Stability for the 2D MHD equations with horizontal dissipation

Abstract

In this paper we consider the following 2D MHD system with horizontal dissipation in a strip domain $\mathit{T}\times \mathbb{R}$ . $$\begin{array}{c}\left\{\begin{array}{c}{\partial }_{\mathit{t}}\mathit{u}+\mathit{u}·\nabla \mathit{u}+{\partial }_{11}\mathit{u}+\nabla \mathit{p}=\mathit{b}·\nabla \mathit{b}\mathrm{,}\hfill \\ {\partial }_{\mathit{t}}\mathit{b}+\mathit{u}·\nabla \mathit{b}+{\partial }_{11}\mathit{b}=\mathit{b}·\nabla \mathit{u}\mathrm{,}\hfill \\ \nabla ·\mathit{u}=\nabla ·\mathit{b}=0\mathrm{,}\hfill \\ \mathit{u}\mathrm{(}0\mathrm{)}={\mathit{u}}_0\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}\mathit{b}\mathrm{(}0\mathrm{)}={\mathit{b}}_0\mathrm{(}\mathit{x}\mathrm{)}.\hfill \end{array}\right.\end{array}$$ A bootstrapping argument together with a more accurate energy functional is employed in order to get the stability for the above system. Moreover, using a suitable transform, we also investigate the 2D MHD system with vertical dissipation in a strip domain $\mathbb{R}\times \mathit{T}$ .