we shall consider here only the case
when we have
e = CQ9, Co > 0, ke (0, oo). (3.71)
In that case, (3.70) reduces to
— + div (uO) - aA9 = ^— (diUj + djm)2 inQx (0, T) (3.72)
ot 2Go
where a = ^- G (0,oo). In the case of Dirichlet boundary conditions
(for the Navier-Stokes part of the system of equations), various boundary
conditions are possible like, for instance, Neumann boundary conditions
89
— = 0 on aft x (0,T). (3.73)
on
Recall that n denotes the unit outward normal to d(l. In the periodic case,
we simply require 9 to be periodic.

Temperature and the Rayleigh-Benard equations 111
At this stage, let us mention that if most of this section is devoted to the
model described above (and a variant of it), we also discuss at the end of the
section an interesting (both physically and mathematically) variant of the
classical homogeneous, incompressible Navier-Stokes equations, namely a
model for a homogeneous incompressible flow with internal degrees of
freedom taken from S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov [17].
Let us first observe that, at least formally, that is when solutions are
smooth, (3.1) and (3.72) are equivalent to (3.1) and
|{tf + C0,} + div(U{J^ + C0,+P})-,A^-^ (374)
= f -u + vdiUjdjUi in fix (0,T),
which is nothing but the "total energy" equation. Recall also that we have
-Ap = diUjdjUi = div ((u • V)u) in ft x (0,T) (3.75)
at least if we normalize / to satisfy, as we can always do as explained in
the preceding sections,
div/ = 0 in ft x(0,T), (3.76)
an assumption we always make from now on without recalling it.
Observe that we have, at least formally, ^ |tx|2 = 0 on dCl x (0,T) in
the case of Dirichlet boundary conditions and thus we deduce from (3.74)
f MQL + Co0(t)dx = [ ^ + Co0Qdx+ [ dsf dxf-u (3.77)
denoting by 0o the initial condition for 0, i.e.
0|t=o = 0o in ft. (3.78)
Of course, in the periodic case, we assume that 6q is given on IR^ and
periodic, and we always assume from now on that #o £ Ll(Q).
From the above considerations, we see that there are two ways to look
at this system of equations (often called the Rayleigh-Benard equations).
Either we decouple the two parts and solve first (3.1)-(3.2), then, given a
weak solution u of (3.1)-(3.2), we attempt to solve (3.72)-(3.73) (with the
initial condition (3.78)). The other possible approach is to build
simultaneously (u,T) solving (3.1)-(3.2), (3.74) and (3.78). The reason why these
two approaches might not yield the same solutions is the fact that we do

112
Navier-Stokes equations
not know if any (or even some) weak solution of (3.1)-(3.2) satisfies the
energy identity
K^)+div(U!£)-„AM! + ,|V«|> = /. i„ nx(0,T).
If it were the case, then both approaches could be reconciled but this is
an open problem that can be solved only if N = 2. Indeed, in the two-
dimensional case, the regularity and uniqueness of weak solutions allow us
to compare the two approaches presented below but we shall not discuss
this point further here.
We begin with the decoupled approach in which u is a given weak solution
of (3.1)-(3.2) as available from Theorems 3.1-3.4. Then, we wish to solve
(3.72)-(3.73) and* (3.78). Let us write D = ^ (diUj + djUi)2; obviously,
D € Ll(Q x (0,T)). Therefore, from heat equation considerations, we
cannot expect a better integrability for 9 than: 6 e Loo(0,T;L1(n)) n
L^TjL^ft)) for all q < jfe. Hence, u0 6 L\QC if N = 2 or N = 3. This
explains why solving (3.72) in the sense of distributions is not adapted to
the problem in hand. Instead, we use the notion of renormalized solutions
which is more flexible (and also more precise)—see R.J. DiPerna and P.L.
Lions [128], P.L. Lions and F. Murat [308].
We recall the definition: we shall say that T is a renormalized solution
of (3.72), (3.73) and (3.78) if 9 6 C([0,T]]Ll(Q)) n Ll(0,T]L^(Q)) for all
Q < 77Z2 satisfies
TR{0) e L2(0,T-,Hl(n)) for all R > 0 and)
rrt
lim i / dxf dt\VTR{6)\2 = 0
(3.79)
/ dxfdx[(3(9)^ + u.Vtp + aA<p} - /J"(0)|V0|V + D0'(9)<p]
+ [dxp(0o)<p(O) = [dx0{O{T))<p{T)
Jn Jn
(3.J0)
for all (3 € C2(1R) such that $ has compact support and for all tp e C°°(Qx
[0,T]) (periodic in the periodic case, with compact support in IR* x [0,T]
if Q = IR ). Some explanations are necessary: indeed—see [308] for more
details and the preceding section for a related argument—(3.79) yields the
fact that V0 € L3(Q x (0,T)) for all s < ^. In particular, V2 6 Lfoc
and VTR{6) = Vdlm<R) a.e. This explains why /3"(0)|V0|2 € Ifoc since
0 € C2(IR) and (3* has compact support. Next, we remark that (3.80) is a

Temperature and the Rayleigh-Benard equations 113
weak formulation of the following equation and conditions
(§i+ u'v"aA)m + a0"Wve\2 = D?W inQx (°'T)
4-P{e) = 0 ondftx(0,T), 0(9)\t=o = Wo) in a
on
The equation for (3(0) follows formally from (3.72) and the notion recalled
above simply consists in requesting that these natural changes of variables
are indeed possible. Finally, the condition on Tr(0) follows, at least
formally, from (3.72) and the integrability of D and 0o since we deduce from
(3.72) upon multiplying by TR(9)
a f dt f dx\VTR(9)\2 = / dt[dxDTR(9) + f dxSR{90).
Jo Jq Jo Jn Jq
Next, we observe that j^TR(9) is bounded by 1 and converges a.e. to
0 while ^ Sr(9q) is bounded by 0o and converges a.e. to 0, and (3.79)
follows. Let us finally recall a few facts from [308]: if |tz|0 € Lloc and 9
is a renormalized solution of (3.72) then 9 satisfies (3.72) in the sense of
distributions. On the other hand, using the fact that u € L2(0,X; Hl(Q)),
it is not difficult to check that if 9 satisfies (3.72) in the sense of distributions
and 9 e L2(Clx (0,X)) then 9 is a renormalized solution.
With this notion, the following result proved in Appendix E holds.
Theorem 3.10. There exists a unique renormalized solution of (3.72),
(3.73) and (3.78).
Remarks 3.7. 1) Recall that u is any weak solution of (3.1)-(3.2) as given
by Theorems 3.1-3.4.
2) Using the results of R.J. DiPerna and RL. Lions [128], we immediately
see that this result also holds if k = 0 (a = 0). Then, we only know that
9eC{[0yT])Ll(JR3)).
3) If fl = R^, we can take 0O to be in L1{1RN) + L°°(IRN) (changing
appropriately the spaces to which 9 belongs).
4) One can show that infessx€^ 9(x,t) (e [-co, +oo)) is a nondecreasing
function of t. This is a simple consequence of the fact that D > 0 a.e.
5) If u satisfies (3.10) and \u\9 6 L\oc(Sl x (0,T))—these two facts hold
if N = 3—then we see that we have
dt
(^+Co,)+divWtf+w+p}).,AM!_fcAT|(38i)
< / • u + udiUj djUi in V'(Q x (0,T)), J

114
Navier-Stokes equations
and in all cases, we only obtain
i«(*,t)i2
L
<-L
+ C09(x1t)dx
M-+C0e0dx +
2 jq
/ ds <f,u
Jo
>H-lxHl
(3.82)
and
TtJ^ + C°9dx<<f,u>H-*xHi in Z>'(0,T).
(3.83)
In particular if / = 0 the total energy is known to be conserved if and only
if u satisfies some energy identity instead of an energy inequality, an open
problem if N > 3 as we saw in section 3.1. □
We now turn to the second approach where we solve simultaneously
(3.1) and (3.74). This will lead to weak solutions (tx, 9) for which the total
energy is conserved (when / = 0), a fact which is physically expected of
course. The price to be paid for this "improvement" is the requirement
that N = 2 or N = 3 because of integrability requirements for |tx|3 or \u\9.
The case when N = 2 being straightforward in view of the regularity of
weak solutions, we consider only the case when N = 3, and we begin with
the case when Q = H3, the periodic case being completely similar.
Theorem 3.11. Let 90 e L^IR3). Then there exists (u,0) such that u is
a weak solution of the Navier-Stokes equations (3.1)-(3.2) (as in Theorem
3.2) satisfying (3.8)-(3.10), 9 e L^TjL^lR3)) n L*'°°(IR3 x (0,T)) n
Ll(0,T; L*(1R3)) for all q < 3, sup*>0 £ JR,dr • fidt \V9\2 1W<R < oo,
and (u,9) solves (3.74), (3.78) in a weak form, namely we have for all
<^eC^(]R3x[o,r))
4- v -—- A(p + k9Aip + f -u<p + i/diUj djU{(p >
+ J dx(^-+Coeo\(p(x,0) = 0. D
(3.84)
In the case of Dirichlet boundary conditions, we have the
Theorem 3.12. Let 90 e L1^), let u0 € W2^^4(Q) n L2(Q) be such
that divuo = 0 in Q and u0 ■ n = 0 on dQ, let f € L5/4{Q x (0,T)) be

Temperature and the Rayleigh-Benard equations 115
such that div/ = 0 in V'{Q x (0,T)). Tien tiere exists (u,0) such that
u € L5/4(0,T;W2>${Q)), Jf, Vp € I5/4(ftx (0,T)), u is a wealcsoiution of
(3.1)-(3.2) as in Theorem 3.4 satisfying (3.8)-(3.10), 0 € L°°(0,r;Il(a))n
L^O.TjL^fl)) for all q< -fa, sxipR>0 \Sndx!ldt lV*l2 ^Hk*) < °°
and (u,6) solves (3.74), (3.73), (3.78) in a weak form, namelyJ^M) holds
(with IR3 replaced by Q and C£°(IR3 x [0,T)) replaced by C£°(ft x (0, T))).
Remarks 3.8. 1) Remark 3.7 (3) also holds for Theorems 3.11 and 3.12.
2) Remark 3.7 (4) holds in the contexts of Theorems 3.11 and 3.12.
3) We deduce from the above results the following facts
Jtj 2+C°edx = J f-udx (=<f,u>H-ixH^
— +div(ue)~aA6 > -^-(diUj+djUi)2 in Z>'(ft x (0,T)).
ut 2C7q
4) Prom the weak formulation, one deduces easily that ^J- + Co0 is
continuous in t with values into .M&(fi) endowed with the weak * topology
(weak topology of measures).
5) In Theorem 3.12, the weak formulation incorporates the Neumann
boundary condition (3.73) together with the observation already made
above, namely: ^^ = 0 on dfi x (0,T) since u = 0 on dti x (0,T)
(at least formally).
6) The only (new) term in (3.84) whose meaning has to be explained is
the term u9. Since u 6 L°°(0,T;L2) n £2(0,T; L6) (Sobolev embeddings),
ueL1? while 9 e Lq for all q < §. Hence 6u e Ll{Sl x (0,T)). D
The proof of Theorem 3.11 being similar (and in fact simpler) to the one
of Theorem 3.12, we only present the latter.
Proof of Theorem 3.12. With the notation of section 2.4, we consider
the solution u£ of
du£
— +u£ • Vu£-v&u£+Vp£ = f£ infix (0,T),
ut
u€\t=o = u£0 in fi, u£ = 0 on dfi x (0, T),
u£ eC2(fix[0,T]), divu£ = 0 infix (0,T). J
(3.85)
We already know, extracting subsequences if necessary, that, as e goes
to 0+, u£,u€ converge weakly in L2(0,T; H^(Ct)) D L°°(0,T; L2(fi)) (weak
*) to a weak solution u of (3.1)-(3.2) satisfying (3.8), (3.9). In addition,
f£ and u£ • Vuc (same proof as for u • Vtx) being bounded in L5/4(fi x

116
Navier-Stokes equations
(0,T)), we deduce as in the proof of Theorem 3.9 that uc is bounded
in L5/4(0,T;^2't(fi)), %£ and Vpff are bounded in L*(fl x (0,T)): in
particular, D2xue, %£, Vp£ are bounded in L5/4(fi x (0,T)) and (3.10)
holds. Finally, let us recall that ue converges to u as e goes to 0+ in
L*(0,T;L2(ft)) for all p e [l,oo), and in L2(0,T;L«(fi)) for all 1 < q < 6.
Next, we introduce the solution 6£ of
fine v
^ + uc • W - aA6* = — (ftuj + dX)2 in ft x (0, T)
^-=0 ondflx(0,T), 0£|t=o = 0S in Q.
on
} (3.86)
where 0§ € Co°(fl), 0q converges to 9q in Ll(Q) as e goes to 0+.
Since ue is smooth and (drf + dju\)2 e C^ft x (0,T)) (for example),
this is nothing but a standard linear parabolic problem and we know there
exists 0e in, say, C2,1(?T x (0,T)), i.e. u,Dxu,Dlu, fjf € C(?7 x (0,T)).
Since (fljjuj + djU?)2 is bounded in L1(H x (0,T)), we deduce when ft =
IR^ in the periodic case from estimates on solutions of heat equations (via,
for instance, symmetrization results due to [8],[9]) that 6C is bounded in
CtfO.TljL1^)) n L§'°°(ft x (0,T)) n L^O.TjL^fl)) for all q < 3. In
the case of Dirichlet boundary conditions (ft ^ IR ), we deduce from
the results shown in Appendix E that $c is bounded in C([0,T];L1(fl)) n
L1(0,T\Lq(Q)) for all q < 3. Finally, as in the proof of Theorem 3.10,
we deduce that TR(6e)R~1/2 is bounded in L2(0,T;ff1(fi)) uniformly in
R,e. Therefore, without loss of generality we may assume that 6e converges
weakly in L°(ft x (0,T)) for all a € (l, f) to some 9 € L°°(0,T',Ll(Q)) n
L1(0,T;L9(n)) (V q < 3). Also, as in the proof of Remark 3.6 (see also
P.L. Lions and F. Murat [308]), V0£ is bounded in Lr(ft x (0,T)) and thus
V0 € Lr(Q x (0,T)) for all r < §.
Next, we deduce from (3.85) and (3.86) that we have
|(M! + Co,.) + div{u,(^+c0r)+uv}
-i/A
u
c|2
2
.£|2
k£6e = fe • u£ + v diu) djul in ft x (0, T),
c /\u \ \
_(L_L + Cos<) =o on an.
> (3.87)
In addition, we know that ^, 0C are bounded in C([0,T];L1(f2)) n
LHO.TjL^)) (V g < 3) and that V^, V0e are bounded in Lr(ft x
(0,T)) (Vr < |). From these bounds and equation (3.85), observing that

Temperature and the Rayleigh-Benard equations 117
>
diU£jdjU* = di((u£ • V)u£) and using classical compactness theorems, we
deduce easily that J^Ili + CQ9e converges to ^1 + c06 in Lr(0, T; Ll(Sl)) n
L^OjT; Ig(fl)) (for all 1 < r < oo, 1 < q < 3). Therefore, 0ff converges to 6
in 2/(0, T; 2^(0)) nLl(Q,T;L*(Q)) (for all 1 < r < oo, 1 < q < 3). Then,
deducing (3.84) from (3.87) is an easy exercise using these convergences
and the bounds collected above.
Remark 3.9. Combining the methods developed in this chapter and those
introduced in chapter 2, it is possible to study density-dependent models
with temperature such as
^ + div(pu) = 0, divtx = 0
ot
?^ + div(puui)^^dj{fi(Ple)(diuj
Co ^ + Co div (pu6) - div (fc(p, 9)VB) = ^^ (^ +djui)2 J
(3.88)
where /i,k e C([0,oo) x IR), inf{n(t,s) / \t\ < R, s e 1R}, inf {k(t,s) /
\t\ < i2, s G R} > 0 for all R > 0 (for instance).
However, we shall not attempt to present here precise results on such a
system of equations. □
Finally, we conclude this section and this chapter with a model for an
incompressible, homogeneous, newtonian fluid taking into account internal
degrees of freedom (for more details see S.N. Antontsev, A.V. Kazhikov and
V.M. Monakhov [17]). We only describe the three-dimensional situation
with Dirichlet boundary conditions and we look for u(x,t),Lj(x,t) 6 IR3
solutions of
du
— +(u- V)u-w\u+Vp = f+(u> x u),
divu = 0 in Q, x (0, T), u = 0 on dQ x (0, T),
— + div (ucj) + F(p)uj = m in Q x (0,T),
ot
L
(3.89)
pdx = 0 in(0,T)
ft
where F is a continuous, non-negative function on IR satisfying
1^(01 < C(l + \t\a) onIR, for some a e [o, |) (3.90)
and for some non-negative constant C. Finally, we keep the initial condition
(3.2) where uq € L2(Q) satisfies (3.3) and uq • n = 0 on dfi, and we add an

118
Navier-Stokes equations
initial condition for uj
w\t=o = ^o in f2,
(3.91)
and we assume that m € L°°(Q x (0,T))3, w0 € L°°(ft)3, u0 6 W*'ti(fl) n
L2(fi) (divu0 = 0 in fi, u0 -n = 0 on 3H), / € L^O.TjJf-^fl^nL^ft x
(0,T))3. Then, we can prove
Theorem 3.13. There exists a solution (u,p,u>) of (3.89) (in the sense of
distributions), (3.2) and (3.91) such that u € L2(0,T; !#(!*)) n C([0,T];
^l(n))nc([o>r];L1(n))nii*(o>r;H^.«(n))f § € l*(o,T; x«(ft))f
p € lt(0,r; Wrl--H(fi))l w € L°°(fi x (0,T)), a; € C([0,T]; £«(«)) for all
l<q<oo.
Remark 3.10. The proof below also shows that u satisfies the energy
inequalities (3.8)-(3.10) and that w satisfies for all (3 € C^IR^IR)
/ (3{u(x,t))dx + I ds [ dxF(p)uj • V0(w)
Jet Jo Jci
= f P(uj°)dx + [ ds I dxm- V/?(w) for all t € [0,T]. D
Jo Jo Jn
Proof of Theorem 3.13. Following the arguments developed in chapter
2 (the situation being somewhat easier here), we introduce the following
approximated system of equations
^-+ue-Vuc-^Aue+Vpe = /e + (u/£xiO,
ot
divu£ = 0 infix (0,T)
dt
L
+div (ueu£)+F(p£)ue = m£ in Q x (0,T),
pe dx = 0 in (0, T)
(3.92)
with uc = 0 on 80, x (0,T), uc|t=o = U(j> <*>e|t=o = ^6» where u£,fs,vrQ have
been defined previously (see chapter 2 in particular) and u/g € Co°(n),
me € Co°(fi x (0,T)), u/<5,mc are bounded uniformly in e respectively in
L°°(Q), L°°(Q x (0,T)) and Wo,me converge respectively to u>o,m a.e. and
in L«(Q), L*(n x (0,T)) for allj. < q < oo.
The existence of smooth (on Q, x [0,T]) solutions (t/,pe,w£) of (3.92) is
an easy adaptation of the argument introduced in section 2.4 and of the

Temperature and the Rayleigh-Benard equations 119
bounds we obtain now. First of all, multiplying the equation satisfied by
us by itself, we find the usual energy identity valid for all t G [0, T)
[ hu£(x,t)\2dx+ ( ds[dxu\Vu£\2{x,s) 1
Jq2 Jo Jq I
= i / \u£Q\2dx + [ ds [dxf£-u£
2 Jq Jo Jq
(3.93)
which yields a bound (uniform in e) on ue in C([0,T];L2(ft)) n £2(0,T;
Hq(Q)). Next, using the equation satisfied by txc, we obtain easily for all
te[0,T)Jora\\peCl(JR3]JR)
(3.94)
/ p(u£(x,t))dx + f [ F{p£)u£ ■ V(3(ue)dxds
Jq JqJo I
= / P(u%)dx+ [ [ m£ - V(3(lj£)dxds.
Jq Jo Jq )
In particular choosing /?(x) = |xi|m:r; for m > 0, i = 1,2,3, we obtain
sup sup / \u£\qdx < oo for all 1 < q < oo (3.95)
and keeping track of the precise bounds as q —► +oo (or applying directly
the maximum principle), we deduce
sup sup {\u£\ J x € ft, * € [0,T]} < oo. (3.96)
€>0
Then, going back to the equation satisfied by txc, we deduce using the
preceding bounds (and Sobolev embeddings) that f£ + u£ x ue - (u£ • V)uc
is bounded in L3(0,T;L"i4 (ft)) uniformly in e. Therefore, u£,^,p£ are
bounded uniformly in e respectively in L% (0,T; W2'H(ft)), L$ (0,T;
LT*(ft)), Li(0,T; Wl'Ti{Ci)). Prom these bounds, we deduce easily that,
extracting subsequences if necessary, u£ converges to some u € C([0,T];
Ll(Q)) n L2(0, T\ Hi (ft)) n C([0, T]; L^ft)) n L? (0, T; W'2'!? (ft)) and the
convergence is a weak convergence in L°°(0,T;L2(ft)) (weak *) nL2(0,T;
Hi(Q)) n it (0,T; W2>" (ft)) and a strong convergence in C([0, T]; L*(ft))
(V 1 < p < 2), in L*(0,T; W^(ft)) (V1<?<2) and in L2(0,T;L*(ft))
(V 1 < q < 6). Similarly, ^, Vpc converge weakly respectively in
l4(0,r;ltt(n)),LJ(0,T;^»(fl)) to ^ and Vp for some p which sat-
isfies: fQpdx = 0 in (0,T). In addition, we may assume that u£ converges
weakly in L°°(ft x (0,T)) (weak *) and strongly in C([0,T];W-s>*(Q))

120
Navier-Stokes equations
(V s > 0, V 1 < p < oo) to some u e L°°(Q x (0,T) n C([09T\;L%(Q))
(V q <oo) which satisfies a;|t=o = ^o on Q. Observe indeed that, because of
(3.90), F(p£) is bounded in Lr(Cl x (0, T)) where r = ^. Finally, we assume
without loss of generality that F(pe) converges weakly in Lr(Q x (0,T)) to
some F > 0.
Obviously, we can pass to the limit in the equation satisfied by uc. We
also recover the energy inequalities (3.8)-(3.10) mentioned in Remark 3.10
from (3.93) and its local variant, namely Jj ^- + div (uc l2LL + uepe) -
j/A ^— + j/|Vtxc|2 = fe - ue. In order to complete the proof of Theorem
3.13, it only remains to pass into the limit in the equation satisfied by ue
and to show that u G C([0,T]; Lq(Q)) for all 1 < q < oo.
In fact, we are first-going to show that ue converges to u in C([0,T];
Lq(Q)) for all 1 < q < oo and that u satisfies the desired equation with,
however, F(p) replaced by F. Then we shall show that F = F(p).
The second step is easy: indeed, once we know that uj£ converges to u in,
say, Lq(Q x (0, T)) for all 1 < q < oo, then we deduce from the convergences
of txc, W, f€ listed above that /c + ue x nc — (ue • V)tzc converges to / +
uxu-(u-V)u in Lqi(Q,T]Lq2(Q)) for alll < ^ < |, 1 < q2 < ±f. Hence,
using the results of Y. Giga and H. Sohr [183] on Stokes equations, we
deduce that ue converges to u in Lq(Q,T] W2^(Q)), 2jg- converges to §f in
Lqi (0, T; Lq2 (ft)) and Vpff converges to Vp in Lqi (0, T; L*2 (ft)) for all qx <
f j 1 < 92 < yf - Since we normalize pc and p to satisfy /^ pc dx = /Qp dx =
0 on (0,T), we deduce that p£ converges to p in Lqi(Q,T;WiM(Q)) for all
9i < f, 1 < 92 < if and in particular in L*(ft x (0,T)) for all 1 < q < 5/3.
Since F satisfies (3.90) and F is continuous, we deduce easily that F(p€)
converges in Lr'(ft x (0, T)) to F(p) for 1 < r' < r = ^. Hence F = F(p)
and we conclude.
Finally, the above claim on lj€ is proven by a convenient adaptation of the
method introduced in steps 1-3 of the proof of part 1 of Theorem 2.4
(chapter 2, section 2.3). More precisely, we claim that if F(pe)u€1 F(p£)\uj£\2
converge weakly in Lr(Q x (0,T)) respectively to Flj and Fu)2—where U2
is the weak * L°°(ft x (0,T)) limit of \u£\2—then the convergence of ue to
lj in C([0,T]; L2(Q)) and thus in C([0,T]; Lq(Q)) for all 1 < q < oo follows
easily Indeed, if this claim is shown, then u and UJ2 solve respectively:
u,u2 e L~(ft x (0,T)) nC([0,Tl;L!(n)) (V K 9 < oo)
5cj
— + div (ucj) + Fa; = m in ft x (0,T), a;|t=o = ^o in ft (3.97)
— + div (txwa) + 2Fu2 = 2m • w in ft x (0, T), . gg
^2|t=o = M)|2 in ft.

Temperature and the Rayleigh-Benard equations 121
In addition, the proof of Theorem 2.4 mentioned above adapts easily to
show that \u\2 also solves (3.98) and that U2 = |u>|2 (uniqueness of transport
equations, recall that divu = 0 and F > 0). Hence, uj£ converges (strongly)
to u; in L2(fi x (0,T)). Finally, the convergence in C([0,T];L2(fl)) of uc
also follows from the adaptation of the arguments of section 2.3: indeed,
we deduce from (3.97) (and the uniqueness) that u € C{[0,T];L2(Q)) and
from (3.98) that we have for all t € [0, T]
[ \u(x,t)\2dx + f f 2F\U\2 dxds = f |u/0(x)|2dx,
Jci Jo Jn Jn
while we deduce from (3.94) taking /3(u) = \u\2 for all s € [0,T]
[ \ue(x,s)\2dx + [' f 2F(pe)\ujc\2dxd(T = [ \ul{x)\2dx.
Jn Jo Jn Jn
Then, if sn->t (sn € (0,T]) and £„-►(), we already know that ojSn(sn)
n n
converges weakly in L2(Q) to u(t). The above equalities together with the
fact that, as claimed above, /0Jn/n 2F(pe*)\ue*12 dx dv^ /0'/o 2Fu2 dx da =
/o/n 2F|a;|2 dx da, show that cuSn (sn) converges (strongly) in L2(Q) to cj(t),
and we conclude.
The only claim remaining to prove is the weak convergence of F(p€)uj€,
F(pe)\LJe\2 respectively to Tuj,Fuj2- Since the proofs are entirely similar,
we only detail them in the case of F(p€)uj£. First of all, since p€ is bounded
in Li(0,T;Wl>fi{n)), there exists, for all 6 > 0, pe6 e L*(0,TiCl(G))
(for instance) such that pe6 is bounded in £s(0,T; Wl^(ft)) uniformly in
£, 8 > 0 and
lb*-pSllLV3(nx(0,T)) ^ 6' (3-99)
Then, we introduce Fn € C^(IR,IR) (Fn and F„ are bounded and
continuous on IR) such that (3.90) holds uniformly in n with F replaced by
Fn and Fn converges to F uniformly on compact sets of IR. Obviously,
Fn{pe6) is bounded in Lr(Cl x (0, T)) uniformly in £, <5, n and without loss of
generality, we may assume that Fn(jp£s) converges weakly to Fs as £ goes
to 0. Next, we estimate Fn{pes) - F(pe) in L1^ x (0,T)) and we have for

\>R
122 Navier-Stokes equations
all i*e(0,oo), 7 6(0,1),
rp
f dt[dx\Fn(pe6)-F(j><)\
Jo Jci
rp
< I cft/dr{F"(pS) + F(pS)}lW|y
Jo Jci
rp
+ C{sup |F»(J,)-F(3)|}+ / dt f dx\F(pe6)-F(pe)\
K\a\<R J Jo Jci
< CJsup|F"(5)-F(S)|}
K\3\<R J
rp
— +cL *■//** ^|o(+i) - (i,p^*+^-^i^)
+c J dtjdx (ipr+i) (i|pi>A+hP'6-p<\>t)
+ C sup {\F{x)-F{y)\ I \x-y\ < 7, |*|, \y\ < R}
L4(f2x(0,T)
< en(R) +o;H(7) + Ci?-(--a) + C7-(*-°) <5*~a
where we used (3.90) and (3.99) and en(R) -» 0 for i? > 0 fixed, w*(7) -> 0
n
as 7 —► 0+ for R > 0 fixed. Hence, letting first n go to +00 and 6 go to 0,
then 7 go to 0 and finally R go to +00, we deduce
In particular, we deduce that F6 converges in Ll(Q x (0,T)) to? as
n goes to +00 and 6 goes to 0. Therefore, we have only to show that
F^lpDu* converges weakly in L?(Q x (0, T)) (say) to T^lj. But this means
we can now assume without loss of generality that F(p€) is replaced by
Fn{pFs) which is bounded onfix (0,T) and satisfies: Fn(p€s) is bounded
in L"3(0,T;Cl(ft)). In other words, we may assume that F(p€) = Fe is
bounded in L°°(fi x (0,T)) n L^O.TjC1^)). Repeating the above
argument, i.e. approximating Fe (in L*(Q,T;Cl(Q))), we may in fact assume
that Fe is bounded in L°°(Q x (0,T)) n L3(orT;C*(?2)) for an arbitrary
A; > 0 and thus in particular Fe is bounded in Lq(0,T;C2(£l)) for any
Q € [l,oo).
Next, from the equation satisfied by u/c, we deduce that
-j£ is bounded in Lr(Q x (0,T)) + Loo(0,r;#-1(n)). (3.100)

Temperature and the Rayleigh-Benard equations 123
We then write (in the sense of distributions)
and we conclude easily since we can use the above bounds on ^j- and
on F£ to deduce that (/0 Fe ds) converges uniformly on fi x [0, T] and in
Li&T-C\U)) to (fiFds). D

4
EULER EQUATIONS AND OTHER
INCOMPRESSIBLE MODELS
This chapter is essentially devoted to the study of incompressible
(homogeneous) Euler equations, namely
-^- + (u-V)u + Vp = 0, divu = 0 inft x (0,oo); (4.1)
at
u\t=o = uo in ft (4.2)
with uq given on ft satisfying
divuo = 0 in ft. (4.3)
(Substracting a gradient term from tzo, we can always make such an
assumption.) Of course, we have to prescribe boundary conditions (unless ft = IR^
or in the periodic case) which take here the following form
u-n = 0 on dftx(0,oo) (4.4)
and we assume that uq satisfies
u0 • n = 0 on dft. (4.5)
Recall that we assume that ft, in the case of "Dirichlet boundary
conditions" (4.4), is a bounded, smooth, connected open set of IR^ (N > 2) and
n denotes the unit outward normal to 9ft. Let us also mention that we
could as well consider extensions of (4.1) with a right-hand side (a force
term) but we shall not do so here to simplify the presentation.
In fact, sections 4.1-4.4 are devoted to the above system of equations
while two variants are considered in the final two sections of this chapter
(sections 4.5-4.6).

A brief review of known results
125
4.1 A brief review of known results
The situation is completely different in two dimensions (i.e. N = 2) and
in dimensions N > 3. This is due to the following fact: if N = 2 (and
only if N = 2), u = curlu (a scalar if N = 2, v = |^ - |^) satisfies the
following equation, deduced from (4.1) by taking the curl of the equation
and observing that if N = 2, curl [(u • V)u] = u • V(curlu) when diva = 0:
|£ + (u • V)u/ = 0. (4.6)
(This fact was also used in chapter 3 in the context of Navier-Stokes
equations.)
When-N > 3, the only results which are available concern the existence
and uniqueness of smooth solutions (say continuous in t with values in Hs
for s > £ + 1, or in Cl,a for a € (0,1) in the case of a bounded domain)
on a maximal time interval [0,7q) where Xb € (0, +oo] and if To < oo the
solution's norm blows up as t goes to Tb_. In fact, it is even known—see
J.T. Beale, T. Kato and A. Majda [28], G. Ponce [391]—that ||u/(*)|U~
has to blow up (at a "certain integral rate") when t goes to To. It is
not known whether To can be finite or in other words if smooth solutions
become singular in finite time. We shall come back to this fundamental
issue in sections 4.3 and 4.4.
If N = 2, the Cauchy problem for incompressible Euler equations is much
better understood and we refer the reader to various existing surveys on
the question: see A. Majda [316], J.Y. Chemin [90].
Before we state results on the above problem, let us first define
precisely what we mean by solution of (4.1)-(4.2) ((4.4) in the case of Dirich-
let boundary conditions): we consider u € L°°(0, oo;L2(fi))N, satisfying
divu = 0 in V'(Q x (0, oo)) and u • n = 0 on dQ x (0, oo), such that we
have for all <p e CCG(Q x [0, oo))^ (for instance) vanishing on fi for t large
J dtldxu-(^ + (u-V)P(p\+ J dxu0-<p(x,0) = 0. (4.7)
Let us recall that we denote by P the projection on divergence-free vector
fields (div(Pv?) = 0 in fi, (P<p) • n = 0 on dQ in the case of Dirichlet
boundary conditions, curl P(p = curl^ in Q) that we used several times in
chapters 2 and 3.
If Q = IRN, the above formulation is replaced by the (equivalent) usual
weak formulation of (4.1), namely
r*^u• $&+(u• vw+kxu° • v(x'o)="■) (4.8)
for all <p € C%°(n x [0, oo)), div<p = 0 in Q x (0, oo), J

126 Euler equations and other incompressible models
and in the periodic case, we impose (4.7) for all (p G C°°(JRN x [0, oo)),
periodic in x, vanishing on IR^ for t large and satisfying div<p = 0 in
JRN x [0,oo). In general, (4.8) is contained in (4.7) but the converse might
not be always true (in the case of Dirichlet boundary conditions).
We may now state a few typical results that are available when N = 2.
Theorem 4.1. Let u0 G L2(Q)2 satisfy (4.3) (and (4.5) in the case of
Dirichlet boundary conditions). We assume that curltxo GXr(f2) for some
r G (l,+oo]. Then, there exists u G C([0, oo); Wl'r) in the periodic or in
the case of Dirichlet boundary conditions, u G tx0+C([0, oo); LqnWl'r) with
g = max(l,^) ifft = IR2, u e C([0,oo);L2 DW1^) for aU s e (l,+oo)
ifr = +oo and Q is bounded, u G C([0, oo); L2 D W*£) for all s G (1, +oo)
ifr = +oo and Q = 1R2, curltx G £°°(ft x (0,oo)) ifr = +oo.
Rirthermore, such a solution is unique when r = +oo, and if uq g
Wk*p(Q) where A; G IN, 1 < p < oo, A; > 1 + 2/p, resp. u0 6 Ck>a(Ti) where
k e IN, A: > 1, a e (0,1), then u G C([0, oo); >7^), ut e C([0, oo); W^^"1^),
resp. u e C([0,oo);C^a), txt € C([0,c»);Cfc-1'a).
Remarks 4.1. 1) We shall see below (Corollary 4.1) additional properties
of solutions when r € (l,oo).
2) It is possible to consider cases when Q = IR2 and curltxo € Lr(IR2)
(1 < r < oo) but we shall not do so here. We shall come back to the specific
case Q = IR2 in the next section.
3) Also, in the next section, we shall discuss the important borderline case
r = 1, different (and more precise) formulations of the equation (including
the vorticity equation (4.6)).
4) When 1 < r < oo, the uniqueness of the above solutions is not known.
We shall see in the next section that, for "generic" uq el2, there exists a
unique solution u € C([0, oo);L2).
5) The existence and regularity properties of the pressure are discussed
below in the case when r G (l,oo). If uq G Wk'p (resp. Ck'a) then the
pressure lies in C([0, oo); Wk>p) (resp. C[0, oo); Ck>a)).
6) In the above result, one could add in the existence of solutions the
conservation of energy, namely the fact that fQ \u(t)\2 dx is independent of
t
7) The growth of high order estimates of solutions as t goes to +oc is an
interesting open problem: for instance, if uq G H3(1R~), how does the H3
norm of u(t) behaves as t goes to +oo ? Only an upper bound of the form
ee is known.
8) We shall briefly sketch below parts of the above result leaving aside the
regularity results which follow in a direct way from the L°° bound on curl u
and the uniqueness in the case r = +oo originally shown in V.I. Yudovich
[494] and extensively studied (among other topics) in J.Y. Chemin [90].
Again, the crucial bound is the L°° bound on curltx which then implies

A brief review of known results
127
that u is, uniformly in r, "almost" Lipschitz (i.e. admits a t\ log t\ modulus
of continuity). Q
Corollary 4.1. Under the assumptions of Theorem 4.1 and if p < +oo,
for any solution satisfying the properties listed in Theorem 4.1, there exists
p € C([0, oo); Lq) with 1 < q < ~ if r < 2, 1 <q< oo if r = 2,1 < q < oo
ifr > 2 (replacing L°° by Co ifr — +ooj such that (4.1) holds (in the sense
of distributions). In addition, in the periodic case or in the case ofDirichlet
boundary conditions, we have: §j* € C([0,oo)',Lq), p € C([0,oo); Wl>q)
where q = r if r > 2, q € [1, r) if r = 2, <? = ^ if | < r < 2 ; in
tie periodic case, p 6 C([0,oo); W2'$) if r > 2 ; fjf- € C([0,oo); W1-'),
p e C([0,oo);Lq) where q = ^ if l< r < f.
Finalfrrtf-ft =-R2, we have: % € C([0,oo);L«), p € CflO.oofcW1*)
where ^ < q < r if r > 2, 1 < q < r if r = 2, 1 < q < ^ if
| < r < 2, D2p 6 CflO.oofcL1) ifr = 2, p € C([0,oo); W2-r/2) ifr > 2 ;
f£ € C([0, oo); W~l>q), p € C([0, oo); Lq) where l<^<5£7ifl<r<|
andPF-1-9 = {T€5,/(-A)-1/2T6i9}. D
Remarks 4.2. 1) The proof of Corollary 4.1, given below after the proof
of Theorem 4.1, shows in fact a bit more. When r = |, q = ^l. becomes 1
and we may replace L1 by the Hardy space H1 (see section 3.2 in chapter
3); similarly, we may replace W1,1 by {p € L1 , Vp € H1}. The same
remark holds when r = 2 or when q = 1, Q = 1R2 replacing L1 by H1,
W1'1 by {/ € L1, V/ € ft1} or W2-1 by {/ € Wl>1 , D2f € H1}. Finally,
using the results of R. Coifman, P.L. Lions, Y. Meyer and S. Semmes [95]
in the proofs below, we shall see that, in the case when Q, = IR2 (or in the
periodic case), ^ (and thus u - u0) € C([0, oo); W) where ^ < q < ^
if r < 2 where W = Lq if q > 1—indeed observe that §j* = -P((u ■ V)u),
(u • V)u € C([0,oo);W«) and P maps W into ft9 (if <? > §)—and £>2p €
C*([0,oo);Wr/2), Dp € C([0,oo);W«) where ^<g<^ifr<2. This
last observation on D2p follows from the fact that we have
-Ap = det(D24>) where u = V±4>=( jgM (4.9)
and D24> € C([0,oo);Lr). Indeed, by the results of [95], det (D2<j>) €
C([0,co);^/2) and thus ^- = (-A)"1 *££*> € C([0,oo);^/2),
since r/2 > 1/2.
2) The proof of Corollary 4.1 also shows that §* e C([0,oo); W~x<q)
when ft = ]R2andl<<?<^ifl<r<2, l<<?<ooifr = 2, 1<<?<
+oo if r > 2. We can even replace W~1^ by {/ € S' / (-A)"1/2/ € Ca}
where a = 1 — 2 in the case when r > 2. D

128 Euler equations and other incompressible models
Proof of Theorem 4.1. The proof is divided into several steps. Let us
recall that, for the reasons mentioned in Remark 4.1 (8)), we only prove
here the case when 1 < r < oo.
Step 1. Fundamental a priori bounds. We first wish to explain the
heart of the matter. First of all, multiplying (4.1) by u, we expect u to
satisfy the following local form of the energy identity (or energy conservation),
namely
i(?)W#-H} = ° «■«»
which implies, at least formally, the conservation of energy, i.e. the fact
that /Q \u\2 dx should be independent of t and thus a bound on u in
C([0,oo);Z,2).
Next, using (4.6), we deduce formally
2-{W} + dW{u\u>\r} = 0 (4.11)
hence integrating in x
\uj\t dx is independent of t. (4.12)
i
In particular, this yields a bound on u in C([0, oo);Lr) hence a bound
on Vu in C([0,oo);L2): indeed observe that divtx = 0 and u • n = 0 in
the case of Dirichlet boundary conditions. This bound, combined with the
C([0, oo);L2) bound on tx, yields a bound on u in C([0, oo); Wl>r) in the
periodic case, in the case of Dirichlet boundary conditions or if Q = IR2,
r > 2. Finally, if Cl = IR2, we observe that (4.1) yields
|t + P[(u . V)u] = 0. (4.13)
Since (u - V)u is bounded in C([0,oo);L7+5) in view of the preceding
bounds, |j*r is also bounded in C([0, oo); L7**) and thus u — u0 is bounded
in C([0,T\;L&) for all T e (0,oo).
Step 2. Navier-Stokes approximation. In the periodic case or in the
case when Q = IR2, there is no difficulty and we simply approximate (4.1)
by the Navier-Stokes equation
du
— + (u- V)u - i/Au + Vp = 0, divu = 0 (4.14)
keeping the same initial condition (4.2), where v > 0.

A brief review of known results
129
However, in the case of Dirichlet boundary conditions, a fundamental
diflSculty arises with boundary conditions. It is of course tempting to use
again (4.14) with Dirichlet boundary conditions, namely: u = 0 on dQ x
(0, oo). The difficulty which then appears is emphasized in the following
remark.
Remark 4.3. (2D Navier-Stokes —► 2D Euler, with Dirichlet boundary
conditions?). We consider (4.14) in the case of Dirichlet boundary
conditions: u = 0 on dCl x (0,oo). It is quite clear that a boundary layer will
form since solutions of Euler equations only satisfy u • n = 0 on d£l and
there is no reason why the tangential part of u should vanish (and they do
not in general!). It is not known if the "L2 strength" of this "layer" goes
to 0 and, more precisely, the limit as r goes to 0 of solutions of Navier-
Stokes equations with Dirichlet boundary conditions to solutions of Euler
equations is an important open problem. Equivalent formulations of this
problem can be found in T. Kato [238]. D
To circumvent that difficulty, we modify the Dirichlet boundary
conditions associated to (4.14) in the following way
u-n = 0, u; = 0 on dfix(0,oo). (4.15)
Then (4.14)-(4.15) is equivalent to
(4.16)
— + (u • V)uj - vAu = 0 in fi x (0, oo),
ot
uj = 0 on dQ x (0, oo), curltz = a;,
divu = 0 infix (0,oo), u-n = Q on dQ x (0,oo). )
Notice also that since we are in two dimensions
Au = Vdivu + V^curlu where V"1 = ( ^ J
hence, if divu = 0, Au -n = (n- V1)curlu and we deduce from (4.15)
Au-n = 0 on dfix(0,oo), (4.17)
and we may rewrite (4.14) as
du
— + P((u • V)ti) - vAu = 0 in fi x (0, oo). (4.18)
We then claim that the system (4.14)-(4.15) (or equivalently (4.16))
can be solved exactly as the usual Navier-Stokes equations (i.e. with the

130 Euler equations and other incompressible models
usual full Dirichlet boundary conditions)—see section 3.1—and thus that
we obtain (in all cases) for each i/>0a unique solution uu 6 L2(0,T; Hl)d
C([0,oo);L2), SgM. € L2(0,T;H~l) (for all T 6 (0foo)). AU these facts
follow from a modified energy identity that we briefly sketch (and from
(4.18) to obtain the regularity information on Off). Multiplying by u,
integrating by parts over dQ and using the fact that u-n = 0 on dfl x (0, oo),
we obtain
Using the boundary conditions, namely u • n = 0, cj = 0 on dQ, it is easy
to check that |^ • u = k\u\2 on 5Q, where /c is the curvature of dQ. Hence,
we obtain for some Co > 0 that depends only on ft
11 f \u\2dx + v f \Vu\2dx < Q>v f \u\2dS.
2 dt yn Jq Jdn
(Recall that in the periodic case, or if Q = 1R2, or in the usual Dirichlet
case we obtain an equality with a right-hand side that vanishes.) Then,
using a classical trace inequality, we deduce from the preceding inequality
\jt f \u\2dx + u f |Vu|2dx < cJf |Vti|2dx) (I \u\2di!\
where, here and below, C denotes various positive constants that depend
only on Q. Hence
1
dt
[ \u\2dx + v f \Vu\2dx < Cv f \u\2dx, (4.19)
Jn Jn Jo,
and our claim is shown. We have obtained in fact the following bound
/ \u(x,t)\2dx < eCut f \u0\2dx for all t > 0. (4.20)
Jo Jo
Let us finally observe that, for each v > 0, uu is in fact smooth for
t > 0: for instance, multiplying (4.16) by tuu and integrating by parts, we
immediately obtain for all T G (0, oo):
*k,(*)li»+ / s\Vuu(s)\2L2ds < C{T,v) for 0<t<T
Jo
or
*KII/fi+ / s\M\i*ds ^ C(T>u) for o<*<r n
Jo

A brief review of known results
131
Step 3. Existence when r > 2. We treat here the case when r > 2 and
we first consider either periodic or Dirichlet boundary conditions. Then, we
explain how to modify the proof when ft = IR2. First of all, we obtain an
estimate on uu in C([0, oo);Lr). We simply multiply (4.16) by \uju\r"m2(^u
and obtain in all cases (either periodic or Dirichlet boundary conditions)
for all t > 0
- [ Mt)\r dx + v(r-l) [ ds [ \Vuu\2 \uu\r-2 dx }
T Jn Jo Jq I
= - / \cuvlu0\rdx.
rJn
(4.21)
This identity yields the desired bound. Of course, the preceding calculation
has to be justified. This is not difficult and we leave it to the reader
(multiply (4.16) by \TR(uu)\r-2TR{uju) where Tr(lju) = max(mm(uju,R), -R)
for R > 0 and let R go to +oo, observing that for instance uu 6 £2(0, oo;
Hl) nC([0,oo);L1)). Observe also that y/uu)u is bounded in L2(0,oo; Hl)
(take r = 2 in (3.21)).
The bound on uu yields a bound on uu in C([0,oo); W1,r) exactly as in
step 1. Then, from the equation (4.16), we deduce that ^f- is bounded in
L2(0, T; W-1*) for all T € (0, oo), 1 < q < 2 for instance. This is enough to
ensure that uju is relatively compact in C([0, T)\ Lr - w) (recall that Lr — w
means Lr endowed with the weak topology) by the observation detailed in
Appendix C. Hence, uu is relatively compact in C([0,T]; Wl'r - w) by the
same argument as in step 1 and uu is relatively compact in C([0, T]; Lq) for
all 1 < q < oo, T 6 (0, oo) by the Rellich-Kondrakov theorem. Extracting
subsequences if necessary, we may thus assume that, as v goes to 0, uu
converges to somen in C([Q,T]\Wl*r -w)C\C{[Q,T\\Lq) for all 1 < q < oo
while u)u converges to some u in C([0,T];Lr - w) (for all T e (0,oo)). In
particular, we have on passing to the limit: u(0) = tz0, ^(0) = curltx0 in fi,
fQ \u(t)\2dx < JQ \u0\2 dx for all* > 0 in view of (4.20),
u 6C([0,T];I2) , divtx = 0, curltz = o; infix (0,oo) (4.22)
-^ + div {uu) = 0 in ft x (0, oo) (4.23)
at
and u • n = 0 on dQ x (0, oo) in the case of Dirichlet boundary conditions.
Furthermore, we may pass to the limit in (4.18) and we obtain
du
— + P((u-V)u) = 0 in flx(0,oo). (4.24)
We claim that this yields (4.7) and we explain why (4.7) holds in the
case of Dirichlet boundary conditions: indeed, Vu 6 £°°(0,co;Lr), u 6

132 Euler equations and other incompressible models
L°°(0,oo; Wl>r) and thus we have for all tp € C°°(fi x [0,oo)) (vanishing
for t large)
0 = / dt /dz{-u-^ + (u-V)u-JV}- Idxu0-<p(x,Q)
= / dt I dxj-tx- -^ + div(tx <g) tx) • JfyJ - / dxuo -<p(x,0)
= - / dt f dxfu~ + u- (u-V)[P<p]} - /dxtxo-v?(x,0)
since divtx = 0 and tx • n = 0 on #f2 x (0, oo).
In order to complete the proof of Theorem 4.1 in this case, it only remains
to show that u € C([0,oo);Lr) and thus tx € C([0,oo); Wx>2)7 This fact
is a consequence of (4.23). Indeed, by general results due to R.J. DiPerna
and P.L. Lions [128] and recalled in section 2.3 on transport equations
with divergence-free vector fields, we see that u is necessarily equal to the
unique renormalized solution of (4.23) which satisfies u 6 C([0, oo);Lr)
(and Jn \u(t) \r dx is independent of t > 0). We have used here the fact that
r > 2 and thus, in particular, tx € C([0, oo); W1*2) while u 6 C([0, oo); L2).
Remarks 4.4. 1) Let us observe that since u 6 C([0,oo);Lr) and
/n \u(t)\r dx is independent oft, the identity (4.21) immediately yields the
convergence of uu to u in C([0, T]; Lr) and thus of uu to tx in C([0, T]; W1>r)
for all T 6 (0,oo).
2) (4.24) shows that %fi e C([0,oo);Z/*) for all 1 < q < r, and,
multiplying (4.24) by tx, one can easily justify the energy conservation since we
have for all £ > 0
f\u{t)\2dx- f\u0\2dx = -2! ds[dx[(u-V)u\-Pu
Jn Jn Jo Jn
= -2 dsl dx[{u- V)tx] -tx
Jo Jn
= - [ dsf dx(u-V)\u\2 = 0.
Jo Jn
This verification only requires |tx|2|Vtx| to be integrable and this is the
case as soon as tx 6 Lco(0,oo; VV1,r) with r > § since, in this case, |tx|2 6
L°°(0,00; L3) by Sobolev inequalities. □
We now briefly explain how to modify the above proof in the case when
Q, = IR . All the steps of the proof are easily adapted, replacing C7([0,T];
L«) or C([0,T};L2) by C([0,T];Lfoc) or C([0,T]; Lj^): in particular we
obtain tx 6 Loo(0,oo;L2)nCr([0,oo);Ll2oc), Vtx 6 L°°(0,oo;Lr), u 6 C([0,oo);

