# American Institute of Mathematical Sciences

• Previous Article
Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum
• KRM Home
• This Issue
• Next Article
The two dimensional Vlasov-Poisson system with steady spatial asymptotics
December  2017, 10(4): 1011-1033. doi: 10.3934/krm.2017040

## Cucker-Smale model with normalized communication weights and time delay

 1 Department of Mathematics, Inha University, Incheon, 402-751, Republic of Korea 2 Computer, Electrical and Mathematical Sciences & Engineering, King Abdullah University of Science and Technology, 23955 Thuwal, KSA

Received  August 2016 Revised  January 2017 Published  March 2017

Fund Project: YPC was supported by Engineering and Physical Sciences Research Council (EP/K00804/1) and ERC-Starting grant HDSPCONTR "High-Dimensional Sparse Optimal Control". He was also supported by the Alexander Humboldt Foundation through the Humboldt Research Fellowship for Postdoctoral Researchers. JH was supported by KAUST baseline funds and KAUST grant no. 1000000193.

We study a Cucker-Smale-type system with time delay in which agents interact with each other through normalized communication weights. We construct a Lyapunov functional for the system and provide sufficient conditions for asymptotic flocking, i.e., convergence to a common velocity vector. We also carry out a rigorous limit passage to the mean-field limit of the particle system as the number of particles tends to infinity. For the resulting Vlasov-type equation we prove the existence, stability and large-time behavior of measure-valued solutions. This is, to our best knowledge, the first such result for a Vlasov-type equation with time delay. We also present numerical simulations of the discrete system with few particles that provide further insights into the flocking and oscillatory behaviors of the particle velocities depending on the size of the time delay.

Citation: Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic and Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040
##### References:
 [1] S. Ahn and S.-Y Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp.  doi: 10.1063/1.3496895. [2] J. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, ESAIM Proc., 47 (2014), 17-35.  doi: 10.1051/proc/201447002. [3] J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation (eds. A. Muntean and F. Toschi), Springer Series: CISM International Centre for Mechanical Sciences, 533 (2014), 1–46. doi: 10.1007/978-3-7091-1785-9_1. [4] J. Carrillo, Y. -P. Choi, M. Hauray and S. Salem, Mean-field limit for collective behavior models with sharp sensitivity regions, preprint, arXiv: 1510.02315. [5] J. Carrillo, Y. -P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, preprint, arXiv: 1609.03447. [6] J. Carrillo, Y. -P. Choi and S. Pérez, A review an attractive-repulsive hydrodynamics for consensus in collective behavior, in Active Particles Vol. Ⅰ -Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser-Springer Series: Modelling and Simulation in Science and Technology, (2017). [7] J. Cañizo, J. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131. [8] J. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic Flocking Dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290. [9] J. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi and G. Toscani), Birkhäuser Series: Modelling and Simulation in Science and Technology, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12. [10] Y.-P. Choi, Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.  doi: 10.1088/0951-7715/29/7/1887. [11] Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser-Springer Series: Modelling and Simulation in Science and Technology, 2017. [12] F. Cucker and S. Smale, Emergent behaviour in flocks, IEEE T. on Automat. Contr., 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [13] F. Cucker and S. Smale, On the mathematics of emergence, Jap. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x. [14] R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z. [15] R. Erban, J. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.  doi: 10.1137/15M1030467. [16] S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Comm. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9. [17] S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Comm. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2. [18] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415. [19] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags Academic Press, New York London, 1966. [20] J. Haskovec, Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Physica D, 261 (2013), 42-51.  doi: 10.1016/j.physd.2013.06.006. [21] P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinetic and Related models, 7 (2014), 661-711.  doi: 10.3934/krm.2014.7.661. [22] Y. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl., 415 (2014), 53-61.  doi: 10.1016/j.jmaa.2014.01.036. [23] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behaviour, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9. [24] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods Oxford University Press, 2014. [25] J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differ. Equat., 257 (2014), 2900-2925.  doi: 10.1016/j.jde.2014.06.003. [26] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences Springer, New York Dordrecht Heidelberg London, 2011. doi: 10.1007/978-1-4419-7646-8. [27] D. Sumpter, Collective Animal Behavior Princeton University Press, 2010. doi: 10.1515/9781400837106. [28] T. Ton, N. Linh and A. Yagi, Flocking and non-flocking behaviour in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.  doi: 10.1142/S0219530513500255. [29] T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.

