Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings

Collision avoidance is an interesting feature of the Cucker-Smale (CS) model of flocking that has been studied in many works, e.g. [2,1,4,6,7,20,21,22]. In particular, in the case of singular interactions between agents, as is the case of the CS model with communication weights of the type $ψ(s) = s^{-α}$ for $α ≥ 1$ , it is important for showing global well-posedness of the underlying particle dynamics. In [4], a proof of the non-collision property for singular interactions is given in the case of the linear CS model, i.e. when the velocity coupling between agents $i,j$ is $v_{j}-v_{i}$ . This paper can be seen as an extension of the analysis in [4]. We show that particles avoid collisions even when the linear coupling in the CS system has been substituted with the nonlinear term $Γ(·)$ introduced in [12] (typical examples being $Γ(v) = v|v|^{2(γ -1)}$ for $γ ∈ (\frac{1}{2},\frac{3}{2})$ ), and prove that no collisions can happen in finite time when $α ≥ 1$ . We also show uniform estimates for the minimum inter-particle distance, for a communication weight with expanded singularity $ψ_{δ}(s) = (s-δ)^{-α}$ , when $α ≥ 2γ$ , $δ ≥ 0$ .