Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

Authors:
Gui-Qiang Chen and Mikhail Feldman

Journal:
J. Amer. Math. Soc. **16** (2003), 461-494

MSC (2000):
Primary 35M10, 35J65, 35R35, 76H05, 76L05, 35B45

Published electronically:
January 24, 2003

MathSciNet review:
1969202

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

**1.**H. W. Alt and L. A. Caffarelli,*Existence and regularity for a minimum problem with free boundary*, J. Reine Angew. Math.**325**(1981), 105–144. MR**618549****2.**Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman,*A free boundary problem for quasilinear elliptic equations*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**11**(1984), no. 1, 1–44. MR**752578****3.**Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman,*Compressible flows of jets and cavities*, J. Differential Equations**56**(1985), no. 1, 82–141. MR**772122**, 10.1016/0022-0396(85)90101-9**4.**Thierry Aubin,*Some nonlinear problems in Riemannian geometry*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR**1636569****5.**Lipman Bers,*Existence and uniqueness of a subsonic flow past a given profile*, Comm. Pure Appl. Math.**7**(1954), 441–504. MR**0065334****6.**Sergio Campanato,*A bound for the solutions of a basic elliptic system with nonlinearity 𝑞≥2*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**80**(1986), no. 3, 81–88 (1987) (English, with Italian summary). MR**976693**

Luis A. Caffarelli,*A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz*, Comm. Pure Appl. Math.**42**(1989), no. 1, 55–78. MR**973745**, 10.1002/cpa.3160420105

Luis A. Caffarelli,*A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on 𝑋*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**15**(1988), no. 4, 583–602 (1989). MR**1029856****7.**Luis A. Caffarelli and Xavier Cabré,*Fully nonlinear elliptic equations*, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR**1351007****8.**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**9.**G.-Q. Chen and M. Feldman,

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations,

Preprint, Northwestern University, May 2002;

Transonic shocks and one-phase free boundary problems for nonlinear equations of mixed elliptic-hyperbolic type in unbounded domains,

Preprint, Northwestern University, November 2002.**10.**Gui-Qiang Chen and James Glimm,*Global solutions to the compressible Euler equations with geometrical structure*, Comm. Math. Phys.**180**(1996), no. 1, 153–193. MR**1403862****11.**Shuxing Chen,*Existence of stationary supersonic flows past a pointed body*, Arch. Ration. Mech. Anal.**156**(2001), no. 2, 141–181. MR**1814974**, 10.1007/s002050100121**12.**Shuxing Chen,*Asymptotic behaviour of supersonic flow past a convex combined wedge*, Chinese Ann. Math. Ser. B**19**(1998), no. 3, 255–264. A Chinese summary appears in Chinese Ann. Math. Ser. A 19 (1998), no. 4, 533. MR**1667344****13.**R. Courant and K. O. Friedrichs,*Supersonic Flow and Shock Waves*, Interscience Publishers, Inc., New York, N. Y., 1948. MR**0029615****14.**C. M. Dafermos,*Hyperbolic Conservation Laws in Continuum Physics*, Springer-Verlag: Berlin, 2000. MR**20001m:35212****15.**Guang Chang Dong,*Nonlinear partial differential equations of second order*, Translations of Mathematical Monographs, vol. 95, American Mathematical Society, Providence, RI, 1991. Translated from the Chinese by Kai Seng Chou [Kaising Tso]. MR**1134129****16.**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****17.**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****18.**James Glimm and Andrew J. Majda (eds.),*Multidimensional hyperbolic problems and computations*, The IMA Volumes in Mathematics and its Applications, vol. 29, Springer-Verlag, New York, 1991. Papers from the IMA Workshop held in Minneapolis, Minnesota, April 3–14, 1989. MR**1087068****19.**C.-H. Gu,

A method for solving the supersonic flow past a curved wedge (in Chinese),*Fudan J.***7**(1962), 11-14.**20.**Carlos E. Kenig,*Harmonic analysis techniques for second order elliptic boundary value problems*, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR**1282720****21.**Peter D. Lax,*Hyperbolic systems of conservation laws and the mathematical theory of shock waves*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR**0350216****22.**Da Qian Li,*On a free boundary problem*, Chinese Ann. Math.**1**(1980), no. 3-4, 351–358 (English, with Chinese summary). MR**619582****23.**Gary M. Lieberman,*Regularity of solutions of nonlinear elliptic boundary value problems*, J. Reine Angew. Math.**369**(1986), 1–13. MR**850625**, 10.1515/crll.1986.369.1**24.**Gary M. Lieberman and Neil S. Trudinger,*Nonlinear oblique boundary value problems for nonlinear elliptic equations*, Trans. Amer. Math. Soc.**295**(1986), no. 2, 509–546. MR**833695**, 10.1090/S0002-9947-1986-0833695-6**25.**V. I. Ivanov,*A mathematical model of the Stokes soliton*, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza**1**(1999), 174–180 (Russian, with Russian summary); English transl., Fluid Dynam.**34**(1999), no. 1, 147–152. MR**1697990**, 10.1007/BF02698766**26.**Andrew Majda,*The stability of multidimensional shock fronts*, Mem. Amer. Math. Soc.**41**(1983), no. 275, iv+95. MR**683422**, 10.1090/memo/0275

Andrew Majda,*The existence of multidimensional shock fronts*, Mem. Amer. Math. Soc.**43**(1983), no. 281, v+93. MR**699241**, 10.1090/memo/0281**27.**Cathleen S. Morawetz,*On the non-existence of continuous transonic flows past profiles. I*, Comm. Pure Appl. Math.**9**(1956), 45–68. MR**0078130**

Cathleen S. Morawetz,*On the non-existence of continuous transonic flows past profiles. II*, Comm. Pure Appl. Math.**10**(1957), 107–131. MR**0088253**

Cathleen S. Morawetz,*On the non-existence of continuous transonic flows past profiles. III.*, Comm. Pure Appl. Math.**11**(1958), 129–144. MR**0096478****28.**Cathleen S. Morawetz,*Potential theory for regular and Mach reflection of a shock at a wedge*, Comm. Pure Appl. Math.**47**(1994), no. 5, 593–624. MR**1278346**, 10.1002/cpa.3160470502**29.**David G. Schaeffer,*Supersonic flow past a nearly straight wedge*, Duke Math. J.**43**(1976), no. 3, 637–670. MR**0413736****30.**Max Shiffman,*On the existence of subsonic flows of a compressible fluid*, J. Rational Mech. Anal.**1**(1952), 605–652. MR**0051651****31.**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****32.**Yongqian Zhang,*Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary*, SIAM J. Math. Anal.**31**(1999), no. 1, 166–183 (electronic). MR**1742300**, 10.1137/S0036141097331056

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
35M10,
35J65,
35R35,
76H05,
76L05,
35B45

Retrieve articles in all journals with MSC (2000): 35M10, 35J65, 35R35, 76H05, 76L05, 35B45

Additional Information

**Gui-Qiang Chen**

Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730

Email:
gqchen@math.northwestern.edu

**Mikhail Feldman**

Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Email:
feldman@math.wisc.edu

DOI:
http://dx.doi.org/10.1090/S0894-0347-03-00422-3

Keywords:
Transonic shocks,
free boundary problems,
elliptic-hyperbolic,
nonlinear equations,
second-order,
mixed type,
existence,
uniqueness,
stability,
entropy solutions,
Euler equations,
compressible flow

Received by editor(s):
October 17, 2001

Published electronically:
January 24, 2003

Article copyright:
© Copyright 2003
American Mathematical Society