The Goursat problem for the Einstein–Vlasov system: (II) The evolution of initial data

Abstract

We solve, locally in time, the evolution problem associated with the Einstein–Vlasov (EV) system, the initial data being specified on two intersecting smooth null hypersurfaces. The proof of the obtained result relies heavily on a fixed point method deployed in appropriate weighted Sobolev spaces. The main tools of this method consist of adequate Sobolev inequalities and Moser estimates combined with energy inequalities for first-order and second-order linear hyperbolic partial differential equations.