We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity,
, with Gaussian whitenoise initial data. This system was originally proposed by Burgers1 as a crude model of hydrodynamic turbulence, and more recently by Zel'dovichet al..12 to describe the evolution of gravitational matter at large spatio-temporal scales, with shocks playing the role of mass clusters. We present here a rigorous proof of the scaling relationP(s)∞s1/2,s≪1 whereP(s) is the cumulative probability distribution of shock strengths. We also show that the set of spatial locations of shocks is discrete, i.e. has no accumulation points; and establish an upper bound on the tails of the shock-strength distribution, namely 1−P(s)≤exp−Cs3 fors≫1. Our method draws on a remarkable connection existing between the structure of Burgers turbulence and classical probabilistic work on the convex envelope of Brownian motion and related diffusion processes.
Description
SpringerLink - Communications in Mathematical Physics, Volume 172, Number 1
%0 Journal Article
%1 springerlink:10.1007/BF02104509
%A Avellaneda, Marco
%A Weinan, E
%C Berlin / Heidelberg
%D 1995
%I Springer
%J Communications in Mathematical Physics
%K convex_minorant
%P 13-38
%R 10.1007/BF02104509
%T Statistical properties of shocks in Burgers turbulence
%U http://dx.doi.org/10.1007/BF02104509
%V 172
%X We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity,
, with Gaussian whitenoise initial data. This system was originally proposed by Burgers1 as a crude model of hydrodynamic turbulence, and more recently by Zel'dovichet al..12 to describe the evolution of gravitational matter at large spatio-temporal scales, with shocks playing the role of mass clusters. We present here a rigorous proof of the scaling relationP(s)∞s1/2,s≪1 whereP(s) is the cumulative probability distribution of shock strengths. We also show that the set of spatial locations of shocks is discrete, i.e. has no accumulation points; and establish an upper bound on the tails of the shock-strength distribution, namely 1−P(s)≤exp−Cs3 fors≫1. Our method draws on a remarkable connection existing between the structure of Burgers turbulence and classical probabilistic work on the convex envelope of Brownian motion and related diffusion processes.
@article{springerlink:10.1007/BF02104509,
abstract = {We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity,
, with Gaussian whitenoise initial data. This system was originally proposed by Burgers[1] as a crude model of hydrodynamic turbulence, and more recently by Zel'dovichet al..[12] to describe the evolution of gravitational matter at large spatio-temporal scales, with shocks playing the role of mass clusters. We present here a rigorous proof of the scaling relationP(s)∞s1/2,s≪1 whereP(s) is the cumulative probability distribution of shock strengths. We also show that the set of spatial locations of shocks is discrete, i.e. has no accumulation points; and establish an upper bound on the tails of the shock-strength distribution, namely 1−P(s)≤exp{−Cs3} fors≫1. Our method draws on a remarkable connection existing between the structure of Burgers turbulence and classical probabilistic work on the convex envelope of Brownian motion and related diffusion processes.},
added-at = {2010-11-09T20:56:37.000+0100},
address = {Berlin / Heidelberg},
affiliation = {Courant Institute of Mathematical Sciences New York University 10012 New York N.Y. USA},
author = {Avellaneda, Marco and Weinan, E},
biburl = {https://www.bibsonomy.org/bibtex/27187ba525f3487f4ef1f64c9950d2a0e/pitman},
description = {SpringerLink - Communications in Mathematical Physics, Volume 172, Number 1},
doi = {10.1007/BF02104509},
interhash = {53b401ff6f40ea29c402546d072010bd},
intrahash = {7187ba525f3487f4ef1f64c9950d2a0e},
issn = {0010-3616},
issue = {1},
journal = {Communications in Mathematical Physics},
keyword = {Physics and Astronomy},
keywords = {convex_minorant},
pages = {13-38},
publisher = {Springer},
timestamp = {2010-11-09T20:56:37.000+0100},
title = {Statistical properties of shocks in Burgers turbulence},
url = {http://dx.doi.org/10.1007/BF02104509},
volume = 172,
year = 1995
}