Diffusion process in a perforated domain around a vanishing suspension

Abstract

Our work deals with the homogenization of a diffusion process which take place in a domain formed by an ambient connected phase surrounding an ε-periodical network of small spherical particles and holes, ε is a small parameter $ε > 0$ . The asymptotic behavior is determined as $ε \to 0$ , assuming that the total volume of the holes and particles vanishes as $ε \to 0$ , while the total mass of the particles remains of the unity order.

Keywords

Diffusion homogenization perforated domains particles fine-scale substructure

1. Introduction and notations

The aim of this work is the study of the asymptotic behavior as $ε \to 0$ of a diffusion process in a domain formed by an ambient connected phase ε-periodically perforated by small holes and surrounding an ε-periodical suspension of small spherical particles.

More precisely, for $N ⩾ 3$ , let $Ω \subseteq R^{N}$ a bounded Lipschitz domain occupied by this mixture, let us denote $\begin{array}{l} (1.1) & Y : = {(- \frac{1}{2}, + \frac{1}{2})}^{N}, \\ (1.2) & Y_{ε}^{k} : = ε k + ε Y, k \in Z^{N}, \\ (1.3) & Z_{ε} : = {k \in Z^{N}, Y_{ε}^{k} \subset Ω}, \\ (1.4) & Ω_{Y_{ε}} : = int ⋃_{k \in Z_{ε}} \overline{Y_{ε}^{k}} . \end{array}$ Let $y_{1}, y_{2} \in Y$ with $y_{1} \neq y_{2}$ , and $l > 0$ such that $\begin{matrix} (1.5) & B (y_{1}, l) \subset Y, B (y_{2}, l) \subset Y and \overline{B (y_{1}, l)} \cap \overline{B (y_{2}, l)} = \emptyset, \end{matrix}$ where $B (a, r)$ is the ball of radius r and centered at a.

For two small parameters $δ_{ε}, α_{ε}$ such that $\begin{matrix} (1.6) & \{\begin{matrix} 0 < α_{ε} < l, & 0 < δ_{ε} < l, \\ α_{ε} \to 0, δ_{ε} \to 0 & as ε \to 0, \end{matrix} \end{matrix}$ we define $\begin{matrix} (1.7) & D_{ε} : = ⋃_{k \in Z_{ε}} {B (ε y_{1}, ε δ_{ε}) + ε k}, T_{ε} : = ⋃_{k \in Z_{ε}} {B (ε y_{2}, ε α_{ε}) + ε k} . \end{matrix}$

The sets $D_{ε}$ and $T_{ε}$ are respectively, the suspension and the holes, they have obviously a vanishing volume, that is $\begin{matrix} (1.8) & | D_{ε} | ⟶ 0, | T_{ε} | ⟶ 0 as ε ⟶ 0 . \end{matrix}$

Denote by $Ω_{ε}$ the domain obtained by removing from Ω the set ${\overline{T}}_{ε} and Ω_{ε}^{'} = Ω_{ε} ∖ \overline{D_{ε}}$ the ambient connected domain.

We also use the following notation for the cylindrical time-domain $\begin{matrix} (1.9) & Ω^{T} : = Ω \times] 0, T [, \end{matrix}$ with similar definitions for $Ω_{ε}^{T}$ , $Ω_{Y_{ε}}^{T}$ and $D_{ε}^{T}$ .

We consider the problem which governs the diffusion process throughout our mixture. Denoting by $a_{ε} > 0$ and $b_{ε} > 0$ the relative mass density and diffusivity of the suspension and $f_{ε}$ a source term, then assuming without loss of generality that $| Ω | = 1$ , its non-dimensional form is the following: to find $u_{ε}$ solution of $\begin{matrix} (1.10) & \{\begin{matrix} ρ_{ε} \frac{\partial u_{ε}}{\partial t} - div (ν_{ε} \nabla u_{ε}) = f_{ε} & in Ω_{ε}^{T}, \\ {[u_{ε}]}_{ε} = 0 & on \partial D_{ε} \times] 0, T [, \\ {[ν_{ε} \nabla u_{ε}]}_{ε} n = 0 & on \partial D_{ε} \times] 0, T [, \\ u_{ε} = 0 & on (\partial Ω \cup \partial T_{ε}) \times] 0, T [, \\ u_{ε} (0) = u_{ε}^{0} & in Ω_{ε}, \end{matrix} \end{matrix}$ where ${[\cdot]}_{ε}$ is the jump across the interface $\partial D_{ε}$ , n is the normal on $\partial D_{ε}$ in the outward direction, and $\begin{matrix} (1.11) & ρ_{ε} (x) = \{\begin{matrix} 1 & if x \in Ω_{ε}^{'}, \\ a_{ε} & if x \in D_{ε}, \\ 0 & if x \in T_{ε}, \end{matrix} ν_{ε} (x) = \{\begin{matrix} 1 & if x \in Ω_{ε}^{'}, \\ b_{ε} & if x \in D_{ε}, \\ 0 & if x \in T_{ε} . \end{matrix} \end{matrix}$ Denote by $\tilde{u_{ε}}$ the extension by zero of $u_{ε}$ to the whole of Ω, defined by $\begin{matrix} (1.12) & \tilde{u_{ε}} = \{\begin{matrix} u_{ε} & in Ω_{ε}, \\ 0 & in T_{ε} . \end{matrix} \end{matrix}$

We assume $\begin{matrix} (1.13) & lim_{ε \to 0} a_{ε} | D_{ε} | = a > 0, \end{matrix}$ which means that although the volume of the suspension is vanishing, its mass stays of unity order; in this case the density of the particles is much higher than that of the surrounding phase. We also suppose that the diffusivity of the particles is such that $\begin{matrix} (1.14) & b_{ε} ⩾ b > 0, \forall ε > 0 . \end{matrix}$

As for the data, we assume that $\begin{matrix} (1.15) & f_{ε} \in L^{2} (0, T; L^{2} (Ω_{ε})), u_{ε}^{0} \in L^{2} (Ω_{ε}), \end{matrix}$ and $\begin{matrix} (1.16) & \{\begin{matrix} {\tilde{f}}_{ε} ⇀ f & in L^{2} (0, T; L^{2} (Ω)), \\ {\tilde{u}}_{ε}^{0} ⇀ u_{0} & in L^{2} (Ω), \\ {\int -}_{D_{ε}} | u_{ε}^{0} |^{2} d x ⩽ C, \\ \frac{1}{| D_{ε} |} u_{ε}^{0} χ_{D_{ε}} ⇀ v_{0} & in D^{'} (Ω), with v_{0} \in L^{2} (Ω), \end{matrix} \end{matrix}$ where for any $D \subset Ω$ , we used the notation $\begin{matrix} {\int -}_{D} \cdot d x = \frac{1}{| D |} \int_{D} \cdot d x . \end{matrix}$

In this paper, the homogenization of the problem (1.10)–(1.16) is studied by means of an energy method adapted to fine scale substructures, based on the Cioranescu–Murat method [11] combined with the control zone method [5]. Our study is provided for a periodically perforated domain with small holes verifying the Cioranescu–Murat hypotheses (3.22).

Several cases are distinguished according to the value of the rarefied coefficient $\begin{matrix} (1.17) & γ = lim_{ε \to 0} \frac{{δ_{ε}}^{N - 2}}{ε^{2}}, 0 ⩽ γ ⩽ + \infty . \end{matrix}$

The asymptotic behavior of the homogeneous Dirichlet problem in perforated domain with small holes, that is holes of size $β_{ε}$ such that $\frac{β_{ε}}{ε} ⟶ 0 when ε \to 0$ , was studied first by Cioranescu and Murat [11]; they showed that for each dimension $N (N > 2)$ of the space, when the size of the holes is “critical” in the sense that they verify the Cioranescu–Murat conditions (3.22), an additional term called “strange term”, related to the capacity of the holes, appears in the limit problem.

