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 (R 2) × H s (R 2) 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 (R 2) , m ∈ N and in H ˙ m − 1 (R 2) with m ∈ N , m ≥ 2 , respectively. [ABSTRACT FROM AUTHOR]