Imperial College London

Professor Grigorios A. Pavliotis

Faculty of Natural SciencesDepartment of Mathematics

Professor of Applied Mathematics



+44 (0)20 7594 8564g.pavliotis Website




736aHuxley BuildingSouth Kensington Campus






BibTex format

author = {Carrillo, de la Plata JA and Gvalani, R and Pavliotis, G and Schlichting, A},
journal = {Archive for Rational Mechanics and Analysis},
title = {Long-time behaviour and phase transitions for the McKean—Vlasov equation on the torus},
url = {},

RIS format (EndNote, RefMan)

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
SN - 0003-9527
TI - Long-time behaviour and phase transitions for the McKean—Vlasov equation on the torus
T2 - Archive for Rational Mechanics and Analysis
UR -
ER -