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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), 135-153. Google Scholar

[2]

B. Birnir, An ODE model of the motion of pelagic fish, J. Stat. Phys., 128 (2007), 535-568. doi: 10.1007/s10955-007-9292-2.  Google Scholar

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

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F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math, 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.  Google Scholar

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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

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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

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P. Degond and S. Motsch, Large-scale dynamics of the persistent turing walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1022. doi: 10.1007/s10955-008-9529-8.  Google Scholar

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

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R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.  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), 1043021. doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

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A. Dragurlescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jou., B, 17 (2000), 723-729. Google Scholar

[17]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete and Continuous Dynamical Systems B, 12 (2009), 77-103. doi: 10.3934/dcdsb.2009.12.77.  Google Scholar

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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), 297-325. 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), 415-435. 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), 1067-1141. doi: 10.1103/RevModPhys.73.1067.  Google Scholar

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H. Levine W.-J. Rappel, Self-organization in systems of self-propelled particles, Phys. Rev. E, 63 (2000), 017101. Google Scholar

[22]

H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles, Trans. Fluid Dynamics, 18 (1977), 663-678. Google Scholar

[23]

J. K. Parrish and W.-J. Rappel, Self-orgainzed fish schools: an examination of emergent properties, The Biological Bulletin, 202 (2002), 296-305. 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, Control, and Dynamics, 32 (2009), 527-537. doi: 10.2514/1.36269.  Google Scholar

[25]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719. 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), 152-174. 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), 1601-1623. 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), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[29]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Amer. Math. Soc, Providence, vol.58, 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), 135-153. Google Scholar

[2]

B. Birnir, An ODE model of the motion of pelagic fish, J. Stat. Phys., 128 (2007), 535-568. 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), 515-539. 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), 218-236. doi: 10.1137/090757290.  Google Scholar

[5]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys., 13 (2002), 1315-1321. 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), 33-47. 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), 513-516. doi: 10.1038/nature03236.  Google Scholar

[8]

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

[9]

F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math, 2 (2007), 197-227. 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), 852-862. 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), 555-560. 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), 989-1022. 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), 1193-1215. doi: 10.1142/S0218202508003005.  Google Scholar

[14]

R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.  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), 1043021. doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[16]

A. Dragurlescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jou., B, 17 (2000), 723-729. Google Scholar

[17]

S.-Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete and Continuous Dynamical Systems B, 12 (2009), 77-103. 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), 297-325. 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), 415-435. 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), 1067-1141. 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), 017101. Google Scholar

[22]

H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles, Trans. Fluid Dynamics, 18 (1977), 663-678. Google Scholar

[23]

J. K. Parrish and W.-J. Rappel, Self-orgainzed fish schools: an examination of emergent properties, The Biological Bulletin, 202 (2002), 296-305. 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, Control, and Dynamics, 32 (2009), 527-537. doi: 10.2514/1.36269.  Google Scholar

[25]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719. 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), 152-174. 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), 1601-1623. 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), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[29]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Amer. Math. Soc, Providence, vol.58, 2003. doi: 10.1007/b12016.  Google Scholar

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