  This paper concerns the structural stability of smooth cylindrically
symmetric transonic flows in a concentric cylinder. Both cylindrical and
axi-symmetric perturbations are considered. The governing system here is of
mixed elliptic-hyperbolic and changes type and the suitable formulation of
boundary conditions at the boundaries is of great importance. First, we
establish the existence and uniqueness of smooth cylindrical transonic spiral
solutions with nonzero angular velocity and vorticity which are close to the
background transonic flow with small perturbations of the Bernoulli's function
and the entropy at the outer cylinder and the flow angles at both the inner and
outer cylinders independent of the symmetric axis, and it is shown that in this
case, the sonic points of the flow are nonexceptional and noncharacteristically
degenerate, and form a cylindrical surface. Second, we also prove the existence
and uniqueness of axi-symmetric smooth transonic rotational flows which are
adjacent to the background transonic flow, whose sonic points form an
axi-symmetric surface. The key elements in our analysis are to utilize the
deformation-curl decomposition for the steady Euler system introduced in
\cite{WengXin19} to deal with the hyperbolicity in subsonic regions and to find
an appropriate multiplier for the linearized second order mixed type equations
which are crucial to identify the suitable boundary conditions and to yield the
important basic energy estimates.
