In this talk, we investigate mathematically how capillary-driven viscous thin fluid films evolve on microscopic length scales, in which case thermal noise due to fluctuations of the fluid particles comes into play. The underlying stochastic partial differential equation (SPDE) is a stochastic thin-film equation, a fourth-order degenerate-paraolic PDE driven by nonlinear gradient noise. This equation was first suggested in the physical literature approximately 20 years ago and existence of solutions for nonlinear noise was only established very recently. The key observation is that the Stratonovich formulation of the equation is the physically correct mathematical formulation, leading to a suitable balance of fluctuations and dissipation of the underlying physics and the correct balance in the energy-entropy dissipation relations. Specifically we are able to establish existence of nonnegative martingale solutions for nonlinear mobilities and we further prove existence of measure-valued solutions for initial values with non-full support. The latter forms a first step towards proving finite speed of propagation and for investigating contact-line dynamics on microscopic length scales.

This talk is based on joint works with Konstantinos Dareiotis (Leeds), Benjamin Gess (TU Berlin and Max Planck Institute MiS, Leipzig), Günther Grün (Erlangen), and Max Sauerbrey (formerly TU Delft, now Max Planck Institute MiS, Leipzig).