  In this paper, we study the global existence of steady subsonic Euler flows
through infinitely long nozzles without the assumption of irrotationality. It
is shown that when the variation of Bernoulli's function in the upstream is
sufficiently small and mass flux is in a suitable regime with an upper critical
value, then there exists a unique global subsonic solution in a suitable class
for a general variable nozzle. One of the main difficulties for the general
steady Euler flows, the governing equations are a mixed elliptic-hyperbolic
system even for uniformly subsonic flows. A key point in our theory is to use a
stream function formulation for compressible Euler equations. By this
formulation, Euler equations are equivalent to a quasilinear second order
equation for a stream function so that the hyperbolicity of the particle path
is already involved. The existence of solution to the boundary value problem
for stream function is obtained with the help of the estimate for elliptic
equation of two variables. The asymptotic behavior for the stream function is
obtained via a blow up argument and energy estimates. This asymptotic behavior,
together with some refined estimates on the stream function, yields the
consistency of the stream function formulation and thus the original Euler
equations.
