The sonic line as a free boundary
Authors:
Barbara Lee Keyfitz, Allen M. Tesdall, Kevin R. Payne and Nedyu I. Popivanov
Journal:
Quart. Appl. Math. 71 (2013), 119-133
MSC (2010):
Primary 35L65, 35M30, 76H05; Secondary 35R35, 42A38.
DOI:
https://doi.org/10.1090/S0033-569X-2012-01283-6
Published electronically:
October 2, 2012
MathSciNet review:
3075538
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider the steady transonic small disturbance equations on a domain and with data that lead to a solution that depends on a single variable. After writing down the solution, we show that it can also be found by using a hodograph transformation followed by a partial Fourier transform. This motivates considering perturbed problems that can be solved with the same technique. We identify a class of such problems.
References
- Sunčica Čanić, Barbara Lee Keyfitz, and Eun Heui Kim, A free boundary problem for a quasi-linear degenerate elliptic equation: regular reflection of weak shocks, Comm. Pure Appl. Math. 55 (2002), no. 1, 71–92. MR 1857880, DOI https://doi.org/10.1002/cpa.10013
- Sunčica Čanić, Barbara Lee Keyfitz, and Gary M. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. 53 (2000), no. 4, 484–511. MR 1733695, DOI https://doi.org/10.1002/%28SICI%291097-0312%28200004%2953%3A4%3C484%3A%3AAID-CPA3%3E3.3.CO%3B2-B
- Gui-Qiang Chen and Mikhail Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc. 16 (2003), no. 3, 461–494. MR 1969202, DOI https://doi.org/10.1090/S0894-0347-03-00422-3
- Volker Elling and Tai-Ping Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl. Math. 61 (2008), no. 10, 1347–1448. MR 2436185, DOI https://doi.org/10.1002/cpa.20231
- F. G. Friedlander, Introduction to the theory of distributions, Cambridge University Press, Cambridge, 1982. MR 779092
- Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
- E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
- P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427–455. MR 118949, DOI https://doi.org/10.1002/cpa.3160130307
- Cathleen S. Morawetz, A weak solution for a system of equations of elliptic-hyperbolic type, Comm. Pure Appl. Math. 11 (1958), 315–331. MR 96893, DOI https://doi.org/10.1002/cpa.3160110305
- Cathleen Synge Morawetz, Mixed equations and transonic flow, J. Hyperbolic Differ. Equ. 1 (2004), no. 1, 1–26. MR 2052469, DOI https://doi.org/10.1142/S0219891604000081
- Allen M. Tesdall and John K. Hunter, Self-similar solutions for weak shock reflection, SIAM J. Appl. Math. 63 (2002), no. 1, 42–61. MR 1952886, DOI https://doi.org/10.1137/S0036139901383826
- Allen M. Tesdall and Barbara L. Keyfitz, A continuous, two-way free boundary in the unsteady transonic small disturbance equations, J. Hyperbolic Differ. Equ. 7 (2010), no. 2, 317–338. MR 2659739, DOI https://doi.org/10.1142/S0219891610002153
- Allen M. Tesdall, Richard Sanders, and Barbara L. Keyfitz, The triple point paradox for the nonlinear wave system, SIAM J. Appl. Math. 67 (2006/07), no. 2, 321–336. MR 2285865, DOI https://doi.org/10.1137/060660758
- Allen M. Tesdall, Richard Sanders, and Barbara L. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM J. Appl. Math. 68 (2008), no. 5, 1360–1377. MR 2407128, DOI https://doi.org/10.1137/070698567
References
- S. Čanić, B. L. Keyfitz, and E. H. Kim. A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks. Communications on Pure and Applied Mathematics, LV:71–92, 2002. MR 1857880 (2003a:35206)
- S. Čanić, B. L. Keyfitz, and G. M. Lieberman. A proof of existence of perturbed steady transonic shocks via a free boundary problem. Communications on Pure and Applied Mathematics, LIII:484–511, 2000. MR 1733695 (2001m:76056)
