Stability and sharp decay for 3D incompressible MHD system with fractional horizontal dissipation and magnetic diffusion

This paper aims as the stability and large-time behavior of 3D incompressible magnetohydrodynamic (MHD) equations with fractional horizontal dissipation and magnetic diffusion. By using the energy methods, we obtain that if the initial data are small enough 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}, then this system possesses a global solution, and whose horizontal derivatives decay at least at the rate of (1+t)-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+t)^{-\frac{1}{2}}$$\end{document}. Moreover, if we control the initial data further small in H3(R3)∩Hh-1(R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^3(\mathbb {R}^3)\cap H_h^{-1}(\mathbb {R}^3)$$\end{document}, the sharp decay of this solution and its first-order derivatives is established.