A bound on the mixing rate of 2d perfect fluid flows

TL;DR

This paper establishes an exponential upper bound on the mixing rate of passive scalars in 2D Euler flows on a torus, linking it to the vorticity's BMO norm and providing bounds on scalar gradient growth.

Contribution

It introduces a new bound on the mixing rate of 2D Euler flows using the $H^{-1}$ norm and relates it to the vorticity's BMO norm, offering insights into fluid mixing limitations.

Findings

Mixing rate is at most exponential in 2D Euler flows.

The mixing rate is linearly bounded by the BMO norm of vorticity.

Provides bounds on the growth rate of scalar gradients.

Abstract

Using the $H^{-1}$ norm as a measure of mixing, we prove that 2d Euler flows on the torus mix passive scalars at most exponentially. The mixing rate is bounded linearly by the BMO norm of the vorticity (and thus by its $L^\infty$ norm). We also give an analogous bound on the growth rate of scalar gradients.