We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an Lp-setting with p∈(1,∞). The Sobolev space Wps(R)
Nearby in the stack
with
s=1+1/p
is a critical space for this problem. We prove, for
s∈(1+1/p,2),
that the Rayleigh-Taylor condition identifies an open subset of
Wps(R)
within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.