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The Geometry of Diffusing and Self-Attracting Particles in a One-Dimensional Fair-Competition Regime

  • Vincent Calvez
  • José Antonio CarrilloEmail author
  • Franca Hoffmann
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2186)

Abstract

We consider an aggregation-diffusion equation modelling particle interaction with non-linear diffusion and non-local attractive interaction using a homogeneous kernel (singular and non-singular) leading to variants of the Keller-Segel model of chemotaxis. We analyse the fair-competition regime in which both homogeneities scale the same with respect to dilations. Our analysis here deals with the one-dimensional case, building on the work in Calvez et al. (Equilibria of homogeneous functionals in the fair-competition regime), and provides an almost complete classification. In the singular kernel case and for critical interaction strength, we prove uniqueness of stationary states via a variant of the Hardy-Littlewood-Sobolev inequality. Using the same methods, we show uniqueness of self-similar profiles in the sub-critical case by proving a new type of functional inequality. Surprisingly, the same results hold true for any interaction strength in the non-singular kernel case. Further, we investigate the asymptotic behaviour of solutions, proving convergence to equilibrium in Wasserstein distance in the critical singular kernel case, and convergence to self-similarity for sub-critical interaction strength, both under a uniform stability condition. Moreover, solutions converge to a unique self-similar profile in the non-singular kernel case. Finally, we provide a numerical overview for the asymptotic behaviour of solutions in the full parameter space demonstrating the above results. We also discuss a number of phenomena appearing in the numerical explorations for the diffusion-dominated and attraction-dominated regimes.

Notes

Acknowledgements

VC received funding for this project from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 639638). JAC was partially supported by the Royal Society via a Wolfson Research Merit Award. FH acknowledges support from the EPSRC grant number EP/H023348/1 for the Cambridge Centre for Analysis. The authors are very grateful to the Mittag-Leffler Institute for providing a fruitful working environment during the special semester Interactions between Partial Differential Equations & Functional Inequalities.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Vincent Calvez
    • 1
  • José Antonio Carrillo
    • 2
    Email author
  • Franca Hoffmann
    • 3
    • 2
  1. 1.Unité de Mathématiques Pures et Appliquées, CNRS UMR 5669 and équipe-projet INRIA NUMEDÉcole Normale Supérieure de LyonLyonFrance
  2. 2.Department of MathematicsImperial College LondonLondonUK
  3. 3.DPMMS, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

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