Self-similarity in a thin film Muskat problem

TL;DR

This paper investigates the long-term behavior of solutions to a thin film Muskat problem, classifying self-similar solutions and proving convergence of solutions to these self-similar profiles.

Contribution

It provides a comprehensive classification of self-similar solutions and establishes convergence results for all non-negative weak solutions.

Findings

Unique even self-similar solution exists.

Continuum of non-symmetric solutions for certain configurations.

All solutions converge to a self-similar profile.

Abstract

The large time behavior of non-negative weak solutions to a thin film approximation of the two-phase Muskat problem is studied. A classification of self-similar solutions is first provided: there is always a unique even self-similar solution while a continuum of non-symmetric self-similar solutions exist for certain fluid configurations. Despite this non-uniqueness, convergence of all non-negative weak solutions towards a self-similar solution is proved.