Structural Stability of the Transonic Shock Problem in a Divergent Three-Dimensional Axisymmetric Perturbed Nozzle

In this paper, we prove the structural stability of transonic shocks for three-dimensional axisymmetric Euler system with swirl velocity under perturbations incoming supersonic flow, nozzle boundary, and exit pressure. Compared known results on shocks, one major difficulties flows swirls is that corner singularities near intersection point shock surface boundary artificial singularity axis appear simultaneously. One key points in analysis paper introduction an invertible modified Lagrangian transformation which can straighten streamlines whole help to represent solutions transport equations explicitly. Furthermore, simple but useful makes treatment easy clean. Such a technique may be helpful studies other problems flows.