Hyperbolicity and nonconservativity of a hydrodynamic model of swarming rigid bodies

Degond, P.; Frouvelle, A.; Merino-Aceituno, S.; Trescases, A.

doi:10.1090/qam/1651

Authors: P. Degond, A. Frouvelle, S. Merino-Aceituno and A. Trescases
Journal: Quart. Appl. Math. 82 (2024), 35-64
MSC (2020): Primary 35L60, 35L65, 35L67, 76L05
DOI: https://doi.org/10.1090/qam/1651
Published electronically: March 21, 2023
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a nonlinear system of first order partial differential equations describing the macroscopic behavior of an ensemble of interacting self-propelled rigid bodies. Such system may be relevant for the modelling of bird flocks, fish schools or fleets of drones. We show that the system is hyperbolic and can be approximated by a conservative system through relaxation. We also derive viscous corrections to the model from the hydrodynamic limit of a kinetic model. This analysis prepares the future development of numerical approximations of this system.

References

I. Aoki, A simulation study on the schooling mechanism in fish, Bull. Japan. Soc. Sci. Fish 48 (1982), 1081–1088.
S. Bazazi, J. Buhl, J. J. Hale, M. L. Anstey, G. .A Sword, S. J. Simpson, and I. D. Couzin, Collective motion and cannibalism in locust migratory bands, Current Biology 18 (2008), no. 10, 735–739.
E. Bertin, M. Droz, and G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles, Phys. Rev. E 74 (2006), no. 2, 022101.
E. Bertin, M. Droz, and G. Grégoire, Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis, J. Phys. A 42 (2009), no. 44, 445001.
Andrea L. Bertozzi, Theodore Kolokolnikov, Hui Sun, David Uminsky, and James von Brecht, Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci. 13 (2015), no. 4, 955–985. MR 3325085, DOI 10.4310/CMS.2015.v13.n4.a6
François Bolley, José A. Cañizo, and José A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett. 25 (2012), no. 3, 339–343. MR 2855983, DOI 10.1016/j.aml.2011.09.011
Mihai Bostan and Jose Antonio Carrillo, Asymptotic fixed-speed reduced dynamics for kinetic equations in swarming, Math. Models Methods Appl. Sci. 23 (2013), no. 13, 2353–2393. MR 3109433, DOI 10.1142/S0218202513500346
Marc Briant, Antoine Diez, and Sara Merino-Aceituno, Cauchy theory for general kinetic Vicsek models in collective dynamics and mean-field limit approximations, SIAM J. Math. Anal. 54 (2022), no. 1, 1131–1168. MR 4381950, DOI 10.1137/21M1405885
Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620, DOI 10.1007/978-1-4419-8524-8
H. Chaté, F. Ginelli, G. Grégoire, and F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E 77 (2008), no. 4, 046113.
Floraine Cordier, Pierre Degond, and Anela Kumbaro, Phase appearance or disappearance in two-phase flows, J. Sci. Comput. 58 (2014), no. 1, 115–148. MR 3147651, DOI 10.1007/s10915-013-9725-9
Felipe Cucker and Steve Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007), no. 5, 852–862. MR 2324245, DOI 10.1109/TAC.2007.895842
A. Czirók, E. Ben-Jacob, I. Cohen, and T. Vicsek, Formation of complex bacterial colonies via self-generated vortices, Phys. Rev. E 54 (1996), no. 2, 1791.
Pierre Degond, Macroscopic limits of the Boltzmann equation: a review, Modeling and computational methods for kinetic equations, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2004, pp. 3–57. MR 2068579
P. Degond, A. Diez, and A. Frouvelle, Body-attitude coordination in arbitrary dimension, Preprint, arXiv:2111.05614, 2021.
P. Degond, A. Diez, A. Frouvelle, and S. Merino-Aceituno, Phase transitions and macroscopic limits in a BGK model of body-attitude coordination, J. Nonlinear Sci. 30 (2020), no. 6, 2671–2736. MR 4170309, DOI 10.1007/s00332-020-09632-x
Pierre Degond, Antoine Diez, and Mingye Na, Bulk topological states in a new collective dynamics model, SIAM J. Appl. Dyn. Syst. 21 (2022), no. 2, 1455–1494. MR 4434350, DOI 10.1137/21M1393935
Pierre Degond, Amic Frouvelle, and Jian-Guo Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci. 23 (2013), no. 3, 427–456. MR 3067586, DOI 10.1007/s00332-012-9157-y
