From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge
Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 247-296.

We study the asymptotic regime of strong electric fields that leads from the Vlasov–Poisson system to the Incompressible Euler equations. We also deal with the Vlasov–Poisson–Fokker–Planck system which induces dissipative effects. The originality consists in considering a situation with a finite total charge confined by a strong external field. In turn, the limiting equation is set in a bounded domain, the shape of which is determined by the external confining potential. The analysis extends to the situation where the limiting density is non–homogeneous and where the Euler equation is replaced by the Lake Equation, also called Anelastic Equation.

Nous étudions le régime asymptotique de forts champs électriques qui conduit du système de Vlasov-Poisson aux équations d’Euler incompressibles. Nous abordons aussi le système de Vlasov-Poisson-Fokker-Planck qui induit des effets dissipatifs additionnels. L’originalité de cette étude réside dans le fait qu’on suppose la charge totale finie et confinée par un fort champ extérieur. En conséquence, l’équation limite est posée dans un domaine borné dont la géométrie est déterminée par ce champ confinant. L’analyse s’étend au cas où la densité limite est inhomogène ; l’équation d’Euler est alors remplacée par l’équation des lacs (ou modèle anélastique).

Received:
Accepted:
DOI: 10.5802/jep.24
Classification: 82D10, 35Q35, 82C40
Keywords: Plasma physics, Vlasov–Poisson system, Vlasov–Poisson–Fokker–Planck system, incompressible Euler equations, lake equations, quasi–neutral regime, modulated energy, relative entropy
Mot clés : Physique des plasmas, système de Vlasov-Poisson, système de Vlasov-Poisson-Fokker-Planck, équations d’Euler incompressibles, équation des lacs, régime quasi-neutre, énergie modulée, entropie relative

Julien Barré 1; David Chiron 1; Thierry Goudon 2; Nader Masmoudi 3

1 Université de Nice Sophia Antipolis, CNRS UMR 7351, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice, France
2 Université de Nice Sophia Antipolis, CNRS UMR 7351, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice, France & Inria, Sophia Antipolis Méditerranée Research Centre, Project COFFEE
3 Courant Institute for Math. Sciences, New York University 251 Mercer St., New York, NY 10012, U.S.A.
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JEP_2015__2__247_0,
     author = {Julien Barr\'e and David Chiron and Thierry Goudon and Nader Masmoudi},
     title = {From {Vlasov{\textendash}Poisson} and {Vlasov{\textendash}Poisson{\textendash}Fokker{\textendash}Planck} systems to incompressible {Euler} equations: the~case~with finite charge},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {247--296},
     publisher = {\'Ecole polytechnique},
     volume = {2},
     year = {2015},
     doi = {10.5802/jep.24},
     mrnumber = {3426749},
     zbl = {1339.82010},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.24/}
}
TY  - JOUR
AU  - Julien Barré
AU  - David Chiron
AU  - Thierry Goudon
AU  - Nader Masmoudi
TI  - From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2015
SP  - 247
EP  - 296
VL  - 2
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.24/
DO  - 10.5802/jep.24
LA  - en
ID  - JEP_2015__2__247_0
ER  - 
%0 Journal Article
%A Julien Barré
%A David Chiron
%A Thierry Goudon
%A Nader Masmoudi
%T From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge
%J Journal de l’École polytechnique — Mathématiques
%D 2015
%P 247-296
%V 2
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.24/
%R 10.5802/jep.24
%G en
%F JEP_2015__2__247_0
Julien Barré; David Chiron; Thierry Goudon; Nader Masmoudi. From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge. Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 247-296. doi : 10.5802/jep.24. https://jep.centre-mersenne.org/articles/10.5802/jep.24/

[1] A. A. Arsenev - “Global existence of a weak solution of Vlasov’s system of equations”, U.S.S.R. Comput. Math. and Math. Phys. 15 (1975), p. 131-143 | DOI

[2] G. Ben Arous & A. Guionnet - “Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy”, Probab. Theory Relat. Fields 108 (1997) no. 4, p. 517-542 | MR | Zbl

[3] F. Bouchut - “Existence and uniqueness of a global smooth solution for the VPFP system in three dimensions”, J. Funct. Anal. 111 (1993), p. 239-258 | DOI

