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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 $\Bbb R^3$. It is proved that the solution is regular if the associate pressure satisfies $$p\in L^{\frac{2}{2-r}} \big((0,T);\dot{\cal M}_{2,\frac3r}(\Bbb R^3)\big) \quad\text{or}\quad \nabla p\in L^{\frac{2}{3-r}} \big((0,T);\dot{\cal M}_{2,\frac3r}(\Bbb R^3)\big),$$ for $0<r<1$, where $\dot{\cal M}_{2,\frac3r}(\Bbb R^3)$ is the critical Morrey-Campanto space. Regularity criteria for the 3D MHD equations are also given.

MSC:
35Q30Stokes and Navier-Stokes equations
35B65Smoothness and regularity of solutions of PDE
76D05Navier-Stokes equations (fluid dynamics)
76D03Existence, uniqueness, and regularity theory
76W05Magnetohydrodynamics and electrohydrodynamics
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References:
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