Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T2-symmetric vacuum spacetimes

Abstract

We set up the singular initial value problem for quasilinear hyperbolic Fuchsian systems of first order and establish an existence and uniqueness theory for this problem with smooth data and smooth coefficients (and with even lower regularity). We apply this theory in order to show the existence of smooth (generally not analytic) T2-symmetric solutions to the vacuum Einstein equations, which exhibit AVTD (asymptotically velocity term dominated) behavior in the neighborhood of their singularities and are polarized or half-polarized.

78 pages

References in corpus (4)

• Analytic description of singularities in Gowdy spacetimes
• Spikes in the Mixmaster regime of G_2 cosmologies
• Asymptotic Behavior in Polarized {\bf T}${}^2$-symmetric Vacuum Spacetimes
• On polynomially integrable domains in Euclidean spaces

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• The maximal development of near-FLRW data for the Einstein-scalar field system with spatial topology ${\mathbb{S}}^3$
• A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges
• Numerical solutions of Einstein's equations for cosmological spacetimes with spatial topology S3 and symmetry group U(1)
• An exact smooth Gowdy-symmetric generalized Taub-NUT solution
• Self-gravitating fluid flows with Gowdy symmetry near cosmological singularities
• Wave equations with initial data on compact Cauchy horizons
• A New Class of Asymptotically Non-Chaotic Vacuum Singularities
• Localized big bang stability for the Einstein-scalar field equations
• A numerical algorithm for Fuchsian equations and fluid flows on cosmological spacetimes
• On the initial geometry of a vacuum cosmological spacetime
• Stability of AVTD Behavior within the Polarized $T^2$-symmetric vacuum spacetimes
• The annoying null boundaries
• Gowdy-symmetric cosmological models with Cauchy horizons ruled by non-closed null generators
• Stability Within $T^2$-Symmetric Expanding Spacetimes
• Contracting asymptotics of the linearized lapse-scalar field sub-system of the Einstein-scalar field equations
• Stability of Asymptotic Behavior Within Polarised $T^2$-Symmetric Vacuum Solutions with Cosmological Constant
• Analysis of a Bianchi-like equation satisfied by the Mars-Simon tensor