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Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation
1. | Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China |
2. | Universität Mannheim, Lehrstuhl für Mathematik IV, 68131, Mannheim, Germany |
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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