Comptes Rendus
Partial differential equations/Mathematical problems in mechanics
Inviscid symmetry breaking with non-increasing energy
Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 907-910.

In a recent article, C. Bardos et al. constructed weak solutions of the three-dimensional incompressible Euler equations which emerge from two-dimensional initial data yet become fully three-dimensional at positive times. They asked whether such symmetry-breaking solutions could also be constructed under the additional condition that they should have non-increasing energy. In this note, we give a positive answer to this question and show that such a construction is possible for a large class of initial data. We use convex integration techniques as developed by De Lellis and Székelyhidi.

Récemment, C. Bardos et al. ont construit des solutions faibles de lʼéquation dʼEuler incompressible en dimension trois qui sont vraiment tridimensionnelles aux temps positifs, bien quʼelles émergent dʼune donnée initiale bidimensionnelle. Les auteurs se sont demandé si une telle construction était possible sous la condition additionnelle que les solutions aient une énergie non croissante. Dans cette note, on résout cette question en montrant quʼune telle construction est en fait possible pour une grande famille de données initiales. On utilise la méthode dʼintégration convexe de De Lellis et Székelyhidi.

Published online:
DOI: 10.1016/j.crma.2013.10.021

Emil Wiedemann 1

1 Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical Sciences, Vancouver, BC, Canada
     author = {Emil Wiedemann},
     title = {Inviscid symmetry breaking with non-increasing energy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {907--910},
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     year = {2013},
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     language = {en},
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Emil Wiedemann. Inviscid symmetry breaking with non-increasing energy. Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 907-910. doi : 10.1016/j.crma.2013.10.021.

[1] C. Bardos; E.S. Titi; E. Wiedemann The vanishing viscosity as a selection principle for the Euler equations: the case of 3D shear flow, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012), pp. 757-760

[2] C. Bardos; M.C. Lopes Filho; D. Niu; H.J. Nussenzveig Lopes; E.S. Titi Stability of two-dimensional viscous incompressible flows under three-dimensional perturbations and inviscid symmetry breaking, SIAM J. Math. Anal., Volume 45 (2013) no. 3, pp. 1871-1885

[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] P.-L. Lions Mathematical Topics in Fluid Mechanics, vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and Its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996

[5] L. Székelyhidi Weak solutions to the incompressible Euler equations with vortex sheet initial data, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 19–20, pp. 1063-1066

[6] L. Székelyhidi; E. Wiedemann Young measures generated by ideal incompressible fluid flows, Arch. Ration. Mech. Anal., Volume 206 (2012), pp. 333-366

[7] E. Wiedemann Existence of weak solutions for the incompressible Euler equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011), pp. 727-730

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