There were afterwards many works treating the same framework, for instance Allaire [1], Casado-Diaz [9]. When the small holes are filled up with a different material (“small particles”) and when this later has a very contrast behavior with respect to the ambient phase, which is the case in [4–6], it has been shown that, under hypotheses on the size of the particles, the limit problem has a different structure due to the appearance of non local terms. The notion of non local effects has been developed in the case of fibres by many authors in [2,3,7,8], and developed in the case of particles in [4–6].

In our case we associate the two situations, that is we consider a domain with small holes and small particles. This mixture aspect induced the main difficulty of our study, related to the choice of test functions to be used in the variational formulation of problem (1.10).

The paper is organized as follows. In Section 2 we recall briefly the theorem of existence and uniqueness of a solution, and then establish a priori estimates. In Section 3 we introduce the specific tools of the control zone method and give also some hypotheses on the holes, in order to handle the limiting process. Section 4 contains the study of the homogenization process.

2. A priori estimates

2.1. Existence and uniqueness

Under the above assumptions, problem (1.10)–(1.16) has a unique weak solution $u_{ε}$ in the following sense see [12]:

Theorem 2.1.
Under the above assumptions, there exists a unique $u_{ε}$ such that $\begin{matrix} (2.1) & \{\begin{matrix} u_{ε} \in L^{2} (0, T; H_{0}^{1} (Ω_{ε})) \cap C^{0} ([0, T]; L^{2} (Ω_{ε})) and \frac{\partial u_{ε}}{\partial t} \in L^{2} (0, T; H^{- 1} (Ω_{ε})), \\ \frac{d}{d t} {(ρ_{ε} u_{ε}, w)}_{Ω_{ε}} + {(ν_{ε} \nabla u_{ε}, \nabla w)}_{Ω_{ε}} = {(f_{ε}, w)}_{Ω_{ε}} in D^{'} (0, T), \forall w \in H_{0}^{1} (Ω_{ε}), \\ u_{ε} (0) = u_{ε}^{0} in Ω_{ε} . \end{matrix} \end{matrix}$

For the proof, we use the Faedo–Galerkin method (see [12]). Throughout the paper C denotes a positive constant the value of which may be changed from line to another.
2.2. A priori estimates

Proposition 2.2.
Under hypotheses ( 1.13 ) to ( 1.16 ) $\begin{matrix} (2.2) & \tilde{u_{ε}} is bounded in L^{\infty} (0, T; L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω)) . \end{matrix}$ Moreover, there exists $C > 0$ independent of ε, such that $\begin{array}{l} (2.3) & {\int -}_{D_{ε}} | u_{ε} |^{2} d x ⩽ C a.e. in [0, T], \\ (2.4) & b_{ε} | \nabla u_{ε} |_{L^{2} ({D_{ε}}^{T})}^{2} ⩽ C . \end{array}$
Proof.
Substituting $w = u_{ε}$ in the variational problem (2.1) and integrating it over $(0, t)$ for any $t \in] 0, T [$ , we get successively $\begin{array}{rcl} \frac{1}{2} | \sqrt{ρ_{ε}} u_{ε} |_{Ω_{ε}}^{2} + \int_{0}^{t} | \sqrt{ν_{ε}} \nabla u_{ε} |_{Ω_{ε}}^{2} = \int_{0}^{t} \int_{Ω_{ε}} f_{ε} u_{ε} d x d s + \frac{1}{2} {| \sqrt{ρ_{ε}} u_{ε}^{0} |}_{Ω_{ε}}^{2}, \end{array}$ and $\begin{array}{l} \frac{1}{2} (| u_{ε} |_{Ω_{ε}^{'}}^{2} + a_{ε} | u_{ε} |_{D_{ε}}^{2}) + b_{ε} \int_{0}^{t} | \nabla u_{ε} |_{D_{ε}}^{2} d s + \int_{0}^{t} | \nabla u_{ε} |_{Ω_{ε}^{'}}^{2} d s \\ (2.5) & = \int_{0}^{t} \int_{Ω_{ε}} f_{ε} u_{ε} d x d s + \frac{1}{2} ({| u_{ε}^{0} |}_{Ω_{ε}^{'}}^{2} + a_{ε} {| u_{ε}^{0} |}_{D_{ε}}^{2}) . \end{array}$

According to hypotheses (1.13)–(1.16), we have $\begin{matrix} {| u_{ε}^{0} |}_{Ω_{ε}^{'}}^{2} + a_{ε} {| u_{ε}^{0} |}_{D_{ε}}^{2} ⩽ {| {\tilde{u}}_{ε}^{0} |}_{Ω}^{2} + a_{ε} | D_{ε} | {\int -}_{D_{ε}} {| u_{ε}^{0} |}^{2} d x ⩽ C . \end{matrix}$

Moreover, $\begin{matrix} \int_{0}^{t} \int_{Ω_{ε}} f_{ε} u_{ε} d x d s = \int_{0}^{t} \int_{Ω} {\tilde{f}}_{ε} {\tilde{u}}_{ε} d x d s ⩽ \int_{0}^{t} | {\tilde{f}}_{ε} |_{Ω} | {\tilde{u}}_{ε} |_{Ω} d s . \end{matrix}$

Using the Poincaré and Young inequalities, it follows that $\begin{array}{l} \int_{0}^{t} | {\tilde{f}}_{ε} |_{Ω} | {\tilde{u}}_{ε} |_{Ω} d s ⩽ & C (Ω) \int_{0}^{t} | {\tilde{f}}_{ε} |_{Ω} | \nabla {\tilde{u}}_{ε} |_{Ω} = \int_{0}^{t} {| C (Ω) f_{ε} |}_{Ω_{ε}} | \nabla u_{ε} |_{Ω_{ε}} \\ ⩽ & \int_{0}^{t} {| C (Ω) f_{ε} |}_{Ω_{ε}} | \nabla u_{ε} |_{Ω_{ε}^{'}} d s + \int_{0}^{t} {| C (Ω) f_{ε} |}_{Ω_{ε}} | \nabla u_{ε} |_{D_{ε}} d s \\ ⩽ & \frac{1}{2} \int_{0}^{t} {| C (Ω) f_{ε} |}_{Ω_{ε}}^{2} d s + \frac{1}{2} \int_{0}^{t} | \nabla u_{ε} |_{Ω_{ε}^{'}}^{2} d s \\ + \frac{1}{2 b_{ε}} \int_{0}^{T} {| C (Ω) f_{ε} |}_{Ω_{ε}}^{2} d s + \frac{b_{ε}}{2} \int_{0}^{t} | \nabla u_{ε} |_{D_{ε}}^{2} d s . \end{array}$ Recalling (2.5) and using (1.14)–(1.16), we get $\begin{matrix} (2.6) & \frac{1}{2} (| u_{ε} |_{Ω_{ε}^{'}}^{2} + a_{ε} | u_{ε} |_{D_{ε}}^{2}) + \frac{1}{2} \int_{0}^{T} | \nabla u_{ε} |_{Ω_{ε}^{'}}^{2} d s + \frac{b_{ε}}{2} \int_{0}^{T} | \nabla u_{ε} |_{D_{ε}}^{2} d s ⩽ C, \end{matrix}$ which ends the proof. □
3. Specific tools

