Improved regularity criteria for the MHD equations

Abstract

In this paper, we establish new regularity criteria for the three-dimensional (3D) viscous incompressible magnetohydrodynamic (MHD) equations. It is proved that if the solution of the MHD equations satisfies u3∈Lp(0,T;Lq(R3)),j3∈Lr(0,T;Ls(R3)),2p+3q=1324,7213≤q≤∞;2r+3s=2,32<s≤∞ or u3∈Lp(0,T;Lq(R3)),w3∈Lr(0,T;Ls(R3)),2p+3q=1324,7213≤q≤∞;2r+3s=2,32<s≤∞, then the regularity of the solution on (0, T), where u3, j3 and ω3 are the third component of velocity u, current density ∇ × b and vorticity ∇ × u, respectively. These results give new improvements of regularity theory of weak solutions.