IN A RAREFIED BINARY STRUCTURE

A DIFFUSION PROCESS

IN A RAREFIED BINARY STRUCTURE

FADILA BENTALHA, ISABELLE GRUAIS and DAN POLIˇSEVSKI

We study the homogenization of a diffusion process which takes place in a binary structure formed by an ambiental connected phase surrounding a suspension of very small spheres distributed in anε-periodic network. The asymptotic distri- bution of the concentration is determined for both phases as ε → 0, assuming that the suspension has mass of unity order and vanishing volume. Three cases are distinguished according to the values of a certain limit capacity. When it is positive and finite, the macroscopic system involves a two-concentration system, coupled through a term accounting for the non local effects. In the other two cases, where the capacity is either infinite or going to zero, although the form of the system is much simpler, some peculiar effects still account for the presence of the suspension.

AMS 2000 Subject Classification: 35B27, 35K57, 76R50.

Key words: diffusion, homogenization, fine-scale substructure.

1. INTRODUCTION

Diffusion occurs naturally and is important in many industrial and geo- physical problems, particularly in oil recovery, earth pollution, phase transi- tion, chemical and nuclear processes. When one comes to a rational study of binary structures, a crucial point lies in the interaction between the micro- scopic and macroscopic levels and particularly the way the former influences the latter. Once the distribution is assumed to be ε-periodic, this kind of study can be accomplished by the homogenization theory.

The present study reveals the basic mechanism which governs diffusion in both phases of such a binary structure, formed by an ambiental connected phase surrounding a periodical suspension of small particles. For simplicity, the particles are considered here to be spheres of radius r_ε ε, that is,

ε→0limrε

ε = 0. We balance this assumption, which obviously means that the sus- pension has vanishing volume, by imposing the total mass of the suspension to be always of unity order. This simplified structure permits the accurate estab- lishment of the macroscopic equations by means of a multiple scale method of the homogenization theory adapted for fine-scale substructures. It allows to

REV. ROUMAINE MATH. PURES APPL.,52(2007),2, 129–149

have a general view on the specific macroscopic effects which arise in every pos- sible case. As we use the non-dimensional framework, the discussion is made in fact with respect to only two parameters: r_ε and b_ε, the latter standing for the ratio of suspension/ambiental phase diffusivities. As the diffusivities of the two components can differ by orders of magnitude, the interfacial conditions play an important role.

It happens that the following cases have different treatments: r_ε ε³, ε³ r_εεand r_ε =O(ε³).

To give a flavor of what may be considered as an appropriate choice of the relative scales, we refer to the pioneering work [7] where the appearance of an extra term in the limit procedure is responsible for a change in the nature of the mathematical problem and is linked to a critical size of the inclusions.

Later [6] showed how this could be generalized to theN-dimensional case for non linear operators satisfying classical properties of polynomial growth and coercivity. Since then, the notion of non local effects has been developed in a way that is closer to the present point of view in [4], [5] and [8].

In dealing with our problem, the main difficulty was due to the choice of test functions to be used in the associated variational formulation and which are classically some perturbation of the solution to the so-called cellular prob- lem. Indeed, proceeding as usual in homogenization theory, we use energy arguments based on a priori estimates where direct limiting procedure ap- parently leads to singular behavior. Non local effects appear when these sin- gularities can be overcome, which is usually achieved by using adequate test functions in the variational formulation. Since the fundamental work [7], an important step was accomplished in this direction in [3]. A slightly different approach [5] uses Dirichlet forms involving non classical measures in the spirit of [10]. However, the main drawback of this method lies in its essential use of the Maximum Principle, which was avoided in [4] for elastic fibers, and later in [8] where the case of spherical symmetry is solved. The asymptotic behav- ior of highly heterogeneous media has also been considered in the framework of homogenization when the coefficient of one component is vanishing and both components have volumes of unity order: see the derivation of a double porosity model for a single phase flow by [2] and the application of two-scale convergence in order to model diffusion processes in [1].

