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Zbl 1210.35182
Zhou, Yong; Gala, Sadek
Regularity criteria in terms of the pressure for the Navier-Stokes equations in the critical Morrey-Campanato space. (English)
[J] Z. Anal. Anwend. 30, No. 1, 83-93 (2011). ISSN 0232-2064; ISSN 1661-4534/e

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 2000:: *35Q30 Stokes and Navier-Stokes equations
35B65 Smoothness of solutions of PDE
76D05 Navier-Stokes equations (fluid dynamics)
76D03 Existence, uniqueness, and regularity theory
76W05 Flows in presence of electromagnetic forces

Keywords: Navier-Stokes equations; Morrey-Campanato space; weak solution; regularity criterion

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