Stability for a System of the 2D Incompressible MHD Equations with Fractional Dissipation

Several fundamental problems on the 2D magnetohydrodynamic (MHD) equations with only magnetic diffusion (no velocity dissipation) remain open, especialy in the case when the spatial domain is the whole space \({\mathbb {R}}^2\) ${\mathbb{R}}^2$ . This paper establishes that, near a background magnetic field, any fractional dissipation in one direction in the velocity equation would allow us to establish the global existence and stability for perturbations near the background. The magnetic diffusion here is not required to be given by the standard Laplacian operator but any general fractional Laplacian with positive power.