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Ground states in the diffusion-dominated regime

Carrillo, José A. and Hoffmann, Franca and Mainini, Edoardo and Volzone, Bruno (2018) Ground states in the diffusion-dominated regime. Calculus of Variations and Partial Differential Equations, 57 (5). Art. No. 127. ISSN 0944-2669. PMCID PMC6190998. doi:10.1007/s00526-018-1402-2.

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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 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.

Item Type:Article
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URLURL TypeDescription CentralArticle Paper
Carrillo, José A.0000-0001-8819-4660
Hoffmann, Franca0000-0002-1182-5521
Mainini, Edoardo0000-0003-0379-1728
Volzone, Bruno0000-0001-8028-2017
Additional Information:© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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. Received: 9 May 2017 / Accepted: 15 June 2018 / First Online: 11 August 2018. Communicated by L. Ambrosio. We thank Y. Yao and F. Brock for useful discussion about the continuous Steiner symmetrisation. We thank X. Ros-Otón, P. R. Stinga and P. Mironescu for some fruitful explanations concerning the regularity properties of fractional elliptic equations used in this work. We are grateful to R. Frank for suggesting the alternative proof for the existence of minimisers in Remark 1. JAC was partially supported by the Royal Society via a Wolfson Research Merit Award and by the EPSRC grant number EP/P031587/1. FH acknowledges support from the EPSRC grant number EP/H023348/1 for the Cambridge Centre for Analysis. EM was partially supported by the FWF project M1733-N20. BV was partially supported by GNAMPA of INdAM, “Programma triennale della Ricerca dell’Università degli Studi di Napoli “Parthenope”- Sostegno alla ricerca individuale 2015-2017”. EM and BV are members of the GNAMPA group of the Istituto Nazionale di Alta Matematica (INdAM). This work was partially supported by the Simons-Foundation grant 346300 and the Polish Government MNiSW 2015-2019 matching fund. 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
Engineering and Physical Sciences Research Council (EPSRC)EP/P031587/1
Engineering and Physical Sciences Research Council (EPSRC)EP/H023348/1
FWF Der WissenschaftsfondsM1733-N20
Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro ApplicazioniUNSPECIFIED
Simons Foundation346300
Ministerstwo Nauki i Szkolnictwa Wyższego (MNiSW)UNSPECIFIED
Issue or Number:5
Classification Code:Mathematics Subject Classification 35K55 35K65 49K20
PubMed Central ID:PMC6190998
Record Number:CaltechAUTHORS:20180813-091037905
Persistent URL:
Official Citation:Carrillo, J.A., Hoffmann, F., Mainini, E. et al. Calc. Var. (2018) 57: 127.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:88786
Deposited By: George Porter
Deposited On:13 Aug 2018 16:59
Last Modified:16 Nov 2021 00:29

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