Existence and weak-strong uniqueness of measure solutions to Euler-alignment/Aw--Rascle--Zhang model of collective behaviour

TL;DR

This paper proves the existence and uniqueness of measure solutions for a multi-dimensional Euler-alignment system with nonlocal interactions, linking it rigorously to a nonlocal Aw--Rascle--Zhang traffic model.

Contribution

It establishes global measure solutions and a weak-strong uniqueness principle for the nonlocal Euler-alignment and ARZ models, connecting them through a common inviscid limit.

Findings

Proved global-in-time existence of measure solutions.

Established weak-strong uniqueness principle.

Linked Euler-alignment and ARZ models via inviscid limit.

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.