Scattering for the Equivariant U(1) Problem

TL;DR

This paper proves that for small data, the linear dynamics dominate the nonlinear behavior in the equivariant Einstein-wave map system in 2+1 dimensions, using a nonlinear Morawetz estimate to analyze the coupled equations.

Contribution

It extends previous work by establishing linear dominance in the coupled Einstein-wave map system with a new nonlinear Morawetz estimate for small data.

Findings

Linear part dominates nonlinear part for small data

Established nonlinear Morawetz estimate for coupled system

Applicable to U(1) symmetric 3+1 vacuum Einstein equations

Abstract

Extending our previous works on the Cauchy problem for the $2+1$ equivariant Einstein-wave map system, we prove that the linear part dominates the nonlinear part of the wave maps equation coupled to the full set of the Einstein equations, for small data. A key ingredient in the proof is a nonlinear Morawetz estimate for the fully coupled equivariant Einstein-wave maps. The $2+1$ dimensional Einstein-wave map system occurs naturally in the U(1) symmetric $3+1$ dimensional vacuum Einstein equations of general relativity.