Sharp conditions to avoid collisions in singular Cucker-Smale interactions

We consider the Cucker-Smale flocking model with a singular communication weight $\psi (s)=s^{-\alpha }$ with $\alpha >0$. We provide a critical value of the exponent $\alpha$ in the communication weight leading to global regularity of solutions or finite-time collision between particles. For $\alpha \geq 1$, we show that there is no collision between particles in finite time if they are placed in different positions initially. For $\alpha \ge 2$ we investigate a version of the Cucker-Smale model with expanded singularity i.e. with weight ${\psi }_{\delta }(s)=(s-\delta {)}^{-\alpha }$, $\delta \ge 0$. For such model we provide a uniform with respect to the number of particles estimate that controls the $\delta$-distance between particles. In case of $\delta =0$ it reduces to the estimate of non-collisioness.