Some Aspects of the Thin Film Equation

2001 ◽  
pp. 291-301 ◽  
Author(s):  
Josephus Hulshof
Keyword(s):  
Thin Film ◽  
Thin Film Equation

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

2021 ◽  
Author(s):  
Konstantinos Dareiotis ◽  
Benjamin Gess ◽  
Manuel V. Gnann ◽  
Günther Grün
Keyword(s):  
Thin Film ◽  
Energy Estimates ◽  
Approximation Procedure ◽  
Thin Film Flow ◽  
Thin Film Equations ◽  
Scalar Conservation Law ◽  
Thin Film Equation ◽  
Degenerate Parabolic ◽  
Scalar Conservation ◽  
Martingale Solutions

AbstractWe 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.

scholarly journals Global existence for a thin film equation with subcritical mass

2017 ◽  
Vol 22 (4) ◽  
pp. 1461-1492 ◽  
Author(s):  
Jian-Guo Liu ◽  
◽  
Jinhuan Wang ◽  
Keyword(s):  
Thin Film ◽  
Global Existence ◽  
Thin Film Equation

Unstable sixth-order thin film equation: II. Global similarity patterns

Nonlinearity ◽  
2007 ◽  
Vol 20 (8) ◽  
pp. 1843-1881 ◽  
Author(s):  
J D Evans ◽  
V A Galaktionov ◽  
J R King
Keyword(s):  
Thin Film ◽  
Sixth Order ◽  
Thin Film Equation ◽  
Global Similarity

scholarly journals Intermediate asymptotics of the capillary-driven thin-film equation

2013 ◽  
Vol 36 (8) ◽  
Author(s):  
Michael Benzaquen ◽  
Thomas Salez ◽  
Elie Raphaël
Keyword(s):  
Thin Film ◽  
Intermediate Asymptotics ◽  
Thin Film Equation

Asymptotic behaviour of the thin film equation in bounded domains

2001 ◽  
Vol 12 (2) ◽  
pp. 135-157 ◽  
Author(s):  
M. BOWEN ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Contact Angle ◽  
Early Time ◽  
Similarity Solutions ◽  
Bounded Domains ◽  
Extinction Time ◽  
Degenerate Diffusion ◽  
Thin Film Equation ◽  
Order Degenerate ◽  
Mass Increase

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases, an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included.

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