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## Optimal control and homogenization in a mixture of fluids separated by a rapidly oscillating

## interface ^{∗}

### Hakima Zoubairi

Abstract

We study the limiting behaviour of the solution to optimal control problems in a mathematical mixture of two homogeneous viscous fluids.

These fluids are separated by a rapidly oscillating periodic interface with constant amplitude. We show that the limit of the optimal control is the optimal control for the limiting problem

### 1 Introduction

The aim of this paper is to study the optimal control problem in a mixture of
fluids. More precisely, we consider a mixture of two viscous, homogeneous and
incompressible fluids occupying sub-domains of a bounded domain Ω ⊂ R^{n+1}
(n = 1 or 2). These fluids are separated by a given interface whose form is
determined using a rapidly oscillating function of period ε > 0 and constant
amplitudeh_{1}>0.

We assume that the velocity and pressure of both fluids satisfy the Stokes equations. On the interface, we assume that the velocity is continuous and that the normal forces that the fluids exert each other are equal in magnitude and opposite in direction (hence, surface tension effects are neglected).

We associate an optimal control problem to these equations and our aim is to study the limiting behaviour of the solutions when the oscillating period ε tends to 0. To do so, we use some homogenization tools (see Bensoussan-Lions- Papanicolaou [4] and Sanchez-Palencia [15]) and Murat’s compactness result [11].

This work is based on the mathematical framework of Baffico & Conca [3]

for the Stokes problem, of Brizzi [5] for the transmission problem, and of Baffico

& Conca [1, 2] for the transmission problem in elasticity.

The plan of this paper is as follows. In Section 2, we present the domain with the rapidly oscillating interface, the Stokes problem posed in this domain

∗Mathematics Subject Classifications: 35B27, 35B37, 49J20, 76D07.

Key words: Optimal control, homogenization, Stokes equations.

2002 Southwest Texas State University.c

Submitted January 23, 2002. Published March 5, 2002.

1

and the definition of the associated optimal control problem. In Section 3, we introduce the adjoint problem and present related convergence results. In Section 4, we prove the results announced in the previous section. In Section 5, we sketch the proof of the convergence results concerning the optimal control.

In Section 6, we present the case Ω⊂R^{2}.

### 2 Setting of the problem

For n= 1 or 2, letY =]0,1[^{n} and Ω =]0, Le i[^{n} with Li >0,i = 1, . . . , n. Let
h: ¯Y →R, h≥0 be a smooth function such that

i) h|∂Y =h1 whereh1= max{h(y) :y∈Y¯} andh1>0.

ii) Existsy_{0}∈Y such that h(y_{0}) = 0 and∇yh(y_{0}) = 0.

Forzo∈R^{+}, define Ω⊂R^{n+1} by Ω =Ωe×]−z0, z0[ and Γ the boundary Ω by
Γ =Ωe× {−z0} ∪∂Ωe×]−z0, z0[∪Ωe× {z0}.

To define the reference cell, we introduce the sub-domains:

Ω^{1}_{1}={(y, z)∈Y ×R:h(y)< z < z_{0}}
Ω^{1}_{2}={(y, z)∈Y ×R:−z0< z < h(y)},
which are separated by the interface

Γ^{1}={(y, z)∈Y ×R:h(y) =z}.

So that, we have the decomposition of the reference cell Λ (as in figure 2)
Λ = Ω^{1}_{1}∪Γ^{1}∪Ω^{1}_{2}= (Y×]−z_{0}, z_{0}[)

When we intersect Λ with the hyperplane {Z = z}(0 < z < h_{1}) we obtain
Y × {z} and the following decomposition forY (as in figure 2)

Y =Y^{?}(z)∪γ(z)∪O(z),
where

Y^{?}(z) ={y∈Y :h(y)> z}, O(z) ={y∈Y :h(y)< z},
γ(z) ={y∈Y :h(y) =z}.

Letε >0 be a small positive parameter. We extendhbyY-periodicity toR^{n},
we restrict this function toΩ (this function is still denoted bye h). Let

h^{ε}(x) =h(x

ε) x∈Ω.e Now we introduce

Ω^{ε}_{1}={(x, z)∈Ωe×R:h^{ε}(x)< z < z0}

Figure 1: The reference cell Λ
Ω^{ε}_{2}={(x, z)∈Ωe×R:−z0< z < h^{ε}(x)}
and the rapidly oscillating interface is therefore defined by

Γ^{ε}={(x, z)∈Ωe×R:h^{ε}(x) =z}.

So that, we obtain the following decomposition of Ω (see figure 2):

Ω = Ω^{ε}_{1}∪Γ^{ε}∪Ω^{ε}_{2}.

Finally, as in figure 2, we set Ω1=Ωe×]h1, z0[, Ωm=Ωe×]0, h1[, Ω2=Ωe×]−z0,0[.

We notice that Ω = Ω1∪Ωm∪Ω2.

### The Stokes Problem

Let the viscosity of the problem defined by
µ^{ε}=µ1χΩ^{ε}_{1}+µ2χΩ^{ε}_{2},

where µ1, µ2 >0, µ16=µ2, andχΩ^{ε}_{i} correspond to the characteristic functions
of Ω^{ε}_{i} (i= 1,2).

We denote by~v = (v, vn+1), a vector of R^{n+1}. Throughout this paper, C
denotes various real positive constants independent ofε. We also denote by| · |
the n-dimensional Lebesgue measure and by (e_{k})1≤k≤n the canonical basis of
R^{n} (yk is thek-th component ofy∈R^{n} in this basis).

Figure 2: Decomposition ofY whenn= 1 (left) and whenn= 2 (rigut)

Figure 3: Rapidly oscillating interface (left) and its homogenized version (right)

We define the optimal control as follows. LetUad⊂L^{2}(Ω)^{n+1} be a closed
non empty convex subset. LetN >0 be a given constant. For~θ∈ Uadthe state
equation is given by the Stokes problem

−div(2µ^{ε}e(~u^{ε})) =f~^{ε}− ∇p^{ε}+θ~ in Ω
div~u^{ε}= 0 in Ω

~

u^{ε}= 0 on∂Ω.

