Journal of Statistical Physics

, Volume 140, Issue 6, pp 1195–1223

Kinetic Theory and Lax Equations for Shock Clustering and Burgers Turbulence


DOI: 10.1007/s10955-010-0028-3

Cite this article as:
Menon, G. & Srinivasan, R. J Stat Phys (2010) 140: 1195. doi:10.1007/s10955-010-0028-3


We study shock statistics in the scalar conservation law tu+xf(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u2/2). This suggests the kinetic equations of shock clustering are completely integrable.


Shock clusteringStochastic coalescenceKinetic theoryIntegrable systemsBurgers turbulence

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Division of Applied Mathematics, Box FBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsThe University of Texas at AustinAustinUSA