The transonic shock in a nozzle, 2-D and 3-D complete Euler systems

In this paper, we study a transonic shock problem for the Euler flows through a class of 2-D or 3-D nozzles. The nozzle is assumed to be symmetric in the diverging (or converging) part. If the supersonic incoming flow is symmetric near the divergent (or convergent) part of the nozzle, then, as indicated in Section 147 of [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publ., New York, 1948], there exist two constant pressures P1 and P2 with P1