GRK-Kolloquium, Dr. Manuel Gnann

Feb 10
10-02-2023 16:45 Uhr bis 17:45 Uhr

Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with
Nonlinear Gradient Noise

Abstract: We prove the 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 Grün, 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 passage to solutions with non-full support of the initial
data using alpha entropies in a range of sub-cubic mobilities will be
The talk is based on joint works with Konstantinos Dareiotis (University
of Leeds), Benjamin Gess (Bielefeld University/MPI Leipzig), Günther
Grün (University of Erlangen-Nuremberg), and Max Sauerbrey (TU Delft).