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.
Bibliographical noteFunding Information:
J.A.C. was partially supported by the Royal Society via a Wolfson Research Merit and by the EPSRC under the Grant EP/P031587/1. Y.P.C. was supported by the ERC-Starting grant HDSPCONTR ?High-Dimensional Sparse Optimal Control?. J.A.C. and Y.P.C. were partially supported by EPSRC grant EP/K008404/1. YPC is also supported by the Alexander von Humboldt Foundation through the Humboldt Research Fellowship for Postdoctoral Researchers. J.P. was supported by the Polish NCN grant PRELUDIUM 2013/09/N/ST1/04113.
© 2017 Elsevier Ltd
All Science Journal Classification (ASJC) codes
- Economics, Econometrics and Finance(all)
- Computational Mathematics
- Applied Mathematics