含引力的定常 Euler 方程组球对称解的适定性

Abstract

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

CLC Number:

O175.2

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.

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

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