Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–Stokes

Isabelle Gallagher; Dragoş Iftimie; Fabrice Planchon

doi:10.1016/S1631-073X(02)02255-0

Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–Stokes

Isabelle Gallagher ¹ ; Dragoş Iftimie ^1,² ; Fabrice Planchon ³

¹ Centre de mathématiques, UMR 7640, École polytechnique, 91128 Palaiseau, France
² IRMAR, UMR 6625, Université de Rennes 1, campus de Beaulieu, 35042 Rennes, France
³ Laboratoire d'analyse numérique, UMR 7598, boîte 187, Université Paris-VI, 4, place Jussieu, 75252 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 289-292.

Résumé (VO)
Abstract

Nous nous donnons a priori une solution globale des équations de Navier–Stokes incompressibles dans $ℝ^{3}$ , dans la classe $C_{t} ({\dot{H}}^{1 / 2})$ . Nous montrons successivement que la norme ${\dot{H}}^{1 / 2}$ tend vers 0 à l'infini, que cette norme contrôle la norme $L_{t}^{2} ({\dot{H}}^{3 / 2})$ , et qu'une telle solution globale est stable.

Suppose there exists a global solution u to the incompressible Navier–Stokes equations, such that $u \in C_{t} ({\dot{H}}^{1 / 2})$ . We prove that its ${\dot{H}}^{1 / 2}$ norm goes to 0 at infinity. We next use this fact to control the $L_{t}^{2} ({\dot{H}}^{3 / 2})$ norm of u, and finally we prove that such a solution is stable.

Accepté le : 2001-12-17
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02255-0

Affiliations des auteurs :

Isabelle Gallagher ¹ ; Dragoş Iftimie ^{1, 2} ; Fabrice Planchon ³

¹ Centre de mathématiques, UMR 7640, École polytechnique, 91128 Palaiseau, France
² IRMAR, UMR 6625, Université de Rennes 1, campus de Beaulieu, 35042 Rennes, France
³ Laboratoire d'analyse numérique, UMR 7598, boîte 187, Université Paris-VI, 4, place Jussieu, 75252 Paris cedex 05, France

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Isabelle Gallagher; Dragoş Iftimie; Fabrice Planchon. Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–Stokes. Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 289-292. doi : 10.1016/S1631-073X(02)02255-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02255-0/

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