Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



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
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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.

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Additional Information

Myoungjean Bae
Affiliation: Department of Mathematics, POSTECH, San 31, Hyojadong, Namgu, Pohang, Gyungbuk, Korea
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
Email: chengq@maths.ox.ac.uk

Mikhail Feldman
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
Email: feldman@math.wisc.edu

DOI: https://doi.org/10.1090/S0033-569X-2013-01335-2
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.

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