$[t,T]$.

7.  If at some point $t$ we have $v^n_i(t)=v^n_j(t)$ for all $i,j=1,...,N$, then $v^n$ is constant on $[t,T]$.

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1.  Since $x^n$ is continuous, if at some point $t$ all the particles have different positions i.e. $|x^n_i(t)-x^n_j(t)|>0$ for all $i,j=1,...,N$ then it is also true in some neighborhood of $t$. Moreover in this neighborhood of $t$ the right-hand side of $(\ref{n})_2$ is differentiable, which by iteration implies that $x^n$ is smooth at $t$.

2.  This part was already done in section \[int\].

3.  Let $r_n(t):=\sum_{i,j=1}^N(v_i^n(t)-v_j^n(t))^2$. By $(\ref{n})_2$, we have $$\begin{aligned}
    r_n^{'}=2\sum_{i,j=1}^N(v_i^n-v_j^n)\left(\frac{1}{N}\sum_{k=1}^N(v_k^n-v_i^n)\psi_n(|x_i^n-x_k^n|)- \frac{1}{N}\sum_{k=1}^N(v_k^n-v_j^n)\psi_n(|x_j^n-x_k^n|)\right)=\\
    = \frac{2}{N}\sum_{i,j,k=1}^N(v_i^n-v_j^n)(v_k^n-v_i^n)\psi_n(|x_i^n-x_k^n|)- \frac{2}{N}\sum_{i,j,k=1}^N(v_i^n-v_j^n)(v_k^n-v_j^n)\psi_n(|x_j^n-x_k^n|).\end{aligned}$$ Again, we substitute $i$ and $k$ in the first summant and $j$ and $k$ in the second summant to obtain $$\begin{aligned}
    r_n^{'}=\frac{1}{N}\sum_{i,j,k=1}^N(v_i^n-v_j^n)(v_k^n-v_i^n)\psi_n(|x_i^n-x_k^n|) +\frac{1}{N}\sum_{i,j,k=1}^N(v_k^n-v_j^n)(v_i^n-v_k^n)\psi_n(|x_i^n-x_k^n|)\\
    -\frac{1}{N}\sum_{i,j,k=1}^N(v_i^n-v_j^n)(v_k^n-v_j^n)\psi_n(|x_j^n-x_k^n|) -\frac{1}{N}\sum_{i,j,k=1}^N(v_i^n-v_k^n)(v_j^n-v_k^n)\psi_n(|x_j^n-x_k^n|)=\\
    -\frac{1}{N}\sum_{i,j,k=1}^N(v_i^n-v_k^n)^2\psi_n(|x_i^n-x_k^n|) -\frac{1}{N}\sum_{i,j,k=1}^N(v_j^n-v_k^n)^2\psi_n(|x_j^n-x_k^n|)=\\
    =-2\sum_{i,j=1}^N(v_i^n-v_j^n)^2\psi_n(|x_i^n-x_j^n|)\leq 0.\end{aligned}$$ Thus for each $n$, function $r_n$ is nonincreasing with it’s maximum at $0$ i.e. $r_n(t)\leq r_n(0)$. Now let $\bar{v}^n$ be the avarage velocity, which as we know from property 2 is a constant. We have [$$\begin{aligned}
    \sum_{i=1}^N(\bar{v}^n-v^n_i)^2=\sum_{i=1}^N\left(\frac{1}{N}\sum_{j=1}^Nv_j^n-v_i^n\right)^2=\frac{1}{N^2} \sum_{i=1}^N\left(\sum_{j=1}^N(v_j^n-v_i^n)\right)^2
    \leq \frac{1}{N}\sum_{i,j=1}^N(v_j^n-v_i^n)^2=\frac{1}{N}r_n(0).\end{aligned}$$ ]{} Lastly we have $$\begin{aligned}
    |v^n_i|\leq |v^n_i-\bar{v}^n|+|\bar{v}^n|
    \leq C(N)\sqrt{r_n(0)}+|\bar{v}^n|\leq C(N)\sqrt{r_n(0)}=:M(n),\end{aligned}$$ where $C(N)$ is a generic constant depending on $N$.

4.  We simply note that if initial velocity is uniformly bounded, then $M(n)\leq M$ for 