Existence of nonnegative energy-dissipating solutions to a class of stochastic thin-film equations under weak slippage: Part I -- positive solutions

TL;DR

This paper proves the existence of positive solutions to stochastic thin-film equations with singular potentials for mobility exponents between 2 and 3, introducing a novel discretization method to control energy decay and pave the way for future analysis.

Contribution

It introduces a new finite-element discretization technique ensuring nonnegativity of key integral terms in stochastic thin-film equations with exponents in (2,3).

Findings

Existence of strictly positive solutions for the specified stochastic thin-film equations.

A discretization method that guarantees nonnegativity of a critical integral under periodic boundary conditions.

Decay estimates for surface-tension energy and interface potential without dependence on initial data.

Abstract

For mobility exponents $n \in (2,3)$, we prove existence of strictly positive solutions to stochastic thin-film equations with singular effective interface potential and Stratonovich-type lower-order terms. With the perspective of using these solutions in Part II to construct surface-tension-energy dissipating solutions to stochastic thin-film equations with compactly supported initial data, for which finite speed of propagation is shown in future work, we establish decay estimates on the sum of surface-tension energy and effective interface potential -- without relying on further functionals involving initial data. Besides an extension of earlier techniques used in the case $n=2$ and a refinement of oscillation estimates for discrete solutions, the main analytical novelty of this paper is a discretization method which shows nonnegativity for a finite-element counterpart of the integral $\int_{\mathcal O} (u^{n-2}u_{x})_x u_{xx} dx$ under periodic boundary conditions in the parameter regime $n \in (2,3)$. This nonnegativity property serves to control It\^o-correction terms in the estimate for the decay of the surface-tension energy. This way, it is the key to obtain the desired decay estimates for the sum of surface-tension energy and effective interface potential which permit to establish the singular limit of vanishing effective interface potential in Part II.