Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of nonnegative solutions to stochastic thin-film equations in two space dimensions

Stefan Metzger, Günther Grün

Abstract

We prove the existence of martingale solutions to stochastic thin-film equations in the physically relevant space dimension $d=2$. Conceptually, we rely on a stochastic Faedo-Galerkin approach using tensor-product linear finite elements in space. Augmenting the physical energy on the approximate level by a curvature term weighted by positive powers of the spatial discretization parameter $h$, we combine Ito's formula with inverse estimates and appropriate stopping time arguments to derive stochastic counterparts of the energy and entropy estimates known from the deterministic setting. In the limit $h\searrow 0$, we prove our strictly positive finite element solutions to converge towards nonnegative martingale 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.

Existence of nonnegative solutions to stochastic thin-film equations in two space dimensions

Abstract

We prove the existence of martingale solutions to stochastic thin-film equations in the physically relevant space dimension . Conceptually, we rely on a stochastic Faedo-Galerkin approach using tensor-product linear finite elements in space. Augmenting the physical energy on the approximate level by a curvature term weighted by positive powers of the spatial discretization parameter , we combine Ito's formula with inverse estimates and appropriate stopping time arguments to derive stochastic counterparts of the energy and entropy estimates known from the deterministic setting. In the limit , we prove our strictly positive finite element solutions to converge towards nonnegative martingale 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.

Paper Structure

This paper contains 12 sections, 29 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Notation, technical preliminaries, and basic assumptions on the data
  3. The semi-discrete scheme
  4. A priori estimates
  5. The combined energy-entropy estimate
  6. Hölder continuity in time
  7. Passage to the limit
  8. Compactness
  9. Convergence of the deterministic terms
  10. Convergence of the stochastic integral
  11. Conclusion
  12. Auxiliary results

Key Result

Theorem 3.5

Let Assumptions item:S, item:initial, item:potential, item:stoch, item:regularization, item:stochbasis:boundthirdderivative, and item:stochbasis:convergence be satisfied and let $T_{\text{max}}>0$ be given. Furthermore, let ${(u_{h},p_{h})}_{h\searrow0}$ be a sequence of solutions to the regularized with ${\overline{p}}<\infty$. In particular, $\tilde{\mathds{P}}$-almost surely, $\tilde{u}(\cdot,t

Theorems & Definitions (59)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 3.4
  • Theorem 3.5
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • proof
  • ...and 49 more