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Asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions

Ho Lee

Abstract

We study the asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions. The standard argument of Glassey and Strauss \cite{GS87} for studying small solutions to the Vlasov--Maxwell system does not apply to the Vlasov--Klein--Gordon system due to the massiveness of the Klein--Gordon field. In this paper we use the vector field method and consider solutions in dimensions $ n \geq 4 $ with the hyperboloidal foliation of the Minkowski spacetime to obtain the asymptotic properties for the Vlasov--Klein--Gordon system.

Asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions

Abstract

We study the asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions. The standard argument of Glassey and Strauss \cite{GS87} for studying small solutions to the Vlasov--Maxwell system does not apply to the Vlasov--Klein--Gordon system due to the massiveness of the Klein--Gordon field. In this paper we use the vector field method and consider solutions in dimensions with the hyperboloidal foliation of the Minkowski spacetime to obtain the asymptotic properties for the Vlasov--Klein--Gordon system.

Paper Structure

This paper contains 25 sections, 20 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Structure of the paper
  4. Preliminaries
  5. Hyperboloidal foliation
  6. Killing fields
  7. Energy-momentum tensor
  8. Energies
  9. Basic estimates
  10. Estimates of the energy densities
  11. Energy density for the distribution function
  12. Energy density for the Klein--Gordon field
  13. Klainerman--Sobolev inequalities
  14. Commuted equations
  15. Vlasov equation
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $n \geq 4$ and $N \geq 5 n + 2$. Let $( \phi , f )$ be a regular solution of the VKG system V--KG with initial data given at the unit hyperboloid $H_1$. Then, there exists $\varepsilon > 0$ such that if initial data satisfy then the corresponding solution satisfies the following estimates: Moreover, for each $( t , x )$ in the future of the unit hyperboloid, we have the following estimates:

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 31 more