g$ in $L^1(\Omega)$, then $$\begin{aligned}
f_ng_n\rightharpoonup fg\ {\rm in\ } L^1(\Omega).\end{aligned}$$

Lastly we will say that particles $x_i$ and $x_j$ [*collide*]{} at the time $t$ if and only if $x_i(t)=x_j(t)$ but $x_i\neq x_j$ in some left-sided neighborhood of $t$ and we will say that they [*stick*]{} at the time $t$ if and only if they collide and $v_i(t)=v_j(t)$.

Main result
===========

Outline
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In this section we prove existence for the discrete C-S model (\[cs1\]) and (\[cs2\]) with a singular communication weight given by (\[psi\]). Our strategy is based on the observation that the function $t\mapsto \psi(|x_i(t)-x_j(t)|)$ is Lipschitz continuous in a neighborhood of each time $t_0$, such that for all $i,j$, we have $x_i(t_0)\neq x_j(t_0)$, which makes local existence in such points trivial. The idea is that if we can prove that the particles collide in some sense rarely, then the only difficulty will be to establish existence in a neighborhood of each point of collision of some particles. Technicaly we will obtain existence of solutions by approximating them with solutions of C-S model with bounded weights.

Approximate solutions {#ap}
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In this section we define the approximate solutions and present some of their most important properties. For each 