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Rigorous Remarks about Scaling Laws in Turbulent Fluids

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Abstract

A definition of scaling law for suitable families of measures is given and investigated. First, a number of necessary conditions are proved. They imply the absence of scaling laws for 2D stochastic Navier-Stokes equations and for the stochastic Stokes (linear) problem in any dimension, while they imply a lower bound on the mean vortex stretching in 3D. Second, for the 3D stochastic Navier-Stokes equations, necessary and sufficient conditions for scaling laws to hold are given, translating the problem into bounds for energy and enstrophy of high and low modes respectively. Unlike in the 2D case, the validity or invalidity of such conditions in 3D remains open.

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Authors and Affiliations

  1. Dipartimento di Matematica Applicata, Università di Pisa, via Buonarroti 1, 56127, Pisa, Italia

    F. Flandoli

  2. Equipe de probabilités, statistique et modélisation, Université de Paris-Sud, 91405, Orsay Cedex, France

    M. Gubinelli

  3. Department of Mathematics, The University of Warwick, Coventry, CV4 7AL, United Kingdom

    M. Hairer

  4. Dipartimento di Matematica, Università di Firenze, viale Morgagni 67/a, 50134, Firenze, Italia

    M. Romito

Authors
  1. F. Flandoli
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  2. M. Gubinelli
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  3. M. Hairer
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  4. M. Romito
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Correspondence to M. Romito.

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Communicated by A. Kupiainen

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Flandoli, F., Gubinelli, M., Hairer, M. et al. Rigorous Remarks about Scaling Laws in Turbulent Fluids. Commun. Math. Phys. 278, 1–29 (2008). https://doi.org/10.1007/s00220-007-0398-9

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  • DOI: https://doi.org/10.1007/s00220-007-0398-9

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