The 2D Incompressible Magnetohydrodynamics Equations with only Magnetic Diffusion

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 (−Δ)β . We establish the global regularity for the case when β > 1. This result significantly improves previous work which requires β > 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 β = 1.