Well-posedness and stability results for some periodic Muskat problems

TL;DR

This paper analyzes the well-posedness and stability of the periodic Muskat problem, considering effects of surface tension and density differences, and establishes conditions for existence, uniqueness, and stability of solutions.

Contribution

It provides new well-posedness results for the Muskat problem with and without surface tension in periodic settings, and characterizes equilibrium solutions and their stability.

Findings

Well-posedness in Sobolev spaces for surface tension case

Well-posedness under Rayleigh-Taylor condition without surface tension

Identification and stability analysis of equilibrium solutions

Abstract

We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic problem which is well-posed in the Sobolev space $H^r(\mathbb{S})$ for each $r\in(2,3)$. When neglecting surface tension effects, the Muskat problem is a fully nonlinear evolution equation and of parabolic type in the regime where the Rayleigh-Taylor condition is satisfied. We then establish the well-posedness of the Muskat problem in the open subset of $H^2(\mathbb{S})$ defined by the Rayleigh-Taylor condition. Besides, we identify all equilibrium solutions and study the stability properties of trivial and of small finger-shaped equilibria. Also other qualitative properties of solutions such as parabolic smoothing, blow-up behavior, and criteria for global existence are outlined.