##
Steady Transonic Shocks and Free Boundary Problems in Infinite Cylinders
for the Euler Equations

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

#### Title:
Steady Transonic Shocks and Free Boundary Problems in Infinite Cylinders
for the Euler Equations

**Abstract**

We establish the existence and stability of multidimensional transonic
shocks (hyperbolic-elliptic shocks) for the Euler equations for steady
compressible potential fluids in infinite cylinders.
The Euler equations, consisting of the conservation law of mass and
the Bernoulli law for velocity, can be written as a second order
nonlinear equation of mixed elliptic-hyperbolic type for the velocity
potential.
The transonic shock problem in an infinite cylinder can be formulated
into the following free boundary problem:
The free boundary is the location of the multidimensional transonic shock
which divides two regions of $C^{1,\alpha}$ flow in the infinite cylinder,
and the equation is hyperbolic in the upstream region where
the $C^{1,\alpha}$ 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 in unbounded
domains.
Our results indicate that there exists a solution of the free
boundary problem such that the equation is always elliptic
in the unbounded downstream region,
the uniform velocity state at infinity in the downstream direction
is uniquely determined by the given hyperbolic phase,
and the free boundary is $C^{1,\alpha}$,
provided that the hyperbolic phase is close in $C^{1,\alpha}$ to a
uniform flow. We further prove that,
if the steady perturbation of the hyperbolic phase
is $C^{2,\alpha}$, the
free boundary is $C^{2,\alpha}$ and stable
under the steady perturbation.
\end{abstract}

This article has appeared in:

* Communications on Pure and Applied Mathematics
*, **Vol. ?**, pages ?? (2004).

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