[4] Y. Brenier - “A homogenized model for vortex sheets”, Arch. Rational Mech. Anal. 138 (1997) no. 4, p. 319-353 | DOI | MR | Zbl

[5] Y. Brenier - “Convergence of the Vlasov-Poisson system to the incompressible Euler equations”, Comm. Partial Differential Equations 25 (2000) no. 3-4, p. 737-754 | DOI | MR | Zbl

[6] L. Caffarelli - “The obstacle problem revisited”, J. Fourier Anal. Appl. 4 (1998) no. 3&4, p. 383-402 | MR | Zbl

[7] A. Camara, R. Kaiser & G. Labeyrie - “Behavior of a very large magneto-optical trap”, Phys. Rev. A 90 (2014) no. 6, 063404 pages | DOI

[8] D. Chafaï, N. Gozlan & P.-A. Zitt - “First order asymptotics for confined particles with singular pair repulsions”, Ann. Appl. Probab. 24 (2014) no. 6, p. 2371-2413 | DOI | MR

[9] G.-Q. Chen & H. Frid - “Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics”, Comm. Math. Phys. 236 (2003), p. 251-280 | DOI | MR

[10] J. Dalibard - “Laser cooling of an optically thick gas: The simplest radiation pressure trap?”, Optics Comm. 68 (1988), p. 203-208 | DOI

[11] J. Dalibard & C. Cohen-Tannoudji - “Atomic motion in laser light: connection between semiclassical and quantum descriptions”, J. Phys. B 18 (1985) no. 8, 1661 pages | MR

[12] P. Degond - “Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimension”, Ann. Sci. École Norm. Sup. (4) 19 (1986), p. 519-542 | DOI | Numdam | MR | Zbl

[13] J. Dolbeault - “Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (Large time behavior and steady states)”, J. Math. Pures Appl. 78 (1999) no. 2, p. 121-157 | DOI | MR | Zbl

[14] D. Dubin & T. O’Neil - “Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states)”, Rev. Modern Phys. 71 (1999) no. 1, p. 87-172

[15] J. Frehse - “On the regularity of the solution of a second order variational inequality”, Boll. Un. Mat. Ital. (4) 6 (1972), p. 312-315 | MR | Zbl

[16] A. Friedman - Variational principles and free-boundary problems, Pure and Applied Math., John Wiley & Sons, Inc., New York, 1982, A Wiley-Interscience Publication | Zbl

[17] A. Friedman & D. Phillips - “The free boundary of a semilinear elliptic equation”, Trans. Amer. Math. Soc. 282 (1984) no. 1, p. 153-182 | DOI | MR

[18] O. Frostman - “Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions”, Meddelanden Mat. Sem. Univ. Lund 3 (1935), p. 1-118 | Zbl

[19] M. A. Furman - “Compact complex expressions for the electric field of two-dimensional elliptical charge distributions”, Amer. J. Phys. 62 (12) (1994), p. 1134-1140 | DOI

[20] R. T. Glassey - The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996 | Zbl

[21] E. Grenier - “Defect measures of the Vlasov-Poisson system in the quasineutral regime”, Comm. Partial Differential Equations 20 (1995) no. 7-8, p. 1189-1215 | DOI | MR | Zbl

[22] E. Grenier - “Oscillations in quasineutral plasmas”, Comm. Partial Differential Equations 21 (1996) no. 3-4, p. 363-394 | DOI | MR | Zbl

[23] D. Han-Kwan - “Quasineutral limit of the Vlasov-Poisson system with massless electrons”, Comm. Partial Differential Equations 36 (2011) no. 8, p. 1385-1425 | DOI | MR | Zbl

[24] D. Han-Kwan & M. Hauray - “Stability issues in the quasineutral limit of the one-dimensional Vlasov-Poisson equation”, Comm. Math. Phys. 334 (2015) no. 2, p. 1101-1152 | DOI | MR | Zbl

[25] D. Han-Kwan & M. Iacobelli - “Quasineutral limit for Vlasov–Poisson via Wasserstein stability estimates in higher dimension” (2015), Preprint École polytechnique, arXiv:1503.06097 | Zbl

[26] H. Hedenmalm & N. Makarov - “Coulomb gas ensembles and Laplacian growth”, Proc. London Math. Soc. (3) 106 (2013) no. 4, p. 859-907 | DOI | MR | Zbl

