We study nonlocal conservation laws with a discontinuous flux function of regularity in the spatial variable and show existence and uniqueness of weak solutions in , as well as related maximum principles. We achieve this well-posedness by a proper reformulation in terms of a fixed-point problem. This fixed-point problem itself necessitates the study of existence, uniqueness and stability of a class of discontinuous ordinary differential equations. On the ODE level, we compare the solution type defined here with the well-known Carathéodory and Filippov solutions.
Nous étudions des lois de conservation non locales avec une fonction de flux discontinue de régularité spatiale dans la variable spatiale et montrons l’existence et l’unicité de solutions faibles dans , ainsi que les principes de maxima connexes des principes de maximum correspondants. Nous obtenons ce caractère bien posé par une reformulation appropriée en termes d’un problème de point fixe. Nous obtenons ce caractère bien posé en reformulant de façon appropriée le problème comme un problème de point fixe. Ce problème de point fixe nécessite lui-même l’étude de l’existence, de l’unicité et de la stabilité d’une classe d’équations différentielles ordinaires discontinues. Au niveau des ODE EDO, nous comparons le type de solution défini ici avec les solutions bien connues de Carathéodory et de Filippov.
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Alexander Keimer 1, 2; Lukas Pflug 3, 4
@article{CRMATH_2023__361_G11_1723_0, author = {Alexander Keimer and Lukas Pflug}, title = {Discontinuous nonlocal conservation laws and related discontinuous {ODEs} {\textendash} {Existence,} {Uniqueness,} {Stability} and {Regularity}}, journal = {Comptes Rendus. Math\'ematique}, pages = {1723--1760}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.490}, language = {en}, }
TY - JOUR AU - Alexander Keimer AU - Lukas Pflug TI - Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity JO - Comptes Rendus. Mathématique PY - 2023 SP - 1723 EP - 1760 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.490 LA - en ID - CRMATH_2023__361_G11_1723_0 ER -
%0 Journal Article %A Alexander Keimer %A Lukas Pflug %T Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity %J Comptes Rendus. Mathématique %D 2023 %P 1723-1760 %V 361 %I Académie des sciences, Paris %R 10.5802/crmath.490 %G en %F CRMATH_2023__361_G11_1723_0
Alexander Keimer; Lukas Pflug. Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1723-1760. doi : 10.5802/crmath.490. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.490/
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