Source-type solutions to thin-film equations in higher dimensions

1997 ◽  
Vol 8 (5) ◽  
pp. 507-524 ◽  
Author(s):  
RAUL FERREIRA ◽  
FRANCISCO BERNIS
Keyword(s):  
Thin Film ◽  
Negative Solution ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
First Order ◽  
Source Type ◽  
Thin Film Equations ◽  
Thin Film Equation ◽  
Self Similar ◽  
Speed Of Propagation

We prove that the thin film equation ht+div (hn grad (Δh))=0 in dimension d[ges ]2 has a unique C1 source-type radial self-similar non-negative solution if 0<n<3 and has no solution of this type if n[ges ]3. When 0<n3 the solution h has finite speed of propagation and we obtain the first order asymptotic behaviour of h at the interface or free boundary separating the regions where h>0 and h=0. (The case d=1 was already known [1]).

scholarly journals Finite speed of propagation and waiting time for a thin-film Muskat problem

2017 ◽  
Vol 147 (4) ◽  
pp. 813-830 ◽  
Author(s):  
Philippe Laurençot ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Thin Film ◽  
Initial Data ◽  
Waiting Time ◽  
Scale Invariance ◽  
Sufficient Conditions ◽  
Two Phase ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Muskat Problem ◽  
Speed Of Propagation

Propagation at a finite speed is established for non-negative weak solutions to a thin-film approximation of the two-phase Muskat problem. The expansion rate of the support matches the scale invariance of the system. Moreover, we determine sufficient conditions on the initial data for the occurrence of waiting time phenomena.

On continuous branches of very singular similarity solutions of a stable thin film equation. I – The Cauchy problem

2011 ◽  
Vol 22 (3) ◽  
pp. 217-243 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV
Keyword(s):  
Thin Film ◽  
Cauchy Problem ◽  
Similarity Solutions ◽  
Diffusion Term ◽  
Source Type ◽  
Thin Film Equation ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Image Position ◽  
Harmonic Equation

We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p0, the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n-branching approach. As such, a continuous path as n → 0+ can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation ut = −Δ2u. Numerics are used, wherever appropriate, to support the analysis.

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.

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