A fuzzy $q$-closest alignment model

TL;DR

This paper introduces a fuzzy $q$-closest alignment model to address well-posedness issues in the Cucker-Smale model with limited communication, providing stability estimates and kinetic mean-field limits.

Contribution

The paper proposes the fuzzy $q$-closest system as a novel approach to ensure well-posedness and analyzes its stability and kinetic limits.

Findings

Established stability estimates for measure-valued solutions.

Proved the kinetic mean-field limit for the fuzzy $q$-closest system.

Addressed well-posedness issues in limited communication models.

Abstract

The paper examines the problems related to the well-posedness of the Cucker-Smale model with communication restricted to the $q$-closest neighbors, known also as the Cucker-Dong model. With agents oscillating on the boundary of different clusters, the system becomes difficult to precisely define, which leads to further problems with kinetic limits as the number of agents tends to infinity. We introduce the fuzzy $q$-closest system, which circumvents the issues with well-posedness. For such a system we prove a stability estimate for measure-valued solutions and perform the kinetic mean-field limit.