Asymptotic stability of two-dimensional Couette flow in a viscous fluid

TL;DR

This paper investigates the nonlinear asymptotic stability of 2D Couette flow in viscous fluids, establishing conditions on initial perturbation size and regularity that guarantee stability as viscosity approaches zero.

Contribution

It provides a precise relationship between initial perturbation regularity and size, and the stability threshold for 2D Couette flow in the Navier-Stokes equations with small viscosity.

Findings

Stability holds for initial perturbations in Gevrey classes with specific size bounds.

The stability threshold depends on the viscosity raised to a power related to regularity.

The results clarify the interplay between perturbation size, regularity, and viscosity in flow stability.

Abstract

In this paper, we study the nonlinear asymptotic stability of Couette flow for the two-dimensional Navier-Stokes equation with small viscosity $\nu>0$ in $\mathbb{T}\times\mathbb{R}$. It's generally known the nonlinear asymptotic stability of the Couette flow depends closely on the size and regularity of the initial perturbation, which yields the stability threshold problem. This work studies the relationship between the size and the regularity of the initial perturbation that makes the nonlinear asymptotic stability holds. More precisely, we proved that if the initial perturbation is in some Gevrey-$\frac{1}{s}$ class with size $\epsilon\nu^{\beta}$ where $s\geq \frac{1-3\beta}{2-3\beta}$ and $\beta\in [0,\frac{1}{3}]$, then the nonlinear asymptotic stability holds.