(2.1)

where (~u^{ε}, p^{ε}) are respectively the velocity and the pressure of the fluid,~θis the
control and f~^{ε} is a density of external forces defined by f~^{ε} =f~^{1}χΩ^{ε}_{1}+f~^{2}χΩ^{ε}_{2}

withf~^{i}∈L^{2}(Ω)^{n+1}(i= 1,2). The rate-of-strain tensore(~u^{ε}) is
e(~u^{ε}) = 1

2 ∇~u^{ε}+^{t}∇~u^{ε}
.
The cost function is

J_{ε}(~θ) = 1
2
Z

Ω

e(~u^{ε}) :e(~u^{ε})dx+N
2

Z

Ω

|~θ|^{2}dx. (2.2)
The optimal control~θ^{ε}_{?}is the function inUadwhich minimizesJε(~θ) for~θ∈ Uad,
in other words

Jε(~θ^{ε}_{?}) = min

~θ∈Uad

Jε(~θ). (2.3)

This problem is standard and admits an unique optimal solution~θ_{?}^{ε}∈ Uad (see
Lions [10]). Our aim is to study the limiting behaviour of ~θ_{?}^{ε} as ε → 0. In
particular, it can be shown that (for a subsequence)

~θ_{?}^{ε}* ~θ? weakly inL^{2}(Ω)^{n+1}.

Our objective is to characterizeθ~?as the optimal control of a similar problem with limiting tensorsAandBand to identify these tensors. The homogenization of the problem (2.1) is thanks to Baffico & Conca [3]. About the optimal control, Saint Jean Paulin & Zoubairi [12] studied the problem of a mixture of two fluids periodically distributed one in the other. Also this type of problem has been studied by Kesavan & Vanninathan [9] in the periodic case for a problem which the state equation is a second order elliptic problem with rapidly oscillating coefficients and by Kesavan & Saint Jean Paulin [7] and [8] in the general case (with H-convergence).

In this paper, we adapt these methods to the Stokes problem following the technique used by Baffico & Conca [2] and [3], by Kesavan & Saint Jean Paulin [7] and [8], and by Saint Jean Paulin & Zoubairi [12].

Definition We defineL^{2}_{0}(Ω) by
L^{2}_{0}(Ω) =

g∈L^{2}(Ω) :
Z

Ω

g(y)dy= 0 .

The problem (2.1)–(2.3) can be reduced to a system of equations by introducing
the adjoint state (~v^{ε}, p^{0}^{ε})∈H^{1}(Ω)^{n+1}×L^{2}_{0}(Ω). Thus we get

−div(2µ^{ε}e(~u^{ε})) =f~^{ε}− ∇p^{ε}+~θ in Ω
div(2µ^{ε}e(~v^{ε})−e(~u^{ε})) =−∇p^{0}^{ε} in Ω

div~u^{ε}= div~v^{ε}= 0 in Ω

~

u^{ε}=~v^{ε}= 0 on∂Ω,

(2.4)

the optimal control~θ^{ε}_{?} is characterized by the variationnal inequality

~θ^{ε}_{?}∈ Uad and
Z

Ω

(~v^{ε}+N ~θ^{ε}_{?}).(~θ−~θ_{?}^{ε})dx≥0∀~θ∈ Uad.

### 3 Convergence results

### The homogenized adjoint problem

Letµ=µ(y, z) (wherey∈Y andz∈]0, h_{1}[), be the variable viscosity given by
µ(y, z) =µ1χ_{O(z)}(y) +µ2χ_{Y}?(z)(y).

Let us introduce some functions as solution of the Stokes problem defined onY. These functions, introduced by Baffico & Conca [3], are associated to the state equation.

Let 1≤k, l≤n, and let (χ^{kl}, r^{kl}_{1}) be the solution of

−div

y (2µe_{y}(−χ^{kl}+P^{kl})) =−∇yr^{kl}_{1} in Y
divy χ^{kl}= 0 in Y

χ^{kl}, r_{1}^{kl} Y-periodic,

(3.1)

whereP^{kl}= ^{1}_{2} yke_{l}+yle_{k}

. We defineM^{kl}then×nmatrix byM^{kl}=ey(P^{kl}).

We notice that [M^{kl}]ij = ^{1}_{2}(δikδjl+δilδjk) for all 1 ≤ i, j, k, l ≤ n. We also
define then+ 1×n+ 1 matrix

[E^{kl}]ij =1

2(δikδjl+δilδjk) ∀1≤i, j, k, l≤n+ 1.

For eachz∈]0, h_{1}[, problem (3.1) admits an unique solution in H_{]}^{1}(Y)^{n}/R

×
L^{2}_{0}(Y) (see Sanchez-Palencia [15] or Baffico & Conca [6]).

Now, we consider the periodic problem

−div

y (µ∇y(−ϕ^{k}+ 2y_{k})) = 0 in Y
ϕ^{k} Y-periodic.

(3.2)
For eachz∈]0, h1[ fixed, problem (3.2) has an unique solution inH^{1}(Y) up to
an additive constant.

LetA(z) be the fourth-order tensor whose coefficients are defined by

a_{ijkl}=

2µ_{1}[E^{kl}]_{ij} h_{1}< z < z_{0}
agijkl 0< z < h1

2µ_{2}[E^{kl}]_{ij} −z_{0}< z < h_{1}

1≤i, j, k, l≤n+ 1 (3.3)

with

ag_{ijkl} =

R

Y 2µ

e_{y}(−χ^{kl}+P^{kl})

ijdy 1≤i, j, k, l≤n

1 2

R

Y µ^{∂(}^{−}^{ϕ}_{∂y}^{k}^{+2y}^{k}^{)}

i dy 1≤i, k≤n j, l=n+ 1

1 2

R

Y µ^{∂(}^{−}^{ϕ}_{∂y}^{l}^{+2y}^{l}^{)}

i dy 1≤i, l≤n j, k=n+ 1

1 2

R

Y µ^{∂(}^{−}^{ϕ}_{∂y}^{k}^{+2y}^{k}^{)}

j dy 1≤j, k≤n i, l=n+ 1

1 2

R

Y µ^{∂(}^{−}^{ϕ}_{∂y}^{l}^{+2y}^{l}^{)}

j dy 1≤j, l≤n i, k=n+ 1

1 2

R

Y 2µdy i, j, k, l=n+ 1

0 otherwise.

(3.4)

This tensor (introduced by Baffico & Conca [3]) is the homogenized tensor associated to the state equation. By the same way, we introduce other test functions which will be associated to the adjoint state.

