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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 and 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.

[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.

[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.

[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.

[5]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys., 13 (2002), 1315-1321. doi: 10.1142/S0129183102003905.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.

[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.

[16]

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

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

J. K. Parrish and W.-J. Rappel, Self-orgainzed fish schools: an examination of emergent properties, The Biological Bulletin, 202 (2002), 296-305.

[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.

[25]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719. doi: 10.1137/060673254.

[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.

[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.

[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.

[29]

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

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.

[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.

[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.

[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.

[5]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys., 13 (2002), 1315-1321. doi: 10.1142/S0129183102003905.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.

[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.

[16]

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

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

J. K. Parrish and W.-J. Rappel, Self-orgainzed fish schools: an examination of emergent properties, The Biological Bulletin, 202 (2002), 296-305.

[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.

[25]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719. doi: 10.1137/060673254.

[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.

[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.

[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.

[29]

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

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