Gravitational effect on transonic shocks in finite divergent nozzle

Peikang Wang; Xuemei Deng; Min Wang; Peikang Wang; Xuemei Deng; Min Wang

doi:10.3934/era.2026225

Electronic Research Archive

2026, Volume 34, Issue 7: 5087-5101. doi: 10.3934/era.2026225

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Research article

Gravitational effect on transonic shocks in finite divergent nozzle

1.
College of Science, China Three Gorges University, Yichang 443002, China
2.
Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China

Received: 22 April 2026 Revised: 30 May 2026 Accepted: 05 June 2026 Published: 11 June 2026

In this paper, we adopted the steady compressible Euler system with a gravitational term as the governing equations, and focused on the effect of gravity on the transonic shock position in a three-dimensional spherically symmetric divergent nozzle. For a fixed supersonic inflow at the nozzle inlet, we proved that if the outlet pressure fell within an appropriate interval and the gravitational parameter $ K $ was sufficiently small, the shock position was uniquely determined by $ K $ and the outlet pressure, and the shock position was a strictly decreasing and continuously differentiable function of $ K $.
- steady Euler equations,
- transonic shock wave,
- spherically symmetric solutions,
- gravitational effect,
- implicit function differentiability theorem
Citation: Peikang Wang, Xuemei Deng, Min Wang. Gravitational effect on transonic shocks in finite divergent nozzle[J]. Electronic Research Archive, 2026, 34(7): 5087-5101. doi: 10.3934/era.2026225

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Abstract

In this paper, we adopted the steady compressible Euler system with a gravitational term as the governing equations, and focused on the effect of gravity on the transonic shock position in a three-dimensional spherically symmetric divergent nozzle. For a fixed supersonic inflow at the nozzle inlet, we proved that if the outlet pressure fell within an appropriate interval and the gravitational parameter $ K $ was sufficiently small, the shock position was uniquely determined by $ K $ and the outlet pressure, and the shock position was a strictly decreasing and continuously differentiable function of $ K $.

References

[1]	G. Q. Chen, M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Am. Math. Soc., 16 (2003), 461–494. https://doi.org/10.1090/S0894-0347-03-00422-3 doi: 10.1090/S0894-0347-03-00422-3
[2]	S. X. Chen, Stability of transonic shock front in two-dimensional Euler system, Trans. Am. Math. Soc., 357 (2005), 287–308. https://doi.org/10.1090/S0002-9947-04-03698-0 doi: 10.1090/S0002-9947-04-03698-0
[3]	S. X. Chen, H. R. Yuan, Transonic shocks in compressible flow passing a duct for three-dimensional Euler systems, Arch. Ration. Mech. Anal., 187 (2008), 523–556. https://doi.org/10.1007/s00205-007-0079-z doi: 10.1007/s00205-007-0079-z
[4]	H. R. Yuan, A remark on determination of transonic shocks in divergent nozzles for steady compressible Euler flows, Nonlinear Anal-Real., 9 (2008), 316–325. https://doi.org/10.1016/j.nonrwa.2006.10.006 doi: 10.1016/j.nonrwa.2006.10.006
[5]	Z. P. Xin, H. C. Yin, Transonic shock in a nozzle Ⅰ: two-dimensional case, Commun. Pure Appl. Math., 58 (2005), 999–1050. https://doi.org/10.1002/cpa.20025 doi: 10.1002/cpa.20025
[6]	R. Courant, K. O. Friedrichs, Supersonic Flow and Shock Fronts, Springer-Verlag, Heidelberg, 1976.
[7]	J. Li, Z. P. Xin, H. C. Yin, On transonic shocks in a nozzle with variable end pressures, Commun. Math. Phys., 291 (2009), 111–150. https://doi.org/10.1007/s00220-009-0870-9 doi: 10.1007/s00220-009-0870-9
[8]	M. Bae, G. Q. Chen, M. Feldman, Regularity of solutions to regular shock reflection for potential flow, Invent. Math., 175 (2009), 505–543. https://doi.org/10.1007/s00222-008-0156-4 doi: 10.1007/s00222-008-0156-4
[9]	M. Bae, M. Feldman, Transonic shock in multidimensional divergent nozzles, Arch. Ration. Mech. Anal., 201 (2011), 777–840. https://doi.org/10.1007/s00205-011-0424-0 doi: 10.1007/s00205-011-0424-0
[10]	B. X. Fang, W. Xiang, The uniqueness of transonic shocks in supersonic flow past a 2-D wedge, J. Math. Anal. Appl., 437 (2016), 194–213. https://doi.org/10.1016/j.jmaa.2015.11.067 doi: 10.1016/j.jmaa.2015.11.067
[11]	F. Xie, C. P. Wang, Transonic shock wave in an infinite nozzle asymptotically converging to a cylinder, J. Differ. Equations, 242 (2007), 86–120. https://doi.org/10.1016/j.jde.2007.06.015 doi: 10.1016/j.jde.2007.06.015
[12]	Q. M. Wang, X. M. Deng, The well-posedness of spherically symmetric solutions to the steady Euler equations with gravitation, Acta Math. Sci., 45 (2025), 359–370. https://doi.org/10.3969/j.issn.1003-3998.2025.02.005 doi: 10.3969/j.issn.1003-3998.2025.02.005
[13]	Z. Q. Wu, Y. Li, S. Y. Shi, Ordinary Differential Equations, 2$^{nd}$ edition, Higher Education Press, Beijing, 2023.

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