Rigorous thin film approximations of the one-phase unstable Muskat problem

Abstract

This paper studies the one-phase Muskat problem driven by gravity and surface tension. The regime considered here is unstable with the fluid on top of a dry region. By a novel approach using a depth-averaged formulation, we derive two asymptotic approximations for this scenario. The lower order approximation is the classical thin film equation, while the higher order approximation provides a new refined thin film equation. We prove optimal order of convergence in the shallowness parameter to the original Muskat solutions for both models with lowregular initial data.

A new reformulation of the Muskat problem with surface tension

“…e.g. [4,5,7,13,14,16,17,19,20,26,[37][38][39], the existence of global (weak or strong) solutions [9,23,24,27], the stability properties of the stationary solutions [11,13,14,16,22,23,31,34,37,38], the zero surface tension limit of the problems [4,21], and the singular limit when the thickness of the layers (or a nondimensional parameter) vanishes [6,15,25,32].…”