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

We demonstrate that solutions to scalar balance laws,in one space dimension, which exhibit bounded variation, must be functions of special bounded variation (SBV). This case study illustrates the strategy ap- plied in [Ancona-Caravenna-Marson, preprint, University of Padova, 2024] to systems of balance laws, extending the methodologies developed in pioneering previous works by several authors. While for a single balance law a more general work is already available, generalizing the first breakthrough related to a conserva- tion law, 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 known that entropy solutions can really present nasty fractal Cantor-like behaviours, although f′(u) is still SBV: We thus discuss SBV-like regularity, as SBV-regularity fails.