Corrector for a diffusion process in a rarefied binary structure

Abstract

The aim of this paper is to provide corrector results associated to 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 a ε-periodic network. The suspension has a total mass of unity order and a vanishing volume. The results obtained here complete the earlier study by Bentalha et al. [Revue Roumaine de Math. Pures Appl. 52(2) (2007), 129–149] on the asymptotic behavior of this problem.

Keywords

diffusion periodic homogenization corrector non-local effects fine-scale substructure

1. Introduction and setting of the problem

In this paper we study the corrector results for the following problem: $\begin{matrix} (1.1) & \{\begin{matrix} ρ^{ε} \frac{\partial u^{ε}}{\partial t} - div (k^{ε} \nabla u^{ε}) = f^{ε} & in Ω^{T}, \\ {[u^{ε}]}_{ε} = 0 & on \partial D_{ε} \times] 0, T [, \\ {[k^{ε} \nabla u^{ε}]}_{ε} n = 0 & on \partial D_{ε} \times] 0, T [, \\ u^{ε} = 0 & on \partial Ω \times] 0, T [, \\ u^{ε} (0) = u_{0}^{ε} & in Ω, \end{matrix} \end{matrix}$ where $Ω \subseteq R^{3}$ is a bounded Lipschitz domain such that $| Ω | = 1$ , $f^{ε} \in L^{2} (Ω^{T})$ , $u_{0}^{ε} \in L^{2} (Ω)$ , $\begin{matrix} (1.2) & ρ^{ε} (x) = \{\begin{matrix} 1 & if x \in Ω_{ε}, \\ a_{ε} & if x \in D_{ε}, \end{matrix} k^{ε} (x) = \{\begin{matrix} 1 & if x \in Ω_{ε}, \\ b_{ε} & if x \in D_{ε}, \end{matrix} \end{matrix}$ and $Ω_{ε}$ , $D_{ε}$ are subsets of Ω defined as follows.

Let $\begin{matrix} Y : = {(- \frac{1}{2}, + \frac{1}{2})}^{3} \end{matrix}$ be the periodicity representative cell. For two small parameters ε and $r_{ε}$ such that $0 < r_{ε} ≪ ε$ , we consider $\begin{matrix} \begin{matrix} Y_{ε}^{k} : = ε k + ε Y, k \in Z^{3}, \\ Ω_{Y_{ε}} : = interior ⋃_{k \in Z_{ε}} \overline{Y_{ε}^{k}}, D_{ε} : = ⋃_{k \in Z_{ε}} B (ε k, r_{ε}), \end{matrix} \end{matrix}$ where $Z_{ε} : = {k \in Z^{3}, Y_{ε}^{k} \subset Ω}$ , and $B (ε k, r_{ε})$ is the ball of radius $r_{ε}$ centered at $ε k$ .

The set $D_{ε}$ is a suspension $ε Y$ -periodical distributed in Ω, and has obviously a vanishing volume as $ε \to 0$ ; $Ω_{ε} : = Ω ∖ \overline{D_{ε}}$ is the ambiental connected domain; $a_{ε} > 0$ and $b_{ε} > 0$ are respectively, the relative mass density and diffusivity of the suspension; $Ω^{T} : = Ω \times] 0, T [$ (similar definitions for $Ω_{ε}^{T}$ , $Ω_{Y_{ε}}^{T}$ and $D_{ε}^{T}$ ); ${[\cdot]}_{ε}$ is the jump across the interface $\partial D_{ε}$ ; n is the normal on $\partial D_{ε}$ in the outward direction.

The variational formulation of the problem (1.1) is:

Find $u^{ε} \in L^{2} (0, T; H_{0}^{1} (Ω)) \cap L^{\infty} (0, T; L^{2} (Ω))$ satisfying (in some sense) the initial condition ( 1.1 )⁵ and the equation, $\begin{matrix} (1.3) & \frac{d}{d t} {(ρ^{ε} u^{ε}, w)}_{Ω} + {(k^{ε} \nabla u^{ε}, \nabla w)}_{Ω} = {(f^{ε}, w)}_{Ω} in D^{'} (0, T), \forall w \in H_{0}^{1} (Ω) . \end{matrix}$

Under the above hypotheses, classical results (see [14]) provide the existence of a unique solution $u^{ε}$ of (1.3). Moreover $\frac{\partial u^{ε}}{\partial t} \in L^{2} (0, T; H^{- 1} (Ω))$ .

In the sequel we assume the following hypotheses: $\begin{matrix} (1.4) & \begin{matrix} lim_{ε \to 0} a_{ε} | D_{ε} | = a > 0, \\ b_{ε} ⩾ b_{0} > 0, \forall ε > 0, \\ f^{ε} ⇀ f in L^{2} (Ω^{T}), \end{matrix} \end{matrix}$ and $\begin{matrix} (1.5) & \begin{matrix} u_{0}^{ε} ⇀ u_{0} in L^{2} (Ω), \\ ⨍_{D_{ε}} {| u_{0}^{ε} |}^{2} d x ⩽ C, \\ \frac{1}{| D_{ε} |} u_{0}^{ε} χ_{D_{ε}} ⇀ v_{0} in D^{'} (Ω) with v_{0} \in L^{2} (Ω), \end{matrix} \end{matrix}$ where for any $D \subset Ω$ , we used the notation $\begin{matrix} ⨍_{D} \cdot d x = \frac{1}{| D |} \int_{D} \cdot d x . \end{matrix}$

Hypothesis (1.4)¹ says that the total mass of the suspension is of unity order.

The homogenization of this problem was studied by Bentalha et al. [7], by means of the energy method adapted for fine-scale substructures, called the control-zone method. Under the previous hypotheses three cases were distinguished according to the values of the rarefaction number $\begin{matrix} γ = lim_{ε \to 0} \frac{r_{ε}}{ε^{3}} . \end{matrix}$ When $γ > 0$ and ${lim}_{ε \to 0} b_{ε} = + \infty$ , the sequence ${(u^{ε})}_{ε > 0}$ satisfies $\begin{matrix} \begin{matrix} u^{ε} ⇀ u weakly in L^{2} (0, T; H_{0}^{1} (Ω)), \\ \frac{1}{| D_{ε} |} u^{ε} χ_{D_{ε}} ⇀ v weakly in L^{2} (0, T; H^{- 1} (Ω)), \end{matrix} \end{matrix}$ where the limit $(u, v)$ is the unique solution of the coupled system $\begin{matrix} (1.6) & \{\begin{matrix} \frac{\partial u}{\partial t} - Δ u + 4 π γ (u - v) = f & in Ω^{T}, \\ a \frac{\partial v}{\partial t} + 4 π γ (v - u) = 0 & in Ω^{T}, \\ u = 0 & on \partial Ω \times] 0, T [, \\ u (0) = u_{0}, v (0) = v_{0} & in Ω . \end{matrix} \end{matrix}$

In the other two cases, where $γ = + \infty$ or $γ = 0$ , and with only condition (1.4)² on $b_{ε}$ , we also have $\begin{matrix} u^{ε} ⇀ u weakly in L^{2} (0, T; H_{0}^{1} (Ω)) . \end{matrix}$ The limit problems are much simpler. For $γ = + \infty$ , the homogenized problem is $\begin{matrix} \{\begin{matrix} (1 + a) \frac{\partial u}{\partial t} - Δ u = f & in Ω^{T}, \\ u = 0 & on \partial Ω \times] 0, T [, \\ u (0) = \frac{1}{(1 + a)} u_{0} + \frac{a}{(1 + a)} v_{0} & in Ω, \end{matrix} \end{matrix}$ while for $γ = 0$ , it is $\begin{matrix} \{\begin{matrix} \frac{\partial u}{\partial t} - Δ u = f & in Ω^{T}, \\ u = 0 & on \partial Ω \times] 0, T [, \\ u (0) = u_{0} & in Ω . \end{matrix} \end{matrix}$

The state of the art in homogenization theory deals with problems where the limit problem has a different mathematical structure than the original one. The pioneers in this field are Murat and Cioranescu [13] in the case of perforated domains with small holes (holes of size $r_{ε}$ such that $0 < r_{ε} ≪ ε$ ), who showed the existence of a “strange term” in the limit problem, they were followed later on by many authors, for instance Allaire [1], Zuazua et al. [12] and Casado-Diaz [10]. When the small holes are filled with a different material that has a very contrast behavior with respect to the ambiental phase (which is the interest of our study), it has been shown by Bentalha et al. [5–7], and Bellieud [2], that the limit problem has a completely different structure with the appearance of non-local terms. The notion of non-local effects has been developed in the case of fibers by Bellieud and Bouchitté [3], Bellieud and Gruais [4], Briane and Tchou [9], and Briane [8].

The study of correctors plays an important role in the understanding of homogenization problems and has been the subject of many mathematical papers, the aim being to get sharp and easily computable approximations for the solution of the microscopic problem.

In the case of small holes with the appearance of a “strange term”, the correctors were studied first by Murat and Cioranescu in [13] and later on by other authors, for instance in the case of the wave equation, by Zuazua et al. [12]. These studies concern homogenized problems with only one concentration. In the present article, we consider in the same spirit, the correctors for the problem (1.1). This problem is interesting since the homogenized problem (1.6) involves two concentrations u and v. Under stronger assumptions on the data and using the tools of the control-zone method, three cases need to be treated separately, according to the values of γ.

When $γ > 0$ , we are in the critical case, the most interesting one, and if $b_{ε} \to + \infty$ , than the corrector result (see Theorem 3.2) reads $\begin{matrix} u^{ε} ⟶ u strongly in C^{0} ([0, T]; L^{2} (Ω)), \end{matrix}$ and $\begin{matrix} \{\begin{matrix} u^{ε} = Φ_{ε} (u, v) + R_{ε} with \\ R_{ε} ⟶ 0 strongly in L^{2} (0, T; W_{0}^{1, 1} (Ω)), \end{matrix} \end{matrix}$ where $u^{ε}$ and $(u, v)$ are respectively, the solutions of (1.1) and (1.6), and $\begin{matrix} (1.7) & Φ_{ε} (u, v) = (1 - w_{R_{ε}}) u + w_{R_{ε}} G_{r_{ε}} (v), \end{matrix}$ with $w_{R_{ε}}$ and $G_{r_{ε}}$ defined respectively, by (2.1) and (2.3).

In the other two cases, we obtain a similar corrector results, with $v = u$ when $γ = + \infty$ respectively, $v = v_{0}$ when $γ = 0$ .

