Stability in exponential time of Minkowski space-time with a space-like translation symmetry

Cécile Huneau[1]

  • [1] DMA, École Normale Supérieure 45, rue d’Ulm 75005 Paris

Séminaire Laurent Schwartz — EDP et applications (2014-2015)

  • page 1-14
  • ISSN: 2266-0607

Abstract

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In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the 3 + 1 vacuum Einstein equations reduce to the 2 + 1 Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in 1 / t of free solutions to the wave equation in 2 dimensions, which is weaker than in 3 dimensions. As in [21], we have to rely on the particular structure of Einstein equations in wave coordinates. We also have to carefully choose an approximate solution with a non trivial behaviour at space-like infinity to enforce convergence to Minkowski space-time at time-like infinity.

How to cite

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Huneau, Cécile. "Stability in exponential time of Minkowski space-time with a space-like translation symmetry." Séminaire Laurent Schwartz — EDP et applications (2014-2015): 1-14. <http://eudml.org/doc/275746>.

@article{Huneau2014-2015,
abstract = {In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the $3+1$ vacuum Einstein equations reduce to the $2+1$ Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in $\{1\}/\{\sqrt\{t\}\}$ of free solutions to the wave equation in $2$ dimensions, which is weaker than in $3$ dimensions. As in [21], we have to rely on the particular structure of Einstein equations in wave coordinates. We also have to carefully choose an approximate solution with a non trivial behaviour at space-like infinity to enforce convergence to Minkowski space-time at time-like infinity.},
affiliation = {DMA, École Normale Supérieure 45, rue d’Ulm 75005 Paris},
author = {Huneau, Cécile},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {eng},
pages = {1-14},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Stability in exponential time of Minkowski space-time with a space-like translation symmetry},
url = {http://eudml.org/doc/275746},
year = {2014-2015},
}

TY - JOUR
AU - Huneau, Cécile
TI - Stability in exponential time of Minkowski space-time with a space-like translation symmetry
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2014-2015
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 14
AB - In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the $3+1$ vacuum Einstein equations reduce to the $2+1$ Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in ${1}/{\sqrt{t}}$ of free solutions to the wave equation in $2$ dimensions, which is weaker than in $3$ dimensions. As in [21], we have to rely on the particular structure of Einstein equations in wave coordinates. We also have to carefully choose an approximate solution with a non trivial behaviour at space-like infinity to enforce convergence to Minkowski space-time at time-like infinity.
LA - eng
UR - http://eudml.org/doc/275746
ER -

References

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