Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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
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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.


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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