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Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations. (English) Zbl 1214.35043

Summary: We establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in ${ℝ}^{d}$. It is known that if a Leray weak solution $u$ belongs to

${L}^{\frac{2}{1-r}}\left(\left(0,T\right);{L}^{\frac{d}{r}}\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}0\le r\le 1,$

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

${L}^{\frac{2}{1-r}}\left(\left(0,T\right);{\stackrel{˙}{ℳ}}_{2,\frac{d}{r}}{\left({ℝ}^{d}\right)}^{d}\right),\phantom{\rule{2.em}{0ex}}\left(*\right)$

where ${\stackrel{˙}{ℳ}}_{2,\frac{d}{r}}{\left({ℝ}^{d}\right)}^{d}\right)$ is the critical Morrey-Campanato space (a definition is given in the text) for $0, then the weak solution is actually regular. Since this space ${\stackrel{˙}{ℳ}}_{2,\frac{d}{r}}$ is wider than ${L}^{\frac{d}{r}}$ and ${\stackrel{˙}{X}}_{r}$, the above regularity criterion $\left(*\right)$ is an improvement of Zhou’s result.

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 76D05 Navier-Stokes equations (fluid dynamics) 76D03 Existence, uniqueness, and regularity theory 35B65 Smoothness and regularity of solutions of PDE