On Global Regularity of 2D Generalized Magnetohydrodynamic Equations

In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are $- \nu (- \triangle)^{\alpha} u$and$- \kappa (-\triangle)^{\beta} b$. We show that smooth solutions are global in the following three cases: $\alpha \geqslant 1 / 2, \beta \geqslant 1$;$0 \leqslant \alpha < 1 / 2, 2 \alpha + \beta > 2$;$\alpha \geqslant 2, \beta = 0$. We also show that in the inviscid case $\nu =0$, if $\beta >1$, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.