Vol. 1, No. 4, 2019

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

Christian Zillinger

Vol. 1 (2019), No. 4, 543–570
DOI: 10.2140/paa.2019.1.543
Abstract

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.

Keywords
mixing, damping, transport, Walsh–Fourier
Mathematical Subject Classification 2010
Primary: 76F25, 35Q35
Secondary: 42C10
Milestones
Received: 8 May 2018
Revised: 4 April 2019
Accepted: 6 June 2019
Published: 12 October 2019
Authors
Christian Zillinger
Department of Mathematics
University of Southern California
Los Angeles, CA
United States