Homogenization of the compressible Navier-Stokes equations in a porous medium

URL:http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002053

HOMOGENIZATION OF THE COMPRESSIBLE NAVIER–STOKES EQUATIONS IN A POROUS MEDIUM

Nader Masmoudi

Abstract. We study the homogenization of the compressible Navier–Stokes system in a periodic porous medium (of periodε) with Dirichlet boundary conditions. At the limit, we recover different systems depending on the scaling we take. In particular, we rigorously derive the so-called “porous medium equation”.

Mathematics Subject Classification. 76M50.

Received February 5, 2002. Revised March 5, 2002.

1. Introduction

The homogenization of the Stokes and of the incompressible Navier–Stokes equations in a porous medium (open set perforated with tiny holes) has been studied in many works from the formal point of view as well as the rigorous one. We refer the interested reader to [4, 12, 20] for some formal developments and to [1, 17, 21]

for some rigorous mathematical results. In this paper, we try to extend some of the methods developed in the incompressible case to study the case we start from different compressible models built on the compressible Navier–Stokes system. One of the major difficulties we will encounter here is the passage to the limit in the non linear terms. It is worth noticing that in the incompressible case there are many open problems related to the passage to the limit in the non linear terms due to the presence of boundary layers.

Before stating the system, let us recall the domain we consider. A porous medium is defined as the periodic repetition of an elementary cell of sizeεin a bounded domain Ω ofR^N whereN = 2,or 3 (all the results given below also hold for N ≥2). The solid part of the porous medium is also taken of sizeε. The domain Ω_ε is then defined as the intersection of Ω with the fluid part. We consider a compressible fluid governed by the compressible Navier–Stokes equation. So, we have the following system of equations written in (0,∞)×Ω_ε





∂_tρ+ div(ρu) = 0, ρ≥0

∂_t(ρu) + div(ρu⊗u)−µ∆u−ξ∇divu+∇pε=ρ_εf +g.

(1)

Here, u_ε and ρ_ε are respectively the velocity and the density of the fluid and the pressure p_ε is given by a barotropic lawp_ε=ρ^γ_ε. The exterior force is given byρ_εf+g and the viscositiesµandξ are such thatµ >0 andµ+ξ >0. For simplicity, we will assume thatξ≥0. The system should be complemented with initial and

Keywords and phrases:Compressible Navier–Stokes, homogenization, porous medium equation.

1Courant Institute, 251 Mercer Street, New York, NY 10012, USA; e-mail: masmoudi@cims.nyu.edu Partially supported by NSF grant and an Alfred Sloan fellowship.

c EDP Sciences, SMAI 2002

boundary conditions





ρ(t= 0) =ρ₀, ρu_ε(t= 0) =m₀ u_ε= 0 on ∂Ω_ε.

(2)

We will study the limit when ε goes to zero of the above system as well as some related ones with different scalings. The results will be stated and then proved for each model in the next sections.

1.1. The domain

Let Ω be a smooth bounded domain ofR^N and defineY =]0,1[^N to be the unit open cube ofR^N. Let Ys

(the solid part) be a closed smooth subset ofY with a strictly positive measure. The fluid part is then given byYf = Y − Ys and we define θ = |Yf| the Lebesgue measure of Yf and we assume that 0 < θ < 1. The constantθ is called the porosity of the porous medium. Repeating the domainYf byY-periodicity we get the fluid domainE_f which can also be defined as

E_f ={y∈R^N | ∃k∈Z^N, such that y−k∈ Yf } · (3) In the same way, we can defineE_s=R^N −E_f

E_s={y∈R^N | ∃k∈Z^N, such that y−k∈ Ys } · (4) It is easy to see that E_f is a connected domain, whileE_s is formed by separate smooth subsets. In the sequel, we denote for allk∈Z^N,Y^k =Y+k the translate of the cellY by the vectork, we also denoteY_s^k =Ys+k and Y_f^k=Yf+k. Hence, for allε, we can define the domain Ω_ε as the intersection of Ω with the fluid domain rescaled by ε, namely Ω_ε = Ω∩εE_f. However, for some technical reasons and to get a smooth connected domain, we will not remove the solid parts of the cells which intersect the boundary of Ω. We define

Ω_ε= Ω−U{εY_s^k, where, k∈Z^N, εY^k⊂Ω}·

We also denoteK_ε={k|k∈Z^N andεY^k ⊂Ω}.