Let (ψ^{kl}, r_{2}^{kl}) be the solution of

−div

y (2µe_{y}(ψ^{kl}) +e_{y}(−χ^{kl}+P^{kl})) =−∇yr^{kl}_{2} inY
divy ψ^{kl}= 0 inY

ψ^{kl}, r_{2}^{kl} Y-periodic,

(3.5)

For each z ∈]0, h_{1}[, problem (3.5) has an unique solution in H_{]}^{1}(Y)^{n}/R

×
L^{2}_{0}(Y).

As we did for the problem (3.2), we introduce the scalar problem

−div

y (2µ∇yψ^{k}+∇y(−ϕ^{k}+ 2yk)) = 0 in Y
ψ^{k} Y-periodic.

(3.6)

Forz∈]0, h_{1}[ fixed, this problem admits an unique solution inH^{1}(Y) up to an
additive constant.

LetB(z) be the fourth order tensor which coefficients are given by

bijkl=

[E^{kl}]ij h1< z < z0

bg_{ijkl} 0< z < h_{1}
[E^{kl}]ij −z0< z < h1

1≤i, j, k, l≤n+ 1 (3.7)

with

bgijkl=

R

Y 2µ[e_{y}(ψ^{kl})]_{ij}+ [e_{y}(−χ^{kl}+P^{kl})]_{ij}

dy 1≤i, j, k, l≤n

1 4

R

Y 2µ^{∂ψ}_{∂y}^{k}

i +^{∂(}^{−}^{ϕ}_{∂y}^{k}^{+2y}^{k}^{)}

i

dy 1≤i, k≤n j, l=n+ 1

1 4

R

Y 2µ^{∂ψ}_{∂y}^{l}

i +^{∂(}^{−}^{ϕ}_{∂y}^{l}^{+2y}^{l}^{)}

i

dy 1≤i, l≤n j, k=n+ 1

1 4

R

Y 2µ^{∂ψ}_{∂y}^{k}

j +^{∂(}^{−}^{ϕ}_{∂y}^{k}^{+2y}^{k}^{)}

j

dy 1≤j, k≤n i, l=n+ 1

1 4

R

Y 2µ^{∂ψ}_{∂y}^{l}

j +^{∂(}^{−}^{ϕ}_{∂y}^{l}^{+2y}^{l}^{)}

j

dy 1≤j, l≤n i, k=n+ 1

1 i, j, k, l=n+ 1

0 otherwise.

(3.8) Now we give a result concerning some properties of tensor A.

Proposition 3.1 (Baffico & Conca [3]) The coefficients of A in (3.3) sat- isfy:

a)aijkl(z) =aklij(z) =aijlk(z) ∀1≤i, j, k, l≤n+ 1, ∀z∈]−z0, z0[ b) there existsα >0 such that for all ξ,n+ 1×n+ 1 symmetric matrix,

A(z)ξ:ξ≥αξ:ξ ∀z∈]−z0, z0[.

Now we give a result concerning some symmetry and ellipticity properties of the tensorB.

Proposition 3.2 The coefficients of B(voir(3.7)) are such that:

a)b_{ijkl}(z) =b_{klij}(z) =b_{ijlk}(z)for1≤i, j, k, l≤n+ 1, for allz∈]−z_{0}, z_{0}[
b) There existsβ >0 such that for all ξ, the n+ 1×n+ 1 symmetric matrix,

B(z)ξ:ξ≥βξ:ξ ∀z∈]−z0, z0[.

Proof. Throughout this proof, we adopt the convention of summation over repeated indices. To prove a), we first study the coefficients of tensorB with indexes 1 ≤ i, j, k, l ≤n. The symmetry of these coefficients is evident when z∈]h1, z0[ and z∈]−z0, h1[.

Let study the case wherez∈]0, h_{1}[. In this case (cf (3.8))
bijkl=bgijkl =

Z

Y

2µ

ey(ψ^{kl})

ij+

ey(−χ^{kl}+P^{kl})

ij

dy. (3.9)
Following the ideas in [7, 12, 14], we transform the above expression to obtain
a symmetric form. Let (Y^{kl}, r^{kl}_{3}) be the solution of

−div

y (2µe_{y}(−Y^{kl}+P^{kl})) =−∇yr_{3}^{kl} inY
divy Y^{kl}= 0 inY

Y^{kl}, r^{kl}_{1} Y-periodic,

(3.10)

IntroducingY^{kl}the solution of the previous local problem, the coefficients (3.9)
can be rewritten as

bg_{ijkl}=
Z

Y

e_{y}(−Y^{kl}+P^{kl})

ijdy+ Z

Y

2µ

e_{y}(ψ^{kl})

ij−

e_{y}(χ^{kl}−Y^{kl})

ij

dy.

(3.11)
The first term of the second integral of the right-hand side of this equation is
evaluated as follows (using the fact that [e_{y}(ψ^{kl})

ij= [e_{y}(ψ^{kl})

ji) Z

Y

2µ

ey(ψ^{kl})

ijdy= Z

Y

2µ

ey(ψ^{kl})

βmδβiδmjdy

= Z

Y

µ

ey(ψ^{kl})

βm(δβiδmj+δβjδmi)dy

= Z

Y

2µ

ey(ψ^{kl})

βm

ey(P^{ij})

βmdy.

Using successively (3.1),(3.5) and (3.10), we have Z

Y

2µ

ey(ψ^{kl})

βm

ey(P^{ij})

βmdy= Z

Y

2µ

ey(ψ^{kl})

βm

ey(χ^{ij})

βmdy

= Z

Y

ey(χ^{kl}−P^{kl})

βm

ey(χ^{ij})

βmdy

= Z

Y

ey(χ^{kl}−Y^{kl})

βm

ey(χ^{ij})

βmdy.

Moreover using (3.10), we can rewrite the last integral as Z

Y

e_{y}(χ^{kl}−Y^{kl})

βm

e_{y}(χ^{ij})

βmdy

= Z

Y

e_{y}(χ^{kl}−Y^{kl})

βm

e_{y}(χ^{ij}−Y^{ij})

βmdy +

Z

Y

ey(χ^{kl}−Y^{kl})

βm

ey(Y^{ij})

βmdy

= Z

Y

ey(χ^{kl}−Y^{kl})

βm

ey(χ^{ij}−Y^{ij})

βmdy +

Z

Y

ey(χ^{kl}−Y^{kl})

βm

ey(P^{ij})

βmdy

= Z

Y

ey(χ^{kl}−Y^{kl})

βm

ey(χ^{ij}−Y^{ij})

βmdy+ Z

Y

ey(χ^{kl}−Y^{kl})

ijdy Thus the second integral of the right-hand side of (3.11) can be rewritten as

Z

Y

2µ

ey(ψ^{kl})

ij−

ey(χ^{kl}−Y^{kl})

ij

dy

= Z

Y

ey(χ^{kl}−Y^{kl})

βm

ey(χ^{ij}−Y^{ij})

βmdy.

Let us now consider the first integral in (3.11). Multiplying the first equation
of (3.10) byY^{ij} and integrating by parts we have

Z

Y

ey(−Y^{kl}+P^{kl})

βm

ey(Y^{ij})

βmdy= 0.

Using the fact that

ey(P^{ij})

βm=^{1}_{2} δβiδmj+δβjδmi

, we obtain Z

Y

ey(−Y^{kl}+P^{kl})

βm

ey(−Y^{ij}+P^{ij})

βmdy

= Z

Y

ey(−Y^{kl}+P^{kl})

βm

ey(P^{ij})

βmdy

= Z

Y

ey(−Y^{kl}+P^{kl})

ijdy.

Then using definition (3.11), we derive
bg_{ijkl} =

Z

Y

e_{y}(−Y^{kl}+P^{kl}) :e_{y}(−Y^{ij}+P^{ij})dy+

Z

Y

e_{y}(χ^{kl}−Y^{kl}) :e_{y}(χ^{ij}−Y^{ij})dy.

(3.12)
It is immediate from the above form that the coefficients ofBsatisfybgijkl=bgklij.
On the other hand, sincee_{y}(P^{kl}) =e_{y}(P^{lk}) then by uniqueness of problem (3.1),
we haveχ^{kl}=χ^{lk} (up to an additive constant) and thenb_{ijkl}=b_{ijlk}.

We now study the coefficientsbijklwithi=k=n+1 and 1≤j, l≤n. From the definition of B (cf (3.7)), these coefficients are symmetric whenz ∈]h1, z0[ and z ∈]−z0, h1[. To prove the symmetry when z ∈]0, h1[, we proceed as we did before. These coefficients are as follows

b_{n+1jn+1l}^ =1
4

Z

Y

∂(−ϕ^{l}+ 2y_{l})

∂yj

+ 2µ∂ψ^{l}

∂yj

dy. (3.13)

Letτ^{k} be the solution of

−∆y(−τ^{k}+ 2yk) = 0 inY

τ^{k} Y-p´eriodic. (3.14)

The expression (3.13) can be rewritten as follows (usingτ^{l}):

bn+1jn+1l^ =1 4

Z

Y

∂(−τ^{l}+ 2yl)

∂yj

dy+1 4

Z

Y

2µ∂ψ^{l}

∂yj −∂(ϕ^{l}−τ^{l})

∂yj

dy. (3.15) Using exactly the same technique used above, we obtain (using (3.2), (3.6) and (3.14))

Z

Y

2µ∂ψ^{l}

∂yj

=1 2

Z

Y

∂(ϕ^{l}−τ^{l})

∂yk

∂(ϕ^{j}−τ^{j})

∂yk

dy+ Z

Y

∂(ϕ^{l}−τ^{l})

∂yj

dy.

Therefore, Z

Y

2µ∂ψ^{l}

∂yj −∂(ϕ^{l}−τ^{l})

∂yj

dy=1 2

Z

Y

∂(ϕ^{l}−τ^{l})

∂yk

∂(ϕ^{j}−τ^{j})

∂yk

dy.

Concerning the first integral in (3.15), we consider τ^{l} the solution of (3.14):

multiplying the first equation byτ^{j} and integrating by parts, we have
Z

Y

∂(−τ^{l}+ 2yl)

∂y_{k}

∂τ^{j}

∂y_{k}dy= 0,
so we derive

Z

Y

∂(−τ^{l}+ 2y_{l})

∂yk

∂(−τ^{j}+ 2y_{j})

∂yk

dy= 2 Z

Y

∂(−τ^{l}+ 2y_{l})

∂yj

dy.

Finally, we obtain the following expression
b_{n+1jn+1l}^ = 1

8 Z

Y

∇(−τ^{l}+ 2y_{l}).∇(−τ^{j}+ 2y_{j})dy+1
8

Z

Y

∇(ϕ^{l}−τ^{l}).∇(ϕ^{j}−τ^{j})dy.

(3.16) From the form of (3.16), it is evident thatbn+1jn+1l^ =bn+1ln+1j^ . By construc- tion ofB, we also havebn+1ln+1j^ =bn+1ljn+1^ . For the other nonzero terms, the same method can be used to obtain

bin+1kn+1^ =bkn+1in+1^ =bin+1n+1k^
bin+1n+1l^ =bin+1ln+1^ =bn+1lin+1^
b_{n+1jkn+1}^ =b_{n+1jn+1k}^ =b_{kn+1n+1j}^ .

To prove part b), we first notice that the coerciveness ofBwhenz∈]h_{1}, z_{0}[
andz∈]−z0, h1[ is evident. Whenz∈]0, h1[, from the form ofbgijkl1≤i, j, k, l≤
n(see (3.12)) and the form ofb_{n+1jn+1l}^ 1≤j, l≤n(see (3.16), we have thatB

is elliptic.

Now we introduce the homogenized problem. Let (~u, p) and (~v, p^{0}) be in
H^{1}(Ω)^{n+1}×L^{2}_{0}(Ω)^{2}

and be the solution of

−div Ae(~u)

=f~− ∇p+~θ in Ω div Ae(~v)− Be(~u)

=−∇p^{0} in Ω
div~u= div~v= 0 in Ω

~

u=~v= 0 on∂Ω,

(3.17)

where f~is the weak limit off~^{ε} inL^{2}(Ω)^{n+1} and we precise it later, (4.10).

Remark 3.3 Since Propositions 3.1 and 3.2 hold, problem (3.17) admits an unique solution.

Now we state the main result of this paper, and we will prove it in the next section.

Theorem 3.3 Under some regularity hypotheses concerning the solutions of
(3.1), (3.2), (3.5), and (3.6) (detailed in Section 4) the solutions (~u^{ε}, p^{ε}) and
(~v^{ε}, p^{0}^{ε}) of (2.4) are such that~u^{ε} * ~u weakly inH^{1}(Ω)^{n+1},~v^{ε} * ~v weakly in
H^{1}(Ω)^{n+1},p^{ε}* pweakly inL^{2}_{0}(Ω),p^{0}^{ε}* p^{0} weakly inL^{2}_{0}(Ω), where(~u, p)and
(~v, p^{0})are the unique solutions of (3.17).

