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

Carrillo, José A.; Delgadino, Matias G.; Frank, Rupert L.; Lewin, Mathieu (2022), Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions, Mathematical Models and Methods in Applied Sciences (M3AS), 32, 4, p. 831-850. 10.1142/S021820252250018X

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Type
Article accepté pour publication ou publié
Date
2022
Journal name
Mathematical Models and Methods in Applied Sciences (M3AS)
Volume
32
Number
4
Publisher
World Scientific
Pages
831-850
Publication identifier
Metadata
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Author(s)
Carrillo, 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
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.
Subjects / Keywords
Keller–Segel; aggregation–diffusion; mass concentration