The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation

This paper studies a special 2D anisotropic incompressible Boussinesq equation in ${\mathbb{T}}^2$ with $\mathbb{T}=[-\frac{1}{2},\frac{1}{2}]$ being a 1D periodic box. The system concerned here possesses vertical dissipation only in the vertical component of the velocity and vertical heat diffusion. When the buoyancy forcing is not present, the 2D Boussinesq equation is a 2D Navier-Stokes equation with vertical dissipation only in the vertical component. The stability and large-time behavior problem on the solutions to the 2D Navier-Stokes equation with only vertical or horizontal dissipation remains unknown. When coupled with the temperature, the global regularity to the system with vertical dissipation and vertical diffusion in ${\mathbb{R}}^2$ has been solved by Cao and Wu (Arch. Ration. Mech. Anal., 208(2013), 985-1004). The stability with horizontal dissipation and horizontal diffusion in the periodic domain $\mathbb{T}\times \mathbb{R}$ has also been established by Dong, Wu, Xu, and Zhu (Calc. Var. Partial Differential Equations, 60(2021)) recently. Now whether the solution of the 2D system remains stable has yet to be solved when the velocity has vertical dissipation only in the $u_2$ equation. This paper aims to solve the problem and investigates the stability and large-time behavior of the solution to the special 2D Boussinesq equations on perturbations near the hydrostatic equilibrium. The basic idea here is to decompose the physical quantity $f$ into its horizontal average, vertical average, and their corresponding oscillations. By establishing the strong Poincaré-type inequalities and several anisotropic inequalities related to the oscillations, we are able to obtain $H^2$-stability of the solution under the assumptions that the initial data is sufficiently small and obeys some symmetries. Furthermore, the exponential decay rates for the oscillation parts in $H^1$ are also established.