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On the stability of vacuum in the screened Vlasov-Poisson equation

Mikaela Iacobelli, Stefano Rossi, Klaus Widmayer

Abstract

We study the asymptotic behavior of small data solutions to the screened Vlasov-Poisson equation on $\mathbb{R}^d\times\mathbb{R}^d$ near vacuum. We show that for dimensions $d\geq 2$, under mild assumptions on localization (in terms of spatial moments) and regularity (in terms of at most three Sobolev derivatives) solutions scatter freely. In dimension $d=1$, we obtain a long time existence result in analytic regularity.

On the stability of vacuum in the screened Vlasov-Poisson equation

Abstract

We study the asymptotic behavior of small data solutions to the screened Vlasov-Poisson equation on near vacuum. We show that for dimensions , under mild assumptions on localization (in terms of spatial moments) and regularity (in terms of at most three Sobolev derivatives) solutions scatter freely. In dimension , we obtain a long time existence result in analytic regularity.

Paper Structure

This paper contains 15 sections, 11 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Discussion of proofs
  3. Linear decay
  4. Nonlinear propagation of localization and regularity
  5. Free scattering in $d\ge 3$
  6. Linear analysis: dispersive decay estimates
  7. Nonlinear analysis: bootstrap estimates and free scattering
  8. Free scattering in $d=2$
  9. Linear analysis
  10. Nonlinear bootstrap
  11. Free scattering
  12. Long-time stability in $d=1$
  13. Linear estimates
  14. Nonlinear analysis
  15. Proof of an auxiliary statement

Key Result

Theorem 1.1

Let $d\geq 2$. There exists $\varepsilon_0>0$ such that if $\mu_0:\mathbb{R}^d\times\mathbb{R}^d\to\mathbb{R}$ satisfies then there exists a unique, global in time solution $\mu \in C_t(\mathbb{R},W^{1,\infty}_{x,v}(\mathbb{R}^d \times \mathbb{R}^d))$, $d\geq 3$, resp. $\mu \in C_t(\mathbb{R},H^3_{x,v}(\mathbb{R}^2 \times \mathbb{R}^2))$ to eq:scrVP for with $\mu(x,v,0)=\mu_0(x,v)$. Moreover, the

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Long-time stability in $d=1$
  • Remark 1.4
  • Lemma 2.1: Linear Decay in $d\ge3$
  • proof
  • Lemma 2.2: Bootstrap estimate in $d \ge 3$
  • proof
  • Remark 2.3
  • Theorem 2.4: Free scattering in $d\ge 3$
  • ...and 14 more