Many-particle limit for a system of interaction equations driven by Newtonian potentials

Abstract

We consider a one-dimensional discrete particle system of two species coupled through nonlocal interactions driven by the Newtonian potential, with repulsive self-interaction and attractive cross-interaction. After providing a suitable existence theory in a finite-dimensional framework, we explore the behaviour of the particle system in case of collisions and analyse the behaviour of the solutions with initial data featuring particle clusters. Subsequently, we prove that the empirical measure associated to the particle system converges to the unique 2-Wasserstein gradient flow solution of a system of two partial differential equations with nonlocal interaction terms in a proper measure sense. The latter result uses uniform estimates of the \(L^m\)-norms of a piecewise constant reconstruction of the density using the particle trajectories.

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Acknowledgements

A considerable part of this work was carried out during the visit of MDF to King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. MDF is deeply grateful for the warm hospitality by people at KAUST, for the excellent scientific environment, and for the support in the development of this work. AE was partially supported by the German Science Foundation (DFG) through CRC TR 154 “Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”. AE and MS gratefully acknowledge the support of the Hausdorff Research Institute for Mathematics (Bonn), through the Junior Trimester Program on Kinetic Theory.

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Correspondence to Marco Di Francesco.

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Di Francesco, M., Esposito, A. & Schmidtchen, M. Many-particle limit for a system of interaction equations driven by Newtonian potentials. Calc. Var. 60, 68 (2021). https://doi.org/10.1007/s00526-021-01960-4

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Mathematics Subject Classification

  • 35A24
  • 35F55
  • 35Q70
  • 35R09
  • 82C22
  • 35A35