Stability of Poiseuille Flow of Navier-Stokes Equations on $\mathbb{R}^2$

We consider solutions to the Navier-Stokes equations on $\mathbb{R}^2$ close to the Poiseuille flow with viscosity $0< \nu < 1$. For the linearized problem, we prove that when the $x$-frequency satisfy $|k| \ge \nu^{-\frac{1}{3}}$, the perturbation decays on a time-scale proportional to $\nu^{-\frac{1}{2}}|k|^{-\frac{1}{2}}$. Since it decays faster than the heat equation, this phenomenon is referred to as enhanced dissipation. Then we concern the non-linear equations. We show that if the initial perturbation $\omega_{in}$ is of size at most $\nu^\frac{7}{12}$ in an anisotropic Sobolev space, then it remains so for all time.