Stability and large time behavior of the 2D Boussinesq equations with velocity supercritical dissipation

This paper studies the 2D Boussinesq equations with velocity supercritical ${\Lambda }^{\alpha }(0<\alpha <1)$ dissipation and temperature damping near the hydrostatic equilibrium. We are able to establish the global stability and the large time behavior of the solution. By introducing a diagonalization process to eliminate the linear terms, the temporal decay rate of the global solution is obtained. Furthermore, when $\alpha =0$, the velocity dissipation term becomes the velocity damping term, and the solution has an exponential decay.