##
Existence and Stability of Multidimensional Transonic Flows through
an Infinite Nozzle of Arbitrary Cross-Sections

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

#### Title:
Existence and Stability of Multidimensional Transonic Flows through
an Infinite Nozzle of Arbitrary Cross-Sections

**Abstract**

We establish the existence and stability of multidimensional steady
transonic flows through an infinite nozzle with arbitrary
cross-sections, including the slowly varying de Lavel nozzles.
The transonic flow is governed by the inviscid potential flow equation
with supersonic upstream at the entrance, uniform subsonic downstream
at infinity, and the slip boundary condition on the nozzle boundary.
The multidimensional transonic nozzle problem is reformulated into
a free boundary problem, for which the free boundary is a transonic
shock dividing two regions of $C^{1,\alpha}$ flow in the infinite
nozzle, and the equation is hyperbolic in the upstream supersonic
region and elliptic in the downstream subsonic region. We further
develop a nonlinear iteration approach and employ its advantages to
deal with such a free boundary problem in the unbounded domain and
to solve the multidimensional transonic nozzle problem in a direct
and simple fashion. Our results indicate that, for the transonic
nozzle problem, there exists a unique transonic flow such that the flow
is divided into a $C^{1,\alpha}$ subsonic flow up to the nozzle boundary
in the unbounded downstream region from the supersonic upstream flow
by a $C^{1,\alpha}$ multidimensional transonic shock that is orthogonal
to the nozzle boundary at every point of their intersection, and the
uniform velocity state at infinity in the downstream direction is
uniquely determined by the supersonic upstream flow at the entrance
which is sufficiently close to a uniform flow. The uniform velocity
state at the infinity can not be apriori prescribed from the
corresponding pressure for such a flow to exist. We further prove that
the transonic flow with a transonic shock is stable with respect to
the supersonic upstream flow at the entrance.

This article has appeared in:

*Arch. Rational Mech. Anal.
* ** vol. 20**, pages (2006)

This paper is available in the following formats:

A closely related paper is Change me.

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