Influence of a background magnetic field on a 2D magnetohydrodynamic flow

Abstract

Physical experiments and numerical simulations have demonstrated that background magnetic fields stabilize electrically conducting fluids. This paper establishes these observations as mathematically rigorous facts on a 2D magnetohydrodynamic (MHD) system. This system is anisotropic with the velocity equation involving only the vertical dissipation. Flows governed by the 2D Navier–Stokes equations with only vertical dissipation are not known to be stable. Under the influence of a background magnetic field, the velocity field is shown here to stabilize and decay in time through the coupling and the interaction. Mathematically we reduce the MHD system concerned here to a system of degenerate and damped wave equations and exploit the smoothing and stabilizing effects of the wave structure. We are able to prove that any perturbation near a background magnetic field remains asymptotically stable. In addition, certain explicit large time behavior is also established.