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 $\|v_{in}-(y,0)\|_{H_{x,y}^2}\leq \e_0 \min\{\nu,\mu\}^{\f12}$ and $\|\rho_{in}-1\|_{H_x^{1}L_y^2}\leq \e_1 \min\{\nu,\mu\}^{\f{11}{12}}$ for some small $\e_0,\e_1$ independent of $\nu, \mu$, then for the solution of the two-dimensional Navier-Stokes Boussinesq system, the velocity remains within $O(\min\{\nu,\mu\}^{\f12})$ of the Couette flow, and approaches to Couette flow as $t\to\infty$; the temperature remains within $O(\min\{\nu,\mu\}^{\f{11}{12}})$ of the constant $1$, and approaches to $1$ as $t\to\infty$.