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

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Studies in Phase Space Analysis with Applications to PDEs

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((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.

2010 Mathematics Subject Classification: 35B40, 35Q53

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Notes

  1. 1.

    Actually we will not use the third-order profiles.

  2. 2.

    See [8, 25, 30] for the Cauchy theory in larger spaces.

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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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Authors and Affiliations

  1. Département de Mathématiques et Applications, École Normale Supérieure, 45, rue d’Ulm, F-75230, Paris Cedex 05, France

    David Lannes

  2. IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, RJ, Brasil

    Felipe Linares

  3. Laboratoire de Mathématiques, UMR 8628, Université Paris-Sud 11 et CNRS, F-91405, Orsay Cedex, France

    Jean-Claude Saut

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  1. David Lannes
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  2. Felipe Linares
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  3. Jean-Claude Saut
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Correspondence to David Lannes .

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Editors and Affiliations

  1. , Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, Bologna, I-40126, Italy

    Massimo Cicognani

  2. , Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, Pisa, I-56127, Italy

    Ferruccio Colombini

  3. , Dipartimento di Matematica e Geoscienze, Università di Trieste, Via Valerio, 12/1 12/1, Trieste, I-34127, Italy

    Daniele Del Santo

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Lannes, D., Linares, F., Saut, JC. (2013). The Cauchy Problem for the Euler–Poisson System and Derivation of the Zakharov–Kuznetsov Equation. In: Cicognani, M., Colombini, F., Del Santo, D. (eds) Studies in Phase Space Analysis with Applications to PDEs. Progress in Nonlinear Differential Equations and Their Applications, vol 84. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6348-1_10

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