Stability of partially congested travelling wave solutions for the extended Aw-Rascle system

TL;DR

This paper proves the non-linear stability of certain travelling wave solutions in an extended Aw-Rasque system, modeling congestion effects, by employing weighted energy estimates to handle singular viscosity cases.

Contribution

It introduces a novel stability proof for viscous shock waves in the extended Aw-Rasque system with singular viscosity, extending previous work to more complex congestion scenarios.

Findings

Established non-linear stability of travelling wave solutions with singular viscosity.

Extended stability results to congestion models with singular offset functions.

Demonstrated effectiveness of weighted energy estimates in complex viscosity cases.

Abstract

We prove the non-linear stability of a class of travelling-wave solutions to the extended Aw-Rascle system with a singular offset function, which is formally equivalent to the compressible pressureless Navier-Stokes system with a singular viscosity. These solutions encode the effect of congestion by connecting a congested left state to an uncongested right state, and may also be viewed as approximations of solutions to the 'hard-congestion model'. By using carefully weighted energy estimates we are able to prove the non-linear stability of viscous shock waves to the Aw-Rascle system under a small zero integral perturbation, which in particular extends previous results that do not handle the case where the viscosity is singular.