Long-time behaviour and phase transitions for the Mckean–Vlasov equation on the torus

Author: 

Carrillo de la Plata, J
Gvalani, R
Pavliotis, G
Schlichting, A

Publication Date: 

26 July 2019

Journal: 

Archive for Rational Mechanics and Analysis

Last Updated: 

2020-11-13T16:13:22.05+00:00

Issue: 

1

Volume: 

235

DOI: 

10.1007/s00205-019-01430-4

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

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article