Remark 1.1. We can also consider more general domains, specially the more physical case where E_s is a connected set of R^N which can be achieved by allowing Ys to be a closed subset of ¯Y (this is not possible in N = 2 since we also want that Ω_εis connected). We refer the interested reader to the paper of Allaire [1] where the so-called “energy method” of Tartar is extended to the case of a connected E_s. In the sequel and for the clarity of the presentation, we will only study the case whereE_sis not connected.

1.2. Some notations and preliminaries

In all the paper, we denote the space-time Lebesgue spaces byL^r(0, T;L^q(X)) where X is either Ω or Ω_ε. Some times we will also denote it by L^r_T(L^q(X)), L^r_T(L^q) or L^r(L^q) if no ambiguity can occur. If r = q, we will also use the notation L^r_T(X). W^s,p(X) will denote the classical Sobolev space built over L^p and H^s(X) =W^s,2(X). Besides, we will use the notationH^s(X)^N (orH^s(X) if no ambiguity can occur) for vector valued functions ofN components. We will also use Sobolev spaces with negative regularity and we recall that

||u||W^−1,p(X)= sup

v∈W₀^1,p0(X),||v||_W1,p0 0 (X)=1

hu, vi_W−1,p,W₀^1,p0

wherep⁰ is the conjugate exponent ofp.

Due to the presence of the holes εY_s^k, the domain Ω_ε depends on εand hence to study the convergence of the sequence (u_ε, ρ_ε, p_ε), we have to extend the functions defined in Ω_ε to the whole domain Ω. This can be done in two different possible ways.

Definition 1.2. For any functionφ∈L¹(Ω_ε), we define φe=

φ in Ω_ε

0 in Ω−Ω_ε (5)

the extension by 0 of φand

φb=

(φ in Ω_ε

ε|Y1f|

R

εY_f^kφdy in εY_s^k ∀k∈K_ε. (6)

We have the following relation between the weak limits of both types of extensions.

Lemma 1.3. For all sequenceg_ε∈L¹(Ω_ε), the following two assertions are equivalent 1) bg_ε * g inL¹(Ω);

2) eg_ε * θg inL¹(Ω).

Proof. For allψ∈ D(Ω), we use the fact thatψ is uniformly continuous to deduce that ω(ε) = sup

|x−y|≤ε|ψ(x)−ψ(y)| → 0 whenε → 0.

Hence, we have forεsmall enough Z

Ωψeg_ε = X

k∈Kε

Z

εY_f^kψg_ε= X

k∈Kε

ψ(εk) Z

εY_f^kg_ε+r(ε)

= X

k∈Kε

ψ(εk)θ Z

εY^kbg_ε+r(ε) =θ Z

Ωψbg_ε+r⁰(ε) where |r(ε)|+|r⁰(ε)| ≤Cω(ε). Sendingεto 0, we conclude easily.

We will also need the restriction operator constructed by Tartar [21] for the case of a solid partYs strictly included in Y and by Allaire [1] for more general conditions on the solid part.

Lemma 1.4. There exists a linear operator R_ε from H₀¹(Ω)^N to H₀¹(Ω_ε)^N (called restriction operator) such that

(i) ∀φ∈H₀¹(Ω_ε)^N, we haveR_εφe=φ;

(ii) ∇ ·u= 0in Ωimplies that∇ ·R_εu= 0 inΩ_ε;

(iii) there exists a constantC such that for allu∈H₀¹(Ω)^N, we have

||Rεu||_L²_(Ωε)+ε||∇(Rεu)||_L²_(Ωε)≤C

h||u||_L²_(Ω)+ε||∇u||_L²_(Ω)i

. (7)

The operatorR_εdefined above also acts fromW₀^1,r(Ω) intoW₀^1,r(Ω_ε) for all 1< r <∞and we have an estimate similar to (7) where theL² norms are replaced byL^rnorms.

