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. [1, 2, 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 $\psi (s)=s^{-\alpha }$ for $\alpha \ge 1$, 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 $\mathrm{\Gamma }(\cdot )$ introduced in [12] (typical examples being $\mathrm{\Gamma }(v)=v\mathrm{\mid }v{\mathrm{\mid }}^{2(\gamma -1)}$ for $\gamma \in (\frac{1}{2},\frac{3}{2})$), and prove that no collisions can happen in finite time when $\alpha \ge 1$. We also show uniform estimates for the minimum inter-particle distance, for a communication weight with expanded singularity ${\psi }_{\delta }(s)=(s-\delta {)}^{-\alpha }$, when $\alpha \ge 2\gamma$, $\delta \geq 0$.