Pierre Degond, Amic Frouvelle, and Jian-Guo Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Arch. Ration. Mech. Anal. 216 (2015), no. 1, 63–115. MR 3305654, DOI 10.1007/s00205-014-0800-7
Pierre Degond, Amic Frouvelle, and Sara Merino-Aceituno, A new flocking model through body attitude coordination, Math. Models Methods Appl. Sci. 27 (2017), no. 6, 1005–1049. MR 3659045, DOI 10.1142/S0218202517400085
Pierre Degond, Amic Frouvelle, Sara Merino-Aceituno, and Ariane Trescases, Quaternions in collective dynamics, Multiscale Model. Simul. 16 (2018), no. 1, 28–77. MR 3743738, DOI 10.1137/17M1135207
Pierre Degond, Amic Frouvelle, Sara Merino-Aceituno, and Ariane Trescases, Alignment of self-propelled rigid bodies: from particle systems to macroscopic equations, Stochastic dynamics out of equilibrium, Springer Proc. Math. Stat., vol. 282, Springer, Cham, 2019, pp. 28–66. MR 3986062, DOI 10.1007/978-3-030-15096-9_{2}
Pierre Degond, Giacomo Dimarco, Thi Bich Ngoc Mac, and Nan Wang, Macroscopic models of collective motion with repulsion, Commun. Math. Sci. 13 (2015), no. 6, 1615–1638. MR 3351444, DOI 10.4310/CMS.2015.v13.n6.a12
Pierre Degond and Jiale Hua, Self-organized hydrodynamics with congestion and path formation in crowds, J. Comput. Phys. 237 (2013), 299–319. MR 3020033, DOI 10.1016/j.jcp.2012.11.033
Pierre Degond, Jian-Guo Liu, Sebastien Motsch, and Vladislav Panferov, Hydrodynamic models of self-organized dynamics: derivation and existence theory, Methods Appl. Anal. 20 (2013), no. 2, 89–114. MR 3119732, DOI 10.4310/MAA.2013.v20.n2.a1
Pierre Degond and Sébastien Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci. 18 (2008), no. suppl., 1193–1215. MR 2438213, DOI 10.1142/S0218202508003005
Pierre Degond, Pierre-François Peyrard, Giovanni Russo, and Philippe Villedieu, Polynomial upwind schemes for hyperbolic systems, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 6, 479–483 (English, with English and French summaries). MR 1680004, DOI 10.1016/S0764-4442(99)80194-3
Antoine Diez, Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles, Electron. J. Probab. 25 (2020), Paper No. 90, 38. MR 4136470, DOI 10.1214/20-ejp496
Giacomo Dimarco and Sebastien Motsch, Self-alignment driven by jump processes: Macroscopic limit and numerical investigation, Math. Models Methods Appl. Sci. 26 (2016), no. 7, 1385–1410. MR 3494681, DOI 10.1142/S0218202516500330
Yao-li Chuang, Maria R. D’Orsogna, Daniel Marthaler, Andrea L. Bertozzi, and Lincoln S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D 232 (2007), no. 1, 33–47. MR 2369988, DOI 10.1016/j.physd.2007.05.007
Razvan C. Fetecau, Seung-Yeal Ha, and Hansol Park, Emergent behaviors of rotation matrix flocks, SIAM J. Appl. Dyn. Syst. 21 (2022), no. 2, 1382–1425. MR 4431956, DOI 10.1137/21M1404569
Alessio Figalli, Moon-Jin Kang, and Javier Morales, Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow, Arch. Ration. Mech. Anal. 227 (2018), no. 3, 869–896. MR 3744377, DOI 10.1007/s00205-017-1176-2
Amic Frouvelle, Body-attitude alignment: first order phase transition, link with rodlike polymers through quaternions, and stability, Recent advances in kinetic equations and applications, Springer INdAM Ser., vol. 48, Springer, Cham, [2021] ©2021, pp. 147–181. MR 4437189, DOI 10.1007/978-3-030-82946-9_{7}
Amic Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci. 22 (2012), no. 7, 1250011, 40. MR 2924786, DOI 10.1142/S021820251250011X
Amic Frouvelle and Jian-Guo Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math. Anal. 44 (2012), no. 2, 791–826. MR 2914250, DOI 10.1137/110823912
Irene M. Gamba, Jeffrey R. Haack, and Sebastien Motsch, Spectral method for a kinetic swarming model, J. Comput. Phys. 297 (2015), 32–46. MR 3361650, DOI 10.1016/j.jcp.2015.04.033
Irene M. Gamba and Moon-Jin Kang, Global weak solutions for Kolmogorov-Vicsek type equations with orientational interactions, Arch. Ration. Mech. Anal. 222 (2016), no. 1, 317–342. MR 3519972, DOI 10.1007/s00205-016-1002-2
Rasa Giniūnaitė, Ruth E. Baker, Paul M. Kulesa, and Philip K. Maini, Modelling collective cell migration: neural crest as a model paradigm, J. Math. Biol. 80 (2020), no. 1-2, 481–504. MR 4062827, DOI 10.1007/s00285-019-01436-2
