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Large data decay of Yang-Mills-Higgs fields on Minkowski and de Sitter spacetimes

Grigalius Taujanskas

Abstract

We extend Eardley and Moncrief's $L^\infty$ estimates for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localise their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global $L^\infty$ estimates on the cylinder, and extend Choquet-Bruhat and Christodoulou's small data well-posedness result to large data. Finally, by employing another conformal transformation, we deduce exponential decay rates for Yang-Mills-Higgs fields on de Sitter space, and inverse polynomial decay rates on Minkowski space.

Large data decay of Yang-Mills-Higgs fields on Minkowski and de Sitter spacetimes

Abstract

We extend Eardley and Moncrief's estimates for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localise their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global estimates on the cylinder, and extend Choquet-Bruhat and Christodoulou's small data well-posedness result to large data. Finally, by employing another conformal transformation, we deduce exponential decay rates for Yang-Mills-Higgs fields on de Sitter space, and inverse polynomial decay rates on Minkowski space.

Paper Structure

This paper contains 16 sections, 8 theorems, 164 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Yang--Mills--Higgs Equations
  3. Conventions and Notation
  4. Localized Linfty Estimates on Minkowski Space
  5. Conservation of Energy
  6. The Cronström Gauge
  7. Integral Representations and Localization
  8. Gluing onto the Einstein Cylinder
  9. Conformal Transport of Estimates
  10. Global Existence on the Einstein Cylinder
  11. Local Existence á la Choquet-Bruhat and Christodoulou
  12. Energy Estimates
  13. Asymptotics
  14. De Sitter Space
  15. Minkowski Space
  16. ...and 1 more sections

Key Result

Lemma 4.4

The $L^\infty$ estimates of Eardley and Moncrief can be localized entirely to the lightcone. Specifically, one has the estimate where and $p(t)$ and $q(t)$ are positive polynomials (perhaps containing positive fractional powers) in $t$, with coefficients depending on the $(H^2(B(t)) \times H^1(B(t)))^2$ norms of the temporal gauge initial data, the local energy $E_{\mathrm{loc}}$ in the lightcon

Figures (2)

  • Figure 1: The embedding of $\mathbb{M}$ into $\mathfrak{E}$.
  • Figure :

Theorems & Definitions (19)

  • Remark 4.1
  • Definition 4.2
  • Remark 4.3
  • Lemma 4.4
  • proof
  • Theorem 4.5
  • Proposition 5.1
  • proof
  • Remark 5.2
  • Theorem 5.3
  • ...and 9 more