Quasilinear Symmetric Hyperbolic Fuchsian Systems in Several Space Dimensions

Abstract

We establish existence and uniqueness results for the singular initial value problem associated with a class of quasilinear, symmetric hyperbolic, partial differential equations of Fuchsian type in several space dimensions. This is an extension of earlier work by the authors for the same problem in one space dimension.

Contracting asymptotics of the linearized lapse-scalar field sub-system of the Einstein-scalar field equations

“…We point out that a very similar idea was originally put forward in [2,3,15] to solve singular initial value problems underlying the Fuchsian method.…”

“…Now in order to check Eqs. (5.19), we exploit Proposition 5.3 applied to the σ 1 -system where f (1) , f (2) , f (3) are replaced by the components of the source term in (5.21). We conclude that ϕ (0) = Ψ (pre) (u * , ϕ * ), where Ψ (pre) is defined in Proposition 5.3, and that for any 5.3 This map must be injective as Eq.…”

“…Before we discuss the proofs of the two lemmas of this subsection, let us introduce some more notation. Let L [1,1] := σ(1 −α), L [1,2] [1,3],b := σt 2γab D aα , L [1,4],b := σt 2γab D aφ , (5.45)…”

“…L [3,1] := A + σ(π − A), L [3,2] := 1 + σ(α − 1), (5.47) and further decompose the above operators as…”

“…The Fuchsian method has been used in a number of cases [2,3,15,20,25,26,29] to iden-tify asymptotic degrees of freedom -asymptotic data -(often determined heuristically by formal expansions at the singularity) and then for rigorously validating this identification (by proving the existence of a solution of the equation which realizes each choice of asymptotic data in some well-defined sense). It typically yields continuous maps from asymptotic data to Cauchy data.…”