An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Conservation Laws Using Shadow Waves and Dafermos Regularization
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918352069328896 |
|---|---|
| author | Culver, Josh Ayres, Aubrey Halloran, Evan Lin, Ryan Peng, Emily Tsikkou, Charis |
| author_facet | Culver, Josh Ayres, Aubrey Halloran, Evan Lin, Ryan Peng, Emily Tsikkou, Charis |
| contents | We consider a system of two conservation laws and provide a detailed description of both classical and non-classical self-similar Riemann solutions. In particular, we demonstrate the existence of overcompressive delta shocks as singular limits of the Dafermos regularization of the system. The system is chosen for its minimal yet representative structure, which captures the essential features of transport dynamics under density constraints. Our analysis is carried out using blow-up techniques within the framework of Geometric Singular Perturbation Theory (GSPT), allowing us to resolve the internal structure of these singular solutions. Despite its simplicity, the system serves as a versatile prototype for crowding-limited transport across a range of applications, including biological aggregation, ecological dispersal, granular compaction, and traffic congestion. Our findings are supported by numerical simulations using the Local Lax-Friedrichs scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05927 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Conservation Laws Using Shadow Waves and Dafermos Regularization Culver, Josh Ayres, Aubrey Halloran, Evan Lin, Ryan Peng, Emily Tsikkou, Charis Analysis of PDEs 34A05, 34C37, 34C45, 34E15, 35L45, 35L65, 35L67, 35L80, 35Q92, 65M06, 74L10, 76A30 We consider a system of two conservation laws and provide a detailed description of both classical and non-classical self-similar Riemann solutions. In particular, we demonstrate the existence of overcompressive delta shocks as singular limits of the Dafermos regularization of the system. The system is chosen for its minimal yet representative structure, which captures the essential features of transport dynamics under density constraints. Our analysis is carried out using blow-up techniques within the framework of Geometric Singular Perturbation Theory (GSPT), allowing us to resolve the internal structure of these singular solutions. Despite its simplicity, the system serves as a versatile prototype for crowding-limited transport across a range of applications, including biological aggregation, ecological dispersal, granular compaction, and traffic congestion. Our findings are supported by numerical simulations using the Local Lax-Friedrichs scheme. |
| title | An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Conservation Laws Using Shadow Waves and Dafermos Regularization |
| topic | Analysis of PDEs 34A05, 34C37, 34C45, 34E15, 35L45, 35L65, 35L67, 35L80, 35Q92, 65M06, 74L10, 76A30 |
| url | https://arxiv.org/abs/2508.05927 |