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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.'
author:
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    Jan Peszek\
    [*Institute of Applied Mathematics and Mechanics, University of Warsaw,*]{}\
    [*ul. Banacha 2, 02-097 Warsaw, Poland*]{}\
    [*E-mail: j.peszek@mimuw.edu.pl*]{}
title: 'Existence of piecewise weak solutions of a discrete Cucker-Smale’s flocking model with a singular communication weight'
---

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Introduction
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We consider the Cucker-Smale’s (C-S) flocking model describing a collective self-driven motion of self-propelled particles, which for some reason have a tendency to flock, such as flockings of birds, schoolings of fishes or concentration of network activity. The purpose of this paper is to prove existence of piecewise weak solutions for the C-S model with an unbounded and non-Lipschitz communication weight. In the classical, discrete C-S model we assume that we have exactly $N$ particles in some unspecified space and that $(x_i,v_i)$ denotes the position and velocity of $i$-th particle, thus for $i=1,...,N$, we have $$\begin{aligned}
\label{cs1}
\frac{dx_i}{dt}=v_i.\end{aligned}$$ Moreover we know that velocity of each particle changes in time 