Conformal scattering for a nonlinear wave equation on a curved background
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.
Cite
@article{arxiv.1004.1464,
title = {Conformal scattering for a nonlinear wave equation on a curved background},
author = {Jérémie Joudioux},
journal= {arXiv preprint arXiv:1004.1464},
year = {2019}
}