Stability and large-time behavior for the 2D Boussineq system with horizontal dissipation and vertical thermal diffusion

Abstract

This paper solves the stability and large-time behavior problem on perturbations near the hydrostatic equilibrium of the two-dimensional Boussinesq system with horizontal dissipation and vertical thermal diffusion. The spatial domain $\Omega$ is ${\mathbb {T}} \times {\mathbb {R}}$ with ${\mathbb {T}}=[0,1]$ being the 1D periodic box and ${\mathbb {R}}$ being the whole line. The results presented in this paper establish the observed stabilizing phenomenon and stratifying patterns of the buoyancy-driven fluids as mathematically rigorous facts. The stability and large-time behavior problem concerned here is difficult due to the lack of the vertical dissipation and horizontal thermal diffusion. To make up for the missing regularization, we exploit the smoothing and stabilizing effect due to the coupling and interaction between the temperature and the fluids. By constructing suitable energy functional and introducing the orthogonal decomposition of the velocity and the temperature into their horizontal averages and oscillation parts, we are able to establish the stability in the Sobolev space $H^2$ and obtain algebraic decay rates for the oscillation parts in the $H^1$-norm.