The paper is organized as follows. Section 2 is devoted to the main notations and to the description of the initial problem. We set the functional framework (16) where the existence and uniqueness of the solution can be established: see [9] and [11] for similar problems. In Section 3, we introduce specific tools to handle the limiting process. This is based on the use of the operators G_r defined by (38) which have a localizing effect: this observation motivates the additional assumption (47) on the external sources when the

radius of the particles is of critical orderε³ withεdenoting the period of the distribution. While passing to the limit, the capacity number γ_ε defined by (33) appears as the main criterium to describe the limit problem, the relative parts played by the radius of the particles and by the period of the network becoming explicit.

Section 4, which is actually the most involving one, deals with the critical case whenγ_ε has a positive and finite limit γ. In this part, where we assume alsob_ε →+∞ the test functions are a convex combination of the elementary solution of the Laplacian and its transformed by the operator G_rε defined by (38) with r = r_ε. This choice, which is inspired from [3], [4] and [8] and has to be compared with [7], allows to overcome the singular behavior of the energy term when the period ε tends to zero. We have to emphasize that this construction highly depends on the geometry of the problem, that is the spherical symmetry . To our knowledge, the generalization to more intricate geometries remains to be done. The resulting model (70)–(73), with the initial value defined after u₀ in (21) and v₀ in (23), involves a pairing (u, v) which is coupled through a linear operator acting on the difference u−v by the factor 4πγ.

The case of the infinite capacity, where ε³ r_ε ε, is worked out in Section 5. The proofs are only sketched because the arguments follow the same lines as in Section 4. Let us mention that the singular behavior of the capacity in this case, that is γ_ε → +∞, forces v to coincide with u. In other words, the infinite capacity prevents the splitting of the distribution, as it did in the critical case. Quite interestingly, the initial value of the global concentration is a convex combination (84) of the initial conditionsu₀ andv₀; moreover, the mass density of the macroscopic diffusion equation (83) takes both components into account, in accordance with the intuition that the limiting process must lead to a binary mixture.

The case of vanishing capacity is handled in Section 6, that is whenr_ε ε³. Here, v remains constant in time, obviously equal to the initial condition v₀, while u satisfies the diffusion equation (88)–(89) with data independent of the initial condition of the suspension. This can be seen as a proof that when the radius of the particles is too small, then the suspension does not present macroscopic effects, although a corresponding residual concentration, constant in time, should be considered.

After this work was completed, we learnt that M. Bellieud obtained a similar result in the case where the diffusivity tends to infinity as an inverse power of the inclusions volume, under more restrictive assumptions on data and with a different method (see [12]).

2. THE DIFFUSION PROBLEM

We consider a bounded Lipschitz domain Ω⊆R³ occupied by a mixture of two different materials, one of them forming the ambiental connected phase and the other being concentrated in a periodical suspension of small spherical particles. Let us denote

(1) Y :=

−1 2,+1

2 ₃

,

(2) Y_ε^k:=εk+εY, k∈Z³,

(3) Z_ε:={k∈Z³, Y_ε^k ⊂Ω}, Ω_Yε :=

k∈Zε

Y_ε^k. The suspension is defined by the union

(4) D_ε :=

k∈Zε

B(εk, r_ε),

where 0< r_εεandB(εk, r_ε) is the ball of radiusr_ε centered atεk,k∈Z_ε. Obviously,

(5) |D_ε| →0 as ε→0.

The fluid domain is given by

(6) Ω_ε= Ω\D_ε.

