An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Conservation Laws Using Shadow Waves and Dafermos Regularization

TL;DR

This paper analyzes classical and non-classical Riemann solutions for a minimal 2x2 conservation law system, demonstrating the existence of overcompressive delta shocks via Dafermos regularization and geometric singular perturbation theory.

Contribution

It introduces a detailed analysis of delta shocks in a simplified system using blow-up techniques and GSPT, providing insights into complex transport phenomena.

Findings

Existence of overcompressive delta shocks as singular limits

Resolution of internal structure of shocks using GSPT

Numerical validation with Local Lax-Friedrichs scheme

Abstract

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.