New energy method in the study of the instability near Couette flow

Hui Li, Nader Masmoudi, Weiren Zhao

Published 2022-03-21Version 1

In this paper, we provide a new energy method to study the growth mechanism of unstable shear flows. As applications, we prove two open questions about the instability of shear flows. First, we obtain the optimal stability threshold of the Couette flow for Navier-Stokes equations with small viscosity $\nu>0$, when the perturbations are in critical spaces. More precisely, we 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. Second, we study the instability of the linearized Euler equations around shear flows that are near Couette flow. We prove the existence of a growing mode for the corresponding Rayleigh operator and give a precise location of the eigenvalues. We think our method has a lot of possible other applications.