Due to the presence of the holes in the domain Ω_ε, the Poincar´e’s inequality reads:

Lemma 1.5. There exists a constantC which depends only on Ys such that for allu∈W₀^1,p(Ω_ε), we have

||u||L^p(Ωε)≤Cε||∇u||L^p(Ωε). (8)

We refer to [21] for a proof of this lemma. By a simple duality argument we also have the following relation for all 1< p <∞

||u||_W^−1,p_(Ωε)≤Cε||u||_L^p_(Ωε). (9) To get some space-timea prioriestimate, we will use the following operator:

Lemma 1.6. For allε >0, there exists a linear operatorB=Bε

B : L^p₀(Ω_ε) =

f ∈L^p(Ω_ε)| Z

Ωε

f = 0

→ W₀^1,p(Ω_ε) (10)

such that v=B(f)solves the equation

div(v) =f inΩ_ε, v = 0 on∂Ω_ε (11)

and the following estimate

||B(f)||_W^1,p

0 (Ωε)≤ C

ε||f||_L^p_(Ωε) (12)

holds for all 1< p <∞. Moreover, iff ∈L^p(Ω_ε) can be written asf = div(g)whereg ∈L^r(Ω_ε)andg.n= 0 on ∂Ω_εfor somer >1 then

||B(f)||_L^r_(Ωε)≤ C||g||_L^r_(Ωε). (13)

Sketch of the Proof. The fact thatB mapsL^p₀(Ω_ε) into W₀^1,p is well known (see for instance [5, 10]). Here we have to explain the presence of the constant ^C_ε in the estimate (12). To this end, we will use the construction of Bogovskii. We have to split our domain in small domains and make the construction on each smaller domain.

Take an open set ˜Y such that ¯Yf ⊂Y ⊂˜ E_f where ¯Yf = [0,1]^N− Ys. We define as above, ˜Y^k = ˜Y+k. Then, there existsφ∈C₀^∞( ˜Y ∪ Ys) such that

• φ= 1 on a neighborhood ofYs;

• P

k∈Z^Nφ(x+k) = 1 ∀x∈R^N.

Moreover, for allkandk⁰ such that|k−k⁰|= 1, we can find a functionφ_k,k⁰ ∈C₀^∞( ˜Y^k∩Y˜^k0) such that Z

Y˜^k∩Y˜^k0φ_k,k⁰ = 1.

Now, for all f ∈L^p₀(Ω_ε), we want to construct the solutionv =B(f) by solving an auxiliary problem in each one of the domains ˜Y^k. For simplicity, we will assume that f has its support in U_k∈KεY_f^k to avoid dealing with the part off close to the boundary of Ω. If we do not make this assumption, we have just to modify the cut-off functions of the cells close to the boundary. Next, we use the partition of the unity to decompose f as f =P

k∈Kεf φ(^x_ε −k). We note thatf(x)φ(^x_ε −k) is supported inεY˜^k but is not necessary of integral equal to 0. Using the functions φ_k,k⁰, we can construct a decompositionf =P

k∈Kεf_k such that suppf_k ∈εY˜^k and R

εY˜^kf_k = 0 and P

k∈Kε||fk||L^p ≤ ^C_ε||f||L^p. Now, for each k ∈ K_ε, there exists v_k ∈ W₀^1,p(εY˜^k) such that divv_k =f_k and||vk|_W1,p(εY˜^k)≤ ||vk|_Lp(εY˜^k). Adding up these estimates, we recover (12). Finally, we point out that in (13) there is no factor ¹_ε since we can decomposeg as g=P

k∈Kεg_k whereg_k=gφ(^x_ε −k) and hence we can takef_k= divg_k.

For all ε > 0, we consider the Stokes problem written in Ω_ε and define the operator S = S_ε by Sf = p where (u, p) solves





−∆u+∇p=f in Ω_ε u= 0 on ∂Ω_ε

div(u) = 0 in Ω_ε and R

Ωεpdx= 0.

