Stability threshold for two-dimensional Boussinesq systems near-Couette shear flow in a finite channel

In this paper, we investigate the stability threshold problem of the two-dimensional Navier-Stokes Boussinesq(NSB) equations in a finite channel $\T \times [-1,1]$, focusing on the stability around the near Couette shear flow $(U(y), 0)$, assuming the Navier slip boundary conditions are satisfied. In particular, when the initial data for the vorticity resides in an anisotropic Sobolev space of size $O(\min \{ \mu^{\frac{1}{2}}, \nu^{\frac{1}{2}}\})$, and the initial perturbation of the temperature resides in an anisotropic Sobolev space of size $O(\min \{ \mu, \nu\})$, we derive the nonlinear enhanced dissipation effect and the inviscid damping effect for the NSB system.