Transition threshold for the 3D Couette flow in Sobolev space

TL;DR

This paper establishes a precise transition threshold for the 3D Couette flow in Sobolev space, showing that initial velocities within a certain Sobolev norm bound lead to global stability at high Reynolds numbers.

Contribution

It proves the transition threshold conjecture in Sobolev space for 3D Couette flow, linking initial perturbation size to flow stability at high Reynolds numbers.

Findings

Global stability for initial perturbations below the threshold

Confirmation of the transition threshold conjecture

Quantitative relation between initial data and flow stability

Abstract

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number $\text{Re}$. It was proved that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0\text{Re}^{-1}$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture in physical literatures.