Stabilizing Effect of the Magnetic Field and Large-Time Behavior of 2D Incompressible MHD System with Vertical Dissipation

The stabilizing and damping phenomenon of a background magnetic field on electrically conducting fluids has been observed in various physical experiments and numerical simulations. This paper establishes this observation as mathematically rigorous decay results on a 2D magnetohydrodynamic (MHD) system with only partial dissipation. Without the magnetic field, the fluid velocity obeys a 2D anisotropic Navier-Stokes equation and is not known to be stable in the Sobolev setting H-2 due to the potential double exponential growth of its H-2-norm in time. However, when coupled with the magnetic field in the MHD system concerned here, we show that the H-2-norm of any perturbation near a background magnetic field actually decays algebraically in time. This result demonstrates that the magnetic field indeed stabilizes and damps the electrically conducting fluids. Mathematically this result along with its proof offers a new and effective approach to the large-time behavior on partially dissipated systems of partial differential equations. Existing methods are mostly designed for systems with full dissipation and do not apply when the dissipation is anisotropic.