Abstract
In this paper, we introduce the second-order Cucker–Smale flocking model with the Hessian communication weight, and its first-order reduction, which is a nonidentical swarming model. Using the equivalent relation between the first- and second-order models, we present the analysis of the emergent flocking behavior of each model. Also, using the gradient flow structure, the uniqueness of the equilibrium of the first-order model is attained. The results on the microscopic models are extended to the kinetic models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Spaces of Probability Measures. Lectures in Mathematics, ETH Zurich, Birkhäuser (2005)
Bae, H.-O., Choi, Y.-P., Ha, S.-Y., Kang, M.-J.: Asymptotic flocking dynamics of Cucker–Smale particles immersed in compressible fluids. Discrete Contin. Dyn. Syst. A 34, 4419–4458 (2014)
Bresch, D., Jabin, P.-E., Wang, Z.: On mean-field limits and quantitative estimates with a large class of singular kernels: application to the Patlak–Keller–Segel model. C. R. Acad. Sci. Paris Ser. I 357, 708–720 (2019)
Carrillo, J.A., Fornasier, M., Rosado, J., Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker–Smale model. SIAM J. Math. Anal. 42, 218–236 (2010)
Carrillo, J.A., Chipot, M., Huang, Y.: On global minimizers of repulsive-attractive power-law interaction energies. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372, 20130399, 13 (2014)
Carrillo, J.A., Choi, Y.-P., Tadmor, E., Tan, C.: Critical thresholds in 1D Euler equations with non-local forces. Math. Models Methods Appl. Sci. 26, 185–206 (2016)
Cattiaux, P., Pédéches, L.: The 2-DstochasticKeller-Segel particlemodel: existence and uniqueness.ALEA Lat. Am. J. Probab. Math. Stat. 13, 447–463 (2016)
Cho, J., Ha, S.-Y., Huang, F., Jin, C., Ko, D.: Emergence of bi-cluster flocking for the Cucker–Smale model. Math. Models Methods Appl. Sci. 26, 1191–1218 (2016)
Choi, Y.-P., Ha, S.-Y., Li, Z.: Emergent dynamics of the Cucker–Smale flocking model and its variants. In: Bellomo, N., Degond, P., Tadmor, E. (eds.) Active Particles—Theory, Models, Applications. Modeling and Simulation in Science and Technology, vol. 1. Springer, Birkhauser (2017)
Cucker, F., Smale, S.: On the mathematics of emergence. Jpn. J. Math. 2, 197–227 (2007a)
Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52, 852–862 (2007b)
Ha, S.-Y., Liu, J.-G.: A simple proof of Cucker–Smale flocking dynamics and mean field limit. Commun. Math. Sci. 7, 297–325 (2009)
Ha, S.-Y., Tadmor, E.: From particle to kinetic and hydrodynamic description of flocking. Kinet. Relat. Models 1, 415–435 (2008)
Ha, S.-Y., Kwon, B., Kang, M.-J.: A hydrodynamic model for the interaction of Cucker–Smale particles and incompressible fluid. Math. Models Methods Appl. Sci. 24, 2311–2359 (2014)
Ha, S.-Y., Kwon, B., Kang, M.-J.: Emergent dynamics for the hydrodynamic Cucker–Smale system in a moving domain. SIAM J. Math. Anal. 47, 3813–3831 (2015)
Ha, S.-Y., Ko, D., Zhang, Y.: Critical coupling strength of the Cucker–Smale model for flocking. Math. Models Methods Appl. Sci. 27, 1051–1087 (2017)
Ha, S.-Y., Kim, J., Zhang, X.: Uniform stability of the Cucker–Smale model and its application to the mean-field limit. Kinet. Relat. Models 11, 1157–1181 (2018a)
Ha, S.-Y., Park, J., Zhang, X.: A first-order reduction of the Cucker–Smale model on the real line and its clustering dynamics. Commun. Math. Sci. 16, 1907–1931 (2018b)
Ha, S.-Y., Kim, J., Park, J., Zhang, X.: Complete cluster predictability of the Cucker-Smale flocking model on the line. Arch. Ration. Mech. Anal. 231, 319–365 (2019)
Karper, M., Mellet, A., Trivisa, K.: Hydrodynamic limit of the kinetic Cucker–Smale flocking model. Math. Models Methods Appl. Sci. 25, 131–163 (2015)
Kim, J.: First-order reduction and emergent behavior of the one-dimensional kinetic Cucker–Smale equation (2021). Submitted
Kuramoto, Y.: International symposium on mathematical problems in mathematical physics. Lect. Notes Theor. Phys. 30, 420 (1975)
Łojasiewicz, S.: Sur les trajectoires du gradient d’une fonction analytique. Geometry seminars, 1982–1983 (Bologna, 1982/1983), pp. 115–117. Univ. Stud. Bologna, Bologna (1984)
Poyato, D., Soler, J.: Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker–Smale models. Math. Models Methods Appl. Sci. 27, 1089–1152 (2017)
Santambrogio, F.: Euclidean, metric, Wasserstein gradient flows: an overview. Bull. Math. Sci. 7, 87–154 (2017)
Shu, R., Tadmor, E.: Anticipation breeds alignment. Arch. Ration. Mech. Anal. 240, 203–241 (2021)
Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Schochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995)
Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin (2009)
Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42 (1967)
Acknowledgements
The work of J. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2020R1A4A3079066).
Author information
Authors and Affiliations
Additional information
Communicated by Eliot Fried.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kim, J. A Cucker–Smale Flocking Model with the Hessian Communication Weight and Its First-Order Reduction. J Nonlinear Sci 32, 20 (2022). https://doi.org/10.1007/s00332-021-09777-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00332-021-09777-3