Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity
We study nonlocal conservation laws with a discontinuous flux function of regularity $\mathsf {L}^{\infty }(\mathbb{R})$ in the spatial variable and show existence and uniqueness of weak solutions in $\mathsf {C}\big ([0,T]; \mathsf {L}^{1}_{\mathrm{loc}}\big )$, as well as related maximum principle...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-12-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.490/ |
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| author | Keimer, Alexander Pflug, Lukas |
| author_facet | Keimer, Alexander Pflug, Lukas |
| author_sort | Keimer, Alexander |
| collection | DOAJ |
| description | We study nonlocal conservation laws with a discontinuous flux function of regularity $\mathsf {L}^{\infty }(\mathbb{R})$ in the spatial variable and show existence and uniqueness of weak solutions in $\mathsf {C}\big ([0,T]; \mathsf {L}^{1}_{\mathrm{loc}}\big )$, 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. |
| format | Article |
| id | doaj-art-3a08162f2802468a8fccc3c8464c89e6 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-12-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-3a08162f2802468a8fccc3c8464c89e62025-02-07T11:12:14ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-12-01361G111723176010.5802/crmath.49010.5802/crmath.490Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and RegularityKeimer, Alexander0https://orcid.org/0000-0003-3825-5853Pflug, Lukas1https://orcid.org/0000-0001-8001-5832Department of Mathematics, FAU Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany; Institute of Transportation Studies (ITS), University of California, Berkeley, 94720 Berkeley, California, USADepartment of Mathematics, Chair of Applied Mathematics (Continuous Optimization), Cauerstraße 11, 91058 Erlangen, Germany; Friedrich-Alexander-Universität Erlangen-Nürnberg, FAU Competence Center Scientific Computing, Martensstr. 5a, 91058 Erlangen, GermanyWe study nonlocal conservation laws with a discontinuous flux function of regularity $\mathsf {L}^{\infty }(\mathbb{R})$ in the spatial variable and show existence and uniqueness of weak solutions in $\mathsf {C}\big ([0,T]; \mathsf {L}^{1}_{\mathrm{loc}}\big )$, 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.490/ |
| spellingShingle | Keimer, Alexander Pflug, Lukas Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity Comptes Rendus. Mathématique |
| title | Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity |
| title_full | Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity |
| title_fullStr | Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity |
| title_full_unstemmed | Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity |
| title_short | Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity |
| title_sort | discontinuous nonlocal conservation laws and related discontinuous odes existence uniqueness stability and regularity |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.490/ |
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