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

Calvez, Vincent and Carrillo, José Antonio and Hoffmann, Franca (2017) The Geometry of Diffusing and Self-Attracting Particles in a One-Dimensional Fair-Competition Regime. In: Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Lecture Notes in Mathematics. No.2186. Springer , Cham, pp. 1-71. ISBN 978-3-319-61493-9. https://resolver.caltech.edu/CaltechAUTHORS:20200331-072541010

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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.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/978-3-319-61494-6_1DOIArticle
https://arxiv.org/abs/1612.08225arXivDiscussion Paper
ORCID:
AuthorORCID
Calvez, Vincent0000-0002-3674-1965
Carrillo, José Antonio0000-0001-8819-4660
Hoffmann, Franca0000-0002-1182-5521
Additional Information:© 2017 Springer International Publishing AG. First Online: 04 October 2017. 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.
Funders:
Funding AgencyGrant Number
European Research Council (ERC)639638
Royal SocietyUNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)EP/H023348/1
Series Name:Lecture Notes in Mathematics
Issue or Number:2186
Record Number:CaltechAUTHORS:20200331-072541010
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200331-072541010
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:102182
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:31 Mar 2020 16:55
Last Modified:11 Nov 2020 00:05

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