The 2D Incompressible Magnetohydrodynamics Equations with only Magnetic Diffusion

Abstract

This paper examines the global (in time) regularity of classical solutions to the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only magnetic diffusion. Here the magnetic diffusion is given by the fractional Laplacian operator $(-\Delta)^\beta$. We establish the global regularity for the case when $\beta>1$. This result significantly improves previous work which requires $\beta>\frac{3}{2}$ and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely, the case when $\beta=1$.