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The Two-Dimensional Euler Equations on Singular Domains

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Abstract

We establish the existence of global weak solutions of the two-dimensional incompressible Euler equations for a large class of non-smooth open sets. Loosely, these open sets are the complements (in a simply connected domain) of a finite number of obstacles with positive Sobolev capacity. Existence of weak solutions with L p vorticity is deduced from a property of domain continuity for the Euler equations that relates to the so-called γ-convergence of open sets. Our results complete those obtained for convex domains in Taylor (Progress in Nonlinear Differential Equations and their Applications, Vol. 42, 2000), or for domains with asymptotically small holes (Iftimie et al. in Commun Partial Differ Equ 28(1–2), 349–379, 2003; Lopes Filho in SIAM J Math Anal 39(2), 422–436, 2007).

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  1. Université Paris-Diderot (Paris 7), Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586-CNRS, Bâtiment Sophie Germain, Case 7012, 75205, Paris Cedex 13, 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 Christophe Lacave.

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Communicated by V. Šverák

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Gérard-Varet, D., Lacave, C. The Two-Dimensional Euler Equations on Singular Domains. Arch Rational Mech Anal 209, 131–170 (2013). https://doi.org/10.1007/s00205-013-0617-9

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