Regularity of Flows in Porous Media: A Survey

Much of the early development of the classical theory of linear partial differential equations was guided by the very detailed body of knowledge accumulated over the years concerning the three model equations: Laplace’s equation, the wave equation, and the equation of heat conduction. Indeed, many of us were indoctrinated at the outset of our research careers with maxims such as: “Whatever is true for Laplace’s equation is also true, (sotto voce) with appropriate modifications, for any elliptic equation.” In this lecture I want to describe some of the results of a continuing project, involving a fairly large number of analyists, concerning a model equation for a class of nonlinear diffusion problems, the so-called porous medium equation. Although the theory is certainly not complete, its general outlines are quite clear and a coherent summary is possible.

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