Statistical properties of shocks in Burgers turbulence

Marco Avellaneda, E. Weinan

Research output: Contribution to journalArticle

Abstract

We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity, {Mathematical expression}, 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'dovich et 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 relation P(s)∞s1/2, s≪1 where P(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} for s≫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.

Original languageEnglish (US)
Pages (from-to)13-38
Number of pages26
JournalCommunications in Mathematical Physics
Volume172
Issue number1
DOIs
StatePublished - Aug 1995

Fingerprint

Statistical property
Turbulence
Shock
turbulence
shock
Convex Envelope
Vanishing Viscosity
Accumulation point
Burger equation
Scaling Relations
Burgers Equation
Diffusion Process
Brownian motion
Tail
Hydrodynamics
Probability Distribution
envelopes
hydrodynamics
viscosity
Upper bound

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

Statistical properties of shocks in Burgers turbulence. / Avellaneda, Marco; Weinan, E.

In: Communications in Mathematical Physics, Vol. 172, No. 1, 08.1995, p. 13-38.

Research output: Contribution to journalArticle

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