Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity

Alexander Keimer; Lukas Pflug

doi:10.5802/crmath.490

Équations aux dérivées partielles

Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity
[Lois de conservation non locales discontinues et ODE EDO discontinues correspondantes connexes – Existence, unicité, stabilité et régularité]

Alexander Keimer ^1,² ; Lukas Pflug ^3,⁴

¹ Institute of Transportation Studies (ITS), University of California, Berkeley, 94720 Berkeley, California, USA
² Department of Mathematics, FAU Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany
³ Friedrich-Alexander-Universität Erlangen-Nürnberg, FAU Competence Center Scientific Computing, Martensstr. 5a, 91058 Erlangen, Germany
⁴ Department of Mathematics, Chair of Applied Mathematics (Continuous Optimization), Cauerstraße 11, 91058 Erlangen, Germany

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1723-1760.

Abstract (VO)
Résumé

We study nonlocal conservation laws with a discontinuous flux function of regularity $L^{\infty} (ℝ)$ in the spatial variable and show existence and uniqueness of weak solutions in $C ([0, T]; L_{loc}^{1})$ , 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 $L^{\infty} (ℝ)$ dans la variable spatiale et montrons l’existence et l’unicité de solutions faibles dans $C ([0, T]; L_{loc}^{1})$ , 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.

Reçu le : 2022-09-02
Révisé le : 2023-01-20
Accepté le : 2023-03-06
Publié le : 2023-12-21

DOI : 10.5802/crmath.490

Classification : 34A12, 34A36, 35L03, 35L65, 35Q99, 35R09, 45K05

Affiliations des auteurs :

Alexander Keimer ^{1, 2} ; Lukas Pflug ^{3, 4}

¹ Institute of Transportation Studies (ITS), University of California, Berkeley, 94720 Berkeley, California, USA
² Department of Mathematics, FAU Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany
³ Friedrich-Alexander-Universität Erlangen-Nürnberg, FAU Competence Center Scientific Computing, Martensstr. 5a, 91058 Erlangen, Germany
⁴ Department of Mathematics, Chair of Applied Mathematics (Continuous Optimization), Cauerstraße 11, 91058 Erlangen, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
}

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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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