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]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[2]

Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167
[3]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[4]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[5]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267
[6]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
[7]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304
[8]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076
[9]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
[10]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353
[11]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303
[12]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[13]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291
[14]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[15]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[16]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[17]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467
[18]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273
[19]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357
[20]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences