Abstract
We give a complete study of the asymptotic behavior of a simple model of alignment of unit vectors, both at the level of particles, which corresponds to a system of coupled differential equations, and at the continuum level, under the form of an aggregation equation on the sphere. We prove unconditional convergence towards an aligned asymptotic state. In the cases of the differential system and of symmetric initial data for the partial differential equation, we provide precise rates of convergence.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ambrosio, L., Crippa, G.: Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In: Transport Equations and Multi-D Hyperbolic Conservation Laws. Lecture Notes of the Unione Matematica Italiana, vol. 5, pp. 3–57. Springer, Berlin (2008)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008)
Aydoğdu, A., McQuade, S.T., Pouradier Duteil, N.: Opinion dynamics on a general compact Riemannian manifold. Netw. Heterog. Media 12(3), 489–523 (2017)
Barbălat, I.: Systèmes d’équations différentielles d’oscillations non linéaires. Rev. Math. Pures Appl. 4, 267–270 (1959)
Benedetto, D., Caglioti, E., Montemagno, U.: On the complete phase synchronization for the Kuramoto model in the mean-field limit. Commun. Math. Sci. 13(7), 1775–1786 (2015)
Bolley, F., Cañizo, J.A., Carrillo, J.A.: Mean-field limit for the stochastic Vicsek model. Appl. Math. Lett. 3(25), 339–343 (2012)
Caponigro, M., Lai, A.C., Piccoli, B.: A nonlinear model of opinion formation on the sphere. Discrete Contin. Dyn. Syst. 35(9), 4241–4268 (2015)
Degond, P., Frouvelle, A., Raoul, G.: Local stability of perfect alignment for a spatially homogeneous kinetic model. J. Stat. Phys. 157(1), 84–112 (2014)
Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18, 1193–1215 (2008)
Fatkullin, I., Slastikov, V.: Critical points of the Onsager functional on a sphere. Nonlinearity 18, 2565–2580 (2005)
Figalli, A., Kang, M.J., Morales, J.: Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow. Arch. Ration. Mech. Anal. 227(3), 869–896 (2018)
Frouvelle, A., Liu, J.G.: Dynamics in a kinetic model of oriented particles with phase transition. SIAM J. Math. Anal. 44(2), 791–826 (2012)
Ha, S.Y., Ko, D., Ryoo, S.W.: On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds. J. Stat. Phys. 172, 1427–1478 (2018)
Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer, Berlin (2001)
Markdahl, J., Thunberg, J., Gonçalves, J.: Almost global consensus on the \(n\)-sphere. IEEE Trans. Autom. Control 63(6), 1664–1675 (2018)
Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics. Springer, Heidelberg (1991)
Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226–1229 (1995)
Acknowledgments
The authors want to thank the hospitality of Athanasios Tzavaras and the University of Crete, back in 2012, where this work was done and supported by the EU FP7-REGPOT project “Archimedes Center for Modeling, Analysis and Computation”.
They also want to thank the anonymous referee for his fast, careful, and efficient reading, despite their very late submission.
A.F. acknowledges support from the EFI project ANR-17-CE40-0030 and the Kibord project ANR-13-BS01-0004 of the French National Research Agency (ANR), from the project Défi S2C3 POSBIO of the interdisciplinary mission of CNRS, and the project SMS co-funded by CNRS and the Royal Society.
J.-G. L. acknowledges support from the National Science Foundation under the NSF Research Network Grant no. RNMS11-07444 (KI-Net) and the grant DMS-1812573.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Frouvelle, A., Liu, JG. (2019). Long-Time Dynamics for a Simple Aggregation Equation on the Sphere. In: Giacomin, G., Olla, S., Saada, E., Spohn, H., Stoltz, G. (eds) Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017. Springer Proceedings in Mathematics & Statistics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-15096-9_16
Download citation
DOI: https://doi.org/10.1007/978-3-030-15096-9_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-15095-2
Online ISBN: 978-3-030-15096-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)