On the relation between enhanced dissipation time-scales and mixing rates
File(s)
Accepted version
Author(s)
Coti Zelati, Michele
Delgadino, Matias G
Elgindi, Tarek M
Type
Journal Article
Abstract
We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a time-scale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation time-scales. The proofs are based on a contradiction argument that takes advantage of the cascading mechanism due to mixing, an estimate of the distance between the inviscid and viscous dynamics, and of an optimization step in the frequency cut-off.
Thanks to the generality and robustness of our approach, we are able to apply our abstract results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation time-scales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation time-scale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion.
Thanks to the generality and robustness of our approach, we are able to apply our abstract results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation time-scales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation time-scale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion.
Date Issued
2020-06-01
Date Acceptance
2018-08-08
Citation
Communications on Pure and Applied Mathematics, 2020, 73 (6), pp.1205-1244
ISSN
0010-3640
Publisher
Wiley
Start Page
1205
End Page
1244
Journal / Book Title
Communications on Pure and Applied Mathematics
Volume
73
Issue
6
Copyright Statement
© 2019 Wiley Periodicals, Inc. This is the pre-peer reviewed version of the article which has been published in final form at https://doi.org/10.1002/cpa.21831
Identifier
http://arxiv.org/abs/1806.03258v1
Subjects
math.AP
math.AP
math.DS
Publication Status
Published
Date Publish Online
2019-05-16