JID:MATPUR AID:2660 /FLA [m3L; v 1.125; Prn:4/02/2014; 15:06] P.1 (1-16)J. Math. Pures Appl. ••• (••••) •••–•••

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Journal de Mathématiques Pures et Appliquées

www.elsevier.com/locate/matpur

Fluid flows through fractured porous media along Beavers–Joseph interfaces

Isabelle Gruais a,∗, Dan Poliševski b

a Université de Rennes 1, I.R.M.A.R., Campus de Beaulieu, 35042 Rennes Cedex, Franceb I.M.A.R., P.O. Box 1-764, Bucharest, Romania

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 May 2013Available online xxxx

MSC:35B2776M5076S0576T99

Keywords:Fractured porous mediaStokes flowBeavers–Joseph interfaceHomogenizationTwo-scale convergence

We study a fluid flow traversing a porous medium and obeying the Darcy’s law in thecase when this medium is fractured in blocks by an ε-periodic (ε > 0) distribution offissures filled with a Stokes fluid. These two flows are coupled by a Beavers–Josephtype interface condition. The existence and uniqueness of this flow in our ε-periodicstructure are proved. As the small period of the distribution shrinks to zero, westudy the asymptotic behaviour of the flow when the permeability and the entirecontribution on the interface of the Beavers–Joseph transfer coefficients are of unityorder. We find the homogenized problem verified by the two-scale limits of thecoupled velocities and pressures. It is well-posed and provides the correspondingclassical homogenized problem.

© 2013 Published by Elsevier Masson SAS.

r é s u m é

On étudie un matériau poreux traversé par un fluide obéissant à la loi de Darcydans le cas où le matériau poreux est fracturé en blocs par un réseau périodiquede fissures emplies d’un fluide de Stokes. Ces deux écoulements sont liés parune loi de type Beavers–Joseph à l’interface. On montre l’existence et l’unicitéde cet écoulement dans la structure périodique considérée. On s’intéresse aucomportement asymptotique de l’écoulement quand la petite période tend vers zéroet la perméabilité ainsi que la contribution à l’interface des coefficients de transfertde Beavers–Joseph sont de l’ordre de l’unité. On détermine le problème homogénéisévérifié par les limites double-échelle couplées des champs de vitesses et de pression.Ce problème est bien posé et permet de retrouver le problème homogénéisé classiqueassocié.

© 2013 Published by Elsevier Masson SAS.

1. Introduction

Issues raised regarding the flow and transport in fractured rock include the requirement for furthercharacterization and construction of coupled structural and hydrogeological models. In this context,homogenization is an efficient modelling tool, yielding the effective equations to use at macroscopic level.A major branch is the so-called periodic homogenization, where the effective properties of a compositemedium are obtained by assuming the periodicity of its microstructure. Although it looks like an idealistic

* Corresponding author.

0021-7824/$ – see front matter © 2013 Published by Elsevier Masson SAS.http://dx.doi.org/10.1016/j.matpur.2013.12.002

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assumption, it usually allows to obtain the effective macroscopic behaviour in a rigorous manner. Similarly,we use periodic homogenization to model the behaviour of a fluid flow through a fractured porous mediumunder the assumption that the porous blocks are periodically distributed and the fractures are filled with aviscous Stokes fluid.

One major achievement of periodic homogenization was the mathematical justification of Darcy’s law [29],considering the Stokes flow around an ε-periodic system of isolated solid obstacles, first with formal asymp-totic expansions [16,17,27]. The first rigorous proof including the construction of a pressure extension wasdue to [29] and was followed by contextual variants [2,18,20]. We refer to [13] and references therein fordevelopments about the physics and mathematics of this subject. Although it is unrealistic, the periodicityassumption allows to cast the problem in a very simple way reflecting the actual process. It was not untilthe non-connectedness assumption could be dropped out that the homogenization of phenomena in frac-tured media could be studied (see [23], [2] and [24]). The models of fluid flows through fractured porousmedia (see [6,7,28,11,25]) are usually obtained by asymptotic methods, from the alteration of a homoge-neous porous medium by a distribution of microscopic fissures. Here, as the process at the microscopic scaletakes place under the assumption of ε-periodicity, the study of its asymptotic behaviour (when ε → 0) isamenable to the procedures of the homogenization theory. The question is how Darcy’s macroscopic lawmay be coupled with Stokes’ flow [30]. The answer is brought in the form of an original model where weassume that the volumes of both media have the same order of magnitude.

Our study is devoted to the mathematical modelisation of the filtration flow through a fractured porousmedia, namely, the coupling between the Stokes and the Darcy flows in a biphasic structure. In accordancewith observed physical laws and underlying mathematical arguments, we consider here that along theinterface of the fracture acts the Saffman variant (see [26]) of the Beavers–Joseph slip boundary condition.

