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On multidimensional nonlocal conservation laws with BV kernels

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arxiv 2408.02423 v2 pith:RRJ4DEI6 submitted 2024-08-05 math.AP

Maria Colombo , Gianluca Crippa , Laura V. Spinolo This is my paper · ORCID
classification math.AP
keywords blow-upconservationkernelnonlocalconvolutiondifferentgenerallaws
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We establish local-in-time existence and uniqueness results for nonlocal conservation laws in several space dimensions under weak (that is, Sobolev or BV) differentiability assumptions on the convolution kernel. In contrast to the case of a smooth kernel, in general the solution experiences finite-time blow-up. We provide an explicit example showing that solutions corresponding to different smooth approximations of the convolution kernel in general converge to different measures after the blow-up time. This rules out a fairly natural strategy for extending the notion of solution of the nonlocal conservation law after the blow-up time.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Singular limit for nonlocal conservation laws with non-locality in density and velocity

    math.AP 2026-07 conditional novelty 7.0 of 10

    Classical solutions of the two-layer nonlocal traffic equation converge to the local conservation law at rate O(ε) before shock formation, for one-sided anisotropic kernels.

  2. A positivity preserving second-order scheme for multi-dimensional system of non-local conservation laws

    math.NA 2024-12 conditional novelty 6.0 of 10

    A MUSCL-Runge-Kutta scheme for multidimensional nonlocal conservation law systems is proven positivity-preserving and L-infinity stable, but its claimed second-order accuracy is not confirmed by the numerical experiments.

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