The Domain of Parabolicity for the Muskat Problem

Abstra t. We address the well-posedness of the Muskat problem in a periodi geometry and in a setting whi h allows us to onsider general initial and boundary data, gravity e e ts, as well as surfa e tension e e ts. In the absen e of surfa e tension we prove that the Rayleigh-Taylor ondition identi es a domain of paraboli ity for the Muskat problem. This property is used to establish the well-posedness of the problem. In the presen e of surfa e tension e e ts the Muskat problem is of paraboli type for general initial and boundary data. As a bi-produ t of our analysis we obtain that Diri hlet-Neumann type operators asso iated with ertain di ra tion problems are negative generators of strongly ontinuous and analyti semigroups in the s ale of small Hölder spa es.