  We derive global weak solutions of Einstein's equations for spherically
symmetric dust-filled space-times which admit shell-crossing singularities. In
the marginally bound case, the solutions are weak solutions of a conservation
law. In the non-marginally bound case, the equations are solved in a
generalized sense involving metric functions of bounded variation. The
solutions are not unique to the future of the shell-crossing singularity, which
is replaced by a shock wave in the present treatment; the metric is bounded but
not continuous.
