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

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

Published electronically:
January 24, 2003

MathSciNet review:
1969202

Full-text PDF

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**83a:49011****2.**H. W. Alt, L. A. Caffarelli, and A. Friedman,

A free boundary problem for quasilinear elliptic equations,*Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)*,**11**(1984), 1-44. MR**86c:49003****3.**H. W. Alt, L. A. Caffarelli, and A. Friedman,

Compressible flows of jets and cavities,*J. Diff. Eqs.***56**(1985), 82-141. MR**86i:35036****4.**T. Aubin,*Some Nonlinear Problems in Riemannian Geometry*, Springer-Verlag: Heidelberg, 1998. MR**99i:58001****5.**L. Bers,

Existence and uniqueness of subsonic flows past a given profile,*Comm. Pure Appl. Math.***7**(1954), 441-504. MR**16:417a****6.**L. A. Caffarelli,

A Harnack inequality approach to the regularity of free boundaries, I. Lipschitz free boundaries are ,*Rev. Mat. Iberoamericana*,**3**(1987), 139-162; II. Flat free boundaries are Lipschitz,*Comm. Pure Appl. Math.***42**(1989), 55-78; III. Existence theory, compactness, and dependence on ,*Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)*,**15**(1989), 583-602. MR**90d:35036**, MR**90b:35246**, MR**91a:35170****7.**L. A. Caffarelli and X. Cabre,*Fully Nonlinear Elliptic Equations*,

Colloquium Publications,**43**, AMS: Providence, RI, 1995. MR**96h:35046****8.**S. Canic, B. L. Keyfitz, and G. Lieberman,

A proof of existence of perturbed steady transonic shocks via a free boundary problem,*Comm. Pure Appl. Math.***53**(2000), 484-511. MR**2001m:76056****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.**G.-Q. Chen and J. Glimm,

Global solutions to the compressible Euler equations with geometrical structure,

Commun. Math. Phys.**180**(1996), 153-193. MR**97j:35120****11.**S.-X. Chen,

Existence of stationary supersonic flows past a point body,*Arch. Rational Mech. Anal.***156**(2001), 141-181. MR**2001m:76058****12.**S.-X. Chen,

Asymptotic behavior of supersonic flow past a convex combined wedge,*Chinese Ann. Math.***19B**(1998), 255-264. MR**2000b:76058****13.**R. Courant and K. O. Friedrichs,*Supersonic Flow and Shock Waves*, Springer-Verlag: New York, 1948. MR**10:637c****14.**C. M. Dafermos,*Hyperbolic Conservation Laws in Continuum Physics*, Springer-Verlag: Berlin, 2000. MR**20001m:35212****15.**G. Dong,*Nonlinear Partial Differential Equations of Second Order*, Transl. Math. Monographs,**95**, AMS: Providence, RI, 1991. MR**93f:35002****16.**R. Finn and D. Gilbarg,

Three-dimensional subsonic flows and asymptotic estimates for elliptic partial differential equations,*Acta Math.***98**(1957), 265-296. MR**19:1179a****17.**D. Gilbarg and N. Trudinger,*Elliptic Partial Differential Equations of Second Order,*

2nd Ed., Springer-Verlag: Berlin, 1983. MR**86c:35035****18.**J. Glimm and A. Majda,*Multidimensional Hyperbolic Problems and Computations*, Springer-Verlag: New York, 1991. MR**91h:00031****19.**C.-H. Gu,

A method for solving the supersonic flow past a curved wedge (in Chinese),*Fudan J.***7**(1962), 11-14.**20.**C. Kenig,*Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems*, CBMS-RCSM,**83**, Amer. Math. Soc.: Providence, RI, 1994. MR**96a:35040****21.**P. D. Lax,*Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves*, CBMS-RCSM, SIAM: Philadelphia, 1973. MR**50:2709****22.**T.-T. Li,

On a free boundary problem,*Chinese Ann. Math.***1**(1980), 351-358. MR**82i:35125****23.**G. Lieberman,

Regularity of solutions of nonlinear elliptic boundary value problems,*J. Reine Angew. Math.***369**(1986), 1-13. MR**87m:35094****24.**G. Lieberman and N. Trudinger,

Nonlinear oblique boundary value problems for nonlinear elliptic equations,*Trans. Amer. Math. Soc.***295**(1986), 509-546. MR**87h:35114****25.**W.-C. Lien and T.-P. Liu,

Nonlinear stability of a self-similar 3-dimensional gas flow,*Commun. Math. Phys.***204**(1999), 525-549. MR**2000f:76016****26.**A. Majda,

The stability of multidimensional shock fronts,*Mem. Amer. Math. Soc.***275**;

The existence of multidimensional shock fronts,*Mem. Amer. Math. Soc.***281**, AMS: Providence, 1983. MR**84e:35100**, MR**85f:35139****27.**C. S. Morawetz,

On the nonexistence of continuous transonic flows past profiles I,II,III,*Comm. Pure Appl. Math.***9**(1956), 45-68;**10**(1957), 107-131;**11**(1958), 129-144. MR**17:1149d**, MR**19:490e**, MR**20:2961****28.**C. S. Morawetz,

Potential theory for regular and Mach reflection of a shock at a wedge,*Comm. Pure Appl. Math.***47**(1994), 593-624. MR**95g:76030****29.**D. G. Schaeffer,

Supersonic flow past a nearly straight wedge,*Duke Math. J.***43**(1976), 637-670. MR**54:1850****30.**M. Shiffman,

On the existence of subsonic flows of a compressible fluid,*J. Rational Mech. Anal.***1**(1952), 605-652. MR**14:510b****31.**E. M. Stein,

*Singular Integrals and Differentiability Properties of Functions,*

Princeton Univ. Press: Princeton, NJ, 1970. MR**44:7280****32.**Y. Zhang,

Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary,*SIAM J. Math. Anal.***31**(1999), 166-183. MR**2000j:76125**

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:
https://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