Some variants of the classical Aubin-Lions Lemma

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by  Ayman Moussa

Some variants of the classical Aubin-Lions Lemma
A. Moussa˚:
E-mail : [email protected]
4 December 9, 2014
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0
2
Abstract

c
e This paper explores two generalizations of the classical Aubin-Lions Lemma. First we give
D
a sufficient condition to commute weak limit and multiplication of two functions. We deduce

7 from this criteria a compactness Theorem for degenerate parabolic equations. Secondly, we
state and prove a compactness Theorem for non-cylindrical domains, includingthe case of dual
]
P estimates involving only divergence-free test functions.
A
. Keywords: evolution equations ; strong compactness ; Aubin-Lions Lemma
h
t
a
m 1 Introduction
[

3 1.1 Aubin-Lions Lemma and beyond
v
1
Inthestudyofnonlinearevolutionequations,theAubin-Lionslemmaisapowerfultoolallowing
3
2 to handle the nonlinear terms, when dealing with an approximation process or asymptotic
7 limit. The standard statement gives sufficient conditions on a sequence of functions u of
n n
. p q
1 two variables t,x (timevariable tbelongs tosomeinterval I, spacevariable xbelongs tosome
0 open boundedpsetqΩ) bounded in Lp I;B where B is some Banach space of functions defined
4 p q
1 on Ω. More precisely, if
:
v (i) u is bounded in Lp I;X ;
n n
i p q p q
X (ii) u is bounded in Lr I;Y ;
t n n
r pB q p q
a (iii) X embeds compactly in B, which in turns embeds continuously in Y,
then u admits a strongly converging subsequence in Lp I;B , provided p or r 1.
n n
p q p q ă 8 ą
This strong convergence allows then to pass to the limit in the approximation procedureor the
asymptotic limit.
Themain purposeof this work is torevisit this classical resultin order tohandlethecase of
estimates arisingfromtwoparticularcases: degenerateparabolicequationsandincompressible
Navier-Stokes equations, the latter being considered in a non-cylindrical domain. These two
types of situations do not allow to apply the usual Aubin-Lions directly (we will explain why
in the sequel).
Of course, these equations have already been well studied in the literature as well as the
difficulties arising from their nonlinearities. Hence, the novelty of this work does not concern
˚SorbonneUniversités,UPMCUnivParis06,UMR7598,LaboratoireJacques-LouisLions,F-75005,Paris,France
:CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005,Paris, France
1
so much the results for their own sake (except one or two improvements) but the strategies of
proof which, as far as we know, are new. In this way, we hope to give simple arguments that
could possibly be reused in different contexts.
Before presenting our results, let us describe the existing literature. The naming “Aubin-
Lions” may be traced back to the seminal papers of Aubin [5] and Lions [23], in the ’60.
However, at the same period, Dubinski˘ı proved a general compactness result (see [16] and the
corrected version [6]) for vector-valued functions which is actually the first nonlinear counter-
partof the Aubin’s result(the vector space X is replaced by acone). Thisis why some authors
refers sometimes to the Aubin-Lions-Dubinksi˘ı Lemma (see for instance [8]). Let us mention
also the result of Kruzhkov in [22] which, though far less general than the previouses, present a
different approach that we will use in section 4. Simon extended the result of Aubin and Lions
to non-reflexive Banach spaces in his highly-cited paper [28]. In this paper, the condition on
the time-derivative was replaced by a more general condition on time translations. The result
of Simon was further sharpened by Amann in [3] to a refined scale of spaces (including Besov
spaces for instance), and broached by Roubíček in a rather general setting, see [26]. At this
stage, we may distinguish three possible directions of generalization (which may overlap)
(a) Nonlinear versions of the Aubin-Lions Lemma
This corresponds to cases in which assumption (i) above is replaced by a nonlinear con-
dition. For instance, Maitre considers in [24] cases in which the space X is replaced by
K X where K is some compact (nonlinear) mapping K : X B. This compactness re-
p q Ñ
sult was motivated by [28, 3] and the nonlinear compactness argument Alt and Luckhaus
used in [2]. In some cases, the compactness result obtained in [16] by Dubinski˘ı may be
seen as a consequence of the Theorem of Maitre (see [6] for more details on that point),
see also [9] for general results of the same flavor.