### 4 Proof of the convergence result

A priori estimates Let

ξ^{ε}= 2µ^{ε}e(~u^{ε}), (4.1)

q^{ε}= 2µ^{ε}e(~v^{ε})−e(~u^{ε}). (4.2)
Proposition 4.1 The sequences (~u^{ε}, p^{ε}),(~v^{ε}, p^{0}^{ε}),ξ^{ε} andq^{ε} are such that (up
to subsequences)

~

u^{ε}* ~uweakly inH^{1}(Ω)^{n+1} ~v^{ε}* ~v weakly in H^{1}(Ω)^{n+1}
p^{ε}* pweakly inL^{2}_{0}(Ω) p^{0}^{ε}* p^{0} weakly inL^{2}_{0}(Ω)

ξ^{ε}* ξ weakly inL^{2}(Ω)^{n+1}^{×}^{n+1} q^{ε}* qweakly in L^{2}(Ω)^{n+1}^{×}^{n+1}.
Proof. Using~u^{ε}as a test function in the first equation of (2.4), we can easily
see that there exists a constantC >0 independent of εsuch that

k~u^{ε}kH^{1}(Ω)^{n+1} ≤C; (4.3)
therefore, we have for a subsequence (still denoted byε)

~

u^{ε}* ~uweakly inH^{1}(Ω)^{n+1}. (4.4)
Similarly, multiplying the second equation of (2.4) by~v^{ε}, integrating by parts
and using (4.3), we obtain

k~v^{ε}kH^{1}(Ω)^{n+1} ≤C,
so we have (for a subsequence)

~

v^{ε}* ~vweakly inH^{1}(Ω)^{n+1}. (4.5)
Now sincekdiv (2µ^{ε}e(~u^{ε}))kH^{−1}(Ω)^{n+1}is bounded, we havek∇p^{ε}kH^{−1}(Ω)^{n+1} ≤C,
this implies (see Temam [16])|p^{ε}|L^{2}_{0}(Ω)≤C, we derive (for a subsequence),

p^{ε}* pweakly inL^{2}_{0}(Ω).

Also by similar arguments, we get

p^{0}^{ε}* p^{0} weakly inL^{2}_{0}(Ω).

The boundedness of kξ^{ε}kL^{2}(Ω)^{(n+1)2} provides from (4.3) and we derive by ex-
traction of subsequences

ξ^{ε}* ξweakly inL^{2}(Ω)^{(n+1)}^{2}.

Similarly, the boundedness ofkq^{ε}k_{L}2(Ω)^{(n+1)2} provides from (4.3) and (4.1.6), so
we can extract a subsequence such that

q^{ε}* q weakly inL^{2}(Ω)^{(n+1)}^{2}. (4.6)
Since (~u^{ε}, p^{ε}) and (~v^{ε}, p^{0}^{ε}) are solutions of (2.4) and since Proposition 4.1
holds, we obtain that (~u, p),(~v, p^{0}), ξ andqsatisfy in the distribution sense

−div ξ

=f~− ∇p+~θ in Ω div q

=−∇p^{0} in Ω
div~u= div~v= 0 in Ω

~

u=~v= 0 on∂Ω,

(4.7)

where f~ is the weak limit of f~^{ε} in L^{2}(Ω)^{n+1}. This limit can be identified
explicitly (cf [3] or [5]). Indeed, the characteristic functions χΩ^{ε}_{1} (i= 1,2) are
such that

χ_{Ω}^{ε}

i * ρandχ_{Ω}^{ε}

2 *(1−ρ) weakly ? in L^{∞}(Ω), (4.8)
where

ρ(x, z) =

1 in Ω_{1}

|O(z)|

|Y| in Ωm

0 in Ω2.

(4.9) Then we have

f~^{ε}* ~f =f~^{1}ρ+f~^{2}(1−ρ) weakly inL^{2}(Ω)^{n+1}. (4.10)
Proposition 4.2 (Baffico & Conca [3]) Under the hypotheses (4.12), (4.15)
and (4.28) (introduced in the next subsections), ξ=Ae(~u), where A is defined
by (3.4).

To prove Theorem 3.3, we have to show thatq, ~uand~v are related by

q=Ae(~v)− Be(~u). (4.11)

Using the same method as Baffico & Conca[3], we show that the identifi- cation of q is carried out in Ω1,Ωmand Ω2 independently. In Ω1 and Ω2, this identification will pose no particular problem. In Ωm, following the ideas of Baffico & Conca ([3]), there is three steps: we first identify the components [q]ij

of q for 1≤ i, j ≤ n, and then [q]n+1j for 1 ≤ j ≤ n and finally we identify [q]n+1n+1. To do so, we use some suitable test functions and the energy method ( cf Bensoussan, Lions & Papanicolaou [4] or Sanchez-Palencia [15]).

### Identification of [q]

_{ij}

### 1 ≤ i, j ≤ n in Ω

_{m}

In what follows, we construct at first the test functions which allow us the
identification of [q]_{ij}, then we introduce the regularity conditions that these
functions must satisfy and finally we establish the identification.

Letw^{kl}=−χ^{kl}+P^{kl} and σ^{kl} = 2µe_{y}(w^{kl}) where (χ^{kl}, r^{kl}_{1}) be the solution
of (3.1). We assume that (χ^{kl}, r_{1}^{kl}), as function of (y, z)∈ Y×]0, h_{1}[, satisfies
the regularity hypothesis

a) χ^{kl}∈L^{2}loc(0, h_{1}, H_{]}^{1}(Y)^{n})∩(L^{2}loc(]0, h_{1}[×R^{n}))^{n}
b) ∂

∂z (χ^{kl})_{i}

∈L^{2}loc(0, h_{1}, L^{2}_{]}(Y)^{n})∩L^{2}loc(]0, h_{1}[×R)

1≤i, j≤n (4.12)
We define the following functions by extension byY-periodicity toR^{n+1}and by
restriction to Ω_{m}:

w^{ε,kl}(x, z) =εw^{kl}(x
ε, z)
r^{ε,kl}_{1} (x, z) =r^{kl}_{1}(x

ε, z)
σ^{ε,kl}(x, z) =σ^{kl}(x

ε, z)

(4.13)

It is easy to check that

divx (σ^{ε,kl}) =−∇xr_{1}^{ε,kl} in Ω_{m}
divx (w^{ε,kl}) = div

x (P^{kl}) =δ_{kl} in Ω_{m}. (4.14)
We also need the Murat’s compactness result [11].

