On the Relation between Enhanced Dissipation Timescales and Mixing Rates

• Published in: Communications on pure and applied mathematics Vol. 73; no. 6; pp. 1205 - 1244
• Main Authors: Zelati, Michele Coti, Delgadino, Matias G., Elgindi, Tarek M.
• Format: Journal Article
• Language: English
• Published: New York John Wiley and Sons, Limited 01.06.2020
• Subjects: Computational fluid dynamics; Diffusion rate; Fluid flow; Incompressible flow; Norms; Optimization; Polynomials; Time
• DOI: 10.1002/cpa.21831
• ISSN: 0010-3640

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 that 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 timescale 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 timescales. 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 an optimization step in the frequency cutoff. Thanks to the generality and robustness of our approach, we are able to apply our results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation timescales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation timescale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion. © 2019 Wiley Periodicals, Inc.