Scalar conservation laws with monotone pure-jump Markov initial conditions

TL;DR

This paper verifies a conjecture about the statistical structure of solutions to scalar conservation laws with specific Markov initial conditions, using a particle system approach for bounded, monotone, piecewise constant initial data.

Contribution

It confirms an analogue of Menon and Srinivasan's conjecture for a class of initial conditions, employing a particle system representation with random boundary conditions.

Findings

Verification of the conjecture for bounded, monotone, piecewise constant initial data.

Development of a particle system representation for the solution.

Use of a random boundary condition at the domain boundary.

Abstract

In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions $\rho$ to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe $\rho(x,t)$ as a stochastic process in $x$ with $t$ fixed. In this article we verify an analogue of the conjecture for initial conditions which are bounded, monotone, and piecewise constant. Our argument uses a particle system representation of $\rho(x,t)$ over $0 \leq x \leq L$ for $L > 0$, with a suitable random boundary condition at $x = L$.