Scalar Conservation Laws with white noise initial data

TL;DR

This paper characterizes the evolution of scalar conservation laws with random initial data, proving the solution's profile is a stationary process, and analyzes the shock structure when initial data is white noise from a Lévy process.

Contribution

It solves a conjecture on the solution profile for general convex Hamiltonians and describes the shock structure for white noise initial data.

Findings

Solution profile is a stationary piecewise-smooth Feller process.

Derived a generalized Chernoff distribution for the argmax of Brownian motion minus convex functions.

Shock structure is almost surely discrete at any fixed time for white noise initial data.

Abstract

The statistical description of the scalar conservation law of the form $\rho_t=H(\rho)_x$ with $H: \mathbb{R} \rightarrow \mathbb{R}$ a smooth convex function has been an object of interest when the initial profile $\rho(\cdot,0)$ is random. The special case when $H(\rho)=\frac{\rho^2}{2}$ (Burgers equation) has in particular received extensive interest in the past and is now understood for various random initial conditions. We solve in this paper a conjecture on the profile of the solution at any time $t>0$ for a general class of hamiltonians $H$ and show that it is a stationary piecewise-smooth Feller process. Along the way, we study the excursion process of the two-sided linear Brownian motion $W$ below any strictly convex function $\phi$ with superlinear growth and derive a generalized Chernoff distribution of the random variable $\text{argmax}_{z \in \mathbb{R}} (W(z)-\phi(z))$. Finally, when $\rho(\cdot,0)$ is a white noise derived from an abrupt L\'evy process, we show that the shocks structure of the solution is a.s discrete at any fixed time $t>0$ under some mild assumptions on $H$.