In this paper, we give a brief survey on the state-of-the-art results on the mean-field limit of the Cucker-Smale(CS) model for flocking. The CS model is one of well-studied collective dynamics models. Collective motions of self-propelled particles often appear in our nature. Some collective motions are often described by different types of partial differential equations. We discuss that they fall down to the special cases of the universal nonlinear consensus model at the microscopic level. We also discuss how an interacting particle system with a large size can be effectively approximated by the corresponding mean-field model by the rigorous justification of the mean-field limit. In particular, we focus on the uniform-in-time mean-field limit of the CS model for flocking using the uniform-in-time stability estimate and asymptotic flocking estimates under some framework which guarantees the exponential flocking.
@article{SLSEDP_2024-2025____A6_0,
author = {Seung-Yeal Ha},
title = {Recent progress on the mean-field limit of {the~Cucker-Smale} model for flocking},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:15},
pages = {1--17},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2024-2025},
doi = {10.5802/slsedp.179},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.179/}
}
TY - JOUR AU - Seung-Yeal Ha TI - Recent progress on the mean-field limit of the Cucker-Smale model for flocking JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:15 PY - 2024-2025 SP - 1 EP - 17 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.179/ DO - 10.5802/slsedp.179 LA - en ID - SLSEDP_2024-2025____A6_0 ER -
%0 Journal Article %A Seung-Yeal Ha %T Recent progress on the mean-field limit of the Cucker-Smale model for flocking %J Séminaire Laurent Schwartz — EDP et applications %Z talk:15 %D 2024-2025 %P 1-17 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.179/ %R 10.5802/slsedp.179 %G en %F SLSEDP_2024-2025____A6_0
Seung-Yeal Ha. Recent progress on the mean-field limit of the Cucker-Smale model for flocking. Séminaire Laurent Schwartz — EDP et applications (2024-2025), Exposé no. 15, 17 p. doi : 10.5802/slsedp.179. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.179/
[1] Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. and Spigler, R., The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77 (2005), 137–185. | DOI
[2] Ahn, H., Ha, S.-Y., Kim, D., Schlöder, F. W. and Shim, W., The mean-field limit of the Cucker-Smale model on complete Riemannian manifolds, Quart. Appl. Math. 80 (2022), 403–450. | DOI | MR | Zbl
[3] Albi, G., Bellomo, N., Fermo, L., Ha, S.-Y., Kim, J., Pareschi, L., Poyato, D. and Soler, J., Vehicular traffic, crowds, and swarms: from kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci. 29 (2019), 1901–2005. | DOI | MR | Zbl
[4] Ahn, S., Choi, H., Ha, S.-Y. and Lee, H., On collision-avoiding initial configurations to Cucker-Smale type flocking models, Comm. Math. Sci. 10 (2012), 625–643. | DOI | Zbl
[5] Aoki, I., A simulation study on the schooling mechanism in fish, Bull. Jpn. Soc. Sci. Fish. 48 (1982), 1081–1088. | DOI
[6] Bellomo, N. and Ha, S.-Y., A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci. 27 (2017), 745–770. | DOI | MR | Zbl
[7] Bellomo, N., Ha, S.-Y. and Outada, N., Towards a mathematical theory of behavioral swarms, ESAIM Control Optim. Calc. Var. 26 (2020), 125. | DOI | MR | Zbl
[8] Buck, J. and Buck, E., Biology of synchronous flashing of fireflies, Nature 211 (1966), 562–564. | DOI
[9] Carrillo, J. A., Fornasier, M., Rosado, J. and Toscani, G., Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal. 42 (2010), 218–236. | DOI | Zbl
[10] Carrillo, J. A., Fornasier, M., Toscani, G. and Vecil, F., Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences, 297–336, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, 2010. | DOI | Zbl
[11] Cho, J., Ha, S.-Y., Huang, F., Jin, C. and Ko, D., Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci. 26 (2016), 1191–1218. | DOI | Zbl
[12] Choi, Y.-P., Ha, S.-Y. and Li, Z., Emergent dynamics of the Cucker-Smale flocking model and its variants, In N. Bellomo, P. Degond, and E. Tadmor (Eds.), Active Particles Vol. 1- Theory, Models, Applications, Series: Modeling and Simulation in Science and Technology, Birkhauser, Cham.
[13] Cucker, F. and Smale, S., Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007), 852–862. | DOI | MR | Zbl
[14] Degond, P. and Motsch, S., Large-scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys. 131 (2008), 989–1022. | DOI | Zbl
[15] Ha, S.-Y., Jin, C. and Zhang, Y., Remarks on the mean-field limit of the Motsch-Tadmor model, Submitted.
