#### Vol. 15, No. 5, 2022

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

### Matthieu Hillairet, Christophe Lacave and Di Wu

Vol. 15 (2022), No. 5, 1131–1167
##### Abstract

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 $\stackrel{~}{d}$, and the fluid fills the exterior. We analyze the asymptotic behavior of the fluid when $\left(a,\stackrel{~}{d}\right)\to \left(0,0\right)$.

If the inclusions are distributed on the unit square, this issue was studied recently when $\stackrel{~}{d}∕a$ tends to zero or infinity, leaving aside the critical case where the volume fraction of the porous medium is below its possible maximal value but nonzero. 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 2-dimensional Euler equations.

##### Keywords
ideal incompressible fluid, porous medium, homogenization, method of reflections
##### Mathematical Subject Classification 2010
Primary: 35B27, 35J25, 76B03
Secondary: 35B25, 35C20, 35Q31
##### Milestones
Revised: 10 September 2020
Accepted: 26 November 2020
Published: 29 September 2022
##### Authors
 Matthieu Hillairet Institut Montpelliérain Alexander Grothendieck CNRS, Université de Montpellier Montpellier France Christophe Lacave Université Grenoble Alpes, CNRS Institut Fourier Grenoble France Di Wu School of Mathematics South China University of Technology Guangzhou China