Long-time behaviour and phase transitions for the McKean—Vlasov equation on the torus
File(s)
Published version
Author(s)
Carrillo de la Plata, JA
Gvalani, Rishabh
Pavliotis, Greg
Schlichting, Andre
Type
Journal Article
Abstract
We study the McKean-Vlasov equation
∂t% = β
−1∆% + κ ∇·(%∇(W ? %)) ,
with periodic boundary conditions on the torus. We first study the global asymptotic stability of the
homogeneous steady state. We then focus our attention on the stationary system, and prove the existence
of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many
bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient
conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these
results by applying them to several examples of interaction potentials such as the noisy Kuramoto model
for synchronisation, the Keller–Segel model for bacterial chemotaxis, and the noisy Hegselmann–Krausse
model for opinion dynamics.
∂t% = β
−1∆% + κ ∇·(%∇(W ? %)) ,
with periodic boundary conditions on the torus. We first study the global asymptotic stability of the
homogeneous steady state. We then focus our attention on the stationary system, and prove the existence
of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many
bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient
conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these
results by applying them to several examples of interaction potentials such as the noisy Kuramoto model
for synchronisation, the Keller–Segel model for bacterial chemotaxis, and the noisy Hegselmann–Krausse
model for opinion dynamics.
Date Issued
2020-01-01
Date Acceptance
2019-07-09
Citation
Archive for Rational Mechanics and Analysis, 2020, 235, pp.635-690
ISSN
0003-9527
Publisher
Springer (part of Springer Nature)
Start Page
635
End Page
690
Journal / Book Title
Archive for Rational Mechanics and Analysis
Volume
235
Copyright Statement
© 2019 The Author(s). 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.
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.
License URL
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Identifier
https://link.springer.com/article/10.1007/s00205-019-01430-4#article-info
Grant Number
EP/P031587/1
EP/L024926/1
EP/L020564/1
Subjects
0101 Pure Mathematics
0102 Applied Mathematics
General Physics
Publication Status
Published
Date Publish Online
2019-07-26