Incompressible, inviscid limit of the compressible Navier-Stokes system

2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0294-1449(00)00123-2/FLA

INCOMPRESSIBLE, INVISCID LIMIT OF THE COMPRESSIBLE NAVIER–STOKES SYSTEM

Nader MASMOUDI¹

Ceremade, URA CNRS 749, Université Paris IX Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

Received in 2 February 2000

ABSTRACT. – We prove some asymptotic results concerning global (weak) solutions of compressible isentropic Navier–Stokes equations. More precisely, we establish the convergence towards solutions of incompressible Euler equations, as the density becomes constant, the Mach number goes to 0 and the Reynolds number goes to infinity._ 2001 Éditions scientifiques et médicales Elsevier SAS

RÉSUMÉ. – Nous prouvons quelques résultats asymptotiques concernant des solutions (faibles) globales des équations de Navier–Stokes (isentropique) compressible. Plus précisément, nous établissons la convergence vers une solution des équations d’Euler incompressible, lorsque la densité devient constante, le nombre de Mach tend vers 0 et le nombre de Reynolds tend vers l’infini.2001 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

From a physical point of view, one can formally derive incompressible models from compressible ones, when the Mach number goes to zero and the density becomes almost constant. In Lions and the author [12], this problem is investigated starting form the global solutions of the compressible Navier–Stokes equations constructed by Lions [11].

We have shown the convergence towards the incompressible Navier–Stokes equations as well as the convergence towards the incompressible Euler equations (if the viscosity coefficients go to zero and if the initial data are “well prepared”). These results have been precised and extended in different works (see [13,2,1,14]).

In this paper, we extend the result shown in [12] concerning the convergence to the Euler system to the case of more general initial data. In fact if the viscosity goes to zero too, we loose spatial compactness properties. In order to circumvent this difficulty, we use energy arguments. Hence, we have to describe (precisely) the oscillations that take place and include them in the energy estimates. Ideas of this type were introduced by

E-mail address: masmoudi@cims.nyu.edu (N. Masmoudi).

1Current adress: Courant Institute, New York University 251 Mercer St, New York, NY 10012, USA.

Schochet [17], (see also [4]) and extended to the case of vanishing viscosity coefficients in [16]. Let us now precise the scalings, we are going to use, which are the same as those used in [12]. The unknowns (ρ, v)˜ are respectively the density and the velocity of the fluid (gaz). We scaleρ˜ andv(and thusp) in the following way

˜

ρ=ρ(x, εt), v=εu(x, εt), (1)

and we assume that the viscosity coefficientsµ,˜ ξ˜ (ξ˜ − ˜µ,µ˜ are called the volumic and dynamical viscosity coefficients) are also small and scale like

˜

µ=εµε, ξ˜=εξε, (2)

where ε∈(0,1) is a “small parameter” and the normalized coefficient µε, ξε satisfy µε>0, ξε+µε>0. Moreover, we assume that

µε→0 asεgoes to 0₊. (3)

With the preceding scalings, the compressible Navier–Stokes system reads









∂ρ

∂t +div(ρu)=0, ρ0,

∂ρu

∂t +div(ρu⊗u)−µεu−ξε∇divu+ a

ε²∇ρ^γ =0.

(4)

One can always assume thata=1/γ by replacingεby√aγ ε. All throughout this paper the domainΩwill be the the whole space R^N or the torusT^N (in this last case, all the functions are defined on R^N and assumed to be periodic with period 2π ai in the ith variable). We recall here that the inviscid limit, namely the convergence of the Navier–

Stokes equations to Euler equations in the case of a domain with boundary is an open problem even in the incompressible case, which seems to be easier (see [15] for a partial result).

