Mixing for generic passive scalars by incompressible flows

TL;DR

This paper investigates the genericity of mixing in incompressible flows, showing classical criteria are fragile under perturbations and developing a Young-measure framework to characterize mixing for typical passive scalars.

Contribution

It introduces a Young-measure theory for $L^ abla$ data and links generic mixing to the non-precompactness of flow maps, advancing the understanding of mixing in fluid dynamics.

Findings

Classical mixing criteria fail under small velocity perturbations.

A Young-measure approach characterizes which passive scalars mix.

Generic mixing is equivalent to the existence of a single mixed density.

Abstract

Mixing by incompressible flows is a ubiquitous yet incompletely understood phenomenon in fluid dynamics. While previous studies have focused on optimal mixing rates, the question of its genericity, i.e., whether mixing occurs for typical incompressible flows and typical initial data, remains mathematically unclear. In this paper, it is shown that classical mixing criteria, e.g. topological mixing or non-precompactness in $L^2$ for all nontrivial densities, fail to persist under arbitrarily small perturbations of velocity fields. A Young-measure theory adapted to $L^\infty$ data is then developed to characterize exactly which passive scalars mix. As a consequence, the existence of a single mixed density is equivalent to mixing for generic bounded data, and this equivalence is further tied to the non-precompactness of the associated measure-preserving flow maps in $L^p$. These results provide a foundation for a general theory of generic mixing in non-autonomous incompressible flows.