show all references

##### References:
 [1] S. Ahn and S.-Y Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp.  doi: 10.1063/1.3496895. [2] J. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, ESAIM Proc., 47 (2014), 17-35.  doi: 10.1051/proc/201447002. [3] J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation (eds. A. Muntean and F. Toschi), Springer Series: CISM International Centre for Mechanical Sciences, 533 (2014), 1–46. doi: 10.1007/978-3-7091-1785-9_1. [4] J. Carrillo, Y. -P. Choi, M. Hauray and S. Salem, Mean-field limit for collective behavior models with sharp sensitivity regions, preprint, arXiv: 1510.02315. [5] J. Carrillo, Y. -P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, preprint, arXiv: 1609.03447. [6] J. Carrillo, Y. -P. Choi and S. Pérez, A review an attractive-repulsive hydrodynamics for consensus in collective behavior, in Active Particles Vol. Ⅰ -Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser-Springer Series: Modelling and Simulation in Science and Technology, (2017). [7] J. Cañizo, J. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131. [8] J. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic Flocking Dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290. [9] J. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi and G. Toscani), Birkhäuser Series: Modelling and Simulation in Science and Technology, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12. [10] Y.-P. Choi, Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.  doi: 10.1088/0951-7715/29/7/1887. [11] Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser-Springer Series: Modelling and Simulation in Science and Technology, 2017. [12] F. Cucker and S. Smale, Emergent behaviour in flocks, IEEE T. on Automat. Contr., 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [13] F. Cucker and S. Smale, On the mathematics of emergence, Jap. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x. [14] R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z. [15] R. Erban, J. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.  doi: 10.1137/15M1030467. [16] S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Comm. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9. [17] S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Comm. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2. [18] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415. [19] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags Academic Press, New York London, 1966. [20] J. Haskovec, Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Physica D, 261 (2013), 42-51.  doi: 10.1016/j.physd.2013.06.006. [21] P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinetic and Related models, 7 (2014), 661-711.  doi: 10.3934/krm.2014.7.661. [22] Y. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl., 415 (2014), 53-61.  doi: 10.1016/j.jmaa.2014.01.036. [23] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behaviour, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9. [24] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods Oxford University Press, 2014. [25] J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differ. Equat., 257 (2014), 2900-2925.  doi: 10.1016/j.jde.2014.06.003. [26] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences Springer, New York Dordrecht Heidelberg London, 2011. doi: 10.1007/978-1-4419-7646-8. [27] D. Sumpter, Collective Animal Behavior Princeton University Press, 2010. doi: 10.1515/9781400837106. [28] T. Ton, N. Linh and A. Yagi, Flocking and non-flocking behaviour in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.  doi: 10.1142/S0219530513500255. [29] T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.
The system with two particles: Particle velocities $v_1(t)$, $v_2(t)$ as solututions of (30) (left panels) and velocity diameters $d_V(t)$ (right panels, logarithmic scale) for $\tau=0.25$ (first row) and $\tau=1$ (second row). The initial condition is constant, $v_1(t)\equiv 1$, $v_2(t)\equiv -1$ for $t\in[-\tau,0]$
The system with three particles: particle velocities $v_1(t)$, $v_2(t), v_3(t)$ as solutions of (5)–(7) (left panels) and velocity diameters $d_V(t)$ (right panels, logarithmic scale) for $\tau=0.25$ (first row) and $\tau=1$ (second row), with exponentially decaying influence function $\psi(s) = e^{-s}$. The initial condition is in both cases given by (33)–(34)
The system with four particles: particle velocities as solutions of (5)–(7) (left panels) and velocity diameters $d_V(t)$ (right panels, logarithmic scale) for $\tau=0.25$ (first row) and $\tau=1$ (second row), with the influence function $\psi(s) = {(1+s^2)^{-4}}$. The initial condition is in both cases given by (35)–(36)
 [1] Young-Pil Choi, Cristina Pignotti. Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays. Networks and Heterogeneous Media, 2019, 14 (4) : 789-804. doi: 10.3934/nhm.2019032 [2] Zili Chen, Xiuxia Yin. The delayed Cucker-Smale model with short range communication weights. Kinetic and Related Models, 2021, 14 (6) : 929-948. doi: 10.3934/krm.2021030 [3] Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2409-2432. doi: 10.3934/dcds.2021195 [4] Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168 [5] Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic and Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027 [6] Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419 [7] Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks and Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017 [8] Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic and Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 [9] Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223 [10] Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062 [11] Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072 [12] Seung-Yeal Ha, Jeongho Kim, Peter Pickl, Xiongtao Zhang. A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication. Kinetic and Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039 [13] Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure and Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028 [14] Jan Haskovec. Cucker-Smale model with finite speed of information propagation: well-posedness, flocking and mean-field limit. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022033 [15] Martin Friesen, Oleksandr Kutoviy. Stochastic Cucker-Smale flocking dynamics of jump-type. Kinetic and Related Models, 2020, 13 (2) : 211-247. doi: 10.3934/krm.2020008 [16] Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023 [17] Laure Pédèches. Asymptotic properties of various stochastic Cucker-Smale dynamics. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2731-2762. doi: 10.3934/dcds.2018115 [18] Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155 [19] Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic and Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023 [20] Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic and Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026

2021 Impact Factor: 1.398