Duality solutions to the hard-congestion model for the dissipative Aw-Rascle system

TL;DR

This paper introduces duality solutions for the hard-congestion model within the dissipative Aw-Rascle system, establishing existence, convergence, and weak solutions, and discussing uniqueness issues in the context of compressive dynamics.

Contribution

It develops a new framework of duality solutions for a generalized Aw-Rascle system with singular functions, proving convergence and existence results, and addressing non-uniqueness.

Findings

Existence of duality solutions for the hard-congestion model.

Convergence of solutions to duality solutions under certain initial conditions.

Discussion of non-uniqueness issues in the limiting system.

Abstract

We introduce the notion of duality solution for the hard-congestion model on the real line, and additionally prove an existence result for this class of solutions. Our study revolves around the analysis of a generalised Aw-Rascle system, where the offset function is replaced by the gradient of a singular function, such as $\rho$ $\gamma$ n , where $\gamma$ $\rightarrow$ $\infty$. We prove that under suitable assumptions on the initial data, solutions to the Aw-Rascle system converge towards the so-called duality solutions, which have previously found applications in other systems which exhibit compressive dynamics. We also prove that one can obtain weak solutions to the limiting system under stricter assumptions on the initial data. Finally, we discuss (non-)uniqueness issues.