Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays

Young-Pil Choi; Cristina Pignotti; Young-Pil Choi; Cristina Pignotti

doi:10.3934/nhm.2019032

Networks and Heterogeneous Media

2019, Volume 14, Issue 4: 789-804. doi: 10.3934/nhm.2019032

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Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays

Young-Pil Choi ^{1
,},
Cristina Pignotti ^{2
,
,}

1.
Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Republic of Korea
2.
Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università di L'Aquila, Via Vetoio, 67010 L'Aquila, Italy

Received: 01 February 2019 Revised: 01 June 2019
Primary: 34A12, 34D05; Secondary: 35Q83

We study a Cucker-Smale-type flocking model with distributed time delay where individuals interact with each other through normalized communication weights. Based on a Lyapunov functional approach, we provide sufficient conditions for the velocity alignment behavior. We then show that as the number of individuals $ N $ tends to infinity, the $ N $-particle system can be well approximated by a delayed Vlasov alignment equation. Furthermore, we also establish the global existence of measure-valued solutions for the delayed Vlasov alignment equation and its large-time asymptotic behavior.
- Cucker-Smale model,
- time delay,
- flocking,
- Vlasov equation
Citation: Young-Pil Choi, Cristina Pignotti. Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays[J]. Networks and Heterogeneous Media, 2019, 14(4): 789-804. doi: 10.3934/nhm.2019032

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Abstract

We study a Cucker-Smale-type flocking model with distributed time delay where individuals interact with each other through normalized communication weights. Based on a Lyapunov functional approach, we provide sufficient conditions for the velocity alignment behavior. We then show that as the number of individuals $ N $ tends to infinity, the $ N $-particle system can be well approximated by a delayed Vlasov alignment equation. Furthermore, we also establish the global existence of measure-valued solutions for the delayed Vlasov alignment equation and its large-time asymptotic behavior.

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