  We derive the porous medium equation from an interacting particle system
which belongs to the family of exclusion processes, with nearest neighbor
exchanges. The particles follow a degenerate dynamics, in the sense that the
jump rates can vanish for certain configurations, and there exist blocked
configurations that cannot evolve. In [Gon\c{c}alves-Landim-Toninelli '09] it
was proved that the macroscopic density profile in the hydrodynamic limit is
governed by the porous medium equation (PME), for initial densities uniformly
bounded away from $0$ and $1$. In this paper we consider the more general case
where the density can take those extreme values. In this context, the PME
solutions display a richer behavior, like moving interfaces, finite speed of
propagation and breaking of regularity. As a consequence, the standard
techniques that are commonly used to prove this hydrodynamic limits cannot be
straightforwardly applied to our case. We present here a way to generalize the
\emph{relative entropy method}, by involving approximations of solutions to the
hydrodynamic equation, instead of exact solutions.