We consider an incompressible viscous fluid flow in a fractured porous media represented by a periodicallystructured domain consisting of two interwoven regions, separated by an interface. The first region representsthe system of fissures which form the fracture, which is connected and where the flow is governed by theStokes system. The second region, which is also connected, stands for the system of porous blocks, whichhave a certain permeability and where the flow is governed by Darcy’s law. These two flows are coupled onthe interface by the Saffman’s variant [26] of the Beavers–Joseph condition [8,15] which was confirmed by[14] as the limit of a homogenization process. Besides the continuity of the normal component of the velocity,it imposes the proportionality of the tangential velocity with the tangential component of the viscous stresson the fluid-side of the interface. We prove here the existence and uniqueness of the solution of this modelin our ε-periodic structure.

The system is rescaled in such a way that it becomes relevant that apart ε the asymptotic behaviourdepends also on two parameters, depending on the permeability of the porous blocks and on the Beavers–Joseph transfer coefficient. As the small period of the distribution shrinks to zero, we study the asymptoticbehaviour of the flow when the permeability of the porous blocks is of unity order and the Beavers–Josephtransfer coefficient is of ε-order, balancing the measure of the interface, which is of ε−1-order.

Our main result is the identification of the homogenized problem. It is a well-posed problem, verifiedby the two-scale limits of the velocity, Darcy’s pressure and a second Stokes velocity responding for themicroscopic exchange of mass through the interface. Moreover, by eliminating the second Stokes velocity,we find the corresponding classical homogenized problem (4.3)–(4.4), see Theorem 4.1.

The paper is organized as follows.In Section 2, we present the ε-periodic structure, set the corresponding problem and prove the existence

and uniqueness of its weak solution.Section 3 is devoted to the a priori estimates, which serve as departure point for adapting the compacity

results of the two-scale convergence theory (see [4], [19] and [22]). Unlike previous works [2,13,29] relying onspecific constructions, the velocity and pressure of the fluid have natural bounded extensions in the porousmedium.

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In Section 4 we find the so-called two-scale homogenized problem (4.3)–(4.4). We prove that it is awell-posed problem. As a corollary, we get the corresponding classical homogenized problem (4.23)–(4.29).

2. The flow through the ε-periodic structure

Let Ω be an open connected bounded set in RN (N � 2), locally located on one side of the boundary ∂Ω,

a Lipschitz manifold composed of a finite number of connected components.Let Yf be a Lipschitz open connected subset of the unit cube Y = ] 0, 1[N , such that the intersections of

∂Yf with ∂Y are reproduced identically on the opposite faces of the cube and 0 /∈ Yf . The outward normalon ∂Yf is denoted by ν. Repeating Y by periodicity, we assume that the reunion of all the Yf parts, denotedby R

Nf , is a connected domain in R

N with a boundary of class C2. Defining Ys = Y \ Yf , we assume alsothat the reunion of all the Ys parts is a connected domain in R

N .For any ε ∈]0, 1[ we denote

Zε ={k ∈ Z

N , εk + εY ⊆ Ω}, (2.1)

Iε = {k ∈ Zε, εk ± εei + εY ⊆ Ω, ∀i ∈ 1, N}, (2.2)

where ei are the unit vectors of the canonical basis in RN .

Finally, we define the system of fissures by

Ωεf = int( ⋃

k∈Iε

(εk + εYf ))

(2.3)

and the porous matrix of our structure by Ωεs = Ω\Ωεf . The interface between the porous blocks and thefluid is denoted by Γε = ∂Ωεf . Its normal is:

νε(x) = ν

(x

ε

), x ∈ Γε, (2.4)

where ν has been periodically extended to RN .

Let us remark that Ωεs and Ωεf are connected and that the fracture ratio of this structure is given by

m = |Yf | ∈ ]0, 1[, as|Ωεf ||Ω| → m when ε → 0. (2.5)

To the previous structure we associate a model of fluid flow through a fractured porous medium byassuming that there is a filtration flow in Ωεs obeying the Darcy’s law and that there is a viscous flowin Ωεf governed by the Stokes system. These two flows are coupled by a Saffman’s variant [26] of theBeavers–Joseph condition [8,15]. This system is completed by an impermeability condition on ∂Ω:

div vεs = 0 in Ωεs, (2.6)

μεvεs = Kε

(gε −∇pεs

)in Ωεs, (2.7)

div vεf = 0 in Ωεf , (2.8)

σεij = −pεfδij + 2μεeij

(vεf

)in Ωεf , (2.9)

− ∂

∂xjσεij = gεi in Ωεf , (2.10)

vεs · νε = vεf · νε on Γε, (2.11)

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−pεsνεi − σεijν

εj = αεμε

(TrKε

)−1/2(vεfi −

(vεf · νε

)νεi)

on Γε, (2.12)

vεs · n = 0 on ∂Ω, n the outward normal on ∂Ω, (2.13)

where vεs, vεf and pεs, pεf stand for the corresponding velocities and pressures, με > 0 is the viscosityof the fluid, Kε is the constant positively-defined tensor of permeability, αε ∈ L∞(Ω) is the positivenon-dimensional Beavers–Joseph number, gε ∈ L2(Ω)N is the exterior force and e(v) denotes the symmetrictensor of the velocity gradient defined by

eij(v) = 12

(∂vi∂xj

+ ∂vj∂xi

).

As usual, we use the notations:

H0(div, Ω) ={v ∈ H(div, Ω), v · ν = 0 on ∂Ω

}, (2.14)

L20(Ω) =

{p ∈ L2(Ω),

∫Ω

p = 0}, (2.15)

V0(div, Ω) ={v ∈ H0(div, Ω), div v = 0 in Ω

}. (2.16)

Next, we define

Hε ={v ∈ H0(div, Ω), v ∈ H1(Ωεf )N

}, (2.17)

the Hilbert space endowed with the scalar product

(u, v)Hε=

∫Ωεs

u · v +∫

Ωεs

div u div v +∫

Ωεf

e(u) : e(v) + ε

∫Γε

(γεu−

(γενu

)νε)· γεv, (2.18)

where γε and γεν denote respectively the trace and the normal trace operators on Γε with respect to Ωεf .

Its corresponding subspace of incompressible velocities is

Vε ={v ∈ V0(div, Ω), v ∈ H1(Ωεf )N

}. (2.19)

A useful property of the present structure is the existence of a bounded extension operator similar tothat introduced in [9,10,3,1] in the case of isolated fractures.

Theorem 2.1. There exists an extension operator Pε : H1(Ωεf ) → H1(Ω) such that

Pεu = u in Ωεf , (2.20)∣∣e(Pεu)∣∣L2(Ω) � C

∣∣e(u)∣∣L2(Ωεf ), ∀u ∈ H1(Ωεf ) (2.21)

where C is independent of ε.

A straightforward consequence, via the corresponding Korn inequality, is

Lemma 2.2. There exists some constant C > 0, independent of ε, such that

|u|H1(Ωεf ) � C|u|Hε, ∀u ∈ Hε. (2.22)

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Denoting

Aε =(TrKε

)(Kε

)−1, (2.23)

βε =(TrKε

)1/2> 0 (2.24)

and using the positivity of Kε, we find that

∃a0 > 0 such that Aεijξiξj � a0|ξ|2, ∀ξ ∈ R

N . (2.25)

Rescaling the velocity by

uε ={uεs in Ωεs

uεf in Ωεf= με

(TrKε)

{vεs in Ωεs,

vεf in Ωεf(2.26)

then, for any u, v ∈ Hε and q ∈ L20(Ω), we define

aε(u, v) =∫

Ωεs

Aεu · v + β2ε

∫Ωεf

e(u) : e(v) + βε

∫Γε

αε

(γεu−

(γενu

)νε)· γεv, (2.27)

bε(q, v) = −∫Ω

q div v. (2.28)

We see that if the pair (uε, pε) is a smooth solution of the problem (2.6)–(2.13), then it is also a solution ofthe following problem: To find (uε, pε) ∈ Hε × L2

0(Ω) such that

aε(uε, v

)+ bε

(pε, v

)=

∫Ω

gε · v, ∀v ∈ Hε, (2.29)

bε(q, uε

)= 0, ∀q ∈ L2

0(Ω). (2.30)

Theorem 2.3. There exists a unique pair (uε, pε) ∈ Hε × L20(Ω) solution of (2.29)–(2.30).

Proof. As H10 (Ω) is obviously included in Hε, the following inf–sup condition is easily satisfied by bε:

∃Cε1 > 0 such that inf

q∈L20(Ω)

supv∈Hε

bε(q, v)|v|Hε

|q|L20(Ω)

� Cε1 .

The positivity conditions (2.24) and (2.25) imply that

∃Cε2 > 0 such that aε(v, v) � Cε

2 |v|2Hε, ∀v ∈ Hε,

that is the Vε-ellipticity of aε. As we also have

Vε ={v ∈ Hε, bε(q, v) = 0, ∀q ∈ L2

0(Ω)}, (2.31)

the proof is completed by Corollary 4.1, Ch. 1 of [12]. �In the rest of the paper we shall study the asymptotic behaviour (when ε → 0) of (uε, pε), the unique

solution of (2.29)–(2.30).

