Stability in exponential time of Minkowski Space-time with a translation space-like Killing field

Cécile Huneau (1)

1 IF - Institut Fourier

In this paper, we prove the nonlinear stability in exponential time of Minkowki space-time with a translation space-like Killing eld. In the presence of such a symmetry, the 3 + 1 vacuum Einstein equations reduce to the 2 + 1 Einstein equations with a scalar eld. We work in generalised wave coordinates. In this gauge the Einstein equations can be written as a system of quasilinear quadratic wave equations. The main diculty in this paper 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 the Einstein equations in wave coordinates. We also have to carefully choose the behaviour of our metric in the exterior region to enforce convergence to Minkowski space-time at time-like innity.

Domaines: Equations aux dérivées partielles [math.AP]

Cécile Huneau. Stability in exponential time of Minkowski Space-time with a translation space-like Killing field. Annals of PDE, 2016. ⟨hal-01709687⟩