Asymptotic stability of transonic steady nozzle flows
Abstract
In the present article, we prove the nonlinear asymptotic stability of stationary solutions to the quasi-one-dimensional barotropic Euler equations in a finite-length nozzle with variable cross-sectional area A(x). We consider all stationary flow configurations arising from subsonic inflow: flows that remain subsonic throughout the nozzle, flows that pass continuously through a sonic state and remain supersonic up to the exit, and flows that subsequently return to the subsonic regime through a stationary shock. We prove that all these stationary solutions are asymptotically stable with respect to sufficiently small perturbations. No smallness restrictions are imposed on the mass flux, the magnitude of Ax, or the prescribed exit pressure, provided that the corresponding stationary solution exists. The only geometrical assumptions are that Axx>0 at a smooth subsonic--supersonic transition, and Ax>0 at a stationary supersonic--subsonic shock. To the best of our knowledge, this gives the first nonlinear asymptotic stability result for a stationary flow passing smoothly from the subsonic to the supersonic regime in a finite-length nozzle. The principal analytical difficulty is the degeneracy of the characteristic structure at the sonic point. We overcome this difficulty by coupling energy structures adapted respectively to the subsonic and supersonic regimes. This coupling provides the dissipation needed to control the smooth sonic transition, and when combined with the dynamics of the shock, yields a unified nonlinear stability framework for smooth and discontinuous transonic nozzle flows.
Related
- Analysis on the steady Euler flows with stagnation points in an infinitely long nozzle
- Stability of steady transonic shocks with constant states in 3D rectangular nozzles under small external forces
- Asymptotic Stability of Steady Prandtl Equations with Suction
- On the singular layer for the 1-D steady Euler system with heat conductivity in a finite nozzle as the heat conduction coefficient vanishes
- Three-dimensional steady transonic shocks for the relativistic Euler equations
BibTeX
@misc{ohnawa2026asymptotic,
title = {{Asymptotic stability of transonic steady nozzle flows}},
author = {Masashi Ohnawa and Masahiro Suzuki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33951},
eprint = {2610.33951},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33951},
}
RIS
TY - UNPB
TI - Asymptotic stability of transonic steady nozzle flows
AU - Masashi Ohnawa
AU - Masahiro Suzuki
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33951
UR - https://arxiv.org/abs/2610.33951
AB - In the present article, we prove the nonlinear asymptotic stability of stationary solutions to the quasi-one-dimensional barotropic Euler equations in a finite-length nozzle with variable cross-sectional area A(x). We consider all stationary flow configurations arising from subsonic inflow: flows that remain subsonic throughout the nozzle, flows that pass continuously through a sonic state and remain supersonic up to the exit, and flows that subsequently return to the subsonic regime through a stationary shock. We prove that all these stationary solutions are asymptotically stable with respect to sufficiently small perturbations. No smallness restrictions are imposed on the mass flux, the magnitude of A_x, or the prescribed exit pressure, provided that the corresponding stationary solution exists. The only geometrical assumptions are that A_xx>0 at a smooth subsonic--supersonic transition, and A_x>0 at a stationary supersonic--subsonic shock. To the best of our knowledge, this gives the first nonlinear asymptotic stability result for a stationary flow passing smoothly from the subsonic to the supersonic regime in a finite-length nozzle. The principal analytical difficulty is the degeneracy of the characteristic structure at the sonic point. We overcome this difficulty by coupling energy structures adapted respectively to the subsonic and supersonic regimes. This coupling provides the dissipation needed to control the smooth sonic transition, and when combined with the dynamics of the shock, yields a unified nonlinear stability framework for smooth and discontinuous transonic nozzle flows.
ER -