(b) Discrete-in-time setting
Quite often, when dealing with approximate solutions of an evolution equation, it is not
straightforward to fulfill assumption (ii), since it could happen that u satisfy only a
n n
p q
discrete (in time) equation. This typically happens when one replaces the operator by
t
B
some finite-difference approximation. Several papers deal with this issue, (based on the
time translations condition of Simon) : see [4, 8, 9, 15, 18].
(c) Non-cylindrical domain
A time/space domain is called cylindrical whenever it may be written I Ω where Ω is
ˆ
some subset of Rd and I some intervall of R, see Figure 1.1 for an illustration. In the
study of PDEs these types of domain are used for evolution problems with a fixed spatial
domain. If one wishes to consider the case of time-dependent or moving spatial domain,
one has to consider a family of domains Ωt , representing the motion of the spatial
tPI
p q
domain and the corresponding non-cylindrical time/space domain
Ωˆ : t Ωt.
“ t uˆ
ďtPI
The study of PDEs in non-cylindrical domains leads to the following difficulty: functions
u: t,x u t,x definedonΩˆ may not anymore beseenasfunctionsofthetime-variable
p q ÞÑ p q
t with value in a fixed space of functions of the x variable. Typically, this forbids the
assumptions (i), (ii) above and even the conclusion of compactness in Lp I;B : the very
p q
statement of the Aubin-Lions lemma is already problematic. As far as we know, the first
proofof acompactness Lemma“àlaAubin-Lions”in thecase ofa non-cylindricaldomain
appeared in a paper by Fujita and Sauer [17], for the treatment of the incompressible
Navier-Stokes equations in moving domain. The method of proof (which was reused in
the framework of fluid/structure interaction, see Conca et. al. [10] for instance) uses the
idea that, under appropriate regularity conditions, the non-cylindrical domain Ωˆ is close
2
Ω
Ωb
b
Ωt
a
Ω pa,bqˆΩ Ωa Ωˆ
Figure 1: A cylindrical domain (left) and a non-cylindrical one (right).
toafiniteunionofcylindricaldomainsonwhichonecouldthenusetheusualAubin-Lions
Lemma.
The purpose of this study is to prove two generalizations of the Aubin-Lions Lemma. The
first one corresponds to cases (a) and (b) above, and the second one to case (c). Both will be
proven using totally different methods than the one developped in the above literature. Before
stating precisely these two results, we introduce a few notations.
1.2 Notations
Thenormof a vector space X will always bedenoted , with an exception for theLp spaces
X
}¨}
for which we will often use the notation if there is no ambiguity. Vectors and vector fields
p
}¨}
are written in boldface. We omit the exponent for the functionnal spaces constituted of vector
fields: wedenote for instance L2 O instead of L2 O d theset of all vector fields O Rd whose
p q p q Ñ
norm is square-integrable.
When O is someopen set of Rd (or R Rd ...), we adopt theusualnotations for the Sobolev
ˆ
spaces W1,p O and Hm O , for p 1, and m N. D O denotes the space test functions:
p q p q P r 8s P p q
smooth functions having a compact support in O, while D O is the restriction to O (closure
p q
of O) of elements in D Rd . H´m O denotes the dual space of Hm O the latter being the
0
p q p q p q
closure of D O under the Hm O norm. If O has a Lipschitz boundary and p d, we denote
by p‹ the exppoqnent of the Sobpoleqv embedding W1,p O ã Lp‹ O , that is p‹ă: dp d p .
p q Ñ p q “ {p ´ q
We adopt the convention p‹ when p d (the previous embedding fails in this case).
“ 8 “
Conjugate coefficient of p is denoted by p1. We denote by M O (resp. M O ) the set of finite
p q p q
Radon measures on O (resp. O) and by BV O the subset of L1 O constituted of functions
p q p q
having all their weak derivatives in M O . If I is an intervall and X some Banach (or Fréchet)
p q
vector space, we denote by Lp I;X the set of all measurable Lp functions from I to X and by
p q
C0 I;X the space of continuous functions from I to X. When I is closed, M I;H´m Rd is
p q p p qq
simply the dual space of C0 I;Hm Rd .
p p qq
We denote by D O (and similarly D O ) the set of divergence-free test function with
div div
p q p q
support in O and by L2 O the subspace L2 O vector fields having a vanishing (weak)
div
p q p q
divergence.
