Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise
- Authors: Konstantinos Dareiotis, Benjamin Gess, Manuel V. Gnann, Günther Grün
- Preprint year: 2020
- First public date: 2020-12-08
- arXiv: 2012.04356
- Status: Published
- Publication type: Journal article
- Publication year: 2021
- Journal: Archive for rational mechanics and analysis, 242 (2021) 1, pp. 179-234
- DOI: 10.1007/s00205-021-01682-z
Abstract
We prove the existence of nonnegative 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.