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From now on we assume that Aε = A where A is a constant positively-defined tensor and moreover

∃g ∈ L2(Ω)N such that gε → g strongly in L2(Ω). (2.32)

Because |Γε| is of ε−1-order, we expect that macroscopic effects of the Beavers–Joseph condition willappear only when αεβε is of ε1-order (see [21]). Therefore, we shall work under the hypothesis:

∃α ∈ C1per(Y ) and α0 > 0 such that ε−1βεαε(x) = α

(x

ε

)� α0, x ∈ Ω. (2.33)

Thus, for the study of the asymptotic behaviour it remains only the order of βε to be taken into account.In the sequel we shall study the case when βε is of unity order. Without loss of generality we can consider

from now on that

∃β > 0 such that βε = β, ∀ε > 0. (2.34)

3. A priori estimates and two-scale convergences

From now on, for any function ϕ defined on Ω × Y we shall use the notations

ϕh = ϕ|Ω×Yh, ϕh = 1

|Yh|

∫Yh

ϕ(·, y) dy, h ∈ {s, f}, (3.1)

ϕ =∫Y

ϕ(·, y) dy, that is ϕ = (1 −m)ϕs + mϕf . (3.2)

Also, for any sequence (ϕε)ε, bounded in L2(Ω × Y ), we denote

ϕε 2⇀ ϕ

iff ϕε is two-scale convergent to ϕ ∈ L2(Ω × Y ) in the sense of [4].As usual, the asymptotic study starts by the search of a priori estimations.Noticing that uε ∈ Vε and setting v = uε in (2.29) we get

∣∣uε∣∣2Hε

� C∣∣uε

∣∣L2(Ω). (3.3)

Applying (2.22) we find that

{uε}ε

is bounded in Vε and in V0(div, Ω), (3.4)∣∣uε∣∣H1(Ωεf ) � C, C being independent of ε. (3.5)

It follows that ∃u ∈ L2(Ω × Y )N such that, on some subsequence

uε 2⇀ u, (3.6)

uε ⇀

∫Y

u(·, y) dy ∈ V0(div, Ω) weakly in L2(Ω)N . (3.7)

Denoting χεf (x) = χf (xε ) and χεs(x) = χs(xε ), where χf and χs are the characteristic functions of Yf

and Ys in Y , we see that (χεsuε)ε, (χεfu

ε)ε and (χεf∂uε

)ε are bounded in (L2(Ω))N , ∀i ∈ {1, 2, . . . , N}.

∂xi

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Using the compacity result of [22], it follows that ∃ηi ∈ L2(Ω × Y )N such that, on some subsequence of(3.6)–(3.7) convergences, we have also

uε 2⇀ u, (3.8)

χεf∇uεi

2⇀ ηi. (3.9)

Denoting by

H1per(Yf ) =

{ϕ ∈ H1

loc(R

Nf

), ϕ is Y -periodic,

∫Yf

ϕ = 0}

(3.10)

we can present a first result.

Lemma 3.1. uf = uf ∈ H10 (Ω)N and there exists w ∈ L2(Ω, (H1

per(Yf ))N ) such that

ηi = χf

(∇uf

i + ∇ywi

). (3.11)

Proof. Let ψ ∈ D(Ω,C∞per(Yf )) with

∫Yf

ψ = 0 in Ω. Let us consider ϕ ∈ D(Ω,H1per(Yf )N ) satisfying

divy ϕ = ψ in Ω × Yf , (3.12)

ϕ · ν = 0 in Ω × Γ. (3.13)

Defining ϕε(x) = ϕ(x, xε ) we find that ϕε ∈ H1(Ωεf )N and γε

νϕε = 0 on Γε. As uε ∈ H1(Ωεf )N with

uε = 0 on ∂Ω ∩ ∂Ωεf it follows

∫Ωεf

uε(x)ψ(x,

x

ε

)dx = −ε

∫Ωεf

∂uε

∂xi(x)ϕi

(x,

x

ε

)dx− ε

∫Ωεf

uε(x)(divx ϕ)(x,

x

ε

)dx. (3.14)

Passing at the limit on the subsequence on which (3.6)–(3.9) hold, we find

∫Ω×Yf

u(x, y)ψ(x, y) dx dy = 0. (3.15)

It follows that ∃v ∈ L2(Ω)N such that uf = v and hence

χεfuε ⇀ muf weakly in L2(Ω)N . (3.16)

But from Lemma A.3 [5] we know that ∃v ∈ H10 (Ω) such that, by extracting a subsequence of (3.16) we have

χεfuε ⇀ mv weakly in L2(Ω)N , (3.17)

that is uf = v ∈ H10 (Ω).

It remains to prove (3.11). First, we remark that ηi|Ω×Ys= 0.

