Global uniqueness of transonic shocks in divergent nozzles for steady poten tial flows

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 (resp., velocity) is appropriately large (resp., small) and also spherically symmetric at the exit, then there exists uniquely one transoni c shock in the nozzle. In addition, the shock-front and the supersonic flow ahead of it, as well as the subsonic flow behind 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.