Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function

TL;DR

This paper analyzes a one-dimensional Aw-Rascle system with a polynomial offset function, establishing global solutions and convergence to a hard-congestion model relevant for traffic and suspension flows.

Contribution

It proves the existence of global classical solutions without approximations and demonstrates convergence to a hybrid free-congested system as the polynomial degree tends to infinity.

Findings

Established global-in-time classical solutions for fixed n.

Proved uniform lower bounds on density using maximum principle.

Demonstrated convergence to a hard-congestion model as n approaches infinity.

Abstract

We study the Aw-Rascle system in a one-dimensional domain with periodic boundary conditions, where the offset function is replaced by the gradient of the function $\rho_{n}^{\gamma}$, where $\gamma \to \infty$. The resulting system resembles the 1D pressureless compressible Navier-Stokes system with a vanishing viscosity coefficient in the momentum equation and can be used to model traffic and suspension flows. We first prove the existence of a unique global-in-time classical solution for $n$ fixed. Unlike the previous result for this system, we obtain global existence without needing to add any approximation terms to the system. This is by virtue of a $n-$uniform lower bound on the density which is attained by carrying out a maximum-principle argument on a suitable potential, $W_{n} = \rho_{n}^{-1}\partial_{x}w_{n}$. Then, we prove the convergence to a weak solution of a hybrid free-congested system as $n \to \infty$, which is known as the hard-congestion model.