Asymptotic stability threshold of the 2-D monotone shear flow with no-slip boundary condition

In this paper, we investigate the asymptotic stability threshold problem for the 2-D Navier-Stokes equations in a finite channel with no-slip boundary conditions, around monotone shear flow $(U(t,y),0)$. We establish that the flow is asymptotically stable under perturbations satisfying $\|u^{\mathrm{in}}\|_{H^2}\leq c\nu^{\frac12}$. To achieve the stability threshold $\nu^{\frac{1}{2}}$, the key ingredients of the proof include: sharp resolvent estimates for the vorticity based on weak-type resolvent bounds; weighted space-time estimates for the vorticity; pointwise estimates for the velocity. Furthermore, we handle the nonlinear term through a divergence formulation, which facilitates the sharp application of the aforementioned space-time estimates.