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^δ(R²)×H^s(R²) for δ∈[s−1,s+1] if s∈R,s>2, for δ∈(1,s+1] if s∈(0,2] and for δ∈[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(R²), m∈N and in H˙^m−1(R²) with m∈N, m≥2, respectively.