October  2019, 12(5): 1045-1067. doi: 10.3934/krm.2019039

A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea

2. 

Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea

3. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea

4. 

Mathematisches Institut, Ludwig-Maximilians-Universiät, Theresienstr. 39, Munich 80333, Germany

5. 

Center for Mathematical Sciences, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China

* Corresponding author

Received  August 2018 Revised  February 2019 Published  July 2019

We present a probabilistic approach for derivation of the kinetic Cucker-Smale (C-S) equation from the particle C-S model with singular communication. For the system we are considering, it is impossible to validate effective description for certain special initial data, thus such a probabilistic approach is the best one can hope for. More precisely, we consider a system in which kinetic trajectories are deviated from a microscopic model and use a suitable probability measure to quantify the randomness in the initial data. We show that the set of "bad initial data" does in fact have small measure and that this small probability decays to zero algebraically, as $ N $ tends to infinity. For this, we introduce an appropriate cut-off in the communication weight. We also provide a relation between the order of the singularity and the order of the cut-off such that the machinery for deriving classical mean-field limits introduced in [3] can be applied to our setting.

Citation: 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 & Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039
References:
[1]

S. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10. Google Scholar

[2]

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. Google Scholar

[3]

N. Boers and P. Pickl, On mean field limits for dynamical systems, J. Stat. Phys., 164 (2016), 1-16. doi: 10.1007/s10955-015-1351-5. Google Scholar

[4]

F. BolleyJ. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702. Google Scholar

[5]

J. A. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, MMCS, Mathematical Modeling of Complex Systems, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 47 (2014), 17–35. doi: 10.1051/proc/201447002. Google Scholar

[6]

J. A. CarrilloY.-P. ChoiP. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328. doi: 10.1016/j.nonrwa.2017.02.017. Google Scholar

[7]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539. doi: 10.1142/S0218202511005131. Google Scholar

[8]

J. A. CarrilloM. R. D' Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378. doi: 10.3934/krm.2009.2.363. Google Scholar

[9]

J. A. CarrilloM. FornasierJ. 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. Google Scholar

[10]

J. A. CarrilloA. KlarS. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684. Google Scholar

[11]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 28 (2016), 1191-1218. doi: 10.1142/S0218202516500287. Google Scholar

[12]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for agent-based models with unit speed constraint, Anal. Appl., 14 (2016), 39-73. doi: 10.1142/S0219530515400023. Google Scholar

[13]

Y.-P. Choi, S.-Y. Ha and Z. Li., Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkh'áuser/Springer, Cham, 1 (2017) 299–331. Google Scholar

[14]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243. doi: 10.1109/TAC.2010.2042355. Google Scholar

[15]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002. Google Scholar

[16]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. Google Scholar

[17]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Math. Acad. Sci. Paris, 345 (2007), 555-560. doi: 10.1016/j.crma.2007.10.024. Google Scholar

[18]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021. doi: 10.1007/s10955-008-9529-8. Google Scholar

[19]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005. Google Scholar

[20]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z. Google Scholar

[21]

M. FornasierJ. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Phys. D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003. Google Scholar

[22]

S.-Y. Ha, T. Ha and J. Kim, Asymptotic dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A, 43 (2010), 315201, 19pp. doi: 10.1088/1751-8113/43/31/315201. Google Scholar

[23]

S.-Y. HaJ. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181. doi: 10.3934/krm.2018045. Google Scholar

[24]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9. Google Scholar

[25]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar

[26]

S.-Y. Ha and M. Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations, 22 (2010), 325-330. doi: 10.1007/s10884-009-9142-9. Google Scholar

[27]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. Google Scholar

[28]

M. Hauray and P.-E. Jabin, $N$-particles Approximation of the Vlasov Equations with Singular Potential, Arch. Ration. Mech. Anal., 183 (2007), 489-524. doi: 10.1007/s00205-006-0021-9. Google Scholar

[29]

M. K.-H. Kiessling, The microscopic foundations of Vlasov theory for jellium-like Newtonian $N$-body systems, J. Stat. Phys., 155 (2014), 1299-1328. doi: 10.1007/s10955-014-0934-x. Google Scholar

[30]

J. Kim and J. Peszek, Cucker-Smale model with a bonding force and singular interaction kernel, preprint.Google Scholar

[31]

D. Lazarovici and P. Pickl, A mean field limit for the Vlasov-Poisson system, Arch. Ration. Mech. Anal., 225 (2017), 1201-1231. doi: 10.1007/s00205-017-1125-0. Google Scholar

[32]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774. Google Scholar

[33]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9. Google Scholar

[34]

P. B. Mucha and J. Peszek, The Cucker-Smale equation: singular communication weight, measure solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308. doi: 10.1007/s00205-017-1160-x. Google Scholar

[35]