The paper is organized as follows. Section 2 is divided into three parts, we first introduce the specific tools of the control-zone method, then we recall the homogenization results obtained by Bentalha et al. [7]. In the third part we present some compactness results. In Section 3 we study the corrector for three specific situations. Thus, Section 3.1 is devoted to the proof of corrector results for $b_{ε} \to + \infty$ and for the critical case $γ > 0$ . In Sections 3.2 and 3.3, we prove respectively, corrector results for the cases $γ = + \infty$ and $γ = 0$ . Finally, the Appendix contains the proof of the proposition which gives an example of a data $u_{0}^{ε}$ which satisfies the special assumptions necessary for proving the corrector results mentioned above.

2. Preliminaries

2.1. Tools of the control-zone method

Let us introduce first the sets $\begin{matrix} R : = {{(R_{ε})}_{ε}, r_{ε} ≪ R_{ε} ≪ ε}, \end{matrix}$ and $\begin{matrix} \begin{matrix} C_{ε}^{k} (a, b) : = ε k + C (a, b), \\ C_{ε} : = ⋃_{k \in Z_{ε}} C_{ε}^{k} (r_{ε}, R_{ε}), \\ C_{ε}^{T} : = C_{ε} \times] 0, T [, \end{matrix} \end{matrix}$ where $C (a, b) : = {x \in R^{3}, a < | x | < b}$ .

Definition 1.
For any ${(R_{ε})}_{ε} \in R$ , we define $w_{R_{ε}} \in H_{0}^{1} (Ω)$ by $\begin{matrix} (2.1) & w_{R_{ε}} (x) : = \{\begin{matrix} 0 & in Ω_{ε} ∖ C_{ε}, \\ W_{R_{ε}} (x - ε k) & in C_{ε}^{k}, \forall k \in Z_{ε}, \\ 1 & in D_{ε}, \end{matrix} \end{matrix}$ where $\begin{matrix} W_{R_{ε}} (y) = \frac{r_{ε}}{(R_{ε} - r_{ε})} (\frac{R_{ε}}{| y |} - 1) for y \in C (r_{ε}, R_{ε}) \end{matrix}$ is the elementary solution of the Laplace equation in $C (r_{ε}, R_{ε})$ .

From now on, we denote $\begin{matrix} γ_{ε} : = \frac{r_{ε}}{ε^{3}} and γ : = lim_{ε \to 0} γ_{ε} . \end{matrix}$
Proposition 2.1 ([7]).

For any ${(R_{ε})}_{ε} \in R$ , we have $\begin{matrix} (2.2) & \begin{matrix} (i) & 0 ⩽ w_{R_{ε}} ⩽ 1, \\ (ii) & | \nabla w_{R_{ε}} |_{L^{2} (Ω)} ⩽ C γ_{ε}^{1 / 2}, \\ (iii) & w_{R_{ε}} ⟶ 0 strongly in L^{2} (Ω) . \end{matrix} \end{matrix}$

Definition 2.
Consider the piecewise constant function $G_{r} : L^{2} (0, T; H_{0}^{1} (Ω)) \to L^{2} (Ω^{T})$ defined for any $r > 0$ by $\begin{matrix} (2.3) & G_{r} (θ) (x, t) : = \sum_{k \in Z_{ε}} (⨍_{S_{r}^{k}} θ (y, t) d σ_{y}) 1_{Y_{ε}^{k}} (x), \end{matrix}$ where $S_{r}^{k} = \partial B (ε k, r)$ .

From [7] we get the following results. Lemma 2.2.
For any ${(R_{ε})}_{ε} \in R$ and $θ \in H_{0}^{1} (Ω)$ , we have $\begin{matrix} \begin{matrix} {| θ - G_{R_{ε}} (θ) |}_{L^{2} (Ω_{Y_{ε}})} ⩽ C {(\frac{ε^{3}}{R_{ε}})}^{1 / 2} | \nabla θ |_{L^{2} (Ω)}, \\ {| θ - G_{r_{ε}} (θ) |}_{L^{2} (D_{ε})} ⩽ C r_{ε} | \nabla θ |_{L^{2} (D_{ε})}, \\ {| G_{R_{ε}} (θ) - G_{r_{ε}} (θ) |}_{L^{2} (Ω)} ⩽ C {(\frac{ε^{3}}{r_{ε}})}^{1 / 2} | \nabla θ |_{L^{2} (C_{ε})} . \end{matrix} \end{matrix}$ Moreover, $\begin{matrix} (2.4) & {| G_{r_{ε}} (θ) |}_{L^{2} (Ω)}^{2} = ⨍_{D_{ε}} {| G_{r_{ε}} (θ) |}^{2} d x . \end{matrix}$
Lemma 2.3.
For any ${(R_{ε})}_{ε} \in R$ and $θ \in L^{\infty} (0, T; C^{1} (\overline{Ω}))$ , we have $\begin{matrix} (2.5) & \begin{matrix} (i) & {| G_{r_{ε}} (θ) - θ |}_{L^{\infty} (Ω^{T})} ⟶ 0, \\ (ii) & {| G_{r_{ε}} (θ) - θ |}_{L^{\infty} (C_{ε}^{T} \cup D_{ε}^{T})} ⩽ 2 R_{ε} | \nabla θ |_{L^{\infty} (Ω^{T})} . \end{matrix} \end{matrix}$
Lemma 2.4.
For any $φ \in C_{c} (Ω)$ , we have $\begin{matrix} lim_{ε \to 0} ⨍_{D_{ε}} {| φ - M_{D_{ε}} (φ) |}^{2} d x = 0, \end{matrix}$ where $M_{D_{ε}} : C_{c} (Ω) \to L^{2} (Ω)$ is defined by $\begin{matrix} M_{D_{ε}} (φ) (x) : = \sum_{k \in Z_{ε}} (⨍_{Y_{ε}^{k}} φ (y) d y) 1_{B (ε k, r_{ε})} (x) . \end{matrix}$
Proposition 2.5.
For any $θ \in L^{2} (0, T; H_{0}^{1} (Ω))$ , we have $\begin{matrix} \int_{0}^{T} ⨍_{D_{ε}} | θ |^{2} d x d t ⩽ C max (1, \frac{1}{γ_{ε}}) | \nabla θ |_{L^{2} (Ω^{T})}^{2} . \end{matrix}$

2.2. Statement of the homogenization results

We recall here the estimates and homogenization results [7].

Proposition 2.6 ([7]).

Under hypotheses ( 1.4 ) and ( 1.5 ), we have $\begin{matrix} (2.6) & u^{ε} is bounded in L^{\infty} (0, T; L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω)) . \end{matrix}$ Moreover, there exists $C > 0$ , independent of ε, such that $\begin{matrix} (2.7) & ⨍_{D_{ε}} {| u^{ε} |}^{2} d x ⩽ C a.e. in [0, T] . \end{matrix}$

In the next theorems, we assume that hypotheses ( 1.4 )–( 1.5 ) hold.

Theorem 2.7 ([7]).

Suppose that $γ \in] 0, + \infty [$ and $b_{ε} \to + \infty$ . Then the sequence ${(u^{ε})}_{ε > 0}$ satisfies $\begin{matrix} (2.8) & \begin{matrix} u^{ε} \overset{*}{⇀} u {weakly}^{*} in L^{\infty} (0, T; L^{2} (Ω)), \\ u^{ε} ⇀ u weakly in L^{2} (0, T; H_{0}^{1} (Ω)), \end{matrix} \end{matrix}$ and $\begin{matrix} (2.9) & \begin{matrix} G_{r_{ε}} (u^{ε}) ⇀ v weakly in L^{2} (Ω^{T}), \\ \frac{1}{| D_{ε} |} u^{ε} χ_{D_{ε}} ⇀ v weakly in L^{2} (0, T; H^{- 1} (Ω)), \end{matrix} \end{matrix}$ where the limit $(u, v)$ is the unique solution of $\begin{matrix} (2.10) & \{\begin{matrix} \frac{\partial u}{\partial t} - Δ u + 4 π γ (u - v) = f & in Ω^{T}, \\ a \frac{\partial v}{\partial t} + 4 π γ (v - u) = 0 & in Ω^{T}, \\ u = 0 & on \partial Ω \times] 0, T [, \\ u (0) = u_{0}, v (0) = v_{0} & in Ω . \end{matrix} \end{matrix}$

Moreover, $(u, v) \in {(C^{0} ([0, T]; L^{2} (Ω)))}^{2}$ .

Theorem 2.8 ([7]).

Suppose $γ = + \infty$ . Then the sequence ${(u^{ε})}_{ε > 0}$ satisfies $\begin{array}{l} (2.11) & u^{ε} \overset{*}{⇀} u in L^{\infty} (0, T; L^{2} (Ω)), u^{ε} ⇀ u in L^{2} (0, T; H_{0}^{1} (Ω)), \\ (2.12) & G_{r_{ε}} (u^{ε}) \to u in L^{2} (Ω^{T}), \frac{1}{| D_{ε} |} u^{ε} χ_{D_{ε}} ⇀ u in L^{2} (0, T; H^{- 1} (Ω)), \end{array}$ where the limit u is the unique solution of the problem $\begin{matrix} (2.13) & \{\begin{matrix} (1 + a) \frac{\partial u}{\partial t} - Δ u = f & in Ω^{T}, \\ u = 0 & on \partial Ω \times] 0, T [, \\ u (0) = \frac{1}{(1 + a)} u_{0} + \frac{a}{(1 + a)} v_{0} & in Ω . \end{matrix} \end{matrix}$

Moreover, $u \in C^{0} ([0, T]; L^{2} (Ω))$ .

Theorem 2.9 ([7]).

Suppose $γ = 0$ . Then the sequence ${(u^{ε})}_{ε > 0}$ satisfies $\begin{matrix} \begin{matrix} u^{ε} \overset{*}{⇀} u {weakly}^{*} in L^{\infty} (0, T; L^{2} (Ω)), \\ u^{ε} ⇀ u weakly in L^{2} (0, T; H_{0}^{1} (Ω)), \end{matrix} \end{matrix}$ and $\begin{matrix} (2.14) & \begin{matrix} \frac{1}{| D_{ε} |} u^{ε} χ_{D_{ε}} ⇀ v_{0} weakly in D^{'} (Ω) a.e. t in [0, T], \end{matrix} \end{matrix}$ where u is the unique solution of the following problem: $\begin{matrix} (2.15) & \{\begin{matrix} \frac{\partial u}{\partial t} - Δ u = f & in Ω^{T}, \\ u = 0 & on \partial Ω \times] 0, T [, \\ u (0) = u_{0} & in Ω . \end{matrix} \end{matrix}$ Moreover, $u \in C^{0} ([0, T]; L^{2} (Ω))$ .

