Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition

TL;DR

This paper establishes quantitative stability results for 2D Couette flow on an infinite channel with non-slip boundary conditions, revealing new frequency division techniques and enhanced dissipation effects.

Contribution

It introduces a novel frequency division at 10 times the viscosity and refines estimates for stability and dissipation in the non-slip boundary context.

Findings

Stability in the low-frequency range $0 o |k|<1$ is established.

Space-time estimates lead to a nonlinear transition threshold of $rac{1}{2}$.

Enhanced dissipation occurs for frequencies $|k| extgreater u^{1-}$.

Abstract

In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel $\mathbb{R}\times [-1,1]$ with non-slip boundary condition. Compared to the case $\mathbb{T}\times [-1,1]$, we establish the stability in the context of long wave associated with the frequency range $0\leq |k|<1$ by developing the resolvent estimate argument. The new ingredient is to discover the key division point at $10\nu$ in the frequency interval $(0,1)$ by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval $(0,1)$, and then we establish the space-time estimates on the low-frequency $0\leq |k|\leq 10 \nu$ and the intermediate-frequency $ 10 \nu\leq |k|<1$, respectively. As an application of the space-time estimates, we obtain the nonlinear transition threshold to be $\gamma\leq\frac{1}{2}$.Meanwhile, we also show that when the frequencies $|k|\geq \nu^{1-}$, the enhanced dissipation effect occurs for the linearized Navier-Stokes equations.