1.1. Statement of the results 1.1.1. The whole space case

Let us begin with the case of the whole space. We consider a sequence of global weak solutions(ρε, uε)of the compressible Navier–Stokes equations (4) and we assume thatρε−1∈L^∞(0,∞;L^γ₂)∩C([0,∞), L^p₂)for all 1p < γ ,whereL^p₂ = {f ∈L¹_loc,

|f|1_|f|1∈L^p, |f|1_|f|1∈L²}, uε ∈L²(0, T; H¹) for all T ∈(0,∞) (with a norm which can explode when ε goes to 0),ρε|uε|²∈L^∞(0,∞; L¹)and ρεuε ∈C([0,∞);

L^{2γ /(γ+1)}−w) i.e. is continuous with respect tot0 with values inL^{2γ /(γ+1)}endowed with its weak topology. We require (4) to hold in the sense of distributions and we impose the following conditions at infinity

ρε→1 as|x| → +∞, uε→0 as|x| → +∞. (5)

Finally, we prescribe initial conditions

ρε|t=0=ρ_ε⁰, ρεuε|t=0=m⁰_ε, (6) whereρ_ε⁰0,ρ_ε⁰−1∈L^γ,m⁰_ε∈L^{2γ /(γ+1)},m⁰_ε=0 a.e. on{ρ_ε⁰=0}andρ_ε⁰|u⁰_ε|² ∈L¹, denoting by u⁰_ε =m⁰_ε/ρ_ε⁰ on {ρ⁰ε >0}, u⁰_ε =0 on {ρε⁰=0}. The initial conditions also satisfy the following uniform bounds

ρ_ε⁰u⁰_ε

2+ 1

ε² ρ_ε⁰

γ −1−γρ_ε⁰−1C, (7)

where, here and below, C denotes various positive constants independent ofε. Let us notice that (7) implies in particular that, roughly speaking, ρ_ε⁰is of order 1+O(ε).In the sequel, we will use the following notation ρε=1+εϕε. Notice that ifγ <2, we cannot deduce any bound forϕε inL^∞(0, T;L²). This is why we prefer to work with the following approximation

Φε=1 ε

2a γ −1

ρ_ε^γ−1−γ (ρε−1).

Furthermore, we assume that

ρ_ε⁰u⁰_ε converges strongly in L² to someu⁰. Then, we denote byu⁰=Pu˜⁰, whereP is the projection on divergence-free vector fields, we also defineQ(the projection on gradient vector fields), henceu˜⁰=Pu˜⁰+Qu˜⁰. Moreover, we assume thatΦ_ε⁰converges strongly inL²to someϕ⁰. This also implies thatϕ_ε⁰converges toϕ⁰inL^γ₂.

Our last requirement on(ρε, uε)concerns the total energy: we assume that we have Eε(t)+

t 0

Dε(s)dsE_ε⁰ a.e.t, dEε

dt +Dε0 inD(0,∞), (8)

where

Eε(t)=

Ω

1

2ρε|uε|²(t)+ a ε²(γ −1)

(ρε)^γ −1−γ (ρε−1)(t),

Dε(t)=

Ω

µε|Duε|²(t)+ξε(divuε)²(t) and

E_ε⁰=

Ω

1

2ρ_ε⁰u⁰_ε²+ a ε²(γ −1)

ρ_ε^0γ −1−γρ⁰_ε−1.

We now wish to emphasize the fact that we assume the existence of a sequence of solutions with the above properties, and we shall also assume that γ > N/2. We recall

the results of Lions [11] which yield the existence of such solutions precisely when γ > N/2 ifN 4,γ 9/5 ifN =3 andγ 3/2 ifN =2. We also refer to [12] for the proof of the uniform bounds.

Whenεgoes to zero andµε goes to 0, we expect thatuεconverges tov, the solution of the Euler system

∂tv+div(v⊗v)+ ∇π=0,

divv=0 v|t=0=u⁰, (9)

inC([0, T^∗);H^s). We show the following theorem

THEOREM 1.1 (The whole space case). – We assume thatµε→ε0 (such thatµε+ ξε>0 for allε) and thatPu˜⁰∈H^s for somes > N/2+1, thenP (√

ρεuε)converges to vinL^∞(0, T;L²)for allT < T^∗, wherevis the unique solution of the Euler system in L^∞_loc([0, T^∗);H^s)andT^∗is the existence time of(9). In addition√ρεuε converges tov inL^p(0, T;L²_loc)for all 1p <+∞and allT < T^∗.

