Enhanced Dissipation and Transition Threshold for the Poiseuille Flow in a Periodic Strip

We consider the solution to the 2D Navier-Stokes equations around the Poiseuille flow $(y^2,0)$ on $\mathbb{T}\times\mathbb{R}$ with small viscosity $\nu>0$. Via a hypocoercivity argument, we prove that the $x-$dependent modes of the solution to the linear problem undergo the enhanced dissipation effect with a rate proportional to $\nu^{\frac{1}{2}}$. Moreover, we study the nonlinear enhanced dissipation effect and we establish a transition threshold of $\nu^{\frac{2}{3}+}$. Namely, when the perturbation of the Poiseuille flow is size at most $\nu^{\frac{2}{3}+}$, its size remains so for all times and the enhanced dissipation persists with a rate proportional to $\nu^{\frac{1}{2}}$.