stated in Proposition \[properties\].

\[properties2\] Properties 1,2,6,7 from Proposition \[properties\] remain true also for the solution $x$ on $[0,T_0)$. Moreover the following version of properties 3 and 4 holds:

[*$(4^{'})$*]{}

:   For all initial data $x(0)$ and $v(0)$ and all $i=1,...,N$, we have $$\begin{aligned}
    \|v_i\|_{L^\infty([0,T_0))}\leq M,\end{aligned}$$ where $M$ is the constant from property 4 from Proposition \[properties\].

Properties 1,2,$4^{'}$ follow by similar argumentation as in the proof of Proposition \[properties\]. Property 6 follows by definition of our system (namely by substituing equation (\[cs2\]) with $(\ref{cs})_2$) and property 7 follows by calculating the derivative of $$\begin{aligned}
r(t)=\sum_{i,j}(v_i-v_j)^2.\end{aligned}$$

Property 5 from Proposition \[properties\] clearly does not hold on $[0,T_0)$ even though it holds on $[0,t]$ for $t\nearrow T_0$ (to see this, we simply substitute in property 5, $n$ with $\lfloor\delta_t\rfloor+2$, where $\delta_t$ is defined in part [**(A)**]{} of the proof of Proposition \[odc\]).

Clustering at the time of collision {#clus}
-----------------------------------

In the previous section we established existence of solutions on the interval $[0,T_0)$, where $T_0$ is time of the first collision of some pair of particles. The solution $x$ belongs to $(W^{2,1}([0,t]))^{Nd}\cap (C^{1}([0,T_0)))^{Nd}\cap (C([0,T_0]))^{Nd}$ for all $0<t<T_0$ and satisfies (\[cs\]) in a classical sense in $(0,T_0)$ and weakly in $(W^{2,1}([0,t]))^{Nd}$. Therefore 