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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
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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.

Information

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

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