Asymptotic flocking dynamics for the kinetic cucker-smale model

SIAM Journal on Mathematical Analysis

Volumn 42, Issue 1, 2010, Pages 218-236

  Carrillot, J A ^a   Fornasier, M ^b   Rosado, J ^a   Toscani, G ^c  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^b JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

^c UNIVERSITY OF PAVIA   (Italy)

Author keywords

Flocking; Mass transportation methods; Nonlinear friction equations

Indexed keywords

A-STABILITY; ASYMPTOTIC BEHAVIOR OF SOLUTIONS; BOLTZMANN; COLLECTIVE BEHAVIOR; CONTINUOUS ANALOGUES; FLOCKING; IN-PHASE; INITIAL CONDITIONS; KINETIC MODELS; LARGE TIME BEHAVIOR; MASS TRANSPORTATION METHODS; MEAN VELOCITIES; NONLINEAR FRICTION EQUATIONS; PARTICLE APPROXIMATIONS; SMALE MODEL;

ANIMALS; ASYMPTOTIC ANALYSIS; BOLTZMANN EQUATION; FRICTION; LATHES; MASS TRANSPORTATION; MULTI AGENT SYSTEMS; PHASE SPACE METHODS; STABILITY;

NONLINEAR EQUATIONS;

EID: 77954545287     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/090757290     Document Type: Article

References (38)

1. A. BARBARO, K. TAYLOR, P. F. TRETHEWEY, L. YOUSEFF, AND B. BIRNIR, Discrete and continuous models of the dynamics of pelagic fish: Application to the capelin, Math. Comput. Simulation, 79 (2009), pp. 3397-3414.
2. D. BENEDETTO, E. CAGLIOTI, AND M. PULVIRENTI, A kinetic equation for granular media, RAIRO Modél. Math. Anal. Numer., 31 (1997), pp. 615-641.
3. A. BERTOZZI, J. A. CARRILLO, AND T. LAURENT, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), pp. 683-710.
4. B. BIRNIR, An ODE model of the motion of pelagic fish, J. Stat. Phys., 128 (2007), pp. 535-568.
5. A. V. BOBYLËV, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules, Soviet Sei. Rev. Sect. C Math. Phys. Rev., 7 (1988), pp. 111-233.
6. M. J. CA;CERES AND G. TOSCANI, Kinetic approach to long time behavior of linearized fast diffusion equations, J. Stat. Phys., 128 (2007), pp. 883-925.
7. J. A. CARRILLO, M. DIFRANCESCO, A. FIGALLI, T. LAURENT, AND D. SLEPCEV, Global in Time Weak Measure Solutions, Finite-Time Aggregation and Confinement for Nonlocal Interaction Equations, preprint.
8. J. A. CARRILLO, M. R. D'ORSOGNA, AND V. PANFEROV, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), pp. 363-378.
9. J. A. CARRILLO, R. J. MCCANN, AND C VILLANI, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), pp. 971-1018.
10. J. A. CARRILLO, R. McCANN, AND C VILLANI, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), pp. 217-263.
11. J. A. CARRILLO AND G. TOSCANI, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma (7), 6 (2007), pp. 75-198.
12. A. CHAKRABORTI, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), pp. 1315-1321.
13. Y.-L. CHUANG, Y. R. HUANG, M. R. D'ORSOGNA, AND A. L. BERTOZZI, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, in Proceedings of the IEEE International Conference on Robotics and Automation, Rome, Italy, 2007, pp. 2292-2299.
14. Y.-L. CHUANG, M. R. D'ORSOGNA, D. MARTHALER, A. L. BERTOZZI, AND L. S. CHAYES, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), pp. 33-47.
15. 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), pp. 513-516.
16. F. CUCKER AND S. SMALE, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), pp. 852-862.
17. F. CUCKER AND S. SMALE, On the mathematics of emergence, Jpn. J. Math., 2 (2007), pp. 197-227.
18. P. DEGOND AND S. MOTSCH, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), pp. 1193-1215.
19. 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), 104302-104305.
20. A. DRAGULESCU AND V. M. YAKOVENKO, Statistical mechanics of money, Eur. Phys. J. B, 17 (2000), pp. 723-729.
21. S. GALAM, Y. GEFEN, AND Y. SHAPIR, Sociophysics: A new approach of sociological collective behavior, J. Math. Sociology, 9 (1982), pp. 1-13.
22. A. JADBABAIE, J. LIN, AND A. S. MORSE, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), pp. 988-1001.
23. 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), pp. 297-325.
24. S.-Y. HA AND E. TADMOR, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), pp. 415-435.
25. D. MATTHES AND G. TOSCANI, Analysis of a model for wealth redistribution, Kinet. Relat. Models, 1 (2008), pp. 1-27.
26. S. MCNAMARA AND W. R. YOUNG, Kinetics of a one-dimensional granular medium in the quasielastic limit, Phys. Fluids A, 5 (1993), pp. 34-45.
27. H. NEUNZERT, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles, Trans. Fluid Dynamics, 18 (1977), pp. 663-678.
28. L. PEREA, G. GÓMEZ, AND P. ELOSEGUI, Extension of the Cucker-Smale control law to space flight formations, AIAA J. Guid. Control Dynam., 32 (2009), pp. 527-537.
29. A.Y. POVZNER, The Boltzmann equation in kinetic theory of gases, Amer. Math. Soc. Transi. Ser. 2, 47 (1962), pp. 193-216.
30. J. SHEN, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), pp. 694-719.
31. H. SPOHN, Large Scale Dynamics of Interacting Particles, Texts Monogr. Phys., Springer, Berlin, 1991.
32. K. SZNAJD-WERON AND J. SZNAJD, Opinion evolution in closed community, Int. J. Mod. Phys. C, 11 (2000), pp. 1157-1165.
33. C M. TOPAZ AND A. L. BERTOZZI, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), pp. 152-174.
34. C M. TOPAZ, A. L. BERTOZZI, AND M. A. LEWIS, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), pp. 1601-1623.
35. G. TOSCANI, Kinetic and hydrodynamic models of nearly elastic granular flow, Monatsh. Math., 142 (2004), pp. 179-192.
36. G. TOSCANI, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), pp. 481-1196
37. T. VlCSEK, A. CZIRK, E. BEN-JACOB, AND O. SHOCHET, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), pp. 1226-1229.
38. C VILLANI, Topics in Optimal Transportation, Grad. Stud. Math. 58, AMS, Providence, RI, 2003.