A Remark On Global Regularity of 2D Generalized Magnetohydrodynamic Equations

Abstract

In this paper we study the global regularity of the following 2D (two-dimensional) generalized magnetohydrodynamic equations \begin{eqnarray*} \left\{\begin{array}{llll} u_t + u \cdot \nabla u & = & - \nabla p + b \cdot \nabla b - \nu (-\triangle)^{\alpha} u b_t + u \cdot \nabla b & = & b \cdot \nabla u - \kappa (-\triangle)^{\beta} b \end{array}\right. \end{eqnarray*} and get global regular solutions when $ 0\leqslant\alpha < 1 / 2,\,\, \beta \geqslant 1, \,\,3\alpha + 2\beta >3 $, which improves the results in \cite{TYZ2013}. In particular, we obtain the global regularity of the 2D generalized MHD when $\alpha=0$ and $\beta>\frac 32$.