On the Thin Film Muskat and the Thin Film Stokes Equations

Ostwald Ripening in Thin Film Equations

Fourth order thin film equations can have late stage dynamics that are analogous to the classical Cahn–Hilliard equation. We undertake a systematic asymptotic analysis of a class of equations that describe partial wetting with a stable precursor film introduced by intermolecular interactions. The limit of small precursor film thickness is considered, leading to explicit expressions for the late...

A thin film approximation of the Muskat problem with gravity and capillary forces

Abstract. Existence of nonnegative weak solutions is shown for a thin film approximation of the Muskat problem with gravity and capillary forces taken into account. The model describes the space-time evolution of the heights of the two fluid layers and is a fully coupled system of two fourth order degenerate parabolic equations. The existence proof relies on the fact that this system can be vie...

Notes on Blowup and Long Wave Unstable Thin Film Equations

(1) ut = −(uuxxx)x − (uux)x. This is the one dimensional version of ut = −∇ · (f(u)∇∆u)−∇ · (g(u)∇u), with f(u) = un and g(u) = un. Such equations have been used to model the dynamics of a thin film of viscous liquid spreading on a flat solid surface. The air/liquid interface is at height z = u(x, y, t) ≥ 0 and the liquid/solid interface is at z = 0. The one dimensional equation (1) applies if ...

[hal-00636647, v1] A gradient flow approach to a thin film approximation of the Muskat problem

Abstract. A fully coupled system of two second-order parabolic degenerate equations arising as a thin film approximation to the Muskat problem is interpreted as a gradient flow for the 2-Wasserstein distance in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of weak solutions. The availability of two ...

A gradient flow approach to a thin film approximation of the Muskat problem