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Equilibria of homogeneous functionals in the fair-competition regime

Calvez, V. and Carrillo, J. A. and Hoffmann, F. (2016) Equilibria of homogeneous functionals in the fair-competition regime. . (Unpublished)

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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/smooth 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 smooth 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 smooth kernel case by numerical experiments.

Item Type:Report or Paper (Discussion Paper)
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: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.
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)EP/H023348/1
Record Number:CaltechAUTHORS:20200331-080736207
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:102188
Deposited By: Tony Diaz
Deposited On:31 Mar 2020 15:59
Last Modified:31 Mar 2021 23:18

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