From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge

Barré, Julien; Chiron, David; Goudon, Thierry; Masmoudi, Nader

doi:10.5802/jep.24

From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge
[Des équations de Vlasov-Poisson et Vlasov-Poisson-Fokker-Planck aux équations d’Euler incompressibles : le cas de charge finie]

Barré, Julien ¹ ; Chiron, David ¹ ; Goudon, Thierry ² ; Masmoudi, Nader 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.

Journal de l’École polytechnique - Mathématiques, Tome 2 (2015), pp. 247-296.

Résumé
Abstract

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).

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.

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

Affiliations des auteurs :

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

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Barré, Julien; Chiron, David; Goudon, Thierry; Masmoudi, Nader. From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge. Journal de l’École polytechnique - Mathématiques, Tome 2 (2015), pp. 247-296. doi : 10.5802/jep.24. http://www.numdam.org/articles/10.5802/jep.24/

Bibliographie
Cité par

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

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

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

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

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

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

[7] Camara, A.; Kaiser, R.; Labeyrie, G. Behavior of a very large magneto-optical trap, Phys. Rev. A, Volume 90 (2014) no. 6, pp. 063404

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

[9] Chen, G.-Q.; Frid, H. Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics, Comm. Math. Phys., Volume 236 (2003), pp. 251-280 | MR | Zbl

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

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

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

[13] Dolbeault, J. 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., Volume 78 (1999) no. 2, pp. 121-157 | MR | Zbl

[14] Dubin, D.H.E.; O’Neil, T.M. Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states), Rev. Modern Phys., Volume 71 (1999) no. 1, pp. 87-172

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

[16] Friedman, A. Variational principles and free-boundary problems, Pure and Applied Math., John Wiley & Sons, Inc., New York, 1982, pp. ix+710 (A Wiley-Interscience Publication) | MR | Zbl

[17] Friedman, A.; Phillips, D. The free boundary of a semilinear elliptic equation, Trans. Amer. Math. Soc., Volume 282 (1984) no. 1, pp. 153-182 | MR | Zbl

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

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

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

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

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

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

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

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

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

[27] Kato, T.; Mitrea, M.; Ponce, G.; Taylor, M. Extension and representation of divergence-free vector fields, Math. Res. Lett., Volume 7 (2000), pp. 643-650 | MR | Zbl

[28] Kellogg, O. D. Foundations of potential theory, Grundlehren Math. Wiss., 31, Springer-Verlag, Berlin, New York, 1967, pp. ix+384 (Reprint from the first edition of 1929) | MR | Zbl

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

[30] Kinderlehrer, D.; Stampacchia, G. An introduction to variational inequalities and their applications, Pure and Applied Math., 88, Academic Press, Inc., New York, London, 1980, pp. xiv+313 | MR | Zbl

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

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

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

[34] Lions, P.-L.; Perthame, B. Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system, Invent. Math., Volume 105 (1991), pp. 415-430 | EuDML | MR | Zbl

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

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

[37] Mendonça, J.T.; Kaiser, R.; Terças, H.; Loureiro, J. Collective oscillations in ultra-cold atomic gas, Phys. Rev. A, Volume 78 (2008) no. 1, pp. 013408

[38] Ogura, Y.; Phillips, N. Scale analysis for deep and shallow convection in the atmosphere, J. Atmospheric Sci., Volume 19 (1962), pp. 173-179

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

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

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

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

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

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

[45] Temam, R. Local existence of $C^{\infty}$ solutions of the Euler equations of incompressible perfect fluids, Turbulence and Navier-Stokes equations (Orsay, 1975) (Lect. Notes in Math.), Volume 565, Springer, Berlin, 1976, pp. 184-194 | MR | Zbl

[46] Walker, T.; Sesko, D.; Wieman, C. Collective behavior of optically trapped neutral atoms, Phys. Rev. Lett., Volume 64 (1990), pp. 408-412

[47] Wineland, D.J.; Bollinger, J.J.; Itano, W.M.; Prestage, J.D. Angular momentum of trapped atomic particles, J. Opt. Soc. Amer., Volume 2 (1985) no. 11, pp. 1721-1730

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