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Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7883-7910. doi: 10.3934/mbe.2019396
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Hydrodynamic limits for kinetic flocking models of Cucker-Smale type

  • 1.
    Department of Mathematics, Imperial College London, London SW7 2AZ, UK
  • 2.
    Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, UMR 7373, Château Gombert 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France
  • Received: 30 January 2019 Accepted: 05 June 2019 Published: 28 August 2019

  • We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in [1], is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.

    Citation: Pedro Aceves-Sánchez, Mihai Bostan, Jose-Antonio Carrillo, Pierre Degond. Hydrodynamic limits for kinetic flocking models of Cucker-Smale type[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7883-7910. doi: 10.3934/mbe.2019396

    Related Papers:

  • We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in [1], is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.


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    [1] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193–1215.
    [2] M. Ballerini, N. Cabibbo, R. Candelier, et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232–1237.
    [3] U. Lopez, J. Gautrais, I. D. Couzin, et al., From behavioural analyses to models of collective motion in fish schools, Interface Focus, 2 (2012), 693–707.
    [4] J. Buhl, D. J. T. Sumpter, I. D. Couzin, et al., From disorder to order in marching locusts, Science, 312 (2006), 1402–1406.
    [5] E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. Math. Theor., 42 (2009), 445001.
    [6] H. Chaté, F. Ginelli, G. Grégoire, et al., Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E, 77 (2008), 046113.
    [7] I. D. Couzin, J. Krause, N. R. Franks, et al., Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513–516.
    [8] D. Saintillan and M. J. Shelley, Instabilities, pattern formation, and mixing in active suspensions, Phys. Fluids, 20 (2008), 123304.
    [9] J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical XY model: How birds fly together, Phys. Rev. Lett., 75 (1995), 4326–4329.
    [10] T. Vicsek, A. Czirok, E. Ben-Jacob, et al., Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226–1229.
    [11] A. Bricard, J. B. Caussin, N. Desreumaux, et al., Emergence of macroscopic directed motion in populations of motile colloids, Nature, 503 (2013), 95.
    [12] A. Creppy, F. Plouraboué, O. Praud, et al., Symmetry-breaking phase transitions in highly concentrated semen, J. R. Soc. Interface, 13 (2016), 20160575.
    [13] D. L. Koch and G. Subramanian, Collective hydrodynamics of swimming microorganisms: Living fluids, Annu. Rev. Fluid Mech., 43 (2011), 637–659.
    [14] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, et al, Hydrodynamics of soft active matter, Rev. Mod. Phys., 85 (2013), 1143–1188.
    [15] T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71–140.
    [16] S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577–621.
    [17] J. A. Carrillo, M. Fornasier, G. Toscani, et al., Particle, kinetic and hydrodynamic models of swarming, Math. Model Collect. Behav. Socio-Econ. Life Sci. (MSSET), (2010), 297–336.
    [18] T. Kolokolnikov, J. A. Carrillo, A. Bertozzi, et al., Emergent behaviour in multi-particle systems with non-local interactions, Phys. D, 260 (2013), 1–4.
    [19] F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-lipschitz forces & swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179–2210.
    [20] F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339–343.
    [21] W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Comm. Math. Phys., 56 (1977), 101–113.
    [22] J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515–539.
    [23] H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles, Trans. Fluid Dyn., 18 (1977), 663–678.
    [24] R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115–123.
    [25] S. Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453–469.
    [26] J. A. Carrillo, Y. P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collect. Dyn. Bact. Crowds, 553 (2014), 1–46.
    [27] J. A. Carrillo, Y. P. Choi, M. Hauray, et al., Mean-field limit for collective behavior models with sharp sensitivity regions, J. Eur. Math. Soc., 21 (2019), 121–161.
    [28] P. Degond, G Dimarco, T. B. N. Mac, et al., Macroscopic models of collective motion with repulsion, Comm. Math. Sci., 13 (2015), 1615–1638.
    [29] P. Degond, A. Manhart and H. Yu, A continuum model for nematic alignment of self-propelled particles, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1295–1327.
    [30] P. Degond, S. Merino-Aceituno, F. Vergnet, et al., Coupled self-organized hydrodynamics and stokes models for suspensions of active particles, J. Math. Fluid Mech., 21 (2019), 6.
    [31] P. Degond, A. Frouvelle, S. Merino-Aceituno, et al., Quaternions in collective dynamics, Multiscale Model Simul., 16 (2018), 28–77.
    [32] M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi, et al., Self-propelled particles with soft-core interactions: Patterns, stability and collapse, Phys. Rev. Lett., 96 (2006), 104302–1/4.
    [33] Y. L. Chuang, Y. R. Huang, M. R. D'Orsogna, et al., Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, IEEE Int. Conf. Rob. Autom., (2007), 2292–2299.
    [34] Y. L. Chuang, M. R. D'Orsogna, D. Marthaler, et al., State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33–47.
    [35] J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in a self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363–378.
    [36] J. A. Carrillo, A. Klar, S. Martin, et al., Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci., 20 (2010), 1533–1552.
    [37] A. B. T. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1249–1278.
    [38] A. B. T. Barbaro, J. A. Cañizo, J. A. Carrillo, et al., Phase transitions in a kinetic flocking model of Cucker-Smale type, Multiscale Model Simul., 14 (2016), 1063–1088.
    [39] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852–862.
    [40] S. Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415–435.
    [41] 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.
    [42] J. A. Carrillo, M. Fornasier, J. Rosado, et al., Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218–236.
    [43] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923–947.
    [44] A. Frouvelle and J. G. Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math. Anal., 44 (2012), 791–826.
    [45] P. Degond, A. Frouvelle and J. G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427–456.
    [46] P. Degond, A. Frouvelle and J. G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Arch. Ration. Mech. Anal., 216 (2015), 63–115.
    [47] M. Bostan and J. A. Carrillo, Asymptotic fixed-speed reduced dynamics for kinetic equations in swarming, Math. Models Methods Appl. Sci., 23 (2013), 2353–2393.
    [48] M. Bostan and J. A. Carrillo, Reduced fluid models for self-propelled populations, interacting through alignment, Math. Models Methods Appl. Sci., 27 (2017), 1255–1299.
    [49] M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91–123.
    [50] M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differ. Eq., 249 (2010), 1620–1663.
    [51] M. Bostan, Gyro-kinetic Vlasov equation in three dimensional setting. Second order approximation, SIAM J. Multiscale Model. Simul., 8 (2010), 1923–1957.
    [52] V. Bonnaillie-Noël, J. A. Carrillo, T. Goudon, et al., Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations, IMA J. Numer. Anal., 36 (2016), 1–34.
    [53] M. Reed and B. Simon, Methods of modern mathematical physics: IV analysis of operators, Academic Press, 1980.
    [54] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. formal derivations, J. Stat. Phys., 63 (1991), 323–344.
    [55] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations II convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math., 46 (1993), 667–753.
    [56] M. Bostan, On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 339–371.
    [57] M. Bostan, High magnetic field equilibria for the Fokker-Planck-Landau equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 899–931.
    [58] D. Levermore, Entropic convergence and the linearized limit for the Boltzmann equation, Commun. Part. Diff. Eq., 18 (1993), 1231–1248.
    [59] D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021–1065.
    [60] A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011.
    [61] P. D. Miller, Applied asymptotic analysis, American Mathematical Society, 2006.
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