Homogenization of Stokes Equations in Perforated Domains: A Unified Approach
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Journal of Mathematical Fluid Mechanics
Homogenization of Stokes Equations in Perforated Domains: A Unified Approach Yong Lu Communicated by E. Feireisl
Abstract. We consider the homogenization of the Stokes equations in a domain perforated with a large number of small holes which are periodically distributed. Allaire (Arch Ration Mech Anal 113(3):209–259, 1990, Arch Ration Mech Anal 113(3):261–298 1990), gave a systematic study on this problem. In this paper, we introduce a unified proof for different sizes of holes for the homogenization of the Stokes equations by employing a generalized cell problem inspired by Tartar (in: S´ anchez-Palencia (ed) Nonhomogeneous media and vibration theory, 1980). Mathematics Subject Classification. 35B27, 35Q35, 76S05. Keywords. Homogenization, Stokes equations, Perforated domain, Generalized cell problem.
1. Introduction Homogenization problems in the framework of fluid mechanics have gain a lot interest both in mathematical analysis and numerical analysis. Such problems represent the study of fluid flows in domains perforated with a large number of tiny holes (obstacles). The goal is to describe the asymptotic behavior of fluid flows (governed by Stokes equations, Navier–Stokes equations, etc.) as the number of holes goes to infinity and the size of holes goes to zero simultaneously. The limit equations that describe the limit behavior of fluid flows are called homogenized equations which are defined in homogeneous domains without holes. The perforated domain under consideration is described as follows. Let Ω ⊂ Rd , d ≥ 2 be a bounded domain of class C 1 . The holes in Ω are denoted by Tε,k which are assumed to satisfy B(εxk , δ1 aε ) ⊂⊂ Tε,k = εxk + aε T ⊂⊂ B(εxk , δ2 aε ) ⊂⊂ B(εxk , δ3 ε) ⊂⊂ εQk ,
(1.1)
where the cube Qk := (− 12 , 12 )d + k and xk = x0 + k with x0 ∈ (− 12 , 12 )d , for each k ∈ Zd ; T is a model hole which is assumed to be closed, bounded, and simply connected, with C 1 boundary; δi , i = 1, 2, 3 are fixed positive numbers. The perforation parameters ε and aε are used to measure the mutual distance of holes and the size of holes, respectively, and εxk = εx0 + εk are the locations of the holes. Without loss of generality, we assume that x0 = 0 and 0 < aε < ε ≤ 1. Otherwise it is sufficient to consider the domain with a shift of εx0 and consider different values of δi , i = 1, 2, 3. The perforated domain (fluid domain) Ωε is then defined as: Ωε := Ω \ Tε,k , where Kε := {k ∈ Zd : εQk ⊂ Ω}. (1.2) k∈Kε
Throughout the paper, we consider the following Dirichlet ⎧ ⎪ ⎨ −Δuε + ∇pε = f , div uε = 0, ⎪ ⎩ u = 0, ε
problem of Stokes equations in Ωε : in Ωε , in Ωε , on ∂Ωε .
(1.3)
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Y. Lu
JMFM
Here we take the external force f ∈ L2 (Ω). For each fixed ε > 0, the domain Ωε is bounded and is of C 1 ; the existence and uniqueness of the weak solution (uε , pε ) ∈ W01,2 (Ωε ; Rd ) × L20 (Ωε ) to (1.3) is known, see for instance [9]. Here W01,2 denotes the Sobolev space with zero trace, and L20 is the collection of all L2 functions with zero average. The behavior
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