Hard congestion limit of the dissipative Aw-Rascle system

TL;DR

This paper investigates the Aw-Rascle system with a singular offset function, proving global smooth solutions and analyzing the hard congestion limit, which models the transition from suspension to granular flow regimes.

Contribution

It establishes the global existence of smooth solutions and rigorously analyzes the convergence towards a weak solution in the hard congestion limit.

Findings

Proved global existence of smooth solutions.

Demonstrated convergence to a weak solution in the congestion limit.

Connected the mathematical model to physical regimes of suspension and granular flows.

Abstract

In this study, we analyse the famous Aw-Rascle system in which the difference between the actual and the desired velocities (the offset function) is a gradient of a singular function of the density. This leads to a dissipation in the momentum equation which vanishes when the density is zero. The resulting system of PDEs can be used to model traffic or suspension flows in one dimension with the maximal packing constraint taken into account. After proving the global existence of smooth solutions, we study the so-called "hard congestion limit", and show the convergence of a subsequence of solutions towards a weak solution of an hybrid freecongested system. In the context of suspension flows, this limit can be seen as the transition from a suspension regime, driven by lubrication forces, towards a granular regime, driven by the contact between the grains.