Sharp conditions to avoid collisions in singular Cucker–Smale interactions
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 col...
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| Published in | Nonlinear analysis: real world applications Vol. 37; pp. 317 - 328 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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01.10.2017
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| 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. |
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| AbstractList | 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. We consider the Cucker-Smale flocking model with a singular communication weight...(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...σ(s) = (s-σ)-α, σ ≥ 0. For such model we provide a uniform with respect to the number of particles estimate that controls the σ-distance between particles. In case of σ = 0 it reduces to the estimate of collision avoidance.(ProQuest: ... denotes formulae/symbols omitted.) |
| Author | Mucha, Piotr B. Carrillo, José A. Choi, Young-Pil Peszek, Jan |
| Author_xml | – sequence: 1 givenname: José A. surname: Carrillo fullname: Carrillo, José A. email: carrillo@imperial.ac.uk organization: Department of Mathematics, Imperial College London, SW7 2AZ, London, United Kingdom – sequence: 2 givenname: Young-Pil surname: Choi fullname: Choi, Young-Pil email: ychoi@ma.tum.de organization: Department of Mathematics, Inha University, Incheon, 402-501, Republic of Korea – sequence: 3 givenname: Piotr B. surname: Mucha fullname: Mucha, Piotr B. email: p.mucha@mimuw.edu.pl organization: Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland – sequence: 4 givenname: Jan surname: Peszek fullname: Peszek, Jan email: j.peszek@mimuw.edu.pl organization: Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland |
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| Keywords | Non-collisioness gases Global in time existence Cucker–Smale system Flocking models Singular communication weight |
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| Snippet | 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... We consider the Cucker-Smale flocking model with a singular communication weight...(s) = s-α with α > 0. We provide a critical value of the exponent α in the... |
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| SubjectTerms | Collision avoidance Cucker–Smale system Flocking models Global in time existence Non-collisioness gases Singular communication weight Weight |
| Title | Sharp conditions to avoid collisions in singular Cucker–Smale interactions |
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