2.3. Some compactness results

In this subsection, we recall some compactness results which will be used in following. Let X and Y be two reflexive Banach spaces such that $X \subset Y$ with continuous, compact and dense imbedding.

Proposition 2.10 ([12]).

Assume that $\begin{matrix} \begin{matrix} g^{ε} ⇀ g weakly in L^{1} (0, T; X), \\ \frac{\partial g^{ε}}{\partial t} ⇀ \frac{\partial g}{\partial t} weakly in L^{1} (0, T; Y) . \end{matrix} \end{matrix}$ Then $\begin{matrix} g^{ε} ⟶ g strongly in C^{0} ([0, T]; Y) . \end{matrix}$

Corollary 2.11 ([12]).

Assume that $\begin{matrix} \begin{matrix} g^{ε} \overset{*}{⇀} g {weakly}^{*} in L^{\infty} (0, T; X), \\ \frac{\partial g^{ε}}{\partial t} ⇀ \frac{\partial g}{\partial t} weakly in L^{1} (0, T; Y) . \end{matrix} \end{matrix}$ Then, for all $w \in X^{'}$ , $\begin{matrix} {⟨ w, g^{ε} (\cdot) ⟩}_{X^{'} \times X} ⟶ {⟨ w, g (\cdot) ⟩}_{X^{'} \times X} strongly in C^{0} ([0, T]) . \end{matrix}$

3. Corrector results for a class of initial data

In this section we give a corrector results for problem (1.1) with initial data $u_{0}^{ε}$ satisfying the following conditions: $\begin{matrix} (3.1) & \begin{matrix} (i) & u_{0}^{ε} \in H^{1} (Ω) and \int_{Ω} k^{ε} {| \nabla u_{0}^{ε} |}^{2} d x ⩽ C, \\ (ii) & u_{0}^{ε} ⇀ u_{0} in L^{2} (Ω), \\ (iii) & ⨍_{D_{ε}} {| u_{0}^{ε} |}^{2} d x ⟶ | v_{0} |_{Ω}^{2}, \\ (iv) & \frac{1}{| D_{ε} |} u_{0}^{ε} χ_{D_{ε}} ⇀ v_{0} in D^{'} (Ω), v_{0} \in H_{0}^{1} (Ω) . \end{matrix} \end{matrix}$

Proposition 3.1 below, which will be proved in the Appendix, shows the existence of such a sequence ${(u^{ε})}_{ε > 0}$ for $(u_{0}, v_{0})$ smoother than ${(L^{2} (Ω))}^{2}$ , satisfying hypotheses (3.1).

Let us set, for any ${(R_{ε})}_{ε} \in R$ and $(φ, ψ) \in {(H^{1} (Ω))}^{2}$ , $\begin{matrix} (3.2) & Φ_{ε} (φ, ψ) : = (1 - w_{R_{ε}}) φ + w_{R_{ε}} G_{r_{ε}} (ψ) . \end{matrix}$

Proposition 3.1.

For $γ \in [0, + \infty [$ and $(u_{0}, v_{0}) \in (L^{\infty} (Ω) \cap H^{1} (Ω)) \times C_{0}^{1} (\overline{Ω})$ , hypotheses ( 3.1 ) are satisfied by the sequence $u_{0}^{ε} : = Φ_{ε} (u_{0}, v_{0})$ .

For $γ = + \infty$ and $u_{0} \in C_{0}^{1} (\overline{Ω})$ and $v_{0} : = u_{0}$ , hypotheses ( 3.1 ) are satisfied by the sequence $u_{0}^{ε} : = Φ_{ε} (u_{0}, u_{0})$ .

3.1. Corrector for the case $r_{ε} = O (ε^{3})$

In this subsection we state and prove a corrector result for problem (1.1) in the critical radius case $\begin{matrix} (3.3) & lim_{ε \to 0} γ_{ε} = γ \in] 0, + \infty [, \end{matrix}$ and under the condition $\begin{matrix} (3.4) & lim_{ε \to 0} b_{ε} = + \infty . \end{matrix}$

The main results of the present paper is the following theorem:

Theorem 3.2.
Under hypotheses ( 3.3 ), ( 3.4 ), ( 1.4 ) and ( 3.1 ), the sequence ${(u^{ε})}_{ε > 0}$ satisfies $\begin{matrix} (3.5) & u^{ε} ⟶ u strongly in C^{0} ([0, T]; L^{2} (Ω)), \end{matrix}$ and $\begin{matrix} (3.6) & \{\begin{matrix} u^{ε} = Φ_{ε} (u, v) + R_{ε} with \\ R_{ε} ⟶ 0 strongly in L^{2} (0, T; W_{0}^{1, 1} (Ω)), \end{matrix} \end{matrix}$ where $(u, v)$ is the solution of ( 2.10 ).

Before proving Theorem 3.2 we need the following technical results whose the proofs are postponed at the end of this subsection.
Proposition 3.3.
Assume that the hypotheses of Theorem 3.2 hold true. Then $\begin{matrix} (3.7) & \begin{matrix} (i) & \sqrt{a_{ε}} \frac{\partial u^{ε}}{\partial t} χ_{D_{ε}} is bounded in L^{2} (0, T; L^{2} (Ω)), \\ (ii) & \nabla u^{ε} χ_{D_{ε}} is bounded in L^{\infty} (0, T; L^{2} (Ω)), \\ (iii) & u^{ε} \overset{}{⇀} u {weakly}^{} in L^{\infty} (0, T; H_{0}^{1} (Ω)), \\ (iv) & \frac{\partial u^{ε}}{\partial t} ⇀ \frac{\partial u}{\partial t} weakly in L^{2} (0, T; L^{2} (Ω)) . \end{matrix} \end{matrix}$

Let now introduce the energies associated respectively, to problems (1.1) and (2.10), $\begin{matrix} \begin{matrix} E_{ε} (t) : = \frac{1}{2} ({| u^{ε} (t) |}_{Ω_{ε}}^{2} + a_{ε} {| u^{ε} (t) |}_{D ε}^{2}) + b_{ε} \int_{0}^{t} {| \nabla u^{ε} |}_{D_{ε}}^{2} d s + \int_{0}^{t} {| \nabla u^{ε} |}_{Ω_{ε}}^{2} d s, \\ E (t) : = \frac{1}{2} ({| u (t) |}_{Ω}^{2} + a {| v (t) |}_{Ω}^{2}) + 4 π γ \int_{0}^{t} | u - v |_{Ω}^{2} d s + \int_{0}^{t} | \nabla u |_{Ω}^{2} d s . \end{matrix} \end{matrix}$ Proposition 3.4.
Assume that the hypotheses of Theorem 3.2 hold true. Then $\begin{matrix} (3.8) & E_{ε} ⟶ E strongly in C^{0} ([0, T]) . \end{matrix}$
Proposition 3.5.
Assume that the hypotheses of Theorem 3.2 hold true. Then for any $(φ, ψ) \in {(C^{\infty} ([0, T]; D (Ω)))}^{2}$ we have $\begin{matrix} (3.9) & e_{ε} (Φ_{ε} (φ, ψ)) \to e (φ, ψ) strongly in C^{0} ([0, T]), \end{matrix}$ where $Φ_{ε}$ is given by ( 3.2 ) and $\begin{matrix} (3.10) & \begin{matrix} e_{ε} (φ) (t) : = \frac{1}{2} ({| φ (t) |}_{Ω_{ε}}^{2} + a_{ε} {| φ (t) |}_{D ε}^{2}) + b_{ε} \int_{0}^{t} | \nabla φ |_{D_{ε}}^{2} d s \\ + \int_{0}^{t} | \nabla φ |_{Ω_{ε}}^{2} d s, \\ e (φ, ψ) (t) : = \frac{1}{2} ({| φ (t) |}_{Ω}^{2} + a {| ψ (t) |}_{Ω}^{2}) + 4 π γ \int_{0}^{t} | φ - ψ |_{Ω}^{2} d s \\ + \int_{0}^{t} | \nabla φ |_{Ω}^{2} d s . \end{matrix} \end{matrix}$
Proposition 3.6.
Assume that the hypotheses of Theorem 3.2 hold true. Then for any $ψ \in C^{\infty} ([0, T]; D (Ω))$ we have $\begin{matrix} (3.11) & ⨍_{D_{ε}} u^{ε} ψ d x \to \int_{Ω} v ψ d x strongly in C^{0} ([0, T]) . \end{matrix}$
Lemma 3.7.
Assume that the hypotheses of Theorem 3.2 hold true. Then for any $(φ, ψ) \in (C^{\infty} ([0, T]; D {(Ω)))}^{2}$ we have $\begin{matrix} (3.12) & e_{ε} (u^{ε} - Φ_{ε} (φ, ψ)) \to e (u - φ, v - ψ) strongly in C^{0} ([0, T]), \end{matrix}$ where $e_{ε}$ and e are defined by ( 3.10 ).
Proof of Theorem 3.2.
The first result, namely convergence (3.5), is obtained by using (2.8)², (3.7)(iv) and Proposition 2.10.

To prove the second result, i.e., (3.6), observe first that from Theorem 2.7 we have $(u, v) \in {(C^{0} ([0, T]; L^{2} (Ω)))}^{2}$ . Furthermore, by solving the ordinary differential equation in (2.10) (see [14]), we have $\begin{matrix} (3.13) & v (t) = e^{- k t} v_{0} + \int_{0}^{t} k e^{- k (t - s)} u (s) d s with k = \frac{4 π γ}{a} . \end{matrix}$

Then, by using (2.8)² and the fact that $v_{0} \in H_{0}^{1} (Ω)$ , we conclude that $v \in L^{2} (0, T; H_{0}^{1} (Ω))$ .

