  The Muskat problem models the filtration of two incompressible immiscible
fluids of different characteristics in porous media. In this paper, we consider
both the 2D and 3D setting of two fluids of different constant densities and
different constant viscosities. In this situation, the related contour
equations are non-local, not only in the evolution system, but also in the
implicit relation between the amplitude of the vorticity and the free
interface. Among other extra difficulties, no maximum principles are available
for the amplitude and the slopes of the interface in $L^\infty$. We prove
global in time existence results for medium size initial stable data in
critical spaces. We also enhance previous methods by showing smoothing (instant
analyticity), improving the medium size constant in 3D, together with sharp
decay rates of analytic norms. The found technique is twofold, giving
ill-posedness in unstable situations for very low regular solutions.
