Well-posedness of the Muskat problem in subcritical <i>L</i>_<i>p</i>-Sobolev spaces

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