A situation where there is more hope of obtaining rigorous results is where the BKL picture suggests that there should be monotone behaviour near the singularity. This is the situation for which Fuchsian techniques can often be applied to prove the existence of large classes of spacetimes having the expected behaviour near the initial singularity (see Section 6.2). It would be desirable to have a stronger statement than these techniques have provided up to now. Ideally, it should be shown that a non-empty open set of solutions of the given class (by which is meant all solutions corresponding to an open set of initial data on a regular Cauchy surface) lead to a singularity of the given type. The only results of this type in the literature concern polarized Gowdy spacetimes , plane symmetric spacetimes with a massless scalar field , spacetimes with collisionless matter and spherical, plane or hyperbolic symmetry , and a subset of general Gowdy spacetimes . The work of Christodoulou  on spherically symmetric solutions of the Einstein equations with a massless scalar field should also be mentioned in this context, although it concerns the singularity inside a black hole rather than singularities in cosmological models. Note that all these spacetimes have at least two Killing vectors so that the PDE problem to be solved reduces to an effective problem in one space dimension.
|Theorems on Existence and Global Dynamics for the
Alan D. Rendall
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