Asymptotic Stability of the two-dimensional Couette flow for the Stokes-transport equation in a finite channel

TL;DR

This paper proves the asymptotic stability of Couette flow in a 2D Stokes-transport system with moving boundaries, showing decay rates for velocity perturbations when initial density is close to constant.

Contribution

It establishes the stability of stratified Couette flow in a finite channel under small Gevrey-3 class perturbations, with explicit decay rates for velocity components.

Findings

Horizontal velocity perturbation decays as 1/⟨t⟩^3

Vertical velocity perturbation decays as 1/⟨t⟩^4

Stability holds for density close to constant in Gevrey-3 class

Abstract

We study the Stokes-transport system in a two-dimensional channel with horizontally moving boundaries, which serves as a reduced model for oceanography and sedimentation. The density is transported by the velocity field, satisfying the momentum balance between viscosity, pressure, and gravity effects, described by the Stokes equation at any given time. Due to the presence of moving boundaries, stratified densities with the Couette flow constitute one class of steady states. In this paper, we investigate the asymptotic stability of these steady states. We prove that if the stratified density is close to a constant density and the perturbation belongs to the Gevrey-3 class with compact support away from the boundary, then the velocity will converge to the Couette flow as time approaches infinity. More precisely, we prove that the horizontal perturbed velocity decays as $\frac{1}{\langle t\rangle^3}$ and the vertical perturbed velocity decays as $\frac{1}{\langle t\rangle^4}$.