Stability and optimal decay for the 3D magnetohydrodynamic equations with only horizontal dissipation

This paper develops an effective approach to establishing the optimal decay estimates on solutions of the 3D anisotropic magnetohydrodynamic (MHD) equations with only horizontal dissipation. As our first step, we prove the global existence and stability of solutions to the aforementioned MHD system emanating from any initial data with small H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document}-norm. Due to the lack of dissipation in the vertical direction, the large-time behavior does not follow from the classical approaches. The analysis of the nonlinear terms are much more difficult than in the case of full dissipation. In particular, we need to represent the MHD equations in an integral form, exploit cancellations and other properties such as the incompressibility in order to control terms involving vertical derivatives.