Let us introduce

V per0 (div, Yf ) =

{ϕ ∈ Hloc

(div,RN

f

), divy ϕ = 0 in R

Nf , ϕ · ν = 0 on Γ, ϕ is Y -periodic

}. (3.18)

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In the classical way (see Th. 2.7, Ch. I [12]), we find that the orthogonal space of V per0 (div, Yf ) in

L2(Ω × Y ) is:

∇H1per(Yf ) =

{∇q, q ∈ H1

per(Yf )}. (3.19)

Let ψ ∈ L2(Ω, V per0 (div, Yf )); denoting ψε(x) = ψ(x, xε ), we have

ψε ∈ H1(Ωεf )N and ψε · νε = 0 on Γε, (3.20)∫Ωεf

∇uεi (x) · ψε(x) dx = −

∫Ωεf

uεi (x)(divx ψ)

(x,

x

ε

)dx. (3.21)

Using the two-scale convergences (3.6)–(3.9) we find

∫Ω×Yf

ηi(x, y) · ψ(x, y) dx dy = −∫Ω

ufi (x) divx

(∫Yf

ψ(x, y) dy)dx

=∫

Ω×Yf

∇ufi (x) · ψ(x, y) dx dy (3.22)

which yields (ηi −∇ufi ) ∈ L2(Ω,∇H1

per(Yf )) for any i ∈ {1, . . . , N} and the proof is completed. �In the following, we sum the convergence results obtained until now.

Theorem 3.2. There exist u ∈ L2(Ω,Hloc(div,RN )) ∩ L2(Ω, V per0 (div, Yf )) with uf ∈ H1

0 (Ω) andw ∈ L2(Ω, H1

per(Yf )N ) such that the following convergences hold on some subsequence:

uε 2⇀ u, (3.23)

χεf∇uεi

2⇀ χf

(∇uf

i + ∇ywi

), ∀i. (3.24)

Moreover, we have

γnus = 0 on ∂Ω, (3.25)

(1 −m) div us + m div uf = 0 in Ω, (3.26)

divy u = 0 in Ω × Y, (3.27)

divy w + div uf = 0 in Ω × Yf , (3.28)

where γn denotes the normal trace on ∂Ω.

Proof. The convergences (3.23)–(3.24) are straight consequences of (3.8)–(3.9) and Lemma 3.1.The properties (3.25)–(3.26) follow from (3.7).

Now let ϕ ∈ D(Ω,H1per(Y )) with

ϕ = 0 in Ω × Yf . (3.29)

Defining ϕε(x) = ϕ(x, x ), we notice that ϕε ∈ D(Ω) and hence

ε

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0 = ε

∫Ω

uε∇ϕε(x) dx =∫Ω

uε(x) · (∇yϕ)(x,

x

ε

)dx + ε

∫Ω

uε(x) · (∇xϕ)(x,

x

ε

)dx. (3.30)

Using (3.23), we pass to the limit and get∫

Ω×Ys

u(x, y) · ∇yϕ(x, y) dx dy = 0. (3.31)

Hence, we have since now:

divy us = 0 in Ω × Ys and divy u

f = 0 in Ω × Yf .

Let ϕ ∈ D(Ω,H1per(Y )). Denoting

ϕε(x) = εϕ

(x,

x

ε

)for a.e. x ∈ Ω, (3.32)

we get ϕε ∈ H10 (Ω). As uε ∈ Vε, we easily obtain

∫Ω

(χεsu

ε)(x) · (∇yϕ)

(x,

x

ε

)dx +

∫Ω

(χεfu

ε)(x) · (∇yϕ)

(x,

x

ε

)dx = O(ε). (3.33)

Using (3.23) we find that∫

Ω×Ys

u(x, y) · (∇yϕ)(x, y) dx dy +∫

Ω×Yf

v(x) · (∇yϕ)(x, y) dx dy = 0, (3.34)

which implies∫

Ω×Γ

(γνv − γνu) · ϕ = 0, (3.35)

where γν denotes the normal trace on Γ ; the property (3.27) is proved.The property (3.28) can be proved similarly. �

Theorem 3.3. There exists p ∈ L20(Ω × Y ) with ps = ps ∈ H1(Ω), such that on some subsequence we have

pε2⇀ p. (3.36)

Proof. As pε ∈ L20(Ω), it follows that there exists ϕε ∈ H1

0 (Ω)N such that

divϕε = pε in Ω, (3.37)∣∣∇ϕε∣∣L2(Ω) � C

∣∣pε∣∣L2(Ω) with C independent of ε. (3.38)

For any ϕ ∈ H10 (Ω)N , we have like in [11]:

ε1/2|ϕ|L2(Γε) � C(ε|∇ϕ|L2(Ωεf ) + |ϕ|L2(Ωεf )

). (3.39)

Combining it with (2.22) we get, via (3.39):

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ε1/2∣∣γεϕε −(γενϕ

ε)νε∣∣L2(Γε)

� C∣∣∇ϕε

∣∣L2(Ωεf ) � C

∣∣pε∣∣L2(Ω). (3.40)

