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 $s \geq 2$ with a threshold of size $\epsilon \nu^{1/3}$ for time smaller than $c_*\nu^{-1}$ with $\epsilon, c_* \ll 1$. Additionally, we demonstrate nonlinear inviscid damping and enhanced dissipation.