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Nonlinear Landau damping for the two-species screened Vlasov-Poisson system with large initial distributions

Yi Wang, Meixia Xiao, Hang Xiong

Abstract

We investigate nonlinear Landau damping for the two-species screened Vlasov-Poisson system with large initial distributions on the phase space $\mathbb{R}^d \times \mathbb{R}^d$ (where $d \geq 3$). Under a structural quasi-neutrality condition, we establish the existence and uniqueness of global strong solutions to the two-species system with arbitrarily large initial distributions. Furthermore, we prove the time-asymptotic stability of Penrose-stable equilibria with an optimal decay rate of $t^{-d}$, thereby verifying the nonlinear Landau damping effect for the two-species screened Vlasov-Poisson system in the whole space. To the best of our knowledge, this represents the first result on Landau damping for the two-species Vlasov-Poisson system with large initial distributions that are significantly far from equilibrium.

Nonlinear Landau damping for the two-species screened Vlasov-Poisson system with large initial distributions

Abstract

We investigate nonlinear Landau damping for the two-species screened Vlasov-Poisson system with large initial distributions on the phase space (where ). Under a structural quasi-neutrality condition, we establish the existence and uniqueness of global strong solutions to the two-species system with arbitrarily large initial distributions. Furthermore, we prove the time-asymptotic stability of Penrose-stable equilibria with an optimal decay rate of , thereby verifying the nonlinear Landau damping effect for the two-species screened Vlasov-Poisson system in the whole space. To the best of our knowledge, this represents the first result on Landau damping for the two-species Vlasov-Poisson system with large initial distributions that are significantly far from equilibrium.
Paper Structure (20 sections, 17 theorems, 394 equations, 1 figure)

This paper contains 20 sections, 17 theorems, 394 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Landau Damping
  3. Main Result
  4. Key Challenges and Proof Strategy
  5. Organization of the paper
  6. Preliminaries
  7. Bootstrap Argument and Estimates of Characteristics
  8. Bootstrap Framework
  9. Estimates for the Characteristic Flow
  10. Difference Estimates and the Inverse Velocity Map
  11. Finite-Time Estimates
  12. Statement of Finite-Time a Priori Bounds
  13. Proof of Lemma \ref{['youjie010']}
  14. Proof of Proposition \ref{['xiaoshijian008']}
  15. Large-Time Decay Estimates
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $d \geq 3$, and suppose both Assumption ass:regularity and ass:penrose hold. For any fixed $M_0 \geq 1$, $k>d$, and $a \in (\frac{\gamma+1}{d-\gamma-1}, 1 )$ with $\gamma>0$ being suitably small constant, if the initial perturbations satisfy then there exists a small constant $\varepsilon_0>0$, depending on $M_0$, $a$, $d$, $k$, $\gamma$, such that if the following quasi-neutrality condition

Figures (1)

  • Figure 1: $\rho_{F, \mathrm{in}}^{\pm}:= \int_{\mathbb{R}^d} F^\pm(0,x,v)\, dv$ can be arbitrarily large with the quasi-neutrality condition

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 16 more