Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-11T00:00:11.180Z Has data issue: false hasContentIssue false
  • English
  • Français

A selfsimilar solution to the focusing problem for the porous medium equation

Published online by Cambridge University Press:  26 September 2008

D. G. Aronson  and
J. Graveleau
D. G. Aronson
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
J. Graveleau
Affiliation:
Centre d'Etudes Nucléaires, Commissariat á l'Energie Atomique, 23700 Pierrelatte, France
Get access Rights & Permissions [Opens in a new window]

Abstract

In the focusing problem we seek a solution to the porous medium equation whose initial distribution is in the exterior of some compact set (e.g. a ball). At a finite time T the gas will reach all points of the initially empty region R. We construct a selfsimilar solution of the radially symmetric focusing problem. This solution is an example of a selfsimilar solution of the second kind, i.e. one in which the similarity variable cannot be determined a priori from dimensional considerations. Our solution also shows that in more than one space dimension, the velocity of the gas is infinite at the centre of R at the focusing time T.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 4 , Issue 1 , March 1993 , pp. 65 - 81
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aronson, D. G. 1969 Regularity properties of flows through porous media. SIAM J. Appl. Math. 17, 461–467.CrossRefGoogle Scholar
Aronson, D. G. 1986 The porous medium equation. In Some Problems in Nonlinear Diffusion (eds. Fasano, A. and Primicerio, M.), Lecture Notes in Math. No. 1224, Springer-Verlag.Google Scholar
Aronson, D. G. & Caffarelli, L. A. 1986 Optimal regularity for one dimensional porous medium flow. Rev. Mat. Iberoamericana 2, 357–366.CrossRefGoogle Scholar
Aronson, D. G. & Vazquez, J. L. 1992 Anomalous exponents in nonlinear diffusion (in preparation).Google Scholar
Barenblatt, G. I. 1979 Similarity, Self-Similarity, and Intermediate Asymptotic. Consultants Bureau, New York.CrossRefGoogle Scholar
Barenblatt, G. I. & Zel'dovich, YA. B. 1971 Intermediate asymptotic in mathematical physics. Russian Math. Surveys 26 (2) 45–61.CrossRefGoogle Scholar
Benilan, PH. 1983 A strong regularity Lp for solutions of the porous medium equation. In Contributions to Nonlinear Partial Differential equations (eds. Bardos, C., Damlamian, A., Diaz, J. I. and Hernandez, J.), Research Notes in Math 89, Pitman.Google Scholar
Caffarelli, L. A. & Friedman, A. 1980 Regularity of the free boundary of a gas flow in an n-dimensional porous medium. Indiana Math. J. 29, 361–391.CrossRefGoogle Scholar
Caffarelli, L. A., Vazquez, J. L. & Wolanski, N. I.Lipschitz continuity of solutions and interfaces of the n-dimensional porous medium equation. Indiana Mat. J. 36, 373–401.CrossRefGoogle Scholar
Graveleau, J. 1972 Quelques solutions auto-semblables pour l'éqation dela chaleur non-linéaire. Rapport Interne C.E.A.Google Scholar
Guckenheimer, J. & Holmes, P. 1986 Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer-Verlag.Google Scholar
Guderley, G. 1942 Starke kugelige und zylindrische Verdichtungsstösse in der Nähe des Kugelmittelpunktes bzw. der Zulinderachse. Luftfahrtforschung 19, 302–312.Google Scholar
Holshof, J. 1991 Similarity solutions of the porous medium equation with sign changes. J. Math. Anal. Appl. 157, 75–111.CrossRefGoogle Scholar
Kevrekidis, I. G. 1987 A numerical study of global bifurcation in chemical dynamics. A.I.Ch.E.J. 33, 1850–1864.CrossRefGoogle Scholar
Lacy, A. A., Ockendon, J. R. & Tayler, A. B. 1982 ‘Waiting-time’ solutions of a nonlinear diffusion equation, SIAM J. Appl. Math. 42, 1252–1264.CrossRefGoogle Scholar
Zel'dovich, YA. B. & Raizer, YU. P. 1966 Physics of Shock-waves and High-temperature Phenomena, vol. II. Academic Press.Google Scholar