Edwige Godlewski and Pierre-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, [2021] ©2021. Second edition [of 1410987 ]. MR 4331351, DOI 10.1007/978-1-0716-1344-3
François Golse and Seung-Yeal Ha, A mean-field limit of the Lohe matrix model and emergent dynamics, Arch. Ration. Mech. Anal. 234 (2019), no. 3, 1445–1491. MR 4011701, DOI 10.1007/s00205-019-01416-2
Quentin Griette and Sebastien Motsch, Kinetic equations and self-organized band formations, Active particles. Vol. 2. Advances in theory, models, and applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2019, pp. 173–199. MR 3932461
Seung-Yeal Ha, Dongnam Ko, and Sang Woo Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys. 168 (2017), no. 1, 171–207. MR 3659983, DOI 10.1007/s10955-017-1797-8
Seung-Yeal Ha and Jian-Guo Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 7 (2009), no. 2, 297–325. MR 2536440, DOI 10.4310/CMS.2009.v7.n2.a2
Seung-Yeal Ha and Eitan Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models 1 (2008), no. 3, 415–435. MR 2425606, DOI 10.3934/krm.2008.1.415
H. Hildenbrandt, C. Carere, and C. K. Hemelrijk, Self-organized aerial displays of thousands of starlings: a model, Behavioral Ecology 21 (2010), no. 6, 1349–1359.
Ning Jiang, Linjie Xiong, and Teng-Fei Zhang, Hydrodynamic limits of the kinetic self-organized models, SIAM J. Math. Anal. 48 (2016), no. 5, 3383–3411. MR 3549879, DOI 10.1137/15M1035665
A. J. Kabla, Collective cell migration: leadership, invasion and segregation, Journal of The Royal Society Interface 9 (2012), no. 77, 3268–3278.
Philippe G. LeFloch, Hyperbolic systems of conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002. The theory of classical and nonclassical shock waves. MR 1927887, DOI 10.1007/978-3-0348-8150-0
Randall J. LeVeque, Numerical methods for conservation laws, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. MR 1153252, DOI 10.1007/978-3-0348-8629-1
U. Lopez, J. Gautrais, I. D. Couzin, and G. Theraulaz, From behavioural analyses to models of collective motion in fish schools, Interface Focus 2 (2012), no. 6, 693–707.
Sebastien Motsch and Laurent Navoret, Numerical simulations of a nonconservative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Model. Simul. 9 (2011), no. 3, 1253–1275. MR 2846932, DOI 10.1137/100794067
Sebastien Motsch and Eitan Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 144 (2011), no. 5, 923–947. MR 2836613, DOI 10.1007/s10955-011-0285-9
F. Peruani, A. Deutsch, and M. Bär, A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms, Eur. Phys. J. Spec. Top. 157 (2008), no. 1, 111–122.
V. Petrolli, T. Boudou, M. Balland, and G. Cappello, Oscillations in collective cell migration, Viscoelasticity and Collective Cell Migration (2021), 157–192.
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146, DOI 10.1007/978-1-4684-0152-3
John Toner and Yuhai Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E (3) 58 (1998), no. 4, 4828–4858. MR 1651324, DOI 10.1103/PhysRevE.58.4828
Tamás Vicsek, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995), no. 6, 1226–1229. MR 3363421, DOI 10.1103/PhysRevLett.75.1226
T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep. 517 (2012), no. 3-4, 71–140.
Wei Wang, Pingwen Zhang, and Zhifei Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation, Comm. Pure Appl. Math. 68 (2015), no. 8, 1326–1398. MR 3366748, DOI 10.1002/cpa.21549
W. H. Warren, Collective motion in human crowds, Current Directions in Psychological Science 27 (2018), no. 4, 232–240.
R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria, Proc. Natl. Acad. Sci. USA 98 (2001), no. 26, 14907–14912.
Weinan E and Pingwen Zhang, A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit, Methods Appl. Anal. 13 (2006), no. 2, 181–198. MR 2381545, DOI 10.4310/MAA.2006.v13.n2.a5
Teng-Fei Zhang and Ning Jiang, A local existence of viscous self-organized hydrodynamic model, Nonlinear Anal. Real World Appl. 34 (2017), 495–506. MR 3567974, DOI 10.1016/j.nonrwa.2016.09.016