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

Qionglei Chen, Zhen Li, Changxing Miao

TL;DR

We study the 2D Navier–Stokes system in the infinite channel with Navier-slip boundaries, perturbing Couette flow. The key strategy is to decompose the vorticity as $\omega=\omega_L+\omega_e$, solve the linear model for $\omega_L$ to obtain enhanced dissipation and inviscid damping, and then control the nonlinear remainder $\omega_e$ with a dyadic time-slicing and an infinite superposition principle. Under the smallness condition $\|\omega^{in}\|_{H^3_{x,y}\cap L^1_x H^3_y}\leq c\nu^{1/3}$, the paper proves global stability and achieves an optimal transition threshold characterized by $\gamma=\tfrac{1}{3}$, with precise decay and damping bounds. The introduction of the dyadic time scales and the infinite superposition framework to manage echo cascades provides a robust toolkit that may extend to other shear flows and boundary conditions.

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 $ω^{in}$ around the Couette flow satisfy $\|ω^{in}\|_{H^3_{x,y}\cap L^1_x H^3_y}\leq cν^{\frac13}$, the solution admits enhanced dissipation at $x$-frequencies $|k|\gg ν$ and inviscid damping effect. The key contributions lie in two parts: (1) we adopt the new decomposition of the vorticity $ω=ω_{L}+ω_e$, where $ω_L$ 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 $ω_e$ in order to control the growth induced by echo cascades, which appears to be novel and may hold independent significance.

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

TL;DR

We study the 2D Navier–Stokes system in the infinite channel with Navier-slip boundaries, perturbing Couette flow. The key strategy is to decompose the vorticity as , solve the linear model for to obtain enhanced dissipation and inviscid damping, and then control the nonlinear remainder with a dyadic time-slicing and an infinite superposition principle. Under the smallness condition , the paper proves global stability and achieves an optimal transition threshold characterized by , with precise decay and damping bounds. The introduction of the dyadic time scales and the infinite superposition framework to manage echo cascades provides a robust toolkit that may extend to other shear flows and boundary conditions.

Abstract

We investigate the stability of the 2-D Navier-Stokes equations in the infinite channel with the Navier-slip boundary condition. We show that if the initial perturbations around the Couette flow satisfy , the solution admits enhanced dissipation at -frequencies and inviscid damping effect. The key contributions lie in two parts: (1) we adopt the new decomposition of the vorticity , where effectively captures a ``weak" enhanced dissipation and the corresponding velocity exhibits the inviscid damping effect; (2) we introduce the dyadic decomposition for the long time scale and apply the ``infinite superposition principle" to the equation for in order to control the growth induced by echo cascades, which appears to be novel and may hold independent significance.
Paper Structure (9 sections, 19 theorems, 303 equations)

This paper contains 9 sections, 19 theorems, 303 equations.

Table of Contents

  1. Introduction
  2. A priori estimates
  3. The proof of Theorem \ref{['Th1']}
  4. The estimates for the linear equation
  5. The estimate for $\omega_{e}$
  6. The estimate of $\omega_e$ for $t\leqslant \nu^{-\frac{1}{6}}$
  7. The estimate of $\omega_e$ for $t\geqslant \nu^{-\frac{1}{6}}$
  8. The proof of Theorems \ref{['Th1']}
  9. some basic estimates

Key Result

Theorem 1.1

Assume that $\omega^{in}\in H^3_{x,y}\cap L^1_x H^3_y$ and $0<\nu< 1$. There exists a small constant $c>0$, independent of $\nu$, such that if the solution to the system om is global in time. Moreover, let $\mathcal{M}=\mathcal{M}(D_x)$ and $\mathcal{M}_1(D_x)$ be the Fourier multipliers defined as where $\chi$ is a cut-off function with $\chi(k)=1$ if $|k|\leqslant \frac{1}{2}$ and $\chi(k)=0$

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 27 more