SBV regularity of Entropy Solutions for Hyperbolic Systems of Balance Laws with General Flux function

TL;DR

This paper establishes that vanishing viscosity solutions to certain hyperbolic balance laws in one dimension are functions of special bounded variation, extending SBV regularity results to systems of balance laws and eigenvalue functions.

Contribution

It proves SBV regularity for vanishing viscosity solutions of hyperbolic balance laws, including systems with multiple equations and eigenvalue functions, generalizing previous conservation law results.

Findings

Vanishing viscosity solutions are SBV functions in one dimension.

SBV regularity of eigenvalue functions extends from conservation to balance laws.

Regularity fails for general smooth strictly hyperbolic systems without additional conditions.

Abstract

We prove that vanishing viscosity solutions to smooth non-degenerate systems of balance laws having small bounded variation, in one space dimension, must be functions of special bounded variation. For more than one equation, this is new also in the case of systems of conservation laws out of the context of genuine nonlinearity. For general smooth strictly hyperbolic systems of balance laws, this regularity fails, as known for systems of balance laws: we generalize the SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux from conservation to balance laws. Proofs are based on extending Oleinink-type balance estimates, with the introduction of new source measures, classical localization arguments, and observations in real analysis.