scholarly journals On a fractional thin film equation

2020 ◽  
Vol 9 (1) ◽  
pp. 1516-1558
Author(s):  
Antonio Segatti ◽  
Juan Luis Vázquez
Keyword(s):  
Thin Film ◽  
Porous Medium ◽  
Porous Medium Equation ◽  
Order Theory ◽  
Compactly Supported ◽  
Limit Value ◽  
Medium Equation ◽  
Thin Film Equation ◽  
Self Similar ◽  
Similar Solutions

Abstract This paper deals with a nonlinear degenerate parabolic equation of order α between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value α = 4 while the Porous Medium Equation is the limit α = 2. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.

Self-similar solutions of the second kind for the modified porous medium equation

1994 ◽  
Vol 5 (3) ◽  
pp. 391-403 ◽  
Author(s):  
Josephus Hulshof ◽  
Juan Luis Vazquez
Keyword(s):  
Porous Medium ◽  
Porous Medium Equation ◽  
Self Similarity ◽  
Compactly Supported ◽  
Medium Equation ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions

We construct compactly supported self-similar solutions of the modified porous medium equation (MPME)They have the formwhere the similarity exponents α and β depend on ε, m and the dimension N. This corresponds to what is known in the literature as anomalous exponents or self-similarity of the second kind, a not completely understood phenomenon. This paper performs a detailed study of the properties of the anomalous exponents of the MPME.

Self-similarity in the post-focussing regime in porous medium flows

1996 ◽  
Vol 7 (3) ◽  
pp. 277-285 ◽  
Author(s):  
S. B. Angenent ◽  
D. G. Aronson
Keyword(s):  
Porous Medium ◽  
Finite Time ◽  
Porous Medium Equation ◽  
Initial Distribution ◽  
Compact Set ◽  
Material Flows ◽  
Self Similarity ◽  
Medium Equation ◽  
Self Similar ◽  
Similar Solutions

In the focussing problem for the porous medium equation, one considers an initial distribution of material outside some compact set K. As time progresses material flows into K, and at some finite time T first covers all of K. For radially symmetric flows, with K a ball centred at the origin, it is known that the intermediate asymptotics of this focussing process is described by a family of self-similar solutions to the porous medium equation. Here we study the postfocussing regime and show that its onset is also described by self-similar solutions, even for nonsymmetric flows.

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.

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.

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.

Maximal viscosity solutions of the modified porous medium equation and their asymptotic behaviour

1996 ◽  
Vol 7 (5) ◽  
pp. 453-471 ◽  
Author(s):  
Josephus Hulshof ◽  
Juan Luis Vazquez
Keyword(s):  
Porous Medium ◽  
Asymptotic Behaviour ◽  
Viscosity Solutions ◽  
Propagation Speed ◽  
Porous Medium Equation ◽  
Main Theme ◽  
Optimal Regularity ◽  
Medium Equation ◽  
The Cauchy Problem ◽  
Similar Solutions

We construct a theory for maximal viscosity solutions of the Cauchy problem for the modified porous medium equation ut + γ|ut| = Δ(um) with γ∈(−1, 1) and m > 1. We investigate the existence, uniqueness, finite propagation speed and optimal regularity of these solutions. As a second main theme, we prove that the asymptotic behaviour is given by a certain family of self-similar solutions of the so-called second kind with anomalous similarity exponents.

scholarly journals Null-controllability of perturbed porous medium gas flow

2020 ◽  
Vol 26 ◽  
pp. 85
Author(s):  
Borjan Geshkovski
Keyword(s):  
Porous Medium ◽  
Gas Flow ◽  
Porous Medium Equation ◽  
Null Controllability ◽  
Spectral Techniques ◽  
Medium Equation ◽  
Self Similar Solution ◽  
Linearized Problem ◽  
Thin Film Equation ◽  
Controllability Result

In this work, we investigate the null-controllability of a nonlinear degenerate parabolic equation, which is the equation satisfied by a perturbation around the self-similar solution of the porous medium equation in Lagrangian-like coordinates. We prove a local null-controllability result for a regularized version of the nonlinear problem, in which singular terms have been removed from the nonlinearity. We use spectral techniques and the source-term method to deal with the linearized problem and the conclusion follows by virtue of a Banach fixed-point argument. The spectral techniques are also used to prove a null-controllability result for the linearized thin-film equation, a degenerate fourth order analog of the problem under consideration.

Sign in / Sign up

Export Citation Format

Share Document