  This paper is concerned with the existence and location of three-dimensional
axisymmetric transonic shock with large swirl velocity for solutions of the
steady compressible full Euler system in a cylindrical nozzle with prescribed
receiver pressure. Special non-trivial shock solutions with large vorticity are
first constructed by considering arbitrarily given non-zero swirl functions.
Then the existence and locations of the transonic shock solutions to the full
Euler equations are achieved under small perturbations on the special shock
solutions with appropriate boundary conditions on the entrance of the nozzle
and the receiver pressure at the exit. Moreover, within the proof, fundamental
contributions of the non-zero swirl velocity in determining the position of the
shock front are investigated. Mathematically, it can be formulated as a free
boundary value problem with the shock front as the free boundary, whose
solution states are a perturbation of non-trivial background solutions, and
with a singularity at the axis-symmetry. Because the shock location can be
arbitrary for the special shock solutions, approximate shock solutions of a
free boundary problem for a specific linearized Euler system are constructed
and an iteration scheme is designed around the approximate shock solution with
a perturbation of higher order. As far as we know, it is the first mathematical
result on the three-dimensional transonic shock with either large vorticity or
large swirl velocity. New ideas and methods introduced in this paper will be
also helpful for other problems with similar difficulties.