Lemma 4.3 If the sequence(gn)n belongs to a bounded subset ofW^{−}^{1,p}(Ω)for
some p >2, and(gn)n ≥0 in the following sense i.e., for allφ ∈ D(Ω) such
that φ ≥ 0 then for all n > 0hgn, φi ≥ 0. Then (gn)n belongs to a compact
subset ofH^{−}^{1}(Ω).

If we suppose thatr^{ε,kl}_{1} satisfy

a) r^{ε,kl}_{1} ∈L^{p}loc(Ωm) for somep >2, locally bounded
b) ∂

∂z r_{1}^{ε,kl}

≥0 in distribution sense,

(4.15) Then using Lemma 4.3 and hypothesis (4.12), we have the following result.

Proposition 4.4 (Baffico & Conca [3]) If (4.12) and (4.15) hold. Then for
allΩ^{0} ⊂⊂Ωm, we have the following convergence

a)w^{ε,kl}* P^{kl} weakly in H^{1}(Ω^{0})^{n}
b) _{∂z}^{∂} (w^{ε,kl})i

→0strongly in L^{2}(Ω^{0})^{n},1≤i≤n
c) r_{1}^{ε,kl}→0weakly in L^{2}(Ω^{0})^{n}

d) _{∂z}^{∂} r^{ε,kl}_{1}

→0 strongly inH^{−}^{1}(Ω^{0})^{n},1≤i≤n
e) σ^{ε,kl}* σ^{kl}=m_{Y}(2µe_{y}(w^{kl})weakly inL^{2}(Ω^{0})^{n}^{×}^{n}.

(4.16)

In the same way, we assume that the solution (ψ^{kl}, r^{kl}_{2}) of (3.5) satisfies the
following convergence hypothesis:

a) ψ^{kl}∈L^{2}loc(0, h_{1}, H_{]}^{1}(Y)^{n})∩(L^{2}loc(]0, h_{1}[×R^{n}))^{n}
b) ∂

∂z (ψ^{kl})_{i}

∈L^{2}loc(0, h_{1}, L^{2}_{]}(Y)^{n})∩L^{2}loc(]0, h_{1}[×R)

1≤i, j≤n (4.17) We define

ψ^{ε,kl}(x, z) =εψ^{kl}(x

ε, z), and r^{ε,kl}_{2} (x, z) =r^{kl}_{2}(x

ε, z) (4.18) so we obtain

−div

x 2µ^{ε}e_{x}(ψ^{ε,kl}) +e_{x}(w^{ε,kl})

=−∇xr^{ε,kl}_{2} in Ω_{m}
divx ψ^{ε,kl}= 0 in Ωm

(4.19)

If we suppose thatr^{ε,kl}_{2} satisfy

a) r_{2}^{ε,kl}∈L^{p}loc(Ωm) for somep >2, locally bounded
b) ∂

∂z r^{ε,kl}_{2}

≥0 in distribution sense,

(4.20)

then using Lemma 4.3, we derive

∂

∂z r^{ε,kl}_{2}

→0 strongly inH^{−}^{1}(Ω^{0})^{n}.1≤i≤n
We have the following result concerning these functions

Proposition 4.5 If (4.12) and (4.15) hold. Then for allΩ^{0}⊂⊂Ωm, we have
a) ψ^{ε,kl}*0 weakly inH^{1}(Ω^{0})^{n}

b) ∂

∂z (ψ^{ε,kl})i

→0 strongly inL^{2}(Ω^{0})^{n}, 1≤i≤n
c) r^{ε,kl}_{2} *0 weakly inL^{2}(Ω^{0})^{n}.

(4.21)

To prove this proposition, we use well-known results concerning the convergence of periodic functions.

We shall prove now the principal result of this section

Proposition 4.6 If (4.12), (4.15), (4.17), and (4.20) hold, then[q^{ε}]kl *[q]kl

weakly in L^{2}(Ωm) (up to a subsequence) for all1≤k, l≤nwhere
[q]kl= 1

|Y|

n

X

i,j=1

Z

Y

2µ[ey(−χ^{ij}+P^{ij})]kldy e(~v)

ij

− 1

|Y|

n

X

i,j=1

Z

Y

2µ[ey(−ψ^{ij})]kl+ [ey(−χ^{ij}+P^{ij})]kl

dy e(~u)

ij

Proof. Letφ∈ D(Ωm) andw~^{ε,kl}= (w^{ε,kl},0). Multiplying the second equation
of (2.4) byφ ~w^{ε,kl}, integrating by parts and using (4.2), we obtain

− Z

Ω_{m}

∇p^{0}^{ε}.(φ ~w^{ε,kl})dxdz

=− Z

Ωm

(q^{ε}∇φ). ~w^{ε,kl}dxdz+
Z

Ωm

e(~u^{ε}) :∇w~^{ε,kl}φdxdz

− Z

Ω_{m}

2µ^{ε}e(~v^{ε}) :∇w~^{ε,kl}φdxdz.

Developing the second and third integral of the right-hand side of the above equation,

− Z

Ω_{m}

∇p^{0}^{ε}.φ ~w^{ε,kl}dxdz

=− Z

Ωm

(q^{ε}∇φ). ~w^{ε,kl}dxdz+
Z

Ωm

ex(u^{ε}) :ex(w^{ε,kl})φdxdz

− Z

Ωm

2µ^{ε}ex(v^{ε}) :ex(w^{ε,kl})φdxdz−
Z

Ωm

n

X

j=1

[q^{ε}]n+1j

∂

∂z (w^{ε,kl})j
φdxdz.

(4.22)
Letψ~^{ε,kl}= (ψ^{ε,kl},0). Multiplying the first equation of (2.4) byφ ~ψ^{ε,kl}, integrat-
ing by parts and using definition (4.1), we obtain (after algebraic developments)

Z

Ω_{m}

(f~^{ε}− ∇p^{ε}+~θ).φ ~ψ^{ε,kl}dxdz

= Z

Ωm

(ξ^{ε}∇φ). ~ψ^{ε,kl}+
Z

Ωm

2µ^{ε}ex(u^{ε}) :ex(ψ^{ε,kl})φdxdz
+

Z

Ω_{m}
n

X

j=1

[ξ^{ε}]n+1j

∂

∂z (ψ^{ε,kl})j

φdxdz.

