A new approach to bounds on mixing

TL;DR

This paper introduces a novel harmonic analysis-based method to establish bounds on mixing by incompressible flows, extending and unifying previous results that used different norms of the velocity field.

Contribution

It presents a new approach that recovers existing bounds and provides fresh insights into the problem of flow mixing, particularly relating to Bressan's conjecture.

Findings

Recovers most existing bounds on mixing using $L^1$ norms.

Introduces a harmonic analysis estimate from Seeger, Smart, and Street.

Provides new perspectives on bounds for incompressible flow mixing.

Abstract

We consider mixing by incompressible flows. In 2003, Bressan stated a conjecture concerning a bound on the mixing achieved by the flow in terms of an $L^1$ norm of the velocity field. Existing results in the literature use an $L^p$ norm with $p>1$. In this paper we introduce a new approach to prove such results. It recovers most of the existing results and offers new perpective on the problem. Our approach makes use of a recent harmonic analysis estimate from Seeger, Smart and Street.