The 2D Euler equation on singular domains

TL;DR

This paper proves the existence of global weak solutions to the 2D incompressible Euler equation in a broad class of non-smooth, singular domains, extending previous results for smoother or simpler geometries.

Contribution

It establishes the existence of weak solutions in domains with complex, non-smooth boundaries using domain approximation and $\Gamma$-convergence techniques, broadening the scope of Euler equation solutions.

Findings

Existence of solutions in domains with finite connected compact sets removed

Weak solutions with $L^p$ vorticity established

Complements previous results for convex and small-hole domains

Abstract

We establish the existence of global weak solutions of the 2D incompressible Euler equation, for a large class of non-smooth open sets. These open sets are the complements (in a simply connected domain) of a finite number of connected compact sets with positive capacity. Existence of weak solutions with $L^p$ vorticity is deduced from an approximation argument, that relates to the so-called $\Gamma$-convergence of domains. Our results complete those obtained for convex domains, or for domains with asymptotically small holes. Connection is made to the recent papers of the second author on the Euler equation in the exterior of a Jordan arc.