Minimization Solutions to Conservation Laws with Non-smooth and Non-strictly Convex Flux

TL;DR

This paper extends classical conservation law results to non-smooth, non-strictly convex flux functions, including piecewise linear cases, by analyzing minimizers and their limits, paving the way for stochastic process applications.

Contribution

It generalizes minimization solutions for conservation laws to non-smooth, non-strictly convex flux functions, including piecewise linear cases, with new existence and uniqueness results.

Findings

Extended classical results to non-smooth flux functions

Proved existence and uniqueness for piecewise linear flux

Analyzed limiting behavior of smoothed approximations

Abstract

Conservation laws are usually studied in the context of sufficient regularity conditions imposed on the flux function, usually $C^{2}$ and uniform convexity. Some results are proven with the aid of variational methods and a unique minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these classical results can be extended to a flux function that is not necessarily smooth or uniformly or strictly convex. Although uniqueness a.e. of the minimizer will generally no longer hold, by considering the greatest (or supremum, where applicable) of all possible minimizers, we can successfully extend the results. One specific nonlinear case is that of a piecewise linear flux function, for which we prove existence and uniqueness results. We also approximate it by a smoothed, superlinearized version parameterized by $\varepsilon$ and consider the characterization of the minimizers for the smooth version and limiting behavior as $\varepsilon\downarrow0$ to that of the sharp, polygonal problem. In proving a key result for the solution in terms of the value of the initial condition, we provide a stepping stone to analyzing the system under stochastic processes, which will be explored further in a future paper.