A brief review of known results
133
W*£) satisfying u(0) = u0 in IR2, divu = 0 in IR2 and
du
jj; + P((u • V)u) = 0 in R2x (0,oo). (4.25)
In particular, f*f € L°°(0,oo;L^) hence u - u0 G C([0,oo);L^). This
yields the fact that u G C([0, oo); L2 n Wl>r) since ^ < 2 < r. D
Remark 4.5. The proofs "given" in Remark 4.4 are easily adapted to
the case when £1 = IR2 and yield the conservation of energy together with
the convergence (up to the extraction of subsequences) of uu in C([0,T];
L2 H Wl*r) for all T G (0, oo). □
Step 4. Existence when 1 < r < 2. We now treat the case when
1 < r < 2 and we begin again by excluding the case when Q = IR2 and
thus considering periodic or Dirichlet boundary conditions. In fact, we shall
give the proof in the case of Dirichlet boundary conditions, the periodic case
being similar and even somewhat simpler.
We shall deduce the existence of a solution when 1 < r < 2 from the
case we just treated. We then introduce Uq G Cl(Q) satisfying Uq • n = 0
in fi, divug = 0 in Q and such that Uq converges in W1%r to uq as e
goes to 0+: the existence of Uq can be obtained by considering Pug where
Uq G C°°(Cl) converges to uo in Wl*r. Next, we denote by ue a solution of
the Euler equation provided by the preceding steps (it is in fact unique) and
by ue = curlu*. We have the following information: u€ G C([0, oo); Wl>q),
u€ G C([0,oo);L«) for all 1 < q < oo. (4.7), (4.13) and (4.6) hold with u,u
replaced respectively by u,lj. Also, we have
/ \u£(t)\rdx = / H\rdx ^ / \u0\rdx
Jq Jq Jq
defining Uq = curlug, ojq = curltx0. In particular, as in step 1, we see that
ue G L°°(0,oo; Wl>r) and without loss of generality we may assume that
u£ converges weakly to some u e L00^, oo; Wl>r) which satisfies u • n = 0
on dVt x (0,oo), divu = 0 in fi x (0,co).
Then we invoke the compactness results shown in R.J. DiPerna and P.L.
Lions [128] essentially recalled and proved in section 2.3—only the L°° case
was established there but the general Lr case follows as well by considering
Tr(u£) instead of u£—and we deduce that lj£ converges in C([0, T]; Lr) (for
all T G (0,oo)) to the unique renormalized solution u of (4.6) satisfying
u\t=Q = ljq in Q. Here, as in [128], renormalized solution means that we
have for all /3 G C£°(IR;1R)
^-^ + div (u(3(tu)) =0 in V'{£1 x (0, oo)). (4.26)

134 Euler equations and other incompressible models
Therefore, by the same argument as in step 1, ue converges in C([0,T];
Wl>r) (for all T e (0,oo)) to u and we can pass to the limit in (4.7)
completing the proof of Theorem 4.1 in this case. Notice, by the way, that,
in particular, uc converges to u in C([0,T]; L2) and thus Remark 4.1 (6) is
deduced in that case from Remark 4.4 (2).
In the case when Q = 1R2, the same proof as above applies (we now
simply regularize uq by convolution) and shows that lj€ converges to u = curlti
in C([Q,T];Lr) (for all T € (0, oo)) which is the unique renormalized
solution of (4.6) in 1R2 x (0,oo) satisfying a^o = vo in 1R2. Since u€
is bounded in L°°(0,oo;L2), we deduce that Vu€ converges to Viz in
C([0,T};Lr), ue converges to u in C([0,T\:W££) for all T e (0,oo) where
u 6 C([0,oo); Wjf) nX°°(0,oo;L2), Vu € C([0,oo);Lr). It only remains
to show that ue converges to u in C([0, T]\ L2) (for all T € (0, oo)) and that
u—uq € C([0, oo); Ll) when 1 < r < 2. In order to do so, we follow Remark
4.2 (1): (u£ • V)u£ is bounded, in view of [95], in C([0,oo), Wnfr). Since
3gL = -P((ue • V)u£) and £fe > § if r > 1, we deduce that Q£ is bounded
in C([0,oo); W&) and thus u€ - ue0 is bounded in C([0,T];H#*) for all
T G (0,oo). On the other hand, since V(txc — Uq) converges to V(tz — uo)
in C([0,T];Lr) and 1 < r < 2, we deduce from Sobolev embeddings that
ue-u% converges to u—uq in C{[0,T]\L^) for all T € (0, oo). Hence, by
interpolation, ue-v,Q converges to u-u0 in C([Q,T]\Lq) for 1 < q < ^r
and our claims are shown since Uq converges to uo in L2 by construction.
D
Remark 4.6. It is possible to give a much more elementary proof of the
convergence of u€ to u in C([0,T]; L2) (V T e (0, oo)) which also yields the
fact that u 6 u0 + C{[Q,oo)\LqC\Wl'r) for £- < q < r (but does not reach
L1!) when 1 < r < 2. One simply observes that
j=i y
and t^u£ is bounded in C([0,oo); L1 D I*^). Hence, we have
where /£ is bounded in C([0,T]; L*) for all T € (0, oo), 1 < q < ^, while
ge is bounded in C([0, oo);Lr). These two facts imply easily that uc—Uq
is bounded in C([0,TJ; Is) for ^ < 5 < ^, 0 < T < oo. Observe that
tf£ = (-A)"1/2^6-^) is bounded in C([0,T]; I9) while -Atf£ is bounded

A brief review of known results
135
in C([0,oo);Lr), and this is enough to show our claims following the end
of the proof of Theorem 4.1. □
We now turn to the
Proof of Corollary 4.1. We begin with the existence of p which is a
straightforward consequence of (4.7). Indeed, we have < |^ + dW(u ®
u),(p >wxv= 0 for all tp € C$°(Q x (0,oo)) such that divtp = 0. This
implies the existence of a distribution p such that (4.1) holds in the sense
of distributions. Next, we need to show the integrability requirements
indicated in Theorem 4.1. Since P is bounded from W^q into Wltq for
all 1 < q < oo, we deduce from the weak formulation (4.7) (using P<p for
arbitrary smooth tp) that §* G C([0, oo); W-l«) for 1< q < ^ if r < 2,
1 < q < ooifr = 2, 1 < q < oo ifr > 2sinceu € C([0,oo);L2nL^) if r <
2 {L2C\Lq if r = 2 for 2 < g < oo, L2nL°° if r > 2) by Sobolev embeddings.
The integrability of p then follows since Vp = — ^ — div (tx <g> tx).
In the periodic case or when Q = H2, the argument is a bit simpler: we
just write
aT " - E s- p(%")' vP = -s - iv (u ® U).
i=i J
All the claims listed in Corollary 4.1 follow easily from the bounds on
u, Sobolev embeddings, the fact that |jj* = -P((u • V)u) when r > | and
that we have
-AP - div{(u.vW =±^=± ^<«.«,)• n
(4.27)
Remarks 4.7. 1) The proof we gave of Theorem 4.1 shows a few additional
properties of at least one weak solution. The first one is the fact that
u = curlu is a renormalized solution of (4.6), that is it satisfies (4.26).
This fact can be recovered a posteriori when r > 2 using a regularization
technique as in Lemma 2.3 (section 2.3, chapter 2) for the equation (4.6)
which is satisfied in the sense of distributions, but it is not clear that this
can be done when 1 < r < 2.
The second property we want to mention is the local form of the
conservation of energy, namely equation (4.10): our existence proof shows it
holds for at least one weak solution if r > |. Indeed, in that case we obtain
the convergence of u*,pE in C([0,T];L3), C([0,T];L^2) respectively for all
T 6 (0, co) and we can recover (4.10).

136 Euler equations and other incompressible models
2) Concerning the convergence of solutions of Navier-Stokes equations
to solutions of Euler equations (with the boundary conditions modification
introduced in step 2 of the proof of Theorem 4.1 in the case of Dirichlet
boundary conditions), let us mention that when 1 < r < 2 it is possible
to show that uu converges to u in C([0,T];Lr) and thus Vuu converges
to Vu in C([0,T];Lr) for all T 6 (0,oo) using the (duality) method of the
last section of R.J. DiPerna and P.L. Lions [128].
3) In the case of Dirichlet boundary conditions, it is possible to say a bit
more about the regularity of p when r > 2. Indeed, we have (4.27) and thus
Ap 6 C([0,oo);Lr/2) (Ll being replaced by H1 when r = 2). In addition,
one can show by the proof of Theorem 4.1 that there exists a weak solution
(u,p) such that, denoting by k the curvature of dQ, we have
|P = -K\u\2 on dftx(0,oo). (4.28)
This is what we expect from (4.1) since (u • V)u • n = — (u • V)n • u (u • n = 0
on dQ and thus (u • V)(u • n) = 0 on dQ). Notice that if Q C TRN,
N > 2, k\u\2 is replaced by the "curvature quadratic form" applied to u.
Combining (4.27) and (4.28), it is not difficult to show by elliptic regularity
that D2p 6 C([0,oo);Lr/2(ft)). D
4.2 Remarks on Euler equations in two dimensions
This section is essentially devoted to a discussion of the Euler equation in
two dimensions when the initial condition uq only lies in L2(Q) or belongs
to L2 and is such that curltxo is a bounded measure. Roughly speaking,
this corresponds to the case when r = 1 in the preceding section, a case
which was of course excluded from our analysis. This borderline situation
is not only very interesting mathematically but also corresponds to various
relevant physical situations. We refer the reader to the fundamental series
of works by R.J. DiPerna and A. Majda [129],[130] on this subject for a
more complete discussion of the background of this issue (and of 'Vortex
sheets"). Only at the end of this section shall we leave this issue to mention
a few other questions of interest on Euler equations in two dimensions.
Let us now describe what we discuss below. First of all, if uq € L2,
divuo = 0 (and uo • n = 0 on dQ in the case of Dirichlet boundary
conditions), the existence, uniqueness and stability of solutions are completely
open. However, using the regularity which is available for smooth Uo (and
a few simple tricks), we shall see that there exists a G& set of initial
conditions in L2 (that is a countable intersection of dense open sets in L2) for
which there exists a unique solution of (4.1)-(4.4) in C([0,oo);Z,2) with a

Remarks on Euler equations in two dimensions 137
conserved energy (i.e. Jn \u{t)\2 dx is independent of t). As we shall see
this is a "cheap" result whose only merit is to indicate that the problem is
well posed for most initial conditions in L2.
The other angle of attack that we discuss in this section consists in
pushing as much as we can towards Ll the arguments developed in the
previous section, which are obviously based upon the transport equation
(4.6). Since (4.6) involves a divergence-free vector field we expect solutions
of (4.6) to preserve the initial distributions function (or in other words,
we expect the decreasing rearrangement of solutions to be independent
of t)—and this is precisely the case with renormalized solutions. This
will lead to two different kinds of results which are essentially optimal for
this type of approach. However, we shall remain rather "far" from Ll
or bounded measures. We shall hot "address here in detail the problem
of vortex sheets (uq € L2, curluo is a bounded measure) and we refer
instead to R.J. DiPerna and A. Majda [129],[130] for a discussion of the
possible phenomena involved—see also the presentation of their results in
L.C. Evans [141]. Let us also mention the existence of global weak solutions
in the case when uo € L2, curltxo is a bounded measure such that (curluo)4"
(or (curluo)-) € L1 which was obtained by J.M. Delort [118]; a simpler
proof was proposed by A. Majda [318].
We now begin with our generic result. We introduce the Hilbert space
H (for the L2 scalar product) defined by: H = {u0 e L2(Q)2,divu0 =
0 in I>'(f2), uo -n = 0 on dQ}. In the case of Dirichlet boundary conditions,
H = {uo e L2(JR2)2 , divuo = 0 in P'(IR2)} if Q = IR2, H = {u0 €
L2oc(IR2)2 , u0 is periodic , divtx0 = 0 in 1?'(IR2)} in the periodic case.
Theorem 4.2. There exists a decreasing sequence of dense open sets On
in H such that, for any uo € Hn>i ^n» there exists a unique solution u €
C([0,oo);L2)2 of (4.1)-(4.2) (and (4.4) in the case of Dirichlet boundary
conditions) such that fQ\u(t)\2 dx is independent oft>0. Furthermore,
for any weak solution u € L°°(0,oo;L2)2 n C([0,oo);L2 - w)2 of (4.1)-
(4.2) (and (4.4) in the case of Dirichlet boundary conditions) such that
fn \u(t)\2 dx < fn \u0\2 dx for all t > 0, then u = u in Q x (0,oo).
Proof of Theorem 4.2. The proof is based upon the fact that if u0 e
L2 D C1,a for some fixed a e (0,1), then, see Theorem 4.1, there exists a
unique solution u e C([0,oo); L2 D C1'*) of (4.1),(4.2) with u0 replaced by
uo (and (4.4)) such that, in particular, ||Vtr||L~(nx(o,T)) < C(uo,T) for all
T € (0, oo), and we can always assume that C(Uo,T) > 0 is nondecreasing
with respect to T.
The second ingredient which is basic for our proofs is the following
(essentially classical) observation. If u is any weak solution of the Euler equation
as in Theorem 4.2 then we claim that we have for all t e [0,T]
\\u(t)-u(t)\\L2 < e^^Wuo-uoh*- (4.29)

138 Euler equations and other incompressible models
Indeed, on the one hand we have for all t > 0
f\u(t)\2dx< f\uQ\2dx, f\u(t)\2dx = [\u0\2dx (4.30)
Jn Jn Jn Jn
and on the other hand we deduce from (4.7) using tp = u (a choice that can
be justified by a simple approximation argument, take tpn = P{$n) where
(fin converges in C1,a to u)
/ u(t) • u(t) dx— \ ds u- J — + (u- V)tZJ dx = uo-u0dx
or equivalently using the equation satisfied by u (and the fact that div u =
0)
/ u(t) • u(t) dx — I ds u- {[(u—u) • V]xZ} dx = I uq -u0 dx.
Jn Jo Jn -/n
Next, we observe that /0 ds fnu • {[(u-u) • V]TZ}cb = /0 rf^ /n(H — il) •
V(^)dx = 0 using the fact that divu = divu = 0 (and (u—u) • n = 0 in
the case of Dirichlet boundary conditions), and thus we obtain for all t > 0
/ u(t) • u(t) dx = uouodx+ ds I dx(u—u)-Vu-(u—u). (4.31)
Jn Jci Jo Jn
Combining (4.30) and (4.31), we deduce finally for t e [0,T]
/ \u(t)-u(t)\2dx < J \u0-UQ\2dx + 2C(u0,T) I ds l \u(s)-u(s)\2 dx
Jn Jn Jo Jn
(4.32)
and we deduce (4.29) from (4.32) using GronwalFs inequality.
We then introduce, for n > 1, the following open set On
On = |n0 e H J 3 u0 G L2 n C1'" , ||tx0—tT0|U> < \ e~c^nA (4.33)
and we wish to check the statements listed in Theorem 4.2 for this choice of
On. Since On contains L2nC1,a(n#), it is not difficult to check that On is
dense in H : indeed, (L2nCl>°)2 is dense in (L2)2 and P((L2n W1'")2) =
(L2ncl>a)2nH.
Next, if uq G fln>i ^n, it is not difficult to show the uniqueness part of
Theorem 4.2. Indeed, if uq € f]n>i ^*, there exists, for each n > 1, some
U%eL2nCl*anH such that \\uq-T%\\L2 <ie-C(S5ln)n# Then, if u1,*!2

Remarks on Evler equations in two dimensions 139
are two weak solutions as in Theorem 4.2, (4.29) implies supt€j0nj ||ii1(t)—
«2(*)IU2 < ^. and the uniqueness is proven.
The existence part also follows from (4.29). It is clearly enough to show
that un is a Cauchy sequence in C([0,T]; I?) (for all T € (0, oo)). Then, if
n, m > T, we deduce from (4.29) that we have
sup W^it)-trwwv
*€(0,T]
< min{ec^^T, ec^^T} [K-u0\\l> + IN -Kh>]
< min{eC(^'n)T , e£W'm>T) rI-g-C(tIJ,n)n + J_ e-C<^,m)m\
si+i.
n 77i
Our claim is shown, thus completing the proof of Theorem 4.2. □
We next discuss some other existence results based upon the fact that
we expect u = curltx to satisfy (4.6), i.e. a transport equation with
a divergence-free vector field (namely u). Thus, the distributions
function of u(t), namely the function n^t) on (0,oo) defined by /^(^(A) =
meas{x / |u;(x,£)| > A} for A > 0, should be independent of t. Notice
that this is precisely the case with renormalized solutions of (4.6). Indeed,
(4.26) yields the fact (integrating in x) that /n /3(u(t)) dx is independent
of t for all /?.
From now on, in order to avoid unnecessary technicalities, we restrict
ourselves to the case of Dirichlet boundary conditions (or to the periodic
case).
The first type of result we wish to discuss consists in pushing the proof
of Theorem 4.1 towards Ll. Looking carefully at the proof of Theorem 4.1,
we see that we only need a bound on u in W1,1. But, if we introduce the
stream function, i.e. the solution of
-AV> = o; in ft, ^ = 0 on dQ (4.34)
in the case of Dirichlet boundary conditions for instance assuming that
Q is simply connected (to simplify the presentation), then u = ( *I* , )•
Therefore, Du G L1 if and only ifD27p e L1, and by classical elliptic theory,
this is the case in two dimensions as soon as u G L1, fn \u\ | log |^(t)|| dx <
oo. Observe in addition that JQ \u(t)\ (log |u;(i)|| is, at least formally,
independent of t. Once these observations are made, it is not hard to copy

140 Euler equations and other incompressible models
the proof of Theorem 4.1 and to show that if uq 6 £2(ft), divtxo = 0 in
fi, tio • n = 0 on dQ, uo = curltxo 6 Ll(Q) and fQ \uo\ | log cj0\dx < oo,
then there exists a solution u 6 C{[0,oo)]Whl(Q)) of (4.1), (4.2), (4.4)
such that Jn \u(t)\2 dx is independent of £, u 6 C([0,oo); Ll(Q)) (and even
"L1 log L1") is a renormalized solution of (4.6), i.e. satisfies (4.26).
Indeed, regularizing cj0, we obtain a sequence of solutions of (4.1) (uc1ue)
and, by the results of R.J. DiPerna and P.L. Lions [128], we check that
u£ converges in C([0,T])Ll) (VT 6 (0,oo)) to some u 6 C^oo);!^).
In addition, u€\\og |u;ff|| also converges in C([0,T];L1) (V T 6 (0,oo)) to
u;|log u\. Hence, uc converges to some u in C([0,T];L2) (V T 6 (0,oo))
and we conclude.
However, if we follow this argument, we can ask for less information on
u and we might simply try to deduce from the invariance in t of /-^(t)
(= flu*) some compactness in L2 (or C([0,X];L2)) for all T 6 (0,oo) of
u€. This leads to the following question: find the optimal distributions (or
rearrangement) invariant class for u such that the corresponding velocity
field u belongs to L2(Q) and then prove the existence of solutions of (4.1)
with such initial conditions. This question can be solved completely using
some symmetrization techniques. In order to do so, we need some notation.
First of all, if a; 6 Ll(Q), we denote by u* the decreasing rearrangement
which is the inverse function of /i^,*. in other words, u* is the unique non-
decreasing function in Lx(0, \Q\) (\Q\ = meas(fi)) such that u* e Ll(Q, \Q\)
and flu* = fiv a.e. In two dimensions, we denote by a;" the Schwarz
symmetrization of u (or spherically symmetric decreasing rearrangement), i.e.
the unique spherically symmetric function in Ll(Q*) which is nondecreas-
ing with respect to r = |x| such that /i^u = ^ a.e., where fi" is the ball
centred at 0 with the same volume as Ct (or in other words with a radius
Rq given by (^)l/2). Obviously, w«(x) = u;*(tt|x|2) a.e. in fi».
From now on, we restrict ourselves (to simplify the presentation) to the
case of Dirichlet boundary conditions, assuming in addition that Q is simply
connected (even though the results below hold in general), and we introduce
the stream function, i.e. the solution of (4.34). We then recall the following
general comparison result due to G. Talenti [462]
t/>B < # a.e. in fi" (4.35)
/ \Vip\2dx < / |Vtf|2dx (4.36)
Jo. Jn*
where \I> is the solution of (4.34) with Q,u replaced respectively by fi^a/",
namely
-A^ = cj" in fiB, # = 0 on dQ*. (4.37)

Remarks on Eider equations in two dimensions 141
Recall that # is given by the explicit formula
rRo
38)
tf(l) = f °- f SJ(s)ds, #'(|x|) = -riy / sJ(s)ds. (4
7|i| r J0 \x\ Jo
Then, if we observe that u = ( ~*B_ J, we see that the optimal
rearrangement invariant class ensuring that u G L2(Q) is given by
lueLl{Sl)/ f |V#|2<te<ooj.
But we have easily
f \V*\2dx = 2tt/ °r(*'(r))2dr = 2tt / Uf su\s)ds\ dr
= 2n f -( [*su;*(7rs2)ds\ dr
In conclusion, the optimal class is given by the space
L\(Q) = lu;eLl(n)/ f U u*{s)ds}2j < ooj. (4.39)
For instance, in view of the observation made by A.B. Mergulis [341],
it contains the Orlicz class of functions u in L1^) such that fQ \u\ •
\hg\uj\\^2dx<oc.
We then have
Theorem 4.3. Let uo = curltzo 6 L^(Q) (with divuo = 0 in f2, uq • n = 0
on dQ). Then there exists a solution u 6 C([0,oo);L2(ft)2) of (4.1), (4.2),
(4.4) such that fQ \u(t)\2 dx is independent oft>0.
Remarks 4.8. 1) The results due to J.M. Delort [118], that we mentioned
above, show the existence of a weaker solution of (4.1), (4.2), (4.4) when
uo e Ll(Q), u0 e L2(Q) since the conservation of energy is not known in
that case and u e L°°(0,oo;L2(ft)).
2) If Q = ft" (is a ball!) and if we consider u(x) = j^ | log |x||~a, then
one can check that u e Ll if and only if a > 1, u e L\ if and only if a > \
while D2x/j (= D2V) e Ll(Q) if and only if a > 2. □

142 Euler equations and other incompressible models
Proof of Theorem 4.3. Let us define Uq = Tn(u>o) for n > 1 and let tij
be the unique element of L2(Q)2 such that: curl u£ = u#, divuj = 0 and
Uq -n = 0 on dd. In view of Theorem 4.1, there exists a unique solution
un e C([0,oo);L2(ft)2) of (4.1), (4.2) and (4.4) with u0 replaced by «£,
such that curltxn € L°°(Q x (0, oo)). Of course, we wish to recover the
existence result stated in Theorem 4.3 by passing to the limit as n goes to
+00. To this end we recall that /uw»(t) = /iw™ for all t > 0, n > 1 and thus
we have, writing by a;J1 = wn(t),
(««T = («o )* = wo A n. (4.40)
In particular, in view of the derivation above of Z^fi), «{> is bounded in
£2(ft)2j_and since |un(t)|L2 = |«o U2> we finally deduce that un is bounded
in_C([0,oo);L2(fi)2).
Next, because of (4.7) and since P maps {(p 6 C1,a(fi)2 , <p = 0 on 9fl}
into C1,a(fi)2 for any 0 < a < 1, %g- is bounded in C([0,oo);X4) for
any 0_< a < 1 where Xa is the closure of C£°(ft)2 in C1,a(fi)2 (= {v €
C1,a(fl)2 , v = Vv = 0 on dQ}). Then, we deduce from Appendix C that,
extracting a subsequence if necessary, we may assume that un converges in
C([0,T];L2(Q)2-w) (for all T G (0,oo)) to some u e C([0,oo);£2(ft)2-u;)
satisfying (4.2) and (4.4). In addition, we have
l \u(t)\2dx < Urn [ \un(t)\2dx = lim f \v%\2 dx. (4.41)
J CI n-»oo JQ n-»oo Jci
We are going to show below that we have
un(*n)->u(r) in L2(Q)2 if 0<tn-+t>0. (4.42)
n n
If this claim were proved, we would deduce on the one hand that Uq
converges in L2(Q)2 to u0 and on the other hand that for all t > 0
/ \u(t)\2dx = liminf f \un(t)\2dx
Jq n Jn
= liminf / \u%\2dx = / \u0\2dx.
n Jn Jn
Hence, in particular, u 6 C([0,oo);L2(£2)2), and this fact combined with
(4.42) would show that un converges to u in C([0,r];L2(fi)2) for all T €
(0, oo). Once this convergence is shown, Theorem 4.3 follows easily.
Therefore, we have only to prove (4.42). Let us first remark that we
already know that un(tn) converges to u(t) weakly in L2(ft2). In addition,
since &4>n(tn) is bounded in Ll(Q) (and VV>n(*n) is bounded in L2(Q)2),

