### 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) |
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Pages (from-to) | 13-38 |

Number of pages | 26 |

Journal | Communications In Mathematical Physics |

Volume | 172 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1 1995 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

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