Partial Differential Equations/Numerical Analysis
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.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.09.005
Bardos, Claude 1; Titi, Edriss S. 2, 3; Wiedemann, Emil 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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Bardos, Claude; Titi, Edriss S.; Wiedemann, Emil. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.09.005/

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