Stabilizing effect of a background magnetic field on the 2D damped wave-type MHD equations

Abstract

The stabilizing effect of a background magnetic field on electrically conducting fluids has been rigorously established for the standard MHD equations. This paper extends this theory to the more physically accurate damped wave-type MHD equations, where the induction equation is hyperbolic-parabolic and the velocity field has only vertical damping with no dissipation. These two features make the stability analysis harder than in the standard MHD setting. To overcome these difficulties, we design an energy functional exploiting the anisotropic structure, and discover a remarkable cancellation between the two most dangerous nonlinear terms by exploiting the full algebraic structure of the coupled system. As a consequence, we prove that any small perturbation near the background magnetic field is globally stable and establish optimal decay rates consistent with the 2D heat equation. To the best of our knowledge, this is the first rigorous stability result for the damped wave-type MHD equations near a background magnetic field.