Burgers equation can be used as a simplified model for hydrodynamic turbulence. The purpose of this paper is to study the structure of the shocks for the inviscid equation in dimension 1 when the initial velocity is given by a stable Lévy noise with index α∈ (1/2,2]. We prove that Lagrangian regular points exist (i.e. there are fluid particles that have not participated in shocks at any time between 0 and t) if and only if α≤ 1 and the noise is not completely asymmetric, and that otherwise the shock structure is discrete. Moreover, in the Cauchy case α= 1, we show that there are no rarefaction intervals, i.e. at time t >0$, there are fluid particles in any non-empty open interval.

Bertoin, Jean (1999). *Structure of shocks in Burgers turbulence with stable noise initial data.* Communications in Mathematical Physics, 203(3):729-741.

## Abstract

Burgers equation can be used as a simplified model for hydrodynamic turbulence. The purpose of this paper is to study the structure of the shocks for the inviscid equation in dimension 1 when the initial velocity is given by a stable Lévy noise with index α∈ (1/2,2]. We prove that Lagrangian regular points exist (i.e. there are fluid particles that have not participated in shocks at any time between 0 and t) if and only if α≤ 1 and the noise is not completely asymmetric, and that otherwise the shock structure is discrete. Moreover, in the Cauchy case α= 1, we show that there are no rarefaction intervals, i.e. at time t >0$, there are fluid particles in any non-empty open interval.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 1999 |

Deposited On: | 25 Jul 2013 06:47 |

Last Modified: | 05 Apr 2016 16:52 |

Publisher: | Springer |

ISSN: | 0010-3616 |

Publisher DOI: | https://doi.org/10.1007/s002200050633 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1700933 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0943.60055 |

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