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Comptes Rendus Mathématique
Volume 349, n° 19-20
pages 1063-1066 (novembre 2011)
Doi : 10.1016/j.crma.2011.09.009
Received : 15 July 2011 ;  accepted : 14 September 2011
Weak solutions to the incompressible Euler equations with vortex sheet initial data
Solutions faibles des équations dʼEuler incompressibles avec nappe de tourbillon comme donnée initiale

László Székelyhidi
Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 62, 53115 Bonn, Germany 


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.

The full text of this article is available in PDF format.

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.

The full text of this article is available in PDF format.

© 2011  Published by Elsevier Masson SAS de la part de Académie des sciences.
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