### 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)∞s^{1/2}, 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{-Cs^{3}} 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 language | English (US) |
---|---|

Pages (from-to) | 13-38 |

Number of pages | 26 |

Journal | Communications in Mathematical Physics |

Volume | 172 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1995 |

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### ASJC Scopus subject areas

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

### Cite this

*Communications in Mathematical Physics*,

*172*(1), 13-38. https://doi.org/10.1007/BF02104509

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

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 172, no. 1, pp. 13-38. https://doi.org/10.1007/BF02104509

}

TY - JOUR

T1 - Statistical properties of shocks in Burgers turbulence

AU - Avellaneda, Marco

AU - Weinan, E.

PY - 1995/8

Y1 - 1995/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0003028082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0003028082&partnerID=8YFLogxK

U2 - 10.1007/BF02104509

DO - 10.1007/BF02104509

M3 - Article

VL - 172

SP - 13

EP - 38

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -