The purpose of this note is to address the stability and large-time behavior for the 2D Boussinesq system with vertical dissipation on $u_1$ and horizontal dissipation on $u_2$ 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.

This paper investigates the long time well-posedness of 2-D MHD boundary layer equation without resistivity. It is proved that if the initial data satisfies \begin{align*} \|(u_0,h_0-1)\|_{H_{\mu}^{3,0}}+\|(u_0,h_0-1)\|_{H_{\mu}^{1,2}}+\|(u_0,h_0-1)\|_{H_{\mu}^{2,1}}\le \varepsilon, \end{align*} then the life span of the solution is at least of order $\varepsilon^{-\f43}$.