3.1. Tools of the control-zone method

First, we introduce $\begin{matrix} R : = {(r_{ε}), δ_{ε} ≪ r_{ε} ≪ 1 and r_{ε} < l}, \end{matrix}$ that is $(r_{ε}) \in R$ iff $r_{ε} \to 0 as ε \to 0$ , $r_{ε} < l$ and $\begin{matrix} (3.1) & lim_{ε \to 0} \frac{δ_{ε}}{r_{ε}} = 0 . \end{matrix}$ We denote the domain confined between the spheres of radius a and b by $\begin{matrix} C (a, b) : = {x \in R^{N}, a < | x | < b} . \end{matrix}$ We use the following notations, $\begin{matrix} C_{T_{ε}}^{k} = ε (y_{2} + k + C (α_{ε}, l)), C_{T_{ε}} : = ⋃_{k \in Z_{ε}} C_{T_{ε}}^{k} . \end{matrix}$ For ${(r_{ε})}_{ε} \in R$ , we also define $\begin{array}{l} C_{D_{ε}}^{k} = ε (y_{1} + k + C (δ_{ε}, r_{ε})), C_{D_{ε}} : = ⋃_{k \in Z_{ε}} C_{D_{ε}}^{k}, \\ C_{ε} : = C_{T_{ε}} \cup C_{D_{ε}} . \end{array}$

Definition 3.1.
For $(r_{ε}) \in R$ , we define the function $w_{r_{ε}} \in H_{0}^{1} (Ω)$ by $\begin{matrix} (3.2) & w_{r_{ε}} (x) : = \{\begin{matrix} 0 & in Ω ∖ (\overline{C_{D_{ε}} \cup D_{ε}}), \\ W_{r_{ε}} (\frac{x}{ε} - y_{1} - k) & in C_{D_{ε}}^{k}, \forall k \in Z_{ε}, \\ 1 & in D_{ε}, \end{matrix} \end{matrix}$ where $W_{r_{ε}} \in H^{1} (C (δ_{ε}, r_{ε}))$ is the fundamental solution of the Laplace problem, $\begin{matrix} (3.3) & \{\begin{matrix} Δ W_{r_{ε}} = 0 & in C (δ_{ε}, r_{ε}), \\ W_{r_{ε}} = 1 & for | y | = δ_{ε}, \\ W_{r_{ε}} = 0 & for | y | = r_{ε}, \end{matrix} \end{matrix}$ and given explicitly by $\begin{matrix} (3.4) & W_{r_{ε}} (y) = \frac{r_{ε}^{- N + 2} - | y |^{- N + 2}}{r_{ε}^{- N + 2} - δ_{ε}^{- N + 2}} for y \in C (δ_{ε}, r_{ε}) . \end{matrix}$

From now on, we denote $\begin{matrix} (3.5) & γ_{ε} : = \frac{{δ_{ε}}^{N - 2}}{ε^{2}} . \end{matrix}$

Acting as in [5] we obtain the next results. Proposition 3.2.
For any $(r_{ε}) \in R$ , we have $\begin{array}{l} (3.6) & 0 ⩽ w_{r_{ε}} ⩽ 1, \\ (3.7) & | \nabla w_{r_{ε}} |_{Ω} ⩽ C {(γ_{ε})}^{1 / 2}, \\ (3.8) & w_{r_{ε}} \to 0 strongly in L^{2} (Ω) . \end{array}$
Definition 3.3.
Consider the piecewise constant function $G_{r} : L^{2} (0, T; H_{0}^{1} (Ω)) ⟶ L^{2} (Ω^{T})$ defined for $ε > 0$ by $\begin{matrix} (3.9) & G_{r} (θ) (x, t) = \sum_{k \in Z_{ε}} ({\int -}_{S_{r}^{k}} θ (y, t) d σ_{y}) 1_{Y_{ε}^{k}} (x), \end{matrix}$ where $r > 0$ and $\begin{matrix} (3.10) & S_{r}^{k} : = \partial B_{r}^{k}, B_{r}^{k} : = ε k + ε B (y_{1}, r) . \end{matrix}$
Lemma 3.4.
If $(r_{ε}) \in R$ , then for every $θ \in L^{2} (0, T; H_{0}^{1} (Ω))$ , $\begin{array}{l} (3.11) & {| θ - G_{r_{ε}} (θ) |}_{L^{2} (Ω_{Y_{ε}}^{T})} ⩽ C {(\frac{ε^{2}}{r_{ε}^{N - 2}})}^{1 / 2} | \nabla θ |_{L^{2} (Ω^{T})}, \\ (3.12) & {| θ - G_{δ_{ε}} (θ) |}_{L^{2} (D_{ε}^{T})} ⩽ C (ε δ_{ε}) | \nabla θ |_{L^{2} (D_{ε}^{T})}, \\ (3.13) & {| G_{r_{ε}} (θ) - G_{δ_{ε}} (θ) |}_{L^{2} (Ω^{T})} ⩽ C {(\frac{ε^{2}}{δ_{ε}^{N - 2}})}^{1 / 2} | \nabla θ |_{L^{2} (C_{D_{ε}}^{T})}, \end{array}$ where $G_{δ_{ε}} (θ)$ and $G_{r_{ε}} (θ)$ are defined following ( 3.9 ). Moreover, $\begin{array}{l} (3.14) & {| G_{δ_{ε}} (θ) |}_{L^{2} (Ω^{T})}^{2} = \int_{0}^{T} {\int -}_{D_{ε}} {| G_{δ_{ε}} (θ) |}^{2} d x d t, \\ (3.15) & {| G_{r_{ε}} (θ) |}_{L^{2} (Ω^{T})}^{2} = \int_{0}^{T} {\int -}_{D_{ε}} {| G_{r_{ε}} (θ) |}^{2} d x d t . \end{array}$
Proposition 3.5.
If $(r_{ε}) \in R$ , then for every $θ \in L^{2} (0, T; H_{0}^{1} (Ω))$ we have $\begin{matrix} (3.16) & \int_{0}^{T} {\int -}_{D_{ε}} | θ |^{2} d x d t ⩽ C max (1, \frac{1}{γ_{ε}}) | \nabla θ |_{L^{2} (Ω^{T})} . \end{matrix}$
Remark 3.6.
Using the Mean Value Theorem, we have the following results:

For any $θ \in D (Ω)$ , $\begin{array}{l} (3.17) & {| G_{δ_{ε}} (θ) - θ |}_{L^{\infty} (Ω^{T})} ⟶ 0, \\ (3.18) & {| G_{δ_{ε}} (θ) - θ |}_{L^{\infty} (C_{D_{ε}}^{T} \cup D_{ε}^{T})} ⩽ 2 ε r_{ε} | \nabla θ |_{L^{\infty} (Ω^{T})} . \end{array}$ For any $φ \in C (\overline{Ω})$ , $\begin{matrix} (3.19) & {\int -}_{D_{ε}} φ d x ⟶ \int_{Ω} φ d x . \end{matrix}$

3.2. Hypotheses on the holes

Definition 3.7.
Let us define $w_{ε}$ by setting $\begin{matrix} (3.20) & w_{ε} (x) : = \{\begin{matrix} 0 & in T_{ε}, \\ {\bar{w}}_{ε} (x / ε - y_{2} - k) & in C_{T_{ε}}^{k}, k \in Z_{ε}, \\ 1 & in Ω ∖ {\overline{C}}_{T_{ε}}, \end{matrix} \end{matrix}$ where ${\bar{w}}_{ε} \in H^{1} (C (α_{ε}, l))$ is the fundamental solution of the Laplace problem, that is $\begin{matrix} (3.21) & \{\begin{matrix} Δ {\bar{w}}_{ε} = 0 & in C (α_{ε}, l), \\ \bar{w_{ε}} = 1 & for | y | = l, \\ \bar{w_{ε}} = 0 & for | y | = α_{ε} . \end{matrix} \end{matrix}$