Remarks on Euler equations in two dimensions 143
we deduce from elliptic regularity that V^n(*n) and thus un(tn) converge
in L*(ft)2 for all 1 < p < 2. Finally, let us observe that (4.40) yields for all
T 6(0, oo)
sup sup I \ I ut (sYds) ► 0 as e-»0+. (4.43)
n>l t€[0,T]J0 \Jo J *
In view of these facts, (4.42) is deduced from the following
Lemma 4.1. Let un be bounded in L1(Q) and satisfy
•|0| / rt \ 2
nfMr
F( fl * .\2 dt
sup / I / c*/* as ) — —► 0 as e —> 0H
n>iyo \Jo / t
< 00,
2
(4.44)
Then, un is weakly relatively compact in Ll(Q) and, extracting a
subsequence if necessary, we may assume that un converges weakly in Lx(fi) to
some u € L\ (Q). Denoting by V>n the solution of (4.34) (with u replaced
by un), we then have
ipnUn^ipuj weakly in Ll (Q), ipn->ip inH£(Q). (4.45)
n n
Proof of Lemma 4.1. The weak compactness in L1 follows from the
following observations
r(M,7*0Ma,o«T'forai1
•mi
MM 1 M I.I. MM \ IJM
e> 0,
and
/ a;*flk = sup / \LJn\dx, for all £ > 0. (4.46)
«/0 \A\<eJA
Hence, we may assume that un converges weakly in Ll(Q) to some u
and, similarly, that u;£ converges weakly in L1(f2") to some u. Since
u;£ is radially increasing, we may assume without loss of generality that
u;£ converges to lj a.e. in ft" and thus u;£ converges to u in L1(fi").
In particular, one deduces easily that u € LK^t) and, since fQu*ds <
iinin So un ds = f0&* ds1 u € L\ (Q). Therefore, as shown above when we
introduced L%(Q), unij;n is bounded in Ll(Q)1 ujip € Ll(Q) (observe indeed
that /n \uip\ dx < /nf a/ty* dx < /Qi a;"*" dx)
f \Vipn\2dx = / un^ndx, f |VV>|2dx = / ^ dx (4.47)
Jn Jn Jq Jq

144 Euler equations and other incompressible models
[ |Vtfn|2dx = / Jj!ndx, f |Vtf|2dx = / L>**dx (4.48)
Jn* Jn» */n« «/n«
with the notation introduced before Theorem 4.3 (and obvious
adaptations).
We first claim that cj£#n, V#n converge, respectively, in L1^"),
L2(Q*)2 to d>\£, V#. Since ^,#n are radially nonincreasing, non-negative
and V#n = -7^7 Jq 5cj* (s) ds, we have only to show in view of (4.48)
sup / |V*n|
n>\J\x\<£
dx —► 0 as € —► 0+ .
But this is immediate in view of (4.44) since we check as in the definition
ofL^Q)
Next, we check that ipnujn is weakly relatively compact in Ll(Q). Indeed,
for any measurable set A
I IpnUndx < f iiJndx < f Vn^dx
J A J A* JA*
and our claim is shown since Stncj^ converges in Ll(Cfl).
It only remains to show that ipnUn converges weakly in Ll(Q) to tyuj.
Recalling that \j)n converges a.e. to t/>, we see that for all R € (0,oo),
TR(il>n)vn converges weakly in Ll(Q) to Tr(iP)lj since un converges weakly
in Ll(Q) to uj. We can then conclude observing that
J \^n-TR^n)\\un\dx = f (\1>n\-R) + \un\dx
< J (1>*n-R)+u*ndx < f (9n-R)+u>*ndx
< / *null(yn>R)dx
Jn*
and thus, since \I>n converges in Ll (Q*) (in Hl (ft") in fact), ^n^n converges
in X1(fitt), these integrals can be made, uniformly in n > 1, arbitrarily small
letting R go to +oo. □
Remark 4.9. Let us mention in passing the so-called "vortex-patches"
problem which was settled recently. One considers the case of an initial
vorticity (in the case when Q = IR2 for instance) ljq which is constant (say

Remarks on Euler equations in two dimensions 145
A) and supported on a bounded, smooth domain D. Then, the
corresponding solution of (4.1), (4.2), (4.4) satisfies for all t > 0: cv(t) = Al^t) for
some measurable set D(t). It is proved in J.Y. Chemin [88] (a simplified
proof can be found in A. Bertozzi and P. Constantin [58], see also Ph.
Serfati [419]) that D(t) is in fact a bounded, smooth domain for all t > 0.
□
We conclude this (two-dimensional) section with a different topic which
concerns the case when Q = IR2 and when we no longer assume that uo G
L2(IR2)2. The first systematic treatment of that question seems to be given
in D. Benedetto, C. Marchioro and M. Pulvirenti [51] where the case of an
initial condition u0 satisfying ((4.3) and), for some C > 0,
Ms)| < C(l + |x|a), 0 < a < min(l,-),
uj0 = curltxo e Lpn L°°(TR2) for some 1 < p < oo,
is treated. We wish to extend this analysis here to the case when 1 < p < oo
and
u0 G W^QR2), Vu0 e LP(1R2), divu0 = 0 a.e. in R2. (4.50)
We shall be mainly using the vorticity formulation of (4.1)
-^ + div (uu;) = 0 in V'(TR2 x (0,oo)) (4.51)
requiring u to belong to C([0, oo); W£CP(IR2)) (i.e. u € C([0, oo); W^{BR))
for all R € (0,oo)), to satisfy (4.1) (in the weak sense defined in section
4.1) and
Vu€C([0,oo);Ip(IR2)); divu = 0,
curlu = o; a.e. in IR2 x (0, oo).
Let us mention that (4.51), as in section 4.1, holds if p > 2 and, if p < 2,
(4.51) holds in the renormalized sense, i.e. (4.51) holds with w replaced by
/?(u/)forall/?<EC6(]R;IR).
We may now state our main existence result.
Theorem 4.4. Let u0 satisfy (4.50). Then, there exists u € C([0,oo);
W£CP(IR2))2, a weak solution of (4.1 )-(4.2), satisfying (4.51)-(4.52).
Remarks 4.10. 1) If p < 2, the proof below shows that one can build
u € C([0,oo)',Lq(M2))2 with q = ^, choosing u0 (up to a constant) in
L*(IR2).
(4.49)
(4.52)

146 Euler equations and other incompressible models
2) It is possible to extend the above result to the case when Vuq e
ivi +1?* where 1 < pi,p2 < oo.
3) If curluo belongs, in addition to the assumption (4.50), to L°°(IR2),
then one can check (by standard arguments) the uniqueness of solutions
(normalized by requesting for instance JB udx = fB u^dx, see the proof
of Theorem 4.4 below for more details).
4) If curl i*o only belongs to L°°(IR2), we do not know if the above result
(or an appropriate modification of it) holds.
5) Let us mention that when p > 2, u grows at infinity at most like
|x|i-2/p More precisely, we have for some C > 0
SUP , '"ffiLi < C-{||VW||LP(1R2) + ||U||LP(Bl)}. (4.53)
Indeed, if u € W^f(HI2), Vu € LP(1R2), then we have, setting a = 1 - J,
sup i^)-yi < Co||Vu|,Lp(]R2) (4.54)
for some Cq> 0 independent of u. Therefore, we have
\u(x)\ < CbHVuH^^jIxr + KO)!
< anvu||LP(]R2)|xr + [uw-ir-^iittiiixflol
< CollVtiH^^ja + W) +ir-1/p\\u\\LP{Bl)
using (4.54). The growth |x|a is essentially optimal since u = (1 + lx)2)^2
satisfies Vu e LP(TR2) for all 0 < 0 < a. D
Before we present the proof of Theorem 4.4, let us first make some
preliminary remarks. We introduce the Banach space 'D1,P(IR2) = {u G
<f(IR2),Vu 6 Z,*(IR2)2} equipped with one of the equivalent norms
IIVti||Lp(in*) + INIIip(Bi) °r I|Vu||lp(ir2) + I {Blu\ (we could as well
replace Bi by any ball Br for 0 < R < oo) and we restrict our attention, as
in Theorem 4.4, to the case when 1 < p < oo. When 1 < p < 2, by Sobolev
embeddings, we see that P^QR2) = 1R + {u € L'QR2), Vu € LP(JR2)2}
where q = -^-. We shall need, in the course of proving Theorem 4.4, some
technical results on this space given by
Lemma 4.2. Let u e P1>P(]R2)2 satisfy: divu = 0 a.e. in R2.
(i) We have
-^- = -(-A)-1-^-V±curlw where V"1 = ( *?) , (4.55)

Remarks on Euler equations in two dimensions 147
or in other words ^ is given in terms of curlu by a singular integral
convolution operator whose kernel is ^ ^(jfp) anc' x± = (-*i)-
(ii) There exists un 6 C^°(1R2)2 such that divtxn = 0 on IR2, Vtxn
converges to Vtx in Z7(IR2) and un converges to u in Lp(Br) (for instance
for all Re (0, oo) ifp > 2, while there exists c G IR2 such that un converges
tou-c in Lq(JR2)2 when l<p<2 (and q = 2p/(2 - p)).
Proof of Lemma 4.2. (i) The proof of (4.55) is straightforward since one
checks easily that jj =-357 + (-A)"1 £. V1- curlu € LP(IR2) (using the
fact that the singular integral which is a composition of appropriate Riesz
transforms is bounded on L*(IR2)) and curl/,- = 0, div£ = 0 in P'(IR2).
Hence, fj is harmonic on IR2 (harmonic gradient in fact) and thus fj = 0
a.e. in IR2.
(ii) We have already shown in Appendix A the case when p = 2. When
1 < p < 2, the proof is rather easy since u = c + v where c € IR2, v €
L*(IR2)2, Vv e L*(IR2), divv = 0 a.e. in IR2 (q = ^). Then, by
"standard" density results (or proofs), we can build vn € Cq°(1R2) such
that divvn = 0 in IR2, vn converges in L*(IR2) to i>, Vvn converges in
27 (IR2) to Vi> and the proof is complete in this case.
One possible proof in the case when p > 2 consists in several layers of
approximations. First of all, we truncate and regularize u) = curlu and
obtain uk e C£°(IR2) which converges, as k goes to +00, to u) in LP(IR2).
Next, we consider uk defined by uk € 27(IR2)2 for all r > 2, uk € C£°(IR2)2
and decays at infinity like A, dxvuk = 0 in IR2, curlu*1 = uk in IR2
(u* = ^F ill7 *^/c)- Using, as in (i) above, the boundedness in L^IR2)
of Riesz transforms, we deduce that Vtx* € 27(IR2) for 1 < r < 00 and
that Vtx* converges in LP(1R2) to -(-A)""1 VV1^ = Vtx in view of (4.55).
Therefore, if we set ck = jBi u-uk dx, uk + ck converges in Z>1>P(1R2) to
u as k goes to +co.
Next (using for instance the case p = 2 already treated in Appendix A),
there exists uk*m e C£°(IR2)2 such that divuk>m = 0 in IR2, ufc»m converges
in t>liP(TR2)2 to uk as m goes to +00, for each fixed k > 1.
Finally, we choose for each k > 1 some <pk € Q°(IR2)2 such that div cpk =
0 in IR2, <pk(0) = c* and we set (pk>m = <pk(%) for m > 1. Observe that for
A; > 1 fixed, <pk>m converges to ck in V1>P(1R2)2 since we have
/ \Vtpk>m\pdx = m2~p f \V<pk\pdx -+ 0 as m -* +00.
JJR2 JjR*
In conclusion, we have shown that txfc,m + <pfc'm converges in X>lj)(IR2) as
m goes to +00 to uk + c* which, in turn, converges to u in Z>1,P(IR2) as k
goes to +00, and this completes the proof of Lemma 4.2. □

148 Euler equations and other incompressible models
Remark 4.11. It is possible to give a different proof of part (i) of Lemma
4.2 (when p > 2) using the classical approach to density results, the fact
that the range of the "divergence-map" from VliP(JR2) into LP(1R2) is
LP(1R2) and the density of C£°(IR2) in P1»P(IR2). This last fact, how-
ever, requires some justification: we can either adapt the argument given
above or argue in a slightly more direct way as follows. First,
approximating if necessary u by Tr(u)—observe that Tr(u) converges to u in
VltP(lR2) as R goes to +oo—we can assume without loss of generality that
u € L°° r\Vl*p(JR2). Then, we consider un = (p(%)u where <p € C£°(IR2),
0 < </? < 1, </? = 1 on Bi, Supptp C I?2-
If p > 2, un converges to u in P1,P(IR2) since we have
Vun-Vtx = |V-W-)lVu + -V<p(-)tx
L vn'J n vn'
and
Pdx < Cn2'p\\u\\pLx .
Then, smoothing un by convolution allows to conclude.
If p = 2, we build in the above way un € Cq°(1R2) such that un converges
to u weakly in P1,2(IR2) and this is enough to conclude. □
Proof of Theorem 4.4
Step 1. The case when 1 < p < 2. This is in fact the easy case since
there exists c € IR2 such that txo — c € L^(IR2)2, and we observe that
because of the galilean invariance of the Euler equation we can always take
c to be 0: indeed, u is a solution of (4.1)-(4.2) if and only if v defined
by v{x,t) = u(x + ct,t) - c on IR2 x [0,oc) is a solution of (4.1)-(4.2)
corresponding to txo — c. Therefore, without loss of generality, we may
assume that u0 e £^(IR2)2, Vtx0 € LP(IR2)2. Then, as in Lemma 4.2
(and its proof), we introduce v% e C§°(1R2)2 such that divtzft = 0 in IR2,
Uq converges to uq in L^(IR2) and Vtxft converges to Vtxo in LP(1R2) as
n goes to +oo. In view of Theorem 4.1, we can solve (4.1)-(4.2) with uq
replaced by Uq and we find a smooth solution un on IR2 x [0, oo) such that
u)n = curl un solves (uniquely)
dujn
— + un • Vwn = 0 in IR2 x (0, oo),
u/n|t=0 = u#(=curlt#) in IR2.
Since, for each n > 1, un is bounded on IR2 x [0,T], we deduce that
ujn € C£°(IR2 x [0,TJ) for all T € (0, oo). In addition, because of (4.12), un
L
-V<p(-)u
(4.56)

Remarks on Euler equations in two dimensions 149
is bounded in C([0,oo);Lp(IR2)) and thus Vun is bounded in C([0,oo);
LP(1R2)) by Sobolev inequalities. This implies that un is bounded in
C([0, oo); L^p(TR2)). Extracting a subsequence if necessary, we may
assume that un converges weakly to some u in L*=p(IR2 x (0,X)) (V T €
(0,oo)). Since cj£ converges in L^IR2) to wq = curltxo, we can now use
the general convergence results of R.J. DiPerna and P.L. Lions [128]—see
section 2.2 in chapter 2 for results of a similar nature—to deduce that un
converges in C([0,T];LP(IR2)) (VT € (0,oo)) to the unique renormalized
solution u of (4.51) and of course u) = curlu. Then exactly as above we
deduce that un, Vun converge respectively to u, Vu in C([0, T]; L&p (El2)),
C([0,T];LP(1R2)) for all T € (0,oo). This is enough to conclude.
Step 2. The case when p > 2. We use Lemma 4.2 to introduce Uq €
C£°(1R2)2 such that t# converges to u0 in P1,P(1R2)2 as n goes to +00. We
then follow the argument given in step 1 and obtain un,u}n, as in step 1,
which satisfy: Vtxn,wn are bounded in C([0,co;Lp(]R2)). We then set
cn(t) = / (un(t)-v$)dx
JBi
and define un by un(x,t) = un(x + f£ (fx(s)ds , t) - cn(*). We next set
pn = pn + ^ • x, u>n = un(x + f*cn(s)ds, t). Then, one checks that
(un,pn) solves (4.1)-(4.2) with u0 replaced by uft and that (4.56) holds for
un with un replaced by un. Let us also remark that Vun and un are still
bounded in C([0,oo);I-p(IR2)), and in addition we have now, because of
the choice of cn(£): for all t>Q,fB un(t) dx = fB 11%dx—>fB uqdx.
Therefore, un is bounded in C([0, oo);P1,p(IR2)), and extracting a
subsequence if necessary, we may always assume that un converges to some
u 6 ^(O.oojP^OR2)) in L^0,R; W^Br)) weakly for all R 6 (0,oo),
1 < q < 00 (and Dun converges weakly to Du in L9(0,T;Lp(IR2)) for all
T € (0,00), 1 < q < 00, and fB u(x, t)dx = fB uo(x) dx for all t > 0).
Next, as in step 1, we wish to deduce the convergence of Qn to u (=
curlu) in C([0,T];LP(JR2)) (V T € (0,oo)) from (4.56) where u is the
unique solution of (4.51) (in C([0,T];Lp(IR2))). In order to apply the
convergence results due to R.J. DiPerna and P.L. Lions [128] (see also
section 2.2 in chapter 2), we simply need to check that we have for all
T€ (0,oo)
luKl+lxl)-1 € ^(O.TjJ^flR2)) + L00^2 x (0,T)). (4.57)
Once this is checked, we conclude as in step 1 that Vwn converges to
Vtx in C([0,T];£P(1R2)) for all T € (0,oo) and thus un converges to u in

150 Euler equations and other incompressible models
C([0, T]; Vl>p(TR2)) for all T e (0, oo), as n goes to +00. These convergences
then yield the conclusions of Theorem 4.4.
When p > 2, (4.57) follows immediately from the fact that u e
L°°(0,oo; Vl>*>(TR2)) and (4.53): indeed, |ix|(l + |x|)-a e £°°(IR2 x (0,oo))
for some 0 < a < 1. When p = 2, we write ix = 1x1+1x2 where 1x1,1x2 are
defined by
divtxi =divtX2 = 0, curl 1x1 = ^l(|u;|<i)>
curltX2 =^l(|u;|>i) a-e- m IR-2 x (0,oo),
4 uidx = 4 uodx, 4 tX2 dx = 0 for t > 0;
JBi JBi </£i
txr^eX^^oojP^IR2)).
Clearly, a; l(|a;|<i) € £°°(0,00; L2nL°°(lR2)), hence, as we have done several
times before, 1x1 e £°°(0,00; VllP(JR2)) for 2 < p < 00 and, as we just saw
above, y^eL~(IR2x (0,00)).
On the other hand, since lj € £°°(0,oo;£2(IR2)), o;l(M>1) e £°°(0,oo;
Ll n L2(R2)) and thus ix2 € Loo(0,oo;P1^(IR2))2 for 1 < p < 2. In
particular, as seen above, 1x2 = c(t) + w where w € L°°(0, oo;L9(IR2))2
for 2 < q < 00 and thus c(t) = ^ (1x2 — w)(x,t)dx = ^ w{x\t)dx €
L°°(0, 00). Therefore, 1x2 satisfies (4.57) and we conclude. □
4.3 Estimates in three dimensions?
The incompressible (homogeneous) Euler equations (4.1)-(4.2) in three
dimensions (N = 3) are far from being understood. Indeed, the only
information that is available concerns short time existence. More precisely,
in the case of Dirichlet boundary conditions or in the periodic case (or
when CI = TRN), it is known that if 1x0 is smooth enough (1x0 € X where
X = H3 with s > |, C1,a with 0 < a < 1), then there exists a maximal
time interval [0,T*) (T* < +00) such that there exists a unique solution
ix e C([0,T\;X) (V T 6 (0,T*)) of (4.1)~(4.2) (and with the appropriate
boundary conditions of course) and if T* < 00, ||u(£)Ha* goes to +00 as t
goes to T*-.
Of course, the crucial question which is still completely open is to decide
whether T* < 00 or not. J.T. Beale, T. Kato and A. Majda [28] (see also
G. Ponce [391] for a variant involving the deformation tensor instead of
the vorticity) have established a fine criterion for the finite time blow-up
of smooth solutions: let T € (0,oo); if there exists a (unique) solution
u e C([0,T);X) such that /0T ||curlu(*)IU~ dt < 00, then T* > T. Or in

Estimates in three dimensions?
151
rp
other words, if Tm < oo, then /0 * ||curlix(t)||£,oo dt = +00. We do not wish
to re-prove this statement here. Let us simply mention that the main idea
behind this criterion is the following: if /0 ||curltz(*)||£,~ dt < 00 then one
can bound ||u(*)||;r uniformly on [0,T) and, using the short time existence
result, one can continue the solution on a larger interval than [0,T].
The appearance of singularities (i.e. the breakdown of smooth
solutions) in finite time is an outstanding open mathematical problem, whose
solution would have a serious impact on three-dimensional incompressible
fluid mechanics. After many years of intensive numerical simulations which
were inconclusive, two recent independent numerical experiments by R.
Grauer and T. Sideris [193] and later by R. Kerr [262] indicate possible
breakdowns of smooth solutions in finite time (and certainly violent growths
of ||curltz(i)||L» that, for all practical mathematical purposes, make it
difficult to believe in a priori estimates on curlix).
The striking difference between the cases N = 2 and N = 3 is also
illustrated by the so-called "2 + 1/2" dimensional flows, used in R.J. DiPerna
and A. Majda [129] to provide examples of weakly convergent sequences
of solutions of 3D Euler equations whose weak limits do not satisfy (4.1),
namely solutions of (4.1) say in the periodic case (for instance) such that
ix is independent of X3. In that case, (1x1,1x2) solve the Euler equation (4.1)
in two dimensions (in the periodic case) while w = U3 simply solves the
following transport equation
-j^ + div(uw) = 0 in IR2 x (0,oo), w\t=o = w0 in H2 (4.58)
and w,wo are periodic in X\^X2 (of periods Ti,T2 > 0).
The decoupling between (1x1,1x2) and w allows us to solve first for (1x1,1x2),
using the results of sections 4.1-4.2, and then to solve (4.58) in a classical
way when u is smooth (or Lipschitz, or almost Lipschitz). When uq =
curl 1x0 (= £-2uM - £- ix2(0)) € LP(Q) (Q = (0,Ti) x (0,T2)) with
p > 1, resp. u;o|logu;o| € Z'1(fi), then we have seen that there exists
at least one solution of the Euler equation u € C([0,oo); WllP(Q))y resp.
Wlfl(Q). In this case, there exists, as soon as 1//0 € Lq(Q) (1 < q < 00),
a unique solution of (4.58) which belongs to C([0, oo);Lq(Q)) if q < 00
(and to C([0,oo);Lr(fi)) for all 1 < r < 00 and to L°°(Q x (0,co)) if
q = +00)—see R.J. DiPerna and RL. Lions [128].
These flows provide examples of global weak solutions of the 3D Euler
equations which are smooth (and thus unique) if the initial conditions are
smooth. But these flows, as was observed by R.J. DiPerna and the
author [124], also show that solutions (even smooth ones) of the 3D Euler
equations cannot be estimated in W1,p, for 1 < p < 00, on any time
interval (0, h) if the initial data is only assumed to be bounded in WliP. In

152 Euler equations and other incompressible models
other words, intermediate a priori estimates or, more precisely, WliP a
priori estimates do not hold in three dimensions—let us emphasize the fact
that they do hold in two dimensions, see sections 4.1-4.2 above. Indeed,
we choose for (111,1*2) stationary flows of the 2D Euler equations namely
(ui(x2),0) where txi is smooth, periodic in X2 (of period T2). Then, the
solution of (4.58) is simply given by wo(xi—tui (22), X2). The lack of a
priori estimates in Wl,p is then clear choosing for instance txf fa) to behave
like (e2 + x^fl2 near X2 = 0 (and smooth elsewhere uniformly in e) and
Wo(xi,x2) to behave like (e2 + |x|2)*~1/2 near 0 (and smooth elsewhere
uniformly in e) where e € (0,1], 9 € (0,1), and we take 6 ^ ^, otherwise
log modifications have to be made. Then, obviously, (u^Wq) are bounded
in W1,p, uniformly in £, for 1 < p < j^e while for q > 1 we have
u
•"t'li'fc*,
nl 8x2
= /f^fo-taffoj.xa) \{u\)'{x2)\qdxldx2
Ja I oxx I
for some 6 > 0 sufficiently small, independent of £, where v = |20 — \\q 6q >
0. Since we have for some constant C(0, q) > 0 (if 1 < q < -£§)
f \xi\q 1 [s 1
JbsW^'M^**1**2 = C{e,q)Jo 7«^rdr>
we deduce that ^p is bounded in Lq uniformly in e (for any fixed t > 0) if
and only if q < 3*i_e\, and in particular it is not bounded if jr^j) < 9 <
1
1-0"
This construction shows that, for each 1 < p < 00, t e (0,00), e € (0,1),
6 6 (0,1), there exists a smooth (periodic) solution u of Euler equations in
three dimensions such that ||tx(0)||w1>p < e and IMOII w1-* ^ j- This also
gives examples of smooth flows such that ||curlti(t)||£,~ > i||curlu(0)||2c«.
Let us conclude by mentioning that there are other known smooth
regimes for three-dimensional Euler equations like the axisymmetric case
without swirl (see A. Majda [316] and Ph. Serfati [420] for more details).
But, even in that case, J.M. Delort [119] observed that the 'Vortex sheet"
problem (see section 4.2 for a slightly more detailed presentation of this
problem) is mathematically quite different from the pure two-dimensional
case.

