An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Balance Laws With a Time-Dependent Source Term
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908676881645568 |
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| 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 consisting of one conservation law and one balance law with a time-dependent source term, and provide a comprehensive analysis of Riemann solutions, including the non-classical overcompressive delta shocks. The minimal yet representative structure of the system captures essential features of transport under density constraints and, despite its simplicity, serves as a versatile prototype for crowd-limited transport processes across diverse contexts, including biological aggregation, ecological dispersal, granular compaction, and traffic congestion. In addition to non-self-similar solutions mentioned above, the associated Riemann problem admits solution structures that traverse vacuum states ($ρ= 0$) and the critical density threshold ($ρ= \barρ$), where mobility vanishes and characteristic speed degenerates. Moreover, the explicit time dependence in the source term leads to the breakdown of self-similarity, resulting in distinct Riemann solutions over successive time intervals and highlighting the dynamic nature of the solution landscape. The theoretical findings are numerically confirmed using the Local Lax-Friedrichs scheme. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Balance Laws With a Time-Dependent Source Term Culver, Josh Ayres, Aubrey Halloran, Evan Lin, Ryan Peng, Emily Tsikkou, Charis Analysis of PDEs Mathematical Physics Classical Analysis and ODEs Dynamical Systems 34A05, 34C37, 34C45, 34E15, 35L45, 35L65, 35L67, 35L80, 35Q92, 65M06, 74L10, 76A30 We consider a system consisting of one conservation law and one balance law with a time-dependent source term, and provide a comprehensive analysis of Riemann solutions, including the non-classical overcompressive delta shocks. The minimal yet representative structure of the system captures essential features of transport under density constraints and, despite its simplicity, serves as a versatile prototype for crowd-limited transport processes across diverse contexts, including biological aggregation, ecological dispersal, granular compaction, and traffic congestion. In addition to non-self-similar solutions mentioned above, the associated Riemann problem admits solution structures that traverse vacuum states ($ρ= 0$) and the critical density threshold ($ρ= \barρ$), where mobility vanishes and characteristic speed degenerates. Moreover, the explicit time dependence in the source term leads to the breakdown of self-similarity, resulting in distinct Riemann solutions over successive time intervals and highlighting the dynamic nature of the solution landscape. The theoretical findings are numerically confirmed using the Local Lax-Friedrichs scheme. |
| title | An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Balance Laws With a Time-Dependent Source Term |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs Dynamical Systems 34A05, 34C37, 34C45, 34E15, 35L45, 35L65, 35L67, 35L80, 35Q92, 65M06, 74L10, 76A30 |
| url | https://arxiv.org/abs/2508.10347 |