Global regularity of 2D generalized incompressible magnetohydrodynamic equations

Abstract

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 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.

31 pages. We have added some materials in this updated version. arXiv admin note: text overlap with arXiv:2205.00688