标题： 对带有时间依赖源项的Keyfitz-Kranzer型$2 \times 2$系统Riemann问题的分析 显示英文标题

标题： An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Balance Laws With a Time-Dependent Source Term

摘要： 我们考虑一个由一个守恒定律和一个具有时变源项的平衡定律组成的系统，并对Riemann解进行了全面分析，包括非经典过压缩δ冲击波。该系统的最小但具有代表性的结构捕捉了密度约束下传输的本质特征，尽管其结构简单，却作为一个多功能原型，适用于各种情境下的有限人群传输过程，包括生物聚集、生态扩散、颗粒压实和交通拥堵。除了上述非自相似解外，相关的Riemann问题还允许穿越真空状态（$\rho = 0$）和临界密度阈值（$\rho = \bar{\rho}$）的解结构，在这里，移动性消失且特征速度退化。此外，源项中的显式时间依赖性导致自相似性的破坏，从而在连续的时间区间内产生不同的Riemann解，并突显了解空间的动态特性。理论结果通过局部Lax-Friedrichs格式进行了数值验证。 显示英文摘要

摘要： 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 ($\rho = 0$) and the critical density threshold ($\rho = \bar{\rho}$), 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.