4 research outputs found
On the area of the symmetry orbits in weakly regular Einstein-Euler spacetimes with Gowdy symmetry
This paper establishes novel bounds for Gowdy-symmetric Einstein-Euler
spacetimes and completes the analysis, initiated by LeFloch and Rendall, of the
global areal foliation for these spacetimes. We thus consider the initial value
problem for the Einstein-Euler equations under the assumption of Gowdy
symmetry. We establish that, for the maximal Cauchy development of future
contracting initial data, the area of the group orbits approaches zero toward
the future. This property holds as one approaches the future boundary of the
spacetime, provided a geometry invariant associated with the Gowdy symmetry
property is initially non-vanishing. Our condition is sharp within the class of
spatially homogeneous spacetimes.Comment: 18 page
Future asymptotics and geodesic completeness of polarized T2-symmetric spacetimes
We investigate the late-time asymptotics of future expanding, polarized
vacuum Einstein spacetimes with T2-symmetry on T3, which, by definition, admit
two spacelike Killing fields. Our main result is the existence of a stable
asymptotic regime within this class, that is, we provide here a full
description of the late-time asymptotics of the solutions to the Einstein
equations when the initial data set is close to the asymptotic regime. Our
proof is based on several energy functionals with lower order corrections (as
is standard for such problems) and the derivation of a simplified model which
we exhibit here. Roughly speaking, the Einstein equations in the symmetry class
under consideration consists of a system of wave equations coupled to
constraint equations plus a system of ordinary differential equations. The
unknowns involved in the system of ordinary equations are blowing up in the
future timelike directions. One of our main contributions is the derivation of
novel effective equations for suitably renormalized unknowns. Interestingly,
this renormalization is not performed with respect to a fixed background, but
does involve the energy of the coupled system of wave equations. In addition,
we construct an open set of initial data which are arbitrarily close to the
expected asymptotic behavior. We emphasize that, in comparison, the class of
Gowdy spacetimes exhibits a very different dynamical behavior to the one we
uncover in the present work for general polarized T2-symmetric spacetimes.
Furthermore, all the conclusions of this paper are valid within the framework
of weakly T2-symmetric spacetimes previously introduced by the authors.Comment: 34 page
Self-gravitating fluid flows with Gowdy symmetry near cosmological singularities
We consider self-gravitating fluids in cosmological spacetimes with Gowdy
symmetry on the torus and, in this class, we solve the singular initial
value problem for the Einstein-Euler system of general relativity, when an
initial data set is prescribed on the hypersurface of singularity. We specify
initial conditions for the geometric and matter variables and identify the
asymptotic behavior of these variables near the cosmological singularity. Our
analysis of this class of nonlinear and singular partial differential equations
exhibits a condition on the sound speed, which leads us to the notion of
sub-critical, critical, and super-critical regimes. Solutions to the
Einstein-Euler systems when the fluid is governed by a linear equation of state
are constructed in the first two regimes, while additional difficulties arise
in the latter one. All previous studies on inhomogeneous spacetimes concerned
vacuum cosmological spacetimes only.Comment: 41 page
The global nonlinear stability of Minkowski space for self-gravitating massive fields
The theory presented in this monograph establishes the first mathematically
rigorous result on the global nonlinear stability of self-gravitating matter
under small perturbations of an asymptotically flat, spacelike hypersurface of
Minkowski spacetime. It allows one to exclude the existence of dynamically
unstable, self-gravitating massive fields and, therefore, solves a
long-standing open problem in General Relativity. By a significant extension of
the Hyperboloidal Foliation Method they introduced in 2014, the authors
establish global-in-time existence for the Einstein equations expressed as a
coupled wave-Klein-Gordon system of partial differential equations. The metric
and matter fields are sought for in Sobolev-type functional spaces, suitably
defined from the translations and the boosts of Minkowski spacetime.Comment: 165 page