The Two Dimensional Euler Equations on Singular Exterior Domains
Résumé
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).