Exponential mixing by shear flows

TL;DR

This paper proves a version of Bressan's mixing conjecture for shear flows and constructs a simple, optimally mixing shear flow that alternates randomly between two shears, advancing understanding of mixing dynamics.

Contribution

It establishes a shear flow version of Bressan's mixing conjecture and provides a simple example of an optimally mixing shear flow with random switching.

Findings

Proved a shear flow version of Bressan's mixing conjecture.

Constructed a simple, optimally mixing shear flow.

Demonstrated random switching between two shears achieves optimal mixing rate.

Abstract

We prove a version of Bressan's mixing conjecture where the advecting field is constrained to be a shear at each time. Also, inspired by recent work of Blumenthal, Coti Zelati and Gvalani, we construct a particularly simple example of a shear flow which mixes at the optimal rate. The constructed vector field alternates randomly in time between just two distinct shears.