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 \(t\to\infty\).