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 \((-\Delta )^{\alpha }\) and magnetic diffusion given by reducing about the square root of 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 a priori bounds for the case \(\alpha =0\). Finally, for the case \(\alpha =0\), it is also shown that the control of the direction of the magnetic field in a suitable norm is enough to guarantee the global regularity. 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.