A stochastic approach to enhanced diffusion

TL;DR

This paper demonstrates the sharpness of enhanced diffusion rates in shear and circular flows by providing initial data examples that saturate these rates, using a probabilistic approach to analyze dissipation.

Contribution

It introduces a probabilistic method to establish the optimality of enhanced diffusion rates in specific flow regimes, confirming the sharpness of previous theoretical results.

Findings

Examples saturate the proven diffusion rates

Probabilistic interpretation clarifies dissipation mechanisms

Results confirm sharpness of existing theorems

Abstract

We provide examples of initial data which saturate the enhanced diffusion rates proved for general shear flows which are H\"{o}lder regular or Lipschitz continuous with critical points, and for regular circular flows, establishing the sharpness of those results. Our proof makes use of a probabilistic interpretation of the dissipation of solutions to the advection diffusion equation.