We also use the notation

(7) Ω^T := Ω×]0, T[

for the cylindrical time-domain; similar definitions for Ω^T_ε, Ω^T_Yε and D^T_ε. We consider the problem which governs the diffusion process throughout our binary mixture. Denoting bya_ε >0 and b_ε>0 the relative mass density and diffusivity of the suspension, then, assuming without loss of generality that|Ω|= 1, its non-dimensional form is to find u^ε, solution of

(8) ρ^ε∂u^ε

∂t −div(k^ε∇u^ε) =f^ε in Ω^T,

(9) [u^ε]_ε= 0 on ∂D_ε^T,

(10) [k^ε∇u^ε]_εn= 0 on ∂D^T_ε,

(11) u^ε= 0 on∂Ω^T,

(12) u^ε(0) =u^ε₀ in Ω,

where [·]_ε is the jump across the interface∂D_ε,nis the normal on ∂D_ε in the outward direction,f^ε∈L²(0, T;H⁻¹(Ω)), u^ε₀∈L²(Ω) and

(13) ρ^ε(x) =

1 ifx∈Ω_ε a_ε ifx∈D_ε,

(14) k^ε(x) =

1 ifx∈Ω_ε b_ε ifx∈D_ε.

LetH_ε be the Hilbert spaceL²(Ω) endowed with the scalar product (15) (u, v)_Hε := (ρ^εu, v)_Ω.

AsH₀¹(Ω) is dense inH_ε for any fixedε >0 we can set (16) H₀¹(Ω)⊆H_εH_ε ⊆H⁻¹(Ω) with continuous embeddings.

Now, we can present the variational formulation of problem (8)–(12):

findu^ε ∈L²(0, T;H₀¹(Ω))∩L^∞(0, T;H_ε) satisfying (in some sense) the initial condition (12) and the equation

(17) d

dt(u^ε, w)_Hε+(k_ε∇u^ε,∇w)_Ω=f^ε, w inD(0, T), ∀w∈H₀¹(Ω), where·,·denotes the duality product betweenH⁻¹(Ω) and H₀¹(Ω).

Theorem 2.1. Under the above hypotheses and notation, problem (17) has a unique solution. Moreover, ^du_dt^ε ∈ L²(0, T;H⁻¹(Ω)), hence u^ε is equal almost everywhere to a function ofC⁰([0, T];H_ε); this is the sense of the initial condition (12).

In what follows we consider that the density of the spherical particles is much greater than that of the surrounding phase. The specific feature of our mixture, which describes the fact that although the volume of the suspension is vanishing its mass is of unity order, is given by

(18) lim

ε→0a_ε|D_ε|=a >0.

Regarding the relative diffusivity, we only assume that (19) b_ε≥b₀ >0, ∀ε >0.

As for the data, we assume that there exist f ∈ L²(0, T;H⁻¹(Ω)) and u₀ ∈L²(Ω) such that

(20) f^ε f inL²(0, T;H⁻¹(Ω)),

(21) u^ε₀ u₀ inL²(Ω).

Also, we assume that there exist C > 0 (independent of ε) and v₀ ∈ L²(Ω) for which

(22)

−Dε

|u^ε₀|²dx≤C,

(23) 1

|D_ε|u^ε₀χ_Dε v₀ inD(Ω), where for anyD⊂Ω we denote

−D·dx= 1

|D|

D·dx.

Remark.As u^ε₀ satisfies (22), condition (23) holds at least on some sub- sequence (see Lemma A-2, [3]).

Proposition 2.2. We have

(24) u^ε is bounded in L^∞(0, T;L²(Ω))∩L²(0, T;H₀¹(Ω)).

Moreover, there existsC >0, independent of ε, such that (25)

−Dε

|u^ε|²dx≤C a.e. in[0, T], (26) b_ε|∇u^ε|²_L2(D^T_ε)≤C.

Proof. Substitutingw=u^ε in the variational problem (17) and integra- ting over (0, t) for any t∈]0, T[, we get

1 2

|u^ε(t)|²_Ωε+a_ε|u^ε(t)|²_Dε +b_ε

_t

0

|∇u^ε|²_Dεds+ _t

0

|∇u^ε|²_Ωεds=

= _t

0

f^ε(s), u^ε(s)ds+1 2

|u^ε₀|²_Ωε+a_ε|u^ε₀|²_Dε . Notice that (21) and (22) yield

|u^ε₀|²_Ωε+a_ε|u^ε₀|²_Dε ≤ |u^ε₀|²_Ω+a_ε|D_ε|

−Dε

|u^ε₀|²dx≤C.

