This article is cited in
8 papers
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.
UDC:
517.958:531.35
MSC: Primary
35Q53; Secondary
35B45 Received: 25.12.2013
DOI:
10.4213/rm9629
Fulltext:
PDF file (871 kB)
