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Uniqueness and Nonuniqueness of Steady States of Aggregation‐Diffusion Equations
Communications on pure and applied mathematics, 2022-01, Vol.75 (1), p.3-59
Delgadino, Matias G.
Yan, Xukai
Yao, Yao
2022
Details
Autor(en) / Beteiligte
Delgadino, Matias G.
Yan, Xukai
Yao, Yao
Titel
Uniqueness and Nonuniqueness of Steady States of Aggregation‐Diffusion Equations
Ist Teil von
Communications on pure and applied mathematics, 2022-01, Vol.75 (1), p.3-59
Ort / Verlag
Melbourne: John Wiley & Sons Australia, Ltd
Erscheinungsjahr
2022
Link zum Volltext
Quelle
Wiley Online Library
Beschreibungen/Notizen
We consider a nonlocal aggregation equation with degenerate diffusion, which describes the mean‐field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all steady states must be radially decreasing up to a translation, but uniqueness (for a given mass) within the radial class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/nonuniqueness criteria are determined by the power of the degenerate diffusion, with the critical power being m = 2. In the case m ≥ 2, we show that for any attractive potential the steady state is unique for a fixed mass. In the case 1 < m < 2, we construct examples of smooth attractive potentials such that there are infinitely many radially decreasing steady states of the same mass. For the uniqueness proof, we develop a novel interpolation curve between two radially decreasing densities, and the key step is to show that the interaction energy is convex along this curve for any attractive interaction potential, which is of independent interest. © 2020 Wiley Periodicals LLC.
Sprache
Englisch
Identifikatoren
ISSN: 0010-3640
eISSN: 1097-0312
DOI: 10.1002/cpa.21950
Titel-ID: cdi_proquest_journals_2598072370
Format
–
Schlagworte
Agglomeration
,
Diffusion
,
Interpolation
,
Steady state
,
Uniqueness
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