Stabilization of a Background Magnetic Field on a 2 Dimensional Magnetohydrodynamic Flow

Abstract

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.