A self-similar solution for the porous medium equation in a two-component domain

2012 ◽  
Vol 75 (2) ◽  
pp. 880-898 ◽  
Author(s):  
Ján Filo ◽  
Volker Pluschke
Keyword(s):  
Porous Medium ◽  
Porous Medium Equation ◽  
Similar Solution ◽  
Medium Equation ◽  
Self Similar Solution ◽  
Two Component ◽  
Self Similar

scholarly journals The porous medium equation in a two-component domain

2009 ◽  
Vol 247 (9) ◽  
pp. 2455-2484 ◽  
Author(s):  
Ján Filo ◽  
Volker Pluschke
Keyword(s):  
Porous Medium ◽  
Porous Medium Equation ◽  
Medium Equation ◽  
Two Component

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 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.

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