Moreover, _t

0

f^ε(s), u^ε(s)ds≤ _t

0

|f^ε|_H⁻¹|∇u^ε|Ωds≤

≤ _t

0

|f^ε|_H⁻¹|∇u^ε|Ωεds+ _t

0

|f^ε|_H⁻¹|∇u^ε|_Dεds≤

≤ 1 2

_T

0

|f^ε|²_H−1ds+1 2

_t

0

|∇u^ε|²_Ωεds+ 1 2b_ε

_T

0

|f^ε|²_H−1ds+b_ε 2

_t

0

|∇u^ε|²_Dεds.

It follows that 1

2

|u^ε(t)|²_Ωε +a_ε|u^ε(t)|²_Dε +b_ε

2 _t

0

|∇u^ε|²_Dεds+1 2

_t

0

|∇u^ε|²_Ωεds≤C, and the proof is complete.

3. THE SPECIFIC TOOLS First, we introduce

R_ε={R, r_εRε} that isR∈ R_ε iff

(27) lim

ε→0

r_ε R = lim

ε→0

R ε = 0.

We remark thatR_ε is an infinite set, this property being insured by the assumption 0< r_ε ε.

We denote the domain confined between the spheres of radiusaandbby C(a, b) :={x∈R³, a <|x|< b}

and, correspondingly,C^k(a, b) :=εk+C(a, b).

For anyR_ε ∈ Rε we use the notation Cε:=

k∈Zε

C^k(r_ε, R_ε), C_ε^T :=Cε×]0, T[.

Definition. For anyR_ε∈ Rε we definew_Rε ∈H₀¹(Ω) by w_Rε(x) :=







0 in Ω_ε\ C_ε,

W_Rε(x−εk) inC_ε^k, ∀k∈Z_ε,

1 inD_ε,

(28) where

(29) W_Rε(y) = r_ε (R_ε−r_ε)

R_ε

|y| −1

for y∈ C(r_ε, R_ε).

We remark thatW_Rε ∈H¹(C(r_ε, R_ε)) and satisfies the system

∆W_Rε = 0 inC(r_ε, R_ε), (30)

W_Rε = 1 for|y|=r_ε, (31)

W_Rε = 0 for|y|=R_ε. (32)

Set

(33) γ_ε:= r_ε

ε³.

Proposition 3.1. For any R_ε∈ Rε we have (34) |∇w_Rε|Ω ≤Cγ_ε^1/2,

(35) w_Rε →0 in L²(Ω).

Proof. First notice that

|w_Rε|Ω=|w_Rε|Cε∪Dε ≤ |Cε∪D_ε|^1/2 ≤C R_ε

ε _3/2 and lim

ε→0 Rε

ε = 0 by assumption (27). As for the rest, plain computation shows that

|∇w_Rε|²_Ω =

k∈Zε

C_rε,Rε^k |∇w_Rε|² dx=

=

k∈Zε

_2π

0

dΦ _π

0

sin Θ dΘ _Rε

rε

dr r²

r_εR_ε R_ε−r_ε

₂

≤

≤ C|Ω| ε³

1 r_ε − 1

R_ε

r_εR_ε R_ε−r_ε

₂

≤C γ_ε (1−_R^rε_ε), and the proof is completed by (27).

Lemmas 3.2 and 3.3 below are stated without proof since they are a three-dimensional versions of Lemmas A.3 and A.4, [3].

Lemma 3.2. For every 0< r₁< r₂ and u∈H¹(C(r₁, r₂))we have (36) |∇u|²_C(r1,r2)≥ 4πr₁r₂

r₂−r₁

−Sr2

udσ−

−Sr1

udσ

2

,

where

−Sr

· dσ:= 1 4πr²

Sr

·dσ.

