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

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  • Whether or not classical solutions of the 3D incompressible MHD equations with full dissipation and magnetic diffusion can develop finite-time singularities is a long standing open problem of fluid dynamics and PDE theory. In this paper, we investigate the Cauchy problem for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. We get a unique global smooth solution under the assumption that $u_\theta$ and $b_r$ are trivial. In absence of some viscosities, there is no smoothing effect on the derivatives of that direction. However, we take full advantage of the structures of MHD system to make up this shortcoming.
    Mathematics Subject Classification: 35B65, 76D03, 76W05.


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