Régularité conditionnelle des équations de Navier–Stokes

Kukavica, Igor; Ziane, Mohammed

doi:10.1016/j.crma.2006.04.024

Équations aux dérivées partielles/Problèmes mathématiques de la mécanique

Régularité conditionnelle des équations de Navier–Stokes

Présenté par : Yves Meyer

Kukavica, Igor ¹ ; Ziane, Mohammed ¹

¹ Department of Mathematics, University of Southern California, Los Angeles, California, États-Unis

Comptes Rendus. Mathématique, Tome 343 (2006) no. 1, pp. 31-36.

Résumé
Abstract

Dans cette Note, nous donnons des conditions suffisantes de régularité des solutions des équations de Navier–Stokes. Nous montrons que si une solution de Leray–Hopf vérifie l'une des trois conditions (i) $\partial u / \partial x_{3} \in L_{t}^{s_{0}} L_{x}^{r_{0}}$ où $2 / s_{0} + 3 / r_{0} ⩽ 2$ et $9 / 4 ⩽ r_{0} ⩽ 3$ , (ii) $\nabla u_{3} \in L_{t}^{s_{1}} L_{x}^{r_{1}}$ où $2 / s_{1} + 3 / r_{1} ⩽ 11 / 6$ et $54 / 23 ⩽ r_{0} ⩽ 18 / 5$ , ou (iii) $u_{3} \in L_{t}^{s_{0}} L_{x}^{r_{0}}$ où $2 / s_{0} + 3 / r_{0} ⩽ 5 / 8$ et $24 / 5 ⩽ r_{0} ⩽ \infty$ , alors elle est régulière. Ces conditions améliorent les résultats existants sur la régularité conditionnelle des équations de Navier–Stokes.

In this Note, we give sufficient conditions for the regularity of Leray–Hopf weak solutions to the Navier–Stokes equation. We prove that, if one of three conditions (i) $\partial u / \partial x_{3} \in L_{t}^{s_{0}} L_{x}^{r_{0}}$ where $2 / s_{0} + 3 / r_{0} ⩽ 2$ and $9 / 4 ⩽ r_{0} ⩽ 3$ , (ii) $\nabla u_{3} \in L_{t}^{s_{1}} L_{x}^{r_{1}}$ where $2 / s_{1} + 3 / r_{1} ⩽ 11 / 6$ and $54 / 23 ⩽ r_{0} ⩽ 18 / 5$ , or (iii) $u_{3} \in L_{t}^{s_{0}} L_{x}^{r_{0}}$ where $2 / s_{0} + 3 / r_{0} ⩽ 5 / 8$ and $24 / 5 ⩽ r_{0} ⩽ \infty$ , is satisfied, then the solution is regular. These conditions improve earlier results on the conditional regularity of the Navier–Stokes equations.

Reçu le : 2005-12-05
Accepté le : 2006-04-21
Publié le : 2006-06-21

DOI : 10.1016/j.crma.2006.04.024

Affiliations des auteurs :

Kukavica, Igor ¹ ; Ziane, Mohammed ¹

¹ Department of Mathematics, University of Southern California, Los Angeles, California, États-Unis

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     title = {R\'egularit\'e conditionnelle des \'equations de {Navier{\textendash}Stokes}},
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Kukavica, Igor; Ziane, Mohammed. Régularité conditionnelle des équations de Navier–Stokes. Comptes Rendus. Mathématique, Tome 343 (2006) no. 1, pp. 31-36. doi : 10.1016/j.crma.2006.04.024. http://www.numdam.org/articles/10.1016/j.crma.2006.04.024/

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