Enhanced dissipation and stability threshold for the Navier--Stokes equations near Poiseuille flow

We study the nonlinear stability of the quadratic plane Poiseuille flow $U(y)=(y^2,0)$ for the two-dimensional incompressible Navier--Stokes equations on $\mathbb T\times\mathbb R$. We prove that for initial perturbations of size $O(ν^{2/3})$ in a $ν$-dependent weighted Sobolev space, the flow is gl