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The Cauchy Problem for the Euler–Poisson System and Derivation of the Zakharov–Kuznetsov Equation

Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 84)

Abstract

We consider in this paper the rigorous justification of the Zakharov–Kuznetsov equation from the Euler–Poisson system for uniformly magnetized plasmas. We first provide a proof of the local well-posedness of the Cauchy problem for the aforementioned system in dimensions two and three. Then we prove that the long-wave small-amplitude limit is described by the Zakharov–Kuznetsov equation. This is done first in the case of cold plasma; we then show how to extend this result in presence of the isothermal pressure term with uniform estimates when this latter goes to zero.

Key words

Zakharov–Kuznetsov Euler–Poisson 

Notes

Acknowledgements

The three authors acknowledge the support of IMPA, the Brazilian-French program in Mathematics, and the MathAmSud Project “Propagation of Nonlinear Dispersive Equations.” D. L. acknowledges support from the project ANR-08-BLAN-0301-01 and J.-C. S. from the project ANR-07-BLAN-0250 of the Agence Nationale de la Recherche.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • David Lannes
    • 1
  • Felipe Linares
    • 2
  • Jean-Claude Saut
    • 3
  1. 1.Département de Mathématiques et ApplicationsÉcole Normale SupérieureParis Cedex 05France
  2. 2.IMPARio de JaneiroBrasil
  3. 3.Laboratoire de Mathématiques, UMR 8628Université Paris-Sud 11 et CNRSOrsay CedexFrance

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