Ground states in the diffusion-dominated regime

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Title: Ground states in the diffusion-dominated regime
Authors: Carrillo de la Plata, J
Hoffmann, F
Mainini, E
Volzone, B
Item Type: Journal Article
Abstract: We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and C∞ inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.
Issue Date: 1-Oct-2018
Date of Acceptance: 1-Jul-2018
URI: http://hdl.handle.net/10044/1/61313
DOI: https://dx.doi.org/10.1007/s00526-018-1402-2
ISSN: 1432-0835
Publisher: Springer Verlag
Journal / Book Title: Calculus of Variations and Partial Differential Equations
Volume: 57
Copyright Statement: © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/P031587/1
Keywords: Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
KELLER-SEGEL MODEL
CRITICAL MASS
FUNCTIONAL INEQUALITIES
CELL-ADHESION
CHEMOTAXIS
AGGREGATION
REGULARITY
PRINCIPLE
ASYMPTOTICS
EXISTENCE
35K55
35K65
49K20
General Mathematics
0101 Pure Mathematics
0102 Applied Mathematics
Publication Status: Published
Article Number: ARTN 127
Online Publication Date: 2018-08-11
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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