# Mean field limit for the one dimensional Vlasov-Poisson equation

Maxime Hauray^{[1]}

- [1] LATP Université d’Aix-Marseille & CNRS UMR 7353 13453 Marseille Cedex 13 France

Séminaire Laurent Schwartz — EDP et applications (2012-2013)

- Volume: 2012-2013, page 1-16
- ISSN: 2266-0607

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topHauray, Maxime. "Mean field limit for the one dimensional Vlasov-Poisson equation." Séminaire Laurent Schwartz — EDP et applications 2012-2013 (2012-2013): 1-16. <http://eudml.org/doc/275774>.

@article{Hauray2012-2013,

abstract = {We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in [Tro86]. Here we shall give a simpler proof of this result, and explain why it implies the so-called “Propagation of molecular chaos”. More precisely, both results will be a direct consequence of a weak-strong stability result on the one dimensional Vlasov-Poisson equation that is interesting by it own. We also prove the existence of global solutions to the $N$ particles dynamic starting from any initial positions and velocities, and the existence of global solutions to the Vlasov-Poisson equation starting from any measures with bounded first moment in velocity.},

affiliation = {LATP Université d’Aix-Marseille & CNRS UMR 7353 13453 Marseille Cedex 13 France},

author = {Hauray, Maxime},

journal = {Séminaire Laurent Schwartz — EDP et applications},

language = {eng},

pages = {1-16},

publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},

title = {Mean field limit for the one dimensional Vlasov-Poisson equation},

url = {http://eudml.org/doc/275774},

volume = {2012-2013},

year = {2012-2013},

}

TY - JOUR

AU - Hauray, Maxime

TI - Mean field limit for the one dimensional Vlasov-Poisson equation

JO - Séminaire Laurent Schwartz — EDP et applications

PY - 2012-2013

PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique

VL - 2012-2013

SP - 1

EP - 16

AB - We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in [Tro86]. Here we shall give a simpler proof of this result, and explain why it implies the so-called “Propagation of molecular chaos”. More precisely, both results will be a direct consequence of a weak-strong stability result on the one dimensional Vlasov-Poisson equation that is interesting by it own. We also prove the existence of global solutions to the $N$ particles dynamic starting from any initial positions and velocities, and the existence of global solutions to the Vlasov-Poisson equation starting from any measures with bounded first moment in velocity.

LA - eng

UR - http://eudml.org/doc/275774

ER -

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