Dissipative solutions
153
Finally, let us observe that the lack of intermediate a priori estimates
and more specifically of bounds that yield the compactness of
appropriate sequences of solutions (or approximated solutions) has made
impossible until now the construction of weak solutions in C([0,oo);L2) or even
Z,°°(0,oo;L2) satisfying (4.1) in the sense of distributions. Weaker notions
of solutions are considered in the next section.
4.4 Dissipative solutions
As we have seen in the preceding section, even the global existence of weak
solutions of the Euler equations is not known in three dimensions. On
the other hand, an obvious bound in C([0, co);L2) follows trivially from
the (formal) conservation of energy. It is then natural to attempt
building up solutions in a weaker sense than in the distributions sense. A
very weak notion (relying on relaxed Young measures or relaxed measure-
valued solutions) is proposed by R.J. DiPerna and A. Majda [129] but
the relevance of this notion is not entirely clear since it is not known that
"solutions" in the sense of [129] coincide with smooth solutions as long as
the latter do exist.
We shall propose here a different notion of "very weak" solution that we
call dissipative solutions, for reasons we shall explain later on. This notion
seems to be new and the idea behind the notion appears in P.L. Lions [306]
in the context of Boltzmann's equation. We wish to emphasize immediately
that we are not convinced that such a notion is neither relevant nor useful.
Its only merits are: 1) such solutions exist, 2) as long as a "smooth"
solution exists with the same initial condition, any such dissipative solution
coincides with it, 3) we shall use it in some small Mach number limits in
chapter 9 to pass to the limit from some compressible models to the Euler
equations.
From now on, we take N > 3 and we only consider the periodic case
in order to simplify the presentation and keep the ideas clear, although
everything we do below can be extended or adapted to the case when
ft = TRN or to the case of Dirichlet boundary conditions. This is why all
the functions considered below are always assumed zo be periodic. The
initial condition uq (see 4.2) is always assumed to belong to L2(Q)N and
to satisfy (4.3) in TRT.
Let us first explain the formal idea of this new notion. Let us consider a
smooth test function v on IR^ x [0, co) such that div v = 0 on JRN x [0, oo).
We define
E(v) = ~ - P((v • V)t/) (4.59)

154 Euler equations and other incompressible models
(recall that P is the projection onto periodic divergence-free vector fields),
and we write Vq = v|t=o-
If u is a solution of (4.1)-(4.2), then we can write
f— + u. v) (u-v) + (u-v) • Vv + Vtt = E{v) in m.N x (0, oo)
for some scalar function ir. Then, multiplying this equality by u - v and
integrating (over the period), we find
4- I \u-v\2dx = -2[(d(u-v),u-v)dx+ f 2E(v) • (u-v)dx (4.60)
dt Jn Ju Jo.
where d(= d(v)) = (\{diVj + djVi))^ We then set (for each t> 0)
and we deduce from (4.60)
4- f\u-v\2dx < 2||<f-||00 /" \u-v\2dx + 2/ E{v)-{u-v)dx. (4.61)
dt Jq Jq Jq
Hence, in particular, we have for all t"> 0
/ \u-v\2(x,t)dx < exp(2/ ||d~||oo^J / \uo-v0\2dx
+ 2 J ds jdxexp(2j ||<r||oo<to ) E{v) ■ (u-v).
(4.62)
The definition of dissipative solutions of (4.1)-(4.2) given below consists
precisely in requesting (4.62) to hold for an appropriate class of v.
The reason why we call these solutions dissipative solutions is the
following: if we take v = 0 then obviously E{y) = 0, d < 0 and (4.60) is
nothing but the (formal) conservation of energy while (4.61) and (4.62) are
standard relaxed energy inequalities which allow for a possible loss
(dissipation) of energy though various losses of L2 compactness (via oscillations,
concentrations, etc.).
We may now give the precise
Definition 4.1. Let u e L™(0,oo;L2)N nC([Q,oo)]L2-w)N. Then u is a
dissipative solution of (4.1)-(4.2) if u(0) = u0, div u = 0 in V'iJRN x (0, oo))
and (4.62) holds for all v € C([0,oo);L2)^ such that d € 1^0,T;!00),
E(v) 6 Ll(0,T;L2) (VT € (0,oo)) and divv = 0 in iy(1RN x (0,oo)).

Dissipative solutions
155
Remark 4.12. 1) It is worth pointing out that the main regularity
requirement for v in the above formulation, namely d € Ll(0,T;L°°), is the
same as in the blow-up criterion obtained by G. Ponce [391]—and, in fact,
for similar reasons.
2) If if 6 LP(0,r;L°°) (or curlv 6 Lp(0,T;L°°)) for some p e [l,oc]
then Dv € £*(0,T;L«) for all 1 < q < oo and Dv e £P(0,T;BMO): in
particular, there exists A(t) > 0 e L*(0,T) such that if \x - y\ < 1/2
\v(*,t)-v(y,t)\ < A(t)\x-y\\log\x-y\\. (4.63)
3) When E{y) = 0 (for instance) and Dv0 e Lq for some 1 < q < oo
then d € Lx{0,T\L°°) implies formally that Dv e L°°(0,T;L*). Indeed,
we observe that for i^ j
hfi+v-V^idjVi-diVj) = djVkdkVi-diVkdkVj
= 2 djVk dik - 2 diVk djk.
Therefore, we deduce
|||curlV||I, < C0ffN|L-||Dt;||L.||curlt;||Kl
< C(g)||d||Loc||curlt;||i,
and our claim follows easily.
4) Let us observe that if d € Ll{Q,T;L°°) then E(v) € Ll(0,T;X2) if
and only if ^ € LX{Q,T-L2). Indeed, §* + £(v) = -P((v • V)v) and
£>v 6 Lx(0, T; L9) for all 1 < q < oo. In particular, (v • V)v and thus P((v •
V)v) € L1(0,T;Lr) for all 1 < r < 2 since v € I°°(0,T;L2). Furthermore,
for each 1 < i < N, we have
(v . V)vi = Vj 0,-Vj = Vj(djVi + diVj) - di(- |v|2J
= 2d-v-di(±\v\2).
Therefore, P((v • V)v) = 2P(d • v) € L^O.T;!2) since d € X^O.TjL00)
andv€L°°(0,T;L2). D
An obvious consequence of the definition is the following
Proposition 4.1. If there exists a solution v 6 C([0,T];L2) o/(4.1)-(4.2)
on JRN x (0,T) satisfying d 6 Ll{0,T\L°°) for some T 6 (0,oo), then any
dissipative solution u of (4.1)-(4.2) is equal to v on JRNx[0,T]. D

156 Euler equations and other incompressible models
Indeed, E(y) = 0 and we conclude!
Before we give and prove our existence result for dissipative solutions,
let us mention that, in Definition 4.1, we can replace the regularity
requirements on v by much stronger ones. In other words, it suffices to check (4.62)
for smooth test functions v. Indeed, if (4.62) holds for smooth i>, then we
can check it also holds for the class of v described in Definition 4.1 and in
Remark 4.12 (4) by a straightforward regularization procedure. More
precisely, let pe = ^ p(j), p e C^(JRN), /RiV pdx = 1, Suppp C Bv We set
v£ = v*p£. Obviously v£ converges to v in C([0,T];L2), ^ € Ll(Q,T\C%)
for all k > 0, divvff = 0 in JRN x (0,oo), vc e C([0,T];Cf) for all k > 0,
T e (0, oo) and
EM = -|-P((vVW
= [~ - P[(V • V)V]] * Pe + [P[(V • V)t/] * p£ - P((V£ • V)t/C)]
= E(v) *Pe+2 [[P(d • V)] * Pe " P(dff • V£)]
= £(v) * pff + 2P{(d • v) * pff - de • vff}
in view of Remark 4.12 (4). Obviously, E{y) * p€ converges to E(v)y as e
goes to 0+, in Ll(Q,T\L2) (V T e (0,oo)) and so does E(v€) provided we
show that (d • v) * pe — dff • i>c converges to 0, as e goes to 0+, in Ll (0, T; L2)
(VT 6 (0,oo)). Since d € L^TjL00), t; € C([0,r];L2) (V T € (0,oo)),
this is straightforward. Observe in addition that ||d7||oo < ||rf ||oo f°r
all e > 0, a.e. * > 0 and that ||d~||oo converges to ||d"||oo in Ll(Q,T)
(V T € (0, oo)). We then apply (4.62) with v replaced by vc and we
conclude letting e go to 0 in view of the convergences collected above. Let
us observe that by a second layer of approximation (regularizing in £), we
can take v smooth on IR^ x [0,oo) in (4.62).
We can prove now the existence of dissipative solutions.
Proposition 4.2. There exists at least one dissipative solution of (4.1)-
(4.2).
Proof of Proposition 4.2. We consider uui a weak solution of the Navier-
Stokes equation (see section 3.1),
-^- + (it„ • V)it„ - i/Auv + Vp„ = 0, divu„=0 in IR* x (0, oo) (4.64)
satisfying (4.2) and the energy inequality
17 / huu(t)\2 dx + v [ \Vuu\2dx < 0 in £>'(0,oo). (4.65)
dt Jn 2 J si

Dissipative solutions
157
Recall that u„ G L2(0, T; tf1) (V T € (0, oo)) n L°°(0, oo; X2) n C([0, oo);
L2 — u>) (in addition, uu{t) goes to u<> in L2(Q) as t goes to 0+).
We next consider v as in Definition 4.1, and, as shown above just before
the statement of Proposition 4.2, we can take v arbitrarily smooth on IR^ x
[0, oo). Then, multiplying (4.64) by v we find
d f f 9v , _. A \
— I uu - v dx — / u„ • — + [uu • V)v -uudx
dtjn Jn dt I
+ ul Vuu- Vvdx = 0 in X^O.oo),
Jn )
and thus, by definition of E(v) and since div uu = div v = 0, we deduce
— / u„'vdx+ uu • E(v)dx — / [(uu-v) • V]v- (uu—v)dx
dt Jn Jn Jn
+ u Vu„-Vvdx = 0 in £>'((), oo).
Jn
(4.66)
In addition, we have for all t > 0
12
d f \v\2 f dv f \
-J L±-dx = J v-^dx = J v[-E(v)-(v-V)v]dx\
= — / v • i?(v) cte.
./n
Combining (4.65), (4.66) and (4.67), we deduce
~dt I 2^U,/~V^dx = ~ [(u»-v) ' V]v - {uu-v)dx
+ u Vuv ■ Vvdx + I E(v) • (uu-v) dx
Jn Jn
(4.67)
or
— / \uu-v\2dx = -2 / (d(uu-v),u„-v)dx
+ 2u Vu„ • Vv + 2 / E(v) • (uu-v) dx
Jn Jn
< 2||d-||00 f\uu-v\2dx + Cu((|Vu|2<£r J
+ 2/ £(v)-(u„-v)dx

158 Euler equations and other incompressible models
and finally for all T 6 (0, oo) and for all t € [0, T]
l\uu-v\2{x,t)dx < e2&ldrUia [ \u0-v0\2dx
+ 2 f'ds [dxe2!^^**E{v)-(u„-v) + CTv1/2, I
Jo Jn '
> (4.68)
for some positive constant Ct which is independent of v. Here, we used
the fact that (4.65) yields a bound, uniform in v, on j//0°°dt- Jndx|Vtz^|2.
Finally, we observe that fy = -djPiu^ju) - vll2djP{ylt2dju) and
thus 9jfr is bounded in L2(0, oo; H'1) + Z°°(0, oo; wM*+«).i) for all s > 0.
Extracting a subsequence if necessary (see Appendix C for more details)
we may assume that uu converges to some u weakly in L°°{Q~oo\L2) — *
and weakly in L2 uniformly in t € [0,T] for all T € (0,oo). And u e
LTO(0,oo;I2)nC([0,oo);L2-ti;),divtx = 0inP/(R^x(0,oo)),^
in JRN and passing to the limit in (4.68), as v goes to 0+, we recover (4.62)
and Proposition 4.2 is shown. □
4.5 Density-dependent Euler equations
In this section we briefly review the "state of the art" concerning the
density-dependent Euler equations or, in other words, the inhomogeneous
incompressible Euler equations.
We look for a non-negative scalar function p(x, t) (the density of the
fluid) and for a divergence-free vector field u{x, t) (the velocity of the fluid)
which are solutions of
— + div (pu) = 0, div u = 0
L. > <4-69>
-£-*■ + div (puui) + Vp = 0, for 1 < i < N
at
for some scalar function p, the so-called hydrostatic pressure. We consider
as usual the periodic case (the unknowns are periodic and the equations
hold in TRN x (0,oo)), the case when CI = JRN or the case of Dirichlet
boundary conditions where the equations are set in a smooth, bounded,
open connected set in IR^ and u • n = 0 on dQ where n denotes the unit
outward normal. Of course, we complement (4.69) with initial conditions
p|t=o = A), pu\t=o = m0 (4.70)
where po > 0. In fact, exactly as in chapter 2 for the inhomogeneous
incompressible Navier-Stokes equations, the precise meaning of the initial

Density-dependent Euler equations
159
condition for pu has to be interpreted correctly but in the simple case when
we assume that po satisfies for some 0 < a < @ < oo
a < Po < P (4.71)
then, we may set uq = &* and we require uq to satisfy (4.3) (in 1R or in
Q in the case of Dirichlet boundary conditions).
Obviously, (4.69) contains the usual incompressible Euler equations as a
particular case: take p = 1 in (4.69). Therefore, we cannot expect to know
more about this system of equations than for the incompressible Euler
equations and in particular the case when N > 3 seems rather hopeless as
far as the understanding of global solutions is concerned. But even if we
concentrate on' N = 2 in the rest of this short section, this will not help
much since very little is known even in this case. Of course, (4.69) is well
posed locally in t provided (4.71) (and (4.3)) holds and we choose pc^o to
be smooth enough but, even when N = 2, it is not known whether smooth
solutions persist for all t > 0 or break down in finite time. Again, there
seems to be some numerical evidence of finite time breakdown but this has
yet to be confirmed systematically.
There are very few known a priori estimates. Of course, (4.69) implies,
at least formally, that we have for all t > 0 and for all 0 < a < b < oo:
meas {x / a < p(x, t) < 6} = meas {x / a < po(x) < 6}. (4.72)
In particular, if po satisfies (4.71) then p(t) also satisfies (4.71) for all t > 0.
The conservation of energy is obtained by multiplying the second
equation of (4.69) by u and integrating by parts using the first equation, and
reads
— Ip\u\2dx = 0 for t>0. (4.73)
When (4.71) holds, this yields an estimate on u in C([0,oo);L2).
To the best of our knowledge, these are the only known a priori
estimates even when N = 2. It is also worth remarking that the failure of
intermediate a priori estimates that we showed in section 4.3 on the
incompressible Euler equations when N > 3 can be established for (4.69) when
N = 2 using in fact the same examples: u\ = 1x1(2:2), ^2 = 0, p = 0 and
p(x,t) = Po{Xi-tUi(x2),X2).
Let us conclude by mentioning a remarkable identity which however does
not help—or at least does not seem to help—to analyse (4.65)
mathematically. Still for N = 2, we consider the vorticity u = #2^1 — #1^2 and we
write at least formally
9u , _x 1 _
— + (u • V)u + - Vp = 0
at p

160 Euler equations and other incompressible models
hence, taking the curl (and using the fact that divtx = 0)
^ + (tx-v)o; + a2(i)a1p-a1(i)a2p = o. (4.74)
In particular, we deduce for all smooth function /? from IR into IR
|{^(p)} + (u.V){a;/3(p)} = /3/(p){a1(i)a2(p) - d2(±)dlP}
= ^ {hpdip - dlPd2p} = a2[7(p)]a1p-a1[7(p)]a2p
= d2{l{p)dip) - dl{-f(p)d2p}
where 7' = ££1. Then, we integrate the resulting equation and, at least
when Q = IR^ or in the periodic case, we deduce immediately
^-[u0(p)dx = Q for all t>0 (4.75)
at J
for any function p from IR into IR.
4.6 Hydrostatic approximations
There exists a huge literature on the so-called hydrostatic approximations
of incompressible models and a considerable number of models have been
proposed (see for instance J. Pedlosky [386] on geophysical flows, and P.
Constantin, A. Majda and E.G. Tabak [103], J.L. Lions, R. Temam and S.
Wang [296],[297],[298]) with applications to oceanography, meteorology,
geophysical flows and the huge variety of quasi-geostrophic models. Some
models have been analysed mathematically or implemented numerically
but, to the best of our knowledge, very little seems to be known on the
model we discuss in this section which is the inviscid version of a very
classical model. Our motivation for restricting our attention to this model
is mostly mathematical since we hope it could help to understand some
of the singular features of the classical three-dimensional incompressible
Euler equations.
Let us first present this "hydrostatic" inviscid model. We consider a
three dimensional strip S = {(x\,x2,z) 6 IR3 / 0 < z < 1} and we
look, to simplify the presentation and the notation, at a situation where all
unknown functions are required to be periodic in x\ and in x2 (of periods
respectively Tx and T2 > 0). We could consider as well the case when

Hydrostatic approximations
161
(xi,X2) € a; where a; is a smooth bounded connected open set in IR2 and
then we impose "Dirichlet boundary conditions" on du x [0,1] but we shall
not do so here.
From now on, all differential operators V, div, A, curl refer to the two-
dimensional operators (acting on xi,X2) and we also define d\ = g|-,
* = 3^, 3* = dz = £.
We look for a velocity field (u, v) = (ui(x\, X2,2), U2(xi, X2, z)1v(x\, X2, 2))
(€ IR3 for all (xi,X2,^) = (x,z) € 5) and for a scalar function p (the
pressure as usual) satisfying
du d dp
-£ + (u-V)u + v — u + Vp = 0, ^ = 0,
divu + -- = 0 in Sx(0,oo)
dz
(4.76)
v\z=o = v\z=i =0 on R2 (4.77)
(recall that all functions are required to be periodic in Xi and in X2).
Observe that (4.72) is nothing but the system of incompressible Euler
equations in S "with v = U3" satisfying Dirichlet conditions on {z = 0,1},
where the third equation on v is replaced by g| = 0. In other words, we
neglect in the third equation for v the term fjf+u- ViH-vfj and simply write
3! = 0 (the so-called hydrostatic approximation; in the presence of gravity
terms we can simply replace |f = 0 by || = a for some fixed constant a
but this does not modify (4.72) since we can then consider p — az).
Let us immediately mention that (4.76)-(4.77) contains as a particular
case the usual two-dimensional incompressible Euler equations: indeed,
take v = 0; then u = tt(xi,X2) solves the 2D Euler equation. Let us also
observe that the energy /nx/0 x\ \u\2 dxdz is conserved (at least formally):
indeed, multiply (4.76) by u and integrate over Q x (0,1) to find
and our claim is proved in view of (4.77).
There are various equivalent formulations of (4.77). Let us mention at
least one. We define Tp = ^(x) = fQ <p(x, z) dz for an arbitrary function (p
(periodic in x) on S. Then, if we integrate (4.76) in z from 0 to 1, we find
using (4.77)
du , , o
— + div {u®u\ + Vp = 0, div u = 0 in IR2 (4.78)

162 Euler equations and other incompressible models
(recall that p = p(x,y) in view of (4.76)). In particular, we deduce taking
the divergence of (4.78)
-Ap = aJ(tZTtZJ) in IR2 (4.79)
and p (which is periodic) can be normalized to satisfy f^pdx = 0 for all
t > 0. Then, it is possible to rewrite (4.76)-(4.77) as follows: u = tx(x, z),
p = p(x) solve
■j£ + (uV)u-( dWu(x,t)d£)uz+Vp = 0 inSx(0,oo)
divu = 0 inIR2, -Ap = dJ(uit2J) inlR2,
pdx = 0 for t > 0.
L
9CI
(4.80)
Indeed, we have checked above that a solution of (4.76)-(4.77) satisfies
(4.80). Conversely, we set v = - /Qz div tx(x, £) d£, and obviously (4.76)
holds, u|2=o = 0, and it just remains to check that v\2=\ = 0 in order to
prove our claim. But, v\z=i = — /0 divtz(x,f) df = —divu = 0 in IR2, and
we conclude.
In fact, it is possible to simplify (4.80) slightly. First of all, we prescribe
initial conditions
tz|*=o = txo in S (4.81)
where uq (periodic in x) satisfies
divtlo =0 in IR2. (4.82)
Then, we claim that (4.80) is equivalent to
-^ + (u- V)tx- ( J dW u(x^) dAu2 + Vp = 0 inSx(0,cc)
- Ap = d?Av~Uj) in IR2, / pdx = 0 for t > 0.
(4.83)
Indeed, we just have to check that v = — /Qz div tx(x, f) d£ satisfies: v\z-i =
0. In order to do so, we write the first equation of (4.83) in conserved form
du
— + div (u <g> u) + d2{vu) + Vp = 0 in 5 x (0, oo).
Then, we integrate this equation in z from z = 0 to z = 1, we take its
divergence and we obtain in view of (4.79)
—(divu)+div{v(x,l)u(x,l)} = 0 in IR2 x (0,oo)

Hydrostatic approximations 163
or, in other words, setting w(x,t) = v(x, l,t) = -(divxi)(x,*)
at
— - div {u(x, 1, *M = 0 in IR2 x (0, oo). (4.84)
Then, (4.82) shows that w\t=o = 0 on IR2. This fact combined with the
transport equation (4.84) allows us to conclude: v(x, l,t) = w{x,t) = 0 on
IR2 x (0,oo).
We continue our formal discussion of this model to propose a heuristic
derivation of it from the three-dimensional Euler equations. We feel that
any rigorous justification of this derivation would be a useful contribution.
This derivation is similar to the one proposed by O. Besson and M.R- Laydi
[61] in some viscous situations.
We consider the usual three-dimensioned incompressible Euler equations
in a shallow strip S€ = {X e IR3 / X = (x, z) e IR2 x (0,6:)} where e > 0:
^ + U-VXU + VXP = 0 in Sffx(0,oo),
at
(4.85)
divx(U) = 0 in S£x(0,oo) J
with the following boundary conditions
U\z=0 = U\2=e = 0 (4.86)
and, for instance, periodic boundary conditions in x.
We then assume that the initial conditions take the following form
U\t=o = (tH>(*,f ),«*(*, f)) = UeQ in S£ (4.87)
where u0,vq are given functions on S with values respectively in R2, IR.
Then, requesting that U§ satisfies the boundary conditions (4.86) and
div* U§ = 0 in Sc amounts to requiring
divuo + dzv0 = 0 in S, v0|z=o = v'oU=i = 0 in IR2. (4.88)
Next, if Ue is a solution of (4.85)-(4.86) corresponding to the initial
condition U$, we may try to write: Uc(x,z,t) = (wc(x, j,t),eve(x, f ,<)),
Pe(x,z,t) = p£(x, |,t) in 5C x (0, co) where ue,ve are now defined on
5 x (0,oo) with values respectively in 1R2,1R. In that case, (4.85) and
(4.86) become
— + (u£ • V)ue + vcdzuc + Vp£ = 0 in S x (0, oo),
divuc + -r- =0
az
'dv'
e2(-£-+(ue-V)ve+vedzve>)+dzpe=0 in Sx(0,oo),
? (4.89)
, dve
divue + —=0 in 5x(0,oo),

164 Euler equations and other incompressible models
v£\z=o = o£U=i =0 in IR2x(0,oc). (4.90)
Then, at least formally, we expect that, as e goes to 0+, {ue,ve,p£)
"converges" to (u,v,p) solving (4.76)-(4.77). This is why we think that
the study of the model (4.76)-(4.77) might shed some light on the three-
dimensional incompressible Euler equations.