1.1.2. The periodic case

Now, we takeΩ=T^N and consider a sequence of solutions(ρε, uε)of (4), satisfying the same conditions as in the whole space case (the functions are now periodic in space and all the integration are performed over T^N). Of course, the conditions at infinity are removed and the spaces L^p₂ can be replaced byL^p. Here, we have to impose more conditions on the oscillating part (acoustic waves), namely we have to assume thatQu˜⁰ is more regular than L². In fact, in the periodic case, we do not have a dispersion phenomenon as in the case of the whole space and the acoustic waves will not go to infinity, but they are going to interact with each other. This is way, we have to include them in the energy estimates to show our convergence result.

For the next theorem, we assume that Qu˜⁰, ϕ⁰ ∈H^s−1 and that there exists a nonnegative constantν such thatµε+ξε2ν >0 for allε. For simplicity, we assume thatµε+ξεconverges to 2ν.

THEOREM 1.2 (The periodic case). – We assume thatµε→ε0 (such thatµε+ξε→ 2ν >0) and thatPu˜⁰∈H^sfor somes > N/2+1, andQu˜⁰, ϕ⁰∈H^s−1thenP (√

ρεuε) converges to v in L^∞(0, T;L²) for all T < T^∗, wherev is the unique solution of the Euler system inL^∞_loc(0, T^∗;H^s)andT^∗ is the existence time of(9). In addition √

ρεuε

converges weakly tovinL^∞(0, T;L²).

In the above theorem, one can remove the condition 2ν >0. In that case, we still have the result of Theorem 1.2 but only on an interval of time(0, T^∗∗)which is the existence interval for the equation governing the oscillating part, which will be given later on.

THEOREM 1.3 (The periodic case, ν =0). – We assume that µε→ε0 (such that µε +ξε →0) and thatu˜⁰∈H^s for some s > N/2+1, and ϕ⁰∈H^s then P (√ρεuε) converges tovinL^∞(0, T;L²)for allT < inf(T^∗, T^∗∗), wherevis the unique solution of the Euler system in L^∞_loc(0, T^∗;H^s), T^∗ is the existence time of (9) and T^∗∗ the existence time of (32), with ν = 0. In addition √

ρεuε converges weakly to v in L^∞(0, T;L²).

2. The whole space case

We recall that in the case of the whole space, we do not assume any extra condition on the viscosityξε, neither do we assume any regularity (more thanL²) for the gradient part of the initial data. The proof relies on the dispersion property of the wave equation [18,1] and the notion of non dissipative solutions for the Euler system [10,12].

First, using the energy bounds, we deduce thatρε−1 converges to 0 inL^∞(0, T;L^γ₂) and that there exists some u∈L^∞(0, T;L²) and a subsequence √ρεuε converging weakly to u. Hence, we also deduce thatρεuε converges weakly to u in L^{2γ /(γ+1)}. We next introduce the following group(L(τ ), τ ∈R)defined by e^{τ L}whereLis the operator defined onD×(D)^N, by

L ϕ

v

= − divv

∇ϕ

. (10)

It is easy to check that e^{τ L} is an isometry on eachH^s×(H^s)^N for alls∈Rand for all τ. This will allow

e^{τ L} ϕ

v

= ϕ(τ )

v(τ )

solves

∂ϕ

∂τ = −divv, ∂v

∂τ = −∇ϕ and thus ^∂_∂τ^2ϕ2 −ϕ=0.

Let(ψε, mε= ∇qε)be the solution of the following system









∂ψε

∂t = −1

εdivmε, ψε(t=0)=Φ_ε⁰,

∂mε

∂t = −1

ε∇ψε, mε(t=0)=Q

ρ⁰_εu⁰_ε.

