### 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 |
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Pages (from-to) | 317-328 |

Number of pages | 12 |

Journal | Nonlinear Analysis: Real World Applications |

Volume | 37 |

DOIs | |

Publication status | Published - 2017 Oct 1 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Nonlinear Analysis: Real World Applications*,

*37*, 317-328. https://doi.org/10.1016/j.nonrwa.2017.02.017

}

*Nonlinear Analysis: Real World Applications*, vol. 37, pp. 317-328. https://doi.org/10.1016/j.nonrwa.2017.02.017

**Sharp conditions to avoid collisions in singular Cucker–Smale interactions.** / Carrillo, José A.; Choi, Young Pil; Mucha, Piotr B.; Peszek, Jan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Sharp conditions to avoid collisions in singular Cucker–Smale interactions

AU - Carrillo, José A.

AU - Choi, Young Pil

AU - Mucha, Piotr B.

AU - Peszek, Jan

PY - 2017/10/1

Y1 - 2017/10/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85015393637&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85015393637&partnerID=8YFLogxK

U2 - 10.1016/j.nonrwa.2017.02.017

DO - 10.1016/j.nonrwa.2017.02.017

M3 - Article

AN - SCOPUS:85015393637

VL - 37

SP - 317

EP - 328

JO - Nonlinear Analysis: Real World Applications

JF - Nonlinear Analysis: Real World Applications

SN - 1468-1218

ER -