Asymptotic stability of the three-dimensional Couette flow for the Stokes-transport equation

TL;DR

This paper proves the asymptotic stability of the three-dimensional Couette flow in a stratified fluid governed by the Stokes-transport equation, showing decay of perturbations under certain initial conditions.

Contribution

It demonstrates the stabilizing effect of combined inviscid damping and density damping in a stratified fluid, extending stability results to the Stokes-transport context.

Findings

Perturbed density remains close to a linearly decreasing function.

Velocity field converges to Couette flow with a rate of 1/⟨t⟩^3.

Stability holds for initial density close to -Y in Gevrey-$rac{1}{s}$ class.

Abstract

In this paper, we investigate the asymptotic stability of the three-dimensional Couette flow in a stratified fluid governed by the Stokes-transport equation. We observe that a similar lift-up effect to the three-dimensional Navier-Stokes equation near Couette flow destabilizes the system. We find that the inviscid damping type decay due to the Couette flow together with the damping structure caused by the decreasing background density stabilizes the system. More precisely, we prove that if the initial density is close to a linearly decreasing function in the Gevrey-$\frac{1}{s}$ class with $\frac{1}{2}< s\leq 1$, namely, $\|\varrho_{\mathrm{in}}(X,Y,Z)-(-Y)\|_{\mathcal{G}^{s}}\leq \epsilon$, then the perturbed density remains close to $-Y$. Moreover, the associated velocity field converges to Couette flow $(Y, 0, 0)^{\top}$ with a convergence rate of $\frac{1}{\langle t\rangle^3}$.