Stability threshold of Couette flow for 2D Boussinesq equations in Sobolev spaces

Consider the nonlinear stability of the Couette flow in the Boussinesq equations with vertical dissipation on $T\times R$. We prove that if the initial perturbations $u_{\mathrm{in}}$ and ${\theta }_{\mathrm{in}}$ to the Couette flow $v_s={(y,0)}^{\top }$ and ${\theta }_s=1$, respectively, satisfy ${\Vert u_{\mathrm{in}}\Vert }_{H^{N+1}}+{\nu }^{-\frac{1}{2}}{\Vert {\theta }_{\mathrm{in}}\Vert }_{H^N}+{\nu }^{-\frac{1}{3}}{\Vert |{\partial }_x{|}^{\frac{1}{3}}\theta \Vert }_{H^N}\ll {\nu }^{\frac{1}{3}}$, $N>7$, then the resulting solution remains close to the Couette flow in $L^2$ at the same order for all time.