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 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.
Categories: math.AP
