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Self-similarity in the post-focussing regime in porous medium flows

Published online by Cambridge University Press:  26 September 2008

S. B. Angenent  and
D. G. Aronson
S. B. Angenent
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
D. G. Aronson
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
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Abstract

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.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 3 , June 1996 , pp. 277 - 285
Copyright
Copyright © Cambridge University Press 1996

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References

[1] Angenent, S. B. & Aronson, D. G. 1996 The focussing problem for the radially symmetric porous medium equation. Comm. P.D.E. 20, 1217–1240.Google Scholar
[2] Angenent, S. B. & Aronson, D. G. 1995 Intermediate asymptotics for convergent viscous gravity currents. Phys. Fluids 7, 223–225.Google Scholar
[3] Aronson, D. G. 1986 The porous medium equation. In Some Problems in Nonlinear Diffusion, Fasano, A. and Primicerio, M., eds., Lecture Notes in Mathematics 1224, Springer Verlag.Google Scholar
[4] Aronson, D. G. & Graveleau, J. 1993 A selfsimilar solution to the focusing problem for the porous medium equation. Euro. J. Appl. Math. 4, 65–81.Google Scholar
[5] Barenblatt, G. I. 1956 On automodel solutions of the Cauchy problem for non-linear parabolic equations of non-stationary filtration of a gas in a porous medium. Prikl. Mat. Mekh. 20, 761–763.Google Scholar
[6] Barenblatt, G. I. 1978 Dynamics of turbulent spots and the intrusions in a stably stratified fluid. Izvestiya Atmospheric and Oceanic Phys. 14, 139–145.Google Scholar
[7] Barenblatt, G. I. 1978 Similarity, Self-similarity and Intermediate Asymptotics. Consultants Bureau, New York.Google Scholar
[8] Brushlinskii, K. V. & Kazhdan, Ja. M. 1963 On auto-models in the solution of certain problems in gas dynamics. Russian Math. Surveys 18, 1–22.Google Scholar
[9] Bénilan, Ph., Crandall, M. G. & Pierre, M. 1984 Solutions of the porous medium equation in RN under optimal conditions on initial values. Indiana U. Math. J. 33, 51–87.CrossRefGoogle Scholar
[10] Caffarelli, L. A. & Friedman, A. 1980 Regularity of the free boundary of a gas flow in an n-dimensional porous medium. Indiana U. Math. J. 29, 361–391.Google Scholar
[11] Caffarelli, L. A., Vazquez, J. L. & Wolanski, N. I. 1987 Lipschitz continuity of solutions and interfaces in the n-dimensional porous medium equation. Indiana U. Math. J. 36, 373–401.CrossRefGoogle Scholar
[12] Dahlberg, B. E. J. & Kenig, C. E. 1984 Non-negative solutions of the porous medium equation. Comm. P.D.E. 9, 409–437.Google Scholar
[13] Diez, J. A., Gratton, J. & Minotti, F. 1992 Self-similar solutions of the second kind of nonlinear diffusion type equations. Quart. Appl. Math. 50, 401–414.CrossRefGoogle Scholar
[14] Diez, J. A., Gratton, R. & Gratton, J. 1992 Self-similar solution of the second kind for a convergent viscous gravity current. Phys. Fluids A 4, 1148–1155.Google Scholar
[15] Guderley, G. 1942 Starke kugelige und zylindrische Verdichtungsstösse in der Nähe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforschung 19, 302–312.Google Scholar
[16] Gratton, J. & Minnot, F. 1990 Self similar viscous gravity currents: Phase plane formalism. J. Fluid Mech. 210, 155–182.Google Scholar
[17] Graveleau, J. 1972 Quélques solutions auto-semblables pour l'éqation dela chaleur non-linéaire. Rapport Interne C.E.A.Google Scholar
[18] Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
[19] Zel'dovich, Ya. B. & Raizer, Yu. P. 1966 Physics of Shock-waves and High-temperature Phenomena. Academic Press.Google Scholar