Regularity criteria of the magnetohydrodynamic equations in bounded domains or a half space

Abstract

We study three-dimensional incompressible magnetohydrodynamic equations in bounded domains or a half space. We present new regularity criteria of weak solutions: a pair of weak solutions, (u,b), become regular if u satisfies Serrinʼs type conditions when we consider no-slip or slip boundary conditions for the velocity field, u, and slip boundary conditions for the magnetic field, b, in either bounded domains or a half space. In addition, in the case of a half-space with no-slip boundary conditions for u and slip boundary conditions for b, we demonstrate that, if tangential components of u and normal component of b satisfy Serrinʼs type conditions, then a pair of weak solutions, (u,b), become regular.