On the Exponential Stability of a Stratified Flow to the 2D Ideal MHD Equations with Damping

We study the stability of a kind of stratified flow of the two-dimensional inviscid incompressible magnetohydrodynamics (MHD) equations with velocity damping. The exponential stability for the perturbation near certain stratified flow is investigated in a strip-type area R×[0,1]. Although the magnetic field potential is governed by a transport equation, by using the algebraic structure of the incompressible condition, it turns out that the linearized MHD equations around the given stratified flow retain a nonlocal damping mechanism. Carefully analyzing the nonlinear structure and introducing some suitable weighted energy norm, we get the exponential stability by combining the exponential decay in time in the lower order energy with that in the high order energy.