SBV-like regularity of Entropy Solutions for a Scalar Balance Law

TL;DR

This paper explores the SBV-regularity of entropy solutions for scalar balance laws in one dimension, highlighting new behaviors in systems with linear degeneracies and generalizing previous SBV regularity results.

Contribution

It introduces a new strategy for analyzing SBV-regularity in 1D systems of balance laws, extending prior work to more complex behaviors with linear degeneracies.

Findings

Entropy solutions can exhibit fractal Cantor-like behaviors.

SBV-regularity can be generalized even when it fails.

New analytical tools are developed for systems with multiple equations.

Abstract

In this note we discuss the SBV-regularity for a scalar balance law in one space dimension as a case study in order to explain the strategy that we apply in a separate paper to general hyperbolic systems of balance laws in one space dimension. While for a single balance law the more general work by Robyr is already available, the case of 1d-systems presents new behaviors that require a different strategy. This is why in this note we make the effort to introduce the notation and tools that are required for the case of more equations. When the flux presents linear degeneracies, it is know that entropy solutions can present nasty fractal Cantor-like behaviors, although f'(u) is still SBV: we thus discuss SBV-like regularity generalizing the work by Bianchini-Yu as SBV-regularity fails.