Future Dynamics of T2 symmetric polarized spacetimes

Joint Work with Philippe G. LeFloch. We consider vacuum
spacetimes with two spatial Killing vectors and with initial data
prescribed on $T^3$. The main results that we will present concern the
future asymptotic behaviour of the so-called polarized solutions. Under
a smallness assumption, we derive a full set of asymptotics for these
solutions. Within this symetry class, the Einstein equations reduce to a
system of wave equations coupled to a system of ordinary differential
equations. The main difficulty, not present in previous study of similar
systems, is that, even in the limit of large times, the two systems do
not directly decouple. We overcome this problem by the introduction of a
new system of ordinary differential equations, whose unknown are
renormalized variables with renormalization depending on the solution of
the non-linear wave equations.