Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 301 - 322, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.301

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Quansen Jiu - School of Mathematical Sciences & Beijing Center of Mathematics and Information Sciences, Capital Normal University, Beijing, 100048, China (email)
Jitao Liu - The Institute of Mathematical Sciences, The Chinese University of Hong Kong, China (email)

Abstract: 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.

Keywords:  MHD equations, global regularity, 3D axisymmetric, horizontal dissipation, vertical magnetic diffusion.
Mathematics Subject Classification:  35B65, 76D03, 76W05.

Received: January 2014;      Revised: July 2014;      Available Online: August 2014.