We suppose that $w_{ε}$ satisfies the Cioranescu–Murat hypotheses, that is $\begin{array}{l} (3.22) & \begin{aligned} (H .1) w_{ε} \in H^{1} (Ω), \\ (H .2) w_{ε} = 0 on T_{ε}, \\ (H .3) w_{ε} ⇀ 1 weakly in H^{1} (Ω) and a.e. in Ω, \\ (H .4) μ_{ε}, λ_{ε} \in H^{- 1} (Ω) such that \end{aligned} \\ (3.23) & \{\begin{matrix} - Δ w_{ε} = μ_{ε} - λ_{ε}, & μ_{ε} \to μ in H^{- 1} (Ω), \\ ⟨ λ_{ε}, v_{ε} ⟩ = 0 & for any v_{ε} \in H_{0}^{1} (Ω) such that v_{ε} = 0 on T_{ε} . \end{matrix} \end{array}$ Remark 3.8.
If $α_{ε} = C_{0} ε^{2 / (N - 2)}$ , then $w_{ε}$ satisfies hypotheses (H.1) to (H.4), see [11].

4. The homogenization result

4.1. Homogenization in the case $δ_{ε} = O (ε^{2 / (N - 2)})$

The size of the particles is exactly of the order of $ε δ_{ε}$ , that is $ε^{N / (N - 2)}$ , it means that $\begin{matrix} (4.1) & γ = lim_{ε \to 0} γ_{ε} \in] 0, + \infty [. \end{matrix}$ We start the homogenization process with this case the most involved, under the condition $\begin{matrix} (4.2) & lim_{ε \to 0} b_{ε} = + \infty . \end{matrix}$

Remark 4.1.
Notice that Proposition 3.5 reads $\begin{matrix} (4.3) & \forall φ \in L^{2} (0, T; H_{0}^{1} (Ω)), \int_{0}^{T} {\int -}_{D_{ε}} | φ |^{2} d x d t ⩽ C | \nabla φ |_{L^{2} (Ω^{T})}^{2} . \end{matrix}$
Proposition 4.2.
Under hypotheses ( 1.13 )–( 1.16 ), there exist $\begin{matrix} u \in L^{\infty} (0, T; L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω)) and v \in L^{2} (0, T; L^{2} (Ω)), \end{matrix}$ such that, for some subsequence, $\begin{array}{l} (4.4) & \tilde{u_{ε}} \overset{}{⇀} u in L^{\infty} (0, T; L^{2} (Ω)), \\ (4.5) & \tilde{u_{ε}} ⇀ u in L^{2} (0, T; H_{0}^{1} (Ω)), \\ (4.6) & G_{r_{ε}} ({\tilde{u}}_{ε}) \to u in L^{2} (Ω^{T}), \\ (4.7) & G_{δ_{ε}} ({\tilde{u}}_{ε}) ⇀ v in L^{2} (Ω^{T}), \\ (4.8) & lim_{ε \to 0} \int_{0}^{T} {\int -}_{D_{ε}} {| {\tilde{u}}_{ε} - G_{δ_{ε}} ({\tilde{u}}_{ε}) |}^{2} d x d t = 0 . \end{array}$
Proof.
According to (2.2), for some subsequence, we have convergences (4.4) and (4.5).

Let us prove (4.6). We have for $(r_{ε}) \in R$ , $\begin{array}{rcl} {| u - G_{r_{ε}} ({\tilde{u}}_{ε}) |}_{Ω_{Y_{ε}}^{T}}^{2} ⩽ 2 | u - {\tilde{u}}_{ε} |_{Ω^{T}}^{2} + 2 {| {\tilde{u}}_{ε} - G_{r_{ε}} ({\tilde{u}}_{ε}) |}_{Ω_{Y_{ε}}^{T}}^{2} . \end{array}$

Taking into account Lemma 3.4 and since ${lim}_{ε \to 0} \frac{δ_{ε}}{r_{ε}} = 0$ , we have $\begin{array}{l} {| {\tilde{u}}_{ε} - G_{r_{ε}} ({\tilde{u}}_{ε}) |}_{Ω_{Y_{ε}}^{T}}^{2} & ⩽ C (\frac{ε^{2}}{r_{ε}^{N - 2}}) | \nabla {\tilde{u}}_{ε} |_{Ω^{T}}^{2} ⩽ C (\frac{ε^{2}}{δ_{ε}^{N - 2}}) (\frac{δ_{ε}^{N - 2}}{r_{ε}^{N - 2}}) \\ (4.9) & ⩽ C {(\frac{δ_{ε}}{r_{ε}})}^{N - 2} \to 0 . \end{array}$ Then, in view of (4.5) and since $| Ω ∖ Ω_{Y_{ε}} | \to 0$ , we have $\begin{matrix} lim_{ε \to 0} {| u - G_{r_{ε}} ({\tilde{u}}_{ε}) |}_{Ω^{T}} ⟶ 0 . \end{matrix}$ Now, for proving (4.7), we use Lemma 3.4 and (4.6) to get $\begin{array}{l} {| G_{δ_{ε}} ({\tilde{u}}_{ε}) |}_{L^{2} (Ω^{T})} & ⩽ {| G_{δ_{ε}} ({\tilde{u}}_{ε}) - G_{r_{ε}} ({\tilde{u}}_{ε}) |}_{L^{2} (Ω^{T})} + {| G_{r_{ε}} ({\tilde{u}}_{ε}) |}_{L^{2} (Ω^{T})} \\ ⩽ \frac{C}{{(γ_{ε})}^{1 / 2}} | \nabla {\tilde{u}}_{ε} |_{L^{2} (Ω^{T})} + C ⩽ C . \end{array}$

Finally, using Lemma 3.4 and (2.4), we obtain $\begin{array}{l} \int_{0}^{T} {\int -}_{D_{ε}} {| {\tilde{u}}_{ε} - G_{δ_{ε}} ({\tilde{u}}_{ε}) |}^{2} d x d t & ⩽ C {(ε δ_{ε})}^{2} \int_{0}^{T} {\int -}_{D_{ε}} | \nabla {\tilde{u}}_{ε} |^{2} d x d t \\ ⩽ \frac{C}{γ_{ε} b_{ε}} ⟶ 0 . \end{array}$ □

Let us mention an important result which is an adaptation of Proposition 4.3 [5].
Proposition 4.3.
Under the hypotheses of Proposition* 4.2 , for any $φ \in L^{2} (0, T; H_{0}^{1} (Ω))$ , $\begin{matrix} (4.10) & \int_{0}^{T} {\int -}_{D_{ε}} {\tilde{u}}_{ε} φ d x d t ⟶ \int_{Ω^{T}} v φ d x d t, \end{matrix}$ on the subsequence for which ( 4.4 )–( 4.8 ) hold.

