scholarly journals The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions

2015 ◽  
Vol 35 (3) ◽  
pp. 807-827
Author(s):  
P. Álvarez-Caudevilla ◽  
◽  
J. D. Evans ◽  
V. A. Galaktionov ◽  
Keyword(s):  
Thin Film ◽  
Cauchy Problem ◽  
Similarity Solutions ◽  
Source Type ◽  
Thin Film Equation ◽  
The Cauchy Problem

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.

On continuous branches of very singular similarity solutions of the stable thin film equation. II – Free-boundary problems

2011 ◽  
Vol 22 (3) ◽  
pp. 245-265 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV
Keyword(s):  
Thin Film ◽  
Cauchy Problem ◽  
Free Boundary ◽  
Boundary Problem ◽  
Similarity Solutions ◽  
Thin Film Equation ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions

We discuss the fourth-order thin film equation with a stable second-order diffusion term, in the context of a standard free-boundary problem with zero height, zero contact angle and zero-flux conditions imposed at an interface. For the first critical exponent where N ≥ 1 is the space dimension, there are continuous sets (branches) of source-type very singular self-similar solutions of the form For p ≠ p0, the set of very singular self-similar solutions is shown to be finite and consists 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 second kind similarity solutions of the pure thin film equation Such solutions are detected by a combination of linear and non-linear ‘Hermitian spectral theory’, which allows the application of an analytical n-branching approach. In order to connect with the Cauchy problem in Part I, we identify the cauchy problem solutions as suitable ‘limits’ of the free-boundary problem solutions.

Source-type solutions of the fourth-order unstable thin film equation

2007 ◽  
Vol 18 (3) ◽  
pp. 273-321 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Cauchy Problem ◽  
Fourth Order ◽  
Similarity Solutions ◽  
Diffusion Term ◽  
Thin Film Equation ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Cahn Hilliard Equation ◽  
Image Position

We consider the fourth-order thin film equation (TFE) with the unstable second-order diffusion term. We show that, for the first critical exponent where N ≥ 1 is the space dimension, the free-boundary problem the with zero contact angle and zero-flux conditions admits continuous sets (branches) of self-similar similarity solutions of the form For the Cauchy problem, we describe families of self-similar patterns, which admit a regular limit as n → 0+ and converge to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equation studied earlier in [12]. Using both analytic and numerical evidence, we show that such solutions of the TFE are oscillatory and of changing sign near interfaces for all n ∈ (0,nh), where the value characterizes a heteroclinic bifurcation of periodic solutions in a certain rescaled ODE. We also discuss the cases p ⧧ = p0, the interface equation, and regular analytic approximations for such TFEs as an approach to the Cauchy problem.

scholarly journals The Cauchy Problem for a Tenth-Order Thin Film Equation I. Bifurcation of Oscillatory Fundamental Solutions

2013 ◽  
Vol 10 (4) ◽  
pp. 1761-1792
Author(s):  
Pablo Álvarez-Caudevilla ◽  
Jonathan D. Evans ◽  
Victor A. Galaktionov
Keyword(s):  
Thin Film ◽  
Cauchy Problem ◽  
Fundamental Solutions ◽  
Thin Film Equation ◽  
The Cauchy Problem

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.

scholarly journals Blowup and dissipation in a critical-case unstable thin film equation

2004 ◽  
Vol 15 (2) ◽  
pp. 223-256 ◽  
Author(s):  
T. P. WITELSKI ◽  
A. J. BERNOFF ◽  
A. L. BERTOZZI
Keyword(s):  
Thin Film ◽  
Fourth Order ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Similarity Solutions ◽  
Fluid Layers ◽  
Parabolic Pde ◽  
Thin Film Equation ◽  
Order Degenerate

We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. %For a special balance between %destabilizing second-order terms and regularizing fourth-order terms, There is a critical mass for blow-up and a rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied.

scholarly journals Dipoles and similarity solutions of the thin film equation in the half-line

Nonlinearity ◽  
2000 ◽  
Vol 13 (2) ◽  
pp. 413-439 ◽  
Author(s):  
Francisco Bernis ◽  
Josephus Hulshof ◽  
John R King
Keyword(s):  
Thin Film ◽  
Similarity Solutions ◽  
Half Line ◽  
Thin Film Equation
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