A priori positivity of solutions to a non-conservative stochastic thin-film equation

Abstract

Stochastic conservation laws are often challenging when it comes to proving existence of non-negative solutions. In a recent work by J. Fischer and G. Gr\"un (2018, Existence of positive solutions to stochastic thin-film equations, SIAM J. Math. Anal.), existence of positive martingale solutions to a conservative stochastic thin-film equation is established in the case of quadratic mobility. In this work, we focus on a larger class of mobilities (including the linear one) for the thin-film model. In order to do so, we need to introduce nonlinear source potentials, thus obtaining a non-conservative version of the thin-film equation. For this model, we assume the existence of a sufficiently regular local solution (i.e., defined up to a stopping time $\tau$) and, by providing suitable conditions on the source potentials and the noise, we prove that such solution can be extended up to any $T>0$ and that it is positive with probability one. A thorough comparison with the aforementioned reference work is provided.