Partial Differential Equations
Weak solutions to the incompressible Euler equations with vortex sheet initial data
Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1063-1066.

We construct infinitely many admissible weak solutions to the incompressible Euler equations with initial data given by the classical vortex sheet. The construction is based on the method introduced recently in De Lellis and Székelyhidi Jr. (2009, 2010) [2,3] using convex integration. In particular, the vorticity is not a bounded measure. Instead, the energy decreases in time due to a linearly expanding turbulent zone around the vortex sheet.

Nous construisons une infinité de solutions faibles admissibles des équations dʼEuler incompressibles avec nappes de tourbillons classiques pour données initiales. La construction repose sur la méthode introduite récemment dans De Lellis et Székelyhidi Jr. (2009, 2010) [2,3] faisant appel à lʼintégration convexe. En particulier, la vorticité nʼest pas une mesure bornée. Au lieu de cela, lʼénergie décroît en temps, à cause dʼune zone turbulente, entourant la nappe de tourbillon et augmentant linéairement en temps.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.09.009
Székelyhidi, László Jr. 1

1 Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 62, 53115 Bonn, Germany
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     title = {Weak solutions to the incompressible {Euler} equations with vortex sheet initial data},
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Székelyhidi, László Jr. Weak solutions to the incompressible Euler equations with vortex sheet initial data. Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1063-1066. doi : 10.1016/j.crma.2011.09.009. http://www.numdam.org/articles/10.1016/j.crma.2011.09.009/

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