Subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder

This paper concerns the structural stability of subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder. We establish the existence and uniqueness of axisymmetric subsonic flows with a contact discontinuity by prescribing the horizontal mass flux distribution, the swirl velocity, the entropy and the Bernoulli's quantity at the entrance and the radial velocity at the exit. It can be formulated as a free boundary problem with the contact discontinuity to be determined simultaneously with the flows. Compared with the two-dimensional case, a new difficulty arises due to the singularity near the axis. One of the key points in the analysis is the introduction of an invertible modified Lagrangian transformation which can overcome this difficulty and straighten the contact discontinuity. Another one is to utilize the deformation-curl decomposition for the steady Euler system introduced in \cite{WX19} to effectively decouple the hyperbolic and elliptic modes. Finally, the contact discontinuity will be located by using the implicit function theorem.