[Solutions faibles des équations dʼEuler incompressibles avec nappe de tourbillon comme donnée initiale]
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.
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.
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@article{CRMATH_2011__349_19-20_1063_0,
author = {Sz\'ekelyhidi, L\'aszl\'o Jr.},
title = {Weak solutions to the incompressible {Euler} equations with vortex sheet initial data},
journal = {Comptes Rendus. Math\'ematique},
pages = {1063--1066},
publisher = {Elsevier},
volume = {349},
number = {19-20},
year = {2011},
doi = {10.1016/j.crma.2011.09.009},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.09.009/}
}
TY - JOUR AU - Székelyhidi, László Jr. TI - Weak solutions to the incompressible Euler equations with vortex sheet initial data JO - Comptes Rendus. Mathématique PY - 2011 SP - 1063 EP - 1066 VL - 349 IS - 19-20 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.09.009/ DO - 10.1016/j.crma.2011.09.009 LA - en ID - CRMATH_2011__349_19-20_1063_0 ER -
%0 Journal Article %A Székelyhidi, László Jr. %T Weak solutions to the incompressible Euler equations with vortex sheet initial data %J Comptes Rendus. Mathématique %D 2011 %P 1063-1066 %V 349 %N 19-20 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.09.009/ %R 10.1016/j.crma.2011.09.009 %G en %F CRMATH_2011__349_19-20_1063_0
Székelyhidi, László Jr. Weak solutions to the incompressible Euler equations with vortex sheet initial data. Comptes Rendus. Mathématique, Tome 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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