The optimal transition threshold for the 2D Couette flow in the infinite channel

We investigate the stability of the 2-D Navier-Stokes equations in the infinite channel $\mathbb{R}\times [-1,1]$ with the Navier-slip boundary condition. We show that if the initial perturbations $\omega^{in}$ around the Couette flow satisfy $\|\omega^{in}\|_{H^3_{x,y}\cap L^1_x H^3_y}\leq c\nu^{\frac13}$, the solution admits enhanced dissipation at $x$-frequencies $|k|\gg \nu$ and inviscid damping effect. The key contributions lie in two parts: (1) we adopt the new decomposition of the vorticity $\omega=\omega_{L}+\omega_e$, where $\omega_L$ effectively captures a ``weak" enhanced dissipation $(1+\nu^{\frac13} t)^{-\frac14}e^{-\nu t}$ and the corresponding velocity exhibits the inviscid damping effect; (2) we introduce the dyadic decomposition for the long time scale $t\geq \nu^{-\frac16}$ and apply the ``infinite superposition principle" to the equation for $\omega_e$ in order to control the growth induced by echo cascades, which appears to be novel and may hold independent significance.