Entropy solutions for a traffic model with phase transitions

TL;DR

This paper establishes the existence and properties of entropy solutions for a two-phase macroscopic traffic model with phase transitions, using wave-front tracking and Kruzhkov's entropy criteria, including cases with vacuum states.

Contribution

It applies wave-front tracking to prove existence and bounds, and characterizes entropy solutions for a traffic model with phase transitions, including vacuum states.

Findings

Existence of weak solutions with a priori bounds.

Weak solutions are entropy solutions under certain conditions.

Numerical example illustrating solution behavior.

Abstract

In this paper, we consider the two phases macroscopic traffic model introduced in [P. Goatin, The Aw-Rascle vehicular traffic flow with phase transitions, Mathematical and Computer Modeling 44 (2006) 287-303]. We first apply the wave-front tracking method to prove existence and a priori bounds for weak solutions. Then, in the case the characteristic field corresponding to the free phase is linearly degenerate, we prove that the obtained weak solutions are in fact entropy solutions \`a la Kruzhkov. The case of solutions attaining values at the vacuum is considered. We also present an explicit numerical example to describe some qualitative features of the solutions.