Transition Threshold for Strictly Monotone Shear Flows in Sobolev Spaces

Abstract

We study the stability of spectrally stable, strictly monotone, smooth shear flows in the 2D Navier-Stokes equations on $\mathbb{T}\times \mathbb{R}$ with small viscosity $\hat{I}½$. We establish nonlinear stability in $H^s$ for $s\ge 2$ with a threshold of size $\hat{I}\mu \hat{I}{½}^{1/3}$ for time smaller than $c_{\ast }\hat{I}{½}^{-1}$ with $\hat{I}\mu ,c_{\ast }\ll 1$. Additionally, we demonstrate nonlinear inviscid damping and enhanced dissipation.