American Institute of Mathematical Sciences

Advanced
June  2014, 7(2): 381-400. doi: 10.3934/krm.2014.7.381

Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation

Rong Yang 1, and  Li Chen 2,

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China
2. 

Universität Mannheim, Lehrstuhl für Mathematik IV, 68131, Mannheim, Germany

Received  July 2013 Revised  November 2013 Published  March 2014

A Collision-Avoiding flocking particle system proposed in [8] is studied in this paper. The global wellposedness of its corresponding Vlasov-type kinetic equation is proved. As a corollary of the global stability result, the mean field limit of the particle system is obtained. Furthermore, the time-asymptotic flocking behavior of the solution to the kinetic equation is also derived. The technics used for local wellposedness and stability follow from similar ideas to those have been used in [3,14,22]. While in order to extend the local result globally, the main contribution here is to generate a series of new estimates for this Vlasov type equation, which imply that the growing of the characteristics can be controlled globally. Further estimates also show the long time flocking phenomena.
Keywords: mean-field limit, kinetic equation, flocking., Particle system, Monge-Kantorovich-Rubinstein distance.
Mathematics Subject Classification: Primary: 35A05, 35B40, 35D99; Secondary: 92D1.
Citation: Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381
References:
[1]

M. Aldana and C. Huepe, Phase transition in self-driven many-particle systems and related non-equilibrium models: network approach,, J. Stat. Phys., 112 (2003), 135.   Google Scholar

[2]

B. Birnir, An ODE model of the motion of pelagic fish,, J. Stat. Phys., 128 (2007), 535.  doi: 10.1007/s10955-007-9292-2.  Google Scholar

[3]

J. A Canizo, J. A Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion,, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515.  doi: 10.1142/S0218202511005131.  Google Scholar

[4]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic cucker-smale model,, SIAM Journal on Mathematical Analysis, 42 (2010), 218.  doi: 10.1137/090757290.  Google Scholar

[5]

A. Chakraborti, Distributions of money in models of market economy,, Int. J. Modern Phys., 13 (2002), 1315.  doi: 10.1142/S0129183102003905.  Google Scholar

[6]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. Chayes, State transitions and continuum limit for a 2D interacting, self-propelles particle system,, Physica D, 232 (2007), 33.  doi: 10.1016/j.physd.2007.05.007.  Google Scholar

[7]

I. D. Couzin, J. Krause, N. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move,, Nature, 433 (2005), 513.  doi: 10.1038/nature03236.  Google Scholar

[8]

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

[9]

F. Cucker and S. Smale, On the mathematics of emergence,, Japan. J. Math, 2 (2007), 197.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1193.  doi: 10.1142/S0218202508003005.  Google Scholar

[14]

R. Dobrushin, Vlasov equations,, Funct. Anal. Appl., 13 (1979), 115.   Google Scholar

[15]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006).  doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[16]

A. Dragurlescu and V. M. Yakovenko, Statistical mechanics of money,, Eur. Phys. Jou., 17 (2000), 723.   Google Scholar

[17]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis,, Discrete and Continuous Dynamical Systems B, 12 (2009), 77.  doi: 10.3934/dcdsb.2009.12.77.  Google Scholar

[18]

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

[19]

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

[20]

D. Helbing, Traffic and related self-driven many particle systems,, Review of Modern Physics, 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[21]

H. Levine W.-J. Rappel, Self-organization in systems of self-propelled particles,, Phys. Rev. E, 63 (2000).   Google Scholar

[22]

H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles,, Trans. Fluid Dynamics, 18 (1977), 663.   Google Scholar

[23]

J. K. Parrish and W.-J. Rappel, Self-orgainzed fish schools: an examination of emergent properties,, The Biological Bulletin, 202 (2002), 296.   Google Scholar

[24]

L. Perea, G. Gomez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formations,, AIAA Journal of Guidance, 32 (2009), 527.  doi: 10.2514/1.36269.  Google Scholar

[25]

J. Shen, Cucker-Smale flocking under hierarchical leadership,, SIAM J. Appl. Math., 68 (2008), 694.  doi: 10.1137/060673254.  Google Scholar

[26]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups,, SIAM J. Appl. Math., 65 (2004), 152.  doi: 10.1137/S0036139903437424.  Google Scholar

[27]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bulletin of Mathematical Biology, 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[28]

T. Vicsek, Czirok, E. Beb-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[29]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (2003).  doi: 10.1007/b12016.  Google Scholar

show all references

References:
[1]

M. Aldana and C. Huepe, Phase transition in self-driven many-particle systems and related non-equilibrium models: network approach,, J. Stat. Phys., 112 (2003), 135.   Google Scholar

