Asymptotic behavior of the Stokes problem with Robin condition in a domain with small holes

Abstract

We consider the Stokes problem in a domain $Ω_{ε δ_{1} δ_{2}}$ of $R^{N}$ , $N ⩾ 3$ ε-periodically perforated by holes of size $r_{1} (ε δ_{1})$ and $r_{2} (ε δ_{2})$ (ε, $δ_{1} = δ_{1} (ε)$ , $δ_{2} = δ_{2} (ε)$ being small parameters), with $r_{1} (ε δ_{1}) / ε \to 0$ and $r_{2} (ε δ_{2}) / ε \to 0$ as $ε \to 0$ . Our aim is to describe the asymptotic behavior of the velocity and the pressure of the fluid as $ε \to 0$ and give, if possible, a limit (“homogenized”) problem. To do so, we use the periodic unfolding method introduced by Cioranescu, Damlamian and Griso (C. R. Acad. Sci. Paris, Ser. I 335 (2002) 99–104; SIAM J. of Math. Anal. 40(4) (2008) 1585–1620). It allow us to consider a general geometric framework and so, to extend the results from Cioranescu, Donato and Ene (Math. Models and Methods in Appl. Sciences 19 (1996) 857–881) and Capatina and Ene (European J. of Math. 22 (2011) 333–345).

Keywords

Stokes problem Robin condition unfolding method

1. Introduction

The aim of this work is to apply the unfolding method to perforated domains by holes of size $ε δ$ , distributed periodically with period ε. We consider here the case where $δ = δ (ε)$ is such that $δ \to 0$ as $ε \to 0$ . Such kind of holes are known in literature as “small holes” (see e.g. [12,13]), or “tiny holes” (see e.g. [1,18]).

Let Ω be a bounded open domain of $R^{N}$ , $N ⩾ 3$ with $| \partial Ω | = 0$ . The reference (periodicity) cell Y is given by $\begin{matrix} Y =] - \frac{1}{2}, + \frac{1}{2} [^{N} . \end{matrix}$

Introduce now the compact subsets sets T and B of Y (the holes in Y), such that $T \cap B = \emptyset$ . We assume that B and T have Lipschitz boundary. For two small positive parameters $δ_{1}$ and $δ_{2}$ with $δ_{2} \neq δ_{1}$ , the perforated (periodicity) cell $Y_{δ_{1} δ_{2}}^{*}$ is $\begin{matrix} (1.1) & Y_{δ_{1} δ_{2}}^{*} = Y ∖ (δ_{1} \overline{B} \cup δ_{2} \overline{T}) . \end{matrix}$ Then the holes in Ω are the following sets: $\begin{matrix} (1.2) & \begin{matrix} T_{ε δ_{1}} = ⋃_{ξ \in Ξ_{ε}} (ε δ_{1} T + ε ξ), B_{ε δ_{2}} = ⋃_{ξ \in Ξ_{ε}} (ε δ_{2} B + ε ξ), \end{matrix} \end{matrix}$ where $\begin{matrix} Ξ_{ε} = {ξ \in Z^{N} : ε (ξ + Y) \subset Ω} . \end{matrix}$ By construction, the size of the holes in $T_{ε δ_{1}}$ is of order $ε δ_{1}$ , and the size of those in $B_{ε δ_{2}}$ is of order $ε δ_{2}$ . Notice also that by construction, the boundary $\partial Ω$ do not intersect that of the holes.

Then, the perforated domain $Ω_{ε δ_{1} δ_{2}}$ is defined as $\begin{matrix} Ω_{ε δ_{1} δ_{2}} = Ω ∖ (T_{ε δ_{1}} \cup B_{ε δ_{2}}) . \end{matrix}$

In the following, we suppose that $δ_{1} = δ_{1} (ε)$ , and $δ_{2} = δ_{2} (ε)$ depend on ε, and satisfy $\begin{matrix} (1.3) & lim_{ε \to 0} δ_{1} (ε) = lim_{ε \to 0} δ_{2} (ε) = 0 . \end{matrix}$ This means that we are typically in the framework of domains with “small” holes mentioned at the beginning of this section.

We now consider the Stokes problem in $Ω_{ε δ_{1} δ_{2}}$ with a Robin-type condition on the boundary the set of holes $T_{ε δ_{1}}$ , and a homogeneous Dirichlet one on the external boundary $\partial Ω$ of the domain and on the boundary of the holes in $B_{ε δ_{1}}$ . The unknowns $u_{ε δ_{1} δ_{2}}$ the velocity field, and $p_{ε}$ the (scalar) pressure, are characterized by the system, $\begin{matrix} (1.4) & \{\begin{matrix} - ν Δ u_{ε δ_{1} δ_{2}} + \nabla p_{ε} = f_{ε} & in Ω_{ε δ_{1} δ_{2}}, \\ div u_{ε δ_{1} δ_{2}} = 0 & in Ω_{ε δ_{1} δ_{2}}, \\ u_{ε δ_{1} δ_{2}} = 0 & on \partial Ω \cup \partial B_{ε δ_{2}}, \\ ν \frac{\partial u_{ε δ_{1} δ_{2}, τ}}{\partial n} + α ε^{γ} u_{ε δ_{1} δ_{2}} = g_{ε} & on \partial T_{ε δ_{1}}, \end{matrix} \end{matrix}$ where $f_{ε}$ is the field of exterior body forces and $g_{ε}$ is the field of exterior surface forces. The constants $α ⩾ 0$ and γ are given, as well as $ν > 0$ which is the viscosity of the fluid. The outward normal to $T_{ε δ}$ is n, while τ is the tangent vector to $T_{ε δ}$ , so that for a function v, $v_{τ} = v - v_{n} \cdot n$ where $v_{n} = v \cdot n$ .

There is an extensive literature on the homogenization in perforated domains in $R^{N}$ . For the case of “small” holes of size $ε^{α}$ , $α > 0$ , let us mention the works of Cioranescu and Murat [12,13], concerning the homogeneous Dirichlet problem for the Poisson equation. They showed that the size $ε^{N / N - 2}$ ( $N > 2$ ) is “critical” in the sense that the limit problem not only contains the laplacian but also an additional zero order term, called by the authors “strange term”, it depends on the capacity of the set of holes at the limit.

The non homogeneous Neumann problem for the laplacian in the same geometrical framework, was studied by Conca and Donato [16]. In this case, the critical size of the holes is of order $ε^{N / N - 1}$ , and the contribution of the holes at the limit is an additional right-hand side integral term.

As concerning the Stokes problem, a pioneering work is that of Ene and Sanchez Palencia [17] who considered a periodic porous medium (with ε-size holes) with a Dirichlet condition on the boundary. They obtained at the limit the Darcy law, by applying the multiple scale method (introduced in [4]), together with sharp error estimates. This is the first mathematical justification of the experimental Darcy’s law.

The Stokes problem with non homogeneous slip boundary condition depending on a parameter $γ \in R^{N}$ (with still ε-size holes) was studied by Cioranescu, Donato and Ene in [10] by energy methods. They obtained at the limit, for different values of γ, either a Darcy-type law, or a Brinkman equation or a Stokes-type system.

In the framework of small holes, Allaire considered the Stokes system in [2] and the Navier–Stokes equation in [1,3]. He obtained at the limit the same laws (Darcy, Brinkman or Stokes) but now it is the order of the size of holes with respect to ε who determines the type of the homogenized problem.

When working in perforated domains, the main difficulties are related to the fact that the equations and their solutions are defined on domains which strongly depend on ε. In order to speak about convergence when $ε \to 0$ (that is to “homogenize”), one needs to introduce extension operators to a fixed domain and to construct test functions, specific for each situation. To do that, several restrictions on the geometry of the holes and of the domain are necessary.

These difficulties are solved by the periodic unfolding method ([7] and [8]), as can be seen for example, in Cioranescu et al. [6]. The advantage of the unfolding method, it is the fact that it separates the scales. So, one can treat in particular, holes at different scales in the same period. Such kind of problems, because of their complexity, cannot be solved by the classical methods in homogenization. For the Stokes problem, the method was applied by Capatina and Ene [5] and by Zaki [22] in the case of ε-size holes. This method was also extended to domains with small holes in Cioranescu et al. [9] and used by Ould Hammouda in [19] and [20]. Our aim here is to apply it for the Stokes problem (1.4).

The paper is organized as follows. In Section 2 we give the variational formulation of the problem (1.4). In the Section 3, for the reader’s convenience, we recall briefly the different definitions of the periodic unfolding operators and their properties. So, in Section 3.1, we list some notations. Section 3.2 is devoted to the periodic unfolding operator for fixed domains introduced in [7]. Section 3.3 recalls the definition and the properties of the unfolding operator for perforated domains with holes of size ε and finally Section 3.4 those of the unfolding operator corresponding to the case of volume-distributed small holes. In Section 3.5 we introduce the boundary operator related to the case of small holes, and prove some of its properties.

Finally, the homogenization result concerning problem (1.4) is stated and proved in Section 4.

2. Variational formulation

From now on, we make the following hypotheses on the data $f_{ε}$ and $g_{ε}$ : $\begin{matrix} (2.1) & \begin{matrix} f_{ε} \in {(L^{2} (Ω_{ε δ_{1} δ_{2}}))}^{N}, ε \tilde{f_{ε}} ⇀ f weakly in {(L^{2} (Ω))}^{N} \end{matrix} \end{matrix}$ and $\begin{matrix} (2.2) & \begin{matrix} g_{ε} (x) = g (\frac{1}{δ_{1}} {\frac{x}{ε}}), g \in {(L^{2} (\partial T))}^{N}, Y -periodic function . \end{matrix} \end{matrix}$

Let use introduce the functional space $\begin{matrix} V_{ε δ_{1} δ_{2}} = {φ ∣ φ \in {(H^{1} (Ω_{ε δ_{1} δ_{2}}))}^{N}, φ = 0 on \partial Ω \cup \partial B_{ε δ_{2}}, φ \cdot n = 0 on \partial T_{ε δ_{1}}}, \end{matrix}$ which is Hilbert for scalar product $\begin{matrix} ⟨ φ, ψ ⟩ = \int_{Ω_{ε δ_{1} δ_{2}}} \nabla φ \nabla ψ d x = \sum_{i, j = 1}^{N} \int_{Ω_{ε δ_{1} δ_{2}}} \frac{\partial φ_{i}}{\partial x_{j}} \frac{\partial ψ_{i}}{\partial x_{j}} d x . \end{matrix}$

The variational formulation of system (1.4) is then the following: $\begin{matrix} (2.3) & \{\begin{matrix} Find u_{ε δ_{1} δ_{2}} \in V_{ε δ_{1} δ_{2}}, p_{ε} \in L^{2} (Ω_{ε δ_{1} δ_{2}}) satisfying \\ \int_{Ω_{ε δ_{1} δ_{2}}} \nabla u_{ε δ_{1} δ_{2}} \nabla φ d x + α ε^{γ} \int_{\partial T_{ε δ_{1}}} u_{ε δ_{1} δ_{2}} φ d σ (x) - \int_{Ω_{ε δ_{1} δ_{2}}} p_{ε} div φ d x \\ = \int_{Ω_{ε δ_{1} δ_{2}}} f_{ε} φ d x + \int_{\partial T_{ε δ_{1}}} g_{ε} φ d σ (x), \forall φ \in V_{ε δ_{1} δ_{2}}, \\ \int_{Ω_{ε δ_{1} δ_{2}}} u_{ε δ_{1} δ_{2}} \nabla φ d x = 0, \forall φ \in V_{ε δ_{1} δ_{2}} . \end{matrix} \end{matrix}$ It is classical that this problem (see e.g. [21]) is well-defined, it has a unique solution.

3. The periodic unfolding operators

3.1. Some notations

In the sequel, we will use the following notations: $\begin{matrix} {\hat{Ω}}_{ε} = interior {⋃_{ξ \in Ξ_{ε}} ε (ξ + \overline{Y})}, Λ_{ε} = Ω ∖ {\hat{Ω}}_{ε}, \\ Y_{δ} = Y ∖ δ \overline{T} for δ > 0, \\ \tilde{φ} = the extension by 0 to Ω, \forall φ \in L^{p} (O), O \subset Ω . \end{matrix}$

For every z in $R^{N}$ , we denote by $[z]$ the unique integer combination of periods such that ${z}_{Y} = z - {[z]}_{Y}$ belongs to Y. It follows that any $x \in R^{N}$ can be written in the (unique) form $\begin{matrix} x = ε ({[\frac{x}{ε}]}_{Y} + {\frac{x}{ε}}_{Y}) with {\frac{x}{ε}}_{Y} \in Y . \end{matrix}$

For any subset K of Y, the average of φ over K and on $\partial K$ , are given respectiely, by $\begin{matrix} M_{K} (φ) = \frac{1}{| K |} \int_{K} φ d y, M_{\partial K} (φ) = \frac{1}{| \partial K |} \int_{\partial K} φ d σ . \end{matrix}$

3.2. The unfolding operator $T_{ε}$ for fixed domains

We now recall from [19] and [14], the definition and some properties of a variant of the periodic unfolding operator introduced in [7] and [8].

Definition 3.1.
For $ϕ \in L^{p} (Ω)$ , $p \in [1, + \infty]$ , the unfolding operator $\begin{matrix} T_{ε} : L^{p} (Ω) ⟶ L^{p} (Ω \times Y_{δ}), \end{matrix}$ is defined as follows: $\begin{matrix} T_{ε} (ϕ) (x, y) = \{\begin{matrix} ϕ (ε {[\frac{x}{ε}]}_{Y} + ε y) & a.e. for (x, y) \in {\hat{Ω}}_{ε} \times Y_{δ}, \\ 0 & a.e. for (x, y) \in Λ_{ε} \times Y_{δ} . \end{matrix} \end{matrix}$
Proposition 3.2.
If ${w_{ε}}_{ε}$ is a sequence in $L^{1} (Ω)$ satisfying $\int_{Λ_{ε}} | w_{ε} | d x \to 0$ , then $\begin{matrix} \int_{Ω} w_{ε} d x - \int_{Ω \times Y_{δ}} T_{ε} (w_{ε}) d x d y \to 0 . \end{matrix}$

Notice that if we replace $Y_{δ}$ by Y, we retrieve the original unfolding operator $T_{ε}$ as defined in [7,8].
Definition 3.3.
The local average $M_{Y}^{ε} : L^{p} (Ω) ⟶ L^{p} (Ω)$ , is defined for any ϕ in $L^{p} (Ω)$ , $1 ⩽ p < \infty$ , by $\begin{matrix} M_{Y}^{ε} (ϕ) (x) = \int_{Y} T_{ε} (ϕ) (x, y) d y . \end{matrix}$

It is known that if ${v_{ε}}$ is a bounded sequence in $L^{p} (Ω)$ , such that $v_{ε} \to v$ strongly in $L^{p} (Ω)$ , then $\begin{matrix} (3.1) & M_{Y}^{ε} (v_{ε}) \to v strongly in L^{p} (Ω) . \end{matrix}$
3.3. The unfolding operator $T_{ε}^{*}$ for perforated domains

Following [6], an unfolding operator $T_{ε}^{*}$ for functions defined on perforated domains $Ω_{ε}^{*}$ with holes of size ε. We will give here some of its main properties, needed later on. For more details and proofs, we refer the reader to [6]. Let first define the geometric settings for this special case.

With the notations from Section 3.1 (recall that T is a closed strict subset of Y), the part occupied by the material in the cell Y, supposed to be connected, is now denoted $Y^{*} = Y ∖ \overline{T}$ . Then the perforated domain $Ω_{ε}^{*}$ is obtained by removing from Ω the set of holes $T_{ε}$ , $\begin{matrix} (3.2) & Ω_{ε}^{*} = Ω ∖ T_{ε} = {x \in Ω | {\frac{x}{ε}} \in Y^{*}} . \end{matrix}$ With this definition, we also set $\begin{matrix} (3.3) & \begin{matrix} {\hat{Ω}}_{ε}^{*} ≐ {\hat{Ω}}_{ε} ∖ \overline{T_{ε}}, \\ Λ_{ε}^{*} ≐ Ω_{ε}^{*} ∖ {\hat{Ω}}_{ε}^{*} . \end{matrix} \end{matrix}$

Definition 3.4.
For ϕ Lebesgue-measurable on $Ω_{ε}^{}$ , the linear and continuous unfolding operator $\begin{matrix} T_{ε}^{} : L^{p} (Ω_{ε}^{}) ⟶ L^{p} (Ω \times Y^{}), p \in [1, + \infty [, \end{matrix}$ is defined by $\begin{matrix} T_{ε}^{} (ϕ) (x, y) = \{\begin{matrix} ϕ (ε {[\frac{x}{ε}]}_{Y} + ε y) & a.e. for (x, y) \in {\hat{Ω}}_{ε} \times Y^{}, \\ 0 & a.e. for (x, y) \in Λ_{ε} \times Y^{} . \end{matrix} \end{matrix}$

The main characteristic of this operator is that it maps functions defined on the oscillating domain $Ω_{ε}^{}$ , into functions defined on the fixed domain $Ω \times Y^{}$ .

