[La viscosité évanescente comme critère de sélection pour les solutions de lʼéquation dʼEuler : Le cas du flot de cisaillement]
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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@article{CRMATH_2012__350_15-16_757_0,
author = {Bardos, Claude and Titi, Edriss S. and Wiedemann, Emil},
title = {The vanishing viscosity as a selection principle for the {Euler} equations: {The} case of {3D} shear flow},
journal = {Comptes Rendus. Math\'ematique},
pages = {757--760},
publisher = {Elsevier},
volume = {350},
number = {15-16},
year = {2012},
doi = {10.1016/j.crma.2012.09.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.09.005/}
}
TY - JOUR AU - Bardos, Claude AU - Titi, Edriss S. AU - Wiedemann, Emil TI - The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow JO - Comptes Rendus. Mathématique PY - 2012 SP - 757 EP - 760 VL - 350 IS - 15-16 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.09.005/ DO - 10.1016/j.crma.2012.09.005 LA - en ID - CRMATH_2012__350_15-16_757_0 ER -
%0 Journal Article %A Bardos, Claude %A Titi, Edriss S. %A Wiedemann, Emil %T The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow %J Comptes Rendus. Mathématique %D 2012 %P 757-760 %V 350 %N 15-16 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.09.005/ %R 10.1016/j.crma.2012.09.005 %G en %F CRMATH_2012__350_15-16_757_0
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, Tome 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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