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

Abstract

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 $\Vert {\omega }^{in}{\Vert }_{H_{x,y}^3\cap L_x^1{H}_y^3}\le c{\nu }^{\frac{1}{3}}$, the solution admits enhanced dissipation at $x$-frequencies $|k|\gg \nu$ 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 $(1+{\nu }^{\frac{1}{3}}t{)}^{-\frac{1}{4}}{e}^{-\nu t}$ and the corresponding velocity exhibits the inviscid damping effect; (2) we introduce the dyadic decomposition for the long time scale $t\ge {\nu }^{-\frac{1}{6}}$ 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.

arXiv comment33 pages. This paper has submitted in July 25, 2025