[16] Ha, S.-Y., Kang, M., Park, H., Ruggeri, T. and Shim, W., Uniform stability and uniform-in-time mean-field limit of the thermodynamic Kuramoto model, Quart. Appl. Math. 79 (2021), 445–478. | DOI | Zbl
[17] Ha, S.-Y., Kim, D. and Schlöder, F. W., Emergent behaviors of Cucker-Smale flocks on Riemannian manifolds, IEEE Trans. Automat. Control 66 (2020), 3020–3035. | DOI | Zbl
[18] Ha, S.-Y., Kim, J., Min, C. H., Ruggeri, T. and Zhang, X., Uniform stability and mean-field limit of a thermodynamic Cucker-Smale model, Quart. Appl. Math.77 (2019), 131–176. | DOI | Zbl
[19] Ha, S.-Y., Kim, J., Park, J. and Zhang, X., Uniform stability and mean-field limit for the augmented Kuramoto model, Netw. Heterog. Media 13 (2018), 297–322. | DOI | MR | Zbl
[20] Ha, S.-Y, Kim, J., Park, J. and Zhang, X., Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal. 231 (2019), 319–365. | DOI | MR | Zbl
[21] Ha, S.-Y., Ko, D., Park, J. and Zhang, X., Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci. 3 (2016), 209–267. | DOI | MR | Zbl
[22] Ha, S.-Y., Ko, D. and Zhang, Y., Critical coupling strength of the Cucker-Smale model for flocking, Math. Models Methods Appl. Sci. 27 (2017), 1051–1087. | DOI | MR | Zbl
[23] Ha, S.-Y., Kim, J. and Zhang, X., Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models 11 (2018), 1157–1181. | Zbl
[24] Ha, S.-Y. and Liu, J.-G., A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci. 7 (2009), 297–325. | DOI | MR | Zbl
[25] Ha, S.-Y. and Tadmor, E., From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models 1 (2008), 415–435. | DOI | MR | Zbl
[26] Ha, S.-Y., Park, J. and Zhang, X., A first-order reduction of the Cucker-Smale model on the real line and its clustering dynamics, Commun. Math. Sci. 16 (2018), 1907–1931. | DOI | MR | Zbl
[27] Ha S.-Y., Wang X., and Xue X., On the exponential weak flocking behavior of the kinetic Cucker-Smale model with non-compact support, Math. Models Methods Appl. Sci. 35 (2025), 781–824. | DOI | MR | Zbl
[28] Hellwig, M. F., On the aggregation of information in competitive markets, J. Econom. Theory 22 (1980), 477–498. | DOI | Zbl
[29] Horstmann, D., On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol. 44 (2002), 463–478. | DOI | Zbl
[30] Horstmann, D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003), 103–165. | MR | Zbl
[31] Horstmann, D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein. 106 (2004), 51–69. | Zbl
[32] Keller, E. F. and Segel, L. A., Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415. | DOI | MR | Zbl
[33] Keller, E. F. and Segel, L. A., Model for Chemotaxis, J. Theor. Biol. 30 (1971), 225–234. | DOI | Zbl
[34] Kuramoto, Y., Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Mathematical Physics, Lect. Notes Theor. Phys. 30 (1975), 420–422. | DOI | MR | Zbl
[35] Leonard, N. E., Paley, D. A., Lekien, F., Sepulchre, R., Fratantoni, D. M. and Davis, R. E., Collective motion, sensor networks and ocean sampling, Proc. IEEE 95 (2007), 48–74. | DOI
[36] Motsch, S. and Tadmor, E., Heterophilious dynamics: Enhanced consensus, SIAM Rev. 56 (2014), 577–621. | DOI | MR | Zbl
[37] Motsch, S. and Tadmor, E., A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 144 (2011), 923–947. | DOI | Zbl
[38] Mucha, P. B. and Peszek, J., The Cucker-Smale equation: singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal. 227 (2018), 273–308. | DOI | MR | Zbl
[39] Natalini, R. and Paul, T., On the mean field limit for Cucker-Smale models, Discrete Contin. Dyn. Syst. Ser. B 27 (2022), 2873–2889. | DOI | Zbl
[40] Neunzert, H., An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic theories and the Boltzmann equation, Lect. Notes Math. 1048, Springer, Berlin, Heidelberg. | DOI | MR
[41] Paley, D. A., Leonard, N. E., Sepulchre, R., Grunbaum, D. and Parrish, J. K., Oscillator models and collective motion, IEEE Control Syst. Mag. 27 (2007), 89–105. | DOI
[42] Perea, L., Elosegui, P. and Gómez, G., Extension of the Cucker-Smale control law to space flight formation, J. Guid. Control Dyn. 32 (2009), 527–537. | DOI
[43] Peskin, C. S., Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences, New York 1975. | MR | Zbl
[44] Pikovsky, A., Rosenblum, M. and Kurths, J., Synchronization: A universal concept in nonlinear sciences, Cambridge: Cambridge University Press 2001. | DOI | MR | Zbl
[45] Reynolds, C. W., Flocks, herds and schools: A distributed behavioral model, Comput. Graph. 21 (1987), 25–34. | DOI | Zbl
[46] Shaw, E., Shooling fishes, Am. Sci. 66 (1978), 166–175.
[47] Topaz, C. M. and Bertozzi, A. L., Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), 152–174. | DOI | MR | Zbl
[48] Toner, J. and Tu, Y., Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E 58 (1988), 4828–4858. | DOI | MR
[49] Vicsek, T., and Zeiris, A., Collective motion, Phys. Rep. 517 (2012), 71–140. | DOI
[50] Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I and Schochet, O., Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995), 1226–1229. | DOI | MR
[51] Villani, C., Optimal transport, old and new, Springer 2008. | MR | Zbl
[52] Wang, X. and Xue, X., Formation behaviour of the kinetic Cucker-Smale model with non-compact support, Proc. Roy. Soc. Edinburgh Sect. A 153 (2023), 1315–1346. | DOI | MR | Zbl
[53] Wang, X. and Xue, X., The flocking behavior of the infinite-particle Cucker-Smale model, Proc. Amer. Math. Soc. 150 (2022), 2165–2179. | DOI | MR | Zbl
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