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

TL;DR

This paper analyzes the Riemann problem for a 2x2 system of balance laws with a time-dependent source, revealing complex solution structures including delta shocks and vacuum states, with numerical confirmation.

Contribution

It provides a comprehensive analysis of Riemann solutions for a keyfitz-kranz type system with explicit time dependence, including non-classical shocks and solution breakdowns.

Findings

Existence of non-classical delta shocks.

Solutions can traverse vacuum and critical density states.

Time dependence causes non-self-similar, dynamic solutions.

Abstract

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.