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