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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

 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.
Citation: D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete and 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), 1217-1240. doi: 10.1080/03605309508821130. [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), 277-285. [3] D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Mathematics, 1224, Springer, Berlin, (1986), 1-46. [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), 65-81. [5] G. I. Barenblatt, "Scaling, Self-Similarity, and Intermediate Asymptotics," With a foreword by Ya. B. Zeldovich, Cambridge Texts in Applied Mathematics, 14, Cambridge University Press, Cambridge, 1996. [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), 67-78. [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), 51-87. doi: 10.1512/iumj.1984.33.33003. [8] B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of the porous medium equation, Comm. PDE, 9 (1984), 409-437.

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##### References:
 [1] S. B. Angenent and D. G. Aronson, The focussing problem for the radially symmetric porous medium equation, Comm. PDE, 20 (1995), 1217-1240. doi: 10.1080/03605309508821130. [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), 277-285. [3] D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Mathematics, 1224, Springer, Berlin, (1986), 1-46. [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), 65-81. [5] G. I. Barenblatt, "Scaling, Self-Similarity, and Intermediate Asymptotics," With a foreword by Ya. B. Zeldovich, Cambridge Texts in Applied Mathematics, 14, Cambridge University Press, Cambridge, 1996. [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), 67-78. [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), 51-87. doi: 10.1512/iumj.1984.33.33003. [8] B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of the porous medium equation, Comm. PDE, 9 (1984), 409-437.
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