Zero-contact angle solutions to stochastic thin-film equations

TL;DR

This paper proves the existence of nonnegative solutions to stochastic thin-film equations with zero-contact angles at the boundary, using advanced probabilistic and entropy methods.

Contribution

It introduces a novel approach to establish existence and boundary behavior of solutions to stochastic thin-film equations with Stratonovich noise.

Findings

Solutions are almost surely classically differentiable in space.

Solutions exhibit zero-contact angle at the boundary.

Existence of solutions is proven using entropy estimates and stochastic calculus.

Abstract

We establish existence of nonnegative martingale solutions to stochastic thin-film equations with compactly supported initial data under Stratonovich noise. Based on so called $\alpha$-entropy estimates, we show that almost surely these solutions are classically differentiable in space almost everywhere in time and that their derivative attains the value zero at the boundary of the solution's support. I.e., from a physics perspective, they exhibit a zero-contact angle at the three-phase contact line between liquid, solid, and ambient fluid. These $\alpha$-entropy estimates are first derived for almost surely strictly positive solutions to a family of stochastic thin-film equations augmented by second-order linear diffusion terms. Using It\^o's formula together with stopping time arguments, the Jakubowski/Skorokhod calculus, and martingale identification techniques, the passage to the limit of vanishing regularization terms gives the desired existence result.