On the relation between enhanced dissipation time-scales and mixing rates

TL;DR

This paper establishes a precise link between mixing rates and enhanced dissipation time-scales in fluid dynamics, demonstrating that certain flows can induce faster-than-polynomial dissipation, with broad applications.

Contribution

It introduces a general framework connecting mixing decay rates to enhanced dissipation time-scales and applies it to novel flow examples like contact Anosov flows.

Findings

Contact Anosov flows exhibit logarithmically fast dissipation.

Enhanced dissipation can occur faster than polynomial time.

The approach applies to passive scalar evolution and fractional diffusion.

Abstract

We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a time-scale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation time-scales. The proofs are based on a contradiction argument that takes advantage of the cascading mechanism due to mixing, an estimate of the distance between the inviscid and viscous dynamics, and of an optimization step in the frequency cut-off. Thanks to the generality and robustness of our approach, we are able to apply our abstract results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation time-scales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation time-scale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion.