Recent advances in the analysis of the dissipative Aw-Rascle system
Ewelina Zatorska
TL;DR
This review surveys recent advances in the analysis of the dissipative Aw–Rascle system, emphasizing its multidimensional formulation, the hard-congestion limit, and the well/ill-posedness landscape. It highlights four main results for the multi-dimensional system with $p(\varrho)=\varrho^\gamma$: (i) short-time existence of regular solutions via a two-step iterative scheme with $L^p$ transport estimates, (ii) global-in-time existence of measure-valued weak solutions endowed with a dissipation defect and a weak–strong uniqueness principle, and (iii) ill-posedness of weak solutions proven by convex integration, indicating non-uniqueness in general. The paper also discusses non-local and lubrication-type extensions, and clarifies the links to Brenner-type two-velocity models, Euler–alignment with nonlocal kernels, and the hard-congestion limit, including two rigorous approaches for passing to the limit. These results advance the mathematical understanding of multi-velocity, dissipative traffic and crowd models and provide rigorous frameworks for convergence and non-uniqueness phenomena. Mathematical notation is consistently used to connect density, momentum, and pressure-like offsets across the various formulations.
Abstract
The one-dimensional Aw-Rascle (AR) system has become a cornerstone of macroscopic models for single-lane vehicular traffic. A possible generalization of this model to a multi-dimensional setting is the so-called dissipative AR model, which is more suited to capturing crowd dynamics. This review summarizes recent studies that analyze the dissipative AR model, its hard congestion limit, the non-uniqueness of weak solutions, the existence and asymptotics of solutions within the duality framework, non-local interactions, and the existence of regular solutions.