Global existence  for the Euler-Maxwell system

Germain, Pierre; Masmoudi, Nader

doi:10.24033/asens.2219

Global existence for the Euler-Maxwell system
[Existence globale pour le système d'Euler-Maxwell]

Germain, Pierre ; Masmoudi, Nader

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 3, pp. 469-503.

Résumé
Abstract

Le système d'Euler-Maxwell décrit l'évolution d'un plasma quand les collisions sont suffisamment importantes pour que chaque espèce soit dans un état d'équilibre hydrodynamique. On prouve dans cet article l'existence globale de petites solutions à ce système, posé en dimension 3 d'espace, en combinant la méthode des résonances en espace-temps (pour obtenir la décroissance des solutions) et des estimations d'énergie (pour contrôler la régularité des solutions). La décroissance non intégrable des solutions impose de combiner étroitement ces deux arguments en examinant le rôle des résonances au sein des estimations d'énergie.

The Euler-Maxwell system describes the evolution of a plasma when the collisions are important enough that each species is in a hydrodynamic equilibrium. In this paper we prove global existence of small solutions to this system set in the whole three-dimensional space, by combining the space-time resonance method (to obtain decay) and energy estimates (to control high frequencies). The non-integrable decay of the solutions makes it necessary to examine resonances within the energy estimate argument.

Publié le : 2014-09-24

MR Zbl

DOI : 10.24033/asens.2219

Classification : 35L03, 35L60, 35Q31, 35Q60, 37L50.
Keywords: Euler-Maxwell equations, global existence, space-time resonances.
Mot clés : Équations d'Euler-Maxwell, existence globale, résonances en espace-temps.

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     author = {Germain, Pierre and Masmoudi, Nader},
     title = {Global existence  for the {Euler-Maxwell} system},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {469--503},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 47},
     number = {3},
     year = {2014},
     doi = {10.24033/asens.2219},
     mrnumber = {3239096},
     zbl = {1311.35195},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2219/}
}

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Germain, Pierre; Masmoudi, Nader. Global existence  for the Euler-Maxwell system. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 3, pp. 469-503. doi : 10.24033/asens.2219. https://www.numdam.org/articles/10.24033/asens.2219/

Bibliographie
Cité par

Besse, C.; Degond, P.; Deluzet, F.; Claudel, J.; Gallice, G.; Tessieras, C. A model hierarchy for ionospheric plasma modeling, Math. Models Methods Appl. Sci., Volume 14 (2004), pp. 393-415 (ISSN: 0218-2025) | DOI | MR | Zbl

Bellan, P. M., Cambridge Univ. Press, 2006

Bony, J.-M. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., Volume 14 (1981), pp. 209-246 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Boyd, T. J. M.; Sanderson, J. J., Cambridge Univ. Press, Cambridge, 2003, 532 pages (ISBN: 0-521-45290-2; 0-521-45912-5) | DOI | MR | Zbl

Christodoulou, D., EMS Monographs in Math., European Mathematical Society (EMS), Zürich, 2007, 992 pages (ISBN: 978-3-03719-031-9) | DOI | MR | Zbl

Chen, G.-Q.; Jerome, J. W.; Wang, D. Compressible Euler-Maxwell equations, Transport Theory Statist. Phys. (Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998)), Volume 29 (2000), pp. 311-331 (ISSN: 0041-1450) | DOI | MR | Zbl

Coifman, R. R.; Meyer, Y., Astérisque, 57, Soc. Math. France, Paris, 1978, 185 pages | Numdam | MR | Zbl

Dendy, R.-O., Oxford Univ. Press, 1990

Duan, R. Global smooth flows for the compressible Euler-Maxwell system. The relaxation case, J. Hyperbolic Differ. Equ., Volume 8 (2011), pp. 375-413 (ISSN: 0219-8916) | DOI | MR | Zbl

Germain, P. Space-time resonances, Journées Equations aux Dérivées Partielles (2010) (exp. n^o 8) | DOI

Germain, P. Global existence for coupled Klein-Gordon equations with different speeds, Ann. Inst. Fourier (Grenoble), Volume 61 (2011), pp. 2463-2506 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Germain, P.; Masmoudi, N.; Pausader, B. Nonneutral global solutions for the electron Euler-Poisson system in three dimensions, SIAM J. Math. Anal., Volume 45 (2013), pp. 267-278 (ISSN: 0036-1410) | DOI | MR | Zbl

Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not., Volume 2009 (2009), pp. 414-432 (ISSN: 1073-7928) | DOI | MR | Zbl

Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for the gravity water waves equation in dimension 3, C. R. Math. Acad. Sci. Paris, Volume 347 (2009), pp. 897-902 (ISSN: 1631-073X) | DOI | MR | Zbl

Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for the gravity water waves equation in dimension 3, Ann. of Math., Volume 175 (2012), pp. 691-754 (ISSN: 0003-486X) | DOI | MR | Zbl

Guo, Y.; Pausader, B. Global smooth ion dynamics in the Euler-Poisson system, Comm. Math. Phys., Volume 303 (2011), pp. 89-125 (ISSN: 0010-3616) | DOI | MR | Zbl

Golse, F.; Saint-Raymond, L. The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., Volume 155 (2004), pp. 81-161 (ISSN: 0020-9910) | DOI | MR | Zbl

Guo, Y.; Tahvildar-Zadeh, A. S., Nonlinear partial differential equations (Evanston, IL, 1998) (Contemp. Math.), Volume 238, Amer. Math. Soc., Providence, RI, 1999, pp. 151-161 | DOI | MR | Zbl

Guo, Y. Smooth irrotational flows in the large to the Euler-Poisson system in $𝐑^{3 + 1}$ , Comm. Math. Phys., Volume 195 (1998), pp. 249-265 (ISSN: 0010-3616) | DOI | MR | Zbl

Ginibre, J.; Velo, G. Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., Volume 43 (1985), pp. 399-442 (ISSN: 0246-0211) | Numdam | MR | Zbl

Ibrahim, S.; Masmoudi, N.; Nakanishi, K. Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, Volume 4 (2011), pp. 405-460 (ISSN: 1948-206X) | DOI | MR | Zbl

Jang, J.; Masmoudi, N. Derivation of Ohm's law from the kinetic equations, SIAM J. Math. Anal., Volume 44 (2012), pp. 3649-3669 (ISSN: 0036-1410) | DOI | MR | Zbl

Lions, P.-L.; Masmoudi, N. From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II, Arch. Ration. Mech. Anal., Volume 158 (2001) (ISSN: 0003-9527) | DOI | MR | Zbl

Levermore, C. D.; Masmoudi, N. From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal., Volume 196 (2010), pp. 753-809 (ISSN: 0003-9527) | DOI | MR | Zbl

Masmoudi, N. Global well posedness for the Maxwell-Navier-Stokes system in 2D, J. Math. Pures Appl., Volume 93 (2010), pp. 559-571 (ISSN: 0021-7824) | DOI | MR | Zbl

Makino, T.; Perthame, B. Sur les solutions à symétrie sphérique de l'équation d'Euler-Poisson pour l'évolution d'étoiles gazeuses, Japan J. Appl. Math., Volume 7 (1990), pp. 165-170 (ISSN: 0910-2043) | DOI | MR | Zbl

Muscalu, C.; Schlag, W., Cambridge Studies in Advanced Math., 138, Cambridge Univ. Press, 2013 | Zbl

Makino, T.; Ukai, S.; Kawashima, S. Sur la solution à support compact de l'équation d'Euler compressible, Japan J. Appl. Math., Volume 3 (1986), pp. 249-257 (ISSN: 0910-2043) | DOI | MR | Zbl

Perthame, B. Nonexistence of global solutions to Euler-Poisson equations for repulsive forces, Japan J. Appl. Math., Volume 7 (1990), pp. 363-367 (ISSN: 0910-2043) | DOI | MR | Zbl

Pan, R.; Smoller, J. A. Blowup of smooth solutions for relativistic Euler equations, Comm. Math. Phys., Volume 262 (2006), pp. 729-755 (ISSN: 0010-3616) | DOI | MR | Zbl

Peng, Y.-J.; Wang, S. Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., Volume 40 (2008), pp. 540-565 (ISSN: 0036-1410) | DOI | MR | Zbl

Sideris, T. C. Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., Volume 101 (1985), pp. 475-485 http://projecteuclid.org/euclid.cmp/1104114244 (ISSN: 0010-3616) | DOI | MR | Zbl

Texier, B. WKB asymptotics for the Euler-Maxwell equations, Asymptot. Anal., Volume 42 (2005), pp. 211-250 (ISSN: 0921-7134) | MR | Zbl

Yang, J.; Wang, S. Non-relativistic limit of two-fluid Euler-Maxwell equations arising from plasma physics, ZAMM Z. Angew. Math. Mech., Volume 89 (2009), pp. 981-994 (ISSN: 0044-2267) | DOI | MR | Zbl

Yang, J.; Wang, S.; Li, Y.; Luo, D. Rigorous derivation of incompressible type Euler equations from non-isentropic Euler-Maxwell equations, Nonlinear Anal., Volume 73 (2010), pp. 3613-3625 (ISSN: 0362-546X) | DOI | MR | Zbl

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