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

Abstract

In this article we extend Eardley and Moncrief's L ∞ estimates [5] for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localize their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global L ∞ estimates on the cylinder, and extend Choquet-Bruhat and Christodoulou's [1] 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. the backward lightcone of p. Their key observation is that these lightcone integrals can be estimated by expressions of the formwhich implies, via Grönwall's inequality, that the L ∞ norms cannot blow up in finite time. Part of the trick is to define the L ∞ norms in a gauge-independent manner, and use the Crönstrom gauge in intermediate calculations. Equipped with this estimate, it is then straightforward to show that the H 2 × H 1 norm of the solution does not blow up in finite time. An incarnation of this method has been adapted, for pure Yang-Mills equations, to arbitrary smooth globally hyperbolic four dimensional spacetimes by Chruściel and Shatah [3], by replacing the lightcone integrals with Friedlander's representation formula [6] for the covariant wave equation. However, Chruściel and Shatah require effectively H 3 × H 2 data to deal with a term that causes difficulties in curved space 1 . Though the system has been well-studied, Eardley and Moncrief's method with H 2 × H 1 data for coupled Yang-Mills and Higgs equations does not seem to have been explicitly adapted to curved space, even in the case of the Einstein cylinder. The scalar field part scales differently under a conformal transformation, putting it on unequal footing with the Yang-Mills potential. In particular, this upsets the conformal invariance of the system somewhat, breaking the invariance of the canonical energy-momentum tensor. And although formally the field equations remain conformally invariant, the scalar field introduces a boundary term in the conformal variation of the action that has a non-trivial dependence on the decay of the scalar field. This is expected to be of some importance in path integral formulations of interacting quantum field theories.In this article we extend the L ∞ estimates of Eardley and Moncrief to the Einstein cylinder. Our method is inspired by and combines the techniques of [1,5,9]: we first work on Minkowski space and localize Eardley and Moncrief's estimates, removing the requirement of the global finiteness of the energy. Then, using a conformal transformation, we glue a small conical patch of Minkowski space onto the Einstein cylinder, and show that L ∞ estimates in the Minkowskian patch imply local L ∞ estimates on the cylinder. By patching a finite number of such cones all the way around the Einstein cylinder, we deduce L ∞ bounds ...

On the global dynamics of Yang-Mills-Higgs equations

“…The theorem extends the classical results of [7] and [4] (also see recent work [41]). Compared to Theorem 1.4, these results required faster fall off of the initial data or quantitatively γ 1 ≥ 2.…”

“…In view of the estimate (41) for Dφ and the definition for M 1 (t), by taking the supreme over all x 0 , we derive that…”

“…It is widely used to study the long time dynamics of solutions to linear and nonlinear fields [5], [8]. Combining with the classical global existence result of Eardley-Moncrief, it has been shown that solution of (YMH) in R 1+3 asymptotically behaves like linear wave for sufficiently smooth and rapid decaying data [4], [6], [7], [41]. However, the decay assumptions on the initial data, such as the weighted energy with weights |x| 2 , exclude the existence of charges of the fields, hence exclude the most interesting physical solutions such as solutions with dipole or quadrapole moments 2 .…”