Author
Carrillo de la Plata, J
Gvalani, R
Pavliotis, G
Schlichting, A
Journal title
Archive for Rational Mechanics and Analysis
DOI
10.1007/s00205-019-01430-4
Issue
1
Volume
235
Last updated
2022-08-04T22:20:37.813+01:00
Page
635-690
Abstract
We study the McKean–Vlasov equation ∂tϱ=β-1Δϱ+κ∇·(ϱ∇(W⋆ϱ)),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.
Symplectic ID
1098164
Favourite
On
Publication type
Journal Article
Publication date
26 Jul 2019
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