Maximal mixing by incompressible fluid flows

TL;DR

This paper proves that perfect mixing of binary viscous fluids cannot be achieved in finite time under flows with finite viscous dissipation, establishing exponential lower bounds on mixing rates using mathematical norms.

Contribution

It introduces rigorous lower bounds on mixing measures, demonstrating the impossibility of finite-time perfect mixing for certain fluid flows.

Findings

Mixing cannot be faster than exponential in time.

Lower bounds are uniform across initial data.

Perfect mixing in finite time is impossible under the model.

Abstract

We consider a model for mixing binary viscous fluids under an incompressible flow. We proof the impossibility of perfect mixing in finite time for flows with finite viscous dissipation. As measures of mixedness we consider a Monge--Kantorovich--Rubinstein transportation distance and, more classically, the $H^{-1}$ norm. We derive rigorous a priori lower bounds on these mixing norms which show that mixing cannot proceed faster than exponentially in time. The rate of the exponential decay is uniform in the initial data.