Enhanced dissipation and stability of Poiseuille flow for two-dimensional Boussinesq system

We investigate the nonlinear stability problem for the two-dimensional Boussinesq system around the Poiseuille flow in a finite channel. The system has the characteristic of Navier-slip boundary condition for the velocity and Dirichlet boundary condition for the temperature, with a small viscosity $\nu$ and small thermal diffusion $\mu,$ respectively. More precisely, we prove that if the initial velocity and initial temperature satisfies$$||u_{0}-(1-y^2,0) ||_{H^{\frac{7}{2}+}}\leq c_0\min\left\lbrace \mu,\nu\right\rbrace ^{\frac{2}{3}}$$ and $$ ||\theta_{0}||_{H^1}+|||D_x|^{\frac{1}{8}}\theta_{0}||_{H^1}\leq c_1\min\left\lbrace \mu,\nu\right\rbrace ^{{\frac{31}{24}}}$$ for some small constants $c_0$ and $c_1$ which are both independent of $\mu,\nu$, then we can reach the conclusion that the velocity remains within $O\left( \min\left\lbrace \mu,\nu\right\rbrace ^{\frac{2}{3}}\right) $ of the Poiseuille flow; the temperature remains $O\left( \min\left\lbrace \mu,\nu\right\rbrace ^{\frac{31}{24}}\right) $ of the constant 0, and approaches to 0 as $t\rightarrow\infty.$