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 ≥ 1, we show that there is no collision between particles in finite time if they are placed in different positions initially. For α≥ 2 we investigate a version of the Cucker-Smale model with expanded singularity i.e. with weight ψ_δ(s) = (s-δ)-α, δ≥ 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.