The domain of parabolicity for the Muskat problem

TL;DR

This paper investigates the well-posedness of the Muskat problem in periodic settings, establishing conditions under which it is parabolic, and analyzing related operators as generators of analytic semigroups.

Contribution

It identifies the Rayleigh-Taylor condition as a domain of parabolicity for the Muskat problem without surface tension and proves well-posedness; also studies Dirichlet-Neumann operators as semigroup generators.

Findings

Rayleigh-Taylor condition defines parabolic domain for Muskat without surface tension.

Muskat problem is parabolic with surface tension for general data.

Dirichlet-Neumann operators are negative generators of analytic semigroups.

Abstract

We address the well-posedness of the Muskat problem in a periodic geometry and in a setting which allows us to consider general initial and boundary data, gravity effects, as well as surface tension effects. In the absence of surface tension we prove that the Rayleigh-Taylor condition identifies a domain of parabolicity for the Muskat problem. This property is used to establish the well-posedness of the problem. In the presence of surface tension effects the Muskat problem is of parabolic type for general initial and boundary data. As a bi-product of our analysis we obtain that Dirichlet-Neumann type operators associated with certain diffraction problems are negative generators of strongly continuous and analytic semigroups in the scale of small H\"older spaces.