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International workshop on Stochastic Dynamics out of Equilibrium

IHPStochDyn 2017: Stochastic Dynamics Out of Equilibrium pp 457–479Cite as

Long-Time Dynamics for a Simple Aggregation Equation on the Sphere

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 282))

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.

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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

  1. CEREMADE, CNRS, Université Paris-Dauphine, Université PSL, 75016, Paris, France

    Amic Frouvelle

  2. Department of Physics and Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Jian-Guo Liu

Authors
  1. Amic Frouvelle
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  2. Jian-Guo Liu
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Corresponding author

Correspondence to Amic Frouvelle .

Editor information

Editors and Affiliations

  1. Laboratoire de Probabilités, Statistiques et Modélisation (UMR 8001), Université Paris Diderot, Paris, France

    Giambattista Giacomin

  2. CEREMADE, CNRS UMR 7534, Université Paris Dauphine - PSL, Paris, France

    Stefano Olla

  3. CNRS UMR 8145, MAP5, Université Paris Descartes, Paris, France

    Ellen Saada

  4. Zentrum Mathematik, Technische Universität München, Garching, Bayern, Germany

    Herbert Spohn

  5. CERMICS, Ecole des Ponts, Marne-La-Vallée, France

    Gabriel Stoltz

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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

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