Lemma 3.3. There exists a positive constant C >0 such that ∀(R, α)∈ R⁺×(0,1), ∀u∈H¹(B(0, R)), we have

(37)

B(0,R)

u−

−SαR

udσ

² dx≤CR²

α |∇u|²_B(0,R).

Definition.Consider the piecewise constant functionsG_r :L²(0, T;H₀¹(Ω))

→L²(Ω^T) defined for anyr >0 by

(38) G_r(θ)(x, t) =

k∈Zε

−S^k_rθ(y, t) dσ_y

1_Yk ε(x),

where

(39) S_r^k=∂B(εk, r).

Lemma 3.4. If R_ε∈ Rε then for every θ∈L²(0, T;H₀¹(Ω)) we have

|θ−G_Rε(θ)|_L2(Ω^T_Yε) ≤ C ε³

R_{ε 1/2}

|∇θ|_L²_(Ω^T₎, (40)

|θ−G_rε(θ)|_L²_(D^T_ε) ≤ Cr_ε|∇θ|_L²_(Dε^T₎, (41)

|G_Rε(θ)−G_rε(θ)|_L²_(Ω^T₎ ≤ C ε³

r_{ε 1/2}

|∇θ|_L²_(Cε^T₎, (42)

where G_Rε(θ) andG_rε(θ) are defined by (38). Moreover, (43) |G_Rε(θ)|²_L2(Ω^T) =

_T

0

−Dε

|G_Rε(θ)|²dxdt,

|G_rε(θ)|²_L2(Ω^T)= _T

0

−Dε

|G_rε(θ)|²dxdt.

Proof. Notice that by definition

k∈Zε

_T

0

Y_ε^k

θ−

−S^k_Rεθdσ

² dxdt≤

k∈Zε

_T

0

B(εk,^ε√₂³)

θ−

−S^k_Rεθdσ ² dxdt, where we have used the fact that

Y_ε^k⊂B

εk,ε√ 3 2

for everyk∈Z_ε. Apply Lemma 3.3 with R= ε√

3

2 , α= 2R_ε ε√

3 to deduce that

Ω^T_Yε

|θ−G_Rε(θ)|² dxdt≤C ε√

3 2

₂ ε√

3 2R_ε

k∈Zε

_T

0

B(εk,^ε√₂³)

|∇θ|²dxdt≤

≤Cε³ R_ε

k∈Zε

_T

0

B(εk,^ε√₂³)

|∇θ|² dxdt≤Cε³ R_ε

Ω^T

|∇θ|²dxdt, which shows (40).

To establish (41), we recall the definition

D^T_ε |θ−G_rε(θ)|²dxdt=

k∈Zε

_T

0

B(εk,rε)

θ−

−S^k_rεθdσ ² dxdt.

Applying Lemma 3.3 withR =r_ε and α= 1, we get

D^Tε

|θ−G_rε(θ)|²dxdt≤Cr²_ε

k∈Zε

_T

0

B(εk,rε)

|∇θ|² dxdt≤Cr²_ε

Dε^T

|∇θ|²dxdt, that is, (41).

We come now to (42). Applying Lemma 3.2 and (27), we get

Ω^T

|G_Rε(θ)−G_rε(θ)|²dxdt=

k∈Zε

_T

0

Y_ε^k

−S^k_Rεθdσ−

−S^k_rεθdσ

² dydt≤

≤

k∈Zε

Y_ε^k

(R_ε−r_ε) 4πR_εr_ε dy

_T

0

C_rε,Rε^k |∇θ|² dxdt=

= (R_ε−r_ε) 4πr_εR_ε

k∈Zε

ε³ _T

0

C_rε,Rε^k |∇θ|²dxdt=

=Cε³(R_ε−r_ε) 4πr_εR_ε

C_ε^T|∇θ|² dxdt≤Cε³ r_ε

C^T_ε |∇θ|²dxdt.

Finally, a plain computation yields (43).

