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On the singular limit problem for a discontinuous nonlocal conservation law

Alexander KeimerLukas Pflug
Dec 2022
摘要
In this contribution we study the singular limit problem of a nonlocalconservation law with a discontinuity in space. The specific choice of thenonlocal kernel involving the spatial discontinuity as well enables it toobtain a maximum principle for the nonlocal equation. The corresponding localequation can be transformed diffeomorphically to a classical scalarconservation 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 studyof convergence is interesting to pursue. For exponential kernels in thenonlocal operator, we establish the converge to the corresponding localequation under mild conditions on the involved discontinuous velocity. Weillustrate our results with some numerical examples.
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