Mean-field limit of collective dynamics with time-varying weights

Nastassia Pouradier Duteil; Nastassia Pouradier Duteil

doi:10.3934/nhm.2022001

Networks and Heterogeneous Media

2022, Volume 17, Issue 2: 129-161. doi: 10.3934/nhm.2022001

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Mean-field limit of collective dynamics with time-varying weights

Nastassia Pouradier Duteil

Sorbonne Université, Inria, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France

Received: 01 April 2021 Revised: 01 October 2021 Published: 18 February 2022
Primary: 35Q70, 34A34; Secondary: 35L60

In this paper, we derive the mean-field limit of a collective dynamics model with time-varying weights, for weight dynamics that preserve the total mass of the system as well as indistinguishability of the agents. The limit equation is a transport equation with source, where the (non-local) transport term corresponds to the position dynamics, and the (non-local) source term comes from the weight redistribution among the agents. We show existence and uniqueness of the solution for both microscopic and macroscopic models and introduce a new empirical measure taking into account the weights. We obtain the convergence of the microscopic model to the macroscopic one by showing continuity of the macroscopic solution with respect to the initial data, in the Wasserstein and Bounded Lipschitz topologies.
- Interacting weighted particle system,
- mean-field limit,
- indistinguishability,
- transport equation with source,
- existence and uniqueness
Citation: Nastassia Pouradier Duteil. Mean-field limit of collective dynamics with time-varying weights[J]. Networks and Heterogeneous Media, 2022, 17(2): 129-161. doi: 10.3934/nhm.2022001

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Abstract

In this paper, we derive the mean-field limit of a collective dynamics model with time-varying weights, for weight dynamics that preserve the total mass of the system as well as indistinguishability of the agents. The limit equation is a transport equation with source, where the (non-local) transport term corresponds to the position dynamics, and the (non-local) source term comes from the weight redistribution among the agents. We show existence and uniqueness of the solution for both microscopic and macroscopic models and introduce a new empirical measure taking into account the weights. We obtain the convergence of the microscopic model to the macroscopic one by showing continuity of the macroscopic solution with respect to the initial data, in the Wasserstein and Bounded Lipschitz topologies.

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