Communications in Mathematical Physics

, Volume 203, Issue 3, pp 729–741 | Cite as

Structure of Shocks in Burgers Turbulence¶with Stable Noise Initial Data

  • Jean Bertoin


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.


Initial Data Initial Velocity Open Interval Burger Equation Regular Point 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Jean Bertoin
    • 1
  1. 1.Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu, F-75252 Paris Cedex 05, France. E-mail: jbe@ccr.jussieu.frFR

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