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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2015-05-15
Accepted: 2015-11-04
Published online: 2016-01-22

DOI: 10.1016/j.crma.2015.10.020

Author's affiliations:

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

@article{CRMATH_2016__354_2_167_0,
     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},
}

TY  - JOUR
AU  - Hyeong-Ohk Bae
AU  - Jörg Wolf
TI  - A local regularity condition involving two velocity components of Serrin-type for the Navier–Stokes equations
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 167
EP  - 174
VL  - 354
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2015.10.020
LA  - en
ID  - CRMATH_2016__354_2_167_0
ER  -

%0 Journal Article
%A Hyeong-Ohk Bae
%A Jörg Wolf
%T A local regularity condition involving two velocity components of Serrin-type for the Navier–Stokes equations
%J Comptes Rendus. Mathématique
%D 2016
%P 167-174
%V 354
%N 2
%I Elsevier
%R 10.1016/j.crma.2015.10.020
%G en
%F CRMATH_2016__354_2_167_0

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/

References
Cited by

[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

Cited by Sources:

Comments - Policy