[27] T. Kato, M. Mitrea, G. Ponce & M. Taylor - “Extension and representation of divergence-free vector fields”, Math. Res. Lett. 7 (2000), p. 643-650 | DOI | MR

[28] O. D. Kellogg - Foundations of potential theory, Grundlehren Math. Wiss., vol. 31, Springer-Verlag, Berlin, New York, 1967, Reprint from the first edition of 1929 | MR | Zbl

[29] D. Kinderlehrer & L. Nirenberg - “Regularity in free boundary problems”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977) no. 2, p. 373-391 | Numdam | MR | Zbl

[30] D. Kinderlehrer & G. Stampacchia - An introduction to variational inequalities and their applications, Pure and Applied Math., vol. 88, Academic Press, Inc., New York, London, 1980 | MR | Zbl

[31] C. D. Levermore, M. Oliver & E. S. Titi - “Global well-posedness for models of shallow water in a basin with a varying bottom”, Indiana Univ. Math. J. 45 (1996) no. 2, p. 479-510 | DOI | MR | Zbl

[32] C. D. Levermore, M. Oliver & E. S. Titi - “Global Well-Posedness for the Lake Equations”, Phys. D 98 (1996) no. 2-4, p. 492-509 | Zbl

[33] J.-L. Lions & E. Magenes - Problèmes aux limites non homogènes et applications (Volume 1), Travaux et recherches mathématiques, Dunod, Paris, 1968 | Zbl

[34] P.-L. Lions & B. Perthame - “Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system”, Invent. Math. 105 (1991), p. 415-430 | DOI | MR | Zbl

[35] N. Masmoudi - “From Vlasov-Poisson system to the incompressible Euler system”, Comm. Partial Differential Equations 26 (2001) no. 9-10, p. 1913-1928 | DOI | MR | Zbl

[36] N. Masmoudi - “Rigorous derivation of the anelastic approximation”, J. Math. Pures Appl. 88 (2007) no. 3, p. 230-240 | DOI | MR | Zbl

[37] J. Mendonça, R. Kaiser, H. Terças & J. Loureiro - “Collective oscillations in ultra-cold atomic gas”, Phys. Rev. A 78 (2008) no. 1, 013408 pages | DOI

[38] Y. Ogura & N. Phillips - “Scale analysis for deep and shallow convection in the atmosphere”, J. Atmospheric Sci. 19 (1962), p. 173-179 | DOI

[39] M. Oliver - “Classical solutions for a generalized Euler equation in two dimensions”, J. Math. Anal. Appl. 215 (1997) no. 2, p. 471-484 | DOI | MR | Zbl

[40] A. Olivetti - Effets collectifs et particules en interaction : des systèmes à longue portée aux atomes froids, Univ. Nice Sophia Antipolis, 2011, Ph. D. Thesis

[41] K. Pfaffelmoser - “Global existence of the Vlasov-Poisson system in three dimensions for general initial data”, J. Differential Equations 95 (1992), p. 281-303 | DOI | MR | Zbl

[42] E. Saff & V. Totik - Logarithmic potentials with external fields, Grundlehren Math. Wiss., Springer, Berlin, 1997 | Zbl

[43] S. Serfaty - “Coulomb Gases and Ginzburg–Landau Vortices”, 2014, Lecture Notes of the “nachdiplom vorleisung” course, ETH Zürich, Forschungsinstitut für Mathematik

[44] R. Temam - “On the Euler equations of incompressible perfect fluids”, J. Functional Analysis 20 (1975) no. 1, p. 32-43 | MR

[45] R. Temam - “Local existence of C solutions of the Euler equations of incompressible perfect fluids”, in Turbulence and Navier-Stokes equations (Orsay, 1975), Lect. Notes in Math., vol. 565, Springer, Berlin, 1976, p. 184-194 | DOI | MR | Zbl

[46] T. Walker, D. Sesko & C. Wieman - “Collective behavior of optically trapped neutral atoms”, Phys. Rev. Lett. 64 (1990), p. 408-412 | DOI

[47] D. Wineland, J. Bollinger, W. Itano & J. Prestage - “Angular momentum of trapped atomic particles”, J. Opt. Soc. Amer. 2 (1985) no. 11, p. 1721-1730 | DOI

Cited by Sources: