Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics
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Author(s)
Carrillo de la Plata, Jose Antonio
Hittmeir, Sabine
Volzone, Bruno
Yao, Yao
Type
Journal Article
Abstract
We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs.
This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially
symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations
tools allow us to show the existence of global minimizers among these equilibria. Finally, in the
particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria
for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations
as t → ∞.
This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially
symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations
tools allow us to show the existence of global minimizers among these equilibria. Finally, in the
particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria
for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations
as t → ∞.
Date Issued
2019-12-01
Date Acceptance
2019-06-10
Citation
Inventiones Mathematicae, 2019, 218 (3), pp.889-977
ISSN
0020-9910
Publisher
Springer Verlag
Start Page
889
End Page
977
Journal / Book Title
Inventiones Mathematicae
Volume
218
Issue
3
Copyright Statement
© 2019 The Authors. 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.
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.
Sponsor
The Royal Society
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Grant Number
WM120001
EP/K008404/1
EP/P031587/1
Subjects
0101 Pure Mathematics
General Mathematics
Publication Status
Published
Date Publish Online
2019-07-25