Communications on Pure and Applied Analysis (CPAA)

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

Pages: 2515 - 2542, Volume 12, Issue 6, November 2013      doi:10.3934/cpaa.2013.12.2515

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Gui-Qiang G. Chen - School of Mathematical Sciences, Fudan University, Shanghai 200433, China (email)
Hairong Yuan - Department of Mathematics, East China Normal University, Shanghai 200241, China (email)

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.

Keywords:  Uniqueness, transonic shock-front, Euler system, three-dimensional, free boundary problem, mixed-composite elliptic-hyperbolic type, iteration mapping, decomposition, Venttsel problem, nonlocal elliptic operators, intrinsic structure, subtle interactions.
Mathematics Subject Classification:  5M20, 35J65, 35R35, 35B45, 76H05, 76L05.

Received: June 2012;      Revised: December 2012;      Available Online: May 2013.