Long-time behaviour and phase transitions for the McKean—Vlasov equation on the torus

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Title: Long-time behaviour and phase transitions for the McKean—Vlasov equation on the torus
Authors: Gvalani, R
Carrillo de la Plata, JA
Pavliotis, G
Schlichting, A
Item 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.
Date of Acceptance: 5-Jul-2019
URI: http://hdl.handle.net/10044/1/71872
ISSN: 0003-9527
Publisher: Springer (part of Springer Nature)
Journal / Book Title: Archive for Rational Mechanics and Analysis
Copyright Statement: This paper is embargoed until publication. Once published it will be available fully open access.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/P031587/1
EP/L024926/1
EP/L020564/1
Keywords: 0101 Pure Mathematics
0102 Applied Mathematics
General Physics
Publication Status: Accepted
Embargo Date: publication subject to indefinite embargo
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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