This lecture will outline the developments over the past two years of Todd Dupont and the author on problems of flow of fluids in porous media arising from petroleum reservoir engineering. Two specific problems will be discussed and a third mentioned. The first is the single-phase flow of a non-ideal gas. This flow can be described by a nonlinear parabolic equation. The second is the two-phase ...

Every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem. The free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both Dirichlet and Neumann data. We establish a rigorous stability analysis of such equilibria, including the construction...