American Institute of Mathematical Sciences

Advanced
  • 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

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

[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

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

[1]

Rogelio Valdez. Self-similarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 897-922. doi: 10.3934/dcds.2006.16.897
[2]

Edoardo Mainini. On the signed porous medium flow. Networks & Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525
[3]

José Ignacio Alvarez-Hamelin, Luca Dall'Asta, Alain Barrat, Alessandro Vespignani. K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases. Networks & Heterogeneous Media, 2008, 3 (2) : 371-393. doi: 10.3934/nhm.2008.3.371
[4]

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 & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
[5]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[6]

Anna Marciniak-Czochra, Andro Mikelić. A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1065-1077. doi: 10.3934/dcdss.2014.7.1065
[7]

Yaqing Liu, Liancun Zheng. Second-order slip flow of a generalized Oldroyd-B fluid through porous medium. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2031-2046. doi: 10.3934/dcdss.2016083
[8]

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 & Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93
[9]

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 & Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093
[10]

Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 377-387. doi: 10.3934/dcdss.2020021
[11]

María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012
[12]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337
[13]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
[14]

R.E. Showalter, Ning Su. Partially saturated flow in a poroelastic medium. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 403-420. doi: 10.3934/dcdsb.2001.1.403
[15]

Michiel Bertsch, Carlo Nitsch. Groundwater flow in a fissurised porous stratum. Networks & Heterogeneous Media, 2010, 5 (4) : 765-782. doi: 10.3934/nhm.2010.5.765
[16]

Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783
[17]

Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure & Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623
[18]

Goro Akagi. Energy solutions of the Cauchy-Neumann problem for porous medium equations. Conference Publications, 2009, 2009 (Special) : 1-10. doi: 10.3934/proc.2009.2009.1
[19]

Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23
[20]

Xinfu Chen, Jong-Shenq Guo, Bei Hu. Dead-core rates for the porous medium equation with a strong absorption. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1761-1774. doi: 10.3934/dcdsb.2012.17.1761

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences