自相似欧拉流中径向对称跨音速激波的稳定性

Acta mathematica scientia,Series A ›› 2026, Vol. 46 ›› Issue (3): 1114-1131.

Previous Articles Next Articles

Stability of Radially Symmetric Transonic Shocks in Self-Similar Euler Flows

Ziang Wang¹(), Ling Gao²^,^*(), Xuemei Deng²()

¹ College of Mathematics and Physics, China Three Gorges University, Hubei Yichang 443002
² Mathematics Research Center of China Three Gorges University, Hubei Yichang 443002

Received:2025-12-10 Revised:2026-03-24 Online:2026-06-26 Published:2026-06-16
Contact: Ling Gao E-mail:ziang.wang@ctgu.edu.cn;ring024@126.com;dxuemei@ctgu.edu.cn
Supported by:
HuBei Provincial Natural Science Foundation of China(2025AFB530);Key Laboratory of Nonlinear Analysis and Applications, Ministry of Education(Central China Normal University)

Abstract

Abstract:

In this work, we investigate the stability of transonic shock phenomena with self-similar structure for the compressible, isentropic Euler system in a finite expanding nozzle or an annular domain. Assuming radial symmetry and prescribing small perturbations of the background solution at the inlet as the supersonic initial condition, we prove that a unique transonic shock solution exists for exit pseudo-velocities within a suitable range. Moreover, the shock location is a monotonically increasing function of the pseudo-velocity at the exit.

Key words: Euler equations, self-similar solutions, transonic shocks, stability.

CLC Number:

O175.2

Ziang Wang, Ling Gao, Xuemei Deng. Stability of Radially Symmetric Transonic Shocks in Self-Similar Euler Flows[J].Acta mathematica scientia,Series A, 2026, 46(3): 1114-1131.

Trendmd

Figures/Tables 5

References 9

[1]	Bae M, Feldman M. Transonic shocks in multidimensional divergent nozzles. Arch Ration Mech Anal, 2011, 201 (3): 777-840 doi: 10.1007/s00205-011-0424-0
[2]	Chen G Q, Chen J, Feldman M. Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles. J Math Pures Appl, 2007, 88 (2): 191-218 doi: 10.1016/j.matpur.2007.04.008
[3]	Chen G Q, Feldman M. Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J Amer Math Soc, 2003, 16 (3): 461-494 doi: 10.1090/jams/2003-16-03
[4]	Chen G Q, Feldman M. Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations. Comm Pure Appl Math, 2004, 57 (3): 310-356 doi: 10.1002/cpa.v57:3
[5]	Chen S X. Stability of transonic shock fronts in two-dimensional Euler systems. Trans Amer Math Soc, 2005, 357 (1): 287-308 doi: 10.1090/tran/2005-357-01
[6]	Chen S, Yuan H. Transonic shocks in compressible flow passing a duct for three-dimensional Euler systems. Arch Ration Mech Anal, 2008, 187 (3): 523-556 doi: 10.1007/s00205-007-0079-z
[7]	Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. Volume 21. New York: SpringerScience & Business Media, 1999
[8]	Li J, Xin Z P, Yin H C. On transonic shocks in a nozzle with variable end pressures. Comm Math Phys, 2009, 291 (1): 111-150
[9]	Yuan H R. A remark on determination of transonic shocks in divergent nozzles for steady compressible euler flows. Nonlinear Anal: RWA, 2008, 9 (2): 316-325 doi: 10.1016/j.nonrwa.2006.10.006

[1]	Hao Liu, Yun Wang, Chunjing Xie. Existence of Discretely Self-Similar Solutions to Four-Dimensional Steady Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2026, 46(2): 709-723.
[2]	Tianyi Wang, Zhengyang Yu. Rigidity of Steady Inhomogeneous Incompressible Euler Equations in Two-Dimensional Annular Domains [J]. Acta mathematica scientia,Series A, 2026, 46(2): 819-839.
[3]	Wang Qiming,Deng Xuemei. The Well-Posedness of Spherically Symmetric Solutions to the Steady Euler Equations with Gravitation [J]. Acta mathematica scientia,Series A, 2025, 45(2): 359-370.
[4]	Liu Jia, Bao Xiongxiong. Asymptotic Stability of Pyramidal Traveling Front for Nonlocal Delayed Diffusion Equation [J]. Acta mathematica scientia,Series A, 2025, 45(1): 44-53.
[5]	Zhang Chunguo, Sun Baonan, Fu Yuzhi, Yu Xin. Stabilization of 2-D Mindlin Timoshenko Plate Systems with Local Damping [J]. Acta mathematica scientia,Series A, 2024, 44(4): 946-959.
[6]	Zhang Haiqun. Bounded Rationality and Stability of Weakly Efficient Nash Equilibria for a Class of Population Games [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1311-1320.
[7]	Lin Jie, Wang Tianyi. Well-Posedness and Convergence Rates of Three-Dimensional Incompressible Euler Flows in Axisymmetric Nozzles with Symmetric Body [J]. Acta mathematica scientia,Series A, 2023, 43(1): 219-237.
[8]	Zhiqiang Shao. Concentration and Cavitation in the Pressureless Limit of Euler Equations of Compressible Fluid Flow with Damping and Friction [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1150-1172.
[9]	Qingling Zhang,Ying Ba. The Perturbed Riemann Problem for the Pressureless Euler Equations with a Flocking Dissipation [J]. Acta mathematica scientia,Series A, 2020, 40(1): 49-62.
[10]	Huang Shoujun, Wang Rui. Plane Wave Solutions to the Euler Equations for Chaplygin Gases [J]. Acta mathematica scientia,Series A, 2016, 36(4): 681-689.
[11]	ZHU Xu-Sheng, LI Fang-E. The Global Solutions of the Compressible Euler Equations With Nonlinear Damping in 3-Dimensions [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1111-1122.
[12]	LI Jian-Ye, SHAO Zhi-Qiang. The Riemann Problem with Delta Initial Data for the Zero-Pressure Relativistic Euler Equations Hyperbolic Systems of Conservation Laws [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1083-1092.
[13]	GENG Yong-Cai. Steady State Solutions of Relativistic Euler Equations with Spherical Symmetry [J]. Acta mathematica scientia,Series A, 2014, 34(4): 841-850.
[14]	Xia Wenhua;Deng Feiqi;Luo Yiping. Global Exponential Stability of Neural Networks’ Periodic Solution Involving Finite Distributed Delays with Periodic Inputs [J]. Acta mathematica scientia,Series A, 2009, 29(1): 170-178.
[15]	Zhang Jing;Zhang Luming;Chen Juan. A Numerical Simulation Method of Nonlinear Schrodinger equation [J]. Acta mathematica scientia,Series A, 2007, 27(6): 1111-1117.

Stability of Radially Symmetric Transonic Shocks in Self-Similar Euler Flows

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 5

References 9

Related Articles 15

Metrics

Comments

Recommended 0