Long-Time Behaviour and Phase Transitions for the Mckean-Vlasov Equation on the Torus
Long-Time Behaviour and Phase Transitions for the Mckean-Vlasov Equation on the Torus
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DOI：
10.1007/s00205-019-01430-4
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发表时间：
2020-01-01
影响因子：
2.5
通讯作者：
Schlichting, A.
中科院分区：
文献类型：
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作者：
Carrillo, A.;Gvalani, R. S.;Schlichting, A.
我们研究了McKean-Vlasov方程偏导数(iota) = (-1)Delta + kappa del。（W星）)，在环面上有周期边界条件。首先研究了齐次稳态的全局渐近稳定性。然后，我们将注意力集中在平稳系统上，并在适当的相互作用势假设下，证明了非平凡解的存在性，从齐次稳态分支，通过可能无限多个分支。并给出了连续相变和不连续相变存在的充分条件。最后，我们通过将这些结果应用于几个相互作用潜力的例子来展示这些结果，例如用于同步的嘈杂Kuramoto模型，用于细菌趋化的Keller-Segel模型，以及用于意见动态的嘈杂Hegselmann-Krausse模型。
We study the McKean-Vlasov equationpartial derivative(iota)rho = beta(-1)Delta rho + kappa del.(rho del (W star rho)),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.