  We consider the Muskat problem describing the viscous displacement in a
two-phase fluid system located in an unbounded two-dimensional porous medium or
Hele-Shaw cell. After formulating the mathematical model as an evolution
problem for the sharp interface between the fluids, we show that Muskat problem
with surface tension is a quasilinear parabolic problem, whereas, in the
absence of surface tension effects, the Rayleigh-Taylor condition identifies a
domain of parabolicity for the fully nonlinear problem. Based upon these
aspects, we then establish the local well-posedness for arbitrary large initial
data in $H^s$, $s>2$, if surface tension is taken into account, respectively
for arbitrary large initial data in $H^2$ that additionally satisfy the
Rayleigh-Taylor condition if surface tension effects are neglected. We also
show that the problem exhibits the parabolic smoothing effect and we provide
criteria for the global existence of solutions.
