International Conference on Applications of Mathematics to Nonlinear Sciences. Electron. J. Diff. Eqns., Conference 24 (2017), pp. 35-36.

Global regularity results for four systems of 2D MHD equations with partial dissipation

Jingna Li, Bo-Qing Dong, Jiahong Wu

This article examines the global well-posedness problem on four closely related systems of the 2D magnetohydrodynamic (MHD) equations with partial dissipation. They all share the same partial dissipation in the equation of the magnetic field b, only the vertical magnetic diffusion in the horizontal component and the horizontal magnetic diffusion in the vertical component. When the velocity equation has no fluid viscosity, the global regularity problem is an outstanding open problem. We prove a weak-sensed small data global existence result for the case when there is no fluid viscosity. When the velocity equation involves partial dissipation of the same structure as in the equation of b, we show that any L^2 initial datum leads to a unique global solution, which becomes smooth instantaneously. When the partial dissipation in the velocity equation is either in the horizontal or vertical direction, we prove that any $H^1$ initial datum generates a unique global solution.

Published November 15, 2017.
Math Subject Classifications: 35Q35, 35B65, 76B03.
Key Words: 2D MHD equations; global well-posedness; partial dissipation.

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Jingna Li
Department of Mathematics
Jinan University
Guangzhou 510632, China
Bo-Qing Dong
College of Mathematics and Statistics
Shenzhen University
Shenzhen 518060, China
Jiahong Wu
Department of Mathematics
Oklahoma State University
Stillwater, OK 74078, USA

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