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We consider solutions to the 2d Navier-Stokes equations on$\mathbb{T}\times\mathbb{R}$ close to the Poiseuille flow, with small viscosity$\nu>0$. Our first result concerns a semigroup estimate for the linearizedproblem. Here we show that the $x$-dependent modes of linear solutions decay ona time-scale proportional to $\nu^{-1/2}|\log\nu|$. This effect is oftenreferred to as \emph{enhanced dissipation} or \emph{metastability} since itgives a much faster decay than the regular dissipative time-scale $\nu^{-1}$(this is also the time-scale on which the $x$-independent mode naturallydecays). We achieve this using an adaptation of the method of hypocoercivity.  Our second result concerns the full nonlinear equations. We show that whenthe perturbation from the Poiseuille flow is initially of size at most$\nu^{3/4+}$, then it remains so for all time. Moreover, the enhanceddissipation also persists in this scenario, so that the $x$-dependent modes ofthe solution are dissipated on a time scale of order $\nu^{-1/2}|\log\nu|$.This transition threshold is established by a bootstrap argument using thesemigroup estimate and a careful analysis of the nonlinear term in order todeal with the unboundedness of the domain and the Poiseuille flow itself.

Enhanced dissipation in the Navier-Stokes equations near the Poiseuille flow