Kinetic description of scalar conservation laws with Markovian data

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.