Ground states in the diffusion-dominated regime
File(s)
Published version
Author(s)
Carrillo de la Plata, J
Hoffmann, Franca
Mainini, Edoardo
Volzone, Bruno
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.
Date Issued
2018-10-01
Date Acceptance
2018-07-01
Citation
Calculus of Variations and Partial Differential Equations, 2018, 57
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
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/P031587/1
Subjects
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
Date Publish Online
2018-08-11