Global regularity of 2D generalized incompressible magnetohydrodynamic equations

TL;DR

This paper proves the global regularity of 2D incompressible MHD equations with fractional velocity dissipation and reduced magnetic diffusion, advancing understanding of the conditions ensuring smooth solutions.

Contribution

It establishes global regularity for 2D MHD with fractional dissipation and improves previous results, especially for cases with minimal magnetic diffusion.

Findings

Global regularity for 2D MHD with positive fractional dissipation.

New a priori bounds for the case with no magnetic diffusion.

Progress towards resolving the full regularity problem for 2D resistive MHD.

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.