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 uniq...

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Bibliographic Details
Published inarXiv.org
Main Authors Chen, Gui-Qiang G, Yuan, Hairong
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 08.12.2011
Subjects
Divergent nozzles
Euler-Lagrange equation
Eulers equations
Fixed points (mathematics)
Free boundaries
Iterative methods
Mapping
Mathematical analysis
Operators (mathematics)
Riemann manifold
Uniqueness
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Summary: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.
ISSN:2331-8422