Special Session 19: Stochastic Partial Differential Equations

Martingale solutions to the stochastic thin film equation

Manuel V Gnann
TU Delft
Co-Author(s):    Konstantinos Dareiotis, Benjamin Gess, Guenther Gruen, Max Sauerbrey
We prove existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Gruen, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise. The construction of solutions with non-full support for the initial data using alpha-entropies shall be discussed.