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.

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.

Existence and blow-up of solutions for a non-local filtration and porous medium problem

2010 ◽  
Vol 53 (1) ◽  
pp. 195-209 ◽  
Author(s):  
E. A. Latos ◽  
D. E. Tzanetis
Keyword(s):  
Porous Medium ◽  
Initial Data ◽  
Blow Up ◽  
Porous Medium Equation ◽  
Filtration Equation ◽  
Medium Equation ◽  
Non Local ◽  
Image Position

AbstractWe consider a non-local filtration equation of the formand a porous medium equation, in this case K(u) = um, with some boundary and initial data u0, where 0 < p < 1 and f, f′, f″ > 0. We prove blow-up of solutions for sufficiently large values of the parameter λ > 0 and for any u0 > 0, or for sufficiently large values of u0 > 0 and for any λ λ 0.

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.

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