A local regularity condition involving two velocity components of Serrin-type for the Navier–Stokes equations

Hyeong-Ohk Bae; Jörg Wolf

doi:10.1016/j.crma.2015.10.020

Partial differential equations

A local regularity condition involving two velocity components of Serrin-type for the Navier–Stokes equations
[Une condition de la régularité locale impliquant deux composantes de la vitesse de type Serrin pour les équations de Navier–Stokes]

Présenté par : Haïm Brézis

Hyeong-Ohk Bae ¹ ; Jörg Wolf ²

¹ Department of Financial Engineering, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon-si, Gyeonggi-do, 443-749, Republic of Korea
² Department of Mathematics, Humboldt University Berlin, Unter den Linden 6, 10099 Berlin, Germany

Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 167-174.

Résumé
Abstract

Le présent papier traite le problème de la régularité locale de solutions faibles à l'équation de Navier–Stokes en $Ω \times (0, T)$ de terme de force f en $L^{2}$ . Nous prouvons que u est forte dans un sous-cylindre $Q_{r} \subset Ω \times (0, T)$ si deux composantes de la vitesse $u^{1}$ , $u^{2}$ satisfont une condition de type Serrin.

The present paper deals with the problem of local regularity of weak solutions to the Navier–Stokes equation in $Ω \times (0, T)$ with forcing term f in $L^{2}$ . We prove that u is strong in a sub-cylinder $Q_{r} \subset Ω \times (0, T)$ if two velocity components $u^{1}$ , $u^{2}$ satisfy a Serrin-type condition.

Reçu le : 2015-05-15
Accepté le : 2015-11-04
Publié le : 2016-01-22

DOI : 10.1016/j.crma.2015.10.020

Affiliations des auteurs :

Hyeong-Ohk Bae ¹ ; Jörg Wolf ²

¹ Department of Financial Engineering, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon-si, Gyeonggi-do, 443-749, Republic of Korea
² Department of Mathematics, Humboldt University Berlin, Unter den Linden 6, 10099 Berlin, Germany

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     author = {Hyeong-Ohk Bae and J\"org Wolf},
     title = {A local regularity condition involving two velocity components of {Serrin-type} for the {Navier{\textendash}Stokes} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {167--174},
     publisher = {Elsevier},
     volume = {354},
     number = {2},
     year = {2016},
     doi = {10.1016/j.crma.2015.10.020},
     language = {en},
}

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Hyeong-Ohk Bae; Jörg Wolf. A local regularity condition involving two velocity components of Serrin-type for the Navier–Stokes equations. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 167-174. doi : 10.1016/j.crma.2015.10.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.020/

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