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BibTex format

@article{Carrillo:2020:10.1007/s00205-019-01430-4,
author = {Carrillo, de la Plata JA and Gvalani, R and Pavliotis, G and Schlichting, A},
doi = {10.1007/s00205-019-01430-4},
journal = {Archive for Rational Mechanics and Analysis},
pages = {635--690},
title = {Long-time behaviour and phase transitions for the McKean—Vlasov equation on the torus},
url = {http://dx.doi.org/10.1007/s00205-019-01430-4},
volume = {235},
year = {2020}
}

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TY  - JOUR
AB  - 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 thehomogeneous steady state. We then focus our attention on the stationary system, and prove the existenceof nontrivial solutions branching from the homogeneous steady state, through possibly infinitely manybifurcations, under appropriate assumptions on the interaction potential. We also provide sufficientconditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase theseresults by applying them to several examples of interaction potentials such as the noisy Kuramoto modelfor synchronisation, the Keller–Segel model for bacterial chemotaxis, and the noisy Hegselmann–Kraussemodel for opinion dynamics.
AU  - Carrillo,de la Plata JA
AU  - Gvalani,R
AU  - Pavliotis,G
AU  - Schlichting,A
DO  - 10.1007/s00205-019-01430-4
EP  - 690
PY  - 2020///
SN  - 0003-9527
SP  - 635
TI  - Long-time behaviour and phase transitions for the McKean—Vlasov equation on the torus
T2  - Archive for Rational Mechanics and Analysis
UR  - http://dx.doi.org/10.1007/s00205-019-01430-4
UR  - https://link.springer.com/article/10.1007/s00205-019-01430-4#article-info
UR  - http://hdl.handle.net/10044/1/71872
VL  - 235
ER  -

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Professor Grigorios A. Pavliotis

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