# A homogenized limit for the 2D Euler equations in a perforated domain

## TL;DR

This paper investigates the asymptotic behavior of 2D Euler equations in a perforated domain with small inclusions, deriving a homogenized Euler equation with a new elliptic term in the critical volume fraction regime.

## Contribution

It provides the first homogenization result for the 2D Euler equations in the critical volume fraction regime with a homogenized elliptic term, using the method of reflections.

## Key findings

- Derivation of a homogenized Euler equation with an elliptic term.
- Novel estimates on solutions to the div-curl problem.
- Analysis of the critical volume fraction regime in perforated domains.

## 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 $\tilde d$ 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 below its possible maximal value but 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.