Nonlinear interaction of three impulsive gravitational waves II: the wave estimates

TL;DR

This paper develops advanced wave estimates for scalar fields arising from the nonlinear interaction of impulsive gravitational waves under symmetry constraints, demonstrating localized singularity behavior and enhanced regularity properties.

Contribution

It introduces novel energy estimates and anisotropic Sobolev embeddings for low-regularity scalar fields in Einstein vacuum solutions with impulsive waves.

Findings

Scalar field is Lipschitz everywhere

Scalar field exhibits $C^{1, heta}$ regularity away from singularities

Energy estimates reveal localized singularities

Abstract

This is the second and last paper of a series aimed at solving the local Cauchy problem for polarized $\mathbb U(1)$ symmetric solutions to the Einstein vacuum equations featuring the nonlinear interaction of three small amplitude impulsive gravitational waves. Such solutions are characterized by their three singular "wave-fronts" across which the curvature tensor is allowed to admit a delta singularity. Under polarized $\mathbb U(1)$ symmetry, the Einstein vacuum equations reduce to the Einstein-scalar field system in $(2+1)$ dimensions. In this paper, we focus on the wave estimates for the scalar field in the reduced system. The scalar field terms are the most singular ones in the problem, with the scalar field only being Lipschitz initially. We use geometric commutators to prove energy estimates which reflect that the singularities are localized, and that the scalar field obeys additional fractional-derivative regularity, as well as regularity along appropriately defined "good directions". The main challenge is to carry out all these estimates using only the low-regularity properties of the metric. Finally, we prove an anisotropic Sobolev embedding lemma, which when combined with our energy estimates shows that the scalar field is everywhere Lipschitz, and that it obeys additional $C^{1,\theta}$ estimates away from the most singular region.