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Scattering map for the Vlasov--Poisson system with a repulsive harmonic potential

Wenrui Huang, Hyunwoo Kwon

Abstract

We consider the Vlasov--Poisson system with a repulsive harmonic potential and prove the (modified) scattering of solutions, as well as the existence of wave operators, in any spatial dimension $d\geq 2$. The main novelty of this work is the construction of the wave operators and the introduction of the lens transform for the Vlasov--Poisson system. In addition, we provide a new and simpler proof that relaxes the assumptions on the initial data compared with those in Bigorgne, Velozo Ruiz, and Velozo Ruiz (2025) and Velozo Ruiz, and Velozo Ruiz (2024).

Scattering map for the Vlasov--Poisson system with a repulsive harmonic potential

Abstract

We consider the Vlasov--Poisson system with a repulsive harmonic potential and prove the (modified) scattering of solutions, as well as the existence of wave operators, in any spatial dimension . The main novelty of this work is the construction of the wave operators and the introduction of the lens transform for the Vlasov--Poisson system. In addition, we provide a new and simpler proof that relaxes the assumptions on the initial data compared with those in Bigorgne, Velozo Ruiz, and Velozo Ruiz (2025) and Velozo Ruiz, and Velozo Ruiz (2024).
Paper Structure (14 sections, 15 theorems, 184 equations)

This paper contains 14 sections, 15 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Idea of the proofs
  4. Organization
  5. Preliminaries
  6. Hyperbolic lens transform
  7. Estimates on the electric fields
  8. Interpolation inequalities
  9. Local well-posedness of the Vlasov--Poisson system with the repulsive potential
  10. Scattering of solutions
  11. Construction of the wave operator
  12. Commutative relations
  13. Closing the bootstrap
  14. Construction of local solutions

Key Result

Theorem 1.1

There exists $\varepsilon_0>0$ such that the following result holds: given $\mu_0 \in C^1_{x,v}$ satisfying there exists a unique global solution $\mu$ of eq:VP-rep-potential-reformulate with $\mu(t=0)=\mu_0$. Moreover, there exist $\mu_\infty\in L^{2}_{x,v}\cap L^{\infty}_{x,v}$ and $E_\infty=E[\mu_\infty]\in L^\infty_x$ such that we have where

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 27 more