Quasi-one-dimensional Riemann problems and their role in self-similar two-dimensional problems

We study two-dimensional Riemann problems with piecewise constant data. We identify a class of two-dimensional systems, including many standard equati ons of compressible flow which are simplified by a transformation to simila rity variables. For equations in this class, a two-dimensional Riemann prob lem with sectorially constant data becomes a boundary-value problem in the finite plane. For data leading to shock interactions, this problem separate s into two parts: a quasi-one-dimensiona1 problem in supersonic regions, an d an equation of mixed type in subsonic regions. We prove a theorem on local existence of solutions of quasi-one-dimensional Riemann problems. For 2 x 2 systems, we generalize a theorem of COURANT & FRIEDRICHS, that any hyperbolic state adjacent to a constant state must be a simple wave. In the subsonic regions, where the governing equation is of mixed hyperboli c-elliptic type, we show that the elliptic part is degenerate at the bounda ry, with a nonlinear variant of a degeneracy first described by KELDYSH.