Abstract
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.
Highlights
The study of the Euler equations for compressible fluids is one of the central topics in the mathematical fluid dynamics, and the analysis of solutions to the system is of particular interest in applications
We are concerned with the local uniqueness of transonic shock solutions with spherical symmetry for the three-dimensional, steady, full Euler system of polytropic gases
Such a study helps us to understand transonic shock phenomena occurred in divergent nozzles, which have many important applications, and provides new insights for the theory of free boundary problems of partial differential equations of composite–mixed elliptic–hyperbolic type
Summary
The study of the Euler equations for compressible fluids is one of the central topics in the mathematical fluid dynamics, and the analysis of solutions to the system is of particular interest in applications. We are concerned with the local uniqueness of transonic shock solutions with spherical symmetry for the three-dimensional, steady, full Euler system of polytropic gases. Such a study helps us to understand transonic shock phenomena occurred in divergent nozzles, which have many important applications, and provides new insights for the theory of free boundary problems of partial differential equations of composite–mixed elliptic–hyperbolic type. We remark that the existence and uniqueness of supersonic flow U − in M subject to the initial data U −|Σ0 satisfying (1.7) follow directly from the theory of semi-global classical solutions of the Cauchy problem of quasilinear symmetric hyperbolic systems if ε0 is sufficiently small (cf [14, 18]). We remark in passing that the analysis and results developed here should be straightforward extended to the higher dimensional case, even general Riemannian manifolds for most of them
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