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

TL;DR

This paper analyzes the stability of 2D Navier-Stokes equations in an infinite channel with Navier-slip boundary conditions, identifying the optimal transition threshold for enhanced dissipation and inviscid damping effects.

Contribution

It introduces a new vorticity decomposition and a dyadic long-time scale analysis to control echo cascades, advancing understanding of flow stability thresholds.

Findings

Enhanced dissipation at high frequencies for small initial perturbations

Effective inviscid damping observed under certain conditions

Novel decomposition and superposition methods for long-time behavior

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 $\|\omega^{in}\|_{H^3_{x,y}\cap L^1_x H^3_y}\leq c\nu^{\frac13}$, 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.