Global well-posedness and stability of the 2D Boussinesq equations with partial dissipation near a hydrostatic equilibrium

The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the case of partial viscosity and without thermal diffusion for the initial data belonging to $H^{\delta }({\mathbb{R}}^2)\times H^s({\mathbb{R}}^2)$ for $\delta \in [s-1,s+1]$ if $s\in \mathbb{R},s>2$, for $\delta \in (1,s+1]$ if $s\in (0,2]$ and for $\delta \in [0,1]$ if $s=0$. In addition, if one has either horizontal or vertical thermal diffusion then the stability and large-time behavior are provided in $H^m({\mathbb{R}}^2)$, $m\in \mathbb{N}$ and in ${\dot{H}}^{m-1}({\mathbb{R}}^2)$ with $m\in \mathbb{N}$, $m\ge 2$, respectively.