(11)

We recall that for all v∈ H^s, we define Qv = ∇⁻¹divv and P v=v −Qv. We also introduce the following regularizations ψε,δ=ψε∗χδ, ∇qε,δ= ∇qε ∗χδ, where χ ∈C₀^∞(R^N) such that χ =1 and χδ(x)=(1/δ^N)χ (x/δ). Since (11) is linear it is easy to see that (ψε,δ,∇qε,δ) is a solution of the same system with regularized initial data. Using (as in [1]) Strichartz type inequality, we get

ψε,δ

∇qε,δ

L^p(R;W^s,q(R^N)))

Cε^1/p

Φ_ε⁰ Q(

ρ_ε⁰u⁰_ε)

∗χδ

H^s+σ

(12) for allp, q >2 andσ >0 such that

2

q =(N−1) 1

2− 1 p

, σ=N+1 N−1.

This yields that for all fixedδand for alls∈R, we have asεgoes to 0

(ψε,δ,∇qε,δ)→0 inL^pR;W^s,qR^N. (13) Now, we turn to the energy estimates. Let us rewrite the energy inequality for almost allt

1 2

Ω

ρε|uε|²(t)+Φ_ε²(t)+ t

0

Ω

µε|Duε|²+ξε(divuε)²1 2

Ω

ρ_ε⁰u⁰_ε²+Φ_ε²(0). (14) Then, the conservation of energy forvreads

Ω

1

2|v|²(t)=

Ω

1

2|u0|², (15)

and using the fact thatLis an isometry onL², we obtain for allt

Ω

1

2ψ_ε,δ² (t)+1

2|∇qε,δ|²(t)=

Ω

1

2ψ_ε,δ² (0)+1

2|∇qε,δ|²(0). (16) Next, the weak formulation of the conservation of mass yields for almost allt

Ω

ψε,δϕε(t)+1 ε

t 0

Ω

div(ρεuε)ψε,δ+div(∇qε,δ)ϕε=

Ω

ψε,δϕε(0), (17) while the weak formulation of (4) implies that we have for almost allt

Ω

ρεuε.v(t)+ t

0

Ω

ρεuε.(v.∇v+ ∇p)

− t 0

Ω

ρεuεuε.∇v+µε

t 0

Ω

∇uε.∇v=

Ω

ρ_ε⁰u⁰_ε.u⁰, (18)

Ω

ρεuε.∇qε,δ(t)+ t 0

Ω

ρεuε. 1

ε∇ψε,δ

− t 0

Ω

ρεuεuε.∇mε,δ

+ t 0

Ω

µε∇uε.∇mε,δ+ξεdiv(uε)div(mε,δ)

− t 0

Ω

1

εϕε+γ −1 2 Φ_ε²

div(∇qε,δ)=

Ω

ρ_ε⁰u⁰_ε.m⁰_ε,δ. (19) Summing up (14), (15), (16) and subtracting (17), (18), (19), we deduce from straightforward computations the following inequality

1 2

Ω

√

ρεuε−v−mε,δ²(t)+(Φε−ψε,δ)²(t)+ t 0

Ω

µε|∇uε|²+ξεdiv(uε)²

−

Ω

(Φε−ϕε)ψε,δ(t)+

Ω

(Φε−ϕε)ψε,δ(0) +

Ω

√

ρε−1√

ρεuε.(v+mε,δ)(t)

−ρ_ε⁰−1ρ_ε⁰u⁰_ε.v⁰+m⁰_ε,δ− t

0

Ω

µε|∇uε|²+ξεdiv(uε)²

+ t 0

Ω

µε∇uε.∇(v+mε,δ)+ξεdiv(uε)div(mε,δ)

+ t 0

Ω

ρεuε.(v.∇v+ ∇p)−ρεuε(uε.∇v) (∗)

− t 0

Ω

ρεuε(uε.∇mε,δ)−

γ −1 2 Φ_ε²

div(∇qε,δ) +1

2

Ω

ρ_ε⁰u⁰_ε−v⁰−m⁰_ε,δ²+(Φε−ψε,δ)²(0). (20) Only the term marked by (∗) in the second hand side of (20) needs some special treatment, we are going to compute it below. In the sequel, we denote by wε,δ =

√ρεuε−v−mε,δ. Then, we have easily t

0

Ω

ρεuε.(v.∇v+ ∇p)−ρεuε(uε.∇v)

