Local uniqueness of steady sphericaltransonic shock-fronts for the three--dimensional full Eulerequations

We establish the local uniqueness of steady transonic shock solutionswith spherical symmetry for the three-dimensional full Eulerequations. These transonic shock-fronts are important forunderstanding transonic shock phenomena in divergent nozzles. Frommathematical point of view, we show the uniqueness of solutions of a freeboundary problem for a multidimensional quasilinear system ofmixed-composite elliptic--hyperbolic type. To this end, we develop adecomposition of the Euler system which works in a generalRiemannian manifold, a method to study a Venttsel problem ofnonclassical nonlocal elliptic operators, and an iteration mappingwhich possesses locally a unique fixed point. The approach revealsan intrinsic structure of the steady Euler system and subtleinteractions of its elliptic and hyperbolic part.