Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Inviscid limits for a stochastically forced shell model of turbulent flow

Susan Friedlander, Nathan Glatt-Holtz, and Vlad Vicol

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Abstract

We establish the anomalous mean dissipation rate of energy in the inviscid limit for a stochastic shell model of turbulent fluid flow. The proof relies on viscosity independent bounds for stationary solutions and on establishing ergodic and mixing properties for the viscous model. The shell model is subject to a degenerate stochastic forcing in the sense that noise acts directly only through one wavenumber. We show that it is hypo-elliptic (in the sense of Hörmander) and use this property to prove a gradient bound on the Markov semigroup.

Résumé

Nous étudions le taux anormal de la dissipation moyenne de l’énergie dans la limite non visqueuse d’un modèle en couche de fluide turbulent. La preuve se base sur des estimations indépendantes de la viscosité pour des solutions stationnaires, ainsi que sur des propriétés ergodiques et de mélange pour le modèle visqueux. Le modèle en couche subit un forçage aléatoire dégénéré, c’est à dire que le bruit n’agit seulement que sur un mode. Nous montrons que le système est hypoelliptique au sens d’Hörmander et utilisons cette propriété pour prouver une borne sur le gradient du semigroupe de Markov.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1217-1247.

Dates
Received: 4 April 2014
Revised: 1 December 2014
Accepted: 8 December 2014
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723518

Digital Object Identifier
doi:10.1214/14-AIHP663

Mathematical Reviews number (MathSciNet)
MR3531707

Zentralblatt MATH identifier
1348.76081

Subjects
Primary: 76F05: Isotropic turbulence; homogeneous turbulence 35Q35: PDEs in connection with fluid mechanics 37L40: Invariant measures 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15]

Keywords
Inviscid limits Invariant measures Dissipation anomaly Shell models Ergodicity

Citation

Friedlander, Susan; Glatt-Holtz, Nathan; Vicol, Vlad. Inviscid limits for a stochastically forced shell model of turbulent flow. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1217--1247. doi:10.1214/14-AIHP663. https://projecteuclid.org/euclid.aihp/1469723518


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