Controlled regularity at future null infinity from past asymptotic initial data: the wave equation

TL;DR

This paper establishes a quantitative link between initial data at past null infinity and the regularity of wave equation solutions at future null infinity in Minkowski space, using conformal methods and Gr"onwall estimates.

Contribution

It introduces a new approach to analyze the regularity of wave solutions at null infinity based on asymptotic initial data, including non-compact support cases.

Findings

Solutions exhibit peeling behaviour near spatial infinity.

Regularity of solutions is quantitatively controlled by initial data.

Method applies to data with non-compact support at past null infinity.

Abstract

We study the relationship between asymptotic characteristic initial data for the wave equation at past null infinity and the regularity of the solution at future null infinity on the Minkowski spacetime. By constructing estimates on a causal rectangle reaching the conformal boundary, we prove that the solution admits an asymptotic expansion near null and spatial infinity whose regularity is controlled quantitatively in terms of the regularity of the data at past null infinity. In particular, our method gives rise to solutions to the wave equation in a neighbourhood of spatial infinity satisfying the peeling behaviour, for data on past null infinity with non-compact support. Our approach makes use of Friedrich's conformal representation of spatial infinity in which we prove delicate non-degenerate Gr\"onwall estimates. We describe the relationship between the solution and the data both in terms of Friedrich's conformal coordinates and the usual physical coordinates on Minkowski space.