Proposition 3.5. If R_ε∈ R_ε then for any θ∈L²(0, T;H₀¹(Ω)) we have _T

0

−Dε

|θ|² dxdt≤Cmax

1,ε³ r_ε

|∇θ|²_L2(Ω^T). Proof. We have

_T

0

−Dε

|θ|² dxdt≤2 _T

0

−Dε

|θ−G_rε(θ)|² dxdt+ 2 _T

0

−Dε

|G_rε(θ)|²dxdt=

= 2 _T

0

−Dε

|θ−G_rε(θ)|² dxdt+ 2

Ω^T

|G_rε(θ)|² dxdt≤

≤Cr²_{ε T}

0

−Dε

|∇θ|² dxdt+ 4

Ω^T

|G_rε(θ)−G_Rε(θ)|² dxdt+

+8

Ω^T

|G_Rε(θ)−θ|²dxdt+ 8

Ω^T

|θ|² dxdt≤

≤Cr_ε² _T

0

−Dε

|∇θ|²dxdt+Cε³ r_ε

C^T_ε |∇θ|² dxdt+

+Cε³ R_ε

Ω^T

|∇θ|²dxdt+C

Ω^T

|∇θ|²dxdt≤

≤C ε³

r_ε + ε³ R_ε + 1

Ω^T

|∇θ|²dxdt≤Cmax

1,ε³

r_{ε Ω}T |∇θ|²dxdt.

Remark. Using the Mean Value Theorem, we easily find that (44) |G_rε(ϕ)−ϕ|_L^∞_(Cε∪Dε)≤2R_ε|∇ϕ|_L^∞_(Ω), ∀ϕ∈ D(Ω).

Definition. Let M_Dε :L²(0, T;C_c(Ω))→L²(Ω^T) be defined by M_Dε(ϕ)(x, t) :=

k∈Zε

−Y_ε^kϕ(y, t) dy

1_B(εk,rε)(x).

Lemma 3.6. For any ϕ∈L²(0, T;C_c(Ω)) we have

ε→0lim _T

0

−Dε

ϕ−M_Dε(ϕ)

²dxdt= 0.

Proof. Notice that

−Dε

|ϕ−M_Dε(ϕ)|²dx= 1

|D_ε|

k∈Zε

B(εk,rε)

ϕ−

−Y_ε^kϕdy ² dx.

As card(Z_ε) ^|Ω|_ε3, we have|B(0, r_ε)|^card(Z_|Dε|^ε) → |Ω| = 1 and it follows from the uniform continuity of ϕ on Ω that it converges to zero, a.e. on [0, T].

Lebesgue’s dominated convergence theorem concludes the proof.

4. HOMOGENIZATION OF THE CASE r_ε=O(ε³) The present critical radius case is described by

(45) lim

ε→0γ_ε=γ∈]0,+∞[.

Its homogenization process is the most involved one. This is why we start the homogenization study of our problem with this case, under the condition

(46) lim

ε→0b_ε= +∞.

We also assume that f^ε has the additional property (47)

∃R_ε∈ R_ε and g∈L²(0, T;H⁻¹(Ω)) for which f^ε, w_Rεϕ → g, ϕ inD(0, T), ∀ϕ∈ D(Ω) (see [8] for a certain type of functionsf^ε which satisfy (47)).

Remark. Notice that by (45), Proposition 3.5 in this case reads (48) ∀ϕ∈L²(0, T;H₀¹(Ω)),

_T

0

−Dε

|ϕ|²dxdt≤C|∇ϕ|²_L2(Ω^T). A preliminary result is

Proposition 4.1. There exist u ∈ L^∞(0, T;L²(Ω))∩L²(0, T;H₀¹(Ω)) andv∈L²(Ω^T) such that, on some subsequence,

(49) u^ε u in L^∞(0, T;L²(Ω)), (50) u^ε u in L²(0, T;H₀¹(Ω)), (51) G_Rε(u^ε)→u in L²(Ω^T), (52) G_rε(u^ε) v in L²(Ω^T).

