Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight

TL;DR

This paper proves the existence of global piecewise weak solutions for a discrete flocking model with a singular communication weight, and discusses conditions for finite-time velocity alignment.

Contribution

It establishes the existence of solutions for a discrete Cucker-Smale model with a singular weight and explores velocity alignment scenarios.

Findings

Existence of global $C^1$ piecewise weak solutions proven.

Finite-time velocity alignment discussed.

Model with singular communication weight analyzed.

Abstract

We prove existence of global $C^1$ piecewise weak solutions for the discrete Cucker-Smale's flocking model with the communication weight $\psi(s)=s^{-\alpha}, 0<\alpha<1.$ We also discuss the possibility of finite in time alignment of the velocities of the particles.