Enhanced dissipation in the Navier-Stokes equations near the Poiseuille flow

We study an instance of diffusion and mixing in the incompressible 2d Navier-Stokes equations. Broadly put, mixing causes a transfer of energy to high frequencies. This adds to the dissipative forces, giving rise to what we refer to as "enhanced dissipation". While such effects are well known in the physics literature and experimentally studied, mathematically rigorous results are much more scarce.

This talk is based on recent work for the case of the Poiseuille flow. We first demonstrate enhanced dissipation for the Navier-Stokes equations linearized around this flow by identifying a time scale that is faster than the purely diffusive one. Subsequently, we give a transition stability threshold below which this behavior persists in the full, nonlinear setting.