  We consider the evolution of two incompressible, immiscible fluids with
different densities in porous media, known as the Muskat problem [21], which in
two dimensions is analogous to the Hele-Shaw cell [26]. We establish, for a
class of large and monotone initial data, the global existence of weak
solutions. The proof is based on a local well-posedness result for the initial
data with certain specific asymptotics at spatial infinity and a new maximum
principle for the first derivative of the graph function.
