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Future global non-linear stability of surface symmetric solutions of the Einstein-Vlasov system with a cosmological constant

Ernesto Nungesser

Abstract

We show future global non-linear stability of surface symmetric solutions of the Einstein-Vlasov system with a positive cosmological constant. Estimates of higher derivatives of the metric and the matter terms are obtained using an inductive argument. In a recent research monograph Ringström shows future non-linear stability of (not necessarily symmetric) solutions of the Einstein-Vlasov system with a non-linear scalar field if certain local estimates on the geometry and the matter terms are fulfilled. We show that these assumptions are satisfied at late times for the case under consideration here which together with Cauchy stability leads to our main conclusion.

Future global non-linear stability of surface symmetric solutions of the Einstein-Vlasov system with a cosmological constant

Abstract

We show future global non-linear stability of surface symmetric solutions of the Einstein-Vlasov system with a positive cosmological constant. Estimates of higher derivatives of the metric and the matter terms are obtained using an inductive argument. In a recent research monograph Ringström shows future non-linear stability of (not necessarily symmetric) solutions of the Einstein-Vlasov system with a non-linear scalar field if certain local estimates on the geometry and the matter terms are fulfilled. We show that these assumptions are satisfied at late times for the case under consideration here which together with Cauchy stability leads to our main conclusion.

Paper Structure

This paper contains 16 sections, 10 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. The Einstein-Vlasov system with a cosmological constant
  3. The Einstein-Vlasov system with surface symmetry
  4. Basic equations
  5. Asymptotic behaviour of the first derivatives of the metric terms
  6. Characteristics
  7. Higher order estimates
  8. First step
  9. Second step: higher order derivatives of the characteristic system
  10. Geometric consequences and cosmic no-hair theorem
  11. Energy estimates for the distribution function
  12. Energy of order zero
  13. Energies of higher order
  14. Stability of surface symmetric solutions
  15. Outlook
  16. ...and 1 more sections

Key Result

Lemma 1

Consider a solution to the Einstein-Vlasov system with a positive cosmological constant $\Lambda$ and surface symmetry and fix $t_0\in(0,\infty)$ for $K\leq 0$ and fix $t_0\in(\Lambda^{-\frac{1}{2}},\infty)$ for $K=1$. Then there is a positive constant $C$, depending on the solution, such that where and the estimates hold for all $(t,r,w)$ in the support of $f$ and $s\in[t_0,t]$.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 13 more