Comptes Rendus
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, Volume 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

Jiřı́ Neustupa 1 ; Patrick Penel 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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Jiřı́ Neustupa; Patrick Penel. The role of eigenvalues and eigenvectors of the symmetrized gradient of velocity in the theory of the Navier–Stokes equations. Comptes Rendus. Mathématique, Volume 336 (2003) no. 10, pp. 805-810. doi : 10.1016/S1631-073X(03)00174-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00174-2/

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

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

[3] J. Neustupa; P. Penel Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier–Stokes equations (J. Neustupa; P. Penel, 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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