On the singular limit problem for a discontinuous nonlocal conservation law

TL;DR

This paper investigates the convergence of solutions from a nonlocal conservation law with spatial discontinuities to a local conservation law, establishing conditions for convergence and illustrating with numerical examples.

Contribution

It provides the first analysis of the singular limit for nonlocal conservation laws with spatial discontinuities, including convergence results for exponential kernels.

Findings

Convergence of nonlocal to local equations under mild conditions

Maximum principle established for the nonlocal equation

Numerical examples illustrating theoretical results

Abstract

In this contribution we study the singular limit problem of a nonlocal conservation law with a discontinuity in space. The specific choice of the nonlocal kernel involving the spatial discontinuity as well enables it to obtain a maximum principle for the nonlocal equation. The corresponding local equation can be transformed diffeomorphically to a classical scalar conservation law where the well-know Kru\v{z}kov theory can be applied. However, the nonlocal equation does not scale that way which is why the study of convergence is interesting to pursue. For exponential kernels in the nonlocal operator, we establish the converge to the corresponding local equation under mild conditions on the involved discontinuous velocity. We illustrate our results with some numerical examples.