### Abstract

The statistics of the multivalued solutions of the forced Riemann equation, u_{t} + uu_{x}=f, is considered. An exact equation for the signed probability density function of these solutions and their gradient ξ=U_{x} is derived, and some properties of this equation are analyzed. It is shown in particular that the tails of the signed probability density function generally decay as |ξ|^{-3} for large |ξ|. Further considerations give bounds on the cumulative probability density function for the velocity gradient of the solution of Burgers equation.

Original language | English (US) |
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Pages (from-to) | 2149-2153 |

Number of pages | 5 |

Journal | Physics of Fluids |

Volume | 11 |

Issue number | 8 |

DOIs | |

State | Published - Aug 1999 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Computational Mechanics
- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering
- Fluid Flow and Transfer Processes

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## Cite this

Weinan, E., & Eijnden, E. V. (1999). On the statistical solution of the Riemann equation and its implications for Burgers turbulence.

*Physics of Fluids*,*11*(8), 2149-2153. https://doi.org/10.1063/1.870076