[Remarque sur les solutions en dualité pour une loi de conservation scalaire faiblement non-linéaire]
Nous considérons lʼexistence et lʼunicité de solutions en dualité pour une loi de conservation scalaire avec un noyau dʼinteraction non-local. En suivant Bouchut et James (1999) [3], une notion de solution en dualité pour un tel système non-linéaire est proposée pour laquelle nous nʼavons cependant pas dʼunicité. Dans ce travail nous prouvons alors quʼen sélectionnant le flux, nous retrouvons un résultat dʼexistence et dʼunicité des solutions mesures de notre système.
We investigate existence and uniqueness of duality solutions for a scalar conservation law with a nonlocal interaction kernel. Following Bouchut and James (1999) [3], a notion of duality solution for such a nonlinear system is proposed, for which we do not have uniqueness. However we prove that a natural definition of the flux allows to select a solution for which uniqueness holds.
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@article{CRMATH_2011__349_11-12_657_0,
author = {James, Fran\c{c}ois and Vauchelet, Nicolas},
title = {A remark on duality solutions for some weakly nonlinear scalar conservation laws},
journal = {Comptes Rendus. Math\'ematique},
pages = {657--661},
publisher = {Elsevier},
volume = {349},
number = {11-12},
year = {2011},
doi = {10.1016/j.crma.2011.05.004},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.05.004/}
}
TY - JOUR AU - James, François AU - Vauchelet, Nicolas TI - A remark on duality solutions for some weakly nonlinear scalar conservation laws JO - Comptes Rendus. Mathématique PY - 2011 SP - 657 EP - 661 VL - 349 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.05.004/ DO - 10.1016/j.crma.2011.05.004 LA - en ID - CRMATH_2011__349_11-12_657_0 ER -
%0 Journal Article %A James, François %A Vauchelet, Nicolas %T A remark on duality solutions for some weakly nonlinear scalar conservation laws %J Comptes Rendus. Mathématique %D 2011 %P 657-661 %V 349 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.05.004/ %R 10.1016/j.crma.2011.05.004 %G en %F CRMATH_2011__349_11-12_657_0
James, François; Vauchelet, Nicolas. A remark on duality solutions for some weakly nonlinear scalar conservation laws. Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 657-661. doi : 10.1016/j.crma.2011.05.004. http://www.numdam.org/articles/10.1016/j.crma.2011.05.004/
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