Partial Differential Equations
The role of eigenvalues and eigenvectors of the symmetrized gradient of velocity in the theory of the Navier–Stokes equations
[Rôle des valeurs propres et des vecteurs propres du gradient symétrisé des vitesses en théorie des équations de Navier–Stokes]
Comptes Rendus. Mathématique, Tome 336 (2003) no. 10, pp. 805-810.

Dans cette Note, on formule des conditions géométriques suffisantes pour la régularité intérieure des solutions faibles ( « suitable weak ») des équations de Navier–Stokes dans un sous-domaine D du cylindre spatio–temporel QT : ces conditions suffisantes portent sur une des valeurs propres ou bien sur les composantes des vecteurs propres du gradient symétrisé.

In this Note, we formulate sufficient conditions for regularity of a so called suitable weak solution (v;p) in a sub-domain D of the time–space cylinder QT by means of requirements on one of the eigenvalues or on the eigenvectors of the symmetrized gradient of velocity.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00174-2
Neustupa, Jiřı́ 1 ; Penel, Patrick 2

1 Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic
2 Université de Toulon et du Var, Département de mathématique, BP 132, 83957 La Garde, France
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Neustupa, Jiřı́; Penel, Patrick. The role of eigenvalues and eigenvectors of the symmetrized gradient of velocity in the theory of the Navier–Stokes equations. Comptes Rendus. Mathématique, Tome 336 (2003) no. 10, pp. 805-810. doi : 10.1016/S1631-073X(03)00174-2. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00174-2/

[1] Caffarelli, L.; Kohn, R.; Nirenberg, L. Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math., Volume 35 (1982), pp. 771-831

[2] Galdi, G.P. An Introduction to the Navier–Stokes initial-boundary value problem (Galdi, G.P.; Heywood, J.; Rannacher, R., eds.), Fundamental Directions in Mathematical Fluid Mechanics, Birkhäuser, Basel, 2000, pp. 1-98

[3] Neustupa, J.; Penel, P. Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier–Stokes equations (Neustupa, J.; Penel, P., eds.), Mathematical Fluid Mechanics, Recent Results and Open Problems, Birkhäuser, Basel, 2001, pp. 237-268

[4] J. Neustupa, P. Penel, Regularity of weak solutions to the Navier–Stokes equations in dependence on eigenvalues and eigenvectors of the rate of deformation tensor, Preprint, 2002

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