Conformal scattering for a nonlinear wave equation on a curved background

TL;DR

This paper establishes a geometric scattering theory for a conformally invariant nonlinear wave equation on asymptotically simple spacetimes, using trace operators at null infinity and adapted energy estimates.

Contribution

It introduces a new approach to define the conformal scattering operator for nonlinear waves on curved backgrounds, combining geometric and analytical techniques.

Findings

Derived a priori linear estimates using Morawetz vector fields.

Proved well-posedness of the characteristic Cauchy problem at null infinity.

Constructed trace operators to define the scattering operator.

Abstract

The purpose of this paper is to establish a geometric scattering result for a conformally invariant nonlinear wave equation on an asymptotically simple spacetime. The scattering operator is obtained via trace operators at null infinities. The proof is achieved in three steps. A priori linear estimates are obtained via an adaptation of the Morawetz vector field in the Schwarzschild spacetime and a method used by H\"ormander for the Goursat problem. A well-posedness result for the characteristic Cauchy problem on a light cone at infinity is then obtained. This requires a control of the nonlinearity uniform in time which comes from an estimates of the Sobolev constant and a decay assumption on the nonlinearity of the equation. Finally, the trace operators on conformal infinities are built and used to define the conformal scattering operator.