# American Institute of Mathematical Sciences

October  2018, 38(10): 5245-5260. doi: 10.3934/dcds.2018232

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

 Institute of Applied and Computational Mathematics, Foundation for Research and Technology - Hellas, N. Plastira 100, Vassilika Vouton, GR - 700 13, Heraklion, Crete, Greece

Received  March 2018 Revised  June 2018 Published  July 2018

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 $α ≥ 2γ$, $δ ≥ 0$.

Citation: Ioannis Markou. Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5245-5260. doi: 10.3934/dcds.2018232
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