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Uspekhi Mat. Nauk, 2014, Volume 69, Issue 6(420), Pages 3–44 (Mi umn9629)  

This article is cited in 1 scientific paper (total in 1 paper)

Turbulence for the generalised Burgers equation

A. A. Boritchev

Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France

Abstract: This survey reviews rigorous results obtained by A. Biryuk and the author on turbulence for the generalised space-periodic Burgers equation
$$ u_t+f'(u)u_x=\nu u_{xx}+\eta,\qquad x \in S^1=\mathbb{R}/\mathbb{Z}, $$
where $f$ is smooth and strongly convex, and the constant $0<\nu\ll 1$ corresponds to the viscosity coefficient. Both the unforced case ($\eta=0$) and the case when $\eta$ is a random force which is smooth with respect to $x$ and irregular (kick or white noise) with respect to $t$ are considered. In both cases sharp bounds of the form $C\nu^{-\delta}$, $\delta\geqslant 0$, are obtained for the Sobolev norms of $u$ averaged over time and over the ensemble, with the same value of $\delta$ for upper and lower bounds. These results yield sharp bounds for small-scale quantities characterising turbulence, confirming the physical predictions.
Bibliography: 56 titles.

Keywords: Burgers equation, stochastic partial differential equations, turbulence, intermittency, stationary measure.

Funding Agency Grant Number
European Research Council BLOWDISOL
BRIDGES
A part of the present paper was completed during my stays at the AGM of the University of Cergy-Pontoise and at the Section de Physique of the University of Geneva, supported respectively by the grants ERC BLOWDISOL and ERC BRIDGES.


DOI: https://doi.org/10.4213/rm9629

Full text: PDF file (871 kB)
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English version:
Russian Mathematical Surveys, 2014, 69:6, 957–994

Bibliographic databases:

UDC: 517.958:531.35
MSC: Primary 35Q53; Secondary 35B45
Received: 25.12.2013

Citation: A. A. Boritchev, “Turbulence for the generalised Burgers equation”, Uspekhi Mat. Nauk, 69:6(420) (2014), 3–44; Russian Math. Surveys, 69:6 (2014), 957–994

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Boritchev, “Multidimensional potential Burgers turbulence”, Comm. Math. Phys., 342:2 (2016), 441–489  crossref  mathscinet  zmath  isi  scopus
  • Успехи математических наук Russian Mathematical Surveys
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