Global regularity of 2D generalized incompressible magnetohydrodynamic equations

In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by $(-\mathrm{\Delta }{)}^{\alpha }$ and magnetic diffusion given by reducing about logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power $\alpha$ is a positive constant. In addition, we prove several global \emph{a priori} bounds for the case $\alpha =0$. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global regularity issue on the 2D resistive MHD equations, namely, the case when the MHD equations only have standard Laplacian magnetic diffusion.