Comptes Rendus
no. 15-16
Partial Differential Equations/Numerical Analysis
The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow
[La viscosité évanescente comme critère de sélection pour les solutions de lʼéquation dʼEuler : Le cas du flot de cisaillement]

Présenté par : Olivier Pironneau

Claude Bardos1 ; Edriss S. Titi2, 3 ; Emil Wiedemann4
1 Laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, France
2 Department of Mathematics, University of California, Irvine, CA 92697, USA
3 Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel
4 Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical Sciences, Vancouver, BC, Canada
Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 757-760.

On montre que pour une certaine famille de données initiales, il existe plusieurs solutions faibles de lʼéquation dʼEuler incompressible qui satisfont lʼinégalité dʼénergie au sens faible. Cependant toute solution faible de lʼéquation dʼEuler qui de surcroit est limite faible dʼune suite de solutions des équations de Navier–Stokes (au sens de Leray–Hopf) avec les mêmes données initiales et une viscosité évanescente est déterminée de manière unique. Cet exemple simple suggère que, de même, dans des situations plus générales, la limite pour viscosité évanescente des solutions dʼéquations de Navier–Stokes puisse servir de critère dʼunicité pour les solutions faibles des équations dʼEuler.

We show that for a certain family of initial data, there exist non-unique weak solutions to the 3D incompressible Euler equations satisfying the weak energy inequality, whereas the weak limit of every sequence of Leray–Hopf weak solutions for the Navier–Stokes equations, with the same initial data, and as the viscosity tends to zero, is uniquely determined and equals the shear flow solution of the Euler equations corresponding to this initial data. This simple example suggests that, also in more general situations, the vanishing viscosity limit of the Navier–Stokes equations could serve as a uniqueness criterion for weak solutions of the Euler equations.

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DOI : 10.1016/j.crma.2012.09.005
Claude Bardos 1 ; Edriss S. Titi 2, 3 ; Emil Wiedemann 4

1 Laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, France
2 Department of Mathematics, University of California, Irvine, CA 92697, USA
3 Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel
4 Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical Sciences, Vancouver, BC, Canada 
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     journal = {Comptes Rendus. Math\'ematique},
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Claude Bardos; Edriss S. Titi; Emil Wiedemann. The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow. Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 757-760. doi : 10.1016/j.crma.2012.09.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.09.005/

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[5] R.J. DiPerna; P.-L. Lions Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989) no. 3, pp. 511-547

[6] Ronald J. DiPerna; Andrew J. Majda Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., Volume 108 (1987) no. 4, pp. 667-689

[7] Lawrence C. Evans Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010

[8] Dragoş Iftimie; Geneviève Raugel Some results on the Navier–Stokes equations in thin 3D domains, Atlanta, GA/Lisbon, 1998 (J. Differential Equations), Volume 169 (2001) no. 2, pp. 281-331

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[10] László 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

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

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