# Kinetic theory and Lax equations for shock clustering and Burgers turbulence

Source: [Kinetic theory and Lax equations for shock clustering and Burgers turbulence](<https://www.sota2.com/research/paper/68fa2f1d1e209131ce868d76>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:53:34.824Z

Authors: Govind Menon, Ravi Srinivasan

Published: 2009-09-22T16:37:16.000Z

## Abstract

We study shock statistics in the scalar conservation law $\\partial\_t u + \\partial\_x f(u)=0$, $x \\in \\R$, $t\>0$, 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 L\\'evy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process $u(x,t)$, $x\\in \\R$. 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 (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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/0909.4036>)

[PDF](<https://arxiv.org/pdf/0909.4036.pdf>)