$n$ let $$\begin{aligned}
\psi_n(s)=\left\{
\begin{array}{ccc}
\psi(s) & {\rm if}& s\geq (n-1)^{-\frac{1}{\alpha}},\\
smooth\ and\ monotone & {\rm if}& n^{-\frac{1}{\alpha}}\leq s\leq (n-1)^{-\frac{1}{\alpha}},\\
n & {\rm if}& s\leq n^{-\frac{1}{\alpha}}
\end{array}
\right.\end{aligned}$$ for all $s\in [0,\infty)$ with $\psi$ given by (\[psi\]). For all $n$, functions $\psi_n$ are smooth and bounded, thus C-S systems associated with these weights have unique solutions. This can be expressed by the following proposition.

\[psin\] For each positive integer $n$ and for arbitrary initial data, the system $$\begin{aligned}
\label{n}
\left\{
\begin{array}{lll}
\dot{x}_i^n=&v_i^n,\\
\dot{v}_i^n=&\frac{1}{N}\sum_{k=1}^N(v_k^n-v_i^n)\psi_n(|x_i^n-x_k^n|)
\end{array}
\right.\end{aligned}$$ has a unique global classical solution $x^n$ belonging to the class $(C^2([0,T]))^{Nd}$. Moreover, this solution is stable with respect to the initial data i.e. for all $x^{n,1}(0),x^{n,2}(0),v^{n,1}(0),v^{n,2}(0)$, we have $$\begin{aligned}
\|x^{n,1}-x^{n,2}\|_{W^{1,\infty}([0,T])}< L(n)\left(|x^{n,1}(0)-x^{n,2}(0)|+|v^{n,1}(0)-v^{n,2}(0)|\right),\end{aligned}$$ where $L(n)$ is a constant depending on $n$ (and also on $T$, $d$, $\alpha$ and $N$).

The proof of this proposition is standard and we omit it.

\[stab\] Stability with respect to the initial data is uniform in a neighborhood of each time in which no collision occurs. Indeed, if in some interval $[0,t]$ we have $$\begin{aligned}
\inf_{i,j}|x_i^n-x_j^n|>\delta\end{aligned}$$ for all $n$, then C-S systems associated with $\psi_n$ are exactly the same for all $n>\delta^{-\alpha}+1$. It follows from the fact that for $n>\delta^{-\alpha}+1$ and 