On the Thin Film Muskat and the Thin Film Stokes Equations

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.

Global existence and decay to equilibrium for some crystal surface models

“…Our approach is very adaptable and can lead to advances in other systems of PDE. For instance, it has been used Bruell & Granero-Belinchón to study the evolution of thin films in Darcy and Stokes flows [3] by Córdoba and Gancedo [6], Constantin, Córdoba, Gancedo, Rodriguez-Piazza, & Strain [5] for the Muskat problem (see also [7] and [17]), by Burczak & Granero-Belinchón [4] to analyze the Keller-Segel system of PDE with diffusion given by a nonlocal operator and by Bae, Granero-Belinchón & Lazar [2] to prove several global existence results (with infinite L p energy) for nonlocal transport equations.…”

Non-Newtonian two-phase thin-film problem: Local existence, uniqueness, and stability

“…We refer the reader to the works [16-18, 20, 26, 27] for strong solutions in (weighted) Sobolev or Hölder spaces with different prescribed slip conditions. Concerning stratified two-phase generalisations of the thin film equation (1.4) for Newtonian fluids, we refer to [8,13,14].…”

On a thin film model with insoluble surfactant

“…The aim of the present work is to prove the existence of global weak solutions of (1) under fairly low regularity assumptions with respect to the initial data. Similar as in the companion paper [10], we work in scales of Wiener spaces. Exploiting the algebra inequality verified by the norms of the underlying spaces, we show a priori energy estimates in Wiener algebra, which guarantee the existence of global weak solutions and imply the exponential decay towards the flat equilibrium state.…”

“…Pioneering works in this direction in absence of surfactant effects are due to Greenspan [32], Constantin, Dupont, Goldstein, Kadanoff, Shelley & Zhou [13], Bernis & Friedman [4], Beretta, Bertsch & Dal Passo [3] and Bertozzi & Pugh [5]. Also, Escher, Matioc & Matioc [23] considered the flow in porous media (see also Escher & Matioc [25], Matioc [39], Escher, Laurençot & Matioc [21], Laurençot & Matioc [35][36][37][38] and Bruell & Granero-Belinchón [10]) while the Stokes flow was considered by Escher, Matioc & Matioc [24] (see also Escher & Matioc [26] and Bruell & Granero-Belinchón [10]). A more recent reference is Pernas-Castaño & Velázquez [40], where the authors study the evolution of the interface between two different fluids in two concentric cylinders when the velocity is given by the Navier-Stokes equation and one of the fluids is thin.…”