S. Canic et Bl. Keyfitz, Quasi-one-dimensional Riemann problems and their role in self-similar two-dimensional problems, ARCH R MECH, 144(3), 1998, pp. 233-258
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.