Let us consider a sequence $(φ_{n}, ψ_{n})$ in ${(C^{\infty} ([0, T]; D (Ω)))}^{2}$ (see Proposition 3.60 [11]) such that, as $n \to \infty$ , $\begin{array}{l} (3.14) & φ_{n} \to u strongly in L^{2} (0, T; H_{0}^{1} (Ω)) \cap C^{0} ([0, T]; L^{2} (Ω)), \\ (3.15) & ψ_{n} \to v strongly in L^{2} (0, T; H_{0}^{1} (Ω)) \cap C^{0} ([0, T]; L^{2} (Ω)) . \end{array}$

Since $b_{ε} ⩾ b_{0} > 0$ , we have $\begin{matrix} min {b_{0}, 1} \int_{0}^{t} {| \nabla (u^{ε} - Φ_{ε} (φ_{n}, ψ_{n})) |}_{Ω}^{2} d s ⩽ e_{ε} (u^{ε} - Φ_{ε} (φ_{n}, ψ_{n})) (t) . \end{matrix}$

Using Lemma 3.7, it follows that $\begin{matrix} lim_{ε \to 0} min {b_{0}, 1} \int_{0}^{T} {| \nabla (u^{ε} - Φ_{ε} (φ_{n}, ψ_{n})) |}_{Ω}^{2} d s ⩽ {‖ e (u - φ_{n}, v - ψ_{n}) ‖}_{C^{0} ([0, T])} . \end{matrix}$

Therefore, from (3.14) and (3.15), $\begin{matrix} (3.16) & lim_{n \to \infty} lim_{ε \to 0} \int_{0}^{T} {| \nabla (u^{ε} - Φ_{ε} (φ_{n}, ψ_{n})) |}_{Ω}^{2} d s = 0 . \end{matrix}$

By writing $\begin{matrix} \nabla (u^{ε} - Φ_{ε} (u, v)) = \nabla (u^{ε} - Φ_{ε} (φ_{n}, ψ_{n})) + \nabla (Φ_{ε} (φ_{n}, ψ_{n}) - Φ_{ε} (u, v)), \end{matrix}$ we obtain $\begin{matrix} (3.17) & \begin{matrix} {‖ \nabla (u^{ε} - Φ_{ε} (u, v)) ‖}_{L^{2} (0, T; L^{1} (Ω))} ⩽ & C {‖ \nabla (u^{ε} - Φ_{ε} (φ_{n}, ψ_{n})) ‖}_{L^{2} (Ω^{T})} \\ + {‖ (1 - w_{R_{ε}}) \nabla (φ_{n} - u) ‖}_{L^{2} (0, T; L^{1} (Ω))} \\ + {‖ (φ_{n} - u) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω))} \\ + {‖ G_{r_{ε}} (ψ_{n} - v) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω))} . \end{matrix} \end{matrix}$

To conclude, we show that (3.6) holds by proving that each term of the right-hand side of (3.17) tends to zero as $ε \to 0$ .

The first term goes to zero thanks to (3.16).

For the second term, using (3.14) and the estimate $0 ⩽ w_{R_{ε}} ⩽ 1$ , we have $\begin{matrix} {‖ (1 - w_{R_{ε}}) \nabla (φ_{n} - u) ‖}_{L^{2} (0, T; L^{1} (Ω))} ⩽ C {‖ \nabla (φ_{n} - u) ‖}_{L^{2} (Ω^{T})} \to 0 . \end{matrix}$

For the third term, from (3.14), (2.2)(ii) and as $γ_{ε} \to γ$ , we easily find that $\begin{matrix} {‖ (φ_{n} - u) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω))} ⩽ C γ_{ε}^{1 / 2} ‖ φ_{n} - u ‖_{L^{2} (Ω^{T})} \to 0 . \end{matrix}$

Concerning the last term, Hölder’s inequality and estimate (2.2)(ii), yield $\begin{matrix} \begin{matrix} {‖ G_{r_{ε}} (ψ_{n} - v) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω))}^{2} ⩽ & C γ_{ε} {‖ G_{r_{ε}} (ψ_{n} - v) ‖}_{L^{2} (Ω^{T})}^{2} \\ ⩽ & C γ_{ε} (2 {‖ G_{r_{ε}} (ψ_{n} - v) - G_{R_{ε}} (ψ_{n} - v) ‖}_{L^{2} (Ω^{T})}^{2} \\ + 4 {‖ G_{R_{ε}} (ψ_{n} - v) - (ψ_{n} - v) ‖}_{L^{2} (Ω^{T})}^{2}) \\ + 4 C γ_{ε} ‖ ψ_{n} - v ‖_{L^{2} (Ω^{T})}^{2}, \end{matrix} \end{matrix}$ from which, by Poincaré inequality, Lemma 2.2, (3.15), and since $r_{ε} ≪ R_{ε} ≪ ε$ and $γ_{ε} \to γ$ , we get $\begin{matrix} \begin{matrix} {‖ G_{r_{ε}} (ψ_{n} - v) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω^{T}))}^{2} & ⩽ C (1 + \frac{r_{ε}}{R_{ε}} + γ_{ε}) {‖ \nabla (ψ_{n} - v) ‖}_{L^{2} (Ω^{T})}^{2} \\ \to 0 . \end{matrix} \end{matrix}$ This ends the proof of Theorem 3.2. □
Proof of Proposition 3.3.
We have (see [14]) $\begin{matrix} \int_{0}^{t} \int_{Ω} ρ_{ε} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x d s + \frac{1}{2} \int_{Ω} k^{ε} {| \nabla u^{ε} (t) |}^{2} d x = \int_{0}^{t} \int_{Ω} f^{ε} \frac{\partial u^{ε}}{\partial t} d x d s + \frac{1}{2} \int_{Ω} k^{ε} {| \nabla u_{0}^{ε} |}^{2} d x . \end{matrix}$

Furthermore, $\begin{matrix} \begin{matrix} \int_{0}^{t} \int_{Ω} f^{ε} \frac{\partial u^{ε}}{\partial t} d x d s = & \int_{0}^{t} \int_{Ω_{ε}} f^{ε} \frac{\partial u^{ε}}{\partial t} d x d s + \int_{0}^{t} \int_{D_{ε}} f^{ε} \frac{\partial u^{ε}}{\partial t} d x d s \\ ⩽ & \frac{1}{2} {‖ f^{ε} ‖}_{L^{2} (Ω^{T})}^{2} + \frac{1}{2} \int_{0}^{t} \int_{Ω_{ε}} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x d s \\ + \frac{1}{2 a_{ε}} {‖ f^{ε} ‖}_{L^{2} (Ω^{T})}^{2} + \frac{a_{ε}}{2} \int_{0}^{t} \int_{D_{ε}} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x d s . \end{matrix} \end{matrix}$ Notice that $\begin{matrix} \frac{1}{2 a_{ε}} = \frac{| D_{ε} |}{2 a_{ε} | D_{ε} |} . \end{matrix}$ Therefore, by using (1.4)¹ and (1.4)³, as $| D_{ε} | \to 0$ we obtain $\begin{matrix} \int_{0}^{t} \int_{Ω} f^{ε} \frac{\partial u^{ε}}{\partial t} d x d s ⩽ C + \frac{1}{2} \int_{0}^{t} \int_{Ω} ρ_{ε} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x d s . \end{matrix}$

We conclude thanks to (3.1)(i), that the following global estimate holds: $\begin{matrix} (3.18) & \begin{matrix} \frac{1}{2} \int_{0}^{t} \int_{Ω_{ε}} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x d s + \frac{a_{ε}}{2} \int_{0}^{t} \int_{D_{ε}} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x d s \\ + \frac{1}{2} \int_{Ω} k^{ε} {| \nabla u^{ε} (t) |}^{2} d x ⩽ C, \end{matrix} \end{matrix}$ whence estimates (3.7)(i) and (3.7)(ii).

In order to prove (3.7)(iii), observe that $\begin{matrix} min {b_{0}, 1} \int_{Ω} {| \nabla u^{ε} (t) |}^{2} d x ⩽ \int_{Ω} k^{ε} {| \nabla u^{ε} (t) |}^{2} d x ⩽ C . \end{matrix}$ Furthermore, since $| D_{ε} | \to 0$ and $a_{ε} | D_{ε} | \to a$ , using (3.18), we have $\begin{matrix} \int_{0}^{t} \int_{Ω} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x d s = \int_{0}^{t} \int_{Ω_{ε}} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x d s + | D_{ε} | \frac{1}{a_{ε} | D_{ε} |} a_{ε} \int_{0}^{t} \int_{D_{ε}} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x d s ⩽ C, \end{matrix}$ then the above estimate, together with (2.8)², yield (3.7)(iv) and this ends the proof. □
Proof of Proposition 3.4.
We have the following energy identities: $\begin{matrix} E_{ε} (t) = E_{ε} (0) + \int_{0}^{t} \int_{Ω} f^{ε} u^{ε} d x d s, \end{matrix}$ and $\begin{matrix} E (t) = E (0) + \int_{0}^{t} \int_{Ω} f u d x d s, \end{matrix}$ with $\begin{matrix} E_{ε} (0) = \frac{1}{2} ({| u_{0}^{ε} |}_{Ω_{ε}}^{2} + a_{ε} {| u_{0}^{ε} |}_{D_{ε}}^{2}), E (0) = \frac{1}{2} (| u_{0} |_{Ω}^{2} + a | v_{0} |_{Ω}^{2}) . \end{matrix}$

First we prove the pointwise convergence in time $\begin{matrix} (3.19) & E_{ε} (t) ⟶ E (t), for any t \in [0, T] . \end{matrix}$

By using convergences (1.4)³ and (3.5), we have $\begin{matrix} \int_{0}^{t} \int_{Ω} f^{ε} u^{ε} d x d s ⟶ \int_{0}^{t} \int_{Ω} f u d x d s . \end{matrix}$ Furthermore, convergences (3.5) and (3.1)(iii) and as $| D_{ε} | \to 0$ , we have $\begin{matrix} | {| u_{0}^{ε} |}_{Ω_{ε}}^{2} - | u_{0} |_{Ω}^{2} | ⩽ | {| u_{0}^{ε} |}_{Ω}^{2} - | u_{0} |_{Ω}^{2} - {| u_{0}^{ε} |}_{D_{ε}}^{2} | ⩽ | {| u_{0}^{ε} |}_{Ω}^{2} - | u_{0} |_{Ω}^{2} | + {| u_{0}^{ε} |}_{D_{ε}}^{2} \to 0 . \end{matrix}$ Then (3.19) follows, since $\begin{matrix} a_{ε} | D_{ε} | \to a and ⨍_{D_{ε}} {| u_{0}^{ε} |}^{2} d x ⟶ | v_{0} |_{Ω}^{2}, \end{matrix}$ imply $\begin{matrix} a_{ε} {| u_{0}^{ε} |}_{D_{ε}}^{2} = a_{ε} | D_{ε} | ⨍_{D_{ε}} {| u_{0}^{ε} |}^{2} d x ⟶ a | v_{0} |_{Ω}^{2} . \end{matrix}$