= − t 0

Ω

wε,δ.∇vwε,δ+ t 0

Ω

ρε−√ ρε

uε.(v.∇v)+ρεuε.∇p

−√

ρεuε−v.∇v² 2 −

t 0

Ω

mε,δ.∇vwε,δ+√

ρεuε−v.∇v.mε,δ. (21) Finally, we can see that (20) may be rewritten as

wε,δ(t)²_L2+Φε−ψε,δ(t)²_L2

wε,δ(0)²_L2+Φε−ψε,δ(0)²_L2+2A^δ_ε+2 t 0

wε,δ(s)²_L2∇v(s)_L∞ds, (22)

where

A^δ_ε= −

Ω

(Φε−ϕε)ψε,δ(t)+

Ω

(Φε−ϕε)ψε,δ(0) +

Ω

√

ρε−1√

ρεuε.(v+mε,δ)(t)−ρ_ε⁰−1ρ_ε⁰u⁰_ε.v⁰+m⁰_ε,δ

+ t

0

Ω

µε∇uε.∇(v+mε,δ)+ξεdiv(uε)div(mε,δ)

− t

0

Ω

ρεuε(uε.∇mε,δ)−

γ −1 2 Φ_ε²

div(∇qε,δ)

+ t

0

Ω

ρε−√ ρε

uε.(v.∇v)+ρεuε.∇p−√

ρεuε−v.∇v² 2

− t

0

Ω

mε,δ.∇vwε,δ+√

ρεuε−v.∇v.mε,δ. (23) For all fixedδ, it is easy to see thatA^δ_ε(t)converges to 0 for almost allt, uniformly in t whenε goes to 0. Then, by Grönwall’s inequality, we deduce that we have for almost allt∈(0, T )

wε,δ(t)²_L2+Φε−ψε,δ(t)²_L2

wε,δ(0)²_L2 +Φε−ψε,δ(0)²_L2+ sup

0st

A^δ_ε(s)e^C t

0∇v(s)²_L∞. (24) Then, lettingεgo to 0, we obtain

u−v²_L∞(0,T;L²(Ω))lim

ε

wε,δ²_L∞(0,T;L²(Ω))+ Φε−ψε,δ²_L∞(0,T;L²(Ω))

C0u˜⁰−u⁰−Qu˜⁰∗χδ

L²(Ω)+ϕ⁰−ϕ⁰∗χδ

L²(Ω)

, where

C0=e^C T

0 ∇v(s)²_L∞ <+∞.

Then, letting δ go to 0, we deduce that u =v and we obtain also the uniform convergence intofP (√

ρεuε)tovinL², since limε

P√ ρεuε

−v_L∞(0,T;L²)

C0lim

δ u˜⁰−u⁰−Qu˜⁰∗χδ

L²(Ω)+ϕ⁰−ϕ⁰∗χδ

L²(Ω)

=0.

Moreover, we see that√ρεuε−mεconverges uniformly int, strongly inL²tov. In fact √

ρεuε−mε−v_L2√

ρεuε−mε,δ−v_L2+ mε−mε,δL²+ Φε−ψε,δL²,

and since (Φ_ε⁰, m⁰_ε =Q

ρ_ε⁰u⁰_ε) converges strongly to (ϕ⁰, m⁰) and since L is an isometry inL², we deduce that we have

mε−mε,δL²+ Φε−ψε,δL² →0 whenδ→0 uniformly intandε.

Finally, we can also deduce the local strong convergence of √ρεuε to v in L^p(0, T;L²(B)). Indeed let us denote by B a bounded domain ofR^N. Then, we have for allt

√

ρεuε−v_L2(B)√

ρεuε−mε,δ−v_L2(B)+ mε,δL²(B)

√

ρεuε−mε,δ−v_L2(B)+ mε,δL^q(B)

for anyq >2. Then, using the fact that for allδ,mε,δL^p(0,T;L^q(B))converges to 0 asε goes to 0, we conclude easily by taking the limit inεand then inδas above.

