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

Published electronically:
October 2, 2012

MathSciNet review:
3075538

Full-text PDF

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.

**1.**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**, 10.1002/cpa.10013**2.**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**, 10.1002/(SICI)1097-0312(200004)53:4<484::AID-CPA3>3.3.CO;2-B**3.**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 (electronic). MR**1969202**, 10.1090/S0894-0347-03-00422-3**4.**Volker Elling and Tai-Ping Liu,*Supersonic flow onto a solid wedge*, Comm. Pure Appl. Math.**61**(2008), no. 10, 1347–1448. MR**2436185**, 10.1002/cpa.20231**5.**F. G. Friedlander,*Introduction to the theory of distributions*, Cambridge University Press, Cambridge, 1982. MR**779092****6.**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****7.**E. L. Ince,*Ordinary Differential Equations*, Dover Publications, New York, 1944. MR**0010757****8.**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**0118949****9.**Cathleen S. Morawetz,*A weak solution for a system of equations of elliptic-hyperbolic type.*, Comm. Pure Appl. Math.**11**(1958), 315–331. MR**0096893****10.**Cathleen Synge Morawetz,*Mixed equations and transonic flow*, J. Hyperbolic Differ. Equ.**1**(2004), no. 1, 1–26. MR**2052469**, 10.1142/S0219891604000081**11.**Allen M. Tesdall and John K. Hunter,*Self-similar solutions for weak shock reflection*, SIAM J. Appl. Math.**63**(2002), no. 1, 42–61 (electronic). MR**1952886**, 10.1137/S0036139901383826**12.**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**, 10.1142/S0219891610002153**13.**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 (electronic). MR**2285865**, 10.1137/060660758**14.**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**, 10.1137/070698567

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

Email:
kevin.payne@unimi.it

**Nedyu I. Popivanov**

Affiliation:
Department of Mathematics and Informatics, University of Sofia

Email:
nedyu@fmi.uni-sofia.bg

DOI:
http://dx.doi.org/10.1090/S0033-569X-2012-01283-6

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.