Transonic shocks for 3-D axisymmetric compressible inviscid flows in cylinders

We establish the existence of an axisymmetric weak solution to the steady Euler system with a transonic shock, nonzero vorticity, and nonzero swirl in a three-dimensional cylinder. When prescribing the super- sonic solution in the upstream region by axisymmetric functions with variable entropy and variable angular momentum density (=swirl), we construct such a solution by using a Helmholtz decomposition of the ve- locity field and the method of iteration. An iteration scheme is developed using a delicate decomposition of the Rankine-Hugoniot conditions on the transonic shock via Helmholtz decomposition.