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Article Dans Une Revue Analysis & PDE Année : 2022

A homogenized limit for the 2-dimensional Euler equations in a perforated domain

Christophe Lacave
Di Wu
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Résumé

We study the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size $a$ separated by distances $\tilde d$ and the fluid fills the exterior. We analyse the asymptotic behavior of the fluid when $(a,\tilde d) \to (0,0)$. If the inclusions are distributed on the unit square, this issue is studied recently when $\frac{\tilde d}a$ tends to zero or infinity, leaving aside the critical case where the volume fraction of the porous medium is bounded and non-zero. In this paper, we provide the first result in this regime. In contrast with former results, we obtain an Euler type equation where a homogenized term appears in the elliptic problem relating the velocity and the vorticity. Our analysis is based on the so-called method of reflections whose convergence provides novel estimates on the solutions to the div-curl problem which is involved in the 2D-Euler equations.

Dates et versions

hal-02178658 , version 1 (10-07-2019)

Identifiants

Citer

Matthieu Hillairet, Christophe Lacave, Di Wu. A homogenized limit for the 2-dimensional Euler equations in a perforated domain. Analysis & PDE, 2022, 15 (5), pp.1131-1167. ⟨10.2140/apde.2022.15.1131⟩. ⟨hal-02178658⟩
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