  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 linearized
problem. Here we show that the $x$-dependent modes of linear solutions decay on
a time-scale proportional to $\nu^{-1/2}|\log\nu|$. This effect is often
referred to as \emph{enhanced dissipation} or \emph{metastability} since it
gives a much faster decay than the regular dissipative time-scale $\nu^{-1}$
(this is also the time-scale on which the $x$-independent mode naturally
decays). We achieve this using an adaptation of the method of hypocoercivity.
  Our second result concerns the full nonlinear equations. We show that when
the perturbation from the Poiseuille flow is initially of size at most
$\nu^{3/4+}$, then it remains so for all time. Moreover, the enhanced
dissipation also persists in this scenario, so that the $x$-dependent modes of
the solution are dissipated on a time scale of order $\nu^{-1/2}|\log\nu|$.
This transition threshold is established by a bootstrap argument using the
semigroup estimate and a careful analysis of the nonlinear term in order to
deal with the unboundedness of the domain and the Poiseuille flow itself.
