On the relation between enhanced dissipation time-scales and mixing rates
| Autor: | Michele Coti Zelati, Matias G. Delgadino, Tarek M. Elgindi |
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| Rok vydání: | 2018 |
| Předmět: |
FLOW
General Mathematics Mathematics Applied Dynamical Systems (math.DS) 01 natural sciences 0101 Pure Mathematics 010104 statistics & probability Mathematics - Analysis of PDEs Inviscid flow 0102 Applied Mathematics Fluid dynamics FOS: Mathematics Mathematics - Dynamical Systems 0101 mathematics Diffusion (business) Mixing (physics) math.AP Mathematics Science & Technology STABILITY Applied Mathematics 010102 general mathematics Scalar (physics) Mechanics Dissipation DIFFUSION Flow (mathematics) PASSIVE SCALARS Physical Sciences Dissipative system math.DS Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.1806.03258 |
| Popis: | 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. Comment: 28 pages |
| Databáze: | OpenAIRE |
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