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We study the Muskat problem for one fluid or two fluids, with or withoutviscosity jump, with or without rigid boundaries, and in arbitrary spacedimension $d$ of the interface. The Muskat problem is scaling invariant in theSobolev space $H^{s_c}(\mathbb{R}^d)$ where $s_c=1+\frac{d}{2}$. Employing aparadifferential approach, we prove local well-posedness for large data in anysubcritical Sobolev spaces $H^s(\mathbb{R}^d)$, $s>s_c$. Moreover, the rigidboundaries are only required to be Lipschitz and can have arbitrarily largevariation. The Rayleigh-Taylor stability condition is assumed for the case oftwo fluids with viscosity jump but is proved to be automatically satisfied forthe case of one fluid. The starting point of this work is a reformulationsolely in terms of the Drichlet-Neumann operator. The key elements of proofsare new paralinearization and contraction results for the Drichlet-Neumannoperator in rough domains.

A paradifferential approach for well-posedness of the Muskat problem