Prandtl-Meyer reflection for supersonic flow past a solid ramp
Authors:
Myoungjean Bae, Gui-Qiang Chen and Mikhail Feldman
Journal:
Quart. Appl. Math. 71 (2013), 583-600
MSC (2010):
Primary 35M10, 35M12, 35B65, 35L65, 35L70, 35J70, 76H05, 35L67, 35R35; Secondary 35L15, 35L20, 35J67, 76N10, 76L05
DOI:
https://doi.org/10.1090/S0033-569X-2013-01335-2
Published electronically:
May 22, 2013
MathSciNet review:
3112830
Full-text PDF Free Access
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Additional Information
Abstract: We present our recent results on the Prandtl-Meyer reflection for supersonic potential flow past a solid ramp. When a steady supersonic flow passes a solid ramp, there are two possible configurations: the weak shock solution and the strong shock solution. Elling-Liu’s theorem (2008) indicates that the steady supersonic weak shock solution can be regarded as a long-time asymptotic state of an unsteady flow for a class of physical parameters determined by certain assumptions for potential flow. In this paper we discuss our recent progress in removing these assumptions and establishing the stability theorem for steady supersonic weak shock solutions as the long-time asymptotics of unsteady flows for all the physical parameters for potential flow. We apply new mathematical techniques developed in our recent work to obtain monotonicity properties and uniform a priori estimates for weak solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory for the general case.
References
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References
- Bae, M., Chen, G.-Q. and Feldman, M., Regularity of solutions to regular shock reflection for potential flow, Invent. Math. 175 (2009), 505–543. MR 2471595 (2010d:35247)
- Bae, M., Chen, G.-Q. and Feldman, M., Global solutions to the Prandtl-Meyer reflection for supersonic potential flow impinging onto a solid wedge, Preprint, September 2012.
- Busemann, A. Gasdynamik, Handbuch der Experimentalphysik, Vol. IV, Akademische Verlagsgesellschaft, Leipzig, 1931.
- Chen, G.-Q. and Feldman, M., Potential theory for shock reflection by a large-angle wedge, Proc. Nat. Acad. Sci. U.S.A. 102 (2005), 15368–15372. MR 2188921 (2006f:35214)
- Chen, G.-Q. and Feldman, M., Global solutions to shock reflection by large-angle wedges for potential flow, Ann. Math. 171 (2010), 1019–1134. MR 2630061 (2011g:35248)
- Chen, G.-Q. and Feldman, M., Mathematics of Shock Reflection-Diffraction, von Neumann’s Conjectures, and Related Analysis, Monograph, Oxford 2012.
- Chen, G.-Q. and Li, T.-H., Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge, J. Diff. Eqs. 244 (2008), 1521–1550. MR 2396508 (2009b:35264)
- Chen, G.-Q., Zhang, Y., and Zhu, D., Existence and stability of supersonic Euler flows past Lipschitz wedges, Arch. Ration. Mech. Anal. 181 (2006), 261–310. MR 2221208 (2007a:76099)
- Chen, S. and Fang, B., Stability of transonic shocks in supersonic flow past a wedge, J. Diff. Eqs. 233 (2007), 105–135. MR 2290273 (2008i:35158)
- Courant, R. and Friedrichs, K. O., Supersonic Flow and Shock Waves, Springer-Verlag: New York, 1948. MR 0029615 (10:637c)
- Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics, Third edition, Springer-Verlag: Berlin, 2010. MR 2574377 (2011i:35150)
- Elling, V. and Liu, T.-P., The ellipticity principle for self-similar potential flows, J. Hyperbolic Differ. Equ. 2 (2005), 909–917. MR 2195986 (2006j:35193)
- Elling, V and Liu, T.-P., Supersonic flow onto a solid wedge, Comm. Pure Appl. Math. 61 (2008), 1347–1448. MR 2436185 (2010g:76102)
- Lieberman, G. M., Hölder continuity of the gradient at a corner for the capillary problem and related result, Pacific J. Math. 133 (1988), 115–135. MR 936359 (89h:35050)
- Meyer, Th., Über zweidimensionale Bewegungsvorgänge in einem Gas, das mit Überschallgeschwindigkeit strömt. Dissertation, Göttingen, 1908. Forschungsheft des Vereins deutscher Ingenieure, Vol. 62, Berlin, 1908, pp. 31–67.
- Prandtl, L., Allgemeine Überlegungen über die Strömung zusammendrückbarer Flüssigkeiten, Zeitschrift für angewandte Mathematik und Mechanik 16 (1936), 129–142.
- Serre, D., Von Neumann’s comments about existence and uniqueness for the initial-boundary value problem in gas dynamics, Bull. Amer. Math. Soc. (N.S.) 47 (2010), 139–144. MR 2566448 (2010k:35304)
- Whitham, G. B., Linear and Nonlinear Waves, Wiley-Interscience [John Wiley & Sons]: New York-London-Sydney, 1974. MR 0483954 (58:3905)
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Additional Information
Myoungjean Bae
Affiliation:
Department of Mathematics, POSTECH, San 31, Hyojadong, Namgu, Pohang, Gyungbuk, Korea
MR Author ID:
860452
Email:
mjbae@postech.ac.kr
Gui-Qiang Chen
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, England; School of Mathematical Sciences, Fudan University, Shanghai 200433, China; Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208-2734
MR Author ID:
249262
ORCID:
0000-0001-5146-3839
Email:
chengq@maths.ox.ac.uk
Mikhail Feldman
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
MR Author ID:
226925
Email:
feldman@math.wisc.edu
Keywords:
Prandtl-Meyer reflection,
supersonic flow,
unsteady flow,
steady flow,
solid wedge,
weak shock solution,
strong shock solution,
stability,
self-similar,
transonic shock,
sonic boundary,
free boundary,
existence,
regularity,
elliptic-hyperbolic mixed,
monotonicity,
apriori estimates,
uniform estimates,
separation estimates
Received by editor(s):
December 21, 2011
Published electronically:
May 22, 2013
Dedicated:
Dedicated to Costas Dafermos on the occasion of his 70th birthday
Article copyright:
© Copyright 2013
Brown University
The copyright for this article reverts to public domain 28 years after publication.
