Existence and weak-strong uniqueness of measure solutions to Euler-alignment/Aw--Rascle--Zhang model of collective behaviour
Jakub Woźnicki, Ewelina Zatorska
TL;DR
The paper analyzes a multi-dimensional Euler–alignment system with a matrix-valued communication kernel and its formal nonlocal Aw–Rascle–Zhang (ARZ) counterpart, establishing global-in-time measure-valued solutions via a vanishing-viscosity limit of a degenerate pressureless Navier–Stokes system. It proves a weak-strong uniqueness principle in the pressureless, nonlocal setting and demonstrates that Euler–alignment and nonlocal ARZ arise from the same inviscid limit, coinciding with classical solutions when available. The approach combines Bresch–Desjardins entropy and Mellet–Vasseur estimates for compactness with a Wasserstein-based relative entropy method to control nonlocal interactions. This work thus rigorously connects the macroscopic Euler–alignment and ARZ models on the whole space and extends the theory of measure-valued solutions for nonlocal, pressureless systems.
Abstract
We study the multi-dimensional Euler-alignment system with a matrix-valued communication kernel, motivated by models of anticipation dynamics in collective behaviour. A key feature of this system is its formal equivalence to a nonlocal variant of the Aw--Rascle--Zhang (ARZ) traffic model, in which the desired velocity is modified by a nonlocal gradient interaction. We prove the global-in-time existence of measure solutions to both formulations, obtained via a single degenerate pressureless Navier--Stokes approximation. Furthermore, we establish a weak-strong uniqueness principle adapted to the pressureless setting and to nonlocal alignment forces. As a consequence, we rigorously justify the formal correspondence between the nonlocal ARZ and Euler-alignment models: they arise from the same inviscid limit, and the weak-strong uniqueness property ensures that, whenever a classical solution exists, both formulations coincide with it.