If O Rd has a Lipschitz boundary we can equip H1{2 O with the norm
Ă pB q
}g}H1{2pBOq “ vPH1piOnqf,γv“g}v}H1pOq,
whereγ is trace operator on H1 O . We denoteby H´1{2 Ω thetopological dualof H1{2 O .
p q pB q pB q
We recall that in that case (see [20] for instance), there exists a normal trace operator, that
3
we denote by γ , extending the operator C0 O v v n C0 O , where n is the outward
n
p q Q ÞÑ ¨ P pB q
unitnormaldefinedontheboundary O, into alinear andonto mapγ :L2 O H´1{2 O
n div
B p q Ñ pB q
satisfying
}γnu}H´1{2pBΩq ď }u}L2pΩq. (1)
L2 O is a closed subspace of L2 O : it is in fact the closure of D O . It is however
div div
p q p q p q
important to recall that D O is not dense in L2 O , its closure is the subspace Kerγ ,
div div n
p q p q
that we will denote L2 O in the sequel.
div,0
p q
We adapt the previous notations for solenoidal vector fields to the case of functions de-
pending on both time and space, that is when O is an open set of R Rd (first component t,
ˆ
last components x). In that case, when there is no ambiguity on the time variable, we perform
a slight abuse of notation and denote for instance D R Rd the set of all ϕ D R Rd
div
p ˆ q P p ˆ q
such as for, for all t R, ϕ t : x ϕ t,x D Rd . D O is then just the subspace of
div div
P p q ÞÑ p q P p q p q
D R Rd having a compact support in O, while L2 O and L2 O are respectively the
div div div,0
p ˆ q p q p q
closure of D O and D O in L2 O . Notice that in this way we recover the definition we
div div
p q p q p q
had without the time variable.
9 :
Inallthisstudythesymbols and willrespectivelymean“isboundedin”and“isrelatively
P P
compact in”.
If A is a connected open set of Rd, and ε 0 we define ε-interior of A as A : x A :
ε
ě “ t P
d x,Ac ε , while for A denote the ε-exterior of A, that is A : A B 0,ε . One checks
´ε ´ε
p q ą u “ ` p q
easily that A A and A A.
p ε1qε2 “ ε1`ε2 0 “
For σ R and h Rd, we denote by λ and τ the shift operators in time and space
σ h
P P
respectively : if f is some function depending on t,x , then λ f t,x f t σ,x and
σ
p q p q “ p ´ q
τ f t,x f t,x h .
h
p q“ p ´ q
1.3 Main results
The two main results of this paper both generalize the usual Aubin-Lions Lemma in cases in
which it may not be applied.
The first one deals with the case when compactness on the space variable is not known on
the sequence itself, but on some function of it.
Theorem 1 : Consider I R a non-empty closed and bounded interval, and Ω Rd a
Ă Ă
bounded open set with Lipschitz boundary. Consider also a function Φ C1 R,R such as
P p q
z R : Φ1 z 0 is finite, with Φ1 lower bounded by a positive value near . If a
stequPence of Wp 1q,1“I uΩ functions sa|tisfi|es a 9 L2 I Ω , a 9 M I;H´m˘8Ω and
∇ Φ a 9 Ll2ocIp ˆΩ qthen a :L2 I Ωp.nqnP p ˆ q pBt nqnP p p qq
x n n n n
p p qq P p ˆ q p q P p ˆ q
Remark1.1: Thisresultcomesundercases(a)and(b)following thedescription ofsubsection
1.1 : the compactness assumption for the space variable is not known for the sequence a ,
n n
p q
but for some nonlinear function of it ; also note that the assumption on the time derivative
allows Dirac masses in time. In particular, this Theorem applies for step functions in time
with values in H´m Ω .
p q
Let us first explain why there is a real loss of information with respect to the compactness
in the space variable x for a in comparison with the usual Aubin-Lions Lemma. On each
n n
p q
points where Φ1 vanishes we may write
1
∇ a ∇ Φ a ,
x n “ Φ1 a x p nq
n
p q
4
whenever a does not meet the set of critical points and expect an estimate for ∇ a , but
n x n n
p q
when a approaches a critical point, the estimate degenerates : the usual Aubin-Lions Lemma
n
may not be invoked since no estimates for the gradient of a may be obtained generally.
