Stability issues in the quasineutral limit of the one-dimensional Vlasov-Poisson equation

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.

50 pages

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• A new perspective on Wasserstein distances for kinetic problems
• A mean field approach to the quasineutral limit for the Vlasov-Poisson equation
• Instabilities in the mean field limit
• Integrating Quantum Algorithms Into Classical Frameworks: A Predictor-corrector Approach Using HHL
• Nonlinear instability of inhomogeneous steady states solutions to the HMF Model