# Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays

## TL;DR

This paper analyzes a flocking model with normalized weights and distributed delays, establishing conditions for velocity alignment, approximation by a Vlasov equation, and global solutions with long-term behavior.

## Contribution

It introduces a novel analysis of a Cucker-Smale model with distributed delays and normalized weights, including new conditions for flocking and a rigorous continuum limit.

## Key findings

- Velocity alignment conditions derived
- Approximation by delayed Vlasov equation validated
- Global existence and asymptotic behavior established

## Abstract

We study a Cucker-Smale-type flocking model with distributed time delay where individuals interact with each other through normalized communication weights. Based on a Lyapunov functional approach, we provide sufficient conditions for the velocity alignment behavior. We then show that as the number of individuals N tends to infinity, the N-particle system can be well approximated by a delayed Vlasov alignment equation. Furthermore, we also establish the global existence of measure-valued solutions for the delayed Vlasov alignment equation and its large-time asymptotic behavior.