Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations
Author:
Sadek Gala
Journal:
Quart. Appl. Math. 69 (2011), 147-155
MSC (2000):
Primary 35Q30, 35K15, 76D05
DOI:
https://doi.org/10.1090/S0033-569X-2011-01206-0
Published electronically:
January 18, 2011
Full-text PDF Free Access
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Abstract: In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in $\mathbb {R}^{d}$. It is known that if a Leray weak solution $u$ belongs to \begin{equation} L^{\frac {2}{1-r}}\left ( \left ( 0,T\right ) ;L^{\frac {d}{r}}\right ) \text { \ \ for some \ \ }0\leq r\leq 1, \end{equation} then $u$ is regular. It is proved that if the pressure $p$ associated to a Leray weak solution $u$ belongs to \begin{equation} L^{\frac {2}{2-r}}\left ( \left ( 0,T\right ) ;\overset {.}{\mathcal {M}}_{2,\frac { d}{r}}\left ( \mathbb {R}^{d}\right ) ^{d}\right ) , \end{equation} where $\overset {.}{\mathcal {M}}_{2,\frac {d}{r}}\left ( \mathbb {R}^{d}\right )$ is the critical Morrey-Campanato space (a definition is given in the text) for $0<r<1$, then the weak solution is actually regular. Since this space $\overset {.}{\mathcal {M}}_{2,\frac {d}{r}}$ is wider than $L^{\frac {d}{r}}$ and $\overset {.}{X}_{r}$, the above regularity criterion (0.2) is an improvement of Zhou’s result.
References
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References
- H. Beirão da Veiga, A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech. 2 (2000), 99-106. MR 1765772 (2001d:76025)
- S. Benbernou, A note on the regularity criterion in terms of pressure for the Navier-Stokes equations, Appl. Math. Letters 22 (2009), 1438-1443. MR 2536829
- L.C. Berselli and G.P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc. 130 (2002), 3585-3595. MR 1920038 (2003e:35240)
- L. Caffarelli, J. Kohn and L. Nirenberg, Partial regularity for suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831. MR 673830 (84m:35097)
- D. Chae and J. Lee, Regularity criterion in terms of pressure for the Navier-Stokes equations, Nonlinear Anal. 46 (2001), 727-735. MR 1857154 (2002g:76032)
- Y. Giga, Solutions for semilinear parabolic equations in $L^{p}$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations 62 (1986), 186-212. MR 833416 (87h:35157)
- E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213-231. MR 0050423 (14:327b)
- S. Kaniel, A sufficient condition for smoothness of solutions of Navier-Stokes equations, Israel J. Math. 6 (1968), 354-358. MR 0244651 (39:5965)
- T. Kato, Strong $L^{p}-$solutions of the Navier-Stokes equation in $\mathbb {R}^{m}$, with applications to weak solutions, Math. Z. 187 (1984), 471-480. MR 760047 (86b:35171)
- H. Kozono and H. Sohr, Regularity criterion on weak solutions to the Navier-Stokes equations, Adv. Differential Equations 2 (1997), 535-554. MR 1441855 (97m:35206)
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- J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187-195. MR 0136885 (25:346)
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- M. Struwe, On a Serrin-Type Regularity Criterion for the Navier–Stokes Equations in Terms of the Pressure, J. Math. Fluid Mech. 9 (2007), 235-242. MR 2329267 (2008i:35185)
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- Y. Zhou, Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain, Math. Ann. 328 (2004), 173-192. MR 2030374 (2004j:35229)
- Y. Zhou, A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity, Monatsh. Math. 144 (2005), 251–257. MR 2130277 (2006a:35243)
- Y. Zhou, A new regularity criterion for weak solutions to the Navier-Stokes equations. J. Math. Pures Appl. 84 (2005), 1496–1514. MR 2181458 (2006g:35210)
- Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbb {R}^{3}$, Proc. Amer. Math. Soc. 134 (2006), 149-156. MR 2170554 (2006i:35288)
- Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $\mathbb {R}^{n}$, Z. Angew. Math. Phys. 57 (2006), 384-392. MR 2228171 (2007g:35179)
- Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier–Stokes equations in multiplier spaces, J. Math. Anal. Appl. 356 (2009), 498-501. MR 2524284
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Sadek Gala
Affiliation:
Department of Mathematics, University of Mostaganem, Box 227, Mostaganem (27000), Algeria
Received by editor(s):
July 30, 2009
Published electronically:
January 18, 2011
Article copyright:
© Copyright 2011
Brown University