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Exponential self-similar mixing by incompressible flows

Authors: Giovanni Alberti, Gianluca Crippa and Anna L. Mazzucato
Journal: J. Amer. Math. Soc.
MSC (2010): Primary 35Q35, 76F25
Published electronically: November 5, 2018
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Abstract: We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space $ W^{s,p}$, where $ s \geq 0$ and $ 1\leq p\leq \infty $. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm  $ \dot H^{-1}$, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case $ s=1$ and $ 1 \leq p \leq \infty $ (including the case of Lipschitz continuous velocities and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalars that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.

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Additional Information

Giovanni Alberti
Affiliation: Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, I-56127 Pisa, Italy

Gianluca Crippa
Affiliation: Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland

Anna L. Mazzucato
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania, 16802

Keywords: Mixing, continuity equation, negative Sobolev norms, incompressible flows, self-similarity, potentials, regular Lagrangian flows.
Received by editor(s): June 11, 2016
Received by editor(s) in revised form: July 26, 2018, and August 31, 2018
Published electronically: November 5, 2018
Additional Notes: The first and third authors acknowledge the hospitality of the Department of Mathematics and Computer Science at the University of Basel, where this work was started. Their stay was partially supported by the Swiss National Science Foundation grants 140232 and 156112. The visits of the second author to Pisa were supported by the University of Pisa PRA project “Metodi variazionali per problemi geometrici [Variational Methods for Geometric Problems]”. The second author was partially supported by the ERC Starting Grant 676675 FLIRT. The third author was partially supported by the US National Science Foundation grants DMS 1312727 and 1615457.
Dedicated: Dedicated to Alberto Bressan and Charles R. Doering on the occasion of their 60$^{th}$ birthdays
Article copyright: © Copyright 2018 American Mathematical Society

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