For $φ, ψ \in D (Ω)$ , we introduce the test function $\begin{matrix} (4.11) & Φ_{ε} : = (1 - w_{r_{ε}}) φ + (w_{ε} - 1) φ + w_{r_{ε}} G_{δ_{ε}} (ψ) . \end{matrix}$
Remark 4.4.
We have in this case $\begin{matrix} (4.12) & Φ_{ε} ⇀ φ in H_{0}^{1} (Ω) . \end{matrix}$

The main results of the present paper is the following theorem:
Theorem 4.5.
Assume that the hypotheses of Proposition 4.2 and ( 3.22 ) hold true. Then the limit $(u, v)$ defined in Proposition 4.2 is the unique solution of the homogenized problem $\begin{matrix} (4.13) & \{\begin{matrix} \frac{\partial u}{\partial t} - Δ u + μ u + (N - 2) γ S_{N} (u - v) = f & in Ω^{T}, \\ a \frac{\partial v}{\partial t} + (N - 2) γ S_{N} (v - u) = 0 & in Ω^{T}, \\ u (0) = u_{0} & in Ω, \\ v (0) = v_{0} & in Ω, \end{matrix} \end{matrix}$ where $S_{N}$ is the measure of the unit sphere in $R^{N}$ .

Before proving Theorem 4.5, we need the following technical propositions whose proofs are postponed at the end of this subsection. Proposition 4.6.
Under the hypotheses of Theorem 4.5 , for any $ζ \in D ([0, T [)$ , and for a subsequence verifying the convergences of Proposition 4.2 , we have $\begin{matrix} (4.14) & lim_{ε \to 0} \int_{0}^{T} \int_{Ω} ρ_{ε} {\tilde{u}}_{ε} Φ_{ε} ζ^{'} (t) d t d x = \int_{0}^{T} \int_{Ω} u φ ζ^{'} (t) d x d t + a \int_{0}^{T} \int_{Ω} v ψ ζ^{'} (t) d x d t . \end{matrix}$
Proposition 4.7.
Under the hypotheses of Theorem 4.5 , for any $ζ \in D ([0, T [)$ and for a subsequence satisfying the convergences of Proposition 4.2 , we have $\begin{array}{l} lim_{ε \to 0} \int_{0}^{T} \int_{Ω} ν_{ε} \nabla {\tilde{u}}_{ε} \nabla Φ_{ε} ζ d x d t \\ = \int_{0}^{T} \int_{Ω} \nabla u \nabla φ ζ d x d t \\ (4.15) & + \int_{0}^{T} \int_{Ω} φ u ζ d μ d t + (N - 2) γ S_{N} \int_{0}^{T} \int_{Ω} (u - v) (φ - ψ) ζ d x d t . \end{array}$
Proof of the Theorem 4.5.
Choose $w = Φ_{ε}$ in the variational formulation 2.1, integrating it over $[0, T]$ after multiplication by $ζ \in D ([0, T [)$ , we get $\begin{array}{l} - \int_{0}^{T} \int_{Ω_{ε}} ρ_{ε} u_{ε} Φ_{ε} ζ^{'} (t) d x d t + \int_{0}^{T} \int_{Ω_{ε}} ν_{ε} \nabla u_{ε} \nabla Φ_{ε} ζ (t) d x d t \\ (4.16) & = \int_{0}^{T} \int_{Ω_{ε}} f_{ε} Φ_{ε} ζ (t) d x d t + \int_{Ω_{ε}} ρ_{ε} u_{ε}^{0} Φ_{ε} ζ (0) d x . \end{array}$ Extending by zero outside $Ω_{ε}$ , gives $\begin{array}{l} - \int_{0}^{T} \int_{Ω} ρ_{ε} \tilde{u_{ε}} Φ_{ε} ζ^{'} (t) d x d t + \int_{0}^{T} \int_{Ω} ν_{ε} \nabla \tilde{u_{ε}} \nabla Φ_{ε} ζ (t) d x d t \\ (4.17) & = \int_{0}^{T} \int_{Ω} {\tilde{f}}_{ε} Φ_{ε} ζ (t) d x d t + \int_{Ω} ρ_{ε} {\tilde{u}}_{ε}^{0} Φ_{ε} ζ (0) d x . \end{array}$

We now pass successively to the limit in each term of (4.17) by using Propositions 4.6 and 4.7. Thanks to (1.16) and (4.12) $\begin{array}{l} - \int_{0}^{T} \int_{Ω} u φ ζ^{'} (t) d x d t - a \int_{0}^{T} \int_{Ω} v ψ ζ^{'} (t) d x d t + \int_{0}^{T} \int_{Ω} \nabla u \nabla φ ζ (t) d x d t \\ + \int_{0}^{T} \int_{Ω} u φ ζ (t) d μ d t + (N - 2) γ S_{N} \int_{0}^{T} \int_{Ω} (u - v) (φ - ψ) ζ (t) d x d t \\ = \int_{0}^{T} \int_{Ω} f φ ζ d x d t + ζ (0) \int_{Ω} (u_{0} φ + a v_{0} ψ) d x . \end{array}$ □
Proof of Proposition 4.6.
We first write $\begin{array}{l} \int_{0}^{T} \int_{Ω} ρ_{ε} {\tilde{u}}_{ε} Φ_{ε} ζ^{'} (t) d x d t \\ = \int_{0}^{T} \int_{Ω_{ε}^{'} ∖ C_{ε}} {\tilde{u}}_{ε} φ ζ^{'} (t) d x d t \\ + \int_{0}^{T} \int_{C_{T_{ε}}} {\tilde{u}}_{ε} w_{ε} φ ζ^{'} (t) d x d t + a_{ε} \int_{0}^{T} \int_{D_{ε}} {\tilde{u}}_{ε} G_{δ_{ε}} (ψ) ζ^{'} (t) d x d t \\ (4.18) & + \int_{0}^{T} \int_{C_{D_{ε}}} {\tilde{u}}_{ε} ((1 - w_{r_{ε}}) φ + w_{r_{ε}} G_{δ_{ε}} (ψ)) ζ^{'} (t) d x d t . \end{array}$

Furthermore, $\begin{array}{rcl} \int_{0}^{T} \int_{Ω_{ε}^{'} ∖ C_{ε}} {\tilde{u}}_{ε} φ ζ^{'} (t) d x d t = \int_{0}^{T} \int_{Ω_{ε}^{'} ∖ C_{D_{ε}}} {\tilde{u}}_{ε} φ ζ^{'} (t) d x d t - \int_{0}^{T} \int_{C_{T_{ε}}} {\tilde{u}}_{ε} φ ζ^{'} (t) d x d t . \end{array}$

Replacing this later in (4.18), we obtain $\begin{array}{l} \int_{0}^{T} \int_{Ω} ρ_{ε} {\tilde{u}}_{ε} Φ_{ε} ζ^{'} (t) d x d t \\ = \int_{0}^{T} \int_{Ω_{ε}^{'} ∖ C_{D_{ε}}} {\tilde{u}}_{ε} φ ζ^{'} (t) d x d t \\ + \int_{0}^{T} \int_{C_{T_{ε}}} {\tilde{u}}_{ε} (w_{ε} - 1) φ ζ^{'} (t) d x d t + a_{ε} \int_{0}^{T} \int_{D_{ε}} {\tilde{u}}_{ε} G_{δ_{ε}} (ψ) ζ^{'} (t) d x d t \\ (4.19) & + \int_{0}^{T} \int_{C_{D_{ε}}} {\tilde{u}}_{ε} ((1 - w_{r_{ε}}) φ + w_{r_{ε}} G_{δ_{ε}} (ψ)) ζ^{'} (t) d x d t . \end{array}$

