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January 2022 Almost-sure exponential mixing of passive scalars by the stochastic Navier–Stokes equations
Jacob Bedrossian, Alex Blumenthal, Samuel Punshon-Smith
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Ann. Probab. 50(1): 241-303 (January 2022). DOI: 10.1214/21-AOP1533

Abstract

We deduce almost-sure exponentially fast mixing of passive scalars advected by solutions of the stochastically-forced 2D Navier–Stokes equations and 3D hyper-viscous Navier–Stokes equations in Td subjected to nondenegenerate Hσ-regular noise for any σ sufficiently large. That is, for all s>0 there is a deterministic exponential decay rate such that all mean-zero Hs passive scalars decay in Hs at this same rate with probability one. This is equivalent to what is known as quenched correlation decay for the Lagrangian flow in the dynamical systems literature. This is a follow-up to our previous work, which establishes a positive Lyapunov exponent for the Lagrangian flow—in general, almost-sure exponential mixing is much stronger than this. Our methods also apply to velocity fields evolving according to finite-dimensional models, for example, Galerkin truncations of Navier–Stokes or the Stokes equations with very degenerate forcing. For all 0k<, this exhibits many examples of CtkCx random velocity fields that are almost-sure exponentially fast mixers.

Funding Statement

J. Bedrossian was supported by NSF CAREER grant DMS-1552826 and NSF RNMS #1107444 (Ki-Net).
This material was based upon work supported by the National Science Foundation under Award No. DMS-1604805. A. Blumenthal would like Dmitry Dolgopyat for useful insights and helpful discussions.
This material was based upon work supported by the National Science Foundation under Award No. DMS-1803481.

Citation

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Jacob Bedrossian. Alex Blumenthal. Samuel Punshon-Smith. "Almost-sure exponential mixing of passive scalars by the stochastic Navier–Stokes equations." Ann. Probab. 50 (1) 241 - 303, January 2022. https://doi.org/10.1214/21-AOP1533

Information

Received: 1 May 2020; Revised: 1 February 2021; Published: January 2022
First available in Project Euclid: 23 February 2022

Digital Object Identifier: 10.1214/21-AOP1533

Subjects:
Primary: 37A25 , 37A30 , 37N10 , 76D06 , 76F25
Secondary: 37A60 , 37H15 , 60H15

Keywords: Exponential mixing , passive scalars , quenched correlation decay , spectral theory of Markov semigroups , stochastic Navier–Stokes equations

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 1 • January 2022
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