$s\geq \delta$ all functions $\psi_n$ coincide. It implies that if we consider functions $x^n$ restricted to $[0,t]$, then Lipschitz continuity with respect to the initial data mentioned in Proposition \[psin\] holds with an $n$ independent constant $L(\lfloor\delta^{-\alpha}\rfloor+2)$ which corresponds to the C-S system associated with the smallest weight for which $\psi_n$ coincide for $s\geq \delta$.

Next, let us state some properties of solutions of the C-S model associated with $\psi_n$.

\[properties\] Let $x^n$ be a solution of the C-S model associated with weight $\psi_n$. Then $x^n$ has the following properties:

1.  It belongs to the class $C^\infty$ in a neighborhood of every such point $t$, that $|x^n_i(t)-x^n_j(t)|>0$ for all $i,j=1,...,N$.

2.  The avarage velocity of the particles is constant: $$\begin{aligned}
    \frac{1}{N}\sum_{i=1}^Nv_i(t)=const.\end{aligned}$$

3.  Velocity $v^n$ is bounded: there exists a constant $M(n)$ such that for all $i=1,...,N$, we have $$\begin{aligned}
    \|v^n_i\|_{L_\infty([0,T])}\leq M(n).\end{aligned}$$

4.  If the initial data $x^n(0), v^n(0)$ are uniformly bounded, then also $v^n$ is uniformly bounded: there exists a constant $M$ such that for all $i=1,...,N$ and all $n=1,2,...$, we have $$\begin{aligned}
    \|v^n_i\|_{L_\infty([0,T])}\leq M.\end{aligned}$$

5.  Acceleration $\dot{v}_i^n$ is bounded by $2M(n)n$.

6.  If at some point $t$ we have $x_i^n(t)=x_j^n(t)$ and $v_i^n(t)=v^n_j(t)$ for any $i,j=1,...,N$, then $x^n_i\equiv x^n_j$ on 