Self-similar focusing in porous media: An explicit calculation

Previous Article
The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions
DCDS-B Home
This Issue
Next Article
Stabilization of a reaction-diffusion system modelling malaria transmission

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

Abstract
Related Papers

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 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.

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), 1217-1240. 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.

[1]	Edoardo Mainini. On the signed porous medium flow. Networks and Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525
[2]	Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems and Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001
[3]	Rogelio Valdez. Self-similarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 897-922. doi: 10.3934/dcds.2006.16.897
[4]	José Ignacio Alvarez-Hamelin, Luca Dall'Asta, Alain Barrat, Alessandro Vespignani. K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases. Networks and Heterogeneous Media, 2008, 3 (2) : 371-393. doi: 10.3934/nhm.2008.3.371
[5]	Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks and Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
[6]	Hillel Furstenberg. From invariance to self-similarity: The work of Michael Hochman on fractal dimension and its aftermath. Journal of Modern Dynamics, 2019, 15: 437-449. doi: 10.3934/jmd.2019027
[7]	Changming Song, Yun Wang. Nonlocal latent low rank sparse representation for single image super resolution via self-similarity learning. Inverse Problems and Imaging, 2021, 15 (6) : 1347-1362. doi: 10.3934/ipi.2021017
[8]	Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[9]	Anna Marciniak-Czochra, Andro Mikelić. A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 1065-1077. doi: 10.3934/dcdss.2014.7.1065
[10]	Yaqing Liu, Liancun Zheng. Second-order slip flow of a generalized Oldroyd-B fluid through porous medium. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2031-2046. doi: 10.3934/dcdss.2016083
[11]	Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021
[12]	Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93
[13]	María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks and Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012
[14]	Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093
[15]	Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks and Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337
[16]	Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
[17]	R.E. Showalter, Ning Su. Partially saturated flow in a poroelastic medium. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 403-420. doi: 10.3934/dcdsb.2001.1.403
[18]	Michiel Bertsch, Carlo Nitsch. Groundwater flow in a fissurised porous stratum. Networks and Heterogeneous Media, 2010, 5 (4) : 765-782. doi: 10.3934/nhm.2010.5.765
[19]	Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783
[20]	Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure and Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
D. G. Aronson
on Google Scholar
D. G. Aronson

[Back to Top]