Conformal scattering for a nonlinear wave equation on a curved background

Details Conformal scattering for a nonlinear wave equation on a curved background Conformal scattering for a nonlinear wave equation on a curved background

2010-04-09

arxiv.org/abs/1004.1464v2

Mathematics

Analysis of PDEs

Scientific paper

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.

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