Integrating by parts now the second integral of the right-hand side, we have Z

Ωm

(f~^{ε}− ∇p^{ε}+~θ).φ ~ψ^{ε,kl}dxdz

= Z

Ω_{m}

(ξ^{ε}∇φ). ~ψ^{ε,kl}−
Z

Ω_{m}

divx 2µ^{ε}e_{x}(ψ^{ε,kl})

.(u^{ε}φ)dxdz

− Z

Ωm

(2µ^{ε}e_{x}(ψ^{ε,kl})∇xφ).u^{ε}dxdz+
Z

Ωm

n

X

j=1

[ξ^{ε}]_{n+1j} ∂

∂z (ψ^{ε,kl})_{j}
φdxdz.

Using the equation thatψ^{ε,kl} satisfy (see (4.19)),
Z

Ωm

(f~^{ε}− ∇p^{ε}+~θ).φ ~ψ^{ε,kl}dxdz

= Z

Ωm

(ξ^{ε}∇φ). ~ψ^{ε,kl}−
Z

Ωm

u^{ε}div

x e_{x}(w^{ε,kl})

φdxdz+ Z

Ωm

∇xr_{2}^{ε,kl}.(φu^{ε})dxdz

− Z

Ωm

2µ^{ε}ex(ψ^{ε,kl})∇xφ

.u^{ε}dxdz+
Z

Ωm

n

X

j=1

[ξ^{ε}]n+1j

∂

∂z (ψ^{ε,kl})j
φdxdz.

Integrating again by parts the second integral, Z

Ωm

(f~^{ε}+~θ− ∇p^{ε}).φ ~ψ^{ε,kl}dxdz

= Z

Ω_{m}

(ξ^{ε}∇φ). ~ψ^{ε,kl}−
Z

Ω_{m}

e_{x}(u^{ε}) :e_{x}(w^{ε,kl})φdxdz
+

Z

Ωm

∇xr_{2}^{ε,kl}.(φu^{ε})dxdz−
Z

Ωm

2µ^{ε}ex(ψ^{ε,kl})∇xφ
.u^{ε}dxdz

− Z

Ω_{m}

ex(w^{ε,kl})∇xφ

.u^{ε}dxdz+
Z

Ω_{m}
n

X

j=1

[ξ^{ε}]n+1j

∂

∂z (ψ^{ε,kl})j

φdxdz.

Adding (4.22) and the above equation, we obtain Z

Ω_{m}

(f~^{ε}+~θ).φ ~ψ^{ε,kl}dxdz−
Z

Ω_{m}

∇p^{ε}.φ ~ψ^{ε,kl}dxdz−
Z

Ω_{m}

∇p^{0}^{ε}.φ ~w^{ε,kl}dxdz

=− Z

Ωm

(q^{ε}∇φ). ~w^{ε,kl}dxdz−
Z

Ωm

2µ^{ε}ex(v^{ε}) :ex(w^{ε,kl})φdxdz

− Z

Ω_{m}
n

X

j=1

[q^{ε}]n+1j

∂

∂z (w^{ε,kl})j

φdxdz+ Z

Ω_{m}

(ξ^{ε}∇φ). ~ψ^{ε,kl}
+

Z

Ωm

∇xr^{ε,kl}_{2} .(φu^{ε})dxdz−
Z

Ωm

(b^{ε,kl}∇xφ).u^{ε}dxdz
+

Z

Ωm

n

X

j=1

[ξ^{ε}]n+1j

∂

∂z (ψ^{ε,kl})j

φdxdz.

(4.23)

where

b^{ε,kl}= 2µ^{ε}e_{x}(ψ^{ε,kl}) +e_{x}(w^{ε,kl}). (4.24)
We obtain easily that (using Problem (3.5))

divx (b^{ε,kl}) =−∇xr^{ε,kl}_{2} in Ωm.

We now pass to the limit in (4.23) as εtends to 0. In order to do so, we need some preliminaries results.

By Definition (4.24) and classical arguments concerning the convergence of
periodic functions, we conclude that for all Ω^{0} ⊂⊂Ω_{m},

b^{ε,kl}* b^{kl}= m_{Y}(2µe_{y}(ψ^{kl}) +e_{y}(w^{kl})) weakly inL^{2}(Ω^{0})^{n}^{×}^{n}
and div

x (b^{kl}) = 0 in Ωm. (4.25)

By the convergence (4.16) a), we have

~

w^{ε,kl}→P~^{kl}= (P^{kl},0) strongly inL^{2}(Ω^{0})^{n+1}.
Also by (4.21)a), we get

ψ~^{ε,kl}→0 strongly inL^{2}(Ω^{0})^{n+1}.

Now passing to the limit in (4.23) taking into account the precedent convergence results, we obtain

− Z

Ωm

∇p^{0}.φ ~P^{kl}dxdz

=− Z

Ω_{m}

(q∇φ). ~P^{kl}dxdz−
Z

Ω_{m}

σ^{kl} :ex(v)φdxdz−
Z

Ω_{m}

(b^{kl}∇xφ).udxdz
Integrating by parts the right-hand side of the above expression, using the second
equation of (4.1.14) and the expression (4.25), we obtain

0 = Z

Ωm

q:e(P~^{kl})φdxdz−
Z

Ωm

σ^{kl}:ex(v)φdxdz+
Z

Ωm

ex(u) :b^{kl}φdxdz
Since

e(P~^{kl})

ij =
M^{kl}

ij, then we obtain in the distribution sense [q]kl=

n

X

i,j=1

σ^{kl}

ij

ex(v)

ij−

n

X

i,j=1

b^{kl}

ij

ex(u)

ij.

