Kinetic theory and Lax equations for shock clustering and Burgers turbulence

Abstract

We study shock statistics in the scalar conservation law ${\partial }_{t}u+{\partial }_{x}f(u)=0$, $x\in \mathbb{R}$, $t>0$, 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 $u(x,t)$, $x\in \mathbb{R}$. 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.

34 pages, no figures; v2: corrections made, proofs updated

References in corpus (3)

• Universality classes in Burgers turbulence
• Simple Systems with Anomalous Dissipation and Energy Cascade
• Markovian solutions of inviscid Burgers equation

Cited by in corpus (8)

• Complete integrability of shock clustering and Burgers turbulence
• Scalar conservation laws with monotone pure-jump Markov initial conditions
• Scalar Conservation Laws with white noise initial data
• Poles, Shocks and Tygers: The Time-reversible Burgers equation
• Structure of shocks in Burgers turbulence with Lévy noise initial data
• Random Tessellations and Gibbsian solutions of Hamilton-Jacobi Equations
• An invariant in shock clustering and Burgers turbulence
• Hierarchies of N-Point Functions for Nonlinear Conservation Laws with Random Initial Data