(14)

Lemma 1.7. For all ε and 1 < r < ∞, the operator S_ε is bounded from L^r(Ω_ε) onto W^1,r(Ω_ε) and from W^−1,r(Ω_ε)ontoL^r(Ω_ε)(see for instance [6, 10, 11, 22]). Moreover, there exists a C independent ofεsuch that kuk_H0¹_(Ωε)+k∇S_εfk_H⁻¹_(Ωε)+εkS_εfk_L²_(Ωε)≤Ckfk_H⁻¹_(Ωε) (15)

kuk_H²_∩H0¹_(Ωε)+k∇SεfkL²(Ωε)≤CkfkL²(Ωε). (16) Besides, for all r,1< r <∞there exists a constant C

kuk_W^1,r

0 (Ωε)+k∇Sεfk_W^−1,r_(Ωε)+εkSεfk_L^r_(Ωε)≤ C

ε^αkfk_W^−1,r_(Ωε) (17) kuk_W2,r∩W₀^1,r(Ωε)+k∇S_εfkL^r(Ωε)≤ C

ε^αkfkL^r(Ωε) (18) where α=|^N₂ −^N_r|.

Notice here that the factorsε^α associated to rand to the conjugate exponent ofrare the same. It is likely that the presence of the factor 1/ε^αis not really necessary in (17) and (18) but we do not need this refinement here.

Sketch of the Proof. Let us start with (15). By the energy estimate, we get Z

Ωε

|∇u|²≤Ckfk_H⁻¹_(Ωε)kuk_H0¹_(Ωε).

From which we deduce that kuk_H0¹_(Ωε) ≤ Ckfk_H⁻¹_(Ωε) and that k∇pk_H⁻¹_(Ωε) ≤ Ckfk_H⁻¹_(Ωε). Next, using Lemma 1.6, we have

||p||²_L2(Ωε)=h∇p,B(p)i_H−1,H₀¹ ≤ C

εkfk_H⁻¹_(Ωε)||p||_L²_(Ωε).

Hence (15) is proved. To prove (16), we have to argue as in [6, 10] by using interior and boundary regularities and then try to prove that the constants are independent of ε. We start by proving anH₀¹ estimate

Z

Ωε

|∇u|²≤CkfkL²(Ωε)kukL²(Ωε)≤CεkfkL²(Ωε)kuk_H¹_0(Ωε).

From which we deduce that kuk_H¹_0(Ωε) ≤Cεkfk_L²_(Ωε) and kuk_L²_(Ωε) ≤ Cε²kfk_L²_(Ωε). Now, we will explain the idea behind the uniform bounds, namely the fact that the constant appearing in (16) is independent ofε.

Indeed, one can use interior regularity results for each one of the extended cells ˜Y^k. We assume that 0 ∈ Ω and we define U(x) =u(εx),P(x) =p(εx),F(x) =f(εx). Hence−∆U+ε∇P =ε²F. Take a cut-off function φ∈C₀^∞( ˜Y ∪ Ys) such thatφ= 1 on a neighborhood ofY, hence





−∆(φU) +ε∇(φP˜) =ε²φF +∇U· ∇φ+ ∆φU +εP∇φ˜ in Y˜

φU = 0 on ∂Y˜

div(φU) =U· ∇φ in Y˜

(19)

where ˜P =P−_|Y|¹_˜ R

Y˜P. Using classical regularity results for the Stokes system in a bounded domain, we get that

||φU||_H2( ˜Y)+||ε∇(φP˜)||_L2( ˜Y) ≤C||U · ∇φ||_L2( ˜Y)+C||ε²φF||_L2( ˜Y)

+C||∇U· ∇φ+ ∆φ U+εP˜∇φ||_L2( ˜Y) (20)

