  The currently preferred models of the universe undergo accelerated expansion
induced by dark energy. One model for dark energy is a positive cosmological
constant. It is consequently of interest to study Einstein's equations with a
positive cosmological constant coupled to matter satisfying the ordinary energy
conditions; the dominant energy condition etc. Due to the difficulty of
analysing the behaviour of solutions to Einstein's equations in general, it is
common to either study situations with symmetry, or to prove stability results.
In the present paper, we do both. In fact, we analyse, in detail, the future
asymptotic behaviour of T^3-Gowdy symmetric solutions to the Einstein-Vlasov
equations with a positive cosmological constant. In particular, we prove the
cosmic no-hair conjecture in this setting. However, we also prove that the
solutions are future stable (in the class of all solutions). Some of the
results hold in a more general setting. In fact, we obtain conclusions
concerning the causal structure of T^2-symmetric solutions, assuming only the
presence of a positive cosmological constant, matter satisfying various energy
conditions and future global existence. Adding the assumption of T^3-Gowdy
symmetry to this list of requirements, we obtain C^0-estimates for all but one
of the metric components. There is consequently reason to expect that many of
the results presented in this paper can be generalised to other types of
matter.
