Hostname: page-component-f554764f5-nqxm9 Total loading time: 0 Render date: 2025-04-12T13:17:06.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal viscosity solutions of the modified porous medium equation and their asymptotic behaviour

Published online by Cambridge University Press:  26 September 2008

Josephus Hulshof  and
Juan Luis Vazquez
Josephus Hulshof
Affiliation:
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Juan Luis Vazquez
Affiliation:
División de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct a theory for maximal viscosity solutions of the Cauchy problem for the modified porous medium equation ut + γ|ut| = Δ(um) with γ∈(−1, 1) and m > 1. We investigate the existence, uniqueness, finite propagation speed and optimal regularity of these solutions. As a second main theme, we prove that the asymptotic behaviour is given by a certain family of self-similar solutions of the so-called second kind with anomalous similarity exponents.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 5 , October 1996 , pp. 453 - 471
Copyright
Copyright © Cambridge University Press 1996

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.)

Article purchase

Temporarily unavailable

References

[A]Aronson, D. G. (1986) The Porous Medium Equation. In Some Problems in Nonlinear Diffusion, Fasano, A. and Primicerio, M., eds., Lecture Notes in Maths., No. 1224, Springer-Verlag.Google Scholar
[AB]Aronson, D. G. & Bénilan, P. (1979) Régularité des solutions d'équation des milieux poreux dans RN. Comptes Rendu Acad. Sci. Paris A 288, 103–105.Google Scholar
[AV]Aronson, D. G. & Vazquez, J. L. (1994) Calculation of anomalous exponents in nonlinear diffusion. Phys. Rev. Letters 72, 348–351.Google Scholar
[Bl]Barenblatt, G. I. (1987) Dimensional Analysis, Gordon and Breach.Google Scholar
[B2]Barenblatt, G. I. (1968) Similarity, self-similarity and intermediate asymptotics, Consultants Bureau, New York.Google Scholar
[BER]Barenblatt, G. I., Entov, V. M. & Ryzhik, V. M. (1990) Theory of Fluid Flows Through Natural Rocks, Kluwer.CrossRefGoogle Scholar
[Ca]Caffarelli, L. A. (1989) Interior a priori estimates for solutions of fully non-linear equations. Annals Math. 130, 180–213.Google Scholar
[CF]Caffarelli, L. A. & Friedman, A. (1979) Continuity of the density of a gas flow in a porous medium. Trans. Amer. Math. Soc. 252, 99–113.Google Scholar
[CGO]Chen, L.-Y., Goldenfeld, N. & Oono, Y. (1991) Renormalization-group theory for the modified porous-medium equation. Physical Rev. A 44(10), 6544–6550.Google Scholar
[CIL]Crandall, M. G., Ishii, H. & Lions, P. L. (1992) User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27(1), 1–67.Google Scholar
[FK]Friedman, A. & Kamin, S. (1980) The asymptotic behaviour of gas in an n-dimensional porous medium. Trans. Amer. Math. Soc. 262, 551–563.Google Scholar
[GP]Gilding, B. & Peletier, L. A. (1981) Continuity of solutions of the porous medium equation. Annali Sc. Norm. Sup. Pisa IV(8), 659–675.Google Scholar
[G]Goldenfeld, N. (1992) Lectures on Phase Transitions and the Renormalization Group, Frontiers in Physics, No. 85, Addison-Wesley.Google Scholar
[HI]HuLSHOF, J. (1991) Similarity solutions of the porous medium equation with sign changes. J. Math. Anal. Appl. 157, 75–111.CrossRefGoogle Scholar
[H2]HuLSHOF, J. (1994) Recent results on self-similar solutions of degenerate nonlinear diffusion equations. Report W94–09, Math Institute of Leiden University.Google Scholar
[HV]Hulshof, J. & Vazquez, J. L. (1994) Selfsimilar solutions of the second kind for the modified porous medium equation. Euro. J. Appl. Math. 5, 391–403.CrossRefGoogle Scholar
[KPV]Kamin, S., Peletier, L. A. & Vazquez, J. L. (1991) On the Barenblatt equation of elasto plastic filtration. Indiana Univ. Math. J. 40, 1333–1362.Google Scholar
[KV]Kamin, S. & Vazquez, J. L. (1992) The propagation of turbulent bursts. Euro. J. Appl. Math. 3, 263–272.CrossRefGoogle Scholar
[V]Vazquez, J. L. (1992) An introduction to the mathematical theory of the porous medium equation. In Shape Optimization and Free Boundaries, Delfour, M. C., ed., Mathematical and Physical Sciences, Series C 380, Kluwer, 347–389.CrossRefGoogle Scholar
[W1]Wang, L. H. (1992) On the regularity theory of fully nonlinear parabolic equations: I. Comm. Pure Appl. Math. 45, 27–76.Google Scholar
[W2]Wang, L. H. (1992) On the regularity theory of fully nonlinear parabolic equations: II. Comm. Pure Appl. Math. 45, 141–178Google Scholar