n n
p q
The assumptions of Theorem 1 are directly linked with the estimates of the equation of
porous medium u ∆ um 0 (in the case of fast diffusion m 1) and more generally may
t x
B ´ “ ą
be useful to parabolic degenerate equations of the following form:
u div A∇ Φ u 0, (2)
t x x
B ´ r p qs “
where A t,x satisfies some uniform coercivity condition. Indeed if, for instance, Spec A tA
p q p ` q
is lower bounded by λ 0 uniformly in t,x , then, if Ψ is a anti-derivative of Φ, one gets
ą p q
easily
s
Ψ u s t∇ Φ u A∇ Φ u Ψ u 0 ,
x x
ż p qp q`ż ż p q p q “ ż p qp q
Ω 0 Ω Ω
whence
s
Ψ u s λ ∇ Φ u 2 Ψ u 0 ,
x
ż p qp q` ż ż | p q| ď ż p qp q
Ω 0 Ω Ω
which, under appropriate growth condition on Φ will lead to the assumptions used in Theorem
1. Typically, for the porous medium case Φ x xm with m 1, the previous estimate gives
p q “ ą
directly the L2 estimate on the gradient, and that u belongs to L8 I;Lm`1 Ω whence both
p p qq
the L2 estimate for u and the L1 I;H´m Ω estimate for its time-derivative.
p p qq
To prove Theorem 1, we will first give a general criteria to pass to the limit in a prod-
uct a b under assumption of weak convergence for both a and b . This criteria seems
n n n n n
p q
reminiscent of the celebrated compensated compactness phenomenon exhibited by Murat and
Tartar in [25, 30] (see also [19]). However, it is not (as far as we know) a consequence of the
compensated compactness theory, but share this common feature : nonlinearities are handled
without insuring strong convergence for one of the sequences a compactness. As far as we
know, the strategy used in the literature to treat degenerate parabolic equations like (2) is
different of the one we followed, and often relies on the equation’s structure, see [31] and [13]
for instance. Our proof is also different of the quite general approach proposed by Maitre in
[24]. A benefit of our method is that it directly applies to both step-functions (in time) and
continuous functions. The result may hence be used to prove the strong compactness of a
sequence defined by a semi-implicit scheme :
1
u u div A∇ Φ u 0.
n`1 n x x n`1
δp ´ q´ r p qs “
Indeed, if δ 1 N and infI t t supI is a regular discretization of the interval
0 N
“ { “ ă ¨¨¨ ă “
I, one defines
N´1
u˜ t,x : u x 1 t ,
Np q “ kp q ptk,tk`1qp q
kÿ“0
and applies then Theorem 1 to obtain the strong compactness of u˜ . In fact Theorem 1 is
N N
p q
already used by the author and some collaborators in the proof of global weak solutions for a
reaction/cross-diffusion system, approximated by a similar scheme, see [12].
The second main result of this paper comes under case (c) following the description of
subsection 1.1. We consider a family Ωt given by the motion of a Lipschitz, connected
tPra,bs
p q
and bounded reference domain Ω Rd: t a,b , Ωt : A Ω , where for all t, A : Rd Rd
t t
Ă @ P r s “ p q Ñ
is a C1-diffeomorphism. The regularity of the motion is described through the function Θ :
t,x A x , for which we assume the following
t
p q ÞÑ p q
5
Assumption 1 : The function Θ belongs to C0 a,b ;C1 Rd .
pr s p qq
We will need this Assumption for every result of section 4 dealing with the family Ωt
tPI
p q
(including Theorem 2 below) exception made for Theorem 3 (see Remark 4.3).
We work on the non-cylindrical domain
Ωˆ : t Ωt,
“ t uˆ
aăďtăb
for which we have the following compactness result
Theorem 2 : Consider a sequence u L2 R Rd vanishing outside Ωˆ. Assume that
n n div
u , ∇ u 9 L2 Ωˆ , and u 9pL8qR;PL2 Rdp .ˆEveqntually, assume the existence of a
n n x n n n n
pconsqtanpt C q0 aPndpanqintegerpN q 0P sucph as fopr aqlql divergence-free test function ψ D Ωˆ ,
div
ą ą P p q
u ,ψ C αψ . (3)
|xBt n y|ď }Bx }L2pΩˆq
|αÿ|ďN
Then u :L2 Ωˆ .
