On the singular limit problem for a discontinuous nonlocal conservation law

Abstract

In this contribution, we study the singular limit problem of a nonlocal conservation law with a discontinuity in space. The corresponding local equation can be transformed diffeomorphically to a classical scalar conservation law to which the well-known Kružkov 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 convergence to the solution of the corresponding local equation under mild conditions on the discontinuous velocity. We illustrate our results with some numerical examples.