In order to obtain the uniform convergence of the energy, we apply the Ascoli–Arzela’s Theorem. To do so, we need to show that the family of energies ${(E_{ε} (t))}_{ε > 0}$ is uniformly equicontinuous. In fact, given any $t \in [0, T [$ , and $h > 0$ small enough, we have $\begin{array}{rcl} | E_{ε} (t + h) - E_{ε} (t) | & ⩽ & \int_{t}^{t + h} {‖ f^{ε} (s) ‖}_{L^{2} (Ω)} {‖ u^{ε} (s) ‖}_{L^{2} (Ω)} d s \\ ⩽ & {‖ u^{ε} ‖}_{L^{\infty} (0, T; L^{2} (Ω))} \int_{t}^{t + h} {‖ f^{ε} (s) ‖}_{L^{2} (Ω)} d s ⩽ C h^{1 / 2}, \end{array}$ where we have used (1.4)³, (3.7)(iii) and Hölder’s inequality. This inequality implies that $\begin{matrix} (3.20) & | E_{ε} (t + h) - E_{ε} (t) | \to 0 as h ⟶ 0, uniformly in ε . \end{matrix}$ Then, by (3.19), (3.20) and Ascoli–Arzela’s Theorem, we get (3.8). □
Proof of Proposition 3.5.
For $(φ, ψ) \in {(C^{\infty} ([0, T]; D (Ω)))}^{2}$ we have $\begin{matrix} (3.21) & \begin{matrix} e_{ε} (Φ_{ε} (φ, ψ)) (t) = & \frac{1}{2} ({| Φ_{ε} (φ, ψ) (t) |}_{Ω_{ε}}^{2} + a_{ε} {| Φ_{ε} (φ, ψ) (t) |}_{D_{ε}}^{2}) \\ + \int_{0}^{t} {| \nabla Φ_{ε} (φ, ψ) |}_{Ω_{ε}}^{2} d s . \end{matrix} \end{matrix}$

We will pass successively to the limit in each term of the right-hand side of the above equality. Note that $\begin{matrix} | {| Φ_{ε} (φ, ψ) (t) |}_{Ω_{ε}}^{2} - {| φ (t) |}_{Ω}^{2} | ⩽ | {| Φ_{ε} (φ, ψ) (t) |}_{Ω}^{2} - {| φ (t) |}_{Ω}^{2} | + {| Φ_{ε} (φ, ψ) (t) |}_{D_{ε}}^{2} . \end{matrix}$ Then, by using (2.2)(iii) and the uniform boundedness of $G_{r_{ε}} (ψ)$ in $L^{\infty} (Ω^{T})$ (see Lemma 2.3), we get $\begin{array}{l} | {| Φ_{ε} (φ, ψ) (t) |}_{Ω}^{2} - {| φ (t) |}_{Ω}^{2} | & = | {| φ (t) + w_{R_{ε}} (G_{r_{ε}} (ψ) - φ) (t) |}_{Ω}^{2} - {| φ (t) |}_{Ω}^{2} | \\ = | {| w_{R_{ε}} (G_{r_{ε}} (ψ) - φ) (t) |}_{Ω}^{2} + 2 \int_{Ω} w_{R_{ε}} φ (G_{r_{ε}} (ψ) - φ) (t) d x | \\ ⩽ | w_{R_{ε}} |_{Ω}^{2} {| G_{r_{ε}} (ψ) - φ |}_{L^{\infty} (Ω^{T})}^{2} + 2 | w_{R_{ε}} |_{Ω} {| φ (G_{r_{ε}} (ψ) - φ) |}_{L^{\infty} (Ω^{T})} \\ \to 0, \end{array}$ and $\begin{matrix} {| Φ_{ε} (φ, ψ) (t) |}_{D_{ε}}^{2} = {| G_{r_{ε}} (ψ) (t) |}_{D_{ε}}^{2} ⩽ {| G_{r_{ε}} (ψ) |}_{L^{\infty} (Ω^{T})}^{2} | D_{ε} | ⩽ C | D_{ε} | \to 0 . \end{matrix}$

Consequently, $\begin{matrix} (3.22) & {| Φ_{ε} (φ, ψ) (\cdot) |}_{Ω_{ε}}^{2} ⟶ {| φ (\cdot) |}_{Ω}^{2} strongly in C^{0} ([0, T]) . \end{matrix}$

Now, we come to the second term in equality (3.21). We have $\begin{matrix} a_{ε} {| Φ_{ε} (φ, ψ) (t) |}_{D ε}^{2} = a_{ε} {| G_{r_{ε}} (ψ) (t) |}_{D ε}^{2} = a_{ε} | D_{ε} | ⨍_{D_{ε}} {| G_{r_{ε}} (ψ) (t) |}^{2} d x, \end{matrix}$ and with Lemma 2.2, the uniform convergence of $G_{r_{ε}} (ψ)$ to ψ (see Lemma 2.3) and the fact that $a_{ε} | D_{ε} | \to a$ , yield $\begin{matrix} \begin{matrix} a_{ε} {| Φ_{ε} (φ, ψ) |}_{D_{ε}}^{2} & = a_{ε} | D_{ε} | \int_{Ω} {| G_{r_{ε}} (ψ) |}^{2} d x \\ \to a | ψ |_{Ω}^{2} strongly in C^{0} ([0, T]) . \end{matrix} \end{matrix}$

Let treat the last term in equality (3.21). We have $\begin{matrix} \begin{matrix} \int_{0}^{t} {| \nabla Φ_{ε} (φ, ψ) |}_{Ω_{ε}}^{2} d s = & \int_{0}^{t} | \nabla φ |_{Ω_{ε} ∖ C_{ε}}^{2} d s \\ + \int_{0}^{t} {| (1 - w_{R_{ε}}) \nabla φ + (G_{r_{ε}} (ψ) - φ) \nabla w_{R_{ε}} |}_{C_{ε}}^{2} d s = A_{1} + A_{2} . \end{matrix} \end{matrix}$

Observe that for $A_{1}$ , we have $\begin{matrix} \begin{matrix} | \int_{0}^{t} | \nabla φ |_{Ω_{ε} ∖ C_{ε}}^{2} d s - \int_{0}^{t} | \nabla φ |_{Ω}^{2} d s | & = \int_{0}^{t} | \nabla φ |_{D_{ε} \cup C_{ε}}^{2} d s \\ ⩽ T | D_{ε} \cup C_{ε} | ‖ \nabla φ ‖_{L^{\infty} (Ω^{T})}^{2}, \end{matrix} \end{matrix}$ and, since $| D_{ε} \cup C_{ε} | ⟶ 0$ , we obtain $\begin{matrix} (3.23) & A_{1} ⟶ \int_{0}^{t} \int_{Ω} | \nabla φ |^{2} d x d s strongly in C^{0} ([0, T]) . \end{matrix}$

In order to deal with $A_{2}$ , we decompose it into three terms defined below, that will be treated separately $\begin{matrix} \begin{matrix} A_{2} = & \int_{0}^{t} \int_{C_{ε}} {| (1 - w_{R_{ε}}) \nabla φ |}^{2} d x d s + \int_{0}^{t} \int_{C_{ε}} {| (G_{r_{ε}} (ψ) - φ) \nabla w_{R_{ε}} |}^{2} d x d s \\ + 2 \int_{0}^{t} \int_{C_{ε}} (1 - w_{R_{ε}}) \nabla φ (G_{r_{ε}} (ψ) - φ) \nabla w_{R_{ε}} d x d s \\ = & A_{2}^{1} + A_{2}^{2} + A_{2}^{3} . \end{matrix} \end{matrix}$

As $0 ⩽ w_{R_{ε}} ⩽ 1$ , we have $\begin{matrix} \int_{0}^{t} \int_{C_{ε}} {| (1 - w_{R_{ε}}) \nabla φ |}^{2} d x d s ⩽ T | C_{ε} | ‖ \nabla φ ‖_{L^{\infty} (Ω^{T})}^{2} . \end{matrix}$ But $| C_{ε} | ⟶ 0$ , and thus $\begin{matrix} (3.24) & A_{2}^{1} ⟶ 0 strongly in C^{0} ([0, T]) . \end{matrix}$

Using the estimates of Proposition 2.1, the uniform boundedness of $G_{r_{ε}} (ψ)$ in $L^{\infty} (Ω^{T})$ , we have $\begin{matrix} | \int_{0}^{t} \int_{C_{ε}} (1 - w_{R_{ε}}) \nabla φ (G_{r_{ε}} (ψ) - φ) \nabla w_{R_{ε}} d x d s | ⩽ C γ_{ε}^{1 / 2} | C_{ε} |^{1 / 2} . \end{matrix}$ As $γ_{ε} \to γ$ and $| C_{ε} | ⟶ 0$ , it follows that $\begin{matrix} (3.25) & A_{2}^{3} ⟶ 0 strongly in C^{0} ([0, T]) . \end{matrix}$

In order to study $A_{2}^{2}$ , we begin by decomposing it into three terms defined below that will be treated separately $\begin{matrix} \begin{matrix} A_{2}^{2} = & \int_{0}^{t} \int_{C_{ε}} {| (G_{r_{ε}} (ψ) - G_{r_{ε}} (φ)) \nabla w_{R_{ε}} |}^{2} d x d s \\ + \int_{0}^{t} \int_{C_{ε}} {| (G_{r_{ε}} (φ) - φ) \nabla w_{R_{ε}} |}^{2} d x d s \\ + 2 \int_{0}^{t} \int_{C_{ε}} (G_{r_{ε}} (ψ) - G_{r_{ε}} (φ)) (G_{r_{ε}} (φ) - φ) | \nabla w_{R_{ε}} |^{2} d x d s \\ = & A_{2}^{2, 1} + A_{2}^{2, 2} + A_{2}^{2, 3} . \end{matrix} \end{matrix}$

