Stability of steady transonic shocks with constant states in 3D rectangular nozzles under small external forces
Abstract
This paper investigates the stability of steady transonic shocks with constant states in rectangular nozzles under small three dimensional perturbations of the external force and the exit pressure. Mathematically, the problem can be formulated as a nonlinear free boundary problem for a hyperbolic-elliptic coupled PDE system. One of the key difficulties lies in the arbitrary location of the background shock, which may lead to multiple shock solutions for the same external force and exit pressure under perturbations, revealing the instability of the background shock. We reformulate the steady Euler equations in terms of deformation and vorticity to effectively decompose the hyperbolic and elliptic modes. Linearizing the reformulated system and the Rankine-Hugoniot conditions at background transonic shocks, we find that the class of nonzero horizontal external forces helps to uniquely determine the shock location, where the conservation of mass plays a critical role. A delicate iteration is developed to establish the existence and structural stability of transonic shock solutions. Our analysis shows that the stability mechanisms of steady transonic shocks in flat and divergent nozzles are quite different in many aspects.
Related
- On the Well-Posedness of Periodic Strong Transonic Shocks in Divergent Nozzles
- Structural stability of cylindrical transonic shocks to the three-dimensional Zeldovich-von Neumann-Döring combustion equations
- Asymptotic stability of transonic steady nozzle flows
- Wavenumber-explicit stability of three-dimensional Cartesian UPMLs with piecewise-constant profiles for Helmholtz and Maxwell equations
- Rigidity and stability for the Bogovskii constant
BibTeX
@misc{weng2026stability,
title = {{Stability of steady transonic shocks with constant states in 3D rectangular nozzles under small external forces}},
author = {Shangkun Weng and Wengang Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07586},
eprint = {2610.07586},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07586},
}
RIS
TY - UNPB
TI - Stability of steady transonic shocks with constant states in 3D rectangular nozzles under small external forces
AU - Shangkun Weng
AU - Wengang Yang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07586
UR - https://arxiv.org/abs/2610.07586
AB - This paper investigates the stability of steady transonic shocks with constant states in rectangular nozzles under small three dimensional perturbations of the external force and the exit pressure. Mathematically, the problem can be formulated as a nonlinear free boundary problem for a hyperbolic-elliptic coupled PDE system. One of the key difficulties lies in the arbitrary location of the background shock, which may lead to multiple shock solutions for the same external force and exit pressure under perturbations, revealing the instability of the background shock. We reformulate the steady Euler equations in terms of deformation and vorticity to effectively decompose the hyperbolic and elliptic modes. Linearizing the reformulated system and the Rankine-Hugoniot conditions at background transonic shocks, we find that the class of nonzero horizontal external forces helps to uniquely determine the shock location, where the conservation of mass plays a critical role. A delicate iteration is developed to establish the existence and structural stability of transonic shock solutions. Our analysis shows that the stability mechanisms of steady transonic shocks in flat and divergent nozzles are quite different in many aspects.
ER -