Chapter

Metastability and Incompletely Posed Problems

Volume 3 of the series The IMA Volumes in Mathematics and Its Applications pp 325-342

Hyperbolic Aspects in the Theory of the Porous Medium Equation

  • Juan Luis VazquezAffiliated withDivision de Matematicas, Universidad Autonoma

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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].