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

Marco Di Francesco; Antonio Esposito; Markus Schmidtchen

doi:10.1007/s00526-021-01960-4

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

Marco Di Francesco, Antonio Esposito, Markus Schmidtchen

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

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 Lm-norms of a piecewise constant reconstruction of the density using the particle trajectories.

Original language	English (US)
Journal	Calculus of Variations and Partial Differential Equations
Volume	60
Issue number	2
DOIs	https://doi.org/10.1007/s00526-021-01960-4
State	Published - Apr 4 2021
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-15
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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1007/s00526-021-01960-4

Cite this

@article{94df415f7f9e4d76b76f6bcbdc4a54c1,

title = "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 Lm-norms of a piecewise constant reconstruction of the density using the particle trajectories.",

author = "Marco Di Francesco and Antonio Esposito and Markus Schmidtchen",

note = "KAUST Repository Item: Exported on 2022-06-15 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

year = "2021",

month = apr,

day = "4",

doi = "10.1007/s00526-021-01960-4",

language = "English (US)",

volume = "60",

journal = "Calculus of Variations and Partial Differential Equations",

issn = "1432-0835",

publisher = "Springer New York",

number = "2",

}

TY - JOUR

T1 - Many-particle limit for a system of interaction equations driven by Newtonian potentials

AU - Di Francesco, Marco

AU - Esposito, Antonio

AU - Schmidtchen, Markus

N1 - KAUST Repository Item: Exported on 2022-06-15 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2021/4/4

Y1 - 2021/4/4

N2 - 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 Lm-norms of a piecewise constant reconstruction of the density using the particle trajectories.

AB - 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 Lm-norms of a piecewise constant reconstruction of the density using the particle trajectories.

UR - http://hdl.handle.net/10754/679038

UR - https://link.springer.com/10.1007/s00526-021-01960-4

UR - http://www.scopus.com/inward/record.url?scp=85103813009&partnerID=8YFLogxK

U2 - 10.1007/s00526-021-01960-4

DO - 10.1007/s00526-021-01960-4

M3 - Article

SN - 1432-0835

VL - 60

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 2

ER -

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

Abstract

Bibliographical note

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this