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

TL;DR

This paper develops and rigorously justifies thin film approximations for the unstable one-phase Muskat problem, including a classical and a new refined model, with proven convergence rates.

Contribution

It introduces a novel depth-averaged approach to derive and analyze asymptotic thin film models for the unstable Muskat problem, providing the first rigorous convergence results.

Findings

Classical thin film equation as a lower order approximation.

A new refined thin film equation for better accuracy.

Proven optimal convergence rates for both models.

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 the optimal order of convergence in the shallowness parameter to the original Muskat solutions for both models with low-regular initial data.