  We consider the 2D Muskat equation for the interface between two constant
density fluids in an incompressible porous medium, with velocity given by
Darcy's law. We establish that as long as the slope of the interface between
the two fluids remains bounded and uniformly continuous, the solution remains
regular. The proofs exploit the nonlocal nonlinear parabolic nature of the
equations through a series of nonlinear lower bounds for nonlocal operators.
These are used to deduce that as long as the slope of the interface remains
uniformly bounded, the curvature remains bounded. The nonlinear bounds then
allow us to obtain local existence for arbitrarily large initial data in the
class $W^{2,p}$, $1<p\leq \infty$. We provide furthermore a global regularity
result for small initial data: if the initial slope of the interface is
sufficiently small, there exists a unique solution for all time.
