Second-order hyperbolic Fuchsian systems and applications

TL;DR

This paper introduces second-order hyperbolic Fuchsian systems, establishes a solution existence theory with prescribed singularity behavior, and develops a numerical scheme applied to Einstein equations in Gowdy spacetimes.

Contribution

It presents a new class of PDEs, a solution existence theory with asymptotics at the singularity, and a highly accurate numerical algorithm for these systems, applied to Einstein equations.

Findings

Existence of solutions with prescribed asymptotics at the singularity.

Development of a Fuchsian numerical algorithm for accurate approximations.

Identification of Gowdy spacetimes with non-compact Cauchy horizons.

Abstract

We introduce a new class of singular partial differential equations, referred to as the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First, we establish a general existence theory of solutions with asymptotic behavior prescribed on the singularity, which relies on a new approximation scheme, suitable also for numerical purposes. Second, this theory is applied to the (vacuum) Einstein equations for Gowdy spacetimes, and allows us to recover, by more direct arguments, well-posedness results established earlier by Rendall and collaborators. Another main contribution in this paper is the proposed approximation scheme, which we refer to as the Fuchsian numerical algorithm and is shown to provide highly accurate numerical approximations to the singular initial value problem. For the class of Gowdy spacetimes, the numerical experiments presented here show the interest and efficiency of the proposed method and demonstrate the existence of a class of Gowdy spacetimes containing a smooth, incomplete, and non-compact Cauchy horizon.