J. ParkH. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070. Google Scholar

[36]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formations, J. Guid. Control Dynam., 32 (2009), 526-536. doi: 10.2514/1.36269. Google Scholar

[37]

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925. doi: 10.1016/j.jde.2014.06.003. Google Scholar

[38]

J. Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686. doi: 10.1137/15M1009299. Google Scholar

[39]

D. Poyato and J. Soler, Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models, Math. Models Methods Appl. Sci., 27 (2017), 1089-1152. doi: 10.1142/S0218202517400103. Google Scholar

[40]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719. doi: 10.1137/060673254. Google Scholar

[41]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

show all references

References:
[1]

S. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10. Google Scholar

[2]

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. Google Scholar

[3]

N. Boers and P. Pickl, On mean field limits for dynamical systems, J. Stat. Phys., 164 (2016), 1-16. doi: 10.1007/s10955-015-1351-5. Google Scholar

[4]

F. BolleyJ. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702. Google Scholar

[5]

J. A. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, MMCS, Mathematical Modeling of Complex Systems, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 47 (2014), 17–35. doi: 10.1051/proc/201447002. Google Scholar

[6]

J. A. CarrilloY.-P. ChoiP. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328. doi: 10.1016/j.nonrwa.2017.02.017. Google Scholar

[7]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539. doi: 10.1142/S0218202511005131. Google Scholar

[8]

J. A. CarrilloM. R. D' Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378. doi: 10.3934/krm.2009.2.363. Google Scholar

[9]

J. A. CarrilloM. FornasierJ. 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. Google Scholar

[10]

J. A. CarrilloA. KlarS. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684. Google Scholar

[11]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 28 (2016), 1191-1218. doi: 10.1142/S0218202516500287. Google Scholar

[12]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for agent-based models with unit speed constraint, Anal. Appl., 14 (2016), 39-73. doi: 10.1142/S0219530515400023. Google Scholar

[13]

Y.-P. Choi, S.-Y. Ha and Z. Li., Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkh'áuser/Springer, Cham, 1 (2017) 299–331. Google Scholar

[14]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243. doi: 10.1109/TAC.2010.2042355. Google Scholar

[15]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002. Google Scholar

[16]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. Google Scholar

[17]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Math. Acad. Sci. Paris, 345 (2007), 555-560. doi: 10.1016/j.crma.2007.10.024. Google Scholar

[18]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021. doi: 10.1007/s10955-008-9529-8. Google Scholar

[19]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005. Google Scholar

[20]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z. Google Scholar

[21]

M. FornasierJ. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Phys. D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003. Google Scholar

[22]

S.-Y. Ha, T. Ha and J. Kim, Asymptotic dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A, 43 (2010), 315201, 19pp. doi: 10.1088/1751-8113/43/31/315201. Google Scholar

[23]

S.-Y. HaJ. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181. doi: 10.3934/krm.2018045. Google Scholar

[24]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9. Google Scholar

[25]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar

[26]

S.-Y. Ha and M. Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations, 22 (2010), 325-330. doi: 10.1007/s10884-009-9142-9. Google Scholar

[27]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. Google Scholar

[28]

M. Hauray and P.-E. Jabin, $N$-particles Approximation of the Vlasov Equations with Singular Potential, Arch. Ration. Mech. Anal., 183 (2007), 489-524. doi: 10.1007/s00205-006-0021-9. Google Scholar

[29]

M. K.-H. Kiessling, The microscopic foundations of Vlasov theory for jellium-like Newtonian $N$-body systems, J. Stat. Phys., 155 (2014), 1299-1328. doi: 10.1007/s10955-014-0934-x. Google Scholar

[30]

J. Kim and J. Peszek, Cucker-Smale model with a bonding force and singular interaction kernel, preprint.Google Scholar

[31]

D. Lazarovici and P. Pickl, A mean field limit for the Vlasov-Poisson system, Arch. Ration. Mech. Anal., 225 (2017), 1201-1231. doi: 10.1007/s00205-017-1125-0. Google Scholar

[32]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774. Google Scholar

[33]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9. Google Scholar

[34]

P. B. Mucha and J. Peszek, The Cucker-Smale equation: singular communication weight, measure solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308. doi: 10.1007/s00205-017-1160-x. Google Scholar

[35]

J. ParkH. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070. Google Scholar

[36]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formations, J. Guid. Control Dynam., 32 (2009), 526-536. doi: 10.2514/1.36269. Google Scholar

[37]

J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925. doi: 10.1016/j.jde.2014.06.003. Google Scholar

[38]

J. Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686. doi: 10.1137/15M1009299. Google Scholar

[39]

D. Poyato and J. Soler, Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models, Math. Models Methods Appl. Sci., 27 (2017), 1089-1152. doi: 10.1142/S0218202517400103. Google Scholar

[40]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719. doi: 10.1137/060673254. Google Scholar

[41]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

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