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 $m \in \mathbb{N}$, $m \geq 2$, respectively.