Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A Liouville-type theorem for subsonic flows around an infinite long ramp

Authors: Yang Hui and Yin Huicheng
Journal: Quart. Appl. Math. 72 (2014), 253-265
MSC (2010): Primary 35L70, 35L65, 35L67, 76N15
DOI: https://doi.org/10.1090/S0033-569X-2014-01352-0
Published electronically: March 14, 2014
MathSciNet review: 3186235
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we focus on the two-dimensional subsonic flow problem for polytropic gases around an infinite long ramp, which is motivated by a description in Section 111 of Courant-Friedrichs' book Supersonic flow and shock waves. The flow is assumed to be steady, isentropic and irrotational; namely, the movement of the flow is described by a second-order steady potential equation. By the complex methods together with some properties on quasi-conformal mappings, we show that a nontrivial subsonic flow around the infinite long ramp does not exist if the flow is uniformly subsonic.

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  • [1] A. Azzam, Smoothness properties of solutions of mixed boundary value problems for elliptic equations in sectionally smooth $ n$-dimensional domains, Ann. Polon. Math. 40 (1981), no. 1, 81-93. MR 645800 (83i:35055)
  • [2] Lipman Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure Appl. Math. 7 (1954), 441-504. MR 0065334 (16,417a)
  • [3] Lipman Bers, Non-linear elliptic equations without non-linear entire solutions, J. Rational Mech. Anal. 3 (1954), 767-787. MR 0067313 (16,707b)
  • [4] B. Bojarski, Subsonic flow of compressible fluid, Arch. Mech. Stos. 18 (1966), 497-520 (English, with Polish and Russian summaries). MR 0207296 (34 #7112)
  • [5] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243-268. MR 0250205 (40 #3445)
  • [6] Shu Xing Chen, Existence of local solution to supersonic flow past a three-dimensional wing, Adv. in Appl. Math. 13 (1992), no. 3, 273-304 (French). MR 1176578 (93h:35156), https://doi.org/10.1016/0196-8858(92)90013-M
  • [7] Tobias H. Colding and William P. Minicozzi II, The Calabi-Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), no. 1, 211-243. MR 2373154 (2008k:53014), https://doi.org/10.4007/annals.2008.167.211
  • [8] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948. MR 0029615 (10,637c)
  • [9] Guang Chang Dong and Biao Ou, Subsonic flows around a body in space, Comm. Partial Differential Equations 18 (1993), no. 1-2, 355-379. MR 1211737 (94b:35224), https://doi.org/10.1080/03605309308820933
  • [10] R. Finn and D. Gilbarg, Asymptotic behavior and uniqueness of plane subsonic flows, Comm. Pure Appl. Math. 10 (1957), 23-63. MR 0086556 (19,203d)
  • [11] Robert Finn and David Gilbarg, Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math. 98 (1957), 265-296. MR 0092912 (19,1179a)
  • [12] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin-New York, 1998.
  • [13] Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682 (87a:58041)
  • [14] Chin-Chuan Lee, A uniqueness theorem for the minimal surface equation on an unbounded domain in $ {\bf R}^2$, Pacific J. Math. 177 (1997), no. 1, 103-107. MR 1444775 (98e:35072), https://doi.org/10.2140/pjm.1997.177.103
  • [15] Dening Li, Analysis on linear stability of oblique shock waves in steady supersonic flow, J. Differential Equations 207 (2004), no. 1, 195-225. MR 2100818 (2005f:35200), https://doi.org/10.1016/j.jde.2004.08.021
  • [16] Jun Li, HuiCheng Yin, and ChunHui Zhou, On the nonexistence of a global nontrivial subsonic solution in a 3D unbounded angular domain, Sci. China Math. 53 (2010), no. 7, 1753-1766. MR 2665511 (2011e:35227), https://doi.org/10.1007/s11425-010-4028-1
  • [17] Da Qian Li, On a free boundary problem, Chinese Ann. Math. 1 (1980), no. 3-4, 351-358 (English, with Chinese summary). MR 619582 (82i:35125)
  • [18] Gary M. Lieberman, Optimal Hölder regularity for mixed boundary value problems, J. Math. Anal. Appl. 143 (1989), no. 2, 572-586. MR 1022556 (90m:35040), https://doi.org/10.1016/0022-247X(89)90061-9
  • [19] Johannes C. C. Nitsche, On new results in the theory of minimal surfaces, Bull. Amer. Math. Soc. 71 (1965), 195-270. MR 0173993 (30 #4200)
  • [20] Z. Rusak, Subsonic flow around the leading edge of a thin aerofoil with a parabolic nose, European J. Appl. Math. 5 (1994), no. 3, 283-311. MR 1303715 (95i:76055), https://doi.org/10.1017/S0956792500001479
  • [21] David G. Schaeffer, Supersonic flow past a nearly straight wedge, Duke Math. J. 43 (1976), no. 3, 637-670. MR 0413736 (54 #1850)
  • [22] Max Shiffman, On the existence of subsonic flows of a compressible fluid, J. Rational Mech. Anal. 1 (1952), 605-652. MR 0051651 (14,510b)
  • [23] James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105. MR 0233295 (38 #1617)
  • [24] Elias M. Stein and Rami Shakarchi, Complex analysis, Princeton Lectures in Analysis, II, Princeton University Press, Princeton, NJ, 2003. MR 1976398 (2004d:30002)
  • [25] Hui Cheng Yin, Global existence of a shock for the supersonic flow past a curved wedge, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 5, 1425-1432. MR 2251401 (2007g:35151), https://doi.org/10.1007/s10114-005-0611-8
  • [26] L. M. Zigangareeva and O. M. Kiselev, Structure of the long-range fields of plane symmetrical subsonic flows, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza 3 (2000), 132-144 (Russian, with Russian summary); English transl., Fluid Dynam. 35 (2000), no. 3, 421-431. MR 1781314 (2001e:76079), https://doi.org/10.1007/BF02697756

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Additional Information

Yang Hui
Affiliation: School of Mathematics and Physics, Anhui University of Technology, Ma’anshan 243002, People’s Republic of China, and Department of Mathematics and IMS, Nanjing University, Nanjing 210093, People’s Republic of China
Email: yanghsj@ahut.edu.cn

Yin Huicheng
Affiliation: Department of Mathematics and IMS, Nanjing University, Nanjing 210093, People’s Republic of China
Email: huicheng@nju.edu.cn

DOI: https://doi.org/10.1090/S0033-569X-2014-01352-0
Keywords: Subsonic flow, potential equation, quasi-conformal, Riemann metric, ramp
Received by editor(s): May 26, 2012
Published electronically: March 14, 2014
Additional Notes: This research was supported by NSFC (No.11025105, No.10931007), the Doctorial Program Foundation of Ministry of Education of China (No.20090091110005), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
Article copyright: © Copyright 2014 Brown University

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