  This paper concerns smooth supersonic flows with Lipschitz continuous speed
in two-dimensional infinite expanding nozzles, which are governed by a
quasilinear hyperbolic equation being singular at the sonic and vacuum state.
The flow satisfies the slip condition on the walls and the flow velocity is
prescribed at the inlet. First, it is proved that if the incoming flow is away
from the sonic and vacuum state and its streamlines are rarefactive at the
inlet, then a flow in a straight nozzle never approaches the sonic and vacuum
state in any bounded region. Furthermore, a sufficient and necessary condition
of the incoming flow at the inlet is derived for the existence of a global
smooth supersonic flow in a straight nozzle. Then, it is shown that for each
incoming flow satisfying this condition, there exists uniquely a global smooth
supersonic flow in a symmetric nozzle with convex upper wall. It is noted that
such a flow may contain a vacuum. If there is a vacuum for a global smooth
transonic flow in a symmetric nozzle with convex upper wall, it is proved that
for the symmetric upper part of the flow, the first vacuum point along the
symmetric axis must be located at the upper wall and the set of vacuum points
is the closed domain bounded by the tangent half-line of the upper wall at this
point to downstream and the upper wall after this point. Moreover, the flow
speed is globally Lipschitz continuous in the nozzle, and on the boundary
between the gas and the vacuum, the flow velocity is along this boundary and
the normal derivatives of the flow speed and the square of the sound speed both
are zero. As an immediate consequence, the local smooth transonic flow obtained
in [10] can be extended into a global smooth transonic flow in a symmetric
nozzle whose upper wall after the local flow is convex.