n n
p q P p q
The boundsassumed on the sequence u are typical of the incompressible Navier-Stokes
n n
p q
equations, considered in a non-cylindrical domain. Indeed, recall the incompressible Navier-
Stokes equations :
u u ∇ u ∆ u ∇ p 0, (4)
t x x x
B ` ¨ ´ ` “
div u 0. (5)
x
“
If these equations are considered on Ωˆ with appropriate boundary conditions, one has the
following formal energy equality
d t
u t,x 2dx ∇ u s,x 2dxds 0,
x
dtż | p q| `ż ż | p q| “
Ωt 0 Ωs
which explain the assumptions on u and ∇ u . But what about u ? In fact
n n x n n t n n
p q p q pB q
it is not possible to estimate directly u from the equation (as this is usually the case in
t
B
the Aubin-Lions Lemma), because of the pressure term ∇ p, from which usually very less is
x
known. However, testing (4) against a smooth divergence free vector field ψ gives rise to the
estimate (3). This estimate is the equivalent of estimate (ii) in subsection 1.1. Notice, that we
may not write it as u 9 Lr I;Y precisely because the domain is non-cylindrical, whence
t n n
pB q P p q
the dual formulation (3). In fact Theorem 2 has been partially tailored for the study of a
fluid/kinetic coupling in a moving domain, and will soon be used in this context by the author
andsomecollaborators. Ofcourse,normally(4)–(5)arecompletedwithboundary(andinitial)
conditions, and the assumptions of Theorem 2 suggest that we may only handle homogeneous
Dirichlet boundary conditions (that is, u vanishes on the boundary). More general boundary
conditions may in fact be handled, this is the purpose of Corollary 2.
As explained in case (c) of subsection 1.1, the existing proofs of such a compactness result
in non-cylindrical domain are based on the following observation: Ωˆ may be decomposed, up
to some small subset, into a finite union of cylindrical domains. On each one of these, one
may then invoke the usual Aubin-Lions Lemma. We give here a totally different proof which,
gives strong compactness, without using the standard Aubin-Lions lemma. In particular, the
method applies in the case of cylindrical domains and gives hence a new proof of the Aubin-
Lions lemma (but not in a framework as general as in [28]). It avoids also the “slicing” step
for the non-cylindrical domain, which leads in [17] to intricate assumptions for the regularity
of the motion of the domain whereas Assumption 1 is simpler and weaker. The method we
elaborate is far simpler when one replaces (3) by dual estimate against all test function (and
not only divergence-free), see Theorem 3. In fact, the main difficulty of Theorem 2 concerns
this point.
6
1.4 Structure of the paper
Let us now describe the structure of this paper. Section 2 is devoted to results reminiscent
of the celebrated compensated compactness phenomenon exhibited by Murat and Tartar in
[25, 30] (see also [19]). The results of section 2 will be used in the proof of Theorem 1 but are
alsointerestingfortheirownsake. Asabyproduct,weexplainforinstancehowtohandleoneof
the nonlinearities arising in [21] in the study of a hydrodynamic limit, in dimension 2. Section
3 focuses on the proof of Theorem 1. Finally, in section 4 we prove Theorem 2. Subsection 4.1
gives general uniform properties for ε-interior sets, as defined in subsection 1.2. The proof of
Theorem 2 takes it simpler form in the case when the dual estimate on the time derivative (3)
is known for all test functions, we hence dedicate subsection 4.2 to this simplified framework.
In the general case the proof is a bit more involved and relies on properties of divergence-free
vector fieldsthatwedescribeinsubsection 4.3. Thecoreoftheproofof Theorem2iscontained
in subsection 4.4. Section 4 is totally independent of sections 2 and 3. Finally we prove in
appendix section 5 two technical results.
2 Weak convergence of a product
Proposition 1 below is directly inspired from an argument used in [14] (in the L8 L1 frame-
{
work). It was already stated and proved in [7] (in a periodic setting), but we reproduce it here
with the proof, for the sake of completeness and give also two other variants. Let us first treat
the case without boundary, in order to use freely the convolution in the x variable. Since this
work is motivated by evolution equations, t represents here the time variable and I is some
compact intervall of R, but it is quite clear that similar results may be obtained replacing I by
some bounded open set of Rd, with suitable assumptions.
We will use repeatedly a sequence ϕ of nonnegative even mollifiers (in space only) :
k k
p q
ϕ x : kdϕ kx , whereϕ is some smooth even nonnegative function with supportin the unit
k
p q “ p q
ball of Rd. In all this section the convolution has to be understood in the space variable x
‹
only.