A direct computation for the term $A_{2}^{2, 1}$ shows that $\begin{matrix} \begin{matrix} A_{2}^{2, 1} = & \sum_{k \in Z_{ε}} (\int_{0}^{t} {| ⨍_{S_{r_{ε}}^{k}} ψ d σ - ⨍_{S_{r_{ε}}^{k}} φ d σ |}^{2} d s) \\ \times (\int_{0}^{2 π} d Φ \int_{0}^{π} sin Θ d Θ \int_{r_{ε}}^{R_{ε}} {| \frac{\partial W_{R_{ε}}}{\partial r} |}^{2} r^{2} d r) \\ = & 4 π {(\frac{r_{ε} R_{ε}}{R_{ε} - r_{ε}})}^{2} \sum_{k \in Z_{ε}} (\int_{0}^{t} {| ⨍_{S_{r_{ε}}^{k}} ψ d σ - ⨍_{S_{r_{ε}}^{k}} φ d σ |}^{2} d s) (\int_{r_{ε}}^{R_{ε}} \frac{1}{r^{2}} d r) \\ = & 4 π \frac{r_{ε} R_{ε}}{(R_{ε} - r_{ε}) ε^{3}} \sum_{k \in Z_{ε}} (\int_{0}^{t} \int_{Ω} {| ⨍_{S_{r_{ε}}^{k}} ψ d σ - ⨍_{S_{r_{ε}}^{k}} φ d σ |}^{2} χ_{Y_{ε}^{k}} d x d s) \\ = & 4 π γ_{ε} \frac{R_{ε}}{R_{ε} - r_{ε}} (\int_{0}^{t} \int_{Ω} {| G_{r_{ε}} (ψ) - G_{r_{ε}} (φ) |}^{2} d x d s) . \end{matrix} \end{matrix}$

Recalling that $γ_{ε} \to γ$ and $r_{ε} ≪ R_{ε} ≪ ε$ , by (2.5)(i) we obtain $\begin{matrix} (3.26) & A_{2}^{2, 1} ⟶ 4 π γ \int_{0}^{t} \int_{Ω} | ψ - φ |^{2} d x d s strongly in C^{0} ([0, T]) . \end{matrix}$

Using (2.5)(ii) and since $| \nabla w_{R_{ε}} |_{Ω} ⩽ C γ_{ε}^{1 / 2}$ , the term $A_{2}^{2, 2}$ may be estimated as follows: $\begin{matrix} \begin{matrix} \int_{0}^{t} \int_{C_{ε}} {| (G_{r_{ε}} (φ) - φ) \nabla w_{R_{ε}} |}^{2} d x d s & ⩽ C γ_{ε} {| G_{r_{ε}} (φ) - φ |}_{L^{\infty} (C_{ε}^{T})}^{2} \\ ⩽ C γ_{ε} R_{ε}^{2} | \nabla φ |_{L^{\infty} (Ω^{T})}^{2} . \end{matrix} \end{matrix}$ Then, since $R_{ε} \to 0$ and $γ_{ε} \to γ$ , $\begin{matrix} (3.27) & A_{2}^{2, 2} ⟶ 0 strongly in C^{0} ([0, T]) . \end{matrix}$

Concerning $A_{2}^{2, 3}$ , by using the same arguments as those for the proof of (3.27), we infer that $\begin{matrix} (3.28) & A_{2}^{2, 3} ⟶ 0 strongly in C^{0} ([0, T]) . \end{matrix}$

Finally, combining (3.22)–(3.28) we conclude the convergence (3.9). □
Proof of Proposition 3.6.
First we will to prove that there exists a function $Θ : [0, T] \to L^{2} (Ω)$ such that for a subsequence $\begin{matrix} (3.29) & G_{r_{ε}} (u^{ε}) (t) ⇀ Θ (t) weakly in L^{2} (Ω) for any fixed t \in [0, T] . \end{matrix}$

Note that from (3.7)(iii), (3.7)(iv) and Corollary 2.11, we have $\begin{matrix} (3.30) & u^{ε} (t) ⇀ u (t) in H_{0}^{1} (Ω) for any fixed t \in [0, T] . \end{matrix}$

On the other hand, $\begin{matrix} {| G_{r_{ε}} (u^{ε}) (t) |}_{L^{2} (Ω)}^{2} ⩽ 2 {| G_{r_{ε}} (u^{ε}) (t) - G_{R_{ε}} (u^{ε}) (t) |}_{L^{2} (Ω)}^{2} + 2 {| G_{R_{ε}} (u^{ε}) (t) |}_{L^{2} (Ω)}^{2}, \end{matrix}$ and $\begin{matrix} {| G_{R_{ε}} (u^{ε}) (t) |}_{L^{2} (Ω)}^{2} ⩽ 2 {| G_{R_{ε}} (u^{ε}) (t) - u^{ε} (t) |}_{L^{2} (Ω)}^{2} + 2 {| u^{ε} (t) |}_{L^{2} (Ω)}^{2} . \end{matrix}$ From Lemma 2.2 and taking into account (3.30), $\begin{matrix} {| G_{r_{ε}} (u^{ε}) (t) - G_{R_{ε}} (u^{ε}) (t) |}_{L^{2} (Ω)}^{2} ⩽ \frac{C}{γ_{ε}} {| \nabla u^{ε} (t) |}_{L^{2} (C_{ε})}^{2} ⩽ C (t), \end{matrix}$ and $\begin{matrix} {| G_{R_{ε}} (u^{ε}) (t) |}_{L^{2} (Ω)}^{2} ⩽ C \frac{ε^{3}}{R_{ε}} {| \nabla u^{ε} (t) |}_{L^{2} (Ω)}^{2} + 2 {| u^{ε} (t) |}_{L^{2} (Ω)}^{2} ⩽ C (t), \end{matrix}$ and therefore, $\begin{matrix} {| G_{r_{ε}} (u^{ε}) (t) |}_{L^{2} (Ω)}^{2} ⩽ C (t), \end{matrix}$ and this implies (3.29).

Next, we show that $Θ \in L^{2} (Ω^{T})$ an is such that, for the subsequence which gave (3.29), $\begin{matrix} (3.31) & ⨍_{D_{ε}} u^{ε} ψ (\cdot) d x \to \int_{Ω} Θ ψ (\cdot) d x strongly in C^{0} ([0, T]) . \end{matrix}$

We have $\begin{matrix} (3.32) & \begin{matrix} ⨍_{D_{ε}} u^{ε} ψ (t) d x = & ⨍_{D_{ε}} (u^{ε} - G_{r_{ε}} (u^{ε})) ψ (t) d x \\ + ⨍_{D_{ε}} G_{r_{ε}} (u^{ε}) (ψ - M_{D_{ε}} (ψ)) (t) d x \\ + ⨍_{D_{ε}} G_{r_{ε}} (u^{ε}) (t) M_{D_{ε}} (ψ) (t) d x : = I_{1}^{ε} + I_{2}^{ε} + I_{3}^{ε} . \end{matrix} \end{matrix}$

Now, we evaluate the limit of (3.32) term by term for the subsequence which gives (3.29). By using Lemma 2.2 we have $\begin{matrix} \begin{matrix} | I_{1}^{ε} | & ⩽ {(⨍_{D_{ε}} {| u^{ε} (t) - G_{r_{ε}} (u^{ε}) (t) |}^{2} d x)}^{\frac{1}{2}} {(⨍_{D_{ε}} {| ψ (t) |}^{2} d x)}^{\frac{1}{2}} \\ ⩽ C {(\frac{r_{ε}^{2}}{| D_{ε} |})}^{\frac{1}{2}} {| \nabla u^{ε} (t) |}_{L^{2} (D_{ε})} | ψ |_{L^{\infty} (Ω^{T})} \\ ⩽ C \frac{1}{\sqrt{γ_{ε} b_{ε}}} {| \sqrt{b_{ε}} \nabla u^{ε} (t) |}_{L^{2} (D_{ε})} \to 0, \end{matrix} \end{matrix}$ where we have used (3.7)(ii), and the facts that $γ_{ε} \to γ$ and $b_{ε} \to + \infty$ .

The term $I_{2}^{ε}$ tends to zero thanks to Lemma 2.4, (2.4) and (3.29).

By using (3.29), the last term $I_{3}^{ε}$ is handled as follows: $\begin{matrix} \begin{matrix} I_{3}^{ε} = \sum_{k \in Z_{ε}} \int_{Y_{ε}^{k}} (⨍_{S_{r_{ε}}^{k}} u^{ε} (t) d σ_{y}) ψ (t) d y & = \int_{Ω} G_{r_{ε}} (u^{ε}) (t) ψ (t) d x \\ \to \int_{Ω} Θ (t) ψ (t) d x . \end{matrix} \end{matrix}$

Then, for the subsequence which gave the convergence (3.29), we have $\begin{matrix} (3.33) & ⨍_{D_{ε}} u^{ε} (t) ψ (t) d x \to \int_{Ω} Θ (t) ψ (t) d x, for any fixed t \in [0, T] . \end{matrix}$ This convergence takes place in $C^{0} ([0, T])$ due to boundedness of the function $⨍_{D_{ε}} u^{ε} ψ (t) d x$ in $H^{1} (0, T)$ and to the compactness of the injection $H^{1} (0, T) \subset C^{0} ([0, T])$ .

Indeed, from (2.7), we have $\begin{matrix} (3.34) & \begin{matrix} \int_{0}^{T} {| ⨍_{D_{ε}} u^{ε} ψ d x |}^{2} d t & ⩽ \int_{0}^{T} (⨍_{D_{ε}} {| u^{ε} |}^{2} d x) (⨍_{D_{ε}} | ψ |^{2} d x) d t \\ ⩽ C | ψ |_{L^{\infty} (Ω^{T})}^{2} ⩽ C . \end{matrix} \end{matrix}$ Furthermore, $\begin{matrix} \frac{d}{d t} (⨍_{D_{ε}} u^{ε} (t) ψ (t) d x) = ⨍_{D_{ε}} \frac{\partial u^{ε}}{\partial t} (t) ψ (t) d x + ⨍_{D_{ε}} u^{ε} (t) \frac{\partial ψ}{\partial t} (t) d x, \end{matrix}$ so that, recalling that $a_{ε} | D_{ε} | \to a$ , and using (2.7), (3.7), with the same argument as in (3.34), we readily obtain $\begin{matrix} \begin{matrix} \int_{0}^{T} {| \frac{d}{d t} ⨍_{D_{ε}} u^{ε} ψ d x |}^{2} d t \\ ⩽ 2 \int_{0}^{T} (| ψ |_{L^{\infty} (Ω^{T})}^{2} ⨍_{D_{ε}} {| \frac{\partial u^{ε}}{\partial t} |}^{2} d x + {| \frac{\partial ψ}{\partial t} |}_{L^{\infty} (Ω^{T})}^{2} ⨍_{D_{ε}} {| u^{ε} |}^{2} d x) d t \\ ⩽ C . \end{matrix} \end{matrix}$

