New interaction estimates for the Baiti-Jenssen system

TL;DR

This paper develops new interaction estimates for the Baiti-Jenssen system, crucial for wave front-tracking analysis, and demonstrates that Schaeffer's Regularity Theorem does not extend to systems through a constructed counter-example.

Contribution

It introduces novel interaction estimates for the Baiti-Jenssen system and uses them to show the failure of Schaeffer's Regularity Theorem for systems.

Findings

Counter-example showing non-extension of Schaeffer's theorem to systems

Robust wave-pattern with infinitely many shocks under perturbations

Explicit computation of wave fan curves for the system

Abstract

We establish new interaction estimates for a system introduced by Baiti and Jenssen. These estimates are pivotal to the analysis of the wave front-tracking approximation. In a companion paper we use them to construct a counter-example which shows that Schaeffer's Regularity Theorem for scalar conservation laws does not extend to systems. The counter-example we construct shows, furthermore, that a wave-pattern containing infinitely many shocks can be robust with respect to perturbations of the initial data. The proof of the interaction estimates is based on the explicit computation of the wave fan curves and on a perturbation argument.