Three dimensional spherical transonic shock in a hemispherical shell
Shangkun Weng
TL;DR
The paper proves the existence and stability of a three-dimensional spherical transonic shock inside a hemispherical shell under small perturbations of the inflow and exit pressure. It introduces spherical projection coordinates to remove coordinate singularities and develops a deformation-curl decomposition that recasts the steady Euler system into a coupled hyperbolic–elliptic framework, with a nonlocal elliptic equation for a velocity potential. The Rankine–Hugoniot conditions are reformulated to yield an oblique boundary condition on a free shock surface, and a fixed-domain transformation leads to Problem TS, solved by a contraction mapping in a carefully constructed function space. The analysis avoids structural structure conditions and demonstrates that the shock front is uniquely determined by the perturbed data, with improved regularity properties, providing rigorous insight into the stability of spherical transonic shocks in diverse nozzle geometries.
Abstract
The existence and stability of a spherical transonic shock in a hemispherical shell under the three dimensional perturbations of the incoming flows and the exit pressure is established without any further restrictions on the background transonic shock solutions. The perturbed transonic shock are completely free and its strength and position are uniquely determined by the incoming flows and the exit pressure. A key issue in the analysis is the ``spherical projection coordinates" (i.e. the composition of the spherical coordinates and the stereographic projection), which provides an appropriate setting for the spherical transonic shock problem in the sense that the transformed equations have a similar structure as the steady Euler equations and do not contain any coordinates singularities. Then we decompose the hyperbolic and elliptic modes in the steady Euler equations in terms of the deformation and vorticity. An elaborate reformulation of the Rankine-Hugoniot conditions yields an unusual second order differential boundary condition on the shock front to the first order nonlocal deformation-curl system, from which an oblique boundary condition can be derived after homogenizing the curl system and introducing the potential function. The analysis of the compatibility conditions at the intersection of the shock front and the shell boundary is crucial for the optimal regularity of all physical quantities.