On the thin film Muskat and the thin film Stokes equations

TL;DR

This paper analyzes coupled degenerate parabolic PDE systems modeling multiphase thin film flows, proving global existence, exponential decay to equilibrium, and Sobolev regularity propagation, using energy estimates in Wiener and Sobolev spaces.

Contribution

It provides the first rigorous proof of global weak solutions and decay rates for the two-phase thin film Muskat and Stokes equations.

Findings

Existence of global weak solutions for medium initial data.

Exponential decay towards equilibrium with explicit rates.

Propagation of Sobolev regularity under certain initial conditions.

Abstract

The present paper is concerned with the analysis of two strongly coupled systems of degenerate parabolic partial differential equations arising in multiphase thin film flows. In particular, we consider the two-phase thin film Muskat problem and the two-phase thin film approximation of the Stokes flow under the influence of both, capillary and gravitational forces. The existence of global weak solutions for medium size initial data in large function spaces is proved. Moreover, exponential decay results towards the equilibrium state are established, where the decay rate can be estimated by explicit constants depending on the physical parameters of the system. Eventually, it is shown that if the initial datum satisfies additional (low order) Sobolev regularity, we can propagate Sobolev regularity for the corresponding solution. The proofs are based on a priori energy estimates in Wiener and Sobolev spaces.