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 $\mathrm{\parallel }{\omega }^{in}{\mathrm{\parallel }}_{H_{x,y}^3\cap L_x^1{H}_y^3}\le c{\nu }^{\frac{1}{3}}$, the solution admits enhanced dissipation at $x$-frequencies $\mathrm{\mid }k\mathrm{\mid }\gg \nu$ 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 $(1+ν^{\frac13} t)^{-\frac14}e^{-νt}$ and the corresponding velocity exhibits the inviscid damping effect; (2) we introduce the dyadic decomposition for the long time scale $t\geq ν^{-\frac16}$ 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.