Communications in Mathematical Physics

, Volume 172, Issue 1, pp 13–38 | Cite as

Statistical properties of shocks in Burgers turbulence

  • Marco Avellaneda
  • Weinan E 


We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity,\(\frac{\partial }{{\partial t}}u\left( {x,t} \right) + \frac{\partial }{{\partial x}}\left( {\frac{1}{2}u\left( {x,t} \right)^2 } \right) = 0\), 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'dovichet 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 relationP(s)∞s1/2,s≪1 whereP(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} fors≫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.


Initial Data Brownian Motion Spatial Location Quantum Computing Cumulative Probability 
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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Marco Avellaneda
    • 1
  • Weinan E 
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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