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Stability Issues in the Quasineutral Limit of the One-Dimensional Vlasov–Poisson Equation

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Abstract

This work is concerned with the quasineutral limit of the one-dimensional Vlasov–Poisson equation, for initial data close to stationary homogeneous profiles. Our objective is threefold: first, we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the Penrose instability criterion. Second, we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition, together with a symmetry condition. We handle the case of well-prepared as well as ill-prepared data. Last, we study a stationary boundary-value problem for the formal limit, the so-called quasineutral Vlasov equation. We show the existence of numerous stationary states, with a lot of freedom in the construction (compared to that of BGK waves for Vlasov–Poisson): this illustrates the degeneracy of the limit equation.

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Authors and Affiliations

  1. CNRS and École Polytechnique, Centre de Mathématiques Laurent Schwartz UMR7640, 91128, Palaiseau Cedex, France

    Daniel Han-Kwan

  2. Université d’Aix-Marseille, CNRS et École Centrale Marseille, LATP, 13453, Marseille Cedex, France

    Maxime Hauray

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  1. Daniel Han-Kwan
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  2. Maxime Hauray
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Correspondence to Daniel Han-Kwan.

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Communicated by C. Mouhot

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Han-Kwan, D., Hauray, M. Stability Issues in the Quasineutral Limit of the One-Dimensional Vlasov–Poisson Equation. Commun. Math. Phys. 334, 1101–1152 (2015). https://doi.org/10.1007/s00220-014-2217-4

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