The obstacle problem for scalar conservation laws with nonlocal dynamics
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914596912103424 |
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| author | Amorim, Paulo Keimer, Alexander Pflug, Lukas Rodestock, Jakob |
| author_facet | Amorim, Paulo Keimer, Alexander Pflug, Lukas Rodestock, Jakob |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_24677 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The obstacle problem for scalar conservation laws with nonlocal dynamics Amorim, Paulo Keimer, Alexander Pflug, Lukas Rodestock, Jakob Analysis of PDEs 35L65, 76A30 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. |
| title | The obstacle problem for scalar conservation laws with nonlocal dynamics |
| topic | Analysis of PDEs 35L65, 76A30 |
| url | https://arxiv.org/abs/2605.24677 |