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Kinetic Theory and Lax Equations for Shock Clustering and Burgers Turbulence

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Abstract

We study shock statistics in the scalar conservation law ∂ t u+∂ x f(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)=u 2/2). This suggests the kinetic equations of shock clustering are completely integrable.

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Authors and Affiliations

  1. Division of Applied Mathematics, Box F, Brown University, Providence, RI, 02912, USA

    Govind Menon

  2. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA

    Ravi Srinivasan

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  1. Govind Menon
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  2. Ravi Srinivasan
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Correspondence to Govind Menon.

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Menon, G., Srinivasan, R. Kinetic Theory and Lax Equations for Shock Clustering and Burgers Turbulence. J Stat Phys 140, 1195–1223 (2010). https://doi.org/10.1007/s10955-010-0028-3

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