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We develop a method that works in general product Riemannian manifold todecompose the three-dimensional steady full compressible Euler system, which isof elliptic-hyperbolic composite-mixed type for subsonic flows. The method isapplied to show stability of spherically symmetric subsonic flows and transonicshocks in space $\mathbb{R}^3$ under multidimensional perturbations of boundaryconditions.

Decomposition of Three-Dimensional Steady Non-isentropic Compressible  Euler System and Stability of Spherically Symmetric Subsonic Flows and  Transonic Shocks under Multidimensional Perturbations