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

Ioannis Markou
2018 Discrete and Continuous Dynamical Systems. Series A  
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
more » ... e 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 ∈ ( 1 2 , 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. 2010 Mathematics Subject Classification. 82C22, 92D50.
doi:10.3934/dcds.2018232 fatcat:jwujjhso3fc7fakcmufjnbth3e