Then, putting v = ϕε in (2.29) we obtain immediately∣∣pε∣∣

L2(Ω) � C, for some C > 0 independent of ε. (3.41)

Using the compacity result of [22], we find that there exists p ∈ L20(Ω×Y ) such that the convergence (3.36)

holds on some subsequence.Now, choosing v ∈ Hε in (2.29) with v = 0 in Ωεf , we get

−∇pεs = Aεuε − gε in Ωεs, (3.42)

Consider the extension operator of Avellaneda (see [24]) Qεs ∈ L(L2(Ωεs), L2(Ω)) defined by:

Qεsπ ={π(x) in Ωεs,

1ε|Ys|

∫εk+εYs

π(y) dy in Ωεf .

Notice that (3.41) implies that∣∣∣∣∫

Ωεs

pεs dx

∣∣∣∣ � √|Ωεs|

∣∣pεs∣∣L2(Ωεs)

� C, (3.43)

and that (3.42) implies∣∣∇pεs

∣∣H−1(Ωεs)

� Cε. (3.44)

Theorem 3.2 of [24] yields:

∃ps� ∈ L2(Ω) such that Qεspεs → ps� in L2(Ω) and χεsp

εs 2⇀ χs(y)ps�(x) in L2(Ω × Y ).

Combining with the definition of p, we deduce that

ps(x, y) = χs(y)ps�(x) in Ω × Ys

that is ps = ps ∈ L2(Ω).Moreover, (3.20)–(3.21) of [24] reads:

Qεspεs → ps in L2(Ω)/R,

∇(Qεsp

εs)→ ∇ps in H−1(Ω). (3.45)

Noticing that

∣∣∇(Qεsp

εs)∣∣2

L2(Ω) +∣∣∇pεs

∣∣2L2(Ω) � C

we infer that (3.45) implies

∇(Qεsp

εs)→ ∇ps in L2(Ω)

that is ps = ps ∈ H1(Ω). �

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4. The two-scale homogenized problem

Now, we introduce the so-called two-scale homogenized problem, verified by the limits u, w and p, givenby Theorems 3.2–3.3. By proving that this problem is well-posed, it turns that all the convergences inTheorems 3.2–3.3 and so forth hold on the entire sequence. We conclude that the asymptotic behaviour of(uε, pε) is completely described by ((u,w), p), the unique solution of the homogenized problem.

Denoting

H ={u ∈ L2(Ω × Y ), uf = uf ∈ H1

0 (Ω)N , u ∈ H0(div, Ω), divy u = 0 in Ω × Y}, (4.1)

V = {u ∈ H, div u = 0}

we introduce the Hilbert space

X = H × L2(Ω, H1per(Yf )N

)(4.2)

endowed with the scalar product:

((u,w), (ϕ,ψ)

)X

=∫

Ω×Ys

u · ϕ +∫Ω

div u div ϕ +∫

Ω×Yf

(e(u) + ey(w)

):(e(ϕ) + ey(ψ)

),

where the components of ey(ψ) are defined by:

ey,ij(ψ) = 12

(∂ψi

∂yj+ ∂ψj

∂yi

), i, j ∈ {1, . . . , N}.

The last associated spaces are:

M ={q ∈ L2

0(Ω,L2

per(Y )), qs = qs ∈ H1(Ω)

},

X0 ={(u,w) ∈ X, div u = 0 in Ω, divy w + div uf = 0 in Ω × Yf

}.

The two-scale homogenized problem is the following:To find (v, z) ∈ X and q ∈ M such that

a((v, z), (ϕ,ψ)

)+ b

(q, (ϕ,ψ)

)=

∫Ω

g · ϕ, ∀(ϕ,ψ) ∈ X, (4.3)

b(π, (v, z)

)= 0, ∀π ∈ M (4.4)

where a and b are defined by

a((v, z), (ϕ,ψ)

)=

∫Ω×Ys

Av · ϕ + β2∫

Ω×Yf

(e(v) + ey(z)

):(e(ϕ) + ey(ψ)

)+ β

∫Ω×Γ

α(γv − (γνv)ν

)· ϕ,

b(π, (v, z)

)= −

∫Ω×Y

π divx v +∫

Ω×Γ

πs(z · ν) −∫

Ω×Yf

π divy z.

Theorem 4.1. u, w and p introduced by Theorems 3.2–3.3 form a pair ((u,w), p) ∈ X × M which verifiesthe homogenized problem (4.3)–(4.4).

