Year: 2024
Author: Fabio Ancona, Laura Caravenna, Andrea Marson
Communications in Mathematical Analysis and Applications, Vol. 3 (2024), Iss. 3 : pp. 425–449
Abstract
In this note, 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 applied in [Ancona et al., 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 conservation law, the case of 1$D$ 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmaa.2024-0018
Communications in Mathematical Analysis and Applications, Vol. 3 (2024), Iss. 3 : pp. 425–449
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 25
Keywords: Hyperbolic systems vanishing viscosity solutions SBV regularity balance laws.