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 ($u^2{\mathrm{\mid }}_{y=\pm 1}=0$, $[\alpha {\mathrm{\partial }}_n{u}^1+u^1]{\mathrm{\mid }}_{y=\pm 1}=f$), 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 $\mathrm{\parallel }{\omega }_{\mathrm{i}\mathrm{n}}\mathrm{\parallel }\le \epsilon {\nu }^{1\mathrm{/}3}$. Moreover, inviscid damping and enhanced dissipation hold.