Nonlinear stability of spherical self-similar flows to the compressible Euler equations
Authors:
Seung-Yeal Ha, Hsiu-Chuan Huang and Wen-Ching Lien
Journal:
Quart. Appl. Math. 72 (2014), 109-136
MSC (2010):
Primary 35LXX, 76NXX
DOI:
https://doi.org/10.1090/S0033-569X-2013-01329-9
Published electronically:
December 31, 2013
MathSciNet review:
3185135
Full-text PDF Free Access
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Abstract: We present the nonlinear stability of spherical self-similar flows arising from the uniform expansion of a spherical piston toward still gas. If the perturbation of the expansion speed of the piston is sufficiently small compared with the strength of the leading shock, a global weak solution of the isentropic compressible Euler system exists in the region between the spherical piston and the leading shock under the structural condition on the shock Mach number and the nondimensional piston speed. Moreover, we show that the perturbed flow tends to the corresponding self-similar flow time-asymptotically. Our analysis is based on the modified Glimm scheme.
References
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References
- I-Liang Chern, Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities, Comm. Pure Appl. Math. 42 (1989), no. 6, 815โ844. MR 1003436 (90k:65157), DOI https://doi.org/10.1002/cpa.3160420606
- Gui-Qiang Chen and James Glimm, Global solutions to the compressible Euler equations with geometrical structure, Comm. Math. Phys. 180 (1996), no. 1, 153โ193. MR 1403862 (97j:35120)
- Gui-Qiang Chen, Yongqian Zhang, and Dianwen Zhu, Existence and stability of supersonic Euler flows past Lipschitz wedges, Arch. Ration. Mech. Anal. 181 (2006), no. 2, 261โ310. MR 2221208 (2007a:76099), DOI https://doi.org/10.1007/s00205-005-0412-3
- R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948. MR 0029615 (10,637c)
- James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697โ715. MR 0194770 (33 \#2976)
- James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767 (42 \#676)
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- Seung-Yeal Ha and Tong Yang, $L^1$ stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (2003), no. 5, 1226โ1251 (electronic). MR 2001667 (2004i:35216), DOI https://doi.org/10.1137/S0036141001397983
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537โ566. MR 0093653 (20 \#176)
- Wen-Ching Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (1999), no. 9, 1075โ1098. MR 1692156 (2000c:35153), DOI https://doi.org/10.1002/%28SICI%291097-0312%28199909%2952%3A9%24%5Clangle%241075%3A%3AAID-CPA2%24%5Crangle%243.3.CO%3B2-W
- Wen-Ching Lien and Tai-Ping Liu, Nonlinear stability of a self-similar 3-dimensional gas flow, Comm. Math. Phys. 204 (1999), no. 3, 525โ549. MR 1707631 (2000f:76106), DOI https://doi.org/10.1007/s002200050656
- M. J. Lighthill, The position of the shock-wave in certain aerodynamic problems, Quart. J. Mech. Appl. Math. 1 (1948), 309โ318. MR 0028167 (10,413f)
- Tai Ping Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), no. 2, 135โ148. MR 0470508 (57 \#10259)
- Tai Ping Liu, Quasilinear hyperbolic systems, Comm. Math. Phys. 68 (1979), no. 2, 141โ172. MR 543196 (81b:35051)
- Tai-Ping Liu, Hyperbolic and viscous conservation laws, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 72, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. MR 1737025 (2001h:35119)
- Tetu Makino, Kiyoshi Mizohata, and Seiji Ukai, The global weak solutions of compressible Euler equation with spherical symmetry, Japan J. Indust. Appl. Math. 9 (1992), no. 3, 431โ449. MR 1189949 (93k:35205), DOI https://doi.org/10.1007/BF03167276
- Chen-Chang Peng and Wen-Ching Lien, Self-similar solutions of the Euler equations with spherical symmetry, Nonlinear Anal. 75 (2012), no. 17, 6370โ6378. MR 2959812, DOI https://doi.org/10.1016/j.na.2012.07.019
- P. L. Sachdev, Shock waves and explosions, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 132, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2068608 (2006e:76086)
- P. L. Sachdev, K. T. Joseph, and M. Ejanul Haque, Exact solutions of compressible flow equations with spherical symmetry, Stud. Appl. Math. 114 (2005), no. 4, 325โ342. MR 2131550 (2006c:76099), DOI https://doi.org/10.1111/j.0022-2526.2005.01552.x
- M. Slemrod, Resolution of the spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 6, 1309โ1340. MR 1424228 (97k:35211), DOI https://doi.org/10.1017/S0308210500023428
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York, 1983. MR 688146 (84d:35002)
- G. I. Taylor, The air wave surrounding an expanding sphere, Proc. Roy. Soc. London. Ser. A. 186 (1946), 273โ292. MR 0018519 (8,296f)
- Tong Yang, A functional integral approach to shock wave solutions of Euler equations with spherical symmetry, Comm. Math. Phys. 171 (1995), no. 3, 607โ638. MR 1346174 (96h:35158)
- Tong Yang, A functional integral approach to shock wave solutions of the Euler equations with spherical symmetry. II, J. Differential Equations 130 (1996), no. 1, 162โ178. MR 1409028 (97h:35184), DOI https://doi.org/10.1006/jdeq.1996.0137
- Tong Yang, Euler equations with spherical symmetry and an outing [outgoing] absorbing boundary, Comm. Partial Differential Equations 24 (1999), no. 1-2, 1โ23. MR 1671981 (2000j:35185), DOI https://doi.org/10.1080/03605309908821416
- G. B. Whitham, The propagation of weak spherical shocks in stars, Comm. Pure. Appl. Math. 6 (1953), 397โ414. MR 0060940 (15,751b)
- Tung Chang and Ling Hsiao, The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 41, Longman Scientific & Technical, Harlow, 1989. MR 994414 (90m:35122)
- Tong Zhang and Yuxi Zheng, Axisymmetric solutions of the Euler equations for polytropic gases, Arch. Rational Mech. Anal. 142 (1998), no. 3, 253โ279. MR 1636533 (2000d:76103), DOI https://doi.org/10.1007/s002050050092
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Hsiu-Chuan Huang
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan City 70101, Taiwan
Email:
hchuan@mail.ncku.edu.tw
Wen-Ching Lien
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan City 70101, Taiwan
Email:
wlien@mail.ncku.edu.tw
Keywords:
Euler equations,
shock waves,
self-similarity,
Glimm scheme
Received by editor(s):
April 1, 2012
Published electronically:
December 31, 2013
Article copyright:
© Copyright 2013
Brown University
The copyright for this article reverts to public domain 28 years after publication.