Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

TL;DR

This paper proves that fluxes in nonlinear hyperbolic conservation laws are Lipschitz continuous, ensuring weak solutions satisfy the balance law and clarifying the role of fluxes as meaningful objects across boundaries.

Contribution

It establishes the regularity of flux traces in multi-dimensional settings, a key step in validating weak solutions as physically meaningful balance laws.

Findings

Fluxes are Lipschitz continuous with respect to boundary perturbations.

Weak solutions satisfy the integral balance law.

Flux regularity holds in multi-dimensional cases.

Abstract

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the "meaningful objects" are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting. It implies that a weak solution indeed satisfies the balance law. In fact, it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary.