International Conference on Applications of Mathematics to Nonlinear Sciences.
Electron. J. Diff. Eqns., Conference 24 (2017), pp. 3536.
Global regularity results for four systems of 2D MHD equations
with partial dissipation
Jingna Li, BoQing Dong, Jiahong Wu
Abstract:
This article examines the global wellposedness 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 weaksensed 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 wellposedness; partial dissipation.
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Jingna Li
Department of Mathematics
Jinan University
Guangzhou 510632, China
email: jingna8005@hotmail.com


BoQing Dong
College of Mathematics and Statistics
Shenzhen University
Shenzhen 518060, China
email: bqdong@szu.edu.cn


Jiahong Wu
Department of Mathematics
Oklahoma State University
Stillwater, OK 74078, USA
email: jiahong.wu@okstate.edu

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