We will pass successively to the limit in each term of the right-hand side of the above equality. For the first integral, observe that $\begin{matrix} \int_{0}^{T} \int_{Ω_{ε}^{'} ∖ C_{D_{ε}}} {\tilde{u}}_{ε} φ ζ^{'} (t) d x d t = \int_{0}^{T} \int_{Ω} χ_{Ω_{ε}^{'} ∖ C_{D_{ε}}} {\tilde{u}}_{ε} φ ζ^{'} (t) d x d t . \end{matrix}$ Since $χ_{Ω_{ε}^{'} ∖ C_{D_{ε}}} \to 1 in L^{2} (Ω^{T})$ , using (4.5), we conclude that $\begin{matrix} (4.20) & lim_{ε \to 0} \int_{0}^{T} \int_{Ω_{ε}^{'} ∖ C_{D_{ε}}} {\tilde{u}}_{ε} φ ζ^{'} (t) d x d t = \int_{0}^{T} \int_{Ω} u φ ζ^{'} (t) d x d t . \end{matrix}$

We estimate the second term as follows: $\begin{array}{l} | \int_{0}^{T} \int_{C_{T_{ε}}} {\tilde{u}}_{ε} (w_{ε} - 1) φ ζ^{'} d x d t | & ⩽ \int_{0}^{T} \int_{Ω} | {\tilde{u}}_{ε} | | (w_{ε} - 1) | | ζ^{'} φ | d x d t \\ ⩽ {| ζ^{'} φ |}_{L^{\infty} (Ω^{T})} | {\tilde{u}}_{ε} |_{Ω^{T}} | w_{ε} - 1 |_{Ω^{T}} ⟶ 0, \end{array}$ where we have used (H.3) and (4.5).

For the term on $D_{ε}$ in (4.19), we see that $\begin{array}{l} a_{ε} \int_{0}^{T} \int_{D_{ε}} {\tilde{u}}_{ε} G_{δ_{ε}} (ψ) ζ^{'} (t) d x d t = & a_{ε} | D_{ε} | \int_{0}^{T} {\int -}_{D_{ε}} {\tilde{u}}_{ε} (G_{δ_{ε}} (ψ) - ψ) ζ^{'} (t) d x d t \\ + a_{ε} | D_{ε} | \int_{0}^{T} {\int -}_{D_{ε}} {\tilde{u}}_{ε} ψ ζ^{'} (t) d x d t, \end{array}$ so that, by (2.3), (3.17), Proposition 4.3 and the convergence $a_{ε} | D_{ε} | \to a$ , we obtain $\begin{matrix} (4.21) & a_{ε} \int_{0}^{T} \int_{D_{ε}} {\tilde{u}}_{ε} G_{δ_{ε}} (ψ) ζ^{'} (t) d x d t \to a \int_{0}^{T} \int_{Ω} v ψ ζ^{'} (t) d x d t . \end{matrix}$

For the term on $C_{D_{ε}}$ , using (4.5), (3.6), (3.17) and since $| C_{D_{ε}} | \to 0$ , we obtain $\begin{array}{l} | \int_{0}^{T} \int_{C_{D_{ε}}} {\tilde{u}}_{ε} ((1 - w_{r_{ε}}) φ + w_{r_{ε}} G_{δ_{ε}} (ψ)) ζ^{'} (t) d x d t | \\ ⩽ {(\int_{0}^{T} \int_{C_{D_{ε}}} | {\tilde{u}}_{ε} |^{2} d x d t)}^{1 / 2} {(\int_{0}^{T} \int_{C_{D_{ε}}} {| ((1 - w_{r_{ε}}) φ + w_{r_{ε}} G_{δ_{ε}} (ψ)) ζ^{'} (t) |}^{2} d x d t)}^{1 / 2} \\ ⩽ C | {\tilde{u}}_{ε} |_{Ω^{T}} T^{1 / 2} | C_{D_{ε}} |^{1 / 2} ⟶ 0 . \end{array}$ □
Proof of Proposition 4.7.
For any $ζ \in D ([0, T [)$ , we have $\begin{array}{l} \int_{0}^{T} \int_{Ω} ν_{ε} \nabla {\tilde{u}}_{ε} \nabla Φ_{ε} ζ (t) d x d t \\ = \int_{0}^{T} \int_{Ω_{ε}^{'} ∖ C_{ε}} \nabla {\tilde{u}}_{ε} \nabla φ ζ d x d t + \int_{0}^{T} \int_{C_{T_{ε}}} \nabla {\tilde{u}}_{ε} \nabla φ ζ d x d t \\ + \int_{0}^{T} \int_{C_{T_{ε}}} \nabla {\tilde{u}}_{ε} \nabla φ (w_{ε} - 1) ζ d x d t + \int_{0}^{T} \int_{C_{T_{ε}}} \nabla {\tilde{u}}_{ε} \nabla w_{ε} φ ζ d x d t \\ + \int_{0}^{T} \int_{C_{D_{ε}}} \nabla {\tilde{u}}_{ε} \nabla w_{r_{ε}} (G_{δ_{ε}} (ψ) - G_{δ_{ε}} (φ)) ζ d x d t \\ + \int_{0}^{T} \int_{C_{D_{ε}}} \nabla {\tilde{u}}_{ε} \nabla w_{r_{ε}} (G_{δ_{ε}} (φ) - φ) ζ d x d t \\ + \int_{0}^{T} \int_{C_{D_{ε}}} \nabla {\tilde{u}}_{ε} (1 - w_{r_{ε}}) \nabla φ ζ d x d t \\ (4.22) & : = I_{1} + I_{2} + I_{3} + I_{4} + I_{5} + I_{6} + I_{7} . \end{array}$

We will pass successively to the limit in each term of the right-hand side of the above equality. We see that $\begin{array}{l} I_{1} + I_{2} & = \int_{0}^{T} \int_{Ω_{ε}^{'} ∖ C_{ε}} \nabla {\tilde{u}}_{ε} \nabla φ ζ d x d t + \int_{0}^{T} \int_{C_{T_{ε}}} \nabla {\tilde{u}}_{ε} \nabla φ ζ d x d t \\ = \int_{0}^{T} \int_{Ω_{ε}^{'} ∖ C_{D_{ε}}} \nabla {\tilde{u}}_{ε} \nabla φ ζ d x d t . \end{array}$ The Lebesgue dominated theorem gives $\nabla φ 1_{χ_{Ω_{ε}^{'} ∖ C_{D_{ε}}}} \to \nabla φ in L^{2} (Ω)$ , and taking into account (4.5), we obtain $\begin{matrix} I_{1} + I_{2} ⟶ \int_{0}^{T} \int_{Ω} \nabla u \nabla φ ζ d x d t . \end{matrix}$

Now we study the third term in (4.22). We get by (4.5) and (H.3), $\begin{array}{l} | I_{3} | & ⩽ \int_{0}^{T} \int_{Ω} | χ_{C_{T_{ε}}} \nabla {\tilde{u}}_{ε} (w_{ε} - 1) \nabla φ ζ | d x d t \\ ⩽ \sqrt{T} | w_{ε} - 1 |_{Ω} | \nabla φ |_{L^{\infty} (Ω)} | ζ |_{L^{\infty} (0, T)} | \nabla {\tilde{u}}_{ε} |_{Ω^{T}} \\ ⩽ C | w_{ε} - 1 |_{Ω} ⟶ 0 . \end{array}$

As for the fourth term in (4.22), consider the function $U_{ε} \in H_{0}^{1} (Ω)$ defined by $\begin{matrix} U_{ε} : = \int_{0}^{T} {\tilde{u}}_{ε} ζ d t . \end{matrix}$

Convergence (4.5) implies that the sequence $U_{ε}$ is such that $\begin{matrix} (4.23) & \{\begin{matrix} U_{ε} ⇀ U = \int_{0}^{T} u ζ d t & weakly in H_{0}^{1} (Ω) and strongly in L^{2} (Ω), \\ U_{ε} = 0 & on T_{ε} . \end{matrix} \end{matrix}$

