Existence of nonnegative solutions to stochastic thin-film equations in two space dimensions
A talk in the BI.discrete Workshop series by
Stefan Metzger from Erlangen-Nürnberg
| Abstract: | Classical models of continuum mechanics do not always include all effects relevant for a precise description. In particular, thermal effects may lead to qualitatively different results and therefore may not be neglected. To account for thermal fluctuations, deterministic equations need to be augmented by additional stochastic noise terms. In this talk, we discuss the stochastic thin-film equation with conservative noise in the physically relevant space dimension $d=2$ and prove the existence of nonnegative martingale solutions. We start by applying a stochastic Faedo-Galerkin approach based on tensor-product linear finite elements which can also be seen as a first step towards the derivation of a fully discrete scheme. Augmenting the physical energy on the approximative level by a curvature term weighted by positive powers of the discretization parameter allows to establish a discrete version of the combined energy-entropy estimate. Based on this estimate, which serves as the foundation for our analysis, we establish the existence of strictly positive finite element 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, we are able to prove that these solutions converge towards nonnegative martingale solutions of the unregularized stochastic thin-film equation. This talk is based on a joint work with G. Grün (Friedrich-Alexander-Universität Erlangen-Nürnberg).
Within the CRC this talk is associated to the project(s): B7 |