Finally, from (3.31) we have $\begin{matrix} \int_{0}^{T} ⨍_{D_{ε}} u^{ε} ψ d x d t \to \int_{0}^{T} \int_{Ω} Θ ψ d x d t . \end{matrix}$ Then, thanks to (2.9)², we conclude that $Θ = v$ in $D^{'} (Ω^{T})$ and the proof is complete. □
Proof of Lemma 3.7.
For any fixed $t \in [0, T]$ , we have $\begin{matrix} (3.35) & \begin{matrix} e_{ε} (u^{ε} - Φ_{ε} (φ, ψ)) (t) = & e_{ε} (u^{ε}) (t) + e_{ε} (Φ_{ε} (φ, ψ)) (t) \\ - \int_{Ω_{ε}} u^{ε} Φ_{ε} (φ, ψ) (t) d x - a_{ε} \int_{D_{ε}} u^{ε} G_{r_{ε}} (ψ) (t) d x \\ - 2 \int_{0}^{t} \int_{Ω_{ε}} \nabla u^{ε} \nabla Φ_{ε} (φ, ψ) d x d s \\ : = & I_{ε}^{1} + I_{ε}^{2} - I_{ε}^{3} - I_{ε}^{4} - 2 I_{ε}^{5} . \end{matrix} \end{matrix}$

We will pass successively to the limit in each term of the right-hand side of the above equality. Thanks to Proposition 3.4 and Proposition 3.5 we have $\begin{matrix} (3.36) & \begin{matrix} I_{ε}^{1} ⟶ e (u, v) strongly in C^{0} ([0, T]), \\ I_{ε}^{2} ⟶ e (φ, ψ) strongly in C^{0} ([0, T]) . \end{matrix} \end{matrix}$

Now we study the third term in (3.35). We have $\begin{matrix} \begin{matrix} | \int_{Ω_{ε}} u^{ε} Φ_{ε} (φ, ψ) (t) d x - \int_{Ω} u φ (t) d x | \\ ⩽ | \int_{Ω_{ε}} (u^{ε} - u) (t) Φ_{ε} (φ, ψ) (t) d x | \\ + | \int_{D_{ε}} u (t) Φ_{ε} (φ, ψ) (t) d x | + | \int_{Ω} u (t) (Φ_{ε} (φ, ψ) - φ) (t) d x | \\ ⩽ {| (u^{ε} - u) (t) |}_{L^{2} (Ω)} {| (1 - w_{R_{ε}}) φ (t) + w_{R_{ε}} G_{r_{ε}} (ψ) (t) |}_{L^{2} (Ω)} \\ + {| u (t) |}_{L^{2} (Ω)} ({| G_{r_{ε}} (ψ) (t) |}_{L^{2} (D_{ε})} + {| w_{R_{ε}} (G_{r_{ε}} (ψ) - φ) (t) |}_{L^{2} (C_{ε})} \\ + {| (G_{r_{ε}} (ψ) - φ) (t) |}_{L^{2} (D_{ε})}) . \end{matrix} \end{matrix}$ Using (2.2)(i) and the uniform boundedness of $G_{r_{ε}} (ψ)$ in $L^{\infty} (Ω^{T})$ , we get $\begin{matrix} \begin{matrix} sup_{0 ⩽ t ⩽ T} | \int_{Ω_{ε}} u^{ε} Φ_{ε} (φ, ψ) (t) d x - \int_{Ω} u φ (t) d x | \\ ⩽ C {| u^{ε} - u |}_{L^{\infty} (0, T; L^{2} (Ω))} \\ + | u |_{L^{\infty} (0, T; L^{2} (Ω))} ({| G_{r_{ε}} (ψ) |}_{L^{\infty} (Ω^{T})} | D_{ε} |^{1 / 2} + C | w_{R_{ε}} |_{L^{2} (Ω)} \\ + {| G_{r_{ε}} (ψ) - φ |}_{L^{\infty} (Ω^{T})} | D_{ε} |^{1 / 2}) . \end{matrix} \end{matrix}$ Therefore, as $w_{R_{ε}} \to 0$ strongly in $L^{2} (Ω)$ and $| D_{ε} | \to 0$ , thanks to (3.5), we deduce $\begin{matrix} (3.37) & I_{ε}^{3} ⟶ \int_{Ω} u φ d x strongly in C^{0} ([0, T]) . \end{matrix}$

As for the fourth term in (3.35), since $a_{ε} | D_{ε} | \to a$ , using Proposition 3.6 we obtain $\begin{matrix} (3.38) & I_{ε}^{4} = a_{ε} | D_{ε} | ⨍_{D_{ε}} u^{ε} G_{r_{ε}} (ψ) (\cdot) d x \to a \int_{Ω} v ψ d x strongly in C^{0} ([0, T]) . \end{matrix}$

Indeed, we have $\begin{matrix} \begin{matrix} | ⨍_{D_{ε}} u^{ε} G_{r_{ε}} (ψ) (t) d x - \int_{Ω} v ψ (t) d x | ⩽ & | ⨍_{D_{ε}} u^{ε} (G_{r_{ε}} (ψ) - ψ) d x | \\ + | ⨍_{D_{ε}} u^{ε} ψ (t) d x - \int_{Ω} v ψ (t) d x |, \end{matrix} \end{matrix}$ where the right-hand side of this inequality tends to zero strongly in $C^{0} ([0, T])$ thanks to (2.7), to the uniform convergence of $G_{r_{ε}} (ψ)$ to ψ and to Proposition 3.6.

For the last term in (3.35), by using the same arguments as those from Proposition 4.6 [7], we have $\begin{matrix} I_{ε}^{5} (t) \to \int_{0}^{t} \int_{Ω} (\nabla u \nabla φ + 4 π γ (v - u) (ψ - φ)) d x d s, for any t \in [0, T] . \end{matrix}$

Now we want to show that the previous convergence takes place in $C^{0} ([0, T])$ . In fact, given any $t \in [0, T [$ , and $h > 0$ small enough, we obtain from Hölder’s inequality $\begin{matrix} \begin{matrix} | I_{ε}^{5} (t + h) - I_{ε}^{5} (t) | & ⩽ {‖ \nabla u^{ε} ‖}_{L^{\infty} (0, T; L^{2} (Ω))} {‖ \nabla Φ_{ε} (φ, ψ) ‖}_{L^{2} (Ω^{T})} h^{1 / 2} \\ ⩽ C {‖ (1 - w_{R_{ε}}) \nabla φ + (G_{r_{ε}} (ψ) - φ) \nabla w_{R_{ε}} ‖}_{L^{2} (Ω^{T})} h^{1 / 2} \\ ⩽ C h^{1 / 2}, \end{matrix} \end{matrix}$ where we have used (3.7)(iii), the estimates of Proposition 2.1, the uniform boundedness of $G_{r_{ε}} (ψ)$ in $L^{\infty} (Ω^{T})$ (see Lemma 2.3) and the fact that $γ_{ε} \to γ$ .

Hence, by Ascoli–Arzela’s Theorem, we conclude that $\begin{matrix} I_{ε}^{5} \to \int_{0}^{t} \int_{Ω} (\nabla u \nabla φ + 4 π γ (v - u) (ψ - φ)) d x d s strongly in C^{0} ([0, T]) . \end{matrix}$ With this last convergence, and recalling (3.36)–(3.38), we get (3.12), and this ends the proof. □

3.2. Corrector for the case $ε^{3} ≪ r_{ε} ≪ ε$

The corrector result in this case is as follows:

Theorem 3.8.
Under hypotheses ( 1.4 ), ( 3.1 ) with $v_{0} : = u_{0}$ , if $γ = + \infty$ , the sequence ${(u^{ε})}_{ε > 0}$ satisfies $\begin{matrix} u^{ε} ⟶ u strongly in C^{0} ([0, T]; L^{2} (Ω)), \end{matrix}$ and $\begin{matrix} \{\begin{matrix} u^{ε} = Φ_{ε} (u, u) + R_{ε} with \\ R_{ε} ⟶ 0 strongly in L^{2} (0, T; W_{0}^{1, 1} (Ω)), \end{matrix} \end{matrix}$ where u is the solution of ( 2.13 ).

As in the previous subsection, before proving Theorem 3.8, we will need the following technical results. Proposition 3.9.
Assume that the hypotheses of Theorem 3.8 hold true. Then $\begin{matrix} E_{ε} ⟶ E_{1} strongly in C^{0} ([0, T]), \end{matrix}$ where $\begin{matrix} E_{1} (t) : = \frac{1}{2} (1 + a) {| u (t) |}_{Ω}^{2} + \int_{0}^{t} | \nabla u |_{Ω}^{2} d s, \end{matrix}$ is the energy of problem ( 2.13 ).

Moreover, for any $φ \in C^{\infty} ([0, T]; D (Ω))$ , we have the convergences $\begin{array}{l} (3.39) & e_{ε} (Φ_{ε} (φ, φ)) (\cdot) \to e (φ, φ) (\cdot) strongly in C^{0} ([0, T]), \\ ⨍_{D_{ε}} u^{ε} φ (\cdot) d x \to \int_{Ω} u φ (\cdot) d x strongly in C^{0} ([0, T]), \end{array}$ and $\begin{matrix} (3.40) & e_{ε} (u^{ε} - Φ_{ε} (φ, φ)) \to e (u - φ, u - φ) strongly in C^{0} ([0, T]), \end{matrix}$
Remark 1.
Note that in this case Proposition 3.3 is still valid with the assumptions of Theorem 3.8.
Proof of Theorem 3.8.
The proof of the first result is similar to the corresponding one of the Theorem 3.2.

For the second result, from Theorem 2.8 we have $\begin{matrix} u \in L^{2} (0, T; H_{0}^{1} (Ω)) \cap C^{0} ([0, T]; L^{2} (Ω)) . \end{matrix}$

Let us consider a sequence $φ_{n}$ in $C^{\infty} ([0, T]; D (Ω))$ such that, as $n \to + \infty$ , $\begin{matrix} (3.41) & φ_{n} \to u strongly in L^{2} (0, T; H_{0}^{1} (Ω)) \cap C^{0} ([0, T]; L^{2} (Ω)) . \end{matrix}$

Furthermore, we have $\begin{matrix} \begin{matrix} {‖ \nabla (u^{ε} - Φ_{ε} (u, u)) ‖}_{L^{2} (0, T; L^{1} (Ω))} ⩽ & C {‖ \nabla (u^{ε} - Φ_{ε} (φ_{n}, φ_{n})) ‖}_{L^{2} (Ω^{T})} \\ + {‖ (1 - w_{R_{ε}}) \nabla (φ_{n} - u) ‖}_{L^{2} (0, T; L^{1} (Ω))} \\ + {‖ (G_{r_{ε}} (φ_{n} - u) - (φ_{n} - u)) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω))} . \end{matrix} \end{matrix}$

Observe first, that the two first terms on the right-hand side of the above inequality tend to zero, the proof is similar to the corresponding one of Theorem 3.2.

