The stability of Boussinesq equations with partial dissipation around the hydrostatic balance

This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena. The Boussinesq system focused on here is anisotropic, and involves only horizontal dissipation and thermal damping. In the 2D case ℝ², due to the lack of vertical dissipation, the stability and large-time behavior problems have remained open in a Sobolev setting. For the spatial domain \(\mathbb{T} \times \mathbb{R}\) $$\mathbb{T}\times ℝ$$ , this paper solves the stability problem and gives the precise large-time behavior of the perturbation. By decomposing the velocity u and temperature θ into the horizontal average ( \(\bar u,\bar \theta \) $$\overline{u},\overline{\theta }$$ ) and the corresponding oscillation ( \(\tilde u,\tilde \theta \) $$\tilde{u},\tilde{\theta }$$ ), we can derive the global stability in H² and the exponential decay of ( \(\tilde u,\tilde \theta \) $$\tilde{u},\tilde{\theta }$$ ) to zero in H¹. Moreover, we also obtain that ( \({{\bar u}_2},\bar \theta \) $${\overline{u}}_2,\overline{\theta }$$ ) decays exponentially to zero in H¹, and that \({{\bar u}_1}\) $${\overline{u}}_1$$ decays exponentially to \(\bar u(\infty )\) $$\overline{u}(\infty )$$ in H¹ as well; this reflects a strongly stratified phenomenon of buoyancy-driven fluids. In addition, we establish the global stability in H³ for the 3D case ℝ³.