Using the fact that $\begin{array}{rcl} \nabla {\tilde{u}}_{ε} φ = \nabla (φ {\tilde{u}}_{ε}) - {\tilde{u}}_{ε} \nabla φ, \end{array}$ from $I_{4}$ we obtain $\begin{matrix} (4.24) & I_{4} = \int_{0}^{T} \int_{Ω} \nabla (φ {\tilde{u}}_{ε}) \nabla w_{ε} ζ d x d t - \int_{0}^{T} \int_{Ω} {\tilde{u}}_{ε} \nabla φ \nabla w_{ε} ζ d x d t . \end{matrix}$

Applying the Fubini Theorem in (4.24), yields $\begin{array}{l} I_{4} & = \int_{Ω} (\int_{0}^{T} \nabla (φ {\tilde{u}}_{ε}) ζ d t) \nabla w_{ε} d x - \int_{Ω} (\int_{0}^{T} {\tilde{u}}_{ε} ζ d t) \nabla φ \nabla w_{ε} d x \\ = - ⟨ Δ w_{ε}, φ U_{ε} ⟩ + \int_{\partial Ω} \frac{\partial w_{ε}}{\partial n} φ U_{ε} d σ - \int_{Ω} U_{ε} \nabla φ \nabla w_{ε} d x \\ (4.25) & = - ⟨ λ_{ε}, U_{ε} φ ⟩ + ⟨ μ_{ε}, U_{ε} φ ⟩ - \int_{Ω} U_{ε} \nabla φ \nabla w_{ε} d x . \end{array}$

Using (3.22) and (H.4), we have $\begin{matrix} (4.26) & - ⟨ λ_{ε}, φ U_{ε} ⟩ + ⟨ μ_{ε}, φ U_{ε} ⟩ ⟶ ⟨ μ, φ U ⟩ . \end{matrix}$

Again by Fubini’s Theorem we have $\begin{matrix} (4.27) & ⟨ μ, φ U ⟩ = \int_{Ω} (\int_{0}^{T} u ζ d t) φ d μ = \int_{0}^{T} \int_{Ω} u φ ζ d μ d t . \end{matrix}$

As $U_{ε} \to U in L^{2} (Ω)$ and $\nabla w_{ε} ⇀ 0 in L^{2} (Ω)$ , $\begin{matrix} (4.28) & \int_{Ω} U_{ε} \nabla φ \nabla w_{ε} d x ⟶ 0 . \end{matrix}$ Finally, we conclude that $\begin{matrix} (4.29) & I_{4} ⟶ \int_{0}^{T} \int_{Ω} u φ ζ d μ d t . \end{matrix}$

For the term $I_{5}$ , a direct computational gives $\begin{matrix} I_{5} = \sum_{k \in Z_{ε}} ({\int -}_{S_{δ_{ε}}^{k}} ψ d σ - {\int -}_{S_{δ_{ε}}^{k}} φ d σ) \int_{0}^{T} \int_{C_{D_{ε}}^{k}} \nabla u_{ε} \nabla w_{r_{ε}} ζ d x d t . \end{matrix}$

Let us introduce $\begin{matrix} {\bar{u}}_{ε}^{k} (\bar{y}, t) = u_{ε} (x, t) with \bar{y} = \frac{x}{ε} - k - y_{1}, for x \in Y_{ε}^{k}, \end{matrix}$ so, by substitution $\begin{array}{l} \int_{C (δ_{ε}, r_{ε})} \nabla {\bar{u}}_{ε}^{k} (\bar{y}, t) \nabla W_{r_{ε}} (\bar{y}) d \bar{y} \\ = \int_{0}^{2 π} \int_{0}^{π} \dots \int_{0}^{π} {sin}^{N - 2} θ_{1} {sin}^{N - 3} θ_{2} \dots sin θ_{N - 2} d θ_{1} d θ_{2} \dots d θ_{N - 2} d θ_{N - 1} \\ \times \int_{δ_{ε}}^{r_{ε}} \frac{\partial {\bar{u}}_{ε}^{k}}{\partial r} (\bar{y}) \frac{d W_{r_{ε}}}{d r} r^{N - 1} d r . \end{array}$ With this notation, we find that $\begin{array}{l} I_{5} = & ε^{N - 2} \sum_{k \in Z_{ε}} ({\int -}_{S_{δ_{ε}}^{k}} ψ - {\int -}_{S_{δ_{ε}}^{k}} φ) \int_{0}^{2 π} \int_{0}^{π} \dots \int_{0}^{π} {sin}^{N - 2} θ_{1} {sin}^{N - 3} θ_{2} \dots sin θ_{N - 2} d θ_{1} \dots d θ_{N - 1} \\ \times \int_{δ_{ε}}^{r_{ε}} (\int_{0}^{T} \frac{\partial {\bar{u}}_{ε}^{k}}{\partial r} (\bar{y}) ζ (t) d t) \frac{d W_{r_{ε}}}{d r} r^{N - 1} d r \\ = & \frac{ε^{N - 2} (N - 2) {(δ_{ε} r_{ε})}^{N - 2}}{(r_{ε}^{N - 2} - δ_{ε}^{N - 2})} \sum_{k \in Z_{ε}} ({\int -}_{S_{δ_{ε}}^{k}} ψ d σ - {\int -}_{S_{δ_{ε}}^{k}} φ d σ) \\ \times \int_{S_{N}} \int_{0}^{T} ({\bar{u}}_{ε}^{k} |_{| \bar{y} | = δ_{ε}} - {\bar{u}}_{ε}^{k} |_{| \bar{y} | = r_{ε}}) ζ d t d σ \\ = & \frac{S_{N} (N - 2) ε^{N - 2} {(δ_{ε} r_{ε})}^{N - 2}}{ε^{N} (r_{ε}^{N - 2} - δ_{ε}^{N - 2})} \int_{0}^{T} \int_{Ω} (G_{δ_{ε}} ({\tilde{u}}_{ε}) - G_{r_{ε}} ({\tilde{u}}_{ε})) (G_{δ_{ε}} (ψ) - G_{δ_{ε}} (φ)) ζ (t) d x d t . \end{array}$

Then, by using (4.6), (4.7), (4.1) and the uniform convergence of $G_{δ_{ε}} (φ)$ to φ, we get $\begin{matrix} (4.30) & I_{5} ⟶ γ (N - 2) S_{N} \int_{Ω^{T}} (v - u) (ψ - φ) ζ (t) d x d t . \end{matrix}$

Using the boundedness of $| \nabla w_{r_{ε}} |_{L^{2} (Ω)}, | \nabla {\tilde{u}}_{ε} |_{Ω^{T}}$ , thanks to (3.18), we obtain $\begin{matrix} (4.31) & | I_{6} | ⩽ | \nabla {\tilde{u}}_{ε} |_{Ω^{T}} | \nabla w_{r_{ε}} ζ |_{Ω^{T}} {| G_{δ_{ε}} (φ) - φ |}_{L^{\infty} (Ω)} ⟶ 0 . \end{matrix}$

Finally, for the last term, we have $\nabla φ χ_{C_{D_{ε}}} \to 0 in L^{2} (Ω^{T}), \nabla {\tilde{u}}_{ε} ⇀ \nabla u in L^{2} (Ω^{T})$ , since $(1 - w_{r_{ε}})$ is bounded in $L^{\infty} (Ω)$ , $\begin{matrix} I_{7} = \int_{0}^{T} \int_{Ω} \nabla {\tilde{u}}_{ε} \nabla χ_{C_{D_{ε}}} φ (1 - w_{r_{ε}}) ζ (t) d x d t ⟶ 0 . \end{matrix}$ □

4.2. Homogenization in the case $δ_{ε} ≪ ε^{2 / (N - 2)}$

In this section, the order of the size of the particles is smaller than the critical size, this size implies that $r_{ε} \to 0, and w_{r_{ε}} \to 0 in H_{0}^{1} (Ω)$ . We study the homogenization under the Cioranescu–Murat hypotheses (3.22) on the holes.

