Second-order hyperbolic Fuchsian systems. General theory

Florian Beyer; Philippe G. LeFloch

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arXiv · 1004.4885

Second-order hyperbolic Fuchsian systems. General theory

Florian Beyer·Philippe G. LeFloch

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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.

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Florian Beyer, Philippe G. LeFloch. 2011-01-10. Second-order hyperbolic Fuchsian systems. General theory. https://arxiv.org/abs/1004.4885

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