≤C

h||ε²F||_L2( ˜Y)+||∇U||_L2( ˜Y)+||U||_L2( ˜Y)+ε||P||˜ _L2( ˜Y)

i . Besides, we have

||εP˜||_L2( ˜Y)≤C||ε∇P˜||_H−1( ˜Y)≤C||ε²F||_H−1( ˜Y)+C||U||_H1( ˜Y). (21) Hence, the last term in (20) can be estimated by the other terms appearing in the right hand side. Rewriting (20) in the original coordinate system, we get

||u||_H²_(εYf)+||∇p||_L²_(εYf)≤C||f||_L2(εY)˜ +C

ε||∇u||_L2(εY)˜ + C

ε²||u||_L2(εY)˜ . (22) This estimate also hold for any cellεYf such thatεY ⊂˜ Ω_ε. Near the boundary the above argument should be slightly changed. Adding up the above estimates, we infer

||u||H²(Ωε)+||∇p||L²(Ωε)≤C||f||L²(Ωε)+C

ε||∇u||L²(Ωε)+C

ε²||u||L²(Ωε). (23) From which we deduce (16).

To prove (17), we restrict ourselves to the case r > 2 since the case r < 2 can be deduce from the case r >2 by duality. Using that W^−1,r ⊂H⁻¹ sincer >2, we deduce thatkuk_H¹_0(Ωε)≤Ckfk_W^−1,r_(Ωε) and that k∇pk_H⁻¹_(Ωε)≤Ckfk_W^−1,r_(Ωε). Moreover, arguing as above, we have

||φU||_W1,r( ˜Y)+||ε∇(φP˜)||_W−1,r( ˜Y)≤C||U· ∇φ||_W−1,r( ˜Y)C||ε²φF||_W−1,r( ˜Y)

+C||∇U · ∇φ+ ∆φ U+εP˜∇φ||_W−1,r( ˜Y)

≤C

h||ε²F||_W^−1,r +||∇U||_W^−1,r +||U||_W^−1,r +ε||P˜||_W^−1,ri . Besides, we have

||εP˜||_W−1,r( ˜Y)≤C||ε∇P˜||_W−2,r( ˜Y)≤C||ε²F||_W−2,r( ˜Y)+C||U||_Lr( ˜Y) (24) and||∇U||W^−1,r+||U||W^−1,r ≤C||U||_Lr( ˜Y). Adding up all the estimates over the different cells and going back to the initial coordinate system, we get

||u||_W^1,r

0 (Ωε)+||∇p||W^−1,r(Ωε)≤C||f||W^−1,r +C

ε||u||L^r(Ωε). (25) Next, we use that

||u||_L^r_(Ωε)≤C||u||^κ_L2(Ωε)||∇u||^1−κ_Lr(Ωε)

where ¹_r =^κ₂+ (1−κ)(¹_r−_N¹). Hence, we get that

||u||_W^1,r

0 (Ωε)≤C||f||_W^−1,r +C

ε||u||^κ_L2(Ωε)||∇u||^1−κ_Lr(Ωε)

which can be rewritten as

||u||_W^1,r

0 (Ωε)≤C||f||_W^−1,r + C

ε^κ1||u||_L²_(Ωε)≤ C

ε^κ1−1||f||_W^−1,r. And (17) is proved, since _κ¹−1 =α.

Finally, we define the permeability matrix ¯A. For alli, 1≤i≤N, let (v_i, q_i)∈H¹(Yf)^N×L²(Yf)/Rbe the unique solution of the following system

(S_i)





−∆vi+∇qi=e_i inYf

divv_i= 0 inYf

v_i = 0 on∂Ys and v_i, q_i areY −periodic.

(26)

Using regularity results of the Stokes problem, we infer thatv_i andq_i are smooth. We extendv_i to the whole domain Y by setting v_i(y) = 0 if y ∈ Ys. Then, for all y ∈ Yf, A(y) is taken to be the matrix composed of the column vectors v_i(y) and ¯A = R

YfA(y)dy. It is easy to see that ¯A is a symmetric positive definite matrix. Indeed, multiplying the first equation in (S_i) byv_j and the first equation in (S_j) by v_i, we get that R

Yf∇v_i· ∇v_j =R

Yf v_ji= ¯A_ji and R

Yf∇v_j· ∇v_i =R

Yfv_ij = ¯A_ij where we wrotev_i(y) =P_N

j=1v_ji(y)e_j. Then to prove that ¯A is positive definite, we just notice that for all vectorX =P_N

j=1x_ie_i, we have P

ijx_iA¯_ijx_j =

||∇P_N

j=1x_jv_j||²_L2(Yf)and that{vi, 1≤i≤N}is an independent family.

