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 $\nu^{ \frac{ 1}{ 2}}$.
Categories: math.AP