It is easily seen that for any function $w \in L^{p} (Ω_{ε}^{})$ , one has $\begin{matrix} (3.4) & T_{ε}^{} (w) = T_{ε} (\tilde{w}) |_{Ω \times Y^{}} . \end{matrix}$ This implies that the main of properties of $T_{ε}^{}$ are consequences of those of $T_{ε}$ , as in particular, the integration formula written for every ϕ in $L^{p} (Ω_{ε}^{})$ $\begin{matrix} \int_{Ω \times Y^{}} T_{ε}^{} (ϕ) (x, y) d x d y = \int_{Ω_{ε}^{}} ϕ (x) d x - \int_{Λ_{ε}^{}} ϕ (x) d x = \int_{{\hat{Ω}}_{ε}^{}} ϕ (x) d x . \end{matrix}$ An equivalent of Proposition 3.2 holds, allowing to pass from integrals over $Ω_{ε}^{}$ to integrals over the fixed domain $Ω \times Y^{}$ .
Proposition 3.5.
If ${ϕ_{ε}}$ is a sequence in* $L^{1} (Ω_{ε}^{})$ satisfying* $\int_{Λ_{ε}^{}} | ϕ_{ε} | d x \to 0$ , then* $\begin{matrix} \int_{Ω_{ε}^{}} ϕ_{ε} d x - \int_{Ω \times Y^{}} T_{ε}^{} (w_{ε}) d x d y \to 0 . \end{matrix}$
Proposition 3.6.
Let* ${w_{ε}}$ a sequence in $L^{p} (Ω)$ such that $w_{ε} \to w$ strongly in $L^{p} (Ω)$ , then $T_{ε}^{} (w_{ε}) \to w$ strongly in* $L^{p} (Ω \times Y^{})$ .

The main result concerning perforated domains is the next theorem. Theorem 3.7.
Let p be in* $] 1, + \infty]$ . Let ${w_{ε}}$ be a sequence in $W^{1, p} (Ω_{ε}^{})$ such that* $\begin{matrix} ‖ w_{ε} ‖_{W^{1, p} (Ω_{ε}^{})} ⩽ C . \end{matrix}$ Then, there exist a subsequence (still denoted ε), w in* $W^{1, p} (Ω)$ and $\hat{w}$ in $L^{p} (Ω; W_{per}^{1, p} (Y^{}))$ , such that* $\begin{matrix} (3.5) & \begin{matrix} \tilde{w_{ε}} ⇀ | Y^{} | w weakly in L^{p} (Ω), \\ T_{ε}^{} (w_{ε}) \to w strongly in L_{loc}^{p} (Ω; W^{1, p} (Y^{})), \\ T_{ε}^{} (w_{ε}) ⇀ w weakly in L^{p} (Ω; W^{1, p} (Y^{})), \\ T_{ε}^{} (\nabla w_{ε}) ⇀ \nabla w + \nabla_{y} \hat{w} weakly in L^{p} {(Ω \times Y^{*})}^{N} . \end{matrix} \end{matrix}$

3.4. The unfolding operator $T_{ε δ}$ depending on two parameters ε and δ

We start by recalling the definition and the main properties of the periodic unfolding operator with two parameters δ and ε introduced in [9] (see also [20]).

Following [9], the geometry of domains with small holes requires a specific unfolding operator depending on both parameters ε and δ. When considering functions belonging to the space $H_{0}^{1} (Ω_{ε δ})$ , one naturally can extend them by zero to the whole of Ω, and obviously, these extensions belong to $H_{0}^{1} (Ω)$ . It is why one may not distinguish the functions of $H_{0}^{1} (Ω_{ε δ})$ and their extensions in $H_{0}^{1} (Ω)$ .

Definition 3.8.
For $ϕ \in L^{p} (Ω)$ , $p \in [1, + \infty [$ , the linear and continuous unfolding operator $\begin{matrix} T_{ε δ} : L^{p} (Ω) ⟶ L^{p} (Ω \times R^{n}), \end{matrix}$ is defined by $\begin{matrix} T_{ε δ} (ϕ) (x, z) = \{\begin{matrix} T_{ε} (ϕ) (x, δ z) & a.e. for (x, z) \in {\hat{Ω}}_{ε} \times \frac{1}{δ} Y, \\ 0 & otherwise . \end{matrix} \end{matrix}$
Proposition 3.9.

For any $u \in L^{2} (Ω)$ $\begin{matrix} ‖ T_{ε δ} ‖_{L^{2} (Ω \times R^{N})}^{2} ⩽ \frac{1}{δ^{N}} ‖ u ‖_{L^{2} (Ω)}^{2} \end{matrix}$

If ${w_{ε}}$ is a sequence in $L^{1} (Ω)$ satisfying $\int_{Λ_{ε}} | w_{ε} | d x \to 0$ , then $\begin{matrix} \int_{Ω} w_{ε} d x - δ^{N} \int_{Ω \times R^{N}} T_{ε δ} (w_{ε}) d x d z \to 0 . \end{matrix}$

Proposition 3.10.
Suppose $N ⩾ 3$ and denote by $2^{}$ the Sobolev exponent* $\frac{2 N}{N - 2}$ associated to 2. Let ω be an open and bounded set in $R^{N}$ . Then the following estimates hold: $\begin{array}{l} (3.6) & {‖ \nabla_{z} (T_{ε δ} (u)) ‖}_{L^{2} (Ω \times \frac{1}{δ} Y)}^{2} ⩽ \frac{ε^{2}}{δ^{N - 2}} ‖ \nabla u ‖_{L^{2} (Ω)}^{2}, \\ (3.7) & {‖ T_{ε δ} (u - M_{Y}^{ε} (u)) ‖}_{L^{2} (Ω; L^{2^{}} (R^{N}))}^{2} ⩽ \frac{C ε^{2}}{δ^{N - 2}} ‖ \nabla u ‖_{L^{2} (Ω)}^{2}, \\ (3.8) & {‖ T_{ε δ} (u) ‖}_{L^{2} (Ω \times ω)}^{2} ⩽ \frac{2 C ε^{2}}{δ^{N - 2}} | ω |^{\frac{2}{N}} ‖ \nabla u ‖_{L^{2} (Ω)}^{2} + 2 | ω | ‖ u ‖_{L^{2} (Ω)}^{2}, \end{array}$ where C denotes the Sobolev–Poincaré–Wirtinger constant for* $H^{1} (Y)$ and $M_{Y}^{ε}$ is actually given by Definition 3.3 .

We will also make use later on, of the following definition: Definition 3.11.
The local average $M_{Y}^{ε δ} : L^{P} (Ω) ⟶ L^{P} (Ω)$ is given for any ϕ in $L^{p} (Ω)$ , $1 ⩽ p < \infty$ , by $\begin{matrix} M_{Y}^{ε δ} (ϕ) (x) = δ^{N} \int_{\frac{1}{δ} Y} T_{ε δ} (ϕ) (x, y) d y . \end{matrix}$

3.5. The boundary unfolding operator $T_{ε δ}^{b}$

Definition 3.12.
For ϕ in $L^{p} (\partial T_{ε δ})$ , $p \in [1, + \infty [$ , the boundary unfolding operator $T_{ε δ}^{b} : L^{p} (\partial T_{ε δ}) \mapsto L^{p} (R^{N} \times \partial T)$ is defined by $\begin{matrix} (3.9) & T_{ε δ}^{b} (ϕ) (x, z) = ϕ (ε {[\frac{x}{ε}]}_{Y} + ε δ z), x \in R^{N}, z \in \partial T . \end{matrix}$
Remark 3.13.
Observe that if $δ = 1$ , $T_{ε 1}^{b}$ is nothing else than the boundary unfolding operator $T_{ε}^{b}$ defined in [6].
Proposition 3.14.
The linear and continuous boundary unfolding operator $T_{ε δ}^{b}$ has the following property: $\begin{matrix} \int_{\partial T_{ε δ}} ϕ (x) d σ (x) = \frac{δ^{N - 1}}{ε} \int_{R^{N} \times \partial T} T_{ε δ}^{b} (ϕ) (x, z) d x d σ (z), \forall ϕ \in L^{1} (\partial T_{ε δ}) . \end{matrix}$
Proposition 3.15.
Let $φ_{ε}$ in $W^{1, p} (Ω_{ε δ})$ for every ε, and let $φ \in W^{1, p} (\frac{1}{δ} Y)$ such that $\begin{matrix} T_{ε δ} (φ_{ε}) ⇀ φ weakly in L^{p} (Ω; W^{1, p} (\frac{1}{δ} Y)) . \end{matrix}$ Then the following convergence holds true: $\begin{matrix} T_{ε δ}^{b} (φ_{ε}) ⇀ φ weakly in L^{p} (Ω; W^{1 - \frac{1}{p}, p} (\partial T)) . \end{matrix}$
Proposition 3.16.
Let $g \in L^{2} (\partial T)$ and set $\begin{matrix} (3.10) & g_{ε} (x) = g (\frac{1}{δ} {\frac{x}{ε}}), \forall x \in R^{N} ∖ ⋃_{ξ \in Z^{N}} ε (ξ + δ T) . \end{matrix}$ Then, for all $ϕ \in H^{1} (Ω)$ , one has $\begin{matrix} (3.11) & | \int_{\partial T_{ε δ}} g_{ε} (x) ϕ (x) d x | ⩽ C \frac{δ^{\frac{N}{2} - 1}}{ε} (ε δ + | M_{\partial B} (g) |) ‖ \nabla ϕ ‖_{L^{2} {(Ω)}^{N}} . \end{matrix}$
Proposition 3.17.
Let g be in $L^{2} (\partial T)$ and $g_{ε}$ be defined by ( 3.10 ). Then, for all $ϕ \in H^{1} (Ω)$ , one has the convergences $\begin{matrix} (3.12) & \frac{ε}{δ^{N - 1}} \int_{\partial T_{ε δ}} g_{ε} (x) ϕ (x) d s \to | \partial T | M_{\partial T} (g) \int_{Ω} ϕ (x) . \end{matrix}$
Proposition 3.18.
For $g \in L^{2} (\partial T)$ and $v \in H^{1} (Ω)$ , we have the estimate, $\begin{matrix} | \int_{\partial T} g (x) v (x) d σ (x) | ⩽ c (δ^{\frac{N}{2}} ‖ \nabla v ‖_{L^{2} (Ω)} + \frac{δ^{N - 1}}{ε} ‖ v ‖_{L^{2} (Ω)}) \end{matrix}$
Proof.
By using the boundary unfolding operator $T_{ε δ}^{b}$ , we have successively, $\begin{matrix} | \int_{\partial T} g (x) v (x) d σ (x) | & = | \frac{δ^{N - 1}}{ε} \int_{Ω \times \partial T} g (z) T_{ε δ}^{b} (v) d x d σ (z) | \\ ⩽ \frac{δ^{N - 1}}{ε} | \int_{Ω \times \partial T} g (z) (T_{ε δ}^{b} (v) - M_{Y}^{ε δ} (v)) d x d σ (z) | \\ + \frac{δ^{N - 1}}{ε} | \int_{Ω \times \partial T} g (z) M_{Y}^{ε δ} (v) d x d σ (z) | . \end{matrix}$ Let now estimate the two integral terms on the right-hand side of this inequality. On the one hand, by using Cauchy–Schwarz inequality and recalling (3.7), we get $\begin{matrix} (3.13) & \frac{δ^{N - 1}}{ε} | \int_{Ω \times \partial T} g (z) (T_{ε δ}^{b} (v) - M_{Y}^{ε δ} (v)) d x d σ (z) | ⩽ c δ^{\frac{N}{2}} ‖ \nabla v ‖_{L^{2} (Ω)} . \end{matrix}$

On the other hand, $\begin{matrix} \frac{δ^{N - 1}}{ε} | \int_{Ω \times \partial T} g (z) M_{Y}^{ε δ} (v) d x d σ (z) | & ⩽ c \frac{δ^{N - 1}}{ε} ‖ g ‖_{L^{2} (Ω)} {‖ M_{Y}^{ε δ} (v) ‖}_{L^{2} (Ω \times \partial T)} \\ ⩽ c \frac{δ^{N - 1}}{ε} ‖ g ‖_{L^{2} (\partial T)} {(\int_{Ω \times \partial T} {| M_{Y}^{ε δ} (v) |}^{2} d x d σ (z))}^{1 / 2} . \end{matrix}$ Applying the Fubini theorem, we immediately obtain $\begin{matrix} (3.14) & \frac{δ^{N - 1}}{ε} | \int_{Ω \times \partial T} g (z) M_{Y}^{ε δ} (v) d x d σ (z) | ⩽ c \frac{δ^{N - 1}}{ε} ‖ g ‖_{L^{2} (\partial T)} | \partial T |^{\frac{1}{2}} {(\int_{Ω} {| M_{Y}^{ε δ} (v) |}^{2} d x)}^{1 / 2} . \end{matrix}$ Let have a look to the last integral above. By Cauchy–Schwarz inequality and Proposition 3.9(1), it follows that $\begin{matrix} {(\int_{Ω} {| M_{Y}^{ε δ} (v) |}^{2} d x)}^{1 / 2} & = δ^{N} {(\int_{Ω} {(\int_{\frac{1}{δ} Y} T_{ε δ} (v) d z)}^{2} d x)}^{1 / 2} \\ ⩽ δ^{\frac{N}{2}} {‖ T_{ε δ} (v) ‖}_{L^{2} (Ω \times \frac{1}{δ} Y)} ⩽ ‖ v ‖_{L^{2} (Ω)} . \end{matrix}$ This inequality used in (3.14) yields $\begin{matrix} \frac{δ^{N - 1}}{ε} | \int_{Ω \times \partial T} g (z) M_{Y}^{ε δ} (v) d x d σ (z) | ⩽ c \frac{δ^{N - 1}}{ε} ‖ g ‖_{L^{2} (\partial T)} . ‖ v ‖_{L^{2} (Ω)}, \end{matrix}$ which, together with (3.13), gives the result. □
4. Main result

4.1. A priori estimates for $u_{ε δ_{1} δ_{2}}$ and $p_{ε}$

Proposition 4.1.
Let $γ \in R$ and $(u_{ε δ_{1} δ_{2}}, p_{ε})$ be the solution of problem ( 1.4 ). Then there exists a positive constant C independent of ε such that, $\begin{array}{l} (4.1) & ‖ u_{ε δ_{1} δ_{2}} ‖_{H^{1} {(Ω_{ε δ_{1} δ_{2}})}^{N}} ⩽ C, \\ (4.2) & ‖ ε u_{ε δ_{1} δ_{2}} ‖_{H^{1} {(Ω_{ε δ_{1} δ_{2}})}^{N}} ⩽ C, \end{array}$ and $\begin{matrix} (4.3) & {‖ ε^{- 1} u_{ε δ_{1} δ_{2}} ‖}_{L^{2} {(Ω_{ε δ_{1} δ_{2}})}^{N \times N}} ⩽ C . \end{matrix}$
Proof.
Taking $u_{ε δ_{1} δ_{2}}$ as test function in (2.3) (which makes sense), we get $\begin{matrix} ν ‖ \nabla u_{ε δ_{1} δ_{2}} ‖_{L^{2} (Ω_{ε δ_{1} δ_{2}})}^{2} + α ε^{γ} ‖ u_{ε δ_{1} δ_{2}} ‖_{L^{2} (\partial T_{ε δ_{1}})}^{2} \\ ⩽ | \int_{Ω_{ε δ_{1} δ_{2}}} f_{ε} u_{ε δ_{1} δ_{2}} d x | + | \int_{\partial T_{ε δ_{1}}} g_{ε} u_{ε δ_{1} δ_{2}} d σ (x) | . \end{matrix}$

We have successively, by using Cauchy–Schwarz and Poincaré inequalities as well as Proposition 3.18, $\begin{matrix} (4.4) & \begin{matrix} ν ‖ \nabla u_{ε δ_{1} δ_{2}} ‖_{L^{2} (Ω_{ε δ_{1} δ_{2}})}^{2} + α ε^{γ} ‖ u_{ε δ_{1} δ_{2}} ‖_{L^{2} (\partial T_{ε δ_{1}})}^{2} \\ ⩽ C (‖ f ‖_{L^{2} (Ω_{ε δ_{1} δ_{2}})} + δ_{1}^{\frac{N}{2}} + \frac{δ_{1}^{N - 1}}{ε}) ‖ \nabla u_{ε δ_{1} δ_{2}} ‖_{L^{2} (Ω_{ε δ_{1} δ_{2}})} \end{matrix} \end{matrix}$ which implies that $\begin{matrix} ν ‖ \nabla u_{ε δ_{1} δ_{2}} ‖_{L^{2} (Ω_{ε δ_{1} δ_{2}})}^{2} + α ε^{γ} ‖ u_{ε δ_{1} δ_{2}} ‖_{L^{2} (\partial T_{ε δ_{1}})}^{2} ⩽ C ‖ \nabla u_{ε δ_{1} δ_{2}} ‖_{L^{2} (Ω_{ε δ_{1} δ_{2}})} . \end{matrix}$ Using herein the Poincaré inequality yields $\begin{matrix} ‖ u_{ε δ_{1} δ_{2}} ‖_{H^{1} (Ω_{ε δ_{1} δ_{2}})} ⩽ C . \end{matrix}$

On the other hand, by using the Young inequality in (4.4), we have $\begin{matrix} ν ‖ \nabla u_{ε δ_{1} δ_{2}} ‖_{L^{2} (Ω_{ε δ_{1} δ_{2}})}^{2} + α ε^{γ} ‖ u_{ε δ_{1} δ_{2}} ‖_{L^{2} (\partial T_{ε δ_{1}})}^{2} \\ ⩽ C (1 + δ_{1}^{\frac{N}{2}} + η \frac{1}{ε^{2}} + \frac{1}{η} {(δ_{1}^{n - 1})}^{2}) ‖ \nabla u_{ε δ_{1} δ_{2}} ‖_{L^{2} (Ω_{ε δ_{1} δ_{2}})}, \end{matrix}$ which implies, in view of (4.1) and Poincaré’s inequality, $\begin{matrix} (4.5) & ‖ ε u_{ε δ_{1} δ_{2}} ‖_{H^{1} (Ω_{ε δ_{1} δ_{2}})} ⩽ C . \end{matrix}$

Finally, by using estimate (4.1) and the fact that for $v \in H^{1} (Ω_{ε})$ (see for example, [3,16]), $\begin{matrix} ‖ v ‖_{{(L^{2} (Ω_{ε}))}^{N}} ⩽ C ε ‖ \nabla v ‖_{{(L^{2} (Ω_{ε}))}^{N \times N}}, \end{matrix}$ we get estimate (4.3). □

To describe the limit behavior of system (2.3), we need to use some special test functions belonging to $V_{ε δ_{1} δ_{2}}$ . For the construction of these functions, we follow the procedure from [5].

Consider the local periodic problem in $Y^{}$ : $\begin{matrix} (P_{1}) \{\begin{matrix} - Δ_{y} v^{k} + \nabla_{y} q^{k} = e^{k} & in Y^{} \\ {div}_{y} v^{k} = 0 & in Y^{} \\ v^{k} . n = 0 & on \partial T \\ {(\frac{\partial v^{k}}{\partial n})}_{τ} = 0 & on \partial T \\ (v^{k}; q^{k}) Y -periodic \\ M_{Y^{}} (q^{k}) = 0, \end{matrix} \end{matrix}$ where ${(e^{k})}_{k = 1, \dots, N}$ is the canonical basis in $R^{n}$ .

Now, in order to write the local problem $(P_{1})$ in terms of $x = ε y$ , we extend $v^{k}$ and $q^{k}$ by periodicity to $Ω_{ε δ_{1} δ_{2}}$ and set $\begin{matrix} v^{k ε} (x) = v^{k} (\frac{x}{ε}), q^{k ε} = q^{k} (\frac{x}{ε}) for x \in Ω_{ε} . \end{matrix}$ Consequently, problem $(P_{1})$ writes $\begin{matrix} (P_{ε}) \{\begin{matrix} - ε^{2} Δ_{x} v^{k ε} + ε \nabla_{x} q^{k ε} = e^{k} & in Ω_{ε}, \\ {div}_{x} v^{k ε} = 0 & in Ω_{ε}, \end{matrix} \end{matrix}$ and it is easily seen that the following estimates hold: $\begin{matrix} (4.6) & {‖ v^{k ε} ‖}_{{(L^{2} (Ω_{ε δ_{1} δ_{2}}))}^{N}} ⩽ C, \end{matrix}$ as well as $\begin{matrix} (4.7) & {‖ \nabla_{x} v^{k ε} ‖}_{{(L^{2} (Ω_{ε δ_{1} δ_{2}}))}^{N \times N}} ⩽ \frac{C}{ε} . \end{matrix}$

Multiply now $(P_{ε})$ by $v \in V_{ε δ_{1} δ_{2}}$ and integrate on $Ω_{ε δ_{1} δ_{2}}$ , to obtain $\begin{matrix} (4.8) & ε \int_{Ω_{ε δ_{1} δ_{2}}} \nabla v^{k ε} \nabla v d x + \int_{Ω_{ε δ_{1} δ_{2}}} q^{k ε} div v d x = \frac{1}{ε} \int_{Ω_{ε δ_{1} δ_{2}}} e^{k} \cdot v d x, \forall v \in V_{ε δ_{1} δ_{2}} . \end{matrix}$

At this point, we recall the fact ([15,22]) that, if $ϕ \in L^{2} (Ω_{ε δ_{1} δ_{2}})$ , then there exists $φ \in V_{ε δ_{1} δ_{2}}$ , such that $\begin{matrix} div φ = ϕ with ‖ φ ‖_{V_{ε δ_{1} δ_{2}}} ⩽ C ϕ ‖_{L^{2} (Ω_{ε δ_{1} δ_{2}})} . \end{matrix}$ Using this in (4.8), together with (4.6) and (4.7), gives to the estimate $\begin{matrix} (4.9) & {‖ q^{k ε} ‖}_{L^{2} (Ω_{ε δ_{1} δ_{2}})} ⩽ C . \end{matrix}$

Other tools that will be needed below are some results from [9]. To formulate them, define the space $K_{B}$ as $\begin{matrix} K_{B} = {Φ \in L^{2^{*}} (R^{N}) ∣ \nabla Φ \in L^{2} (R^{N}), Φ constant on B} . \end{matrix}$
Lemma 4.2 ([9]).

Let $N ⩾ 3$ . Then, for every $δ_{0} > 0$ , the set $\begin{matrix} ⋃_{0 < δ_{2} < δ_{0}} {ϕ \in H_{per}^{1} (Y) ∣ ϕ = 0 on δ_{2} B}, \end{matrix}$ is dense in $H_{per}^{1} (Y)$ .

Lemma 4.3 ([9]).

Let $v \in D (R^{N}) \cap K_{B}$ and $\begin{matrix} w_{ε δ_{2}} (x) = v (B) - v (\frac{1}{δ_{2}} {\frac{x}{ε}}) for x \in R^{N} . \end{matrix}$ Then $\begin{matrix} (4.10) & w_{ε δ_{2}} ⇀ v (B) weakly in H^{1} (Ω) . \end{matrix}$

Now, as in [9], let the function χ be the solution of the following cell problem: $\begin{matrix} (4.11) & \{\begin{matrix} χ \in L^{\infty} (Ω; K_{B}), \\ \int_{R^{N} ∖ \overline{T}} \nabla_{z} χ (x, z) \nabla_{z} Ψ (z) d z = 0 for a.e. x \in Ω, \forall Ψ \in K_{B} with Ψ (B) = 0, \end{matrix} \end{matrix}$ and introduce the function Θ given by $\begin{matrix} (4.12) & Θ (x) = \int_{R^{N} ∖ \overline{T}} \nabla_{z} χ (x, z) \nabla_{z} χ (x, z) d z . \end{matrix}$ Notice that Θ can be interpreted as a local capacity of the set B since it is positive by definition.

Assume now that there exist $k_{1}$ with $0 ⩽ k_{1} < + \infty$ , and $k_{2}$ with $0 ⩽ k_{2} < + \infty$ , such that $\begin{matrix} (4.13) & k_{1} = lim_{ε \to 0} \frac{δ_{1} {(ε)}^{N - 1}}{ε} and k_{2} = lim_{ε \to 0} \frac{δ_{2} {(ε)}^{\frac{N}{2} - 1}}{ε} . \end{matrix}$

Remark 4.4.
It is easily seen that $k_{1}$ corresponds to the critical size of Dirichlet small holes from [12], that is $C ε^{N / N 2 - 2}$ . As concerning $k_{2}$ , it corresponds to the critical size from [16], that is $C ε^{N - 2}$ . Observe also that, as $ε \to 0$ , since by (4.13), the holes in $T_{ε δ_{1}}$ approach quicker zero, than those from $B_{ε δ_{2}}$ . Consequently, for ε small enough, the holes from $T_{ε δ_{1}}$ do not intersect those from $B_{ε δ_{2}}$ for each ε. It is why that one could make the hypothesis that from now on, in the limiting process $ε \to 0$ , the holes $T_{ε δ_{1}}$ and $B_{ε δ_{2}}$ do not intersect.

We are now in position to give the homogenization result for the Stokes problem (1.4).
Theorem 4.5 (Main Theorem).

Assume that ( 4.13 ) hold true and that the data $f_{ε}$ satisfies the additional property $\begin{matrix} ε \tilde{f_{ε}} ⇀ f weakly in {(L^{2} (Ω))}^{N} \end{matrix}$ and let $(u_{ε δ_{1} δ_{2}}, p_{ε})$ be the solution of ( 1.4 ). Then, there exist $u \in {(L^{2} (Ω))}^{n}$ , $p \in L^{2} (Ω)$ and an extension ${\hat{p}}_{ε}$ of the pressure $p_{ε}$ such that $\begin{matrix} (4.14) & \tilde{u_{ε δ_{1} δ_{2}}} ⇀ u weakly in {(L^{2} (Ω))}^{N}, \end{matrix}$ and $\begin{matrix} (4.15) & ε {\hat{p}}_{ε} ⇀ p weakly in L^{2} (Ω), \end{matrix}$

If $γ < - 1$ , then $u = 0$ .

If $γ = - 1$ , then $(u, p)$ ( $u = (u_{1}, \dots, u_{N})$ is solution in Ω of the problem $\begin{matrix} (4.16) & \begin{matrix} ν u_{k} + α k_{1} | \partial T | M_{\partial T} (v^{k}) u + k_{2}^{2} M_{Y^{*}} (v^{k}) Θ u \\ = M_{Y^{*}} (v^{k}) (f - \nabla p) + | \partial T | M_{\partial T} (g v^{k}), \end{matrix} \end{matrix}$ for $k \in {1, \dots, N}$ .

If $γ > - 1$ , then $(u, p)$ ( $u = (u_{1}, \dots, u_{N})$ is solution in Ω of the problem $\begin{matrix} (4.17) & \begin{matrix} u_{k} + k_{2}^{2} M_{Y^{*}} (v^{k}) Θ u = \frac{1}{ν} M_{Y^{*}} (v^{k}) (f - \nabla p) + \frac{| \partial T |}{ν} M_{\partial T} (g v^{k}), \end{matrix} \end{matrix}$ for $k \in {1, \dots, N}$ .

Remark 4.6.

Observe that in both systems (4.16) and (4.17), we have the “strange term” involving the function Θ, it represents the contribution of the set of small holes $B_{ε δ_{2}}$ with the homogeneous Dirichlet condition on their boundary. This makes the link with the results from [12], where this phenomenon was observed for the Poisson equation, as mentioned in the Introduction.

In turn, the contribution of the holes of the set of small holes $T_{ε δ_{2}}$ with Neumann condition, is reflected by the additional term involving g in the right-hand side of the limit equation. This makes the link with the result from [16] mentioned in the Introduction.

Proof of Theorem 4.5.

Thanks to estimates (4.3) and (4.2), we can apply Proposition 3.10. Thus, there exists some U in $L^{2} (Ω; L_{loc}^{2} (R^{N}))$ such that, up to a subsequence, $\begin{matrix} (4.18) & \frac{δ_{2}^{\frac{N}{2} - 1}}{ε} T_{ε δ_{2}} (ε u_{ε δ_{1} δ_{2}}) ⇀ U weakly in L^{2} (Ω; L_{loc}^{2} (R^{N})) . \end{matrix}$ Using (3.1), one also has the convergence $\begin{matrix} (4.19) & \frac{δ_{2}^{\frac{N}{2} - 1}}{ε} M_{Y}^{ε} (ε u_{ε δ_{1} δ_{2}}) \to k_{2} u strongly in L^{2} (Ω; L_{loc}^{2} (R^{N})) . \end{matrix}$

On the other hand, due to Proposition 3.10, there exists W in $L^{2} (Ω; L^{2^{*}} (R^{N}))$ with $\nabla_{z} W$ in $L^{2} (Ω \times R^{N})$ , such that $\begin{matrix} (4.20) & \frac{δ_{2}^{\frac{N}{2} - 1}}{ε} (T_{ε δ_{2}} (ε u_{ε δ_{1} δ_{2}}) - M_{Y}^{ε} (ε u_{ε δ_{1} δ_{2}})) ⇀ W weakly in L^{2} (Ω; L^{2^{*}} (R^{N})) . \end{matrix}$ From (4.18),(4.19) and (4.20), one conclude that $\begin{matrix} U = W + k_{2} u and \nabla_{z} U = \nabla_{z} W . \end{matrix}$ By Proposition 3.10 again, one also has convergence $\begin{matrix} (4.21) & \frac{δ_{2}^{\frac{N}{2} - 1}}{ε} \nabla_{z} T_{ε δ_{2}} (ε u_{ε δ_{1} δ_{2}}) = δ_{2}^{\frac{N}{2}} T_{ε δ_{2}} (\nabla ε u_{ε δ_{1} δ_{2}}) ⇀ \nabla_{z} U weakly in L^{2} (Ω \times R^{N}) . \end{matrix}$

Let go back to (2.3) where we would like to pass to the limit. To do so, we introduce the function $v_{ε δ_{2}}$ defined by $\begin{matrix} v_{ε δ_{2}} (x) = ε v^{k} (\frac{x}{ε}) w_{ε δ_{2}} φ (x), \end{matrix}$ where $φ \in D (Ω)$ , $v^{k} \in (H_{per}^{1} (Y^{*}))$ is solution of $(P_{1})$ , and $w_{ε δ_{2}}$ was defined in Lemma 4.3. We can choose $v_{ε δ_{2}}$ as test function in (2.3). We get $\begin{array}{l} ε ν \int_{Ω_{ε δ_{1} δ_{2}}} \nabla u_{ε δ_{1} δ_{2}} w_{ε δ_{2}} \nabla (v^{k ε} φ) d x + ε ν \int_{Ω_{ε δ_{1} δ_{2}}} \nabla u_{ε δ_{1} δ_{2}} \nabla w_{ε δ_{2}} (v^{k ε} φ) d x \\ + α ε^{γ + 1} \int_{\partial T_{ε δ_{1}}} u_{ε δ_{1} δ_{2}} v^{k ε} φ w_{ε δ_{2}} d σ (x) - \int_{Ω_{ε δ_{1} δ_{2}}} ε p_{ε} div (v^{k ε} φ w_{ε δ_{2}}) d x \\ (4.22) & = \int_{Ω_{ε δ_{1} δ_{2}}} ε f_{ε} v^{k ε} w_{ε δ_{2}} φ d x + ε \int_{\partial T_{ε δ_{1}}} g_{ε} v^{k ε} w_{ε δ_{2}} φ d σ (x), \end{array}$ where to simplify, we used the notation $\begin{matrix} v^{k ε} (x) = v^{k} (\frac{x}{ε}) . \end{matrix}$

Now, we shall pass to the limit as $ε \to 0$ in (4.22).

Case 1. $γ ⩾ - 1$ . The first term in (4.22) is written as $\begin{matrix} I_{ε}^{1} & = ε ν \int_{Ω_{ε δ_{1} δ_{2}}} \nabla u_{ε δ_{1} δ_{2}} w_{ε δ_{2}} \nabla (φ v^{k ε}) d x \\ = ε ν \int_{Ω_{ε δ_{1} δ_{2}}} \nabla u_{ε δ_{1} δ_{2}} w_{ε δ_{2}} v^{k ε} \nabla φ d x + ε ν \int_{Ω_{ε δ_{1} δ_{2}}} \nabla u_{ε δ_{1} δ_{2}} w_{ε δ_{2}} φ \nabla v^{k ε} d x . \end{matrix}$

On the other hand, considering the integral $\begin{matrix} (4.23) & \begin{matrix} J_{ε} & = ε ν \int_{Ω_{ε δ_{1} δ_{2}}} w_{ε δ_{2}} \nabla v^{k ε} \nabla (φ u_{ε δ_{1} δ_{2}}) d x \\ = ε ν \int_{Ω_{ε δ_{1} δ_{2}}} \nabla v^{k ε} w_{ε δ_{2}} \nabla φ u_{ε δ_{1} δ_{2}} d x + ε ν \int_{Ω_{ε δ_{1} δ_{2}}} \nabla v^{k ε} w_{ε δ_{2}} \nabla u_{ε δ_{1} δ_{2}} φ d x, \end{matrix} \end{matrix}$ and using estimates (4.1), (4.3) and (4.7), we obtain $\begin{matrix} | I_{ε}^{1} - J_{ε} | & ⩽ ε ν | \int_{Ω_{ε δ_{1} δ_{2}}} \nabla u_{ε δ_{1} δ_{2}} w_{ε δ_{2}} \nabla φ v^{k ε}) d x | + ε ν | \int_{Ω_{ε δ_{1} δ_{2}}} \nabla v^{k ε} w_{ε δ_{2}} \nabla φ u_{ε δ_{1} δ_{2}} d x | \\ ⩽ C ε (‖ \nabla u_{ε δ_{1} δ_{2}} ‖_{{(L^{2} (Ω_{ε δ_{1} δ_{2}}))}^{N \times N}} {‖ v^{k ε} ‖}_{{(L^{2} (Ω_{ε δ_{1} δ_{2}}))}^{N}} \\ + {‖ \nabla v^{k ε} ‖}_{{(L^{2} (Ω_{ε δ_{1} δ_{2}}))}^{N \times N}} ‖ u_{ε δ_{1} δ_{2}} ‖_{{(L^{2} (Ω_{ε δ_{1} δ_{2}}))}^{N}}) ⩽ C ε . \end{matrix}$ Consequently, we have $\begin{matrix} (4.24) & lim_{ε \to 0} I_{ε}^{1} = lim_{ε \to 0} J_{ε} . \end{matrix}$

Now take $v = w_{ε δ_{2}} u_{ε δ_{1} δ_{2}} φ$ as test function in (4.8), multiply by ν and use (4.23) to get $\begin{matrix} J_{ε} & = ε ν \int_{Ω_{ε δ_{1} δ_{2}}} \nabla v^{k ε} \nabla (w_{ε δ_{2}}) u_{ε δ_{1} δ_{2}} φ d x + ν \int_{Ω_{ε δ_{1} δ_{2}}} q^{k ε} \nabla (w_{ε δ_{2}}) u_{ε δ_{1} δ_{2}} φ d x \\ + ν \int_{Ω_{ε δ_{1} δ_{2}}} q^{k ε} w_{ε δ_{2}} u_{ε δ_{1} δ_{2}} \nabla φ d x + \frac{1}{ε} ν \int_{Ω_{ε δ_{1} δ_{2}}} e^{k} w_{ε δ_{2}} u_{ε δ_{1} δ_{2}} φ d x . \end{matrix}$ Unfolding with $T_{ε δ_{2}}$ the right-hand term, yields $\begin{matrix} (4.25) & \begin{matrix} J_{ε} & = - ν ε δ_{2}^{N} \int_{Ω \times R^{N}} T_{ε δ_{2}} (\nabla v^{k ε}) T_{ε δ_{2}} \nabla (w_{ε δ_{2}}) T_{ε δ_{2}} (u_{ε δ_{1} δ_{2}} φ) d x \\ + ν δ_{2}^{N} \int_{Ω \times R^{N}} T_{ε δ_{2}} (q^{k ε}) T_{ε δ_{2}} (\nabla w_{ε δ_{2}}) T_{ε δ_{2}} (u_{ε δ_{1} δ_{2}} φ) d x \\ + ν \int_{Ω \times R^{N}} T_{ε} (q^{k ε}) T_{ε} (w_{ε δ_{2}}) T_{ε δ_{2}} (u_{ε δ_{1} δ_{2}} \nabla φ) d x \\ + ν \int_{Ω \times R^{N}} T_{ε} (e^{k}) T_{ε} (w_{ε δ_{2}}) T_{ε} (\frac{u_{ε δ_{1} δ_{2}}}{ε}) T_{ε} (φ) d x d y . \end{matrix} \end{matrix}$ Since ${\tilde{u}}_{ε δ_{1} δ_{2}} \to 0$ strongly in ${(L^{2} (Ω))}^{N}$ , using the boundedness of $q^{k ε}$ , convergences (4.1) and (4.3) from (4.25), as well as (4.24), we deduce that $\begin{matrix} (4.26) & lim_{ε \to 0} I_{ε}^{1} = ν v (B) \int_{Ω} u_{k} φ d x . \end{matrix}$

Unfolding the second term of (4.22) by $T_{ε δ_{2}}$ and using lemma 4.3, we get $\begin{matrix} ν \int_{Ω_{ε δ_{1} δ_{2}}} ε \nabla u_{ε δ_{1} δ_{2}} \nabla w_{ε δ_{2}} (v^{k ε} φ) d x \\ - ν \frac{δ_{2}^{\frac{N}{2} - 1}}{ε} \int_{Ω \times R^{N}} δ_{2}^{\frac{N}{2}} T_{ε δ_{2}} (ε \nabla u_{ε δ_{1} δ_{2}}) (- \nabla_{z} v) v^{k} (y) φ (x) d x d z \to 0 . \end{matrix}$ Then, convergence (4.21) as well as hypothesis (4.13), imply that $\begin{matrix} (4.27) & \begin{matrix} lim_{ε \to 0} ν \int_{Ω_{ε δ_{1} δ_{2}}} ε \nabla u_{ε δ_{1} δ_{2}} \nabla w_{ε δ_{2}} (v^{k ε} φ) d x \\ = - ν k_{2} \int_{Ω \times (R^{N} ∖ B)} \nabla_{z} U (x, z) \nabla_{z} v (z) v^{k} (y) φ (x) d x d z . \end{matrix} \end{matrix}$

For the third term in (4.22), we have two cases.

if $γ > - 1$ , it is clear that this term tends to zero as $ε \to 0$ ,

if $γ = - 1$ , after unfolding with $T_{ε δ_{1}}^{b}$ we have, $\begin{matrix} α ε^{γ + 1} \int_{\partial T_{ε δ_{1}}} u_{ε δ_{1} δ_{2}} w_{ε δ_{2}} v^{k ε} φ (x) d σ (x) \\ = α ε^{γ + 1} \frac{δ_{1}^{N - 1}}{ε} \int_{Ω \times \partial T} T_{ε δ_{1}}^{b} (u_{ε δ_{1} δ_{2}}) T_{ε δ_{1}}^{b} (w_{ε δ_{2}}) v^{k} (y) φ (x) d x d σ (z) . \end{matrix}$ Passing to limit and using hypothesis (4.13), we get $\begin{matrix} (4.28) & \begin{matrix} lim_{ε \to 0} α \int_{\partial T_{ε δ_{1}}} u_{ε δ_{1} δ_{2}} w_{ε δ_{2}} v^{k ε} φ (x) d σ (x) \\ = α k_{1} v (B) \int_{Ω \times \partial T} u v^{k} (z) φ (x) d x d σ (z) . \end{matrix} \end{matrix}$

For the fourth term, we define the extension ${\hat{p}}_{ε}$ of the pressure $p_{ε}$ , by $\begin{matrix} (4.29) & \int_{Ω_{ε δ_{1} δ_{2}}} p_{ε} div (v^{k ε} w_{ε δ_{2}} φ) d x = \int_{Ω} {\hat{p}}_{ε} P (v^{k ε}) \nabla (w_{ε δ_{2}} φ) d x, \forall φ \in D (Ω), \end{matrix}$ where $P (v^{k ε})$ is an extension of $v^{k}$ to Y. It is easy to verify that a possible of such an extension choice is the following one: $\begin{matrix} P (v^{k ε}) = \{\begin{matrix} v^{k} & in Y^{*}, \\ w^{k} & in T, \end{matrix} \end{matrix}$ where $w^{k}$ is the solution of $\begin{matrix} \{\begin{matrix} - Δ_{y} w^{k} + \nabla_{y} q^{k} = 0 & in T \\ {div}_{y} w^{k} = 0 & in T, \\ w^{k} = v^{k} & on \partial T, \\ M_{T} (w^{k}) = 0 . \end{matrix} \end{matrix}$

Arguing as in [10], it can be shown that up to a subsequence, convergence (4.15) holds. Therefore we have, $\begin{matrix} ε \int_{Ω_{ε δ_{1} δ_{2}}} p_{ε} div (v^{k ε} w_{ε δ_{2}} φ) (x) d x & = \int_{Ω} (ε {\hat{p}}_{ε}) P (v^{k ε}) \nabla (w_{ε δ_{2}} φ) d x \\ = \int_{Ω} (ε {\hat{p}}_{ε}) P (v^{k ε}) w_{ε δ_{2}} \nabla φ d x + \int_{Ω} (ε {\hat{p}}_{ε}) P (v^{k ε}) \nabla w_{ε δ_{2}} φ d x . \end{matrix}$

Unfolding by $T_{ε δ_{2}}$ the last integral gives $\begin{matrix} \int_{Ω} (ε {\hat{p}}_{ε}) P (v^{k ε}) \nabla (w_{ε δ_{2}} φ) d x \\ = \int_{Ω} (ε {\hat{p}}_{ε}) P (v^{k ε}) w_{ε δ_{2}} \nabla (φ) d x + δ_{2}^{N} \int_{Ω \times R^{N}} T_{ε δ_{2}} (ε {\hat{p}}_{ε}) T_{ε δ_{2}} (\tilde{v^{k ε}}) T_{ε δ_{2}} (\nabla w_{ε δ_{2}}) T_{ε δ_{2}} (φ) d x d z \\ = \int_{Ω} (ε {\hat{p}}_{ε}) \tilde{v^{k ε}} w_{ε δ_{2}} \nabla (φ) d x + \frac{δ_{2}^{\frac{N}{2} - 1}}{ε} \int_{Ω \times R^{N}} δ_{2}^{\frac{N}{2}} T_{ε δ_{2}} (ε {\hat{p}}_{ε}) T_{ε δ_{2}} (\tilde{v^{k ε}}) (- \nabla_{z} v) T_{ε δ_{2}} (φ) d x d z . \end{matrix}$ Clearly, the last integral goes to 0 as $ε \to 0$ . Consequently, $\begin{matrix} (4.30) & lim_{ε \to 0} ε \int_{Ω_{ε δ_{1} δ_{2}}} p_{ε} div (v^{k ε} w_{ε δ_{2}} φ) d x = v (B) \int_{Ω} p v^{k} \nabla φ d x . \end{matrix}$

