含引力的定常 Euler 方程组球对称解的适定性

数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 359-370.

含引力的定常 Euler 方程组球对称解的适定性

王啟明¹(),邓雪梅^2,^*()

¹三峡大学理学院湖北宜昌 443002
²三峡大学数学研究中心湖北宜昌 443002

收稿日期:2024-04-16 修回日期:2024-10-05 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 邓雪梅 E-mail:qiming.wang@ctgu.edu.cn;dxuemei@ctgu.edu.cn
作者简介:王啟明，E-mail:qiming.wang@ctgu.edu.cn
基金资助:
国家自然科学基金(12061080)

The Well-Posedness of Spherically Symmetric Solutions to the Steady Euler Equations with Gravitation

Qiming Wang¹(),Xuemei Deng^2,^*()

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

Received:2024-04-16 Revised:2024-10-05 Online:2025-04-26 Published:2025-04-09
Contact: Xuemei Deng E-mail:qiming.wang@ctgu.edu.cn;dxuemei@ctgu.edu.cn
Supported by:
NSFC(12061080)

摘要/Abstract

摘要：

以带引力项的可压缩 Euler 方程组为模型, 该文研究了三维球对称扩张管道中跨音速激波解的存在唯一性. 假设流体受引力影响充分小, 在管道入口处给定特殊的超音速初值条件, 当管道出口处的压力 $p$ 在某个确定范围内时, 通过证明出口处压力是激波位置的严格单调函数, 从而证明了管道内跨音速激波解的存在唯一性.

关键词: 跨音速激波解, Bernoulli 函数, Euler 方程组, 球对称流, 引力

Abstract:

This paper studies the existence and uniqueness of transonic shock solutions to the steady compressible Euler equations with gravity in a three-dimensional spherically symmetric divergent nozzle. Assuming that the influence of gravity on the fluid is sufficiently small and the supersonic initial conditions are given at the entrance, it can be proved that when the pressure $p$ at the exit falls in certain range, there exists a unique transonic shock solution within the nozzle by demonstrating that the pressure at the outlet is a strictly monotone function of the shock location.

Key words: transonic solution, Bernoulli's function, Euler equations, spherically symmetrical flow, gravity

中图分类号:

O175.2

王啟明,邓雪梅. 含引力的定常 Euler 方程组球对称解的适定性[J]. 数学物理学报, 2025, 45(2): 359-370.

Qiming Wang,Xuemei Deng. 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.

图/表 2

图1

图2

$u$ 曲线图"

图2

参考文献 10

[1]	Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. Heidelberg: Springer-Verlag, 1976
[2]	Chen G Q, Feldman M. Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. Journal of the American Mathematical Society, 2003, 16(3): 461-494
[3]	Chen G Q, Feldman M. Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations. Communications on Pure and Applied Mathematics, 2004, 57(3): 310-356
[4]	Chen G Q, Chen J, Feldman M. Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles. Journal De Mathematiques Pures Et Appliquees, 2007, 88(2): 191-218
[5]	Chen S X. Compressible flow and transonic shock in a diverging nozzle. Communications In Mathematical Physics, 2009, 289(1): 75-106
[6]	李大潜, 秦铁虎. 物理学与偏微分方程. 上册. 北京: 高等教育出版社, 2005
	Li D Q, Qin T H. Physics and Partial Differential Equations. The First Volume. Beijing: Higher Education Press, 2005
[7]	Li J, Xin Z P, Yin H C. On Transonic shocks in a nozzle with variable end pressures. Communications in Mathematical Physics, 2009, 291(1): 111-150
[8]	Yuan H R. A remark on determination of transonic shocks in divergent nozzles for steady compressible Euler flows. Nonlinear Analysis: Real World Application, 2008, 9(2): 316-325
[9]	Yuan H R. On transonic shocks in two-dimensional variable-area ducts for steady Euler system. SIAM Journal on Mathematical Analysis, 2006, 38(4): 1343-1370
[10]	Yuan H R. Persistence of shocks in ducts. Nonlinear Analysis: Theory, Methods & Applications, 2012, 75(9): 3874-3894

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