Stability in exponential time of Minkowski space-time with a space-like translation symmetry
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 19, 14 p.

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.

@article{SLSEDP_2014-2015____A19_0,
author = {Huneau, C\'ecile},
title = {Stability in exponential time of Minkowski space-time with a space-like translation symmetry},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:19},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2014-2015},
doi = {10.5802/slsedp.77},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/slsedp.77/}
}
Huneau, Cécile. Stability in exponential time of Minkowski space-time with a space-like translation symmetry. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 19, 14 p. doi : 10.5802/slsedp.77. http://archive.numdam.org/articles/10.5802/slsedp.77/

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