Existence of nonnegative energy-dissipating solutions to a class of stochastic thin-film equations under weak slippage: Part II -- compactly supported initial data

TL;DR

This paper establishes the existence of martingale solutions for a class of stochastic thin-film equations with compactly supported initial data, focusing on energy dissipation and free-boundary problem implications.

Contribution

It proves existence results for stochastic thin-film equations with specific mobility exponents and initial conditions, using energy methods and approximation techniques.

Findings

Existence of martingale solutions for n in (2,3)

Energy dissipation can be established in the regime studied

Solutions with compact support are constructed

Abstract

We prove existence of martingale solutions to a class of stochastic thin-film equations for mobility exponents $n \in (2,3)$ and compactly supported initial data. With the perspective to study free-boundary problems related to stochastic thin-film equations in future work, we start from the surface-tension driven stochastic thin-film equation with Stratonovich noise and exploit the regime of coefficients in front of the Stratonovich correction term (which is of porous-media-type) for which energy dissipation can be established. By Bernis inequalities, third order spatial derivatives of appropriate powers of the solution are controlled. Analytically, we rely on approximation with $\mathbb{P}$-almost surely strictly positive solutions and compactness methods based on energy-entropy estimates as well as martingale identification techniques.