Geometric scattering for nonlinear wave equations on the Schwarzschild metric

TL;DR

This paper develops a conformal scattering theory for nonlinear wave equations on Schwarzschild spacetime, connecting initial data to boundary data via energy estimates and well-posedness results.

Contribution

It introduces a new scattering framework for nonlinear waves on Schwarzschild spacetime, combining decay estimates with energy bounds to construct a scattering operator.

Findings

Established energy estimates between initial and boundary fluxes.

Constructed a bounded, Lipschitz continuous scattering operator.

Linked initial data to boundary data through a rigorous mathematical framework.

Abstract

In this paper, we establish a conformal scattering theory for defocusing semilinear wave equations on Schwarzschild spacetime. We combine the energy and pointwise decay results for solutions obtained in \cite{Yang} with a Sobolev embedding on spacelike hypersurfaces to derive two-sided energy estimates between the energy flux of solutions through the Cauchy initial hypersurface $\Sigma_0 = \{ t = 0 \}$ and that through the null conformal boundaries $\mathfrak{H}^+ \cup \scri^+$ (respectively, $\mathfrak{H}^- \cup \scri^-$). By combining these estimates with the well-posedness of the Cauchy and Goursat problems for nonlinear wave equations, we construct a bounded linear and locally Lipschitz scattering operator that maps past scattering data to future scattering data.