For the last term, Hölder’s inequality and the estimate $\begin{matrix} | \nabla w_{R_{ε}} |_{Ω} ⩽ C γ_{ε}^{1 / 2}, \end{matrix}$ give $\begin{matrix} \begin{matrix} {‖ (G_{r_{ε}} (φ_{n} - u) - (φ_{n} - u)) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω))}^{2} \\ ⩽ C γ_{ε} {‖ G_{r_{ε}} (φ_{n} - u) - (φ_{n} - u) ‖}_{L^{2} (Ω^{T})}^{2} \\ ⩽ 2 C γ_{ε} ({‖ G_{r_{ε}} (φ_{n} - u) - G_{R_{ε}} (φ_{n} - u) ‖}_{L^{2} (Ω^{T})}^{2} \\ + {‖ G_{R_{ε}} (φ_{n} - u) - (φ_{n} - u) ‖}_{L^{2} (Ω^{T})}^{2}) . \end{matrix} \end{matrix}$

Applying Lemma 2.2, recalling that $r_{ε} ≪ R_{ε} ≪ ε$ , and using (3.41), we have $\begin{matrix} \begin{matrix} {‖ (G_{r_{ε}} (φ_{n} - u) - (φ_{n} - u)) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω))}^{2} \\ ⩽ C (1 + \frac{r_{ε}}{R_{ε}}) {‖ \nabla (φ_{n} - u) ‖}_{L^{2} (Ω^{T})}^{2} \to 0, \end{matrix} \end{matrix}$ and the proof is complete. □
Proof of Proposition 3.9.
The proof of the result is similar to the corresponding one of the previous subsection but with two convergences which need to be treated differently.

The first is related to the proof of (3.39), specifically when treating $A_{2}$ (see the proof of Proposition 3.5). Indeed, $\begin{array}{rcl} A_{2} & = & \int_{0}^{t} \int_{C_{ε}} {| (1 - w_{R_{ε}}) \nabla φ + (G_{r_{ε}} (φ) - φ) \nabla w_{R_{ε}} |}^{2} d x d s \\ ⩽ & 2 \int_{0}^{t} \int_{C_{ε}} {| (1 - w_{R_{ε}}) \nabla φ |}^{2} d x d s + 2 \int_{0}^{t} \int_{C_{ε}} {| (G_{r_{ε}} (φ) - φ) \nabla w_{R_{ε}} |}^{2} d x d s \\ ⩽ & 2 T ‖ \nabla φ ‖_{L^{\infty} (Ω^{T})}^{2} | C_{ε} | + 2 T {| G_{r_{ε}} (φ) - φ |}_{L^{\infty} (C_{ε}^{T})}^{2} | \nabla w_{R_{ε}} |_{L^{2} (C_{ε})}^{2} \\ ⩽ & C | C_{ε} | + C R_{ε}^{2} γ_{ε} = C | C_{ε} | + C {(\frac{R_{ε}}{ε})}^{2} (\frac{r_{ε}}{ε}) ⟶ 0, \end{array}$ where we have used the estimates of Proposition 2.1, Lemma 2.3, the convergence $| C_{ε} | \to 0$ , and the fact that $r_{ε} ≪ R_{ε} ≪ ε$ .

The second interesting convergence is in the proof of (3.40), specifically the limit of $I_{ε}^{5}$ (see the proof of Lemma 3.7); in this case this is the following one: $\begin{matrix} I_{ε}^{5} = \int_{0}^{t} \int_{Ω_{ε}} \nabla u^{ε} \nabla Φ_{ε} (φ, φ) d x d s \to \int_{0}^{t} \int_{Ω} \nabla u \nabla φ d x d s strongly in C^{0} ([0, T]) . \end{matrix}$ □

3.3. Corrector for the case $r_{ε} ≪ ε^{3}$

For this case we have the following corrector result:

Theorem 3.10.

Under hypotheses ( 1.4 ), ( 3.1 ), if $γ = 0$ , the sequence ${(u^{ε})}_{ε > 0}$ satisfies $\begin{matrix} u^{ε} ⟶ u strongly in C^{0} ([0, T]; L^{2} (Ω)), \end{matrix}$ and $\begin{matrix} \{\begin{matrix} u^{ε} = Φ_{ε} (u, v_{0}) + R_{ε} with \\ R_{ε} ⟶ 0 strongly in L^{2} (0, T; W_{0}^{1, 1} (Ω)), \end{matrix} \end{matrix}$ where u is the solution of ( 2.15 ).

As in the other cases, before proving Theorem 3.10, we will need again some technical results.

Proposition 3.11.

Assume that the hypotheses of Theorem 3.10 hold true. Then $\begin{matrix} E_{ε} ⟶ E_{2} + \frac{a}{2} | v_{0} |_{Ω}^{2} strongly in C^{0} ([0, T]), \end{matrix}$ and for any $(φ, ψ) \in C^{\infty} ([0, T]; D (Ω)) \times D (Ω)$ , we have $\begin{array}{l} (3.42) & e_{ε} (Φ_{ε} (φ, ψ)) \to \overline{e} (φ, ψ) strongly in C^{0} ([0, T]), \\ (3.43) & ⨍_{D_{ε}} u^{ε} ψ d x \to \int_{Ω} v_{0} ψ d x strongly in C^{0} ([0, T]), \end{array}$ and $\begin{matrix} e_{ε} (u^{ε} - Φ_{ε} (φ, ψ)) \to \overline{e} (u - φ, v_{0} - ψ) strongly in C^{0} ([0, T]), \end{matrix}$ where $\begin{matrix} \begin{matrix} E_{2} (t) : = \frac{1}{2} {| u (t) |}_{Ω}^{2} + \int_{0}^{t} | \nabla u |_{Ω}^{2} d s, \\ \overline{e} (φ, ψ) (t) : = \frac{1}{2} ({| φ (t) |}_{Ω}^{2} + a | ψ |_{Ω}^{2}) + \int_{0}^{t} | \nabla φ |_{Ω}^{2} d s, \end{matrix} \end{matrix}$ and $E_{2}$ is the energy of problem ( 2.15 ).

Remark 2.

Note that in this case Proposition 3.3 is still valid with the assumptions of Theorem 3.10.

Proof of Theorem 3.10.

The proof of the first result is similar to the corresponding one of the Theorem 3.2.

For the second result, from Theorem 2.9 we have $\begin{matrix} u \in L^{2} (0, T; H_{0}^{1} (Ω)) \cap C^{0} ([0, T]; L^{2} (Ω)) . \end{matrix}$

Consider a sequence $(φ_{n}, ψ_{n})$ in $C^{\infty} ([0, T]; D (Ω)) \times D (Ω)$ such that, $\begin{array}{l} φ_{n} \to u strongly in L^{2} (0, T; H_{0}^{1} (Ω)) \cap C^{0} ([0, T]; L^{2} (Ω)), \\ ψ_{n} \to v_{0} strongly in H_{0}^{1} (Ω), \end{array}$ as $n \to + \infty$ .

Furthermore, $\begin{matrix} \begin{matrix} {‖ \nabla (u^{ε} - Φ_{ε} (u, v_{0})) ‖}_{L^{2} (0, T; L^{1} (Ω))} ⩽ & C {‖ \nabla (u^{ε} - Φ_{ε} (φ_{n}, ψ_{n})) ‖}_{L^{2} (Ω^{T})} \\ + {‖ (1 - w_{R_{ε}}) \nabla (φ_{n} - u) ‖}_{L^{2} (0, T; L^{1} (Ω))} \\ + {‖ (φ_{n} - u) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω))} \\ + {‖ G_{r_{ε}} (ψ_{n} - v_{0}) \nabla w_{R_{ε}} ‖}_{L^{2} (0, T; L^{1} (Ω))} . \end{matrix} \end{matrix}$

Following along the lines of the proof of (3.6) (see proof of Theorem 3.2), it is easily seen that each term of the right-hand side of the above inequality tends to zero as $ε \to 0$ , which achieves the proof. □

Proof of Proposition 3.11.

The proof of the result is similar to the corresponding one of the first case where $γ \in] 0, + \infty [$ , with a few differences that are shown below.

The proof of convergence (3.42) is much simpler because in this case the term $A_{2}$ (see proof of Proposition 3.5) is handled as follows $\begin{matrix} \begin{matrix} A_{2} & = \int_{0}^{t} \int_{C_{ε}} {| (1 - w_{R_{ε}}) \nabla φ + (G_{r_{ε}} (ψ) - φ) \nabla w_{R_{ε}} |}^{2} d x d s \\ ⩽ 2 \int_{0}^{t} \int_{C_{ε}} {| (1 - w_{R_{ε}}) \nabla φ |}^{2} d x d s + 2 \int_{0}^{t} \int_{C_{ε}} {| (G_{r_{ε}} (ψ) - φ) \nabla w_{R_{ε}} |}^{2} d x d s \\ ⩽ 2 T (‖ \nabla φ ‖_{L^{\infty} (Ω^{T})}^{2} | C_{ε} | + C {| G_{r_{ε}} (ψ) - φ |}_{L^{\infty} (Ω^{T})}^{2} γ_{ε}) \to 0, \end{matrix} \end{matrix}$ where we have used the estimates of Proposition 2.1, the uniform boundedness of $G_{r_{ε}} (ψ)$ in $L^{\infty} (Ω^{T})$ , and the facts that $| C_{ε} | \to 0$ , and $γ_{ε} \to 0$ .

For the convergence (3.43), thanks to (2.14) we have $\begin{matrix} ⨍_{D_{ε}} u^{ε} (t) ψ d x \to \int_{Ω} v_{0} ψ d x a.e. t in [0, T], \end{matrix}$ and this convergence takes place in $C^{0} ([0, T])$ , since the function $⨍_{D_{ε}} u^{ε} (t) ψ d x$ is bounded in $H^{1} (0, T)$ , as in the proof of Proposition 3.6.

Concerning the last convergence, using the same arguments as those from the proof of Lemma 3.7, the difference comes from the limit of $I_{ε}^{5}$ , which now is $\begin{matrix} I_{ε}^{5} = \int_{0}^{t} \int_{Ω_{ε}} \nabla u^{ε} \nabla Φ_{ε} (φ, ψ) d x d s \to \int_{0}^{t} \int_{Ω} \nabla u \nabla φ d x d s strongly in C^{0} ([0, T]) . \end{matrix}$ □

Footnotes

References

Allaire, Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational. Mech. Anal. 113 (1991), 209–259.

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