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Equilibria of homogeneous functionals in the fair-competition regime

Calvez, V. and Carrillo, J. A. and Hoffmann, F. (2017) Equilibria of homogeneous functionals in the fair-competition regime. Nonlinear Analysis, 159 . pp. 85-128. ISSN 0362-546X. doi:10.1016/

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We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular/non-singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which both homogeneities scale the same with respect to dilations, that we coin as fair-competition. In the singular kernel case, we show that existence of global equilibria can only happen at a certain critical value and they are characterised as optimisers of a variant of HLS inequalities. We also study the existence of self-similar solutions for the sub-critical case, or equivalently of optimisers of rescaled free energies. These optimisers are shown to be compactly supported radially symmetric and non-increasing stationary solutions of the non-linear Keller–Segel equation. On the other hand, we show that no radially symmetric non-increasing stationary solutions exist in the non-singular kernel case, implying that there is no criticality. However, we show the existence of positive self-similar solutions for all values of the parameter under the condition that diffusion is not too fast. We finally illustrate some of the open problems in the non-singular kernel case by numerical experiments.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Calvez, V.0000-0002-3674-1965
Carrillo, J. A.0000-0001-8819-4660
Hoffmann, F.0000-0002-1182-5521
Additional Information:© 2017 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ( Received 4 October 2016, Accepted 16 March 2017, Available online 10 April 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 and by EPSRC grant number EP/P031587/1. 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.
Funding AgencyGrant Number
European Research Council (ERC)639638
Engineering and Physical Sciences Research Council (EPSRC)EP/P031587/1
Engineering and Physical Sciences Research Council (EPSRC)EP/H023348/1
Subject Keywords:Aggregation-diffusion equations; Gradient flows; Minimisation
Record Number:CaltechAUTHORS:20200331-070052590
Persistent URL:
Official Citation:V. Calvez, J.A. Carrillo, F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Analysis, Volume 159, 2017, Pages 85-128, ISSN 0362-546X, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:102180
Deposited By: Tony Diaz
Deposited On:31 Mar 2020 17:04
Last Modified:16 Nov 2021 18:09

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