In the next three sections we will study three different types of models. For each model, we will start by a presentation then state the result and finally give the proof of the main result. In Section 2, we study a semi-stationary model and derive in particular the so-called “porous medium” equation. In Section 3, we start from the full compressible system but with a scaling which gives formally the same limit system as in Section 2.

Finally, in Section 4, we deal with an equation describing the acoustics in a porous medium.

2. A semi-stationary model 2.1. The model

We start with the following semi-stationary model





ε²∂_tρ+ div(ρu) = 0,

−µ∆u−ξ∇divu+∇ρ^γ =ρ_εf+g

(27)

complemented with the boundary condition u_ε= 0 on∂Ω_εand the initial conditionρ_ε(t= 0) =ρ_ε0.The force term is such thatf ∈L^∞((0, T)×Ω_ε) andg∈L²((0, T)×Ω_ε). We also assume thatγ≥1 and that||f||L^∞ is small enough if γ= 1.

2.2. Statement of the result

We assume that the initial data is such thatρ_ε0∈L¹∩L^γ(Ω_ε) ifγ >1, thatR

Ωερ_ε0|logρ_ε0|< C ifγ= 1 and thatρb_ε0 converges weakly to ρ₀ in L^γ(Ω).

We consider a sequence of weak solutions (ρ_ε, u_ε) of the semi-stationary model (27) such that for allT >0, ρ_ε∈C([0, T);L¹(Ω_ε))∩L^∞(0, T;L^γ(Ω_ε))∩L^2γ((0, T)×Ωε) andρ_ε|logρε| ∈L^∞(0, T;L¹(Ω_ε)) ifγ= 1. Moreover, u_ε is such that ^u_ε^ε ∈L²(0, T;H₀¹(Ω_ε)) and ^u_ε^ε₂ ∈L²((0, T)×Ω_ε). Finally, we also require thatpb_ε is bounded in L²_T(H¹(Ω)) +εL²_T(L²(Ω)). We assume that the bounds given above are uniform inε. We point out that the

fact that we can consider a sequence of solutions satisfying the above uniform estimates is a consequence of the a prioribounds which will be recalled in Section 2.3.

Before studying the limit of the sequence (u_ε, ρ_ε, p_ε), we have to prolong it to Ω. Let eu_ε,ρe_ε and bp_ε be the extensions of u_ε,ρ_ε andp_ε to the whole domain Ω defined as in Section 1.2.

Theorem 2.1. Under the above assumptions, e

ρ_ε → θρ weakly in L^r_T(L^γ(Ω))∩L^2γ((0, T)×Ω), b

ρ_ε → ρ strongly in L^r_T(L^γ(Ω))∩L^γ+1((0, T)×Ω), e

u_ε

ε² → u weakly in L²_T(L²(Ω))

for all r <∞whereρ∈L^2γ((0, T)×Ω),ρ^γ ∈L²_T(H¹(Ω))andρis the solution of the following system















θ∂_tρ+ 1 µdiv·

ρA(ρf¯ +g− ∇ρ^γ)

= 0 ρA(ρf¯ +g− ∇ρ^γ).n= 0 on ∂Ω ρ(t= 0) =ρ₀

(28)

and uis given by

u= ¯A(ρf+g− ∇ρ^γ) on {ρ >0} · (29) We point out here that even though each one of the termsf,g and∇ρ^γ does not have necessary a trace on the boundary ∂Ω, the combination of them appearing in (28) has a sense. A formal derivation of the system (28) can be found in [8]. The relation (29) giving uas a function of the pressure is a Darcy law [7].

Remark 2.2. if ¯A=αI (which is the case if for instanceYsis a ball) andf =g= 0 then we get the following system















∂_tρ−β∆ρ^γ+1= 0

∂ρ^γ+1

∂n = 0 on ∂Ω ρ(t= 0) =ρ₀

(30)

where β= _θµ(γ+1)^αγ . This system is the so-called “porous medium” equation.