APPENDIX A
Truncation of Divergence-free Vector Fields
in Sobolev Spaces
We sketch here a general procedure to approximate in WliP (1 < p < oo)
divergence-free vector fields, vanishing on the boundary in the case of a
bounded region, by compactly supported divergence-free vector fields.
More precisely, we consider u 6 W01,p(ft) (resp. Wl>p(TRN)) where ft is
a bounded, connected, Lipschitz domain in 1RN (N > 2). We assume
dWu = 0 a.e. in ft (resp. IR^). (A.l)
We then set ft* = {a; e ft, dist (x, 9ft) > 6} (resp. Bl/6 = {x € TRN / \x\ <
1/6}) for S > 0 small enough in case ft is smooth, otherwise we choose ft$
to be a smooth, connected domain satisfying {x 6 ft, dist (x, 3ft) > 6} C
ft* C ft* C ft.
Next, we solve the following (linear) Stokes problem in ft*
— Aus + us + Vps = -Au + u in ft*,
u6 € Wo'p(ft*), divu* = 0 a.e. in ft*.
If ft is bounded, we can of course skip the zero-order terms u and us in
equation (A.2). In view of classical results on Stokes problems (see [472]
for instance), there exists a unique solution (u*,p*) of (A.2) in W01,p(ft*) x
(Lp(ft*)/IR). If we request ft* to be "Lipschitz uniformly in 5", which is
the case if we simply take ft* = {x G ft , dist (x, dd) > 6} or Si/*, we can
in fact normalize p* in such a way that
lb*llz,p(n,) < C\\u\\witp(Qs). (A.3)
On the other hand, we always have
Nllw".p(n,) ^ C\W\\wi'P(ns) > (A-4)
where C denotes various positive constants independent of 6 > 0, u.
(A.2)

166
Appendix A
In fact, if p = 2, (A.4) takes a simpler form since, multiplying (A.l) by
us, we obtain
/ \Vus\2 + \u6\2 dx < / Vu-Vu6 + uu6dx (A.5)
and thus
/ \Vus\2 + \u6\2dx < f \Vu\2 + \u\2dx. (A.6)
Jets Jris
We next claim that, as 6 goes to 0+, u6 converges to u in W01,p(ft) (resp.
WliP(JRN)). In order to make this claim meaningful, we have to extend us
to ft (resp. IR^) by 0: in doing so, we preserve the nullity of div us now
in ft (resp. in IR^). We next prove the (strong) convergence in Wq'p(Q)
by two slightly different arguments, the first one in the case p = 2 where
we use the simple relation (A.6) while the second one will be valid for all
1 < p < oo. First of all, we observe that in all cases, us converges to u
weakly in W01,p(ft) (resp. WliP(JRN)): indeed, extracting a subsequence if
necessary, we may assume that us converges to u weakly in W0,p(ft) (resp.
W1>P(1RN)) while p6 converges to some tt weakly in L^ft) (resp. L^IR^)).
Then we have
-A{u-u) + (u-u) - Vtt = 0 in ft (resp. in Ht^), (A.7)
u-fi 6 WilP(Q) (resp. W^QR*)),
div(tx—u) = 0 a.e. in IR^,
and this implies, by the uniqueness for the Stokes problem, u = u, 7r = 0
(recall that tt € LP) if Q = JRN, tt is a constant otherwise.
Then, if p = 2, (A.6) implies the strong convergence ! In the general
case, using the linearity of the construction and the bound (A.4), we deduce
that it is enough to prove the strong convergence whenever u € WQ'q(Q)
(resp. u € Wl>*(JRN) n Wl>r(1RN)) for some q e (p,+oo) (resp. q 6
(p,oo), r 6 (l,p), r < 2 < q). We use here the density of "smoother"
divergence-free vector fields in W01,p, although it is possible to give slightly
more complicated proofs of the strong convergence which do not use this
fact. We first claim that us^n converges to 0 in C1, say, on compact subsets
of ft. Indeed, taking the divergence of the equation (A.2), we obtain
-Apt = 0 in ft* (A.9)
and we have already shown that ps converges weakly to a constant (resp.
0) in Lp. Hence, Vps converges to 0 in C1, say, on compact subsets of ft.
Since — A(u$— u) + us — u = — Vps and us — u converges weakly to 0 in
(A.8)

Appendix A 167
W1,p, we deduce easily from the regularity results on Laplace's equation
the convergence of us~u to 0 in C1, say, on compact subsets of Q.
On the other hand, u6 is bounded in Witq(Q) (resp. Wl*(TRN) n
Wl>r(TRN)). Then, if ft is bounded, we write, for 6 < <50, using Holder's
inequality and the Wl,q bound
/ \V(u6-u)\p + \u6-u\pdx
< meas(fi*0) sup(|V(u*-u)|p + \u6-u\p) + Cmeas(fi-ft*0)1/(*/p)'
and we conclude letting first 6 go to 0+ then 6q go to 0+.
If Q = IR , we first observe that there exists ~£l,~ nondecreasing, Lips-
chitz on (0,oo), such that fi > 1 on [0,oo), fx(t) —> +oo as t —» +oo and
In* A*(N)(|Vti|2 4- \u\2)dx < oo. Then, multiplying (A.2) by fi(\x\)us, we
easily deduce that we have
/ MM)(|Vti,|2 + |ti,|2) < C. (A.10)
Then, we write for all R € (0,oo), e <E (0,oo), 6 6 (0, l/R), denning
f6 = \V(uS-u)\r + \us-u\P,
I fsdx < meas (BR) sup fs + Cep-r + Ceq~p + f fs 1B% dx
JJRN Br -/(e</«<lA)
where we used the W1,r D W1'*7 bound. Hence, we have, denoting by Ce
various positive constants depending only on e and p
Jib
fsdx < Ce^ + Ce™ + meas(BR) sup fs
m.N br
+ Ct
(f \V(u6-u)\2 + \u6-u\2dx)
'\Jlr
< Ce*-* + Ce^ + meas(BR) sup fs + C£ ju(J?)-1
Br
and we conclude upon letting first 6 go to 0+, then e go to 0+ and finally
R go to +oo.
Remark A.l. The above procedure yields a constructive approximation
of u by Co°(ft) (resp. C^QR^)): indeed, one just needs to smooth u$ by
convolution.
In chapter 2, we use a variant of the preceding truncation in IR that
we describe now. First of all, let V^2(1RN) = {u e L^IR"), Vu €

168 Appendix A
L^IR^)} if N > 3. (Vl>2(TRN) is a Hilbert space for the scalar product
/rat Vu • Vvdx.) Then let P^2(IR2) = {u e H}0C(JR2)} Vu e I2(R2)}.
X>1,2(IR2) is a Hilbert space for the scalar product J^ Vu • Vv 4- 1bx iwdx
and an equivalent norm is given by ||Vtx||£,2(]R2) + \fB udx\. These facts
on P1,2(1R2) are easy consequences of standard inequalities like
(J \u\2dx\ < CR(J \Vu\2 + lBl\u\2dx\
< Cr^\Vu\\LHBr) + \Jb «|J
where Cr > 0 only depends on R which is arbitrary in [1, +00). We denote
by HI • HI the norm in V1,2CRlN) or 2>1,2(3R2). Next, we introduce essentially
as before for u € P^flET) if N > 3, u e Pll2(IR2) if JV = 2 the solution
uR of
-Aur + Vpr = -Au in Br, \
1 r (A. 11)
uR € #0 (#*)» div «« = 0 a.e. in £* J
where il € [1,+co). We shall also use, when JV = 2, the following variant
- Au* + lBl uR + Vpr = -Au + lBlu in BR,}
urEH^(Br), divuR = 0 a.e. in Br. f
If JV > 3, we introduce the linear map Tr(u) = ur and, if JV = 2, we
consider two linear maps
Tr(u) = ur + -[ (u-uR), Tr(u) = ur. (A.13)
Theorem A.l. Assume that (A.l) izo/ds.
1) Tr(u) converges tou, as R goes to +00, in V1'2(1RN) if JV > 3 while
Tr(u) and Tr(u) converge to u, as R goes to +00, in Z>1,2(1R2).
2) We have for all Ro € (0,00)
sup \\u-TR(u)\\L2{B )- 0 as i*-»+oo (A.14)
IIMII<i
sup ||ti-fjR(u)||La(B~) -> 0 as R-++00. (A.15)
IIMII<1
Proof of Theorem A.l. First of all, we observe that, exactly as for (A.5),
we obtain easily
f \VTR(u)\2dx < f \Vu\2dx (A.16)
Jirn Jm."

Appendix A 169
and
/ \VfR(u)\2 + lBl\fR(u)\2dx < f |Vu|2 + lBlH2dx (A.17)
where, of course, we extend ur and ur by 0 to IR . Therefore, Tr(u),
fR(u) are bounded, uniformly in R > 1, in V1*2(1RN), Vl*2(JR2)
respectively. Since, if N = 2, fB Tr(u) dx = fB udx1 we have only to show the
weak convergence to u in view of (A. 16) and (A.17) or, in other words,
extracting subsequences if necessary, that u = u if u is the weak limit in
V1>2(1RN) (or V1>2(1R2)) of TR(u)y fR(u) as R goes to +oo.
We begin with Tr(u) and we observe that, if we normalize pr by f-B pRdx
= 0, we have
diPR=djfiii in V(BR), 1
pRdx = 0, \\ft\\mBR) < C \ (A'18)
'BR )
for some C > 0 independent of R (depending only on |||u|||). This yields
WprUhb*) < C (A.19)
and we may thus assume, extracting a subsequence if necessary, that pr
converges weakly in L2(JRN) to some p. We then obtain, setting w = u —tx,
/.
-At// + Vp = 0 in2?'(lRN), peL2QRN),
div w = 0 a.e. in IR
^ r (A.20)
and w e V1>2(1RN) if N >3,we PX'2(IR2) if iV = 2 and then fBi u; dx = 0.
Taking the divergence of (A.20), we immediately deduce that p = 0 and
thus w = 0ifiV>3, t^ = constant if JV = 2. Finally, ty = 0 if AT = 2 since
^ k; dx = 0.
The proof for Tr(u) is a bit more delicate. Multiplying (A.12) by wr,
where 1 < Rf < i2, w = xT—u and wr is the solution of (A. 11) corresponding
to w (instead of tx), we obtain
/ V(fR(u)-u) -Vwr< dx + J (fR(u)-u)-wR<dx = 0
JBr JBi
and, letting R go to +oo,
/ Vw • Vwr< dx+ w ■ wr' dx = 0. (A.21)
JlR2 JBi

170 Appendix A
If we show that fB wdx = 0, we see that (A.21) holds with wr> replaced
by Tr>{w) and thus, letting R' go to +00, we deduce
/ |Viy|2dx + / \w\2dx = 0
hence w = 0. In order to show our claim on fB w dx, we take the curl of
(A. 12) and we find
-A{cuT\(fR(u)-u)} + cm\(lBl(TR(u)-u)) = 0 in V'{BR)
and, letting R go to +00, we obtain
-Acurliy + curl(lBliy) =0 mV(TR2), cmlw € L2(1R2).
Hence
curitu = — / r—^>w\{y)dy-7r-1 r—-kw2(y)dy
27r Jb! \y-x\ 27r Vsi \y-x\2
as \x\ —► +00.
Since curln; e L2(IR2), we deduce easily that fB wdy = 0, and this
completes the proof of part 1.
Since the embedding of Vl>2(1RN), Plj2(IR2) into L2(SHo) is compact,
we have only to show, in order to prove part 2, that Thti(uti), Tnn(un)
converge weakly in P1,2(IRiV), Z>1,2(1R2) respectively to u whenever un
converges weakly to u in these spaces, and this fact is shown exactly as in
part 1. D
We conclude this appendix with the introduction and the study of some
related truncations. To this end, we consider / € L°°(JRN) D Lt^JR^),
/ > 0, / ^ 0 and we assume in all that follows that N > 3. If u 6
P1*2^^), R e (0,cc), we define Gr(u) = uR to be the solution of
- AuR + fuR + VpR = -Au + fu in V\BR),
uR e H^Br), div ur = 0 a.e. in Br,
and we have
Theorem A.2. Assume that (A.l) holds.
1) &r(u) converges to u in Vl>2(1RN) as R goes to +oo.
(A.22)

Appendix A
171
2) Let un converge weakly in Pli2(IRN) to some u. Let fn > 0, fn €
L%t00(JR.N), fn £ 0 be bounded in L°°(JRN) and assume that y/J*un is
bounded in L2(JRN) and that fn converges to f in Lx{Bm) for all M €
(0,oo). We denote by 6%(un) the solution of (A.22) with u,f replaced by
un,fn respectively Then, QR(un) converges weakly to un in Vlt2(JRN)
uniformly inn as R goes to +oo and, in particular, we have
sup \\un - e£(un)||L2(BRo) -> 0 as R^+co. (A.23)
Proof of Theorem A.2. 1) We have (extending as usual ur = Qr(u) by
0)
/ |Vufl|2 + f\uR\2 dx = / Vit • VuR + fu • uR dx
< [ \Vu\2 + f\u\2 dx.
JjRN
Notice that f\u\2 € L^IR*) since / € L%>°°{1RN), \u\2 € L^^(JRN). It
is thus enough to show that if ur converges weakly to some u in I>1,2(]RiV)
then u = u. Writing w = -A(u-u) + f(u-u) € H'1^.1^), we deduce
from (A.22) that we have
<w,<j»= 0, for all <t> € C^(1RN), div<£ = 0 on IR*. (A.25)
Therefore, by classical results, there exists p € L^'2(]RN)+L2(IRN)such
that w = -Vp: once we know that w = —Vp, the integrability of p is easily
seen by Fourier transforms. Indeed P = n+mlu/a P € L2(TRN) and thus
pl^D € L2(JRN), while pl(|€|<1) = l(|g|<1) <1+lfpt/2 P € L*'2(IR")
since ^ € ^^(E^).
Furthermore, we have clearly
-Ap = divw = div(/(w-u)) in V(1RN)
and f{u-u) € L^'2(1RN) since / € LN^(JRN), (u-ti) 6 L^'2(RK).
Hence, we obtain
p € L2(UN), Vp 6 L^'^IR"). (A.26)
We then set z = (u—u) and multiply iy by z<pn where <pn = y(^), n > 1
and v? € C£°(IRN), Suppv? C B2, 0 < <p < 1, (p = 1 on Bi. We obtain
(A-24)

172 Appendix A
easily
/ (\Vz\2 + f\z\2)<pndx
= J N\z\2(-&<Pn)+PZ'V<Pndx
<£/ \z\2dx + ^f \p\\z\dx
<£[ \z\*dx + c([ \p\2dx) l±[ \z\2dx\.
71 Hn<\x\<2n) \JR" J ln J(n<\x\<2n) J
In view of (A.26), we conclude easily that z = 0 once we observe
4/ tfdxiZff W**)""V)*
U J(n<\x\<2n) n \J(n<\x\<2n) /
J(n<\
2JV
\z\7r^ dx -* 0.
'(n<|x|<2n) n
2) Since, for each n > 1, ©£(un) converges in Vli2(JRN) to un in view
of the proof of part 1, we have only to show that un = @£n(txn) converges
weakly in Vl*2(TRN) to u as n and #n go to +00, and (A.23) then follows
in view of the Rellich-Kondrakov compactness theorems.
Observe first that because of (A.23) and the assumptions made, un is
bounded in Vl*2(JRN) and thus, without loss of generality, we may assume
that un converges weakly in Vli2(TRN) to some u. Next, we simply observe
that /ntin, fnun converge respectively in L\oc (and even Lfoc for 1 < p <
7735) to /u, fu since /n, un and un converge in L2oc. At this stage, we
conclude that u = u exactly as in part 1. □
Let us remark that part 2 of Theorem A.2 yields the following variant of
(A.23)
sup{||un-0£(^
—► 0 as R —> +00
(A.27)
for all un € V^2{JRr). Indeed, we just apply part 2 with un(l+
MUlW) +Hv7^nlli2(IR»)r1/2 instead of «».

APPENDIX B
Two Facts on 2>1,a(]RJ)
Recall (see Appendix A) that P1,2(1R2) = {u € H^QR2) , Vu € L2(fft2)}.
We begin with a remark.
Lemma B.l. We have P^QR2) n (L^TR2) + L2(K2)) ^ tf^IR2), and,
more precisely, there exists a constant C > 0 such that for allue X>1,2(1R2)
satisfying u = U\ -rU2 with u\ 6 L^IR2), U2 € L2(IR2)
Nlw) < ^{lltnll^j ||Vtt||JJ2(Ra) + ||«2||l»(r»)}. (B.l)
Proof of Lemma B.l. We first remark that |u| = ui + 62 with u\ =
min(|ui|,|u|) € L1(1R2), 0 < u2 < \u2\ and u2 € L2(JR2). Next, we also
have: |u| < 2w + |u| 1(;„|<2) where it; = (|u| - 1)+ € Z>li2(]R2), and
/ |u|2 l(:u;<2) <iar < 2 / uf l(fil<2)<k: + 2 / ft2,^
Jir2 Jjr* Jn2
< 4 / v.idx + 2 I u\dx
hence
llUl(|u|<2)!!L2(IR2) < 2(||tZ1||^]R2) + ||tZ2||L2(1R2)). (B.2)
On the other hand, we also have
/ wdx < I \u\ l(|tt|>i)da;
J-b? Jm.*
< / Uldx+ U2\l(ul>l/2)>+l(u2>l/2)jdx
f r li1/2 r ii1/2i
< l|wilUMiR2) + Ilu2!iL2(iR2){meas|ui>-] + meas|u2 >-J J
< IIwiIIlhir2) + !Iu2!Il2(ir2) {v^lluilli^iR2) + 2||u2||l2(ir2)},

174 Appendix B
and we deduce from Gagliardo-Nirenberg type inequalities
Mb(iR>) < c|M|$Ra)\\Vw\\1lw) * cMl^)\\Vu\\%\B?)
therefore we find
IMIlw) < ciivuii^jlllttill^)
+ ii^ni^jiiwiiii^) + imil'or')}
or
IMIlW) < C'UVttll^^Ht.illi^) + IMl>(ir>)}. (B.3)
Combining (B.2) and (B.3) we obtain
NL»(R») < C^IMl^r') + IIVtxIl^jR^Huilli7!2^)} 1
+ ff||«illi/«(Iia) +C\\V^2ijR>)\M\LHJR>y J
(B.4)
Then, replacing u by \u and thus u\,U2 by Aui,Ati2 where A > 0 is
arbitrary, we also deduce
NIl»(r») < c{ll«2||^(iR2) + l|Vtx||^2(R2)||ui||^2(R2)}
Since this inequality holds for all A > 0, we choose A = H^ill/wi^N •
J{m.2)\\U2\\J(iR2) and we find
IMlL*(Ra) <
+ ClltiiH^j llVtill^llttall^,,
and this inequality yields (B.l). □
Remark B.l. A similar (and simpler) proof shows a similar result in
P1,2(1RN). More precisely, if N > 3, u € £»1,2(1RN) with u = ux + u2,
ui € L^JR1*), u2 6 L2(JRN) then u € L2(IRN) (and thus &(!&")) and
we have for some C > 0 depending only on N
Hl'or") < C-llltuH^^NjIIVullil^j + llttall^dtN)} (B.5)

Appendix B
175
where 6 = jfe.
We next turn to weighted IP bounds for elements of Z>1,2(1R2). Recall
(see Appendix A) that we choose the scalar product (J^ Vu • Vv + l^u •
vdx) on P1,2(IR2) for which this space is a Hilbert space, and we denote
by HIu|| | the corresponding norm.
Theorem B.l. For all m € J2,+00), there exists a positive constant C
such that we have for all u € Vl>2{1Rr)
\l/m
«-J < c\\\u\\\
(B.6)
ifde (f+ 1,00).
Proof of Theorem B.l. We begin with the case m = 2 and prove in fact
\l/2
\l/2
(Js\u(rv)\2du^ < 2{Jb |n|2dx) +||Vtz||i/22(1R2)(log2r)1/2, (B.7)
for all r > 1, where we write x = ru, u € S1, r = |x|, and the case m = 2
of (B.5) follows easily upon integrating in Q.
In order to show (B.7), we write for all r > 0
v 1/2
i(/siwmi^) *(/j£H'*,)
< ff \Vu{ru>)\2 dJ\
2 \V2\
7
v 1/2
(B.8)
and observe that there exists ro € [5,1] such that
( / \u{r0u)\2du\ < 2( f N2^
v 1/2
) ■
Hence, integrating (B.8) from ro to r > 1, we deduce
\l/2
< 2U \u\2dx\ + C(J \Vu(t(j)\2dw\ dt
< 2U \u\2dx) +([r[ \Vu(tu)\2tdwdt\ (log^-)
r\i/2

176
Appendix B
and (B.7) follows. The above calculations are obvious if tx is smooth and
(B.7) is thus shown for all u € P1,2(IR2) by density.
Let us also remark that (B.7) implies for all R > 1
/ \u\2dx < C7.R2 log 2JR|||u||i2
JBr
J \u\dx < CR2 (log 2R)1/2 ||\u\
Jbr
(B.9)
(B.10)
We next want to show (B.6) for m € (2,oo). We observe that we have
for all R > 1
0^ \u--f u\mdx)
Br JBr J
< c( f \u--f u\2dx) (f \Vu\2dx)^. \
\JBr JBr J \JBr / J
(B.ll)
This is one form of the Gagliardo-Nirenberg inequalities (the fact that C
does not depend on R can be easily deduced from a simple scaling
argument). Combining (B.9), (B.10) and (B.ll), we deduce for all R > 2
/ \u\mdx < CR2 {log 2R)m'2\\\u\\\m.
JBr
Next, by a simple integration by parts, we obtain for all R > 2
(B.12)
-a
u\mdx\l+R2)-l{\og Vl+R2)~e
Br J
R
+ T±5(logV/I^)-^)I^}dr
< C||H|r(l + ^:;^=(log\/l+^)^-^r)
< c\\\u\\r
ifd>% + l. □

APPENDIX C
Compactness in Time with Values
in Weak Topologies
"Let X be a separable reflexive Banach space and let fn be bounded in
L°°(0,T;X) for some T e (0,00). We assume that fn € C([0,T]; Y) where
Y is a Banach space such that X *-* Y, Y' is separable and dense in A7.
Furthermore, we assume that for all <p € Y', <^,/n(t)>rxy is uniformly
continuous in t € [0,T] uniformly in n > 1.
We then choose a closed ball Br0 of X containing all the values of fn(t)
for t € [0,T], n > 1. The weak topology of X, since Xf is separable, makes
Brq a compact metric space, and we denote by C(\f),T\,X-w) the space
of continuous functions on [0, T] with values in Bj^ C X equipped with
the weak topology.
Lemma C.l. Under the above conditions, fn is relatively compact in
C([0,T];X-tii).
Proof of Lemma C.l. Let (<fk)k>i be a dense sequence in Y'. We
consider the "weak topology" distance on Bj^ given by
<v.9) -Eft
«Pk, f-9>Y'xY
fc>1 - - + l <VJk, f~9>Y'xY
Using the Ascoli-Arzela theorem, we have only to show
sup d(fn(t),fn(s)) — 0 as t,se[0,T], \t-s\ -► 0,
n>l
and this is obvious since, by assumption, we have for each k > 1
sup|«^, /n(*)-/n(sj>vxy| -* ° M K--sl —^ 0.
n>l
Hence, for all k > 1
>l<Vi,rW-r(5)>yxy| + :1
l<j<fc n
d(r(t)jn(s)) < sup sup\«pj,r(t)-fn(s)>Y>*Y\+ 2k
and we conclude upon letting first \t—s\ go to 0 and then k go to +00. □

APPENDIX D
Weak L1 Estimates for Solutions of the
Heat Equation
We wish to prove in this appendix a result concerning solutions of the heat
equation in IR^ that we used in section 3.3. The main result we want to
present in this direction is given by
Theorem D.l. Let f 6 Ll(0,T; L<?'r(IRiV)) where 1< q < oo,l <r <oo.
Let u 6 C([0,T];L9'r(lRiV)) be the solution of
^_Ia« = / in TR.Nx(0,T), it|t=o=0 in BN. (D.l)
at 2
Then, we have
^, D\u 6 L«'r(lRN) a.e. *€(0,T) (D.2)
meas ji € (0,T) / |M|L,.r(1RAr) >>\< C,||/IIl»(o,t;l«.'-<r"))* *
for all A > 0
(D.3)
for some C > 0 independent of f, where (p = -^ , D^-
Remarks D.l. 1) Here and below, L^IR^) denotes the usual Lorentz
spaces (L*«(RN) = !«(£*)).
2) The existence of the solution u of (D.l) and its continuity in t with
values in Lq*r(JRN) follows easily from the representation formula
U(M) = J/SLdyfiy'S) ^ (-Wi>^-S))'m 1 (D.4)
a.e. inIR", Vie[0,T]. J
3) The normalization constant ^ in front of A is irrelevant and the result
is valid if we replace — \ A by — vA with u > 0 or, more generally, by an
arbitrary uniformly elliptic second-order operator with bounded uniformly
continuous coefficients.
4) If we take r = q, the estimate (D.3) together with the classical
^(IR^ x (0,T)) result for D2u and $ shows, by interpolation, that

