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

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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.

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Additional Information

**Gui-Qiang Chen**

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

MR Author ID:
249262

ORCID:
0000-0001-5146-3839

Email:
gqchen@math.northwestern.edu

**Mikhail Feldman**

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

MR Author ID:
226925

Email:
feldman@math.wisc.edu

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