Abstract:In this article, we present a method to find a solution to a one-dimensional nonlocal conservation law that respects a space-dependent mapping, referred to as the obstacle. This is achieved by generalizing existing results for the local conservation law: We consider a relaxation of the velocity, that explicitly depends on the obstacle. We prove existence of solutions to the relaxed problem and show that, as the relaxation mapping converges to a Heaviside-type function, the corresponding solutions converge to a weak solution of a discontinuous nonlocal conservation law. Moreover, we can characterize the limiting flux in several cases.\\ The paper concludes with a numerical study that illustrates the aforementioned convergence.
| Comments: | 17 pages, 5 figures |
| Subjects: | Analysis of PDEs (math.AP) |
| MSC classes: | 35L65, 76A30 |
| Cite as: | arXiv:2605.24677 [math.AP] |
| (or arXiv:2605.24677v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24677 arXiv-issued DOI via DataCite (pending registration) |
From: Alexander Keimer [view email]
[v1]
Sat, 23 May 2026 17:28:23 UTC (5,020 KB)
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