Sharp conditions to avoid collisions in singular Cucker–Smale interactions

José A. Carrillo, Young Pil Choi, Piotr B. Mucha, Jan Peszek

Research output: Contribution to journalArticle

26 Citations (Scopus)


We consider the Cucker–Smale flocking model with a singular communication weight ψ(s)=s−α with α>0. We provide a critical value of the exponent α in the communication weight leading to global regularity of solutions or finite-time collision between particles. For α≥1, we show that there is no collision between particles in finite time if they are placed in different positions initially. For α≥2 we investigate a version of the Cucker–Smale model with expanded singularity i.e. with weight ψδ(s)=(s−δ)−α, δ≥0. For such model we provide a uniform with respect to the number of particles estimate that controls the δ-distance between particles. In case of δ=0 it reduces to the estimate of collision avoidance.

Original languageEnglish
Pages (from-to)317-328
Number of pages12
JournalNonlinear Analysis: Real World Applications
Publication statusPublished - 2017 Oct 1

All Science Journal Classification (ASJC) codes

  • Analysis
  • Engineering(all)
  • Economics, Econometrics and Finance(all)
  • Computational Mathematics
  • Applied Mathematics

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