- G.-Q. Chen and M. Feldman. Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. Journal of the American Mathematical Society, 16:461–494, 2003. MR 1969202 (2004d:35182)
- V. Elling and T.-P. Liu. Supersonic flow onto a solid wedge. Communications on Pure and Applied Mathematics, 61:1347–1448, 2008. MR 2436185 (2010g:76102)
- F. G. Friedlander. Introduction to the Theory of Distributions. Cambridge University Press, Cambridge, 1982. MR 779092 (86h:46002)
- L. Hörmander. The Analysis of Linear Partial Differential Operators I. Springer-Verlag, New York, 1983. MR 717035 (85g:35002a)
- E. L. Ince. Ordinary Differential Equations. Dover, New York, 1956. MR 0010757 (6:65f)
- P. D. Lax and R. S. Phillips. Local boundary conditions for dissipative symmetric linear differential operators. Communications on Pure and Applied Mathematics, XIII:427–455, 1960. MR 0118949 (22:9718)
- C. S. Morawetz. A weak solution for a system of equations of elliptic-hyperbolic type. Communications on Pure and Applied Mathematics, XI:315–331, 1958. MR 0096893 (20:3375)
- C. S. Morawetz. Mixed equations and transonic flow. Journal of Hyperbolic Differential Equations, 1:1–26, 2004. MR 2052469 (2005f:35259)
- A. M. Tesdall and J. K. Hunter. Self-similar solutions for weak shock reflection. SIAM Journal on Applied Mathematics, 63:42–61, 2002. MR 1952886 (2003i:76062)
- A. M. Tesdall and B. L. Keyfitz. A continuous, two-way free boundary in the unsteady transonic small disturbance equations. Journal of Hyperbolic Differential Equations, 7:317–338, 2010. MR 2659739 (2011g:35272)
- A. M. Tesdall, R. Sanders, and B. L. Keyfitz. The triple point paradox for the nonlinear wave system. SIAM Journal on Applied Mathematics, 67:321–336, 2006. MR 2285865 (2008e:35131)
- A. M. Tesdall, R. Sanders, and B. L. Keyfitz. Self-similar solutions for the triple point paradox in gasdynamics. SIAM Journal on Applied Mathematics, 68:1360–1377, 2008. MR 2407128 (2009e:35176)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35L65,
35M30,
76H05,
35R35,
42A38.
Retrieve articles in all journals
with MSC (2010):
35L65,
35M30,
76H05,
35R35,
42A38.
Additional Information
Barbara Lee Keyfitz
Affiliation:
The Fields Institute, Toronto
Address at time of publication:
Department of Mathematics, The Ohio State University
Email:
bkeyfitz@math.ohio-state.edu
Allen M. Tesdall
Affiliation:
Department of Mathematics, College of Staten Island, City University of New York
Email:
allen.tesdall@csi.cuny.edu
Kevin R. Payne
Affiliation:
Dipartimento di Matematica, Universita di Milano
MR Author ID:
306202
Email:
kevin.payne@unimi.it
Nedyu I. Popivanov
Affiliation:
Department of Mathematics and Informatics, University of Sofia
Email:
nedyu@fmi.uni-sofia.bg
Received by editor(s):
April 1, 2011
Published electronically:
October 2, 2012
Additional Notes:
The first author’s research was supported by NSERC of Canada, NSF and the Department of Energy. This project was started during a visit to Loughborough University, funded by the Maxwell Institute.
The second author’s research was supported by NSERC of Canada, NSF, Research Foundation of CUNY, the Fields Institute, and the Department of Energy.
The third author was supported by MIUR project “Equazioni alle derivate parziali e disuguaglianze funzionali: aspetti geometrici, proprietá qualitative e applicazioni”. Additional support from the Fields Institute is acknowledged.
The fourth author was partially supported by the Bulgarian NSF under Grant DO 02-115/2008 “Centre of Excellence on Supercomputer Applications (SuperCA)”. Support from The Ohio State University is acknowledged.
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.
