On the regularity criterion of weak solutions for the 3D MHD equations

The paper deals with the 3D incompressible MHD equations and aims at improving a regularity criterion in terms of the horizontal gradient of velocity and magnetic field. It is proved that the weak solution (u, b) becomes regular provided that (∇ h u,∇ h b)∈L 83 (0,T;B ⋅ −1 ∞,∞ (R 3 )). (∇hu,∇hb)∈L83(0,T;B⋅∞,∞−1(R3)). The result is an extension of regularity criterion for 3D Navier–Stokes equations in Besov space due to Fang and Qian (Commun Pure Appl Anal 13:585–603, 2014) [see also (Ni et al. in J Math Anal Appl 396:108–118, 2012)].