Kinetic theory and Lax equations for shock clustering and Burgers turbulence

TL;DR

This paper develops a kinetic theory for shock clustering in scalar conservation laws with random initial data, revealing integrable structures and exact solutions for Burgers turbulence.

Contribution

It introduces a novel Lax pair formulation for shock statistics evolution, demonstrating complete integrability and deriving exact solutions for Burgers equation.

Findings

Markov property preserved for broad class of initial data

Evolution equation for shock process generator derived

Exact solutions found for Burgers turbulence

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.