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Proof. From Theorem 3.2 we see that (4.3) is readily verified.Let ϕ ∈ D(Ω,C∞

per(Y ))N , ψ ∈ D(Ω,C∞per(Yf ))N such that (ϕ,ψ) ∈ X. Let ψ a prolongation of ψ to

D(Ω, Hper(div, Y )), which can be done, for instance, by considering a certain Neumann problem in Ys.Denoting, as usual, ϕε(x) = ϕ(x, x

ε ) and ψε(x) = ψ(x, xε ), we can set v(x) = ϕε(x) + εψε(x) in (2.29).Passing to the limit with ε → 0 and using the two-scale convergences of Theorems 3.2–3.3, we obtain:

bε(pε, ϕε + εψε

)→ −

∫Ω×Ys

p(divx ϕ + divy ψ) −∫

Ω×Yf

p(divx ϕ + divy ψ) = b(p, (ϕ,ψ)

),

where we have used also ps = ps and ψs · ν = ψf · ν on Γ .Next, we obtain:

aε(uε, ϕε + εψε

)→ a

((u,w), (ϕ,ψ)

),

all the convergences being straightforward, except the one involving Γε; we present it here.Let ϕ ∈ D(Ω)N ; for any i, j ∈ {1, 2, . . . , N}, there exists Gij ∈ D(Ω,C1

per(Yf )) such that

Gij(x, y) =(ϕi(x) − ϕk(x)νk(y)νi(y)

)νj(y) on Γ. (4.5)

This follows from the C1-property on Γ of the right-hand side of (4.5) and from the smoothness of itsprolongation on ∂Yf . Thus, Gε

ij(·) = Gij(·, ·ε ) ∈ C1

0 (Ωεf ) and we have

ε

∫Γε

αε

(γεuε −

(γενu

ε)νε)· γεϕ

= ε

∫∂Ωεf

αεGεijγ

εuεiν

εj dσ

=∫

Ωεf

∂(αGij)∂yj

(x,

x

ε

)uεi (x) dx +

∫Ωεf

εβα

(x

ε

)Gij

(x,

x

ε

)∂uε

i

∂xj(x) dx

with ∣∣∣∣∫

Ωεf

εβα

(x

ε

)Gij

(x,

x

ε

)∂uε

i

∂xj(x) dx

∣∣∣∣� ε

∣∣∣∣α(·ε

)Gij

(·, ·ε

)∣∣∣∣L2(Ωεf )

∣∣∇uε∣∣L2(Ω)

� Cε|αGij |L2(Ω×Yf ) = O(ε)

which implies

limε→0

ε

∫Γε

αε

(γεuε −

(γενu

ε)νε)· γεϕ =

∫Ω

v(x) ·(∫

Yf

∂(αGij)∂yj

(x, y) dy)dx

=∫

Ω×Γ

αγv ·(γϕ− (γνϕ)ν

)(4.6)

and the proof is completed. �

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Let us introduce the so-called local solutions. Denoting

Vf ={ϕ ∈

(H1

per(Yf )/R)N

, divy ϕ = 0}, (4.7)

Kf ={ϕ ∈

(H1

per(Yf )/R)N

, divy ϕ = −1}, (4.8)

we define W kh ∈ Vf , k, h ∈ {1, 2, . . . , N} and W ∈ Kf as the unique solutions of the problems:∫Yf

(δikδjh + ey,ij

(W kh

))ey,ij(ψ) = 0, ∀ψ ∈ Vf , (4.9)

∫Yf

ey(W ) : ey(ψ) = 0, ∀ψ ∈ Vf . (4.10)

The existence and uniqueness results for (4.9) are obtained by the Lax–Milgram Theorem. In order to provethe existence and uniqueness of the solution to (4.10) we notice that W is the projection of 0 on the closedconvex Kf = ∅ in (H1

per(Yf )/R)N and the result follows.

Theorem 4.2. The problem (4.3)–(4.4) is well-posed.

Proof. Let q ∈ M ; as q ∈ L20(Ω), there exists u0 ∈ H1

0 (Ω)N with the properties

div u0 = q, (4.11)

∃c1 > 0 such that |u0|H10 (Ω)N � c1|q|L2(Ω). (4.12)

Similarly, as qf − qf ∈ L2(Ω,L20(Yf )), there exists v0 ∈ L2(Ω,H1

0 (Yf )N ) with the properties

divy v0 = qf − qf , (4.13)

∃c2 > 0 such that |v0|L2(Ω,H10 (Yf )N ) � c2

∣∣qf − qf∣∣L2(Ω×Yf ). (4.14)

Denoting w0 = (v0 + (q − qf )W ) ∈ L2(Ω,H1per(Yf )N ), where W is the local solution defined by (4.10), we

obtain

b(q, (u0, w0)

)= |q|2L2(Ω×Y ), (4.15)∣∣(u0, w0)

∣∣X

� C(|u0|H1

0 (Ω)N + |v0|L2(Ω,H10 (Yf )N ) +

∣∣q − qf∣∣L2(Ω)

)� C|q|2L2(Ω×Y ) (4.16)

and the inf–sup condition of b is obviously satisfied on M ×X.We prove now that

X0 ={(u,w) ∈ X, b

(q, (u,w)

)= 0, ∀q ∈ M

}. (4.17)

If (u,w) ∈ X0, then for any q ∈ M ,

b(q, (u,w)

)=

∫Ω×Ys

qs divx u−∫

Ω×Γ

qs(w · ν) =∫Ω

qs(1 −m) div us −∫Ω

qs∫Yf

divy w =∫Ω

qs div u = 0.

