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.

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.

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.

Blow-up similarity solutions of the fourth-order unstable thin film equation

2007 ◽  
Vol 18 (2) ◽  
pp. 195-231 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Fourth Order ◽  
Blow Up ◽  
Similarity Solutions ◽  
Diffusion Term ◽  
Thin Film Equation ◽  
Self Similar ◽  
Image Position ◽  
General Aspects ◽  
Similar Solutions

We study blow-up behaviour of solutions of the fourth-order thin film equationwhich contains a backward (unstable) diffusion term. Our main goal is a detailed study of the case of the first critical exponentwhereN≥ 1 is the space dimension. We show that the free-boundary problem with zero contact angle and zero-flux conditions admits continuous sets (branches) of blow-up self-similar solutions. For the Cauchy problem inRN×R+, we detect compactly supported blow-up patterns, which have infinitely many oscillations near interfaces and exhibit a “maximal” regularity there. As a key principle, we use the fact that, for small positiven, such solutions are close to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equationwhich are better understood and have been studied earlier [19]. We also discuss some general aspects of formation of self-similar blow-up singularities for other values ofp.

On self-similar solutions of Barenblatt's nonlinear filtration equation

1996 ◽  
Vol 7 (2) ◽  
pp. 151-167 ◽  
Author(s):  
Julian D. Cole ◽  
Barbara A. Wagner
Keyword(s):  
Cauchy Problem ◽  
Linear Problem ◽  
Asymptotic Form ◽  
Similarity Solutions ◽  
The Self ◽  
Nonlinear Filtration ◽  
Filtration Equation ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Similar Solutions

We derive the asymptotic form of the self-similar solutions of the second kind of the Cauchy problem for Barenblatt's nonlinear filtration equation by perturbing the Lie group of the underlying linear problem. We also show that the decay rate, appearing in the similarity solutions, can be found by a simple inspection of the corresponding energy dissipation law.

Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data

2019 ◽  
Vol 150 (2) ◽  
pp. 789-811
Author(s):  
Yūki Naito
Keyword(s):  
Initial Data ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Heat Equations ◽  
Complicated Manner ◽  
The Cauchy Problem ◽  
Self Similar ◽  
The One ◽  
Image Position ◽  
Similar Solutions

AbstractWe consider the Cauchy problem $$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill & {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill & {x\in {\bf R}^N,} \hfill \cr } } \right.$$where N > 2, p > 1, and u0 is a bounded continuous non-negative function in RN. We study the case where u0(x) decays at the rate |x|−2/(p−1) as |x| → ∞, and investigate the convergence property of the global solutions to the forward self-similar solutions. We first give the precise description of the relationship between the spatial decay of initial data and the large time behaviour of solutions, and then we show the existence of solutions with a time decay rate slower than the one of self-similar solutions. We also show the existence of solutions that behave in a complicated manner.

A new free boundary problem for unsteady flows in porous media

1998 ◽  
Vol 9 (1) ◽  
pp. 37-54 ◽  
Author(s):  
G. I. BARENBLATT ◽  
J. L. VAZQUEZ
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Fluid Motion ◽  
Porous Medium Equation ◽  
Careful Analysis ◽  
Medium Equation ◽  
Self Similar ◽  
Similar Solutions ◽  
Theory Of Filtration

We revisit the theory of filtration (slow fluid motion) through a horizontal porous stratum under the usual conditions of gently sloping fluid height profile. We start by considering the model for flooding followed by natural outflow through the endwall of the stratum, which has an explicit dipole solution as generic intermediate asymptotics. We then propose a model for forced drainage which leads to a new kind of free boundary problem for the Boussinesq equation, where the flux is prescribed as well as the height h=0 on the new free boundary. Its qualitative behaviour is described in terms of its self-similar solutions. We point out that such a class of self-similar solutions corresponds to a continuous spectrum, to be compared with the discrete spectrum of the standard Cauchy problem for the porous medium equation. This difference is due to the freedom in the choice of the flux condition allowed in our problem setting. We also consider the modifications introduced in the above models by the consideration of capillary retention of a part of the fluid. In all cases we restrict consideration to one-dimensional geometries for convenience and brevity. It is to be noted however that similar problems can be naturally posed in multi-dimensional geometries. Finally, we propose a number of related control questions, which are most relevant in the application and need a careful analysis.

Pressure-dipole solutions of the thin-film equation

2018 ◽  
Vol 30 (2) ◽  
pp. 358-399
Author(s):  
M. BOWEN ◽  
T. P. WITELSKI
Keyword(s):  
Thin Film ◽  
Accumulation Point ◽  
Full Time ◽  
Infinite Domain ◽  
Compactly Supported ◽  
Sign Changes ◽  
Thin Film Equation ◽  
Self Similar ◽  
Type Boundary ◽  
Similar Solutions

We investigate self-similar sign-changing solutions to the thin-film equation, ht = −(|h|nhxxx)x, on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = hxx = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s(t) = Ltβ) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct first-kind self-similar solutions on the entire half-line x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation point-like behaviour of sign-changes local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full time-dependent partial differential equation (based on a non-local regularisation of the mobility coefficient) and discuss the computational results in relation to the self-similar solutions.

ON THE CAUCHY PROBLEM IN BESOV SPACES FOR A NON-LINEAR SCHRÖDINGER EQUATION

2000 ◽  
Vol 02 (02) ◽  
pp. 243-254 ◽  
Author(s):  
FABRICE PLANCHON
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Initial Value ◽  
Non Linear ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Similar Solutions ◽  
Well Posed

We prove that the initial value problem for a non-linear Schrödinger equation is well-posed in the Besov space [Formula: see text], where the nonlinearity is of type |u|αu. This allows to obtain self-similar solutions, and to recover previous results under weaker smallness assumptions on the data.

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