Self-similar solutions of two-dimensional conservation laws

Self-similar reduction of an important class of two-dimensional conservation laws leads to boundary value problems for equations which change type. We have established a method for solving free boundary problems for quasilinear degenerate elliptic equations which arise when shocks interact with the subsonic (nonhyperbolic) part of the solution. This paper summarizes the principal features of the method. A preliminary version of these notes formed the basis of a series of three lectures at the Newton Institute in April, 2003. They are a report of research carried out jointly with Suncica Canic, Eun Heui Kim and Gray Lieberman.