2018/07/02 by Ioannis N. Markou, Markou, Ioannis
Mathematics · Physics and Astronomy · #82C22 #92D50 #Classical Analysis and ODEs (math.CA) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #Opinion Dynamics and Social Influence #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1807.00485
openalex publication_date 2018/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Collision avoidance is an interesting feature of the Cucker-Smale (CS) model\nof flocking that has been studied in many works, e.g. [1, 2, 4, 6, 7, 20, 21,\n22]. In particular, in the case of singular interactions between agents, as is\nthe case of the CS model with communication weights of the type\n\ψ(s)=s-\α for \α \≥ 1, it is important for showing global\nwell-posedness of the underlying particle dynamics. In [4], a proof of the\nnon-collision property for singular interactions is given in the case of the\nlinear CS model, i.e. when the velocity coupling between agents i,j is\nvj-vi. This paper can be seen as an extension of the analysis in [4].\nWe show that particles avoid collisions even when the linear coupling in the CS\nsystem has been substituted with the nonlinear term \Γ(\⋅) introduced\nin [12] (typical examples being \Γ(v)=v|v|2(\γ -1) for \γ \∈\n(\(1)/(2),\(3)/(2))), and prove that no collisions can happen in finite\ntime when \α \≥ 1. We also show uniform estimates for the minimum\ninter-particle distance, for a communication weight with expanded singularity\n\ψ\δ(s)=(s-\δ)-\α, when \α \≥ 2\γ, \δ\n\≥ 0.\n