We start with a “commutator Lemma” reminiscent of the usual Friedrichs Lemma (both
beingkey elements of [14]), the only difference is that nodifferential operation is involved here,
but the convergence holds uniformly in n.
Lemma 1 : Let q 1, and p 1,d and I R a non-empty closed and bounded interval.
Consider a 9 LqPIr;W81,sp Rd aPndr bs 9 Lq1ĂI;Lα1 Rd withα p‹. Thenthe commutator
n n n n
p q P p p qq p q P p p qq ă
(convolution in x only)
S : a b ϕ a b ϕ
n,k n n k n n k
“ p ‹ q´p q‹
goes to 0 in L1 R Rd as k , uniformly in n.
p ˆ q Ñ `8
Proof. Since a 9 Lq I;W1,p Rd and α p‹, we have
n n
p q P p p qq ă
τ a a 0 in Lq I;Lα Rd ,
h n n n
p ´ q hÝÑÑ0 p p qq
uniformly in n. We now follow [14] and write the following equality for the commutator
S t,x a t,x a t,x y b t,x y ϕ y dy, (6)
n,k n n n k
p q “ ż|y|ď1{k” p q´ p ´ qı p ´ q p q
whence thanks to Fubini’s Theorem, integrating on I Rd
ˆ
}Sn,k}L1pIˆRdq ď }bn}Lq1pI;Lα1pRdqqż |ϕkpyq|}τyan´an}LqpI;LαpRdqqdy,
|y|ď1{k
which yields the desired uniform convergence. ˝
7
Proposition 1 : Let q 1, and p 1,d and I R a non-empty segment. Consider
a 9 Lq I;W1,p Rd anPdr b8s 9 Lq1 I;PLαr1 Rsd respeĂctively weakly or weakly converging
n n n n
pin tqhePse sppaces topa aqnqd b. ApssuqmPe thapt α pp‹.qqIf b 9 M I;H´m Rd fo´r‹some m N
t n n
ă pB q P p p qq P
then, up to a subsequence, we have the following vague convergence in M I Rd (i.e. with
p ˆ q
C0 I Rd test functions) :
c
p ˆ q
a b ab. (7)
n n n
p q nÝÑá`8
Remark 2.1 : As explained in the introduction, we may recognize in this lemma a kind of
compensated compactness flavor since, in the above result, neither a nor b do converge
n n n n
p q p q
strongly: both may oscillate but only in a somehow compatible way. Nevertheless, as far as we
know, this result does not exactly recast in the work of Murat and Tartar.
Remark 2.2 : If a b , one gets strong compactness for a . Of course this situation is
n n n n
“ p q
nothing else than a particular case of the usual Aubin-Lions Lemma.
Proof. Fix N N and denote B : B 0,N Rd (open ball of radius N), O : I B .
N N N
P “ p q Ă “ ˆ
By a standard diagonal argument, it suffices to prove, up to a subsequence, that a b ab
n n n
p q á
in the vague topology of M O , that is, against C0 O test functions.
N c N
p q p q
Let us follow the following routine to conclude the proof.
Step 1. We have clearly
a b ϕ ab, in L1 O strong .
k N
p ‹ qkÑÝÑ`8 p q
Step 2. Since b 9 M R;H´m Rd , we get easily b ϕ 9 BV O so that, we can choose
t n n n k n N
pB q P p p qq p ‹ q P p q
(but we don’t write it explicitly) a common (diagonal) extraction such as, for all fixed k,
b ϕ converges a.e. on O to b ϕ . We hence deduce from the preceding fact that
n k n N k
p ‹ q ‹
(for all fixed k)
a b ϕ a b ϕ in L1 O weak.
n n k n k N
p p ‹ qq nÝÑá`8 p ‹ q p q
Indeed, we have (at least) the weak convergence of a towards a in L1 O , and since
n n N
p q p q
b ϕ converges to b ϕ a.e. on O , theabove convergence follows from the estimate
n k n k N
pb ‹ ϕq 9 L8 O , the‹latter being a direct consequence of b 9 M I;H´m Rd .
n k n N t n n
p ‹ q P p q pB q P p p qq
Step 3. From Lemma 1 we infer
sup a b ϕ a b ϕ 0.
