Structural stability of cylindrical transonic shocks to the three-dimensional Zeldovich-von Neumann-Döring combustion equations
Abstract
We address the structural stability of transonic shock solutions to the Zeldovich-von Neumann-Döring (ZND) combustion equations in a cylindrical domain under three-dimensional perturbations of the incoming supersonic flow and exit pressure. The perturbed transonic shock is completely free and its strength and position are uniquely determined by the incoming flow and the exit pressure. The solution shares the same regularity for the velocity, the pressure, the entropy and the fraction of unburned gas. The main difficulties are that the shock front is a free boundary to be determined simultaneously with the subsonic flow and the equations for all states are coupled with each other due to the combustion process. One of the key ingredients of the analysis is to reformulate the three-dimensional ZND combustion equations into a deformation-curl system and four transport equations in terms of the deformation tensor and vorticity. Another one is an elaborate decomposition of the Rankine-Hugoniot conditions so that the shock front is uniquely determined by an algebraic equation and an unusual second order differential boundary condition is obtained on the shock front for 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.
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BibTeX
@misc{zhang2026structural,
title = {{Structural stability of cylindrical transonic shocks to the three-dimensional Zeldovich-von Neumann-Döring combustion equations}},
author = {Zihao Zhang and Yan Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25614},
eprint = {2610.25614},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25614},
}
RIS
TY - UNPB
TI - Structural stability of cylindrical transonic shocks to the three-dimensional Zeldovich-von Neumann-Döring combustion equations
AU - Zihao Zhang
AU - Yan Zhou
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.25614
UR - https://arxiv.org/abs/2610.25614
AB - We address the structural stability of transonic shock solutions to the Zeldovich-von Neumann-Döring (ZND) combustion equations in a cylindrical domain under three-dimensional perturbations of the incoming supersonic flow and exit pressure. The perturbed transonic shock is completely free and its strength and position are uniquely determined by the incoming flow and the exit pressure. The solution shares the same regularity for the velocity, the pressure, the entropy and the fraction of unburned gas. The main difficulties are that the shock front is a free boundary to be determined simultaneously with the subsonic flow and the equations for all states are coupled with each other due to the combustion process. One of the key ingredients of the analysis is to reformulate the three-dimensional ZND combustion equations into a deformation-curl system and four transport equations in terms of the deformation tensor and vorticity. Another one is an elaborate decomposition of the Rankine-Hugoniot conditions so that the shock front is uniquely determined by an algebraic equation and an unusual second order differential boundary condition is obtained on the shock front for 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.
ER -