Universal Mixers in All Dimensions

TL;DR

This paper constructs universal incompressible flows that achieve near-optimal exponential mixing in all dimensions for solutions of the transport equation, with bounds in specific Sobolev spaces.

Contribution

It introduces universal mixers that work across all dimensions and initial conditions, providing optimal mixing rates and bounds in Sobolev spaces.

Findings

Mixing is exponential in time for regular initial conditions.

Uniform mixing rate for all initial conditions is impossible.

Flows are bounded in Sobolev spaces $W^{s,p}$ for certain $(s,p)$.

Abstract

We construct universal mixers, incompressible flows that mix arbitrarily well general solutions to the corresponding transport equation, in all dimensions. This mixing is exponential in time (i.e., essentially optimal) for any initial condition with at least some regularity, and we also show that a uniform mixing rate for all initial conditions cannot be achieved. The flows are uniformly-in-time bounded in spaces $W^{s,p}$ for a range of $(s,p)$ that includes $s > 1$ and $p>2$.