n n k n n k 1
n } p ‹ q´p q‹ } kÑÝÑ`8
Step 4. For a fixed θ C0 O
c N
P p q
a b ϕ a b ,θ 0,
n n k n n
xp q‹ ´ ykÑÝÑ`8
uniformly in n. Indeed, since ϕ is even, we may write
k
a b ϕ a b ,θ a b ,θ ϕ θ ,
n n k n n n n k
xp q‹ ´ y“ x ‹ ´ y
andtheright-handsidetendsto0withthedesireduniformitybecause a b isbounded
n n n
p q
in L1 I Rd , and θ ϕ θ goes to 0 in L8 I Rd (θ is uniformly continuous ).
k k
p ˆ q p ‹ ´ q p ˆ q
Step 5. Write
ab a b ab a b ϕ
n n k
´ “ ´ p ‹ q
a b ϕ a b ϕ
k n n k
` p ‹ q´ p ‹ q
a b ϕ a b ϕ
n n k n n k
` p ‹ q´p q‹
a b ϕ a b .
n n k n n
`p q‹ ´
8
Fix θ C0 O , multiply the previous equality by θ and integrate over O . In the
c N N
P p q
right-hand side, line number i 1,2,3,4 corresponds to the Step i proven previously.
P t u
We choose first k to handle (uniformly in n) all the lines of the right-hand side, except
the second one. Then, we choose the appropriate n to handle the second line, thanks to
Step 2. This concludes the proof of Proposition 1. ˝
ItisworthnoticingthatintheproofofProposition1,theonlystepinwhichtheassumption
∇ a 9 Lq I;Lp Rd is crucial is Step 3, for the treatment of the “commutator”. In fact one
x n
p qP p p qq
caneasilyrelaxthisassumptioninthefollowingway. IfX denotessomeabstractfunctionspace
of the x variable, a sufficient condition on an n (to handle Step 3) is that τhan an LqpI;Xq
p q } ´ }
goes to 0 with h, provided that b 9 Lq1 I;X1 . Bearing this in mind, one may for instance
n n
p q P p q
prove the following Proposition
Proposition 2 : Consider a non-empty segment I R and two sequences a and b
n n n n
weakly converging in L1 I R2 to a and b. AssuĂme that a 9 Lq I;H1p Rq2 andpthqat
n n
b log b 9 Lq1 I;L1 pR2ˆ andq b 9 M I;H´m R2 fopr soqmPe mp N.p Tqhqen, up to
n n n t n n
p| | | |q P p p qq pB q P p p qq P
a subsequence, we have the following vague convergence in M I Rd (i.e. with C0 I Rd
c
p ˆ q p ˆ q
test functions) : a b ab.
n n n
p q nÝÑá`8
Remark 2.3 : Since the cornerstone in the below proof is the Moser-Trudinger inequality, this
Lemma may of course be generalized to Rd, replacing H1 R2 by W1,d Rd .
p q p q
Proof. Let us sketch the proof briefly. As before we work on B for the x variable and
N
without more precision will denote the Lp B norm. Consider the two convex functions
p N
}¨} p q
Φ :R x ex x 1 and Ψ :R x 1 y log 1 y y. For any measurable function
` `
Q ÞÑ ´ ´ Q ÞÑ p ` q p ` q´
f : R2 R such as Φ f L1 B we recall the Luxemburg gauge
N
Ñ p| |qP p q
f : inf a 0 : Φ f a 1 ,
Φ 1
} } “ ą } p| |{ q} ď
(
and define in a similar way . It is straightforward to check that Φ and Ψ are convex-
Ψ
} ¨}
conjugate of one another, and satisfy the Young inequality xy Φ x Ψ y . One may then
ď p q` p q
deduce the following generalized Hölder inequality, that is for all measurable functions f and
g such as Φ f L1 B and Ψ g L1 B ,
N N
p| |q P p q p| |q P p q
fg f g . (8)
1 Φ Ψ
} } ď } } } }
For more details on the previous inequality, Luxemburg gauge and Orlicz spaces, see [1] for
instance. Since a 9 Lq I;H1 R2 , we deduce from the Moser-Trudinger inequality (see
n n
p q P p p qq
again[1])that eαa2n n 9 Lq I;L1loc R2 , forapositiveconstantαsmallenough(infactα 4π).
In particular, upsing q(8)Pandpthe bopunqdq b log b 9 Lq1 I;L1 R2 , one gets a b 9 Lă1 I
n n n n n n
p| | | |q P p p qq p q P p ˆ
B . This estimate is sufficient to reproduce all Steps of the proof of Proposition 1, except
N
q
the third one, and as mentionned before, only Lemma 1 has to be examined.
