Sharp conditions to avoid collisions in singular Cucker–Smale interactions

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

doi:10.1016/j.nonrwa.2017.02.017

Sharp conditions to avoid collisions in singular Cucker–Smale interactions

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

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

67 Citations (Scopus)

Abstract

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 ψ_δ(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 language	English
Pages (from-to)	317-328
Number of pages	12
Journal	Nonlinear Analysis: Real World Applications
Volume	37
DOIs	https://doi.org/10.1016/j.nonrwa.2017.02.017
Publication status	Published - 2017 Oct 1

Bibliographical note

Funding 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.

Publisher Copyright:
© 2017 Elsevier Ltd

All Science Journal Classification (ASJC) codes

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

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10.1016/j.nonrwa.2017.02.017

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title = "Sharp conditions to avoid collisions in singular Cucker–Smale interactions",

abstract = "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.",

author = "Carrillo, {Jos{\'e} A.} and Choi, {Young Pil} and Mucha, {Piotr B.} and Jan Peszek",

note = "Funding 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. Publisher Copyright: {\textcopyright} 2017 Elsevier Ltd",

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AU - Mucha, Piotr B.

AU - Peszek, Jan

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PY - 2017/10/1

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AB - 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.

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Sharp conditions to avoid collisions in singular Cucker–Smale interactions

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