Comptes Rendus
no. 19-20
Partial Differential Equations
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]

Présenté par : the Editorial Board

László Székelyhidi1
1 Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 62, 53115 Bonn, Germany
Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1063-1066.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.09.009
László Székelyhidi 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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László Székelyhidi. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.09.009/

[1] Y. Brenier; C. De Lellis; L. Székelyhidi Weak-strong uniqueness for measure-valued solutions, Comm. Math. Phys., Volume 305 (2011), pp. 315-361

[2] C. De Lellis; L. Székelyhidi The Euler equations as a differential inclusion, Ann. Math. (2), Volume 170 (2009) no. 3, pp. 1417-1436

[3] C. De Lellis; L. Székelyhidi On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., Volume 195 (2010) no. 1, pp. 225-260

[4] J.-M. Delort Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., Volume 4 (1991) no. 3, pp. 553-586

[5] P.-L. Lions Mathematical Topics in Fluid Mechanics, vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press Oxford University Press, New York, 1996 (Incompressible models, Oxford Science Publications)

[6] V. Scheffer An inviscid flow with compact support in space–time, J. Geom. Anal., Volume 3 (1993) no. 4, pp. 343-401

[7] A. Shnirelman On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math., Volume 50 (1997) no. 12, pp. 1261-1286

[8] L. Székelyhidi Jr., E. Wiedemann, Young measures generated by ideal incompressible fluid flows, Preprint, 2011.

[9] E. Wiedemann Existence of weak solutions for the incompressible Euler equations, Annales de lʼInstitut Henri Poincare (C) Non Linear Analysis, Volume 28 (2011) no. 5, pp. 727-730

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