3. The periodic case

As in the case of the whole space, we can deduce, using the energy bounds, that ρε−1 converges strongly to 0 in L^∞(0, T;L^γ). Then, using the bound on√

ρεuε in L^∞(0, T;L²), we may extract a subsequence which converges weakly to some u. To pass to the limit in the equation, we need to describe the oscillations in time and show that they will not affect the limit equation.

We next introduce, as in the whole space case, the following group (L(τ ), τ ∈R) defined by e^{τ L} where L is the operator defined on D0 ×(D)^N, where D0 = {ϕ ∈ D,ϕ=0}, by

L ϕ

v

= − divv

∇ϕ

. (25)

In the sequel, we will use the following notations

U^ε=ϕε, Q(ρεuε) and V^ε=L(−t/ε)ϕε, Q(ρεuε),

and for some technical reasons related to the L² integrability, we will also use the following approximations

U¯^ε=Φε, Q(√

ρεuε) and V¯^ε=L(−t/ε)Φε, Q(√ ρεuε), which satisfy

U^ε− ¯U^ε_L∞(L2γ /(γ+1))→0 whenε→0.

Let us project Eq. (4) on “gradient vector-fields”

∂

∂tQ(ρεuε)+Qdiv(ρεuε⊗uε)−(µε+ξε)∇divuε

(26) + a

ε²∇ρ_ε^γ−γρε+(γ −1)+ 1

ε²∇(ρε−1)=0.

Combining (26) with the conservation of mass, we obtain ε∂ϕε

∂t +divQ(ρεuε)=0, ε ∂

∂tQ(ρεuε)+ ∇ϕε=εFε, (27) where

Fε=(µε+ξε)∇divuε−Qdiv(ρεuε⊗uε)−a∇ 1

ε²

ρ_ε^γ−γρε+(γ −1).

Eq. (27) yields that ∂tU^ε = ¹_εLU^ε +(0, Fε), which can be rewritten as ∂tV^ε = L(−t/ε)(0, Fε). It is easy to check that Fε is bounded in L²(H^−s) (for some s ∈R), hence V^ε is compact in time (the oscillations have been cancelled). If we had enough compactness in space we could pass to the limit in this equation and recover the following limit system for the oscillating part

∂tV¯ +Q1(u,V )¯ +Q2(V ,¯ V )¯ −νV¯ =0, (28) where Q1 and Q2 are respectively a linear and a bilinear forms in V¯ (which will be defined and computed later on) and the term−νV¯ explained below. In fact, as in [17]

(see also [16]), we consider L(−t/ε)(0, Fε)as an almost periodic function in τ =t/ε and compute its mean value, which yields (28).

DEFINITION 3.1. – For all divergence-free vector field u∈ L²(Ω)^N and all V = (ψ,∇q)∈L²(Ω)^N+1, we define the following linear and bilinear symmetric forms in V

Q1(u, V )= lim

τ→∞

1 τ τ

0

L(−s)

0

div(u⊗L2(s)V +L2(s)V ⊗u)

ds, (29) and

Q2(V , V )= lim

τ→∞

1 τ

τ 0

L(−s)

0

div(L2(s)V ⊗L2(s)V )+^γ−₂¹∇(L1(s)V )²

ds. (30)

The convergences stated above takes place in W^−1,1 and can be shown by using almost-periodic functions (see [16] and the references therein). Indeed the functions inside the integral in (29) and (30) are almost periodic inW^−1,1andQ1(u, V ),Q2(V , V ) are their mean value. We will come back to this issue in the next section. We also remark that in Eq. (28) the viscosity term was divided by 2 and that it applies for both component of the vector V¯. This is due to the following proposition, which can be proved easily using almost periodic functions (see also [5] and [3]).

