  We concern the structural stability of transonic shocks for the steady Euler
system with an external force in an axisymmetric perturbed cylinder. For a
class of external forces, we first prove the existence and uniqueness of the
transonic shock solution to the one-dimensional steady Euler system with an
external force, which shows that the external force has a stabilization effect
on the transonic shock in the flat cylinder and the shock position is uniquely
determined. We then establish the existence and stability of the transonic
shock solution under axisymmetric perturbations of the incoming supersonic
flow, the nozzle boundary, the exit pressure and the external force. Different
from the transonic shock problem in two-dimensional nozzles, there exists a
singularity along the symmetric axis for axisymmetric flows. We introduce an
invertible modified Lagrangian transformation to overcome this difficulty and
straighten the streamline. One of the key elements in the analysis is to
utilize the deformation-curl decomposition to effectively decouple the
hyperbolic and elliptic modes in the steady axisymmetric Euler system with an
external force. Another one is an equivalent reformulation of the
Rankine-Hugoniot conditions so that the shock front is uniquely determined by
an algebraic equation.
