Applications of theorems of Jean Leray to the Einstein-scalar field   equations
  The Einstein-scalar field theory can be used to model gravitational physics with scalar field matter sources. We discuss the initial value formulation of this field theory, and show that the ideas of Leray can be used to show that the Einstein-scalar field system of partial differential equations is well-posed as an evolutionary system. We also show that one can generate solutions of the Einstein-scalar field constraint equations using conformal methods. 
Yvonne Choquet-Bruhat, James Isenberg and Daniel Pollack