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Hyperbolic Aspects in the Theory of the Porous Medium Equation

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Metastability and Incompletely Posed Problems

Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 3))

Abstract

The porous medium equation (PME)

$$ {{\text{u}}_{\text{t}}} = \Delta ({{\text{u}}^{\text{m}}}),\,{\text{m>1}} $$

is one of the simplest models of nonlinear diffusion equations. It arises naturally in the study of a number of problems describing the evolution of a continuous quantity subject to a nonlinear diffusion mechanism, which we can instance explain as caused by a diffusion coefficient of the form

$$ {\text{c(u) = m}}{{\text{u}}^{{{\text{m - 1}}}}} $$
((1.1))

if we write the PME as ut = div(c(u) ∇u). Among the applications of the PME have

  1. (i)

    Percolation of gas through porous media, where m ≥ 2 [M],

  2. (ii)

    Radiative heat transfer in ionized plasmas, where m ≃ 6 [ZR],

  3. (iii)

    Thin liquid films spreading under gravity, where m = 4 [Bu],

  4. (iv)

    Crowd-avoiding population spreading, where m>1 [GM].

Partly supported by USA-Spain Cooperation Agreement under Joint Research Grant CCB-8402023. The paper was written while the author was a member of the Institute for Mathematics and its Applications, University of Minnesota, 1985.

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Author information

Authors and Affiliations

  1. Division de Matematicas, Universidad Autonoma, 28049, Madrid, Spain

    Juan Luis Vazquez

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  1. Juan Luis Vazquez
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Stuart S. Antman

  2. School of Mathematics and Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455, USA

    J. L. Ericksen

  3. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    David Kinderlehrer

  4. FB9-Hermann Föttinger Institut, Technical University, Berlin, Germany

    Ingo Müller

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© 1987 Springer-Verlag New York Inc.

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Vazquez, J.L. (1987). Hyperbolic Aspects in the Theory of the Porous Medium Equation. In: Antman, S.S., Ericksen, J.L., Kinderlehrer, D., Müller, I. (eds) Metastability and Incompletely Posed Problems. The IMA Volumes in Mathematics and Its Applications, vol 3. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8704-6_20

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  • DOI: https://doi.org/10.1007/978-1-4613-8704-6_20

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