Theorem 4.8.
Assume that ( 3.22 ) and ( 1.13 )–( 1.16 ) hold true. Then there exists $\begin{matrix} u \in L^{\infty} (0, T; L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω)) \end{matrix}$ such that, $\begin{array}{l} (4.32) & \tilde{u_{ε}} \overset{}{⇀} u in L^{\infty} (0, T; L^{2} (Ω)), \tilde{u_{ε}} ⇀ u in L^{2} (0, T; H_{0}^{1} (Ω)), \\ (4.33) & \frac{1}{| D_{ε} |} u_{ε} χ_{D_{ε}} \to v_{0} in D^{'} (Ω) a.e. t \in [0, T], \end{array}$ where u is the only solution of the problem* $\begin{matrix} (4.34) & \{\begin{matrix} \frac{\partial u}{\partial t} - Δ u + μ u = f & in Ω^{T}, \\ u (0) = u_{0} & in Ω . \end{matrix} \end{matrix}$
Proof.
According to (2.2), for some subsequences, (4.32) hold true. For (4.33), hypothesis (2.3) and Lemma A-2 in [2] insure the existence of $v \in L^{\infty} (0, T; L^{2} (Ω))$ that satisfies $\begin{matrix} (4.35) & \frac{1}{| D_{ε} |} u_{ε} χ_{D_{ε}} \to v in D^{'} (Ω) a.e. t \in [0, T] . \end{matrix}$ Now, we must prove that $v = v_{0}$ . Taking the test function defined by the formula (4.11) in the variational formulation (2.1), we get $\begin{array}{l} - \int_{0}^{T} \int_{Ω_{ε}} ρ_{ε} u_{ε} Φ_{ε} ζ^{'} d x d t + \int_{0}^{T} \int_{Ω_{ε}} ν_{ε} \nabla u_{ε} \nabla Φ_{ε} ζ d x d t \\ (4.36) & = \int_{0}^{T} \int_{Ω_{ε}} f_{ε} Φ_{ε} ζ d t + \int_{Ω} ρ_{ε} u_{ε}^{0} Φ_{ε} ζ (0) d x . \end{array}$

Since $w_{r_{ε}} \to 0 in H_{0}^{1} (Ω)$ , we pass to the limit and we get in a simple way $\begin{array}{l} - \int_{Ω} (\int_{0}^{T} u ζ^{'} (t) d t) φ d x - a \int_{Ω} (\int_{0}^{T} v ζ^{'} (t) d t) ψ d x \\ + \int_{Ω} (\int_{0}^{T} \nabla u ζ (t) d t) \nabla φ d x + \int_{Ω} (\int_{0}^{T} u ζ d t) φ d μ \\ (4.37) & = \int_{Ω} (\int_{0}^{T} f ζ (t) d t) φ d x + \int_{Ω} u_{0} φ ζ (0) d x + a \int_{Ω} v_{0} ψ ζ (0) d x, \end{array}$ for all $ζ \in D (] 0, T [)$ . Taking herein $φ = 0$ , yields $\begin{matrix} (4.38) & - a \int_{0}^{T} \int_{Ω} v ψ ζ^{'} (t) d x d t = a \int_{Ω} v_{0} ψ ζ (0) d x, \end{matrix}$ which shows that v is independent of t. As $v \in C^{0} ([0, T]; L^{2} (Ω))$ , this means that $v = v_{0}$ . Taking $ψ = 0$ proves (4.34), the last one also holding in the sense of $C^{0} ([0, T]; L^{2} (Ω))$ . □

4.3. Homogenization for the case $δ_{ε} ≫ ε^{2 / (N - 2)}$

In this section, we have $γ = + \infty$ , we suppose that the holes still satisfy the Cioranescu–Murat hypotheses (3.22).

Theorem 4.9.

Assume that ( 3.22 ) and ( 1.13 )–( 1.16 ) hold true. Then there exists $\begin{matrix} u \in L^{\infty} (0, T; L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω)), \end{matrix}$ such that the convergences ( 4.4 )–( 4.5 ) are true for the whole sequence. Moreover, $\begin{array}{l} (4.39) & G_{r_{ε}} ({\tilde{u}}_{ε}) ⟶ u in L^{2} (Ω^{T}), G_{δ_{ε}} ({\tilde{u}}_{ε}) ⟶ u in L^{2} (Ω^{T}), \\ (4.40) & lim_{ε \to 0} \int_{0}^{T} {\int -}_{D_{ε}} {| {\tilde{u}}_{ε} - G_{δ_{ε}} ({\tilde{u}}_{ε}) |}^{2} d x d t = 0, \end{array}$ where the limit u is the unique solution of the following homogenized problem: $\begin{matrix} (4.41) & \{\begin{matrix} (1 + a) \frac{\partial u}{\partial t} - Δ u + μ u = f & in Ω^{T}, \\ u (0) = \frac{1}{(1 + a)} u_{0} + \frac{a}{(1 + a)} v_{0} & in Ω . \end{matrix} \end{matrix}$

Proof.

The proof of the result is similar to the corresponding one for the first case where $γ \in] 0, + \infty [$ , with a few differences that are shown below. The test function $Φ_{ε}$ is given by $\begin{matrix} Φ_{ε} = (1 - w_{r_{ε}}) φ + (w_{ε} - 1) φ + w_{r_{ε}} G_{δ_{ε}} (φ) . \end{matrix}$ In this case, the term $I_{6}$ (see proof of Proposition 4.7 ) is handled as follows: $\begin{array}{l} | \int_{0}^{T} \int_{C_{D_{ε}}} \nabla u_{ε} \nabla w_{r_{ε}} (G_{δ_{ε}} (φ) - φ) ζ d x d t | \\ ⩽ C T^{1 / 2} | \nabla {\tilde{u}}_{ε} |_{Ω^{T}} | \nabla w_{r_{ε}} |_{Ω} | ζ |_{L^{\infty} (0, T)} {| G_{δ_{ε}} (φ) - φ |}_{L^{\infty} (C_{D_{ε}})} \\ ⩽ C {(γ_{ε})}^{\frac{1}{2}} ε r_{ε} = C {(\frac{{δ_{ε}}^{N - 2}}{ε^{2}})}^{1 / 2} (ε r_{ε}) = C {(δ_{ε}^{N - 2})}^{1 / 2} r_{ε} ⟶ 0, \end{array}$ where we have used (3.7), (3.18) and (4.5).

Using a priori estimates of Proposition 3.2, (3.18) and hypotheses (1.13)–(1.16), the same reason as above gives $\begin{matrix} \int_{Ω} ρ_{ε} {\tilde{u}}_{ε}^{0} Φ_{ε} ζ (0) d x ⟶ ζ (0) \int_{Ω} (u_{0} + a v_{0}) φ d x, \end{matrix}$ and this ends the proof. □

References

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Diffusion process in a perforated domain around a vanishing suspension

Abstract

Keywords

1. Introduction and notations

2. A priori estimates

2.1. Existence and uniqueness

3.1. Tools of the control-zone method

4.1. Homogenization in the case δ ε = O ( ε 2 / ( N − 2 ) )

References

4.1. Homogenization in the case $δ_{ε} = O (ε^{2 / (N - 2)})$