A dynamical approach to the study of instability near Couette flow

TL;DR

This paper establishes the sharp instability threshold for Couette flow in Navier-Stokes equations with small viscosity, introducing a new dynamical method that demonstrates transient growth without eigenvalue analysis.

Contribution

The paper presents a novel dynamical approach to determine the optimal instability threshold for Couette flow, improving understanding of flow stability near this classical shear flow.

Findings

Identifies $ u^{1/2}$ as the sharp stability threshold.

Proves transient exponential growth without eigenvalue dependence.

Provides a new method to locate eigenvalues for Rayleigh operator.

Abstract

In this paper, we obtain the optimal instability threshold of the Couette flow for Navier-Stokes equations with small viscosity $\nu>0$, when the perturbations are in the critical spaces $H^1_xL_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size $\nu^{\frac{1}{2}-\delta_0}$ with any small $\delta_0>0$, which implies that $\nu^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo-spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues.