Lectures on Gas Flow in Porous Media
Luis A. Caffarelli, Aram Karakhanyan · Applied and numerical harmonic analysis · 2009
The idea behind the lectures in this chapter is to present, in a relatively simple setting, that of solutions to porous media in one space dimension, several of the main ideas and the main techniques that are at the center of the regularity theory for nonlinear evolution equations and phase transitions. These include exploiting the invariances of the equation to obtain infinitesimal relations and geometric control of the solutions, the role of particular solutions to guide us in our theory and provide us with barriers and asymptotic profiles, the idea of viscosity solutions to a free boundary problem to deduce the geometric properties of the free boundary, and the methods of blowing up solutions and classifying the global profiles to obtain the differentiability properties of a free boundary.