scholarly journals Small- and Waiting-Time Behavior of the Thin-Film Equation

2007 ◽  
Vol 67 (6) ◽  
pp. 1776-1807 ◽  
Author(s):  
James F. Blowey ◽  
John R. King ◽  
Stephen Langdon
Keyword(s):  
Thin Film ◽  
Waiting Time ◽  
Time Behavior ◽  
Thin Film Equation

Uniqueness of the regular waiting-time type solution of the thin film equation

2012 ◽  
Vol 23 (4) ◽  
pp. 537-554 ◽  
Author(s):  
MARINA CHUGUNOVA ◽  
JOHN R. KING ◽  
ROMAN M. TARANETS
Keyword(s):  
Thin Film ◽  
Initial Data ◽  
Waiting Time ◽  
Entropy Condition ◽  
Type Solution ◽  
Energy Equality ◽  
Thin Film Equation ◽  
Gronwall’S Lemma ◽  
Touchdown Point ◽  
Time Type

The main result of this paper is the proof of uniqueness of non-negative entropy solutions of the thin film equation ht + (|h|nhxxx)x = 0 for $\frac{7}{4}$ < n < 4. The uniqueness proved under assumptions that the initial data satisfy a finite β-entropy condition for some negative enough exponent β and that the solution is locally monotone at the touchdown point. The new dissipated functional recently constructed by Laugesen (Commun. Pure Appl. Anal., 4(3):613–634, 2005) is used to prove an auxiliary energy equality, and then Grönwall's lemma leads to uniqueness.

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
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