Now since (4.16) e), (4.25) hold and since ex(v)

ij = e(~v)

ij and ex(u)

ij = e(~u)

ij for all 1≤i, j≤n, we get [q]kl= 1

|Y|

n

X

i,j=1

Z

Y

2µ[ey(−χ^{kl}+P^{kl})]ijdy e(~v)

ij

− 1

|Y|

n

X

i,j=1

Z

Y

2µ[ey(−ψ^{kl})]ij+ [ey(−χ^{kl}+P^{kl})]ij

dy e(~u)

ij. (4.26) Also since the following symmetry property holds (see Proposition 3.1)

Z

Y

2µ[e_{y}(−χ^{kl}+P^{kl})]_{ij}dy=
Z

Y

2µ[e_{y}(−χ^{ij}+P^{ij})]_{kl}dy,
and the following’s one holds too (see Proposition 3.2)

Z

Y

2µ

ey(−ψ^{kl}) +ey(−χ^{kl}+P^{kl})

ij

dy

= Z

Y

2µ

ey(−ψ^{ij}) +ey(−χ^{ij}+P^{ij})

kl

dy, (4.27)

we conclude that
[q]_{kl} = 1

|Y|

n

X

i,j=1

Z

Y

2µ[e_{y}(−χ^{ij}+P^{ij})]_{kl}dy e(~v)

ij

− 1

|Y|

n

X

i,j=1

Z

Y

2µ[e_{y}(−ψ^{ij})]_{kl}+ [e_{y}(−χ^{ij}+P^{ij})]_{kl}

dy e(~u)

ij

This completes the proof.

### Identification of [q]

_{n+1j}

### , 1 ≤ j ≤ n in Ω

_{m}

Letϕ^{k} be the solution of Problem (3.2). We assume thatϕ^{k}=ϕ^{k}(y, z) satisfy
the regularity hypotheses

a) ϕ^{k} ∈L^{2}

loc(0, h_{1}, H_{]}^{1}(Y))∩L^{2}

loc(]0, h_{1}[×R^{n})
b) ^{∂ϕ}_{∂z}^{k} ∈L^{2}loc(0, h1, L^{2}_{]}(Y))∩L^{2}loc(]0, h1[×R^{n})

(4.28)
Let us define ζ^{k} =−ϕ^{k} + 2yk andη^{k} = µ∇yζ^{k}. We also define the following
functions byY-periodicity:

ζ^{ε,k}(x, z) =εζ^{k}(x

ε, z), η^{ε,k}(x, z) =η^{k}(x

ε, z) (4.29)

It is easy to see that −div_{x}η^{ε,k} = 0 in Ω_{m}. We introduce a supplementary
hypothesis concerning η^{ε,k}:

a) { η^{ε,k}

j}ε>0⊂L^{p}

loc(Ωm) for some p >2, locally bounded b) ∂

∂z η^{ε,k}

j≥0 in the distribution sense.

(4.30)

Then we have the following result.

Proposition 4.7 (Baffico & Conca [3]) Assume (4.28) and (4.30). Then
for all Ω^{0} ⊂⊂Ω_{m}, we have

a)ζ^{ε,k}*2yk weakly in H^{1}(Ω^{0})
b) _{∂z}^{∂} ζ^{ε,k}

→0 strongly in L^{2}(Ω^{0})
c) η^{ε,k}* η^{k} =mY(η^{k})weakly in L^{2}(Ω^{0})^{n}
d) _{∂z}^{∂} η^{ε,k}

j → _{∂z}^{∂} η^{k}

j strongly inH^{−}^{1}(Ω^{0}),1≤j≤n.

(4.31)

In view of (4.3.5) c) and (4.3.3), we get−div_{x}η^{k} = 0 in Ω_{m}. Similarly we
assume thatψ^{k}, the solution of (3.6), satisfies

a) ψ^{k} ∈L^{2}loc(0, h1, H_{]}^{1}(Y))∩L^{2}loc(]0, h1[×R^{n})
b) ∂ψ^{k}

∂z ∈L^{2}loc(0, h1, L^{2}_{]}(Y))∩L^{2}loc(]0, h1[×R^{n})

(4.32)

Let

ψ^{ε,k}(x, z) =ψ^{k}(x

ε, z) and d^{ε,k}= 2µ^{ε}∇xψ^{ε,k}+∇xζ^{ε,k}, (4.33)
then using (3.6), (4.29) and (4.33), we get

−div

x d^{ε,k}= 0 in Ωm. (4.34)

We assume thatd^{ε,k}satisfies the regularity conditions:

a)

d^{ε,k}

j ε>0⊂L^{p}loc(Ωm) for somep >2, locally bounded
b) ∂

∂z d^{ε,k}

j≥0 in the distribution sense.

(4.35)

We have the following result.

Proposition 4.8 Assume hypotheses (4.3.7) and (4.35) hold. Then for all
Ω^{0}⊂⊂Ωm, we have

a) ψ^{ε,k}*0 weakly inH^{1}(Ω^{0})
b) ∂

∂z ψ^{ε,k}

→0 strongly inL^{2}(Ω^{0})

c) d^{ε,k}* d^{k} =mY(2µ∇yψ^{k}+∇yζ^{k})weakly in L^{2}(Ω^{0})^{n}
d) ∂

∂z d^{ε,k}

j→ ∂

∂z d^{k}

j strongly inH^{−}^{1}(Ω^{0}), 1≤j≤n.

(4.36)

Remark 4.9 From (4.34) and (4.36) c), we have−divxd^{k}= 0 in Ωm.

Proof of Proposition 4.8 Using classical arguments concerning convergence of periodic functions, we can prove the three first assertions. For the last one, we use the compactness Lemma 4.3 (the hypotheses of this lemma hold since

we suppose that (4.35) is satisfied).

Proposition 4.10 If (4.28), (4.30), (4.32) and (4.35) hold, then up to a sub-
sequence, we have [q^{ε}]_{n+1k} *[q]_{n+1k} weakly inL^{2}(Ω_{m})∀1≤j≤n, where

[q]_{n+1k} = 1

|Y|

n

X

i=1

Z

Y

µ∂(−ϕ^{i}+ 2y_{i})

∂yk

dy e(~v)

n+1i

− 1 2|Y|

n

X

i=1

Z

Y

2µ∂ψ^{i}

∂yk

+∂(−ϕ^{i}+ 2y_{i})

∂yk

dy e(~u)

n+1i.

Proof. Let φ∈ D(Ω_{m}) and ~ζ^{ε,k} = (0, ζ^{ε,k}). Multiplying the second equation
of (2.4) byφ~ζ^{ε,k}, integrating by parts and using (4.2), we obtain

− Z

Ωm

∇p^{0}^{ε}.~ζ^{ε,k}φdxdz=−
Z

Ωm

(q^{ε}∇φ).~ζ^{ε,k}dxdz+
Z

Ωm

e(~u^{ε}) :∇~ζ^{ε,k}φdxdz