Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

TL;DR

This paper establishes the existence of nonnegative martingale solutions for a class of stochastic thin-film equations with nonlinear gradient noise, addressing a long-standing open problem in the mathematical modeling of thin-film flows.

Contribution

It provides the first proof of global-in-time solutions for stochastic thin-film equations with cubic mobility, using novel entropy and energy estimates combined with a specialized approximation method.

Findings

Proves existence of solutions for stochastic thin-film equations with nonlinear noise.

Handles the challenging cubic mobility case, previously unresolved.

Develops new analytical techniques for controlling shocks in stochastic PDEs.

Abstract

We prove the existence of nonnegative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Gr\"un, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.