Appendix D
179
§j*, D2u e Lp(0,T; Lq(lRN)) whenever l<g<oo,l<p<g. Then, by an
easy duality argument, we see that this result is also valid if 1 < g,p < oo
and we recover the "classical" Lp(Lq) regularity result for solutions of the
heat equation, a result that we used in section 4.3. □
Proof of Theorem D.l. We first use and recall the following variant
of the fundamental "covering lemma" due to A.P. Calderon and A. Zyg-
mund [78] (see also A. Zygmund [497] and E. Stein [454]) contained in L.
Hormander [222] (Lemma 3.1, observe that the proof in [222] immediately
extends to Lq,r(TRN) spaces): for each M > 0, we can find g,hk1 disjoint
intervals Ik in [0, T] such that
oo
Jk=l
oo
II5IIl1(0,T;L'i.'-(1R'v)) + 22 IIMIl,1(0>T;.L'>r(IR.'v)) < Hf\\LH0,T;L"-r(m.N))
IMIl^or") < 2M a.e. t 6 (0,T)
I hk(t,x)dt = 0 a.e. in IR^, 1
hk(t,x) = 0 if t*Ifc> V*>1, J
OO -
(D.5)
(D.6)
(D.7)
"" 1
^meas(4) < — \\f\\LH0,T;Li'r(JRN))
fc=l
Then, we write, denning p = (2n)~N/2 exp (-^§|-),
ft p f\ OO et r Q_
-/= ds dy—(t-s,y)g(s,y) + J2ds dy—(t-s,y)hk(s,y)
Jo Jtrn ot ^Jo Jjrn at
du
(D.8)
and denote by v and Wk the first term and the generic term in the series
of the right-hand side. We also write V(t) = \\v(t, OIlL^aR")* Wfc(*) =
\\wk(t, OIIl^or") ae- * e (0>^O- We are going to show the estimate (D.3)
for ^ or equivalently for (p = ff — /, the proof being exactly the same for
D\u (or using elliptic regularity and the equation (D.l)). To this end, we
estimate v and Wk in various ways.
From the classical 1^(11^ x (0,X)) estimates for solutions of the heat
equation (and by interpolation) we obtain, denoting by C various constants
independent of / and M,
v 6 L^r(0,T;L^(JRN)) and \\V\\L,.r{0tT) < C\\G\\L,.ri0,T) (D.9)

180 Appendix D
where we define G(t) = \\g\\L<i>r(iRs) a.e. on (0,T). Notice that (D.6) means
0 < G < 2M a.e. tG(0,T) (D.lO)
while (D.5) yields, normalizing without loss of generality ||/||li(o,t;l*.*-(R"))
to be 1,
II<?||l«(o,t).< 3- P-ll)
Then, ||G||L,.r(0|T) < C(f™ n(t)r/qtr-1 dt)1/r where n(t) = meas{s 6
(0,T) / G(s) > t} (0 < n < T). Notice that n(t) = 0 if * > 2M (by (D.lO))
and that f£°p(t)dt = ||G||Li(o,t) < 3 by (D.ll). Hence, if 1 < r < q, we
deduce, using Holder's inequality,
HG||l..'(o,d < c(jf "*) \[ f(,wl)/Mcft) "
Now, if r > g, we write
Uoo \l/r -i.i
/idn M^^supM*)*}]9 r
Uoo \ 1/9
fidt) Ml~l/q < CMx~xlq.
In both cases we have shown
\\G\\L,.r(o,T) < CM1-1'* (D.12)
and thus, in particular, we deduce from (D.9) that we have for all A > 0
meas{* € [0,T] / V(t) > A} < CMq'lX'q. (D.13)
Next, denoting by 1% the interval with the same centre as Ik and with
a double length (if Ik = (tfcltfc+1), ij = (*i^s±i , ^±^)), we set
/ = Ufc>i Ik- We remark that (D.8) implies
00
meas{/} = 2^meas(/fc) < 2/Af. (D.14)

Appendix D 181
Then, we claim that we have
/ dtWk(t) = \\wk\\Liy2.Lqtr(:R»)) ^ C\\hk\\motT;L^(JRN))' (D.15)
Admitting temporarily this claim, we complete the proof of the theorem.
Indeed, lit £1
oo
IMIl..-<r") < V(t) + W(t) where W(t) = Y,Wk(t)
fc=i
hence, for all A > 0, we deduce using (D.14), (D.13), (D.15) and (D.5)
meas{t e(0,T) / \\<p\\L,,riJRN) > A}
< meas(J) + meas {t £ I / |MIl«.-(1R") > A}
< ± + meas{t *// W(t) > |} + measjt € (0,T) / V(t) > ^}
^li + 2xJIcw^dt + CMq-lx-q
2 C
OO
<^ + jE IIM*(o,t*..-<r">> + CM'"1 A~«
M A
and we conclude choosing M = A.
The only point remaining to check is the claim (D.15). We recall that
we have on IR^
wk(t) = 0 if tg J, t < tk; \
Mt) = / %^s) * M*) <*s if t<£I1t> tk+i J (D'16)
and because of (D.7) if t > tk+i
Wk(t) = I \%{t~s)" %{t~tk)\ *hk{s)ds- (D,17)
We then wish to estimate the norm of §f (£—s) — §jr(*—£fc) as a multiplier in
Lq,r(JRN) (denoting by || • ||a/«.»- this norm) and we shall prove below that
dp dp
dtK ' dt
s-tk
-£(t-8)-%{t-tk)\\ _ < C-^-^r. (D.18)
M,,r (< —^)2

182 Appendix D
If this is the case, we complete the proof of (D.15) easily: indeed, we have
/ Wk(t)dt
J{il)c
rT
-/—di/[|(,-s)-l((-(k)]w,fe(s)ds
2 *
Jtk
L*>r(]RN)
since
s-tk **±iZ** =1 if s€/fc
3*jk+i-*jk—25 3(tjt+i-tjt) 3
Next, in order to prove the estimate (D.18), we compute the Fourier
transform of ff(i-s) - fjr(t-tk), namely {e"^1^ - e^fc)^r}|^|2.
We observe that ^{e""lf|2-e"(1+^)^,2}|^|2 defines a multiplier which
has the same Mq*r norm by the dilations invariance of this norm. Therefore,
(D.18) is proven if we check that {e-M2 - e-(l+x^)\Z\2 = \£\2 e-W2 (I -
e-A|£| ) defines a multiplier whose Mq norm (and thus any Mq%r norm by
interpolation) is bounded by CX for ail A and for all q e (l,oo). This is
straightforward since we have for all \a\ = m > 0
|a°[(1 -c-A'€«8)e-l€l9|^|2] | (l + |ei2)m/a
m
3=0
< C\(l + \Z\2)m+2e-W2 < C(m)X
and we conclude using classical results on Fourier multipliers. □

APPENDIX E
A Short Proof of the Existence and
Uniqueness of Renormalized Solutions
for Parabolic Equations
The goal of this appendix is a short and direct proof of Theorem 3.10. We
thus consider solutions of
?L+u-Vf-Af = F in Qx(0,T) (E.l)
ot
/lt=o = /o in ft, |^=0 on dQ x (0,T) (E.2)
on
where /o € L1(Q)1 F 6 Ll(Q x (0,T)) and Q is, say, a bounded, smooth
open set in IR (N > 2); we could as well treat the case when Q = IR^,
or the periodic case or the case when the Neumann boundary condition
is replaced by a Dirichlet boundary condition. Finally, we assume that u
satisfies
ueL2(ftx(0,r))N, divu = 0 in V(Q x(0,T)),\
u-n = 0 on dftx(0,r)
(E.3)
(recall that u - n € L^O^tf-1/2^))).
More general conditions on u (and divtx) are possible that yield the
same result as below but these conditions are more than enough for the
application we have in mind, namely Theorem 3.10. For the same reason,
we shall assume
F > 0 a.e. in Q x (0,T) (E.4)
even if it is not necessary for the result to follow; however, this condition
allows us to give rather simple proofs.
As explained in chapter 3, section 3.4, we cannot simply use distributions
theory to solve (E.1)-(E.2) since we would have to define the product uf
(writing u • V/ as div (uf) since div u = 0) ; and since we only assume u
to be in L2(ft x (0,r))^, we would need to know that / G L2(ft x (0,T)).
However, we cannot expect / to be in L2(Q x (0,T)) since F and / only
belong to Ll.

184
Appendix E
As we saw in section 3.4, we expect / (and we can obtain corresponding
formal a priori estimates) to satisfy:
f € C([0,T};L1(a))nL1(0,T;L"(n)) for all q€[l,JL) (E.5)
TR(f) € L2(0,T;ff1(fi)) for all J*€(0,oo),
lim i fdxf dt\VTR(f)\2 = 0,
(E.6)
where TR(t) = max(min(*,.R),-.R) for t € IR, R € (0,oo), V denotes the
spatial gradient (in x) and
V,/ 6 Lr(fix(0,T)) for all re[l,^±i). (E.7)
In particular, we know that VTR(f) = V/ l(|/|</i) a.e. in fi x (0,T).
In order to solve (E.1)-(E.2), we shall use the notion of renormalized
soiutions introduced by R.J. DiPerna and P.L. Lions [125] in the context
of the Fokker-Planck-Boltzmann equations, see also L. Boccardo, I. Diaz,
D. Giachetti and F. Murat [67], P.L. Lions and F. Murat [308] for
nonlinear elliptic equations, D. Blanchard [65], D. Blanchard and F. Murat [66]
(and the references therein) for parabolic equations. The idea is simple:
we write down the equation satisfied by /?(/) where (3 € Cq°(IR) using
(E.l); let us just mention that many equivalent formulations are possible.
First of all, we notice that /?(/) € C([0,T]; L1^)) n L°°(fi x (0, T)) since
P is bounded, 0(f) € L^O.Tjtf1^)) since V0(f) = 0'(/)V/ a.e. and
|/5/(/)V/j < supnt \P'(t)\ \VTR(f)\ where R > 0 is chosen in a such a way
that Supp(/3') c [-R,R]. Similarly, /T(/)|V/|2 e L1^ x (0,T)). Then,
formally, we obtain
^p. + dw{up(f)}-A(3(f)+P"(f)\Vf\2 = (3'(f)F infix (0,T).
(E.8)
Obviously, /?(/) should satisfy (E.2) with /o replaced by /5(/o), and this
combined with (E.8) yields the following weak formulation: we have for all
0 € Co°(IR) and for all (feCl(Qx [0,T]) (for instance)
+V(3(f).V<f + p"(f)\Vf\2<p-(3'(f)F<pdx = 0 inV{0,T),} (K9)
W)€C([0,ri;Ll(n)). W)|t=o=Wo) a.e. in O.

Appendix E
185
In conclusion, we say that / is a renormalized solution of (E.1)-(E.2) if
it satisfies (E.5)-(E.7) and (E.9).
Theorem E.l. There exists a unique renormalized solution of (E.1)-(E.2).
Remarks E.l. 1) If we know that u 6 LP(Q x (0,T)) for some p > 2
and that / 6 L^(Q x (0,T)) for some q 6 [1,+ooj such that ± + ± < 1
(thus \u\f 6 L*(ft x (0,T))), then (E.9) combined with (E.6) implies that
/ is a "standard weak" solution of (E.1)-(E.2); we could as well treat cases
where tx, / have different integrabilities in x and in t. Before proving this
claim, let us observe that the integrability of / can be estimated in terms
of integrability requirements upon F and /o. In order to prove the above
fact, we use (E.8) with (3 = fin as given in the proof of Theorem E.l below
and we let n go to +oo using (E.6) to show that /?^(/)[V/|2 converges to
OinL^flx (0,T)).
2) The proof of Theorem E.l below also shows that the unique
renormalized solution of (E.1)-(E.2) is the limit (in C([0,T];L1(tt)) for instance) of
the solutions of regularized problems (regularize tx, F or /0).
3) The uniqueness proof given below also yields the expected fact that
if /i and $2 are renormalized solutions of (E.1)-(E.2) with F, /o replaced
respectively by Fi, i*2> /o,i> /o>2 and if A,/x 6 H then A/i +11/2 is the
renormalized solution of (E.1)-(E.2) with F, /o replaced by XF\ + fiF2,
A/o,i +m/o,2 respectively. This is of course a very natural fact since (E.l)-
(E.2) is a linear problem; however, the notion of renormalized solutions is
a nonlinear one! □
Proof of Theorem E.l
Step 1. Uniqueness. Let /i,/2 be two renormalized solutions of (E.l)-
(E.2). We shall write equations like (E.8) for various quantities involving
/1 and /2 that satisfy the Neumann boundary condition contained in (E.2);
whenever we do so, we really mean that the weak formulation like (E.9)
holds. With this convention, we have by assumption
jt Wi)-/3(/2)} + div [u{/3(fi) -W2)}] - A[0(/i)-Wa)] \
+ /?"(/i)|V/i|2-/3"(/2)|V/2|2 = 2W/i)-Wa)]- J
Notice also that since divu = 0, u € L2(Q x (0,T)), (3(fi) - (3{f2) 6
X2(0,T;if1(f2)) the second-term div[u{P(fi) - /3(/2)}] may be rewritten
as u • V(/3(/i) — /?(/2)) (one simply needs to argue by density on u).
Next, let 7 € C0°°(1R); 9 = 7(/?(/i) - Wa)) € CtfO.T];!1^)) ni2(0,T;
Hx{0)) satisfies g\t=0 = 0 in 17 and

186 Appendix E
^ + div (ug) -Ag + {fi"{h)\Vh\2-0"{f2)\Vf2\2}1'{0{fx)
-W2))+7,,(Wi)-W2))|VWi)-VW2)|2 ) (E-10)
= F{(3f(f1)-0'(f2)W(P(fi)-0(f2)).
This computation is obvious formally: to justify it, we simply observe that
we have as above
div(tzs) = u-Vg = U'V[0(f1)-0(f2)W(0(fi)-P(f2))
while if U - Ah € L1^ x (0,T)), h 6 ^(O.Tjtf1^)), -y(h) satisfies
M0_A7(/l) + 7"(/l)|V/l|2 = {^-Ah)i{h).
This last fact requires some explanations: first of all, regularizing h in t
by convolution, we see it is enough to prove the above claim when h is
smooth in t (h € C1([0,T];ff1(f2))) since we then easily pass to the limit
and prove the claim. When h is smooth in t, we just have to show that if
-A/i e Ll(Q), h e Hl(Q) then -A7(/i) = -y"(h)\Vh\2 + 7'(/i)(-A/i). In
order to check this fact, we use the weak formulation (incorporating the
Neumann boundary condition), namely
/ V/i • V<pdx = / (-Ah)(pdx for all (p e Cl(Q).
Jn Jn
This equality holds by density for (p € if x(ft) n L°°(Ct) (approximate such
a <p by (pn € Cl(Tl) such that (pn -* (p in if^fi) n Lp(fi) for all 1 < p < oo,
n
y?n bounded in L°°(Q)). Then, we take <p = Y(/i)V> where ^ € C1(H) and
we find
/ (-Ah)Y(h)j> dx = / V/i • {tW ViHt"(>0 V W} <£r
= / V7(/»)-V0 + 7"(/i)|V/i|Vrfar,
and this completes the proof of (E.10).
We next use (E.10) with 0 replaced by (3n(t) = n^(^) (n > 1) where
A € C§°(1R), &(*) = * if |t| < 1, &(*) = 0 if |t| > 2 and 7 replaced by
7»W = Tb(OC(*A0 where C € C0°°(1R) > 0, <(*) = 1 if \t\ < 1, <(*) = 0
if |t| > 2, 7£ € C^OR), 70 > 0 and %' > 0 near 0, %(0) = 0, 7o(0) = 0.
Writing gn = 7»(Ai(/i)-A»(/2)), we deduce from (E.10)
^+div(ugn)-A9n < %{\Vfl\2lm<2n) + \Vf2\2lm<2n)}
+ F{Ml)-0nU2))l'Mn{h) - 0n(f2)).

+
rT
Appendix E 187
where we used the fact that |#| < f, 7* = 7^C + H<'(£) + £7o<"(£) >
—— and where C denotes various positive constants independent of n > 1.
We integrate this inequality over Q (using the weak formulation of it
with <p = 1!) and we deduce for all t € [0,T] and for all n > 1
[ Jn(Mfl)-0n(f2)){x,t)dx
Jn
< |jf dtjrdx{|V/i|2l((/l|<2n) + |V/2|2l(|/2|<2n)}
/ dt [ dx\F[P'n(fl)-Pn(f2)Wn(Pn(fl)-Pn(f2))\
Jo Jn
~ n I' dtfadx{\Vfl\2 1(IM<2n) + |V/2|2 l(|/2|<2n)}
+ Cjf*j[<b|F,l«(^)"*(T)|-
Because of (E.6) the first term of the right-hand side goes to 0 as n goes
to +00, and the second term also goes to 0 since 0[ (**•), @[ (**) converge
a.e. in Q x (0, T) to 1 and are bounded uniformly in n. From the definition
of 7„ and /?„ (and the fact that flt f2 € C([0, T]; I1^))) °ne checks easily
that 7»(A»(/i) ~ AiCfr)) converges to 7o(/i-/2) in CtfO.TjjL1^)) as n
goes to +00. Hence, 70 (/1—/2) = 0 and /1—/2 = 0 a.e. in Q x (0,T) since,
by construction, 7o(t) > 0 if \t\ > 0.
Step 2. Approximation, bounds and convergence in C([0,T];Z-1(f2)).
We next want to show the existence of a renormalized solution of (E.l)-
(E.2). To this end, we consider the following regularized problem
^l + uc-V/c-A/£ = Fe in fix(0,T), ]
ir (R11)
-^- = 0 on <9fix(0,T), /£|t=o = /o£ in fij
where e € (0,1], F£ € C$°(ft x (0,T)), F£ — F in Lx(fi x (0,T)), F£ > 0
a.e., /0C € Cg°(fi), /0ff-^/o in ^(fi), uc € C£°(fl x (0,T)), divuc = 0
in Q x (0,T), u£-*u in L2(fi x (0,T)). The equation (E.ll) is now a
standard parabolic problem that admits a unique smooth solution fc (say
C2(fix[0,T])).-
One easily checks that f£ is bounded, uniformly in £ € (0,1], in C([0, T];
L1(fi)). We next claim that (E.6) holds for fc uniformly in e, from which
bound we deduce a bound in L1 (0 ,T\ Lq (fi)) (V q < 7^5) by Theorem

188
Appendix E
E.2 below and a bound upon V/ in Lr(Q x (0,T)) for r e [l, ^±±) by the
arguments developed in chapter 4. Next, in order to prove our claim, we
multiply (E.ll) by Tr(/c), integrate by parts over £2 and find
sup ||5n(/*(*))llL»(n) + / d*/d:r|Vr*(/e)|2]
£[0,T] JO Vfi I
< [dxSR(ft)+ f dtfdxF£TR(fs)
Jn Jo Jet '
(E.12)
where SR{t) = \t2 if \t\ < R, = R\t\ - ^ if |*| > R. Since SR{t) < R\t\
on IR and \TR(fe)\ < R we deduce that TR(fe) is bounded uniformly in e
in l2(0,r; #*($))) for each R € (0,oo). In addition, (E.12) yields for all
M € (0, oo)
ijT dtJjx\vTR(n\2
< [ ^SR(f$)dx+ [ dt[dx\F*\±\TRV<)\
Jet ft Jo Jn ft
(ft)+/ |/oe|^ + ^||Fe||Ll(nx(o)T))
rp
[ dt[dx\F€\l{\n>M).
Jo Jn
M2
< -jr-meas
R
Our claim then follows since on the one hand /q converges to /o in Ll(Q)
and thus L\je\>M\ |/<?|dx Soes to 0 as M goes to +oo uniformly in a €
(0,1], and on the other hand F€ converges to F in Ll(Q), meas{(x,£) e
« x (0,T) / |/*(s,0l > M} < £ sup0<,<T ||/ff(*)||Li(n) hence £dt fndx
\Fe\l(\fc\>M) goes to 0 as M goes to +oo uniformly in e € (0,1]. In
conclusion, we have shown since ^ S#(£) > \ \t\ l\t\>R
rp
±J dtJ^dx\Vf£\2l^n<R) -> 0
as R —*• +oo, uniformly in e € (0,1].
sup ||/c(t)l|/.<t)|>Ji||
t€[0,T]
as .R —► +00, uniformly in £ 6 (0,T]. J
(E.13)
(E.14)
We finally show that /e is a Cauchy sequence in C([0,T];L1(fi)). To
this end, we follow the uniqueness proof with /i,/2 replaced by fe,fv for

Appendix E 189
e,rj e (0,1], and we obtain, setting gn = 7n(/?n(/c) - Pn(fv)),
^■+div(uegn)-Agn
< £{|V/'|2l(|/.,<a») + |Vr|2l(|/n,<2„)}
+ C|tx£-tx^||Vr|l(|/,|<2n)
+ [F<0'n(fe) - F" &{/*)) <y'n(Pn(fc) - Mf))
< f {|V/£|2 l(l/.|<2n) + |Vff l(|/n,<2„)} + Cn\u'-u»\2
+ C\F<-F\ + C\Fr>-F\ + C\F\ l(l/«|>n) + C\F\ l(|/,|>n).
Hence, integrating in x and t, we deduce using (E.13) and the definitions
*C3
SUP bn{Pn(n-Pn(n}\\Li(n) < Sn+Un(e,T)) (E.15)
t€[0,T] V '
where <5n —> 0, un(e, tj) —► 0 as £, 77 —► 0+ for each n > 1 fixed. We also used
n
to derive (E.15) the following observations
rT
sup / dt rfar|F|l(|/c|>n) -+ 0
r€(0,l] JO Jn
as n —► +00
since sup£€(01] measX)t(|/c| > n) —► 0 as n —► +00, and
0 < ln{Pn(fS)-l3n(f2)) < C|tf-/J|.
Next, we observe that we have
ir-fl < l/?n(/C)"/?n(/'7)| + (|/£| + |/T,|)(l(|/'|>n) + l(|/.|>n))
and that for each S e (0,1), there exists Cs > 0 such that \t\ < 6+Cs 7n(*) +
\t\ l|t|>n f°f all t € IR. Hence,
\f-n < 6 + CSln(Pn(fC)-/3n(n)
+ \Pn(fe)-Pn(n\ l<|/M/«HM/i)|>»>
+ (l/£| + l/,,|)(l(|/«|>n)+l(|/''|>n))
< <5 + C,7n(/?n(/£)-i3n(r))
+ C'(l/£| + l/'7|)(1(|/3n(/e)|>-J) + 1(A.(/")|>f)j
(|/'|>n) + 1|/"|

190 Appendix E
Since \(3n{t)\ > § implies /?i(£) > \ and t > §, we obtain finally
\r-r\ <6+csyn(0n(n-0n(n)
+^(i/*i+i/,,i)(i(,/.,>f) + i(„„>f)))-
Combining (E.15) and (E.16), we deduce
(E.16)
sup \\f£(t)-r(t)\\LHa) < 6 + C66n+Cscj(e,7i)+<yn (E.17)
te[o,T]
where 7n —► 0, and we used the following facts
n
sup measx (|/c(*)| > ~) -+ 0 1
t€[0,T] * S,
as n goes to +oo, uniformly in £ € (0,1]. J
sup / \f£(t)\dx < Mmeas(A) + sup /£(t)l(|/«(t)|>M)
=[o,T]Ja [o,t]"
t€[o,
LHQ)
and the last quantity goes to 0 as J? goes to +oo uniformly in £ € (0,1] in
view of (E.14).
Letting e, r\ go to +oo and finally 6 to 0+, we deduce from (E.17) that
fe converges uniformly in t € [0,T] in ^(Q) and in X1(0,T;L«(fi)) to
some / € CftO.r];^^)) D X^O.rjL'^)) for all 1 < ? < -fa such
that /|t=o = /o in ^- In addition, TR(fe) converges to TR(f) weakly in
L2(0,T;Hl(Q)) and (E.6), (E.7) hold. In order to conclude the proof of
Theorem E.l, we only need to show that (E.9) holds or in other words that
we can pass to the limit in the term /?"(/*)|V/C|2 since all the other terms
can be handled easily in view of the convergence of f£ we just established.
Step 3. Convergence in L2(0,r;F1(n)) of the truncations. We
first observe that it is enough to show that TR(fc) converges to TR(f) (as
e - 0+) in I2(0,T\-tf1^)) since /3'f(fe)Nfe\2 = /?"(/<) • |VT*(/«)|2 for
R large enough and aebe —► ab in L1 (Q) if ae —► a, be —* b in L1 (H) and be is
bounded in L°°(12).
Next, we remark that we have for all 6 € (0,T)
[ dtfdx\VTR(fe)\2dx < f dtf dxR\Fe\ + [ R\f£(6)-fZ\dx
Jo Jq Jo Jn Jn
and thus this quantity is small if 6 is small, uniformly in £ € (0,1]. Hence,
we only have to show that TR(fe) converges in £2(<5,T; JJ1^)) to TR(f).

