  We prove the stability of three-dimensional axisymmetric solutions to the
steady Euler system with transonic shocks in divergent nozzles under
perturbations of the exit pressure and the supersonic solution in the upstream
region. We first derive a free boundary problem with the newly introduced
formulation of the Euler system for three-dimensional axisymmetric flows in
divergent nozzles via the method of Helmholtz decomposition. We then construct
an iteration scheme and use the Schauder fixed point theorem and weak implicit
function theorem to solve the problem.
