Regularity Criteria in Terms of the Pressure for the Navier–Stokes Equations in the Critical Morrey–Campanato Space

Yong Zhou; Sadek Gala

doi:10.4171/zaa/1425

JournalszaaVol. 30, No. 1pp. 83–93

Regularity Criteria in Terms of the Pressure for the Navier–Stokes Equations in the Critical Morrey–Campanato Space

Yong Zhou
Shanghai University of Finance and Economics, China
Sadek Gala
Department of Mathematics, Mostaganem, Algeria

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Abstract

In this paper, 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 satifies

p \in L^{\frac{2}{2 - r}} ((0, T); \dot{M}_{2, \frac{3}{r}} (R^{3})) or \nabla p \in L^{\frac{2}{3 - r}} ((0, T); \dot{M}_{2, \frac{3}{r}} (R^{3}))

for $0 < r < 1$ , where $\dot{M}_{2, \frac{3}{r}}$ is the critical Morrey–Campanto space. Regularity criteria for the 3D MHD equations are also given.

Cite this article

Yong Zhou, Sadek Gala, Regularity Criteria in Terms of the Pressure for the Navier–Stokes Equations in the Critical Morrey–Campanato Space. Z. Anal. Anwend. 30 (2011), no. 1, pp. 83–93

DOI 10.4171/ZAA/1425