Second-order hyperbolic Fuchsian systems. General theory

TL;DR

This paper introduces second-order hyperbolic Fuchsian systems, establishing a general theory for their solutions near singularities, with applications to Einstein's equations and numerical methods in general relativity.

Contribution

It develops a comprehensive framework for second-order hyperbolic Fuchsian systems, including existence of solutions with prescribed asymptotics and a new scheme suitable for numerical approximation.

Findings

Constructed asymptotic solutions of arbitrary order.

Proved existence of solutions with prescribed singular behavior.

Framework applicable to Einstein's equations and matter models in relativity.

Abstract

We introduce a class of singular partial differential equations, the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First of all, we analyze a class of equations in which hyperbolicity is not assumed and we construct asymptotic solutions of arbitrary order. Second, for the proposed class of second-order hyperbolic Fuchsian systems, we establish the existence of solutions with prescribed asymptotic behavior on the singularity. Our proof is based on a new scheme which is also suitable to design numerical approximations. Furthermore, as shown in a follow-up paper, the second-order Fuchsian framework is appropriate to handle Einstein's field equations for Gowdy symmetric spacetimes and allows us to recover (and slightly generalize) earlier results by Rendall and collaborators, while providing a direct approach leading to accurate numerical solutions. The proposed framework is also robust enough to encompass matter models arising in general relativity.