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Kolmogorov’s theory of turbulence and its rigorous 1d model

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Abstract

This paper is a synopsis of the recent book [9]. The latter is dedicated to the stochastic Burgers equation as a model for 1d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence.

Résumé

Cet article est un synopsis du livre récent [9]. Le livre est dédié à l’équation be Burgers stochastique comme un modèle du turbulence unidimensionnelle, et l’article discute de son contenu en relation avec la théorie de la turbulence de Kolmogorov.

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Notes

  1. Except those for moments of the \(L_1\)-Sobolev norms.

  2. In this paper we do not deal with the energy range, so we do not define it.

  3. This relation implies that the random field u is not Gaussian since for Gaussian fields the l.h.s. of (3.6) vanishes.

  4. The function above with \(p=4,\, q=2\) is called the flatness of the random variable \(u(x+l) -u(x)\). It equals three for any Gaussian r.v.

  5. The corresponding argument was added by E. Lifschitz to the third Russian edition of the book, after L. Landau passed away. In that version of the book (which corresponds to the second English edition [20]) the part, dedicated to the theory of turbulence, was significantly edited and enlarged.

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Acknowledgements

The author was supported by Agence Nationale de la Recherche through the grant 17-CE40-0006.

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Authors and Affiliations

  1. Institut de Mathémathiques de Jussieu-Paris Rive Gauche, CNRS, Université Paris Diderot, UFR de Mathématiques, Bâtiment Sophie Germain, 75205, Paris Cedex 13, France

    Sergei Kuksin

  2. School of Mathematics, Shandong University, Jinan, 250100, People’s Republic of China

    Sergei Kuksin

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  1. Sergei Kuksin
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Correspondence to Sergei Kuksin.

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Dedicated to my friend Alexander Shnirelman on his 75-th birthday.

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Kuksin, S. Kolmogorov’s theory of turbulence and its rigorous 1d model. Ann. Math. Québec 46, 181–193 (2022). https://doi.org/10.1007/s40316-021-00174-6

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