Transition Threshold for Strictly Monotone Shear Flows in Sobolev Spaces

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 $\nu$. We establish nonlinear stability in $H^s$ for $s\ge 2$ with a threshold of size $\epsilon {\nu }^{1\mathrm{/}3}$ for time smaller than $c_{\ast }{\nu }^{-1}$ with $\epsilon ,c_{\ast }\ll 1$. Additionally, we demonstrate nonlinear inviscid damping and enhanced dissipation.