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arXiv · 2605.24677

The obstacle problem for scalar conservation laws with nonlocal dynamics

Paulo Amorim·Alexander Keimer·Lukas Pflug·Jakob Rodestock
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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.

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Paulo Amorim, Alexander Keimer, Lukas Pflug, Jakob Rodestock. 2026-05-23. The obstacle problem for scalar conservation laws with nonlocal dynamics. https://arxiv.org/abs/2605.24677

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