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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. We then obtain a total variation bound on the nonlocal term and can prove that the (unique) weak solution of the nonlocal problem converges strongly in 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.
@article{AIHPC_2023__40_5_1205_0,
author = {Coclite, Giuseppe Maria and Coron, Jean-Michel and De Nitti, Nicola and Keimer, Alexander and Pflug, Lukas},
title = {A general result on the approximation of local conservation laws by nonlocal conservation laws: {The} singular limit problem for exponential kernels},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1205--1223},
volume = {40},
number = {5},
year = {2023},
doi = {10.4171/aihpc/58},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/58/}
}
TY - JOUR AU - Coclite, Giuseppe Maria AU - Coron, Jean-Michel AU - De Nitti, Nicola AU - Keimer, Alexander AU - Pflug, Lukas TI - A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels JO - Annales de l'I.H.P. Analyse non linéaire PY - 2023 SP - 1205 EP - 1223 VL - 40 IS - 5 UR - http://geodesic.mathdoc.fr/articles/10.4171/aihpc/58/ DO - 10.4171/aihpc/58 LA - en ID - AIHPC_2023__40_5_1205_0 ER -
%0 Journal Article %A Coclite, Giuseppe Maria %A Coron, Jean-Michel %A De Nitti, Nicola %A Keimer, Alexander %A Pflug, Lukas %T A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels %J Annales de l'I.H.P. Analyse non linéaire %D 2023 %P 1205-1223 %V 40 %N 5 %U http://geodesic.mathdoc.fr/articles/10.4171/aihpc/58/ %R 10.4171/aihpc/58 %G en %F AIHPC_2023__40_5_1205_0
Coclite, Giuseppe Maria; Coron, Jean-Michel; De Nitti, Nicola; Keimer, Alexander; Pflug, Lukas. A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels. Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 5, pp. 1205-1223. doi: 10.4171/aihpc/58
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