Exponential mixing by random cellular flows

TL;DR

This paper demonstrates that a passive scalar in a 2D cellular flow with a randomly moving center mixes exponentially fast, independent of diffusivity, using a novel Eulerian hypocoercivity approach.

Contribution

It introduces a new Eulerian hypocoercivity method to prove exponential mixing rates for passive scalars in random cellular flows, independent of diffusivity.

Findings

Passive scalar mixes exponentially fast at a deterministic rate.

Velocity field enhances dissipation and decay rates are uniform in diffusivity.

Method relies on a modified Villani hypocoercivity approach with extended H"ormander commutators.

Abstract

We study a passive scalar equation on the two-dimensional torus, where the advecting velocity field is given by a cellular flow with a randomly moving center. We prove that the passive scalar undergoes mixing at a deterministic exponential rate, independent of any underlying diffusivity. Furthermore, we show that the velocity field enhances dissipation and we establish sharp decay rates that, for large times, are deterministic and remain uniform in the diffusivity constant. Our approach is purely Eulerian and relies on a suitable modification of Villani's hypocoercivity method, which incorporates a larger set of H\"ormander commutators than Villani's original method.