Remarks on regularity criteria for 2D generalized MHD equations

Abstract

In this paper, we establish two regularity criteria for the two-dimensional (2D) incompressible generalized magnetohydrodynamic (GMHD) equations in terms of only one quantity, namely, the current density $j=\nabla \times b$ or the vorticity $\omega =\nabla \times u$. More precisely, it is proved that, if one of the following holds true: \[ \int _{0}^{T}{\|j(t)\|_{\dot {B}_{\infty ,\,\infty }^{0}(\mathbb {R}^{2})}\,dt}\lt \infty , \] \[ \int _{0}^{T}{\|\omega (t)\|_{\dot {B}_{\infty ,\,\infty }^{0}(\mathbb {R}^{2})}\,dt}\lt \infty , \] then the solution $(u,\,b)$ actually remains regular on $[0, T ]$.