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Multidimensional Transonic Shocks and Free Boundary Problems for
Nonlinear Equations of Mixed Type

#### Author: Gui-Qiang Chen and Mikhail Feldman

#### Title:
Multidimensional Transonic Shocks and Free Boundary Problems for
Nonlinear Equations of Mixed Type

**Abstract**

We establish the existence and stability of multidimensional
transonic shocks for the Euler equations for steady potential fluids.
The Euler equations, consisting of the conservation law of mass and
the Bernoulli law for the velocity, can be written into a
second-order, nonlinear equation
of mixed elliptic-hyperbolic type
for the velocity potential.
The transonic shock problem can be formulated as
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.

This article has appeared in:

* Journal of American Mathematical Society (JAMS)
*, **Vol. 16**, pages 461-494 (2003).

This paper is available in the following formats:

A closely related paper is Change me.

**Author Address**
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu