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Desingularization of Vortices for the Euler Equation

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Abstract

We study the existence of stationary classical solutions of the incompressible Euler equation in the planes that approximate singular stationary solutions of this equation. The construction is performed by studying the asymptotics of equation \({-\varepsilon^2 \Delta u^\varepsilon=(u^\varepsilon-q-\frac{\kappa}{2\pi} \log \frac{1}{\varepsilon})_+^p}\) with Dirichlet boundary conditions and q a given function. We also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 place Jussieu, BC 187, 75252, Paris, France

    Didier Smets

  2. Départment de Mathématique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium

    Jean Van Schaftingen

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  1. Didier Smets
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  2. Jean Van Schaftingen
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Correspondence to Didier Smets.

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Communicated by V. Šverák

J. Van Schaftingen was partially supported by the Fonds de la Recherche Scientifique-FNRS (Belgium) and by the Fonds Spéciaux de Recherche (Université catholique de Louvain).

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Smets, D., Van Schaftingen, J. Desingularization of Vortices for the Euler Equation. Arch Rational Mech Anal 198, 869–925 (2010). https://doi.org/10.1007/s00205-010-0293-y

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