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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1778-3569 |