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Global well-posedness for a system of quasilinear wave equations on a product space

Cécile Huneau, Annalaura Stingo

Abstract

We consider a system of quasilinear wave equations on the product space $\mathbb{R}^{1+3}\times \mathbb{S}^1$, which we want to see as a toy model for Einstein equations with additional compact dimensions. We show global existence for small and regular initial data with polynomial decay at infinity. The method combines energy estimates on hyperboloids inside the light cone and weighted energy estimates outside the light cone.

Global well-posedness for a system of quasilinear wave equations on a product space

Abstract

We consider a system of quasilinear wave equations on the product space , which we want to see as a toy model for Einstein equations with additional compact dimensions. We show global existence for small and regular initial data with polynomial decay at infinity. The method combines energy estimates on hyperboloids inside the light cone and weighted energy estimates outside the light cone.

Paper Structure

This paper contains 24 sections, 32 theorems, 282 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and a brief history
  3. The main result
  4. The wave-Klein-Gordon structure
  5. Vector Fields
  6. The null structure
  7. The interior and exterior region
  8. The energy functionals
  9. The quasilinear energies
  10. Higher order norms
  11. Notations
  12. Overview of the proof
  13. The Linearized Equation
  14. Global existence in the exterior region
  15. Weighted Sobolev inequalities
  16. ...and 9 more sections

Key Result

Theorem 1

Assume the initial data $(u[2], v[2])$ for syteme satisfy for some positive fixed $\alpha$ and $\langle x \rangle = \sqrt{1+|x|^2}$. Then the system syteme is globally well-posed in the space $\mathcal{H}^5$.

Figures (2)

  • Figure 1: Vertical section of the region $\mathcal{H}^{\text{in}}_{[2, s]}$ projected onto $\mathbb{R}^{1+3}$
  • Figure 2: Vertical section of the region $\mathcal{H}^{\text{ex}}_{[2, s]}$ and its foliation projected onto $\mathbb{R}^{1+3}$

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • ...and 28 more