Exponentially mixing flows with slow enhanced dissipation

TL;DR

This paper constructs incompressible flows that are exponentially mixing yet do not exhibit the expected enhanced dissipation, challenging previous assumptions about the relationship between mixing properties and dissipation times.

Contribution

It introduces a family of $C^0$ exponentially mixing flows with dissipation times of order $1/$, and provides improved bounds on dissipation times based on mixing rates, including time-inhomogeneous cases.

Findings

Constructed flows with no enhanced dissipation despite exponential mixing.

Derived explicit bounds on dissipation times based on mixing rates.

Extended analysis to time-inhomogeneous mixing flows, including random cases.

Abstract

Consider a passive scalar which is advected by an incompressible flow $u$ and has small molecular diffusivity $\kappa$. Previous results show that if $u$ is exponentially mixing and $C^1$, then the dissipation time is $O(|\log \kappa|^2)$. We produce a family of incompressible flows which are $C^0$ and exponentially mixing, uniformly in $\kappa$; however have a dissipation time of order $1/\kappa$ (i.e. exhibits no enhanced dissipation). We also estimate the dissipation time of mixing flows, and obtain improved bounds in terms of the mixing rate with explicit constants, and allow for a time inhomogeneous mixing rate which is typical for random constructions of mixing flows.