PROPOSITION 3.2. – Under the same hypothesis onV, we have

−νV = lim

τ→∞

1 τ

τ 0

−L(−s)

0 2νL2(s)V

ds. (31)

Nevertheless, the fact that the viscosity applies now for both components ofV is not sufficient to yield compactness in space for V^ε. To recover compactness in space, we will use the regularity of the limit system and extend the method used in [12] to the case of general initial data as was done in [16]. LetV⁰be the solution of the following system

∂tV⁰+Q1(v, V⁰)+Q2(V⁰, V⁰)−νV⁰=0,

V_|⁰_t=0=(ϕ⁰, Qu˜⁰), (32)

wherev is, as in the case of the whole space, the solution of the incompressible Euler equations with initial datau⁰. The existence of global strong solutions for the system (32) (and local solutions if the viscosity term is removed) as well as the exact computations of the two formsQ1andQ2will be detailed in the next section. We only need the following two propositions.

PROPOSITION 3.3. – For all u, V, V1 and V2 (regular enough to define all the products), we have

Q1(u, V )V =0 and

Q2(V , V )V =0, (33)

Q1(u, V1)V2+Q1(u, V2)V1=0, (34)

Q2(V1, V1)V2+2Q2(V1, V2)V1=0. (35) The proof of (33) will be given in the next section, (34) can be shown by applying the first part of (33) toV1+V2and toV1−V2. Finally (35) can be shown by applying the second part of (33) toV1+X V2and identifying the term of degree 1.

The next proposition is a very simple consequence of the theory of almost-periodic functions (see for instance Lemma 2.3 of [16]).

PROPOSITION 3.4. – For all u∈L^p(0, T;L²) and V ∈L^q(0, T;L²), we have the following weak convergences (pandqare such that the product are well defined )

w- lim

ε L

−t ε

0 divu⊗L2

_t

ε

V +L2

_t

ε

V ⊗u

=Q1(u, V ) (36) and

w- lim

ε L

−t ε

0 divL2

_t

ε

V ⊗L2

_t

ε

V+^γ−₂¹∇L1

_t

ε

V²

=Q2(V , V ). (37) Using the symmetry ofQ2, we deduce easily the following proposition.

PROPOSITION 3.5. – Eq.(37)of Proposition 3.4 can be extended to the case where we takeV1andV2using the symmetry ofQ2, namely

w- lim

ε L

−t ε

0 divL2

_t

ε

V1⊗L2

_t

ε

V2+L2

_t

ε

V2⊗L2

_t

ε

V1

+L

−t ε

0

γ−1 2 ∇L1

_t

ε

V1L1

_t

ε

V2

=Q2(V1, V2). (38) Moreover, the above identity holds for V1∈L^q(0, T;H^s)andV2∈L^p(0, T;H^−s)with s∈Rand 1/p+1/q=1. It is also possible to extend it to the case where we replace V2 in the left hand side by a sequence V₂^ε such that V₂^ε converges strongly to V2 in L^p(0, T;H^−s).

In order to show the convergence in Theorem 1.2, we will try to estimate √

ρεuε−v−L2

t ε

V⁰

2

L²

+Φε−L1

t ε

V⁰

2

L²

.

We also introduce the following notations wε = √ρεuε −v−L2(t/ε)V⁰ and βε = Φε−L1(t/ε)V⁰. In the sequel, we also note (ψε, mε)=L(t/ε)V⁰. The proof follows the same lines as the proof in the whole space case apart from the fact that the equation satisfied by the gradient part is not trivial and that we have to use the precise equation satisfied by the oscillating terms. We recall the energy inequality

1 2

Ω

ρε|uε|²(t)+Φ_ε²(t)+ t 0

Ω

µε|Duε|²+ξε(divuε)²1 2

Ω

ρ_ε⁰u⁰_ε²+Φ_ε²(0) (39) as well as the conservation of energy forv

Ω

1

2|v|²(t)=

Ω

1

2|u0|². (40)

Using that

Q1

u, V⁰V⁰=0,

Q2

V⁰, V⁰V⁰=0, we deduce from (32) the following energy identity

Ω

1

2V⁰²(t)+ν t

0

Ω

∇V⁰²(s)ds=

Ω

1

2V⁰(t=0)². (41) Next, using the weak formulation of the conservation of mass, we obtain for almost allt

Ω

ψεϕε(t)+1 ε t

0

Ω

div(ρεuε)ψε+div(∇qε)ϕε