Topological and Purely Topological Alignment Dynamics

TL;DR

This paper investigates the Euler Alignment system with topological interaction protocols, establishing conditions for global solutions and analyzing the decoupling and long-term behavior of purely topological interactions.

Contribution

It provides the first analysis of purely topological alignment dynamics, showing system decoupling and long-term behavior under these protocols.

Findings

Global existence conditions for regular protocols.

Decoupling into velocity and scalar conservation law for purely topological interactions.

Long-time behavior analysis for both regular and singular protocols.

Abstract

We study the Euler Alignment system of collective behavior, equipped with `topological' interaction protocols, which were introduced to the mathematical literature by Shvydkoy and Tadmor. Interactions subject to these protocols may depend on both the Euclidean distance between agents and on the mass distribution between them -- the `topological' component. When the interaction protocol is regular, we prove sufficient conditions for the existence of global-in-time classical solutions, related to the initial nonnegativity of a conserved quantity of the system. The remainder of our results explore the case where the interactions are `purely' topological and the interactions do not depend on the Euclidean distance. We show that in this case, the system decouples into an autonomous velocity equation in mass coordinates together with a scalar conservation law with time-dependent flux determined by the velocity. We analyze the long-time behavior for the dynamics associated to both regular and singular protocols.