Moreover, we have

(53) lim

ε→0

_T

0

−Dε

|u^ε−G_rε(u^ε)|²dxdt= 0.

Proof. From (24) we get, on some subsequence, convergences (49) and (50). Moreover, we have

(54) |u−G_Rε(u^ε)|²_ΩT =|u|²_ΩT\Ω^T_Yε +|u−G_Rε(u^ε)|²_ΩT Yε, where

(55) |u−G_Rε(u^ε)|_Ω^T

Yε ≤ |u−u^ε|_Ω^T

Yε+|u^ε−G_Rε(u^ε)|_Ω^T

Yε ≤

≤ |u−u^ε|_ΩT +|u^ε−G_Rε(u^ε)|_ΩT Yε

and (40) yields

|u^ε−G_Rε(u^ε)|²_ΩT

Yε ≤Cε³

R_ε|∇u^ε|²_ΩT =Cε³ r_ε

r_ε

R_ε|∇u^ε|²_ΩT ≤Cr_ε R_ε, thus

ε→0lim|u^ε−G_Rε(u^ε)|²_ΩT Yε = 0.

As (50) implies

(56) u^ε→u inL²(Ω^T)

the right-hand side of (55) tends to zero asε→0, that is,

ε→0lim|u−G_Rε(u^ε)|_Ω^T

Yε = 0.

After substitution into the right-hand side of (54), and taking into account that

ε→0lim|Ω^T \Ω^T_Yε|= 0, we obtain (51), that is,

(57) G_Rε(u^ε)→u inL²(Ω^T).

In order to prove (52), we see that

|G_rε(u^ε)|_L2(Ω^T)≤ |G_rε(u^ε)−G_Rε(u^ε)|_L2(Ω^T)+|G_Rε(u^ε)|_L2(Ω^T)≤ (58)

≤ ^C

γ_ε^1/2|∇u^ε|_L²_(Ω^T₎+C ≤C.

Moreover, recall that from (41) taking into account (26), we have (59)

_T

0

−Dε

|u^ε−G_rε(u^ε)|² dxdt≤Cr²_{ε T}

0

−Dε

|∇u^ε|² dxdt≤ C γ_εb_ε →0 and the proof is complete.

Proposition 4.2. For any ϕ∈L²(0, T;C_c(Ω)) we have

(60) lim

ε→0

_T

0

−Dε

u^εϕdxdt=

Ω^T

vϕdxdt.

Proof. We have _T

0

−Dε

u^εϕdxdt= _T

0

−Dε

(u^ε−G_rε(u^ε))ϕdxdt+

(61) +

_T

0

−Dε

G_rε(u^ε)(ϕ−M_Dε(ϕ))dxdt+ _T

0

−Dε

G_rε(u^ε)M_Dε(ϕ)dxdt.

The first right-hand term tends to zero by (53) in Proposition 4.1. The second one tends also to zero by Lemma 3.6. The last term can be handled as follows:

_T

0

−Dε

G_rε(u^ε)M_Dε(ϕ)dxdt=λ_ε

k∈Zε

_T

0

Yε^k

−S^kε

u^εdσ

ϕdxdt=

=λ_ε

Ω^T

ϕG_rε(u^ε)dxdt, where

λ_ε:= |B(0, r_ε)|

ε³|D_ε| →1 as|Ω|= 1.

The proof is completed by (52).

Proposition 4.3. For any ϕ∈L²(0, T;H₀¹(Ω)) we have (62)

_T

0

−Dε

u^εϕdxdt→

Ω^T

vϕdxdt.

Proof. On account of Proposition 4.2, we have to prove that the left-hand side term is continuous in the corresponding norm. This can be obtained from

_T

0

−Dε

u^εϕdxdt ≤

_T

0

−Dε

|u^ε|²dxdt

_{1/2 T}

0

−Dε

|ϕ|²dxdt _1/2

≤

≤C|ϕ|²_L2(0,T;H₀¹(Ω)),

where we used (25) and (48).

