Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion

Abstract

Whether or not classical solutions of the 3D incompressible MHDequations with full dissipation and magnetic diffusion can developfinite-time singularities is a long standing open problem of fluiddynamics and PDE theory. In this paper, we investigate the Cauchyproblem for the 3D axisymmetric MHD equations with horizontaldissipation and vertical magnetic diffusion. We get a unique globalsmooth solution under the assumption that $u_\theta$ and $b_r$ aretrivial. In absence of some viscosities, there is no smoothingeffect on the derivatives of that direction. However, we take fulladvantage of the structures of MHD system to make up thisshortcoming.