2.3. A priori estimates

Before proving Theorem 2.1, let us recall how we can get the existence of weak solutions for (27) satisfying the requirement of the last subsection. We will only explain how we can get uniform estimates inεand refer to [14]

(p. 226) for the approximation part. First, integrating the first equation of (27) over the whole domain Ω_ε, we deduce the conservation of mass, namelyR

Ωερ_ε=R

Ωερ_ε0. The fact that we can make this integration over the whole domain Ω_εrigorously comes from theL²bounds we have forρ_ε and∇u_ε. Then, multiplying the second

equation of (27) by u_ε and using the first one, we get (at least formly and in the case γ > 1) the following equality for allt >0

ε² Z

Ωε

ρ^γ_ε(t) γ−1+

Z _t

0

Z

Ωε

µ|∇uε|²+ξ(divu_ε)² =ε² Z

Ωε

ρ^γ_ε0 γ−1+

Z _t

0

Z

Ωε

(ρ_εf +g)·u_ε (31) while ifγ= 1, we get

ε² Z

Ωε

ρ_εlogρ_ε(t) + Z _t

0

Z

Ωε

µ|∇uε|²+ξ(divu_ε)² =ε² Z

Ωε

ρ_ε0logρ_ε0+ Z _t

0

Z

Ωε

(ρ_εf +g)·u_ε. (32) We start with the caseγ≥2, then we can estimate the right hand side

Z

Ωε

|(ρεf+g)·u_ε| ≤ε^2C_µ(||f||²_L∞||ρε||²_L2(Ωε)+||g||²_L2(Ωε)) +_2Cε^µ2||uε||²_L2(Ωε)

≤ε^2C_µ(||f||²_L∞||ρε||²_L2(Ωε)+||g||²_L2(Ωε)) +^µ₂||∇uε||²_L2(Ωε) (33) where Cis the constant appearing in (8). Hence, we deduce that for all T

ε² Z

Ωε

ρ^γ_ε(T) γ−1 +

Z _T

0

Z

Ωε

µ

2|∇u|²≤ε² Z

Ωε

ρ^γ_ε0

γ−1+Cε²

µ (||f||²_L∞||ρε||²_L2((0,T)×Ωε)+||g||²_L2((0,T)×Ωε)). (34) Then using Gronwall lemma and the fact that||ρε||²_L2(Ωε)≤C(1 +||ρε||^γ_Lγ(Ωε)), we deduce that for allT there exists a constant C_T such that

0≤t≤Tsup Z

Ωε

ρ^γ_ε(t) + Z _T

0

Z

Ωε

|∇u_ε

ε |²≤C_T. (35)

Then, from the second equation of (27), we want to deduce a uniform bound for p_ε =ρ^γ_ε in L²((0, T)×Ω_ε).

This can be done using Lemma 1.6, however to get some compactness in space, we will use another method based on the extension of ∇pε to the whole domain Ω. This is done using an extension operator which is the dual of the restriction operatorR_εdefined in Lemma 1.4. LetF_ε∈L²_T(H⁻¹(Ω)) be defined by

Z _T

0 hFε, vi_H⁻¹_,H0¹_(Ω)= Z _T

0 h∇pε, R_ε(v)i_H⁻¹_,H0¹_(Ωε) ∀v∈L²_T(H₀¹(Ω)). (36) Using that

Z _T

0 h∇p_ε, R_ε(v)i_H⁻¹_,H0¹_(Ωε)= Z _T

0 hµ∆u_ε+ξ∇ divu_ε+ρ_εf +g, R_εvi_H⁻¹_,H0¹_(Ωε)

= Z _T

0

Z

Ωε

−µ∇uε· ∇Rε(v)−ξdivu_εdivv+ (ρ_εf +g)R_εv, we obtain

Z _T

0 hFε, vi_H⁻¹_,H0¹_(Ω) ≤C

1

ε||∇uε||_L²_T(Ωε)+||f||L^∞||ρ||_L²_T(Ωε)+||g||_L²_T(Ωε)

×h

||v||_L²_T(Ω)+ε||∇v||_L²_T(Ω)i

. (37)

Hence, we deduce thatF_ε is bounded in L²_T(L²(Ω)) +εL²_T(H⁻¹(Ω)). Moreover property (ii) of Lemma 1.4 implies that there exists a P_ε∈L²_T(L²(Ω)) such that F_ε=∇P_ε and from the bound onF_ε, we get thatP_εis bounded inL²_T(H¹(Ω)) +εL²_T(L²(Ω)). A result of Lipton and Avellaneda [15] (see also Allaire [1]) shows that up to a constant, we have P_ε=pb_ε.

