Skip to main content
SpringerLink
Log in

Asymptotic Limits in Macroscopic Plasma Models

  • Conference paper
Dispersive Transport Equations and Multiscale Models

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 136))

  • 386 Accesses

Abstract

A model hierarchy of macroscopic equations for plasmas consisting of electrons and ions is presented. The model equations are derived from the transient Euler-Poisson system in the zero-relaxation-time, zero-electron-mass and quasineutral limits. These asymptotic limits are performed using entropy estimates and compactness arguments. The resulting limits equations are Euler systems with a nonlinear Poisson equation and nonlinear drift-diffusion equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Refrences

  1. G. Ali, D. Bini, AND S. Rionero. Global existence and relaxation limit for smooth solutions to the Euler-Poisson model for semiconductors. SIAM J. Math. Anal, 32: 572–587, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Artsimowitsch AND R. Sagdejew. Plasmaphysik fü r Physiker. Teubner, Stuttgart, 1983.

    Book  Google Scholar 

  3. H. Brezis, F. Golse, AND R. Sentis. Analyse asymptotique de l’équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité des plasmas. C. R. Acad. Sci. Paris, 321: 953–959, 1995.

    MathSciNet  MATH  Google Scholar 

  4. F. Chen. Introduction to Plasma Physics and Controlled Fusion, Vol. I. Plenum Press, New York, 1984.

    Google Scholar 

  5. G.-Q. Chen, J. Jerome, AND B. Zhang. Particle hydrodynamic models in biology and microelectronics: singular relaxation limits. Nonlin. Anal, 30: 233–244, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  6. G.-Q. Chen, J. Jerome, AND B. Zhang. Existence and the singular relaxation limit for the inviscid hydrodynamic energy model. In J. Jerome, editor, Modelling and Computation for Application in Mathematics, Science, and Engineering, Oxford, 1998. Clarendon Press.

    Google Scholar 

  7. S. Cordier. Global solutions to the isothermal Euler-Poisson plasma model. Appl. Math. Lett, 8: 19–24, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Cordier, P. Degond, P. Markowich, AND C. Schmeiser. Quasineutral limit of travelling waves for the Euler-Poisson model. In G. Cohen, editor, Mathematical and Numerical Aspects of Wave Propagation. Proceedings of the Third International Conference, Mandelieu-La Napoule, France, pp. 724–733, Philadelphia, 1995. SIAM.

    Google Scholar 

  9. S. Cordier, P. Degond, P. Markowich, AND C. Schmeiser. Travelling wave analysis and jump relations for Euler-Poisson model in the quasineutral limit. Asymptotic Anal., 11: 209–224, 1995.

    MathSciNet  MATH  Google Scholar 

  10. S. Cordier AND E. Grenier. Quasineutral limit of two species Euler-Poisson systems. Proceedings of the Workshop “Recent Progress in the Mathematical Theory on Vlasov-Maxwell Equations” (Paris), pp. 95–122, 1997.

    Google Scholar 

  11. S. Cordier AND E. Grenier. Quasineutral limit of an Euler-Poisson system arising from plasma physics. Comm. P.D.E., 25: 1099–1113, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Cordier AND Y.-J. Peng. Système Euler-Poisson non linéaire-existence globale de solutions faibles entropiques. Mod. Math. Anal. Num., 32: 1–23, 1998.

    MathSciNet  MATH  Google Scholar 

  13. J. Delcroix. Plasma physics. John Wiley, London, 1965.

    Google Scholar 

  14. J.I. Diaz, G. Galiano, AND A. Jüngel. On a quasilinear degenerate system arising in semiconductor theory. Part I: existence and uniqueness of solutions. Nonlin. Anal. RWA, 2: 305–336, 2001.

    Article  MATH  Google Scholar 

  15. I. Gasser. The initial time layer problem and the quasi-neutral limit in a nonlinear drift-diffusion model for semiconductors. Nonlin. Diff. Eqs. Appl, 8: 237–249, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  16. I. Gasser, D. Levermore, P. Markowich, AND C. Schmeiser. The initial time layer problem and the quasineutral limit in the drift-diffusion model for semiconductors. Europ. J. Appl. Math., 12: 497–512, 2001.

    MathSciNet  MATH  Google Scholar 

  17. I. Gasser AND P. Marcati. The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors. Math. Methods Appl. Sci., 24: 81–92, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Goudon, A. Jüngel, AND Y.-J. Peng. Zero-electron-mass limits in hydrodynamic models for plasmas. Appl. Math. Lett, 12: 75–79, 1999.

    Article  MATH  Google Scholar 

  19. L. Hsiao AND K.J. Zhang. The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations. J. Diff. Eqs., 165: 315–354, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Junca AND M. Rascle. Strong relaxation of the isothermal Euler-Poisson system to the heat equation. Preprint, Université de Nice, Prance, 1999.

    Google Scholar 

  21. S. Junca AND M. Rascle. Relaxation of the isothermal Euler-Poisson system to the drift-diffusion equations. To appear in Quart. Appl. Math., 2002.

    Google Scholar 

  22. A. Jüngel. On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors. Math. Models Meth. Appl Sci., 4: 677–703, 1994.

