Stability threshold for 2D shear flows near Couette of the Navier-Stokes equation

TL;DR

This paper establishes a stability threshold for 2D shear flows near Couette flow in the Navier-Stokes equations at high Reynolds numbers, showing solutions remain close to shear flows over long times.

Contribution

It provides a quantitative stability threshold in Sobolev norm for 2D shear flows near Couette flow at high Reynolds number, extending understanding of flow stability.

Findings

Solutions stay close to shear flows for long times when initial perturbation is below threshold.

The stability threshold scales as Re^{-1/3} in Sobolev norm.

Flow approaches a nearby shear flow as time progresses.

Abstract

In this paper, we consider the stability threshold of the 2D shear flow $(U(y),0)^{\top}$ of the Navier-Stokes equation at high Reynolds number $Re$. When the shear flow is near in Sobolev norm to the Couette flow $(y,0)^{\top}$ in some sense, we prove that if the initial data $u_0$ satisfies $\|u_0-(U(y),0)^{\top}\|\leq \epsilon Re^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to the Couette flow for $t\gg Re^{1/3}$, as $t\to\infty$.