##
Existence and Stability of Supersonic Euler Flows past
Lipschitz Wedges

#### Authors: Gui-Qiang Chen, Yongqian Zhang, and Dianwen Zhu

#### Title:
Existence and Stability of Supersonic Euler Flows past
Lipschitz Wedges

**Abstract**

It is well known that, when the vertex angle of a straight wedge is
less than the critical angle, there exists a shock-front emanating
from the wedge vertex so that the constant states on both sides of
the shock-front are supersonic. Since the shock-front at the vertex
is usually strong, especially when the vertex angle of the wedge is
large, then such a global flow is physically required to be governed
by the isentropic or adiabatic Euler equations. In this paper, we
systematically study two-dimensional steady supersonic Euler (i.e.
non-potential) flows past Lipschitz wedges and establish the
existence and stability of supersonic Euler flows when the total
variation of the tangent angle functions along the wedge boundaries
is suitably small. We develop a modified Glimm difference scheme and
identify a Glimm-type functional, by incorporating the Lipschitz
wedge boundary and the strong shock-front naturally and by tracing
the interaction not only between the boundary and weak waves but
also between the strong shock front and weak waves, to obtain the
required $BV$ estimates. Then these estimates are employed to
establish the convergence of both approximate solutions to a global
entropy solution and corresponding approximate strong shock-fronts
emanating from the vertex to the strong shock-front of the entropy
solution. The regularity of strong shock-fronts emanating from the
wedge vertex and the asymptotic stability of entropy solutions in
the flow direction are also established.

This article has appeared in:

* Archive Rational Mechanics and Analysis
* ** vol.**, pages (2005)(to appear)

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.northwestern.edu