  We investigate the future asymptotic behavior of Gowdy spacetimes on T3, when
the metric satisfies weak regularity conditions, so that the metric
coefficients (in suitable coordinates) are only in the Sobolev space H1 or have
even weaker regularity. The authors recently introduced this class of
spacetimes in the broader context of T2 symmetric spacetimes and established
the existence of a global foliation by spacelike hypersurfaces when the time
function is chosen to be the area of the surfaces of symmetry. In the present
paper, we identify the global causal geometry of these spacetimes and, in
particular, establish that weakly regular Gowdy spacetimes are future causally
geodesically complete. This result extends a theorem by Ringstr\"om for metrics
with sufficiently high regularity. We emphasize that our proof of the energy
decay is based on an energy functional inspired by the Gowdy-to-Ernst
transformation. In order to establish the geodesic completeness property, we
prove a higher regularity property concerning the metric coefficients along
timelike curves and we provide a novel analysis of the geodesic equation for
Gowdy spacetimes, which does not require high-order regularity estimates. Even
when sufficient regularity is assumed, our proof provides an alternative and
shorter proof of the energy decay and of the geodesic completeness property for
Gowdy spacetimes.
