Kinetic description of scalar conservation laws with Markovian data

TL;DR

This paper develops a kinetic framework for scalar conservation laws with Markov initial data, showing the Markov property persists and deriving kinetic equations for the data's statistical structure under convex Hamiltonians.

Contribution

It introduces a kinetic description of solutions to scalar conservation laws with Markov initial conditions, extending the understanding of their statistical structure and evolution.

Findings

Solutions are Markov processes in space or time under convex Hamiltonians.

Kinetic equations characterize the evolution of drift and jump densities.

Markov property persists along lines with certain slopes when Hamiltonian is not increasing.

Abstract

We derive a kinetic equation to describe the statistical structure of solutions $\rho$ to scalar conservation laws $\rho_t=H(x,t,\rho )_x$, with certain Markov initial conditions. When the Hamiltonian function is convex and increasing in $\rho$, we show that the solution $\rho(x,t)$ is a Markov process in $x$ (respectively $t$) with $t$ (respectively $x$) fixed. Two classes of Markov conditions are considered in this article. In the first class, the initial data is characterize by a drift $b$ which satisfies a linear PDE, and a jump density $f$ which satisfies a kinetic equation as time varies. In the second class, the initial data is a concatenation of fundamental solutions that are characterized by a parameter $y$, which is a Markov jump process with a jump density $g$ satisfying a kinetic equation. When $H$ is not increasing in $\rho$, the restriction of $\rho$ to a line in $(x,t)$ plane is a Markov process of the same type, provided that the slope of the line satisfies an inequality.