Communications in Mathematical Physics

, Volume 303, Issue 1, pp 89–125 | Cite as

Global Smooth Ion Dynamics in the Euler-Poisson System

Article

Abstract

A fundamental two-fluid model for describing dynamics of a plasma is the Euler-Poisson system, in which compressible ion and electron fluids interact with their self-consistent electrostatic force. Global smooth electron dynamics were constructed in Guo (Commun Math Phys 195:249–265, 1998) due to dispersive effect of the electric field. In this paper, we construct global smooth irrotational solutions with small amplitude for ion dynamics in the Euler-Poisson system.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chen, G.Q., Jerome, J.W., Wang, D.: Compressible Euler-Maxwell equations. Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998). Transport Theory Statist. Phys. 29(3–5), 311–331, (2000)Google Scholar
  2. 2.
    Coifman R., Meyer Y.: Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28(3, xi), 177–202 (1978)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cordier S., Grenier E.: Quasineutral limit of an Euler-Poisson system arising from plasma physics. Comm. Part. Diff. Eqs. 25(5–6), 1099–1113 (2000)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Feldman M., Ha S.-Y., Slemrod M.: Self-similar isothermal irrotational motion for the Euler, Euler- Poisson systems and the formation of the plasma sheath. J. Hyp. Diff. Eq. 3(2), 233–246 (2006)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Feldman M., Ha S.-Y., Slemrod M.: A geometric level-set formulation of a plasma-sheath interface. Arch. Ratn. Mech. Anal. 178(1), 81–123 (2005)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Germain P., Masmoudi N., Shatah J.: Global solutions for 3D quadratic Schrödinger equations. Int. Math. Res. Not. 2009(3), 414–432 (2009)MathSciNetMATHGoogle Scholar
  7. 7.
    Germain, P., Masmoudi, N., Shatah, J.: Global solutions for the gravity water waves equation in dimension 3. Preprint, available at http://arxiv.org/abs/1001.5158v1 [math.AP], 2010
  8. 8.
    Germain, P., Masmoudi, N., Shatah, J.: Global solutions for 2D quadratic Schrödinger equations. Preprint, available at http://arxiv.org/abs/0906.5343v1 [math.Ap], 2009
  9. 9.
    Guo Y.: Smooth irrotational Flows in the large to the Euler-Poisson system in R 3+1. Commun. Math. Phys. 195, 249–265 (1998)ADSMATHCrossRefGoogle Scholar
  10. 10.
    Guo, Y., Tahvildar-Zadeh, A.S.: Formation of singularities in relativistic fluid dynamics and in spherically symmetric plasma dynamics. In: Nonlinear partial differential equations (Evanston, IL, 1998), Contemp. Math., 238, Providence, RI: Amer. Math. Soc., 1999, pp. 151–161Google Scholar
  11. 11.
    Guo Z., Peng L., Wang B.: Decay estimates for a class of wave equations. J. Funct. Anal. 254(6), 1642–1660 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gustafson S., Nakanishi K., Tsai T.P.: Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions. Ann. IHP 8(7), 1303–1331 (2007)MathSciNetMATHGoogle Scholar
  13. 13.
    Gustafson S., Nakanishi K., Tsai T.P.: Scattering theory for the Gross-Pitaevskii equation in three dimensions. Commun. Contemp. Math. 11(4), 657–707 (2009)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    John F.: Plane Waves and Spherical Means, Applied to Partial Differential Equations. J. Appl. Math. Mech. 62(7), 285–356 (1982)Google Scholar
  15. 15.
    Kato T.: The Cauchy problem for quasilinear symmetric systems. Arch. Rat. Mech. Anal. 58, 181–205 (1975)MATHCrossRefGoogle Scholar
  16. 16.
    Liu H., Tadmor E.: Critical thresholds in 2D restricted Euler-Poisson equations. SIAM J. Appl. Math. 63(6), 1889–1910 (2003) (electronic)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Liu H., Tadmor E.: Spectral dynamics of the velocity gradient field in restricted flows. Commun Math. Phys. 228(3), 435–466 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Muscalu C.: Paraproducts with flag singularities. I. A case study. Rev. Mat. Iberoam. 23(2), 705–742 (2007)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Muscalu C., Pipher J., Tao T., Thiele C.: Multi-parameter paraproducts. Rev. Mat. Iberoam. 22(3), 963–976 (2006)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Peng Y., Wang S.: Convergence of compressible Euler-Maxwell equations to compressible Euler- Poisson equations. Chin. Ann. Math. Ser. B 28(5), 583–602 (2007)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Peng Y., Wang Y.-G.: Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows. Nonlinearity 17(3), 835–849 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Shatah J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math. 38(5), 685–696 (1985)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Sideris T.: Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys. 101, 475–485 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Stein, E.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Volume 43 of Princeton Mathematical Series. Princeton, NJ: Princeton University Press 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, IIIGoogle Scholar
  25. 25.
    Tao, T.: Nonlinear dispersive equations, local and global analysis. CBMS. Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Science, Washington, DC; Providence, RI: Amer. Math. Soc., 2006Google Scholar
  26. 26.
    Texier B.: WKB asymptotics for the Euler-Maxwell equations. Asymptot. Anal. 42(3-4), 211–250 (2005)MathSciNetMATHGoogle Scholar
  27. 27.
    Texier B.: Derivation of the Zakharov equations. Arch. Ration. Mech. Anal. 184(1), 121–183 (2007)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Wang D.: Global solution to the equations of viscous gas flows. Proc. Roy. Soc. Edinburgh Sect. A 131(2), 437–449 (2001)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Wang D., Wang Z.: Large BV solutions to the compressible isothermal Euler-Poisson equations with spherical symmetry. Nonlinearity 19(8), 1985–2004 (2006)MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Mathematics DepartmentBrown UniversityProvidenceUSA

Personalised recommendations