Existence of nonnegative solutions to stochastic thin-film equations in two space dimensions

TL;DR

This paper establishes the existence of nonnegative martingale solutions for stochastic thin-film equations in two dimensions using a finite element approach, stochastic calculus, and compactness arguments.

Contribution

It introduces a novel finite element scheme with curvature regularization and proves convergence to nonnegative solutions in a stochastic setting.

Findings

Existence of nonnegative martingale solutions in 2D

Convergence of finite element solutions as discretization vanishes

Extension of deterministic energy estimates to stochastic equations

Abstract

We prove the existence of martingale solutions to stochastic thin-film equations in the physically relevant space dimension $d=2$. Conceptually, we rely on a stochastic Faedo-Galerkin approach using tensor-product linear finite elements in space. Augmenting the physical energy on the approximate level by a curvature term weighted by positive powers of the spatial discretization parameter $h$, we combine Ito's formula with inverse estimates and appropriate stopping time arguments to derive stochastic counterparts of the energy and entropy estimates known from the deterministic setting. In the limit $h\searrow 0$, we prove our strictly positive finite element solutions to converge towards nonnegative martingale solutions -- making use of compactness arguments based on Jakubowski's generalization of Skorokhod's theorem and subtle exhaustion arguments to identify third-order spatial derivatives in the flux terms.