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

Abstract

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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.

How to cite

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Hauray, 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 -

References

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  1. Luigi Ambrosio. Transport equation and Cauchy problem for B V vector fields. Invent. Math., 158(2):227–260, 2004. Zbl1075.35087
  2. Luigi Ambrosio. Transport equation and Cauchy problem for non-smooth vector fields. In Calculus of variations and nonlinear partial differential equations, volume 1927 of Lecture Notes in Math., pages 1–41. Springer, Berlin, 2008. Zbl1159.35041
  3. W. Braun and K. Hepp. The Vlasov dynamics and its fluctuations in the 1 / N limit of interacting classical particles. Comm. Math. Phys., 56(2):101–113, 1977. Zbl1155.81383
  4. Patrick Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999. A Wiley-Interscience Publication. Zbl0172.21201
  5. E. Boissard. Problèmes d’interaction discret-continu et distances de Wasserstein. PhD thesis, Université de Toulouse III, 2011. 
  6. M. Bostan. Existence and uniqueness of the mild solution for the 1D Vlasov-Poisson initial-boundary value problem. SIAM J. Math. Anal., 37(1):156–188, 2005. Zbl1099.35149
  7. François Bouchut. Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Ration. Mech. Anal., 157(1):75–90, 2001. Zbl0979.35032
  8. J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent, and D. Slepčev. Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J., 156(2):229–271, 2011. Zbl1215.35045
  9. Jeffery Cooper and Alexander Klimas. Boundary value problems for the Vlasov-Maxwell equation in one dimension. J. Math. Anal. Appl., 75(2):306–329, 1980. Zbl0454.35075
  10. Jean-Marc Delort. Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc., 4(3):553–586, 1991. Zbl0780.35073
  11. F. Delarue, F. Flandoli, and D. Vincenzi. Noise prevents collapse of vlasov-poisson point charges. Commun. Pure Appl. Math., To appear. Zbl1302.35366
  12. Ronald J. DiPerna and Pierre-Louis Lions. Ordinary differential equations. Invent. Math, 98:511–547, 1989. Zbl0696.34049
  13. R. L. Dobrušin. Vlasov equations. Funktsional. Anal. i Prilozhen., 13(2):48–58, 96, 1979. Zbl0422.35068
  14. Alessio Figalli. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal., 254(1):109–153, 2008. Zbl1169.60010
  15. A. F. Filippov. Differential equations with discontinuous righthand sides, volume 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian. Zbl0664.34001
  16. Yan Guo. Singular solutions of the Vlasov-Maxwell system on a half line. Arch. Rational Mech. Anal., 131(3):241–304, 1995. Zbl0840.76097
  17. M. Hénon. Vlasov equation? Astronom. and Astrophys., 114(1):211–212, 1982. 
  18. Maxime Hauray and Stéphane Mischler. On Kac’s chaos and related problems. To appear in JFA. http://www.arxiv.org/abs/1205.4518, 2012. Zbl06326909
  19. Simon Labrunie, Sandrine Marchal, and Jean-Rodolphe Roche. Local existence and uniqueness of the mild solution to the 1D Vlasov-Poisson system with an initial condition of bounded variation. Math. Methods Appl. Sci., 33(17):2132–2142, 2010. Zbl1205.35166
  20. H. P. McKean, Jr. A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. U.S.A., 56:1907–1911, 1966. Zbl0149.13501
  21. H. P. McKean, Jr. Propagation of chaos for a class of non-linear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pages 41–57. Air Force Office Sci. Res., Arlington, Va., 1967. Zbl0181.44401
  22. Sylvie Méléard. Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. In Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), volume 1627 of Lecture Notes in Math., pages 42–95. Springer, Berlin, 1996. Zbl0864.60077
  23. S. Mischler and C. Mouhot. Kac’s Program in Kinetic Theory. http://arxiv.org/abs/1107.3251. Zbl1274.82048
  24. Andrew J. Majda, George Majda, and Yu Xi Zheng. Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case. Phys. D, 74(3-4):268–300, 1994. Zbl0813.35091
  25. Andrew J. Majda, George Majda, and Yu Xi Zheng. Concentrations in the one-dimensional Vlasov-Poisson equations. II. Screening and the necessity for measure-valued solutions in the two component case. Phys. D, 79(1):41–76, 1994. Zbl0839.35104
  26. H. Neunzert and J. Wick. The convergence of simulation methods in plasma physics. In Mathematical methods of plasmaphysics (Oberwolfach, 1979), volume 20 of Methoden Verfahren Math. Phys., pages 271–286. Lang, Frankfurt, 1980. Zbl0563.76125
  27. Elias M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970. Zbl0207.13501
  28. Alain-Sol Sznitman. Topics in propagation of chaos. In École d’Été de Probabilités de Saint-Flour XIX—1989, volume 1464 of Lecture Notes in Math., pages 165–251. Springer, Berlin, 1991. Zbl0732.60114
  29. M. Trocheris. On the derivation of the one-dimensional Vlasov equation. Transport Theory Statist. Phys., 15(5):597–628, 1986. Zbl0617.35120
  30. Cédric Villani. Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003. Zbl1106.90001
  31. Yu Xi Zheng and Andrew Majda. Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data. Comm. Pure Appl. Math., 47(10):1365–1401, 1994. Zbl0809.35088

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