2010/04/27 by Florian Beyer, Philippe G. LeFloch, Beyer, Florian +1 · 1 citation
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #advanced mathematical theories #gr-qc #math.AP #msc:35L10 #msc:83C05 #msc:83C75
paper · pdf · doi:10.48550/arxiv.1004.4885
32 pages. A shortened version of the material presented in this preprint is published in: F. Beyer and P.G. LeFloch, Second-order hyperbolic Fuchsian systems and applications, Class. Quantum Grav. 27 (2010), 245012
arxiv created 2011/01/10 · arxiv updated 2015/03/16
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.