Stability and large-time behavior of the 2D Boussinesq system with mixed partial dissipations near hydrostatic equilibrium

The purpose of this note is to address the stability and large-time behavior for the 2D Boussinesq system with vertical dissipation on u1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_1$$\end{document} and horizontal dissipation on u2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_2$$\end{document} near a hydrostatic equilibrium. Meanwhile, the decay estimates of that system are also presented. Finally, we also obtain the decay rates of the solution to the corresponding linearized equation of the Boussinesq system.