  We show that for steady compressible potential flow in a class of straight
divergent nozzles with arbitrary cross-section, if the flow is supersonic and
spherically symmetric at the entry, and the given pressure (velocity) is
appropriately large (small) and also spherically symmetric at the exit, then
there exists uniquely one transonic shock in the nozzle. In addition, the
shock-front and the supersonic flow ahead of it, as well as the subsonic flow
behind of it, are all spherically symmetric. This is a global uniqueness result
of free boundary problems of elliptic--hyperbolic mixed type equations. The
proof depends on the maximum principles and judicious choices of comparison
functions.
