Homogenization of the 2D Euler system: lakes and porous media

TL;DR

This paper addresses the challenging problem of homogenizing 2D Euler and lake equations with porous media, using advanced mathematical tools to overcome localization issues and establish effective equations.

Contribution

It introduces a novel approach combining homogenization theory, ergodic criteria, and complex analysis to prove homogenization results for 2D fluid flows with inclusions.

Findings

Homogenization towards variants of Euler and lake equations.

Prevention of vorticity localization phenomena.

Application of complex analysis and ergodic theory in homogenization.

Abstract

This work is devoted to the long-standing open problem of homogenization of 2D perfect incompressible fluid flows, such as the 2D Euler equations with impermeable inclusions modeling a porous medium, and such as the lake equations. The main difficulty is the homogenization of the transport equation for the associated fluid vorticity. In particular, a localization phenomenon for the vorticity could in principle occur, which would rule out the separation of scales. Our approach combines classical results from different fields to prevent such phenomena and to prove homogenization towards variants of the Euler and lake equations: we rely in particular on the homogenization theory for elliptic equations with stiff inclusions, on criteria for unique ergodicity of dynamical systems, and on complex analysis in form of extensions of the Rad\'o-Kneser-Choquet theorem.