Couette Flow with Robin Boundary Condition (I): the viscosity-independent friction

Abstract

This article is the first paper in the series. In this series of articles, we will examine the influence of the friction factor $\hat{I}\pm$ 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 ($u^2{|}_{y=\pm 1}=0$, $[\hat{I}\pm {\partial }_n{u}^1+u^1]{|}_{y=\pm 1}=f$), where $\hat{I}\pm$ is the friction factor and $n$ is the unit outer normal vector. In this article, we prove that for a given friction factor $\hat{I}\pm$, as long as the fluid viscosity coefficient $\hat{I}½\ll \hat{I}\pm$ is sufficiently small, the system is asymptotically stable if the initial perturbation satisfies $\Vert \ddot{I}{‰}_{\mathrm{in}}\Vert \le \hat{I}\mu \hat{I}{½}^{1/3}$. Moreover, inviscid damping and enhanced dissipation hold.