Mixing and diffusion for rough shear flows

TL;DR

This paper investigates how rough shear flows influence mixing and diffusion of passive scalars, revealing that roughness limits inviscid mixing enhancement but can significantly accelerate diffusion, challenging smooth-flow heuristics.

Contribution

It demonstrates that rough shear flows do not improve inviscid mixing beyond smooth flows but can greatly enhance diffusion, highlighting complex interactions in low-regularity settings.

Findings

Rough shear flows do not increase inviscid mixing rates beyond smooth flows.

Diffusion can be significantly faster in rough shear flows.

The interplay between mixing and dissipation in low regularity is more complex than in smooth flows.

Abstract

This article addresses mixing and diffusion properties of passive scalars advected by rough ($C^\alpha$) shear flows. We show that in general, one cannot expect a rough shear flow to increase the rate of inviscid mixing to more than that of a smooth shear without critical points. On the other hand, diffusion may be enhanced at a much faster rate. This shows that in the setting of low regularity, the interplay between inviscid mixing properties and enhanced dissipation is more intricate, and in fact contradicts some of the natural heuristics that are valid in the smooth setting.