Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus

作者：José A. Carrillo, Rishabh S. Gvalani, Grigorios A. Pavliotis, André Schlichting · 发表于：Archive for Rational Mechanics and Analysis · 年份：2019 · DOI：10.1007/s00205-019-01430-4 · 被引用次数：114 · 研究领域：Advanced Thermodynamics and Statistical Mechanics、Statistical Mechanics and Entropy、Opinion Dynamics and Social Influence

We study the McKean–Vlasov equation $$\begin{aligned} \partial _t \varrho = \beta ^{-1} \Delta \varrho + \kappa {{\,\mathrm{\nabla \cdot }\,}}(\varrho \nabla (W \star \varrho )), \end{aligned}$$ 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.