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 ∇hu,∇hb∈L830,T;B·∞,∞-1R3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \nabla _{h}u,\nabla _{h}b\right) \in L^{\frac{8}{3}}\left( 0,T;\overset{\cdot }{B}_{\infty ,\infty }^{-1}\left( \mathbb {R}^{3}\right) \right) . \end{aligned}$$\end{document} 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)].