Conversely, if (u,w) ∈ X with the property that b(q, (u,w)) = 0, ∀q ∈ M , then, choosing qs = 0 it followsthat there exists C1 ∈ R such that

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div uf + divw = C1 in Ω × Yf

and consequently

(1 −m) div us −∫Γ

γνw = C1(1 −m) in Ω,

(1 −m) div us + m div uf = C1 in Ω.

Integrating this last relation on Ω we finally get C1 = 0.As we have ∫

Ω×Γ

α(γv − (γνv)ν

)γv � α0

∣∣γv − (γνv)ν∣∣2L2(Ω×Γ ) � 0

the X0-ellipticity of a in X is obvious and the proof is completed by Corollary 4.1, Ch. 1 of [12]. �The following result is a useful preliminary.

Lemma 4.3. If ((u,w), p) ∈ X is the solution of (4.3)–(4.4) then w is uniquely determined with respect touf ∈ H1

0 (Ω) by

w(x, y) = W ij(y)eij(uf

)(x) + W (y) div uf (x). (4.18)

Proof. From the characterization (4.17) of X0 we deduce that (u,w) ∈ X0. Moreover, Eq. (4.3) shows thatw is a linear combination of the family (eij(v))1�i,j�N . We deduce that there exist wij ∈ Vf , w0 ∈ Kf suchthat

w(x, y) = div(v)(x)w0(y) + eij(v)(x)wij(y) in Ω × Yf .

For every (i, ji ∈ {1, . . . , N}2, let ψij ∈ Vf , ϕ ∈ D(Ω) and set

ψ = div(ϕ)w0(y) + eij(ϕ)ψij , in (4.3),

to get: ∀(ψ�m)1��,m�N ∈ Vf ,

0 =∫

Ω×Yf

(ehk(v)

(δihδjk + ey,ij

(whk

)+ div(v)ey,ij

(w0)))e�m(ϕ)ey,ij

(ψ�m

).

There results that w0 = W and wi,j = W i,j defined by (4.9)–(4.10). �Finally, we eliminate w from (4.3)–(4.4) and find the corresponding classical homogenized problem verified

by (u, ps) ∈ H × H1(Ω):

To find (u, ps) ∈ V × H1(Ω) such that∫

Ω×Ys

Aus · ϕs +∫Ω

Be(uf

): e

(ϕf

)+

∫Ω

Cuf · ϕf =∫Ω

(g −∇ps

)· ϕ, ∀ϕ ∈ H, (4.19)

where the so-called effective coefficients which appear in (4.19) are given by

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λ =∫Yf

ey(W ) : ey(W ) > 0, (4.20)

Bijkh = β2∫Yf

(δkiδhj + ey,ij

(W kh

))+ β2λδikδjh

= β2∫Yf

(δ�kδmh + ey,�m

(W kh

))(δ�iδmj + ey,�m

(W ij

))+ λβ2δikδjh, (4.21)

Cij = β

∫Γ

α(y)(δij − νi(y)νj(y)

)dσy. (4.22)

Equivalently, the limit (u, ps) is the solution of the following direct two-scale problem:

u ∈ L2(Ω;L2per(Y )

)and u|Ω×Yf

= uf ∈ H10 (Ω)N , ps ∈ H1(Ω), (4.23)

Aus + ∇yq = g −∇ps in Ω × Ys, q ∈ L2(Ω;L2per(Ys)/R

), (4.24)

divy us = 0 in Ω × Ys, (4.25)

us · ν = uf · ν on Ω × Γ, (4.26)

γnus = 0 on ∂Ω, (4.27)

− div(Be

(uf

))+ Cuf = m

(g −∇ps

)in Ω, (4.28)

(1 −m) div(us)

+ m div(uf

)= 0 in Ω. (4.29)

Remark 4.4. The tensor Bijkh is positive-definite and has the usual symmetry properties Bijkh = Bkhij =Bjikh. Also, we have to notice that

Cij

∫Ω

ϕiϕj =∫

Ω×Γ

α(γϕ− (γνϕ)ν

)2 � 0, ∀ϕ ∈ H10 (Ω)N . (4.30)

Acknowledgements

This work has been accomplished during the visit of D. Polişevschi at the I.R.M.A.R.’s Department ofMechanics (University of Rennes 1), whose support is gratefully acknowledged.

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