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Quarterly of Applied Mathematics
  
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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
Published electronically: January 18, 2011
MathSciNet review: 2807982
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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

$\displaystyle L^{\frac{2}{1-r}}\left( \left( 0,T\right) ;L^{\frac{d}{r}}\right)$    for some  $\displaystyle 0\leq r\leq 1,$ (1)

then $ u$ is regular. It is proved that if the pressure $ p$ associated to a Leray weak solution $ u$ belongs to

$\displaystyle L^{\frac{2}{2-r}}\left( \left( 0,T\right) ;\overset{.}{\mathcal{M}}_{2,\frac{ d}{r}}\left( \mathbb{R}^{d}\right) ^{d}\right) ,$ (2)

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.


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Additional Information

Sadek Gala
Affiliation: Department of Mathematics, University of Mostaganem, Box 227, Mostaganem (27000), Algeria

DOI: http://dx.doi.org/10.1090/S0033-569X-2011-01206-0
PII: S 0033-569X(2011)01206-0
Received by editor(s): July 30, 2009
Published electronically: January 18, 2011
Article copyright: © Copyright 2011 Brown University



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