Using $T_{ε}$ in the fifth integral of (4.22), we get $\begin{matrix} \int_{Ω_{ε δ_{1} δ_{2}}} ε f_{ε} v^{k ε} w_{ε δ_{2}} φ d x = \int_{R^{n} \times Y} T_{ε} (ε f_{ε}) v^{k} (y) T_{ε} (w_{ε δ_{2}}) φ d x d y, \end{matrix}$ which implies that $\begin{matrix} (4.31) & lim_{ε \to 0} \int_{Ω_{ε δ_{1} δ_{2}}} ε f_{ε} v^{k ε} w_{ε δ_{2}} φ d x = v (B) \int_{R^{n} \times Y} f v^{k} (y) φ d x d y . \end{matrix}$

Finally, the sixth term after unfolding by $T_{ε}^{b}$ becomes $\begin{matrix} ε \int_{\partial T_{ε δ_{1}}} g_{ε} w_{ε δ_{2}} v^{k ε} φ d σ (x) = \int_{R^{n} \times \partial T} g (y) T_{ε}^{b} (w_{ε δ_{2}}) v^{k} (y) φ (x) d x d σ (y), \end{matrix}$ so that, passing to the limit yields $\begin{matrix} (4.32) & lim_{ε \to 0} ε \int_{\partial T_{ε δ_{1}}} g_{ε} w_{ε δ_{2}} v^{k ε} φ (x) d σ (x) = v (B) \int_{R^{n} \times \partial T} g (y) v^{k} (y) φ d x d σ (y) . \end{matrix}$

Let now pass to the limit in (4.22). To do so, we use convergences (4.26)–(4.28), (4.30)–(4.32) and assumption (4.13) on $f_{ε}$ . We have to distinguish the three cases mentioned above.

Case $γ = - 1$ . We obtain the equation, $\begin{matrix} (4.33) & \begin{matrix} v (B) \int_{Ω} u_{k} φ (x) d x - k_{2} \int_{Ω \times (R^{N} ∖ B)} \nabla_{z} U (x, z) \nabla_{z} v (z) v^{k} (y) φ d x d z \\ + α k_{1} v (B) \int_{Ω \times \partial T} u v^{k} (z) φ d x d σ (z) + v (B) \int_{Ω} p v^{k} (y) div (φ) d x \\ = v (B) \int_{Ω \times Y} f v^{k} (y) φ d x d y + v (B) \int_{Ω \times \partial T} g (y) v^{k} (y) φ d x d σ (y), \end{matrix} \end{matrix}$ for all $φ \in D (Ω)$ .

Integrating by parts in second term, it is easily seen that $\begin{matrix} (4.34) & \begin{matrix} k_{2} v^{k} (y) \int_{Ω \times (R^{N} ∖ B)} \nabla_{z} U (x, z) \nabla_{z} v (z) φ d x d z \\ = k_{2} v (B) v^{k} (y) \int_{Ω \times \partial B} \nabla_{z} U \cdot ν_{B} φ d x d σ (z) \\ = k_{2} v (B) \int_{Ω} {\int_{\partial B} \nabla_{z} U \cdot ν_{B} d σ (z)} v^{k} (y) φ d x . \end{matrix} \end{matrix}$

On the other hand, from (4.11) by Green’s formula, we have $\begin{matrix} (4.35) & \int_{\partial B} \nabla_{z} U \cdot ν_{B} d σ = \int_{\partial B} \nabla_{z} (U - k_{2} u) \cdot ν_{B} d σ = - k_{2} u \int_{\partial B} \nabla_{z} χ \cdot ν_{B} d σ . \end{matrix}$ Passing to the limit in (4.34), using (4.35) and unfolding with $T_{ε}$ , gives $\begin{matrix} (4.36) & \begin{matrix} k_{2} v^{k} (y) \int_{Ω \times (R^{N} ∖ B)} \nabla_{z} U (x, z) \nabla_{z} v (z) φ (x) d x d z \\ = - k_{2}^{2} v (B) | Y | M_{Y^{*}} (v) \int_{Ω} Θ (x) u (x) φ (x) d x, \end{matrix} \end{matrix}$ where Θ is defined by (4.12). Then equation (4.16) is simply obtained by replacing (4.36) into (4.33).

Case $γ > - 1$ . We now have at the limit $\begin{matrix} ν v (B) \int_{Ω} u_{k} φ (x) d x + k_{2} \int_{Ω \times (R^{N} ∖ B)} \nabla_{z} U (x, z) \nabla_{z} v (z) v^{k} (y) φ (x) d x d z \\ + v (B) \int_{Ω} p v^{k} (y) \nabla φ d x \\ = v (B) \int_{Ω \times Y} f v^{k} (y) φ d x d y + v (B) \int_{Ω \times \partial T} v^{k} (y) g (y) φ (x) d x d σ (y), \end{matrix}$ from which (4.17) follows by using (4.34).

Case $γ < - 1$ . Multiply (4.22) by $ε^{- 1 - γ}$ to get $\begin{matrix} ε^{- 1 - γ} I_{ε}^{1} + ν ε \int_{Ω_{ε δ_{1} δ_{2}}} \nabla u_{ε δ_{1} δ_{2}} \nabla w_{ε δ_{2}} (v^{k ε} φ) d x + α \int_{\partial T_{ε δ_{1}}} u_{ε δ_{1} δ_{2}} v^{k ε} φ w_{ε δ_{2}} d σ (x) \\ - ε^{- 1 - γ} \int_{Ω_{ε δ_{1} δ_{2}}} ε p_{ε} div (v^{k ε} φ w_{ε δ_{2}}) d x - ε^{- 1 - γ} \int_{Ω_{ε δ_{1} δ_{2}}} ε f_{ε} v^{k ε} w_{ε δ_{2}} φ d x \\ - ε^{- 1 - γ} ε \int_{\partial T_{ε δ_{1}}} g_{ε} v^{k ε} w_{ε δ_{2}} φ d σ (x) = 0, \forall φ \in D (Ω) . \end{matrix}$ Passing to the limit we observe that all the integrals vanish, except for the second integral by using (4.28), which gives $\begin{matrix} α k_{1} v (B) \int_{R^{n} \times \partial T} u v^{k} φ (x) d x d σ (z) = 0, \end{matrix}$ and so $u = 0$ . This ends the proof of Theorem 4.5. □

Remark 4.7.

The case $δ_{1} = 1$ and $k_{2} = 0$ , corresponds to the classical homogenization. The results above in this case are those from Capatina and Ene [5].

In the cases listed below, we have various variants of the Darcy law for $γ ⩾ - 1$ .

If $k_{1} = 0$ and $k_{2} = 0$ , then we retrieve the result from [22], $\begin{matrix} u = 0 if γ < - 1, \\ u_{k} = \frac{1}{ν} M_{Y^{*}} (v^{k}) (f - \nabla p) + \frac{| \partial T |}{ν} M_{\partial T} (g v^{k}) if γ ⩾ - 1 . \end{matrix}$

If $k_{1} = 0$ and $k_{2} \neq 0$ , $\begin{matrix} u = 0 if γ < - 1, \\ ν u_{k} + k_{2}^{2} M_{Y^{*}} (v^{k}) Θ u = M_{Y^{*}} (v_{k}) (f - \nabla p) + | \partial T | M_{\partial T} (g v^{k}) if γ ⩾ - 1 . \end{matrix}$

If $k_{1} \neq 0$ and $k_{2} = 0$ , $\begin{matrix} u = 0 if γ < - 1, \\ ν u_{k} + α k_{1} | \partial T | M_{\partial T} (v^{k}) u = M_{Y^{*}} (v_{k}) (f - \nabla p) + | \partial T | M_{\partial T} (g v^{k}) if γ = - 1, \\ u_{k} = \frac{1}{ν} M_{Y^{*}} (v^{k}) (f - \nabla p) + \frac{| \partial T |}{ν} M_{\partial T} (g v^{k}) if γ > - 1 . \end{matrix}$

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Asymptotic behavior of the Stokes problem with Robin condition in a domain with small holes

Abstract

Keywords

1. Introduction

2. Variational formulation

3. The periodic unfolding operators

3.1. Some notations

3.2. The unfolding operator T ε for fixed domains

4.1. A priori estimates for u ε δ 1 δ 2 and p ε

Lemma 4.3 ([9]).

References

3.2. The unfolding operator $T_{ε}$ for fixed domains

4.1. A priori estimates for $u_{ε δ_{1} δ_{2}}$ and $p_{ε}$