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

Dareiotis K, Gess B, Gnann M, Grün G (2021)

Publication Type: Journal article

Publication year: 2021

Journal

Archive for Rational Mechanics and Analysis Springer Verlag (Germany)

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.

Authors with CRIS profile

Günther Grün Professur für Angewandte Mathematik (Analysis und Numerik partieller Differentialgleichungen)

Involved external institutions

University of Leeds

United Kingdom (GB) Max-Planck-Institut für Mathematik (MPIM)

Germany (DE) Delft University of Technology (TU Delft)

Netherlands (NL)

How to cite

APA:

Dareiotis, K., Gess, B., Gnann, M., & Grün, G. (2021). Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise. Archive for Rational Mechanics and Analysis. https://dx.doi.org/10.1007/s00205-021-01682-z

MLA:

Dareiotis, Konstantinos, et al. "Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise." Archive for Rational Mechanics and Analysis (2021).

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