We study diffusion and mixing in the incompressible Navier-Stokes equations and related scalar models. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequency. In turn, mixing acts to enhance the dissipative forces, giving rise to what we refer to as enhanced dissipation: this can be understood by the identification of a time-scale faster than the purely diffusive one. We will present two results:

(1) a general quantitative criterion that links mixing rates (in terms of decay of negative Sobolev norms) to enhanced dissipation time-scales, with several applications including passive scalar evolution in both planar and radial settings, fractional diffusion, and Anosov flows.

(2) a precise identification of the enhanced dissipation time-scale for the Navier-Stokes equations linearized around the Poiseuille flow, along with metastability results and nonlinear transition stability thresholds.