On the asymptotic stability of a stratified flow of the 2D non-resistive incompressible MHD equations

The exponential stability of a stratified flow of the two-dimensional incompressible MHD equations on a periodic domain T2=[0,1]×[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^2=[0,1]\times [0,1]$$\end{document} is presented. The stability mechanism which makes our results possible is due to the dissipation which arose from the mixing effect from the background flow. More precisely, 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 equation around the given stratified flow retains a non-local damping mechanism. We established the stability by combining with the energy estimates and the decay estimates.