SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5001-5035, January 2020.
This paper rigorously establishes the stabilization effect of a background magnetic field on electrically conducting fluids, a phenomenon that has been widely observed in physical experiments and numerical simulations. This study is based on a 2 dimensional (2D) magnetohydrodynamic (MHD) system in which the velocity equation involves no dissipation and there is only damping in the vertical component equation. Without the magnetic field, the corresponding vorticity equation is a 2D Euler-like equation with an extra Riesz transform type term. The global in time regularity and the stability near the trivial solution are well known open problems. When coupled with the magnetic field through the MHD system, the background magnetic field stabilizes the fluid, and the velocity as well as the vorticity remain small if they are initially so and decay algebraically in time. To overcome the difficulties due to the lack of full dissipation or damping, we construct suitable Lyapunov functionals and reduce the system to wave type equations.

中文翻译：

---

二维磁流体动流上背景磁场的稳定

SIAM数学分析杂志，第52卷，第5期，第5001-5035页，2020年1月。
本文严格建立了背景磁场对导电流体的稳定作用，这种现象已在物理实验和数值模拟中广泛观察到。这项研究基于二维（2D）磁流体动力学（MHD）系统，其中速度方程不涉及耗散，垂直分量方程中仅包含阻尼。在没有磁场的情况下，相应的涡旋方程为带有额外Riesz变换类型项的二维Euler式方程。全局时间规律性和琐碎解附近的稳定性是众所周知的开放问题。当通过MHD系统与磁场耦合时，背景磁场使流体稳定，如果速度和旋涡起初是很小的，并且随着时间代数地衰减，则速度和涡度仍然很小。为了克服由于缺乏充分的耗散或阻尼而造成的困难，我们构造了合适的Lyapunov函数并将系统简化为波动型方程。