This paper concerns the
following transonic shock phenomena in a three-dimensional de Laval nozzle
described by Courant and Friedrichs: Given the appropriately large receiver pressure
pr, if the upstream flow is still supersonic behind the throat of the nozzle, then at a
certain place in the widening part of the nozzle a shock front intervenes, and the gas
is compressed and slowed down to subsonic speed. The position and the strength of
the shock front are automatically adjusted so that the end pressure at the exit
becomes pr. We study this problem for the inviscid steady potential equation. In
this case, the transonic shock is a free boundary dividing the hyperbolic
region and the elliptic region in the nozzle. One main result is that for a
general class of nozzles, such a transonic shock solution is unique if the shock
exists and is assumed to pass through a fixed point. We also construct a
class of de Laval nozzles such that the transonic shock phenomena do not
occur for the generally given large pressures at the exit for the potential flow
model.
