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September  2012, 17(6): 1685-1691. doi: 10.3934/dcdsb.2012.17.1685

Self-similar focusing in porous media: An explicit calculation

D. G. Aronson 1,

1. 

Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, United States

Received  October 2010 Revised  December 2011 Published  May 2012

We consider a porous medium flow in which the material is initially distributed in the exterior of an empty region (a hole) and study the final stage of the hole-filling process as well as the initial stage of the post filling regime. It is known that in axially symmetric flow the hole-filling is asymptotically described by a self-similar solution which depends on a constant determined by the initial distribution. The post filling accumulation process is also locally described by a self-similar solution which in turn is characterized by a constant. In general, these constants must be found either experimentally or numerically. Here we present an example of a one-dimensional flow where the constants are obtained explicitly.
Keywords: Porous medium flow, self-similarity, focusing..
Mathematics Subject Classification: Primary: 35K55, 76S05; Secondary: 35K1.
Citation: D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685
References:
[1]

S. B. Angenent and D. G. Aronson, The focussing problem for the radially symmetric porous medium equation,, Comm. PDE, 20 (1995), 1217.  doi: 10.1080/03605309508821130.  Google Scholar

[2]

S. B. Angenent and D. G. Aronson, Self-similarity in the post-focussing regime in porous medium flows,, Euro. J. Appl. Math., 7 (1996), 277.   Google Scholar

[3]

D. G. Aronson, The porous medium equation,, in, (1986), 1.   Google Scholar

[4]

D. G. Aronson and J. Graveleau, A self-similar solution to the focusing problem for the porous medium equation,, Euro. J. Appl. Math., 4 (1993), 65.   Google Scholar

[5]

G. I. Barenblatt, "Scaling, Self-Similarity, and Intermediate Asymptotics,", With a foreword by Ya. B. Zeldovich, 14 (1996).   Google Scholar

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G. I. Barenblatt, On some unsteady motions of a liquid or gas in a porous medium,, Akad. Nauk SSSR Prikl. Mat. Meh., 16 (1952), 67.   Google Scholar

[7]

Ph. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $\mathbfR^N$ under optimal conditions on initial values,, Indiana U. Math. J., 33 (1984), 51.  doi: 10.1512/iumj.1984.33.33003.  Google Scholar

[8]

B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of the porous medium equation,, Comm. PDE, 9 (1984), 409.   Google Scholar

show all references

References:
[1]

S. B. Angenent and D. G. Aronson, The focussing problem for the radially symmetric porous medium equation,, Comm. PDE, 20 (1995), 1217.  doi: 10.1080/03605309508821130.  Google Scholar

[2]

S. B. Angenent and D. G. Aronson, Self-similarity in the post-focussing regime in porous medium flows,, Euro. J. Appl. Math., 7 (1996), 277.   Google Scholar

[3]

D. G. Aronson, The porous medium equation,, in, (1986), 1.   Google Scholar

[4]

D. G. Aronson and J. Graveleau, A self-similar solution to the focusing problem for the porous medium equation,, Euro. J. Appl. Math., 4 (1993), 65.   Google Scholar

[5]

G. I. Barenblatt, "Scaling, Self-Similarity, and Intermediate Asymptotics,", With a foreword by Ya. B. Zeldovich, 14 (1996).   Google Scholar

[6]

G. I. Barenblatt, On some unsteady motions of a liquid or gas in a porous medium,, Akad. Nauk SSSR Prikl. Mat. Meh., 16 (1952), 67.   Google Scholar

[7]

Ph. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $\mathbfR^N$ under optimal conditions on initial values,, Indiana U. Math. J., 33 (1984), 51.  doi: 10.1512/iumj.1984.33.33003.  Google Scholar

[8]

B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of the porous medium equation,, Comm. PDE, 9 (1984), 409.   Google Scholar

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