The obstacle problem for scalar conservation laws with nonlocal dynamics

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.