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Regularity criteria in terms of the pressure for the Navier-Stokes equations in the critical Morrey-Campanato space. (English) Zbl 1210.35182

Summary: We establish a Serrin-type regularity criterion in terms of the pressure for Leray weak solutions to the Navier-Stokes equation in ${ℝ}^{3}$. It is proved that the solution is regular if the associate pressure satisfies

$p\in {L}^{\frac{2}{2-r}}\left(\left(0,T\right);{\stackrel{˙}{ℳ}}_{2,\frac{3}{r}}\left({ℝ}^{3}\right)\right)\phantom{\rule{1.em}{0ex}}\text{or}\phantom{\rule{1.em}{0ex}}\nabla p\in {L}^{\frac{2}{3-r}}\left(\left(0,T\right);{\stackrel{˙}{ℳ}}_{2,\frac{3}{r}}\left({ℝ}^{3}\right)\right),$

for $0, where ${\stackrel{˙}{ℳ}}_{2,\frac{3}{r}}\left({ℝ}^{3}\right)$ is the critical Morrey-Campanto space. Regularity criteria for the 3D MHD equations are also given.

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 35B65 Smoothness and regularity of solutions of PDE 76D05 Navier-Stokes equations (fluid dynamics) 76D03 Existence, uniqueness, and regularity theory 76W05 Magnetohydrodynamics and electrohydrodynamics