We consider the solution to the two-dimensional Navier–Stokes equations around the Poiseuille flow (y²,0)
on T x R
with small viscosity v > 0
. Via a hypocoercivity argument, we prove that the
-dependent modes of the solution to the linear problem undergo the enhanced dissipation effect with a rate proportional to v^1/2


. Moreover, we study the nonlinear enhanced dissipation effect and we establish a transition threshold of v^2/3+


for initial data in L²
. Namely, when the L²
norm of the perturbation of the Poiseuille flow is size at most v^2/3+


, its size remains so for all times and the enhanced dissipation persists with a rate proportional to v^1/2


.