Asymptotic stability for two-dimensional Boussinesq systems around the Couette flow in a finite channel

In this paper, we study the asymptotic stability for the two-dimensional Navier-Stokes Boussinesq system around the Couette flow with small viscosity $\nu$ and small thermal diffusion $\mu$ in a finite channel. In particular, we prove that if the initial velocity and initial temperature $(v_{in},{\rho }_{in})$ satisfies $\mathrm{\parallel }{v}_{in}-(y,0){\mathrm{\parallel }}_{H_{x,y}^2}\le {\text{e}}_0\min \{\nu ,\mu {\}}^{\text{f}12}$ and $\mathrm{\parallel }{\rho }_{in}-1{\mathrm{\parallel }}_{H_x^1{L}_y^2}\le {\text{e}}_1\min \{\nu ,\mu {\}}^{\text{f}1112}$ for some small ${\text{e}}_0,{\text{e}}_1$ independent of $\nu ,\mu$, then for the solution of the two-dimensional Navier-Stokes Boussinesq system, the velocity remains within $O(\min \{\nu ,\mu {\}}^{\text{f}12})$ of the Couette flow, and approaches to Couette flow as $t\to \mathrm{\infty }$; the temperature remains within $O(\min \{\nu ,\mu {\}}^{\text{f}1112})$ of the constant $1$, and approaches to $1$ as $t\to \mathrm{\infty }$.