# A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels

3 CaGE - Control And GEometry
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference between the solution and the nonlocal term.
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Preprints, Working Papers, ...

https://hal.inria.fr/hal-03091618
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Submitted on : Thursday, December 31, 2020 - 11:49:18 AM
Last modification on : Tuesday, January 11, 2022 - 11:16:07 AM

### Identifiers

• HAL Id : hal-03091618, version 1
• ARXIV : 2012.13203

### Citation

Giuseppe Maria Coclite, Jean-Michel Coron, Nicola de Nitti, Alexander Keimer, Lukas Pflug. A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels. 2020. ⟨hal-03091618⟩

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