The 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 \(\Vert \omega ^{in}\Vert _{H^3_{x,y}\cap L^1_x H^3_y}\leqslant c\nu ^{\frac{1}{3}}\), 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 \(t\geqslant \nu ^{-\frac{1}{6}}\) 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.