The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion

This paper studies the global regularity of classical solutions to the 2D incompressible magnetohydrodynamic (MHD) equations with horizontal dissipation and horizontal magnetic diffusion. It is shown here that the horizontal component of any solution admits a global (in time) bound in any Lebesgue space L2r with 1⩽r<∞ and the bound grows no faster than the order of as r increases. In addition, we establish a conditional global regularity in terms of the -norm of the horizontal component and the global regularity of a slightly regularized version of the aforementioned MHD equations.