Stability of hydrostatic equilibrium to the 2D Boussinesq systems with partial dissipation

Abstract

The stability problem on perturbations near the hydrostatic equilibrium of the Boussinesq equations has recently attracted considerable attention and there are substantial developments. This paper establishes the global $H^1$-stability for a 2D Boussinesq system with partial dissipation. When there is no thermal diffusion, the stability of the temperature gradient remains an open problem. This paper assesses the stability of the temperature gradient when there is a horizontal thermal diffusion.