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The Two Dimensional Euler Equations on Singular Exterior Domains

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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).

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  1. Univ Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ Paris 06, 75013, Paris, France

    David Gérard-Varet & Christophe Lacave

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  1. David Gérard-Varet
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  2. Christophe Lacave
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Correspondence to David Gérard-Varet.

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Communicated by L. Saint-Raymond

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Gérard-Varet, D., Lacave, C. The Two Dimensional Euler Equations on Singular Exterior Domains. Arch Rational Mech Anal 218, 1609–1631 (2015). https://doi.org/10.1007/s00205-015-0889-3

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