Global Regularity for the 2D MHD and Tropical Climate Model with Horizontal Dissipation

This paper establishes the global regularity of classical solution to the 2D MHD system with only horizontal dissipation and horizontal magnetic diffusion in a strip domain T×R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}\times {{\mathbb {R}}}$$\end{document} when the initial data are suitable small. To prove this, we combine the Littlewood–Paley decomposition with anisotropic inequalities to establish a crucial commutator estimate. We also analyze the asymptotic behavior of the solution. In addition, the global existence and uniqueness of classical solution is obtained for the 2D simplified tropical climate model with only horizontal dissipation.