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Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions

Carillo, José A.; Delgadino, Matias G.; Frank, Rupert L.; Lewin, Mathieu (2020), Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions. https://basepub.dauphine.psl.eu/handle/123456789/22332

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Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-03079032
Date
2020
Series title
Cahier de recherche du CEREMADE
Pages
25
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Author(s)
Carillo, José A.
Mathematical Institute [Oxford] [MI]
Delgadino, Matias G.
Californian Institute of Technology [Caltech]
Pontifícia Universidade Católica do Rio de Janeiro [PUC-Rio]
Frank, Rupert L.
Mathematisches Institut [München] [LMU]
Lewin, Mathieu cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We show that partial mass concentration can happen for stationary solutions of aggregation-diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions N≥6. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions N≥3, for homogeneous interaction potentials with higher power.