Conformal scattering of the Maxwell-scalar field system on de Sitter space

TL;DR

This paper establishes energy estimates and constructs nonlinear scattering operators for the Maxwell-scalar field system on de Sitter space, revealing decay rates and asymptotic behavior of solutions.

Contribution

It introduces a novel gauge fixing and nonlinear analysis to construct invertible scattering operators for the conformal Maxwell-scalar system on de Sitter space.

Findings

Constructed bounded, invertible nonlinear scattering operators

Proved exponential decay rates for solutions with at least two derivatives

Discovered asymptotic decoupling of scalar field from charge

Abstract

We prove small data energy estimates of all orders of differentiability between past null infinity and future null infinity of de Sitter space for the conformally invariant Maxwell-scalar field system. This allows us to construct bounded and invertible, but nonlinear, scattering operators taking past asymptotic data to future asymptotic data. We also deduce exponential decay rates for solutions with data having at least two derivatives, and for more regular solutions discover an asymptotic decoupling of the scalar field from the charge. The construction involves a carefully chosen complete gauge fixing condition which allows us to control all components of the Maxwell potential, and a nonlinear Gr\"onwall inequality for higher order estimates.