  In this paper, we prove the structural stability of the transonic shocks for
three dimensional axisymmetric Euler system with swirl velocity under the
perturbations for the incoming supersonic flow, the nozzle boundary, and the
exit pressure. Compared with the known results on the stability of transonic
shocks, one of the major difficulties for the axisymmetric flows with swirls is
that corner singularities near the intersection point of the shock surface and
nozzle boundary and the artificial singularity near the axis appear
simultaneously. One of the key points in the analysis for this paper is the
introduction of an invertible Lagrangian transformation which can straighten
the streamlines in the whole nozzle and help to represent the solutions of
transport equations explicitly.
