On the exponential stability of a stratified flow to the 2D IDEAL MHD equations with damping

TL;DR

This paper proves the exponential stability of certain stratified flows in 2D ideal MHD equations with damping, revealing a non-local damping mechanism that ensures decay of perturbations.

Contribution

It demonstrates exponential stability for stratified flows in 2D ideal MHD with damping, utilizing algebraic structures and weighted energy norms to analyze nonlinear stability.

Findings

Exponential decay of perturbations near stratified flow

Identification of a non-local damping mechanism

Stability proven in a strip-type domain

Abstract

We study the stability of a type of stratified flows of the two dimensional inviscid incompressible 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 filed 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 non-local damping mechanism. After carefully analyzing the non-linear structure and introducing some suitable weighted energy norms, we get the exponential stability by combining the exponential decay in time in the lower order energy with that in the high order energy.