TheMoser-Trudingerinequalityaforementionnedisbasedonthefollowingfact: thereexists
a universal constant C such as, for all p ă 8, and all f P H1pR2q, }f}LppRdq ď C?p}f}H1pRdq
(see again [1] for more details). Using this, and expanding the series defining Φ, one may show
easily the existence of a continuous and nondecreasing function ϕ : R R , vanishing in 0,
` `
Ñ
such as, for any f in the unit ball of H1 R2 ,
p q
f τ f ϕ h . (9)
h Φ
} ´ } ď p| |q
Theproofof Proposition 2follows then using(9) andinequality (8) in theexpression giving
the commutator (6). ˝
9
The boundsassumed in Proposition 2 are not coming from nowhere. There are for instance
the one obtained for the density of particles and the fluid velocity in the hydrodynamic limit
studied by Goudon, Jabin and Vasseur in [21] (in dimension 2). We hence recover by compact-
ness one of the nonlinear limit (in fact the easier one) explored in [21]. Let us mention that
in [21] the authors used a relative entropy method, whence a passage to the limit only under
the assumption of preparation of the data, an assumption that we obviously do not need to
apply Proposition 2. An other difference is that our method is always global (in time) whereas
the relative entropy method is usually limited by the existence of regular solutions for the
limit system ; but of course, in dimension 2, global regular solutions for the density-dependent
incompressible Navier-Stokes are known to exist (see [11]), so that the two approaches rejoin
on that point.
Let us conclude this section by giving a version of Proposition 1 in the case of a bounded
domain Ω using a simple localization argument:
Proposition 3 : Let q 1, and p 1,d , I R a non-empty segment and Ω Rd a
bounded open set with LipPscrhit8zsboundaryP.rConssiderĂtwo sequences a 9 Lq I;W1,p ΩĂ and
n n
b 9 Lq1 I;Lα1 Ω respectively weakly or weakly converging inp thqeseP spapces to pa aqqnd b.
n n
pAssqumPe thpat α pp‹q.q If b 9 M I;H´m Ω for´s‹ome m N then, up to a subsequence, we
t n n
ă pB q P p p qq P
have the following weak convergence in M I Ω (i.e. with C0 I Ω test functions) :
´‹ p ˆ q p ˆ q
a b ab. (10)
n n n
p q nÝÑá`8
Proof. First notice that, since α1 p‹ 1, we have by Sobolev embedding a b 9 L1 Lr for
n n n t x
ą p q p q P p q
some r 1, whence uniform absolute continuity in the x variable, in the sense that
ą
sup a b 0,
n n
nPNżIˆE| |µpÝEÑqÑ0
whereE denotes any measurable subsetof Ω and µ the Lebesgue measure on Rd. Now, we just
pick a sequence of functions θ D Ω bounded by 1, and equaling this value on a sequence
k k
p q P p q
of compact sets K such as µ Ω K 1 k.
k k
p z q ď {
Whenk isfixed,thesequences θ a and θ b (extendedby0outsideΩ)verifiesallthe
k n n k n n
p q p q
assumptions of Proposition 1, whence (up to a subsequence) the expected weak convergence
for the product θ2a b in M I Ω . We eventually get (10) by writting, for any test
p k n nqn p ˆ q´‹
function ϕ C0 I Ω
P p ˆ q
a b ,ϕ θ2a b ,ϕ a b , 1 θ2 ϕ ,
n n k n n n n k
x y “x y`x p ´ q y
since the second term of the r.h.s. is going to 0 with 1 k uniformly in n, and one may extract
diagonally allong the k’s to handle the first term of the{ r.h.s. ˝
3 Proof of Theorem 1
As explained in the introduction, the equality
1
∇ a ∇ Φ a ,
x n “ Φ1 a x p nq
n
p q
may not be used to recover an estimate on ∇ a using ∇ Φ a 9 L2 I Ω because
x n n x n n
p q p p qq P p ˆ q
such estimate degenerates when a approaches a a critical point of Φ. Replacing a by a
n n n
p q
truncation like β a where β is some smooth function erasing the critical values is of course
n
p q
a natural strategy. In this way, β a will indeed have compactness in the space variable x
n
p q
through a nice control of its gradient but, all the information on the time variable will be
10