Stability and Optimal Decay for the 3D Anisotropic MHD Equations

This paper focuses on the stability and decay rates of solutions to the three dimensional anisotropic magnetohydrodynamic equations with horizontal velocity dissipation and magnetic damping phenomenon. By fully exploiting the structure of the system, the energy methods and the method of bootstrapping argument, we prove the global stability of solutions to this system with initial data small in H3(R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{3}(\mathbb {R}<^>{3})$$\end{document}. Furthermore, we make use of the integral representation approach to obtain the optimal decay rates of these global solutions and their derivatives. This result along with its proof offers an effective approach to the large-time behavior on partially dissipated systems and reveals the stabilizing phenomenon exhibited by electrically conducting fluids.