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

This article is the first paper in the series. In this series of articles, we will examine the influence of the friction factor $\alpha$ 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 ($u^2|_{y=\pm 1} = 0$, $[\alpha \partial_n u^1 + u^1]|_{y=\pm1} = f$), where $\alpha$ is the friction factor and $n$ is the unit outer normal vector. In this article, we prove that for a given friction factor $\alpha$, as long as the fluid viscosity coefficient $\nu\ll \alpha$ is sufficiently small, the system is asymptotically stable if the initial perturbation satisfies $\|\omega_{\rm in}\| \leq \epsilon \nu^{1/3}$. Moreover, inviscid damping and enhanced dissipation hold.