[2]

B. Birnir, An ODE model of the motion of pelagic fish,, J. Stat. Phys., 128 (2007), 535.  doi: 10.1007/s10955-007-9292-2.  Google Scholar

[3]

J. A Canizo, J. A Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion,, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515.  doi: 10.1142/S0218202511005131.  Google Scholar

[4]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic cucker-smale model,, SIAM Journal on Mathematical Analysis, 42 (2010), 218.  doi: 10.1137/090757290.  Google Scholar

[5]

A. Chakraborti, Distributions of money in models of market economy,, Int. J. Modern Phys., 13 (2002), 1315.  doi: 10.1142/S0129183102003905.  Google Scholar

[6]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. Chayes, State transitions and continuum limit for a 2D interacting, self-propelles particle system,, Physica D, 232 (2007), 33.  doi: 10.1016/j.physd.2007.05.007.  Google Scholar

[7]

I. D. Couzin, J. Krause, N. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move,, Nature, 433 (2005), 513.  doi: 10.1038/nature03236.  Google Scholar

[8]

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

[9]

F. Cucker and S. Smale, On the mathematics of emergence,, Japan. J. Math, 2 (2007), 197.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1193.  doi: 10.1142/S0218202508003005.  Google Scholar

[14]

R. Dobrushin, Vlasov equations,, Funct. Anal. Appl., 13 (1979), 115.   Google Scholar

[15]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006).  doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[16]

A. Dragurlescu and V. M. Yakovenko, Statistical mechanics of money,, Eur. Phys. Jou., 17 (2000), 723.   Google Scholar

[17]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis,, Discrete and Continuous Dynamical Systems B, 12 (2009), 77.  doi: 10.3934/dcdsb.2009.12.77.  Google Scholar

[18]

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

[19]

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

[20]

D. Helbing, Traffic and related self-driven many particle systems,, Review of Modern Physics, 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[21]

H. Levine W.-J. Rappel, Self-organization in systems of self-propelled particles,, Phys. Rev. E, 63 (2000).   Google Scholar

[22]

H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles,, Trans. Fluid Dynamics, 18 (1977), 663.   Google Scholar

[23]

J. K. Parrish and W.-J. Rappel, Self-orgainzed fish schools: an examination of emergent properties,, The Biological Bulletin, 202 (2002), 296.   Google Scholar

[24]

L. Perea, G. Gomez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formations,, AIAA Journal of Guidance, 32 (2009), 527.  doi: 10.2514/1.36269.  Google Scholar

[25]

J. Shen, Cucker-Smale flocking under hierarchical leadership,, SIAM J. Appl. Math., 68 (2008), 694.  doi: 10.1137/060673254.  Google Scholar

[26]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups,, SIAM J. Appl. Math., 65 (2004), 152.  doi: 10.1137/S0036139903437424.  Google Scholar

[27]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bulletin of Mathematical Biology, 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[28]

T. Vicsek, Czirok, E. Beb-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[29]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (2003).  doi: 10.1007/b12016.  Google Scholar

[1]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic & Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385
[2]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
[3]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013
[4]

Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086
[5]

Seung-Yeal Ha, Eitan Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic & Related Models, 2008, 1 (3) : 415-435. doi: 10.3934/krm.2008.1.415
[6]

Charles Bordenave, David R. McDonald, Alexandre Proutière. A particle system in interaction with a rapidly varying environment: Mean field limits and applications. Networks & Heterogeneous Media, 2010, 5 (1) : 31-62. doi: 10.3934/nhm.2010.5.31
[7]

Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045
[8]

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
[9]

Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic & Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052
[10]

Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299
[11]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
[12]

Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086
[13]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929
[14]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303
[15]

Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126
[16]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019111
[17]

Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
[18]

Pierre Degond, Simone Goettlich, Axel Klar, Mohammed Seaid, Andreas Unterreiter. Derivation of a kinetic model from a stochastic particle system. Kinetic & Related Models, 2008, 1 (4) : 557-572. doi: 10.3934/krm.2008.1.557
[19]

Zuo Quan Xu, Jia-An Yan. A note on the Monge-Kantorovich problem in the plane. Communications on Pure & Applied Analysis, 2015, 14 (2) : 517-525. doi: 10.3934/cpaa.2015.14.517
[20]

Jesus Garcia Azorero, Juan J. Manfredi, I. Peral, Julio D. Rossi. Limits for Monge-Kantorovich mass transport problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 853-865. doi: 10.3934/cpaa.2008.7.853

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences