On geometric and analytic mixing scales: comparability and convergence rates for transport problems

TL;DR

This paper compares geometric and analytic mixing scales in passive scalar problems, introduces a dyadic model for clarity, and investigates optimal decay rates for these scales under transport equations.

Contribution

It establishes the comparability of geometric and analytic mixing scales and identifies optimal decay rates for solutions with Sobolev regularity.

Findings

Both mixing scales are comparable after removing large scale projections.

Slightly faster decay rates than algebraic are shown to be optimal.

Introduces a dyadic model problem for clearer analysis.

Abstract

In this article we are interested in the geometric and analytic mixing scales of solutions to passive scalar problems. Here, we show that both notions are comparable after possibly removing large scale projections. In order to discuss our techniques in a transparent way, we further introduce a dyadic model problem. In a second part of our article we consider the question of sharp decay rates for both scales for Sobolev regular initial data when evolving under the transport equation and related active and passive scalar equations. Here, we show that slightly faster rates than the expected algebraic decay rates are optimal.