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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     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},
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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/

Bibliographie
Cité par

[1] H.-O. Bae; H.J. Choe $L^{\infty}$ -bound of weak solutions to Navier–Stokes equations, Taejon, 1996 (Lecture Notes Ser.), Volume vol. 39 (1997) (13 p)

[2] H.-O. Bae; H.J. Choe A regularity criterion for the Navier–Stokes equations, Commun. Partial Differ. Equ., Volume 32 (2007), pp. 1173-1187

[3] H. Beirão da Veiga On the smoothness of a class of weak solutions to the Navier–Stokes equations, J. Math. Fluid Mech., Volume 2 (2000) no. 4, pp. 315-323

[4] D. Chae; H.J. Choe Regularity of solutions to the Navier–Stokes equation, Electron. J. Differ. Equ., Volume 1999 (1999) no. 5, pp. 1-7

[5] D. Chae; K. Kang; J. Lee On the interior regularity of suitable weak solutions to the Navier–Stokes equations, Commun. Partial Differ. Equ., Volume 32 (2007) no. 7–9, pp. 1189-1207

[6] E.B. Fabes; B.F. Jones; N.M. Riviere The initial value problem for the Navier–Stokes equations with data in $L^{p}$ , Arch. Ration. Mech. Anal., Volume 45 (1972), pp. 222-240

[7] G. Galdi An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I, Linearized Steady Problems, vol. 38, Springer-Verlag, New York, 1994

[8] G. Galdi; C. Simader; H. Shor On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl. (4), Volume 167 (1994), pp. 147-1633

[9] I. Kukavica; M. Ziane One-component regularity for the Navier–Stokes equations, Nonlinearity, Volume 19 (2006) no. 2, pp. 453-469

[10] J. Neustupa; A. Novotný; P. Penel Regularity of a suitable weak solution to the Navier–Stokes equations as a consequence of regularity of one velocity component (A. Sequeira; H. Beirão da Veiga; J.H. Videman, eds.), Applied Nonlinear Analysis, Kluwer/Plenum, New York, 1999, pp. 391-402

[11] J. Neustupa; M. Pokorny An interior regularity criterion for an axially symmetric suitable weak solution to the Navier–Stokes equations, J. Math. Fluid Mech., Volume 2 (2000), pp. 381-399

[12] J. Serrin On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal., Volume 9 (1962), pp. 187-195

[13] M. Struwe On partial regularity results for the Navier–Stokes equations, Commun. Pure Appl. Math., Volume 41 (1988), pp. 437-458

[14] J. Wolf On the local regularity of suitable weak solutions to the generalized Navier–Stokes equations, Ann. Univ. Ferrara, Volume 61 (2015) no. 1, pp. 149-171

[15] J. Wolf, On the local pressure of the Navier–Stokes equations and related systems (2015), submitted for publication.

[16] Y. Zhou A new regularity criterion for the Navier–Stokes equations in terms of the gradient of one velocity component, Methods Appl. Anal., Volume 9 (2002) no. 4, pp. 563-578

Jae-Myoung Kim Some regularity criteria of a weak solution to the 3D Navier–Stokes equations in a domain, Archiv der Mathematik, Volume 117 (2021) no. 2, p. 215 | DOI:10.1007/s00013-021-01613-0
D. Chae; J. Wolf On the Serrin-Type Condition on One Velocity Component for the Navier–Stokes Equations, Archive for Rational Mechanics and Analysis, Volume 240 (2021) no. 3, p. 1323 | DOI:10.1007/s00205-021-01636-5
Hugo Beirão da Veiga; Jiaqi Yang Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components, Advances in Nonlinear Analysis, Volume 9 (2019) no. 1, p. 633 | DOI:10.1515/anona-2020-0017
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Yanqing Wang; Gang Wu; Daoguo Zhou Some Interior Regularity Criteria Involving Two Components for Weak Solutions to the 3D Navier–Stokes Equations, Journal of Mathematical Fluid Mechanics, Volume 20 (2018) no. 4, p. 2147 | DOI:10.1007/s00021-018-0402-5
H. Beirão da Veiga On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier–Stokes equations. The half-space case, Journal of Mathematical Analysis and Applications, Volume 453 (2017) no. 1, p. 212 | DOI:10.1016/j.jmaa.2017.03.089
Hyeong-Ohk Bae; Jörg Wolf Sufficient conditions for local regularity to the generalized Newtonian fluid with shear thinning viscosity, Zeitschrift für angewandte Mathematik und Physik, Volume 68 (2017) no. 1 | DOI:10.1007/s00033-016-0751-y

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