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

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.