long as it is a nonnegative function. This is the first piece of information on which we base our hope to obtain some existence for C-S model with singular weight $\psi$. The second piece of information is that Lipschitz continuity and boundedness of $\psi$ fails only at $0$, which means that our main problem will be to prove existence in a neighborhood of each time $t_0$ at which some particles collide. However, heuristically in a neighborhood of each such point we have $$\begin{aligned}
x_i(t)-x_j(t)\approx t(v_i(t_0)-v_j(t_0))\approx t(v_i(t)-v_j(t))\end{aligned}$$ and since in (\[cs2\]) the function $t\mapsto\psi(|x_i(t)-x_j(t)|)$ comes always multiplied by $v_i(t)-v_j(t)$, we have [$$\begin{aligned}
\left(\Psi(|x_i(t)-x_j(t)|)\right)^{'}=\psi(|x_i(t)-x_j(t)|)\frac{(x_i(t)-x_j(t))\cdot(v_i(t)-v_j(t))}{|x_i(t)-x_j(t)|}\approx \psi(|x_i(t)-x_j(t)|)(v_i(t)-v_j(t))\end{aligned}$$ ]{} with $\Psi(s):=\frac{1}{1-\alpha}s^{1-\alpha}$ being a primitive of $\psi$, which is a H" older continuous function, thus there is hope for some better regularity of $v$. These two observations were already used in [@haliu] to obtain asymptotic flocking for C-S model with weight $\psi$. Occurrence of asymptotic flocking is a further clue that a C-S model with singular weight inherits some nice properties from the model with a smooth weight. Lastly in [@haliu] existence for the discrete model served as a mean to obtain existence for the continuous model by defining the solution of the continuous model as 