Publication 25-CNA-015

Exponentially Mixing Flows With Slow Enhanced Dissipation

William Cooperman
Department of Mathematics
ETH Zürich, Switzerland
bill@cprmn.org

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

Keefer Rowan
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012
keefer.rowan@cims.nyu.edu

Seungjae Son
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA, 15213
seungjas@andrew.cmu.edu

Abstract: Consider a passive scalar which is advected by an incompressible flow u and has small molecular diffusivity k. Previous results show that if u is exponentially mixing and C¹, then the dissipation time is O(|log k|²). We produce a family of incompressible flows which are C⁰ and exponentially mixing, uniformly in k; however have a dissipation time of order 1/k (i.e. exhibits no enhanced dissipation). We also estimate the dissipation time of mixing flows, and obtain improved bounds in terms of the mixing rate with explicit constants, and allow for a time inhomogeneous mixing rate which is typical for random constructions of mixing flows.

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