In this lecture we consider the stochastic thin-film equation. The stochastic thin-film equation is a forth-order, degenerate stochastic PDE with nonlinear, conservative noise. This renders the existence of solutions a challenging, largely open problem (since 2006). Due to the forth order nature of the equation, comparison arguments do not apply and the analysis has to rely on integral estimates.
The stochastic thin film equation can be, informally, derived via the lubrication/thin film approximation of the fluctuating Navier-Stokes equations and has been suggested in the physics literature to be an improved mesoscopic model, leading to better predictions for film rupture and expansion.
Besides the specific case of the stochastic thin film equation, this lecture will serve as an introduction to the weak-convergence method to the existence of (weak) solutions to stochastic PDE devised by Flandoli and Gatarek  in the case of stochastic Navier-Stokes equations.
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