Stability and exponential decay for the 2D anisotropic Boussinesq equations near the hydrostatic equilibrium

Abstract

In this paper, we focus on the stability and long-time behavior problem for the 2D Boussinesq equations near the hydrostatic equilibrium with partial dissipation in the velocity and horizontal thermal diffusion. The lack of dissipation in the first component of the velocity and vertical thermal diffusion leads to the main difficulties. We establish the stability in $H^2$, and demonstrate the exponential decay of its oscillatory portion in the $H^1$.