Hyperbolic Aspects in the Theory of the Porous Medium Equation

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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 (1.1) $$ {\text{c(u) = m}}{{\text{u}}^{{{\text{m - 1}}}}} $$ if we write the PME as ut = div(c(u) ∇u). Among the applications of the PME have

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

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

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

  4. 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.