Local Uniqueness of Steady Spherical Transonic Shock-fronts for the Three-Dimensional Full Euler Equations

TL;DR

This paper proves the local uniqueness of steady spherical transonic shock solutions in three-dimensional Euler equations, advancing understanding of shock phenomena in fluid dynamics through a novel mathematical approach.

Contribution

It introduces a new method for proving uniqueness of spherical transonic shocks in 3D Euler equations, involving a decomposition of the system and analysis of a nonlocal elliptic problem.

Findings

Established local uniqueness of spherical transonic shock solutions

Developed a decomposition method applicable to Riemannian manifolds

Analyzed a Venttsel problem with nonlocal elliptic operators

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.