On the critical one component regularity for 3-D Navier-Stokes system

Chemin, Jean-Yves; Zhang, Ping

doi:10.24033/asens.2278

On the critical one component regularity for 3-D Navier-Stokes system
[Autour de la régularité d'une composante critique pour le système de Navier-Stokes tridimensionnel]

Chemin, Jean-Yves ; Zhang, Ping

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 1, pp. 131-167.

Résumé
Abstract

On considère une donnée initiale $v_{0}$ dont la vorticité $Ω_{0} = \nabla \times v_{0}$ appartient à $L^{\frac{3}{2}}$ (ce qui implique que $v_{0}$ appartient à l'espace de Sobolev $H^{\frac{1}{2}}$ ). Nous démontrons que si la solution $v$ de l'équation de Navier-Stokes tridimensionnelle associée à $v_{0}$ par le théorème de Fujita-Kato développe une singularité à l'instant $T^{☆}$ (fini) alors, pour tout $p$ dans l'intervalle $] 4, 6 [$ et tout vecteur unitaire $e$ de $ℝ^{3},$ on a $\int_{0}^{T^{☆}} {∥ v (t) \cdot e ∥}_{H^{\frac{1}{2} + \frac{2}{p}}}^{p} d t = \infty .$ Remarquons que toutes ses quantités sont invariantes par les changements d'échelle de l'équation de Navier-Stokes.

Given an initial data $v_{0}$ with vorticity $Ω_{0} = \nabla \times v_{0}$ in $L^{\frac{3}{2}}$ (which implies that $v_{0}$ belongs to the Sobolev space $H^{\frac{1}{2}}$ ), we prove that the solution $v$ given by the classical Fujita-Kato theorem blows up in a finite time $T^{☆}$ only if, for any $p$ in $] 4, 6 [$ and any unit vector $e$ in $ℝ^{3},$ there holds $\int_{0}^{T^{☆}} {∥ v (t) \cdot e ∥}_{H^{\frac{1}{2} + \frac{2}{p}}}^{p} d t = \infty .$ We remark that all these quantities are scaling invariant under the scaling transformation of Navier-Stokes system.

Publié le : 2016-11-12

MR Zbl

DOI : 10.24033/asens.2278

Classification : 35Q30, 76D03
Keywords: Incompressible Navier-Stokes Equations, Blow-up criteria, Anisotropic, Littlewood-Paley Theory.
Mot clés : Équations de Navier-Stokes incompressibles, critère de l'explosion, théorie de Littlewood-Paley anisotropique.

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     author = {Chemin, Jean-Yves and Zhang, Ping},
     title = {On the critical one component regularity for {3-D} {Navier-Stokes} system},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {131--167},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
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     year = {2016},
     doi = {10.24033/asens.2278},
     mrnumber = {3465978},
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     url = {http://archive.numdam.org/articles/10.24033/asens.2278/}
}

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Chemin, Jean-Yves; Zhang, Ping. On the critical one component regularity for 3-D Navier-Stokes system. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 1, pp. 131-167. doi : 10.24033/asens.2278. http://archive.numdam.org/articles/10.24033/asens.2278/

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