Nonuniqueness of weak solutions to the dissipative Aw-Rascle model

TL;DR

This paper demonstrates that even with dissipation, the multi-dimensional Aw-Rascle traffic model admits infinitely many weak solutions, indicating nonuniqueness and ill-posedness in certain cases.

Contribution

It extends convex integration techniques to a dissipative traffic model, proving nonuniqueness of weak solutions in multi-dimensional settings.

Findings

Infinitely many weak solutions exist despite dissipation.

Nonuniqueness holds for arbitrary initial and final states.

Ill-posedness occurs for specific data choices.

Abstract

We prove nonuniqueness of weak solutions to multi-dimensional generalisation of the Aw-Rascle model of vehicular traffic. Our generalisation includes the velocity offset in a form of gradient of density function, which results in a dissipation effect, similar to viscous dissipation in the compressible viscous fluid models. We show that despite this dissipation, the extension of the method of convex integration can be applied to generate infinitely many weak solutions connecting arbitrary initial and final states. We also show that for certain choice of data, ill posedness holds in the class of admissible weak solutions.