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 $\mathrm{\mid }k\mathrm{\mid }\ge {\nu }^{-\frac{1}{3}}$, the perturbation decays on a time-scale proportional to ${\nu }^{-\frac{1}{2}}\mathrm{\mid }k{\mathrm{\mid }}^{-\frac{1}{2}}$. 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 at most of size ${\nu }^{\frac{7}{3}}$ in an anisotropic Sobolev space, then the size of the perturbation remains no more than twice the size of its initial value.