Appendix E 191
Next, we write fe=ge + he where ge, he are respectively solutions of
dge
+ uc • Vgc - A/ = 0 in ft x (0,T),
^=0 onaftx(0,r), <7£|,=o = /o in«,
an
^-+uff-V/iff-A/iff = F£ in flx(0,T),
-^- = 0 on^x (0,T), /iff|*=o = 0 in a
an
(E.18)
(E.19)
Notice that h€ > 0 in Q x [0, T] and that everything we did in step 2 above
applies to he and ge: in particular, Ji£ and pc converge in C([0,T];X1(f2))
and in L1(0,T;L9(fi)) for all 1 < q < j^ to some ^ anc* # respectively.
In addition, we claim that for each 6 > 0, g£ is bounded in L°°(Q x (<5, T))
uniformly in £ G (0,1]. Indeed, we first observe that for each m > 1,
9m = *mPc solves
*&
m
dt
+ u< • V<& - A<£ = m^_! in ft x (0,T)51
^m _ n £H~i v. /n a-»\ -C I _ xc :_ n I
= 0 on0ftx(O,T), gem\t=o=fo in ft.
5n
Of course, g$ = ge is bounded in L*°(ft x (0,T)) for 1 < q0 < Ej$*.
Next, if t^-i € i9"*-1 for some qm-i > qo (m > 1), we multiply the
equation for g^ by |^|9m_l~2^ and we obtain easily a bound on g^
in C([0,T];L«—»(fl)) and on VO^I^-1 <&] in L2(ft x (0,T)), and
thus in I2(0, T; tf ^ft)) on |pm|~ * p^. Using Sobolev (and Gagliardo-
Nirenberg if N = 2) inequalities and Holder inequalities, we deduce that
gfn is bounded in Lqrn(Q x (0,T)) with qm = ^^ <?m-i. Therefore, for m
large (m> log(l + f)[log(l + ^^^
9m-1 > ^2^- To simplify notation, from now on, we write g in place of
g^ and g in place of p^-i anc^ 9 ^n place of gm-i- Then, for p > 1, we
multiply the equation satisfied by g by \g\v~2g and we obtain easily (where
C denotes various constants independent of p, e^g)
sup ||p(OKP(n) + IIV|y|p/2|li2(nx(o,r))
*€[0,T1
< Cp||p||Lf(nx(o,T))[jf dtjjlx\g\^-V
hence, exactly as above, we deduce
>-i
»'»^<n*<o,r» S ^'M^^^"^)

192 Appendix E
We then observe that ^^ £—■ = A > 1 and we rewrite the preceding
inequality as
l£,p(n
for all p > 1/0 , 9 =
NlL*(nx(o,T)) < (^)1/flpIIPll^x(o)T))IIPlli//(nx(o,T))
9-1
9
hence
°aax{lbllL*p(nx(o,T))»ll5llL»(nx(o,T))}
< (Cp)1/0p max(||y||Lp(nx(o,T)), llsllL«(nx(o,r))) for allP > V*- J
If we choose p > 1/9 and we iterate this inequality with p, Xp,..., A*_1p
we find for all k > 1
i
IMlLAfcP(nx(0,T))
,rt,j.i^i x fl , , A A*-A:A+Jfc-1
<(Cp)^-Texp{-logA_ _
•max{||p||£rp(nx(olr))il|p|U«(nx(olr))}
and letting k go to +00, we finally deduce
IMU~(nx(o,r)) < Cr||fiflU«(nx(o>T))-
In particular, ge is bounded in L°°(Q x (0, T)) uniformly in e 6 (0,1] and
ge(6) converges in L2(Q) to g(6). Furthermore, u£ • Vg€ = div(txcpc) and
uege converges in L2(Q x (0,T)) to up. Therefore, pc converges to g in view
of (E.18) inZ^forj/T1^)). We then claim that the proof of Theorem E.l
is complete as soon as we show that Tr(/ic) converges in L2(0,T;iJ1(f2))
to Tr{K) for all # 6 (0, oo). Indeed, we have on (<5,T) x Q for each 6 > 0
fixed
|vr*(m2 = i(l/«l<jR)|vn2
= 1(|/«|<«){I^£|2 + 2V/.V/1£ + |V/1£|2} .
= 1(I/«I<K){|V<7£|2 + 2Vg< • V7V(/i£) + |VT^(/i£)|2}
where i?' = .R + Cs, \gc\ < Cs on [S,T] x H. The quantity in brackets
converges in L^Q x (6,T)) to |V^ +VT^(/i)|2 (= |V/|2 if |/| < £). Hence
linT / I \VTR(fe)\2dxdt < f f l{m<R)\Vg+VTR,(h)\2dxdt
c-+o+ Jo Jn Jo Ja
\VTR{f)\2dxdt
-II
Jo Jci

Appendix E 193
using Lebesgue's lemma (and its converse), and we conclude.
Step 4. Convergence in L2(0,T]H^(Q)) of TR{hc). It only remains to
show that TR(hc) convergesjo TR(h) in L2(0jT; H1^)). We observe that
hc > 0 since hc\t=o = 0 in ft and Fe > 0 in ft x (0,T). Next, we claim it
is enough to show that, for instance, (l+/ie)-1/2 converges to (l + /i)-1/2
in L2(0,T;.ff1(ft)). Indeed, if it is the case, extracting a subsequence if
necessary, there exists C € Ll{Sl x (0,T)) such that
Jf£L- < C a.e. in ft x (0,T),
V/i£ -> V/i a.e. in ftx (0,T).
In particular, \VTR(he)\2 < C(l+R)3, VTR(hc)-+VTR(h) a.e. inftx(0,T)
and we conclude.
Next, we set /?„(*) = -^ for * > 0, n > 1, 7n(t) = jz(^T)i for
0 < t < (l - £)~ , n > 1. Let us remark that /?n = 7n ° A. /?n is concave
on [0,oo) while 7n is convex on [0, (l — £) ). Next, we notice that we
have for all k > 1, e 6 (0,1].
jj/W) + divK&(/i£)} - A&(/i£)
= 0'k(he)F< + (-&'(/ic)|V/i£|2) in ft x (0,T)
-?-(3k(h£)=0 ondftx(0,T), &(/i£)|t=o=0 in ft.
Letting £ go to 0, we deduce easily that there exists a bounded nonnegative
measure /z* on Jl x [0, X] (V A: > 1) such that
jj AW + div {u0k(h)} - A&(/i) 1
= ^(/i)F + (^(/i)|V/ic|2)+^ in «x(0,T).J
Indeed, -/%'(/i*)|V/iff|2 = ^^ |V/iff|2 = SlVU+Zi')"1/2!2 and (l+h*)-1/2
converges weakly in £2(0,T; Hl(Q)) to (l + Zi)"1/2 in view of the
convergences already shown.
We next claim that fik > fi\ in fix [0, X]. Formally, this is straightforward
since we deduce from the equation satisfied by 0i(h)
^ (3k(h) + div {upk(h)} - A(3k(h)
= ^ (7* ° Pi(h)) + div {ulk o 0i(h)}- A(7fc o ft(/i))
= TiWi W) {fi(h)F-fi(h)\Vh\2+to} - rttfiihWMflVhl2
= P'k(h)F-{3'i;(h)\Vh\e + <yk(01(h))vil.

194 Appendix E
Since Jk(PiW) = i^a > 1, we deduce that /ifc > Mi-
To justify these computations and the conclusion, we have only to show
that we have
^0k(h) + div{u(3k(h)}-A0k(h)
> fi(h)F - K(h)\Vh\2 + m in ft x (0,T).
This is done exactly as in step 1 above, the final step consisting in showing
that if
flTT
-AH = G+m in ft, if € L°°(ft)n#*(«), — = 0 on 5ft, (E.21)
on
where ||#||L9e(n) <' (l - £)~\ H > 0 a.e. in ft, G e ^(ft), m > 0 is a
bounded measure on ft, then
-Alk{H) > yk(H)G-yk'(H)\VH\2 + m in ft,)
~7fc(/f) =0 on aft.
(E.22)
To this end, we use the weak formulation of (E.21) with i'k(Ha)ip where <p is
arbitrary in C^ft), Ha € Gl(U) for a > 0, supa ||#a||L~(n) < (l - i)~\
#a > 0 in ft, Ha converges to H in Jf1(ft) and a.e. in ft as a goes to 0.
We then obtain if (p > 0 in ft
fvH^{^(Ha)VHaip + yk(Ha)VV}dx > [ j'k(Ha)G + m<pdx
Jn Jn
and we recover (E.22) letting a go to 0.
Therefore, we deduce from the weak formulation of (E.20)
I 0k{h{x,T))dx > f dt [ dx0k(h)F + ni(nx[O,T)),
Jn Jo Jn
and letting k go to +00, we obtain
I h{x,T)dx > I dt I dxF + ni(U x [0,T)).
Jn Jo Jn
On the other hand, (E.19) yields
f he(x,T)dx = f dt IdxF€
Jn Jo Jn

Appendix E 195
rp
hence, upon letting e go to 0+, fnh(x1T)dx = /0 dtf^dxF. Therefore,
/xi = 0 and this implies that V(l + /i£)~1/2 converges to V(l + /i)"1/2 in
L2(Q x (0, X)), and the proof of Theorem E.l is complete. □
We conclude this appendix with a simple observation.
Theorem E.2. Let p € [l,oo) and let a e (0,p). We assume that f e
LP(Q x (0,T)), V/ € Ll(Q x (0,r)) (for instance) and that f satisfies for
aUR>l
f dt[dx\Vf\pl(m<R) < CR« (E.23)
Jo Jn
for some C>0. Then, for all (3 e (a,p)
(l + |/|2)i-& 6 If(0,T;Wl*(Q)). (E.24)
Remarks E.2. 1) Many variants and extensions are possible (less
restrictive conditions on /, unbounded domains, time-dependent /, etc.); we skip
these since the argument given below is extremely simple and can be easily
adapted to various situations.
2) Of course, the norm in £p(0, X; WliP(Q)) is estimated in terms of the
constant C in (E.24) and a bound on / in LP(Q x (0,X)), say.
3) Using Sobolev inequalities we deduce easily (at least if p < N) from
(E.24) that / 6 L*-^0,r;L*(n)) where q = ^^. □
Proof of Theorem E.2. It is enough to show that we have
rp
f dtfdx\Vf\p(l + \f\2)-^ < C. (E.25)
Jo Jn
Then, we write
rp
f di/dxiv/ni+i/i2)-!
Jo Jci
< C+J2 f ^/^|V/|P(l + |/|2)-^l(2n<|/|<2n + »)
n>0J° Jn
rp
< C+Y^2-^ f dtfdx\Vf\Pl([n<2n+i)
n>o Jo Ja
< C + C^2_n/32na < C. a
n>0

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INDEX
a priori bounds 128
a priori estimates
density-dependent Navier-Stokes
equations 35-41, 64, 69, 70
Euler equations 152, 153
density-dependent 159
weak solutions of Navier-Stokes
equations 98-107
acoustics limit x
almost Lipschitz 127, 151
approximation
approximated solutions for density-
dependent Navier-Stokes equations
35, 64-7
renormalized solutions for parabolic
equations 187-90
approximated models vii, 9-16
Ascoli-Arzela theorem 177
asymptotic limits vii, x, 14-16
asymptotic problems vii
axisymmetric case 152
barotropic pressure laws 15
Besov space 99
blow-up 150, 155
boundary conditions ix, 1, 16, 19-20
see also Dirichlet boundary conditions;
Neumann boundary conditions;
periodic case; whole space case
breakdown 151, 159
Cauchy problems ix, 16, 125
compactness 117
compensated compactness theory 93
compressible isentropic Navier-Stokes
equations ix
density-dependent Navier-Stokes
equations 35, 41-64
Euler equations 133, 140, 143-4, 153, 154
in time with values in weak topologies
177
compensated compactness theory 93
compressible Euler equations 13
compressible isentropic Euler equations x
compressible isentropic Navier-Stokes
equations ix, 15
compressible isothermal Navier-Stokes
equations 15
compressible models vii, ix-x, 13, 153
compressible Navier-Stokes equations 8-9
concentrations 154
conservation of energy 4
Euler equations 126, 132, 133, 135, 141
density-dependent 159
dissipative solutions 153, 154
conservation of mass 1-2, 9
conservation of momentum 2, 9, 11
convection term 100
convergence
density-dependent Navier-Stokes
equations 41-61
Euler equations 149-50, 156
renormalized solutions for parabolic
equations 187-95
covering lemma 179
P1,2(R2) 55, 57, 168-70, 173-6
decreasing rearrangement 140
see also Schwartz spherically symmetric
decreasing rearrangement
deformation tensor 3, 150
degrees of freedom, internal 117
density 1-2, 158
density-dependent Euler equations ix, 14,
158-60
density-dependent models 117
density-dependent Navier-Stokes
equations viii, 10, 12-13, 19-78
a priori estimates 35-41, 64, 69, 70
compactness results 35, 41-64
existence proofs 35, 41, 64-75
existence results viii, 19-31
regularity results and open problems
31-5
uniqueness see uniqueness
Dirichlet boundary conditions ix, 16, 183
density-dependent Navier-Stokes
equations 19, 35

234
Index
Dirichlet boundary conditions (cont)
density-dependent Navier-Stokes
equations (cont.)
a priori estimates 37
compactness results 42, 46, 47
existence proofs 64, 65, 67, 70-1, 73
regularity results 25, 30, 31
uniqueness 75
Euler equations 124, 125, 127, 128, 136,
153
fundamental difficulty 129-30
two-dimensional 139
Navier-Stokes equations 80, 81, 82, 86,
91, 92
difficulties encountered 83-6
second derivative estimates 107-10
temperature and Rayleigh-Benard
equations 110, 111, 114-15, 116, 117
dissipative solutions viii-ix, x, 153-8
distributions
density-dependent Navier-Stokes
solutions
compactness results 42, 45
existence results 23, 24, 25, 28
Euler equations 127, 135, 153
distributions function 137, 139
Navier-Stokes equations 80, 81, 83, 112,
113, 183
divergence-free vector fields
density-dependent Navier-Stokes
equations 30, 53, 59
Euler equations 125, 132, 137, 139, 154
density-dependent 158
Navier-Stokes equations 84-6
truncation in Sobolev spaces 165-72
elliptic equations 101, 184
elliptic regularity 143, 179
energy 1
conservation see conservation of energy
internal 4, 10-11, 110-23
kinetic 4
total 4, 111, 114
energy identities 71
Euler equations 128, 130
Navier-Stokes equations 112, 114, 119
energy inequalities 154
density-dependent Navier-Stokes
equations 22-3, 73-4
Navier-Stokes equations 86, 114, 118,
120
local energy inequality 82, 88-9, 109-10
entropy 5, 13-14
Euler equations viii-ix, 14, 124-64
density-dependent ix, 14, 158-60
dissipative solutions viii-ix, x, 153-8
hydrostatic approximations ix, 160-4
review of known results 125-36
three-dimensional viii, 150-3, 163-4
two-dimensional viii, 125, 136-50, 161
eulerian form 1
existence of solutions
density-dependent Navier-Stokes
equations
proofs 35, 41, 64-75
results viii, 19-31
Euler equations 125, 126, 131-4
dissipative solutions 156—8
two-dimensional 136-50
Navier-Stokes equations 21, 86-7, 106,
110, 118-19
renormalized solutions for parabolic
equations 187-90
results for compressible models ix-x
first law of thermodynamics 4
fixed point 67-70
Fourier multipliers 182
free boundary problem 34
Gagliardo-Nirenberg inequalities 32, 174,
176, 191
galilean invariance 24
Euler equation 148
geophysical flows 160
Gibbs equation 5
global weak solutions viii, ix
density-dependent Navier-Stokes
equations 21-5, 34, 35
equal to strong solution 31, 75-8
Euler equations 137, 151-2
Navier-Stokes equations viii, 21, 79-82,
86-9, 105-7
Hardy inequalities 46, 74
Hardy spaces 127
refined regularity of weak solutions 92-8
harmonic functions 83-4, 86, 147
heat equations 99, 112, 116
weak L1 estimates 178-82
hilbertian strategy 27
Hodge-de Rham decomposition 25-6
homogeneous fluids 9
homogeneous incompressible Navier-
Stokes equations see Navier-Stokes
equations
homogenization x
hydrostatic approximations ix, 160-4
hydrostatic pressure 2, 10, 126, 158
ideal fluids 9,. 13-14
ideal gases 7-8, 12, 14, 15
incompressible fluids 9

Index
235
incompressible limits 25
incompressible models vii, viii-ix, 9-11, 13
inhomogeneous incompressible (density-
dependent) Euler equations ix, 14,
158-60
inhomogeneous incompressible Navier-
Stokes equations see density-
dependent Navier-Stokes equations
initial conditions 16, 20-1, 25, 83, 162
internal degrees of freedom 117
internal energy 4, 10-11, 110-23
inviscid (non-viscous) case x, 4
inviscid model 160-4
R case see whole space case
isentropic gas dynamics 14
isentropic pressure laws 15
isothermal case ix
Joule's law 7, 8, 14
kinematic viscosity 79
kinetic energy 4
kinetic theory 3, 8
Lame viscosity coefficients 3
Leray, J. viii, 21, 66, 79
Liouville's theorem 35
Lipschitz, almost 127, 151
Lorentz spaces 59, 178
Mach number x, 153
low Mach number expansions 9, 11-13
Mariotte's law 7
mass, conservation of 1-2, 9
measure theory 24
measure-valued solutions 153
momentum, conservation of 2, 9, 11
multi-phase flow 19
Navier-Stokes equations vii, viii, 23, 31,
66, 79-123, 125, 156
density-dependent see density-dependent
Navier-Stokes equations
fundamental equations 10, 11-12
global weak solutions viii, 21, 79-82, 86-
9, 105-7
refined regularity of weak solutions via
Hardy spaces 92-8
review of known results 79-92
second derivative estimates 98-110
stationary 35, 91-2
temperature and Rayleigh-Benard
equations 110-23
Theorem 4.1 128-9, 136
Neumann boundary conditions 16, 110,
115, 183, 186
newtonian fluids vii
fundamental equations vii-viii, 1-9
model of incompressible, homogeneous
newtonian fluid 117-23
non-linear partial differential equations 22
numerical simulations 151
fi = R^ see whole space case
open problems viii, 31-5
ordinary differential equation 67
oscillations 154
propagation ix
parabolkrequations 101
existence and uniqueness of renormalized
solutions 183-95
particle paths 5, 67
passing to the limit 134
density-dependent Navier-Stokes equa-
tions 35, 64, 70-5
Navier-Stokes equations 106-7, 120
perfect fluids see ideal fluids; ideal gases
periodic case 16
density-dependent Navier-Stokes equa-
tions 20, 25, 30-1
a priori estimates 35, 37
compactness 41-7
existence proofs 64, 65, 66, 67, 70
stationary problems 34-5
uniqueness 75
Euler equations 126, 127, 128, 130, 153
Navier-Stokes equations 79, 80, 81-2, 84,
86, 98, 114
pressure 2, 10, 126, 158
pressure field
density-dependent Navier-Stokes
equations 23, 31
Navier-Stokes equations 79, 80, 81-2
pressure laws 14, 15
propagation of oscillations ix
Rayleigh-Benard equations viii, 110-23
rearrangement 137, 140-1
regularity ix
density-dependent Navier-Stokes
equations viii, 31-5
existence proofs 67-70
elliptic 143, 179
Euler equations 126, 130, 136
dissipative solutions 155, 156
Navier-Stokes equations viii, 80, 82, 83,
84, 86, 87, 89-92, 112

236
Index
Navier-Stokes equations (cont)
refined regularity of weak solutions via
Hardy spaces 92-8
second derivative estimates 98-9, 100,
109-10
regularization 135, 147, 156, 187-8
existence proofs for density-dependent
Navier-Stokes equations 65, 66, 70
solutions of transport equations 43-5
Rellich-Kondrakov theorems 57, 131, 172
Remark(s) 2.1 23-5
Remark(s) 2.2 26
Remark(s) 2.3 33
Remark(s) 2.4 42
Remark(s) 2.5 46
Remark(s) 2.6 48-9
Remark(s) 2.7 61-4
Remark(s) 2.8 66-7
Remark(s) 2.9 73-4
Remark(s) 2.10 74-5
Remark(s) 2.11 78
Remark(s) 3.1 82-6
Remark(s) 3.2 89
Remark(s) 3.3 94
Remark(s) 3.4 100-1
Remark(s) 3.5 104
Remark(s) 3.6 105-7
Remark(s) 3.7 113-14, 115
Remark(s) 3.8 115
Remark(s) 3.9 117
Remark(s) 3.10 118
Remark(s) 4.1 126-7
Remark(s) 4.2 127
Remark(s) 4.3 129
Remark(s) 4.4 132
Remark(s) 4.5 133
Remark(s) 4.6 134-5
Remark(s) 4.7 135-6
Remark(s) 4.8 141
Remark(s) 4.9 144-5
Remark(s) 4.10 145-6
Remark(s) 4.11 148
Remark(s) 4.12 155
Remark(s) A.l 167-8
Remark(s) B.l 174-5
Remark(s) D.l 178-9
Remark(s) E.l 185
Remark(s) E.2 195
renormalized solutions 42, 101
Euler equations 132, 133-4, 135, 137,
140, 145
Navier-Stokes equations 112-13
proof of existence for parabolic equations
183-95
Riesz transforms 25, 92, 147
Schwarz spherically symmetric decreasing
rearrangement 95, 106, 140
• second derivative estimates 98-110
second law of thermodynamics 5, 7
sequences of solutions ix, 35, 39, 140
shallow water model ix
shocks (discontinuities) 13-14
short time existence 150
simplified models vii, 9-16
singular integral 147
singularities 83, 151
Sobolev embeddings
density-dependent Navier-Stokes
equations 22, 25, 29, 38, 55, 59
Euler equations 134, 135
Navier-Stokes equations 82, 87, 92, 100
Rayleigh-Benard equations 115, 119
refined regularity of weak solutions 94-5
Sobolev inequalities 24, 78, 132, 149, 191,
195
Sobolev spaces 23, 25, 53, 79, 92
truncation of divergence-free vector fields
165-72
spherically symmetric decreasing
rearrangement 95, 106, 140
stability 136
stationary Navier-Stokes equations 35,
91-2
stationary problems ix
density-dependent Navier-Stokes
equations viii, 34-5
Stokes equations 33, 68, 70, 90, 107, 120
Stokes problem 165-6
Stokes relationship 3-4
stream function 140
stress tensor 2, 9
viscous stress tensor 2
strong solution viii, 75-8
symmetrization 106, 116, 140
temperature 5, 110-23
Theorem 2.1 23, 86
Theorem 2.2 28, 30
Theorem 2.3 31
Theorem 2.4 41-2
Theorem 2.5 48
Theorem 2.6 66
Theorem 2.7 76
Theorem 3.1 81
Theorem 3.2 81-2
Theorem 3.3 82
Theorem 3.4 82
Theorem 3.5 89
Theorem 3.6 93-4
Theorem 3.7 100

Index
237
Theorem 3.8 104
Theorem 3.9 109-10
Theorem 3.10 113, 183
Theorem 3.11 114, 115
Theorem 3.12 114-15
Theorem 3.13 118
Theorem 4.1 126, 136
modification 132-5
Theorem 4.2 137
Theorem 4.3 141
Theorem 4.4 145
Theorem A.l 60, 168
Theorem A.2 58, 170-1, 172
Theorem B.l 175
Theorem D.l 178
Theorem E.l 185
Theorem E.2 195
thermodynamic temperature 5, 110-23
thermodynamics, laws of 4, 5, 7
three-dimensional case
Euler equations viii, 150-3, 163-4
Navier-Stokes equations 117-23
time
compactness in 177
short time existence 150
time continuity 25, 28, 63
time-discretized problems see stationary
problems
total energy 4, 111, 114
transport equations 23, 67, 151, 163
with divergence-free vector fields 132,
137, 139
general regularization for solutions 43-5
uniqueness 121
truncations
convergence in renormalized solutions for
parabolic equations 190-3
divergence-free vector fields in Sobolev
space 165-72
two-dimensional case
Euler equations viii, 125, 136-50, 161
Navier-Stokes equations 105-7, 112
uniformly elliptic second-order operator
178
uniqueness
density-dependent Navier-Stokes
equations viii, 31, 38
compactness results 49, 50-1
equality of weak solution to strong
solution 75-8
existence proofs 67, 69-70
Euler equations viii, 125, 126, 151
two-dimensional 136, 138-9, 146
Navier-Stokes equations 82, 83, 86, 87-8,
92
temperature and Rayleigh-Benard
equations 112, 113
renormalized solutions for parabolic
equations 185-7
transport equations 121
vacuum viii, 20, 34
velocity 158
velocity field 1, 2-4, 140, 161
viscous case 4
viscous stress tensor 2
vortex sheets 136, 137, 144-5, 152
vorticity 126,-144-5, 150
weak formulation 125
density-dependent Navier-Stokes
equations 21, 25, 28
Navier-Stokes equations 112-13, 115
renormalized solutions for parabolic
equations 184, 185, 186, 194
weak limit 70
weak solutions ix, 185
density-dependent Navier-Stokes
equations viii, 21-5, 34, 35
equal to strong solution 31, 75-8
Euler equations viii, 135
global 137, 151-2
three-dimensional 151-2, 153
two-dimensional 145-50
Navier-Stokes equations viii, 79-89, 156
global weak solutions viii, 21, 79-82,
86-9, 105-7
Rayleigh-Benard equations 111-15
refined regularity via Hardy spaces 92-8
regularity 83, 89-92
second derivative estimates 98-110
weak topologies 28, 42, 131
compactness in time 177
weak topology of measures 115
whole space case (fl = EVV) 20, 153
density-dependent Navier-Stokes
equations 24-5, 25, 30-1
a priori estimates 37-8, 40-1
compactness 47-8
existence proofs 64, 75
Navier-Stokes equations 79, 80, 81-2, 84
refined regularity of weak solutions 92-8
second derivative estimates 98-107
Young measures 153