For liquid films with a thickness in the order of 10¹−10³ molecule layers, classical models of continuum mechanics do not always give a precise description of thin-film evolution: While morphologies of film dewetting are captured by thin-film models, discrepancies arise with respect to time-scales of dewetting.
In this talk, we study stochastic thin-film equations. By multiplicative noise inside an additional convective term, these stochastic partial differential equations differ from their deterministic counterparts, which are fourth-order degenerate parabolic. First, we present some numerical simulations which indicate that the aforementioned discrepancies may be overcome under the influence of noise.
In the main part of the talk, we prove existence of almost surely nonnegative martingale solutions. Combining spatial semi-discretization with appropriate stopping time arguments, arbitrary moments of coupled energy/entropy functionals can be controlled.
Having established Hölder regularity of approximate solutions, the convergence proof is then based on compactness arguments - in particular on Jakubowski’s generalization of Skorokhod’s theorem - weak convergence methods, and recent tools for martingale convergence.
The results have been obtained in collaboration with K. Mecke and M. Rauscher and with J. Fischer, respectively
- Partial Differential Equations Seminar