Nonexistence of global weak solution with only one stable supersonic conic shock wave for the steady supersonic Euler flow past a perturbed cone
Authors:
Xu Gang and Yin Huicheng
Journal:
Quart. Appl. Math. 70 (2012), 199-218
MSC (2010):
Primary 35L70, 35L65, 35L67, 76N15
DOI:
https://doi.org/10.1090/S0033-569X-2012-01279-8
Published electronically:
February 2, 2012
MathSciNet review:
2953100
Full-text PDF Free Access
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Abstract: Recently, for the potential equation, a global stable weak solution with only one conic shock wave has been established in some references. However, in contrast to the case of the potential equation, due to the essential influence of the rotations for the Euler flow, in this paper we will show that the global weak solution of the Euler system with one stable supersonic conic shock wave does not exist when a uniform supersonic incoming flow hits an infinitely long and curved sharp conic body.
References
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References
- S. Alinhac, Temps de vie des solutions régulières des équations d’Euler compressibles axisymétriques en dimension deux, Invent. Math. 111, no. 3, 627-670, 1993. MR 1202138 (94a:35089)
- Chen Shuxing, Li Dening, Conical shock waves for an isentropic Euler system, Proc. Roy. Soc. Edinburgh Sect. A 135, no. 6, 1109-1127, 2005. MR 2191891 (2006i:35243)
- Chen Shuxing, Xin Zhouping, Yin Huicheng, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys., 228, 47-84, 2002. MR 1911248 (2003c:76076)
- R.Courant, K.O.Friedrichs, Supersonic flow and shock waves, Interscience Publishers Inc., New York, 1948. MR 0029615 (10:637c)
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- P. Godin, The lifespan of a class of smooth spherically symmetric solutions of the compressible Euler equations with variable entropy in three space dimensions, Arch. Ration. Mech. Anal. 177, no. 3, 479-511, 2005. MR 2187620 (2006i:76092)
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- Xu Gang, Yin Huicheng, Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone, Nagoya J. Math., Vol. 199, 151-181, 2010. MR 2732336
- Yin Huicheng, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math. J. 175, 125-164, 2004. MR 2085314 (2005f:35203)
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Additional Information
Xu Gang
Affiliation:
Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China
Email:
gxu@ujs.edu.cn
Yin Huicheng
Affiliation:
Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China
Email:
huicheng@nju.edu.cn
Keywords:
Supersonic flow,
conic shock,
full Euler system,
stream line,
nonexistence.
Received by editor(s):
February 12, 2010
Published electronically:
February 2, 2012
Additional Notes:
This project is supported by the National Natural Science Foundation of China (Nos.10931007, 10871096, 11025105), the Doctorial Program Foundation of Ministry of Education of China (No.20090091110005) and Natural Science Foundation of Jiangsu province (10KJB110002).
Article copyright:
© Copyright 2012
Brown University
