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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 3 . It is proved that the solution is regular if the associate pressure satisfies

pL 2 2-r (0,T) ; ˙ 2,3 r ( 3 )orpL 2 3-r (0,T) ; ˙ 2,3 r ( 3 ),

for 0<r<1, where ˙ 2,3 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