exists a $(C([0,T]))^{Nd}$ convergent subsequence $x^{n_k}$. From this point we pick one of such convergent subsequences and aim to prove that it has a $(C^1([0,T]))^{Nd}$ convergent subsequence. For simplicity of notation we will assume that $x^n=x^{n_k}$.

In this section we will prove that the approximate solutions converge in every interval $[0,t]\subset[0,T_0)$, where $T_0$ is the time of the first collision of the particles. Additionaly we will prove that their limit is a weak solution in $(W^{2,1}([0,t]))^{Nd}$. Let us begin with defining $T_0$ by means of the approximate solutions: $$\begin{aligned}
T_0:=\inf\{t>0: \min_{\stackrel{i=1,...,N}{j\in B_i(0)}}\lim_{n\to\infty}|x_i^n-x_j^n|=0\}.\end{aligned}$$ Note that the limit in the above definition exists, since we are restricted to a $(C([0,T]))^{Nd}$ convergent subsequence.

\[t0\] Clearly if $t<T_0$ then there exists $\delta>0$, such that $$\begin{aligned}
\min_{\stackrel{i=1,...,N}{j\in B_i(0)}}\lim_{n\to\infty}|x_i^n-x_j^n|>\delta,\end{aligned}$$ which further implies that for all $i,j$, there exists $n_0$ such that for all $n>n_0$, we have $|x_i^n-x_j^n|>\delta$. On the other hand $$\begin{aligned}
\lim_{n\to\infty}|x_i^{n}(T_0)-x_j^{n}(T_0)|=0\end{aligned}$$ and assuming that $x$ is a $(C([0,T_0]))^{Nd}$ limit of $x^{n}$, we have $x_i(T_0)=x_j(T_0)$, which means that $T_0$ is a point of collision for $x$.

\[odc\] For $n=1,2,...$ let $x^n$ be a solution to C-S system on interval $[0,T]$ with weight $\psi_n$ and an independent of $n$ initial data $x(0)$ and $v(0)$. There exists 