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.
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