A Minimization Approach to Conservation Laws With Random Initial Conditions and Non-smooth, Non-strictly Convex Flux

TL;DR

This paper develops an exact solution framework for conservation laws with random initial conditions, including non-smooth, non-convex flux functions, and analyzes shock densities and solution variances.

Contribution

It introduces a minimization approach to solve conservation laws with Gaussian stochastic initial data, extending to non-smooth, non-convex flux functions and deriving explicit formulas for shock densities.

Findings

Exact solutions for discretized Gaussian initial data

Explicit formulas for shock density at given time and position

Analysis of solution variance and special cases like Brownian motion

Abstract

We obtain solutions to conservation laws under any random initial conditions that are described by Gaussian stochastic processes (in some cases discretized). We analyze the generalization of Burgers' equation for a smooth flux function $H\left( p\right) =\left\vert p\right\vert ^{j}$ for $j\geq2$ under random initial data. We then consider a piecewise linear, non-smooth and non-convex flux function paired with general discretized Gaussian stochastic process initial data. By partitioning the real line into a finite number of points, we obtain an exact expression for the solution of this problem. From this we can also find exact and approximate formulae for the density of shocks in the solution profile at a given time $t$ and spatial coordinate $x$. We discuss the simplification of these results in specific cases, including Brownian motion and Brownian bridge, for which the inverse covariance matrix and corresponding eigenvalue spectrum have some special properties. We calculate the transition probabilities between various cases and examine the variance of the solution $w\left(x,t\right)$ in both $x$ and $t$. We also describe how results may be obtained for a non-discretized version of a Gaussian stochastic process by taking the continuum limit as the partition becomes more fine.