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Stability in Quasineutral Plasmas with Thermalized Electrons

Megan Griffin-Pickering, Mikaela Iacobelli

Abstract

In this paper, we establish the stability of the quasineutral limit for the ionic Vlasov-Poisson system under perturbations exponentially small in Wasserstein sense. Notably, we emphasize that exponential smallness is a necessary condition in the electron case, as the presence of instabilities makes polynomial smallness insufficient. The study's quantitative nature introduces unique challenges, primarily arising from the exponential Poisson coupling. These challenges necessitate careful optimization at every step of the proof, whether it be in refining estimates or in the overall approach. Within this paper, we introduce novel tools and approaches to address these challenges. Specifically, we enhance the existing theory concerning the growth of characteristics in Vlasov systems featuring nonlinear couplings. Additionally, we combine stability estimates using kinetic-Wasserstein distances with improved regularity bounds on the elliptic coupling. In the course of demonstrating our central result, we also enhance the moment assumptions associated with the well-posedness of the ionic Vlasov-Poisson system.

Stability in Quasineutral Plasmas with Thermalized Electrons

Abstract

In this paper, we establish the stability of the quasineutral limit for the ionic Vlasov-Poisson system under perturbations exponentially small in Wasserstein sense. Notably, we emphasize that exponential smallness is a necessary condition in the electron case, as the presence of instabilities makes polynomial smallness insufficient. The study's quantitative nature introduces unique challenges, primarily arising from the exponential Poisson coupling. These challenges necessitate careful optimization at every step of the proof, whether it be in refining estimates or in the overall approach. Within this paper, we introduce novel tools and approaches to address these challenges. Specifically, we enhance the existing theory concerning the growth of characteristics in Vlasov systems featuring nonlinear couplings. Additionally, we combine stability estimates using kinetic-Wasserstein distances with improved regularity bounds on the elliptic coupling. In the course of demonstrating our central result, we also enhance the moment assumptions associated with the well-posedness of the ionic Vlasov-Poisson system.
Paper Structure (29 sections, 32 theorems, 314 equations, 3 figures, 2 tables)

This paper contains 29 sections, 32 theorems, 314 equations, 3 figures, 2 tables.

Table of Contents

  1. General Overview
  2. Vlasov-Poisson type systems.
  3. The Quasineutral Limit and Kinetic Euler Equations
  4. Previous results on the quasineutral limit.
  5. The quasineutral limit with rough initial data
  6. Electrons
  7. Ions
  8. Main Result
  9. Spatially Analytic Case
  10. Penrose Stable Case
  11. The long time behaviour of plasmas and the quasineutral limit.
  12. Preliminaries
  13. Representation of the torus $\mathbb{T}^d$
  14. Wasserstein Distances
  15. Density Estimates Using Moments
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $1\leq d \leq3$. Assume that there exists $T_\ast, \varepsilon_\ast > 0$ and, for each $\varepsilon \in (0, \varepsilon_\ast]$, a weak solution $g_\varepsilon$ of the Vlasov-Poisson system for ions eq:vpme on the time interval $[0, T_\ast]$ with initial datum $g_{0,\varepsilon}$ such that the fo Consider rough perturbations $\{ f_{0,\varepsilon} \}_{\varepsilon \leq \varepsilon_\ast}$ of the i

Figures (3)

  • Figure 1: Decomposition of the interval $[a,b]$ into subintervals of length no more than $\Delta(a,b)$.
  • Figure 2: The function $h_\varepsilon(z)$, showing its monotonicity properties for $z$ in different regions.
  • Figure 3: Construction of the times $(\tau_i)_{i=0}^J$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3: Analytic setting
  • Corollary 1.4: Penrose-stable setting
  • Remark 1.5
  • Remark 1.6: Comparison with the electron case
  • Remark 1.7: Sharpness of the result
  • Remark 1.8: Improvements to the well-posedness theory
  • Definition 2.1
  • Lemma 2.2
  • ...and 52 more