Journal article
Ground states in the diffusion-dominated regime
- 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.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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(Preview, Version of record, pdf, 886.9KB, Terms of use)
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- Publisher copy:
- 10.1007/s00526-018-1402-2
Authors
- Publisher:
- Springer
- Journal:
- Calculus of Variations and Partial Differential Equations More from this journal
- Volume:
- 57
- Issue:
- 5
- Article number:
- 127
- Publication date:
- 2018-08-11
- Acceptance date:
- 2018-06-15
- DOI:
- EISSN:
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1432-0835
- ISSN:
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0944-2669
- Pmid:
-
30393443
- Language:
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English
- Keywords:
- Pubs id:
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1098242
- Local pid:
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pubs:1098242
- Deposit date:
-
2020-04-07
- ARK identifier:
Terms of use
- Copyright holder:
- Carrillo, J.A., Hoffmann, F., Mainini, E. et al.
- Copyright date:
- 2018
- Rights 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.
- Licence:
- CC Attribution (CC BY)
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