Proposition 4.4. For any R_ε∈ Rε and ϕ, ψ∈ D(Ω)let (63) Φ^ε= (1−w_Rε)ϕ+w_RεG_rε(ψ).

Then for any η∈ D([0, T[)we have

(64) lim

ε→0|Φ^ε−ϕ|Ω= 0, (65) lim

ε→0

Ω^T

ρ^εu^εΦ^εη(t)dxdt=

Ω^T

uϕη(t)dxdt+a

Ω^T

vψη(t)dxdt.

Proof. Property (64) is an immediate consequence of (35) and of the uniform boundedness ofG_rε(ψ) in L^∞(Ω).

For the second property, let us notice that

Ω^T

ρ^εu^εΦ^εη(t)dxdt= _T

0

Ω

χ_Ωεu^εΦ^ε(x)η(t)dxdt+

+a_{ε T}

0

Dε

u^εG_rε(ψ)η(t)dxdt.

As we obviously have

ε→0lim _T

0

Ω

χ_Ωεu^εΦ^ε(x)η(t)dxdt=

Ω^T

uϕη(t)dxdt, it remains to consider

a_{ε T}

0

Dε

u^εG_rε(ψ)η(t)dxdt=a_ε|D_ε| _T

0

−Dε

u^εG_rε(ψ)η(t)dxdt.

Using (60) and the uniform continuity of ψ, we get

ε→0lima_{ε T}

0

Dε

u^εG_rε(ψ)η(t)dxdt=a

Ω^T

vψηdxdt.

Proposition 4.5. If Φ^ε is defined as in Proposition 4.4, then we have

ε→0lim _T

0

Ω

∇u^ε· ∇Φ^εη(t) dxdt= (66)

=

Ω^T

∇u· ∇ϕη(t) dxdt+ 4πγ

Ω^T

(v−u)(ψ−ϕ)η(t) dxdt.

Proof.First, consider _T

0

Ωε

∇u^ε· ∇Φ^εdxdt, which reduces to

_T

0

Ω_ε\Cε

∇u^ε· ∇ϕηdxdt+ _T

0

Cε

∇u^ε· ∇Φ^εηdxdt.

Lebesgue’s dominated convergence theorem yields∇ϕ1_Ωε\Cε → ∇ϕinL²(Ω).

Thus, taking (50) into account we get _T

0

Ω

∇u^ε· ∇ϕ ηχ_Ωε\Cε dxdt→

Ω^T

∇u· ∇ϕ η dxdt.

Now, let us come to the remaining part, namely _T

0

Cε∇u^ε· ∇Φ^εη(t) dxdt= _T

0

Cε(1−w_Rε)∇u^ε· ∇ϕηdxdt+

(67)

+ _T

0

Cε∇u^ε· ∇w_Rε(G_rε(ψ)−ϕ)dxdt:=I₁+I₂.

In the first integral, as χ_Cε∇ϕ → 0 in L²(Ω^T), ∇u^ε ∇u in L²(Ω^T) and (1−w_Rε) is obviously bounded, we easily find thatI₁ tends to zero. In order to handleI₂, let us notice that

I₂ = _T

0

Cε∇u^ε· ∇w_Rε(G_rε(ϕ)−ϕ)ηdxdt+

(68)

+ _T

0

Cε∇u^ε· ∇w_Rε(G_rε(ψ)−G_rε(ϕ))η dxdt.

The first term in the right-hand side of (68) can be estimated by

_T

0

Cε∇u· ∇^ε w_Rε(ϕ−G_rε(ϕ))η dxdt≤ (69)

≤ |∇u^ε|_Ω^T|∇w_Rεη|_Ω^T|ϕ−G_rε(ϕ)|_L^∞_(Cε).

As (w_Rε) is bounded in H¹(Ω) (see Proposition 3.1), the right hand side of (69) tends to zero by (44). Going back to the second term in the right hand