The Two Dimensional Euler Equations on Singular Exterior Domains

Abstract

This paper is a follow-up of Gérard-Varet and Lacave (Arch Ration Mech Anal 209(1):131–170, 2013), on the existence of global weak solutions to the two dimensional Euler equations in singular domains. In Gérard-Varet and Lacave (Arch Ration Mech Anal 209(1):131–170, 2013), we have established the existence of weak solutions for a large class of bounded domains, with initial vorticity in L p (p > 1). For unbounded domains, we have proved a similar result only when the initial vorticity is in $${L^{p}_{c}}$$ (p > 2) and when the domain is the exterior of a single obstacle. The goal here is to retrieve these two restrictions: we consider general initial vorticity in $${L^{1} {\cap} L^{p}}$$ (p > 1), outside an arbitrary number of obstacles (not reduced to points).