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

Abstract

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 globally nonlinearly stable. In particular, the nonzero streamwise modes experience enhanced dissipation and decay exponentially on the characteristic time scale $O(ν^{-1/2})$, while the zero mode remains uniformly controlled. Thus, in this weighted Sobolev topology, \(O(ν^{2/3})\) is a sufficient stability scale without logarithmic loss.