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

Abstract

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 \begin{document}$ψ(s) = s^{-α}$ \end{document} for \begin{document}$α ≥ 1$ \end{document} , 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 \begin{document}$i,j$ \end{document} is \begin{document}$v_{j}-v_{i}$ \end{document} . 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 \begin{document}$Γ(·)$ \end{document} introduced in [ 12 ] (typical examples being \begin{document}$Γ(v) = v|v|^{2(γ -1)}$ \end{document} for \begin{document}$γ ∈ (\frac{1}{2},\frac{3}{2})$ \end{document} ), and prove that no collisions can happen in finite time when \begin{document}$α ≥ 1$ \end{document} . We also show uniform estimates for the minimum inter-particle distance, for a communication weight with expanded singularity \begin{document}$ψ_{δ}(s) = (s-δ)^{-α}$ \end{document} , when \begin{document}$α ≥ 2γ$ \end{document} , \begin{document}$δ ≥ 0$ \end{document} .