AI 中文总结
研究有界周期通道中罗宾型边界条件下库埃特流稳定性，证明给定摩擦系数α，当流体粘性系数\(\nu\ll \alpha\)足够小且初始扰动满足一定条件时系统渐近稳定，有无粘阻尼和增强耗散。
AI 中文摘要
本文是系列文章中的第一篇。在这一系列文章中，我们将研究固 - 流边界处摩擦系数α对库埃特流稳定性的影响。具体而言，我们考虑在有界周期通道\(\mathbb{T} \times [-1,1]\)中，在罗宾型边界条件（\(u^2|_{y=\pm 1} = 0\)，\([\alpha \partial_n u^1 + u^1]|_{y=\pm1} = f\)）下库埃特流的稳定性，其中α为摩擦系数，n为单位外法向量。本文证明，对于给定的摩擦系数α，只要流体粘性系数\(\nu\ll \alpha\)足够小，当初始扰动满足\(\|\omega_{\rm in}\| \leq \epsilon \nu^{1/3}\)时，系统是渐近稳定的。此外，无粘阻尼和增强耗散成立。
英文摘要
This article is the first paper in the series. In this series of articles, we will examine the influence of the friction factor $α$ at the solid--fluid boundary on the stability of Couette flow. Specifically, we consider the stability of Couette flow in a bounded periodic channel $\mathbb{T} \times [-1,1]$ under Robin-type boundary conditions ($u^2|_{y=\pm 1} = 0$, $[α\partial_n u^1 + u^1]|_{y=\pm1} = f $), where $α$ is the friction factor and $n$ is the unit outer normal vector. In this article, we prove that for a given friction factor $α$, as long as the fluid viscosity coefficient $ν\ll α$ is sufficiently small, the system is asymptotically stable if the initial perturbation satisfies $\|ω_{\rm in}\| \leq εν^{1/3}$. Moreover, inviscid damping and enhanced dissipation hold.
