Stability for a system of the 2D magnetohydrodynamic equations with partial dissipation

Abstract

This paper examines the stability problem on perturbations near a physically important steady state solution of the 2D magnetohydrodynamic (MHD) system with only partial dissipation. We obtain two main results. The first assesses the asymptotic linear stability with explicit decay rates while the second affirms the global stability in the Sobolev space $H^1$ setting.