Abstract
The Cauchy problem plays an important role in both analytical and numerical studies of the Einstein field equations. Here we discuss two particular applications of the Cauchy problem within the framework of General Relativity. In the first of the two problems, we investigate how one can solve the Einstein constraint equations as an initial value problem. For this, our primary focus is on the asymptotic behaviour of the unknowns and how it may be ``controlled''. In particular, we provide analytical and numerical evidence that it is possible to control the asymptotic behaviour of the unknowns. In the second of the two problems, we investigate the asymptotic behaviour of solutions of the Einstein equations with a minimally coupled scalar field near the Big Bang. Our primary focus here is on understanding what effect the addition of a potential has on the asymptotic behaviour of the scalar field.