Local uniqueness of steady spherical transonic shock-fronts for the three-dimensional full euler equations

We establish the local uniqueness of steady transonic shock solutions with spherical symmetry for the three-dimensional full Euler equations. These transonic shock-fronts are important for understanding transonic shock phenomena in divergent nozzles. From mathematical point of view, we show the uniqueness of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite elliptic-hyperbolic type. To this end, we develop a decomposition of the Euler system which works in a general Riemannian manifold, a method to study a Venttsel problem of nonclassical nonlocal elliptic operators, and an iteration mapping which possesses locally a unique fixed point. The approach reveals an intrinsic structure of the steady Euler system and subtle interactions of its elliptic and hyperbolic part.