Now, let us concentrate on the case 1≤γ <2. Iff = 0 then we can argue exactly as above. iff 6= 0, then we have to combine (34) with the estimate based on the space-time integrability of the pressurepb_ε. Arguing as above, we deduce from (37) that

||bp_ε||_L²_T(H¹_(Ω))+εL²_T(L²_(Ω)) ≤C 1

ε||∇uε||_L²_T(Ωε)+||f||L^∞||ρ||_L²_T(Ωε)+||g||_L²_T(Ωε)

. (38)

Combining (38) with (34), we infer that

||bp_ε||_L²_T(L2(Ω)) ≤C h

1 +||f||L^∞||ρ||_L²_T(Ωε)+||g||_L²_(Ωε)i

(39) which can be rewritten

||bp_ε||_L²_T(L2(Ω))≤C_T +C||bp_ε||_L^γ+122

T(L²(Ω)). (40)

Hence, we deduce a bound for ρ^2γ ifγ >1. In the caseγ= 1, we use the smallness condition on f in the L^∞ norm to make the constantC appearing in (40) smaller than 1 and then deduce a bound forρb_εin L²_T(L²(Ω)).

In all cases, namely γ≥1, we deduce that for allT, there exists a constant C_T such that

0≤t≤Tsup Z

Ωε

ρ^γ_ε(t) + Z _T

0

Z

Ωε

ρ^2γ_ε + 1 ε²

Z _T

0

Z

Ωε

|∇uε|²≤C_T (41)

as well as the fact thatpb_εis bounded inL²_T(H¹(Ω)) +εL²_T(L²(Ω)).

2.4. Convergence proof

Using thatρe_ε and ρb_ε are bounded in L^∞(0, T;L^γ(Ω))∩L^2γ((0, T)×Ω), we can extract subsequences (still denoted ρe_εandρb_ε) such that ρb_ε converges weakly to someρwhereρ∈L^∞(0, T;L^γ(Ω))∩L^2γ((0, T)×Ω) and

e

ρ_ε converges weakly to θρ. Besides, using that ||^ue_ε^ε2||_L²_T(L²_(Ω)) ≤ ||^u_ε^ε||_L²_T(H0¹_(Ωε)), we deduce the existence of some u∈L²_T(L²(Ω)) and of a subsequence ^ue_ε2^ε which converges weakly tou.

Finally, from the bound we have on pb_ε, we can deduce the existence of somep∈L²_T(H¹(Ω)) such that p_ε converges weakly to pin L²_T(L²(Ω)). However, we can not deduce strong convergence since we do not have compactness in time. To recover some compactness in time, we will use the conservation of mass equation which provides some compactness in time for ρ_ε. We start by prolongingρ_εin a suitable way.

Lemma 2.3. the extensionρe_ε satisfies the following equation

ε²∂_tρe_ε+ div(eρ_εue_ε) = 0 in Ω. (42) Proof. In this proof,εis supposed to be fixed. For anyδsmall enough, we considerφ_δ ∈ D(Ωε) such that

0≤φ_δ ≤1, in Ω_ε, φ_δ = 1 ifd(x)≥δ

φ_δ = 0 ifd(x)≤^δ₂, |∇φδ| ≤ ^C_δ in Ω_ε (43) whered=d(x, ∂Ω_ε) andCis a constant independent ofδ(but depending onε). We also recall Hardy inequality which implies that for all ε, ^u_d^ε ∈L²((0, T)×Ω_ε) sinceu_ε∈L²(0, T;H₀¹(Ω_ε)). Now, for allψ∈ D((0, T)×Ω),