    Article  MATH  Google Scholar 

  23. A. Jüngel. A nonlinear drift-diffusion system with electric convection arising in semiconductor and electrophoretic modeling. Math. Nachr., 185: 85–110, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Jüngel. Quasi-hydrodynamic Semiconductor Equations. Progress in Nonlinear Differential Equations. Birkhäuser, Basel, 2001.

    Google Scholar 

  25. A. Jüngel AND Y.-J. Peng. A hierarchy of hydrodynamic models for plasmas. Zero-electron-mass limits in the drift-diffusion equations. Ann. Inst. H. Poincaré, Anal, non linéaire, 17: 83–118, 2000.

    Article  MATH  Google Scholar 

  26. A. Jüngel AND Y.-J. Peng. Zero-relaxation-time limits in hydrodynamic models for plasmas revisited. Z. Angew. Math. Phys., 51: 385–396, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Jüngel AND Y.-J. Peng. A hierarchy of hydrodynamic models for plasmas: quasi-neutral limits in the drift-diffusion equations. Asympt. Anal, 28: 49–73, 2001.

    MATH  Google Scholar 

  28. A. Jüngel AND Y.-J. Peng. A hierarchy of hydrodynamic models for plasmas: zero-relaxation-time limits. Comm. P.D.E., 24: 1007–1033, 1999.

    Article  MATH  Google Scholar 

  29. P.L. Lions, B. Perthame, AND E. Souganidis. Existence of entropy solutions for the hyperbolic system of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math., 44: 599–638, 1996.

    MathSciNet  Google Scholar 

  30. P.L. Lions, B. Perthame, AND E. Tadmor. Kinetic formulation for the isentropic gas dynamics and p-system. Comm. Math. Phys., 163: 415–431, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Marcati AND R. Natalini. Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift diffusion equations. Arch. Rat. Mech. Anal, 129: 129–145, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Marcati AND R. Natalini. Weak solutions to a hydrodynamic model for semiconductors: The Cauchy problem. Proc. Roy. Soc. Edinb., Sect. A, 125: 115–131, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  33. P.A. Markowich. On steady state Euler-Poisson models for semiconductors. Z. Angew. Math. Phys., 42: 385–407, 1991.

    Article  MathSciNet  Google Scholar 

  34. P.A. Markowich, C. Ringhofer, AND C. Schmeiser. An asymptotic analysis of one-dimensional models of semiconductur devices. IMA J. Appl. Math., 37: 1–24, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  35. P.A. Markowich AND C.A. Ringhofer. A singularly perturbed boundary value problem modelling a semiconductor device. SIAM J. Appl. Math., 44: 231–256, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  36. R. Natalini. The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations. J. Math. Anal. Appl., 198: 262–281, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  37. Y.-J. Peng. Boundary layer analysis of the quasi-neutral limits in the drift-diffusion equations. Submitted for publication, 2000.

    Google Scholar 

  38. Y.-J. Peng. Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for a nonlinear Euler-Poisson system. Nonlin. Anal, 42: 1033–1054, 2000.

    Article  MATH  Google Scholar 

  39. F. Poupaud, M. Rascle, AND J. Vila. Global solutions to the isothermal Euler-Poisson system with arbitrarily large data. J. Diff. Eqs., 123: 93–121, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Simon. Compact sets in the space LP(0,T;B). Ann. Math. Pura Appl., 146: 65–96, 1987.

    MATH  Google Scholar 

  41. M. Slemrod AND N. Sternberg. Quasi-neutral limit for Euler-Poisson system. J. Nonlin. Sci., 11: 193–209, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  42. L. Tartar. Compensated compactness and applications to partial differential equations. In Nonlinear analysis and mechanics: Heriot-Watt Symp., Vol. 4, Volume 39 of Res. Notes Math., pp. 136–212, 1979.

    Google Scholar 

  43. G.M. Troianiello. Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York, 1987.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik und Statistik, Universität Konstanz, Fach D193, 78457, Konstanz, Germany

    Ansgar Jüngel

Authors
  1. Ansgar Jüngel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire MIP, Université Paul Sabatier, Toulouse Cedex 4, 31062, France

    Naoufel Ben Abdallah  & Pierre Degond  & 

  2. Angewandte Mathematik, Universität des Saarlandes, Saarbrucken, D-66041, Germany

    Anton Arnold

  3. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    Irene M. Gamba

  4. Department of Mathematics, Indiana University, Bloomington, IN, 47405-5701, USA

    Robert T. Glassey

  5. CSCAMM, University of Maryland, College Park, MD, 20742-4015, USA

    C. David Levermore

  6. Department of Mathematics, Arizona State University, Tempe, AZ, 85287, USA

    Christian Ringhofer

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this paper

Cite this paper

Jüngel, A. (2004). Asymptotic Limits in Macroscopic Plasma Models. In: Abdallah, N.B., et al. Dispersive Transport Equations and Multiscale Models. The IMA Volumes in Mathematics and its Applications, vol 136. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8935-2_10

Download citation

  • DOI: https://doi.org/10.1007/978-1-4419-8935-2_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6473-6

  • Online ISBN: 978-1-4419-8935-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation