Stability and large-time behavior of the 2D Boussinesq equations with partial dissipation

Abstract This paper focuses on a special two-dimensional (2D) Boussinesq system modeling buoyancy driven fluids. It governs the motion of the velocity and temperature perturbations near the hydrostatic balance. This is a partially dissipated system with the velocity involving only the vertical dissipation. We are able to establish the global stability and the large-time behavior of the solutions. In particular, our results reveal that the buoyancy force actually stabilizes the fluids through the coupling and interaction. Without the coupling, the 2D Navier-Stokes equation with only vertical dissipation is not known to be stable. Mathematically the coupling allows us to deduce that both the velocity and the temperature obey degenerate damped wave equations, which generates the stabilization effect.