  We are concerned with globally defined entropy solutions to the Euler
equations for compressible fluid flows in transonic nozzles with general
cross-sectional areas. Such nozzles include the de Laval nozzles and other more
general nozzles whose cross-sectional area functions are allowed at the nozzle
ends to be either zero (closed ends) or infinity (unbounded ends). To achieve
this, in this paper, we develop a vanishing viscosity method to construct
globally defined approximate solutions and then establish essential uniform
estimates in weighted $L^p$ norms for the whole range of physical adiabatic
exponents $\gamma\in (1, \infty)$, so that the viscosity approximate solutions
satisfy the general $L^p$ compensated compactness framework. The viscosity
method is designed to incorporate artificial viscosity terms with the natural
Dirichlet boundary conditions to ensure the uniform estimates. Then such
estimates lead to both the convergence of the approximate solutions and the
existence theory of globally defined finite-energy entropy solutions to the
Euler equations for transonic flows that may have different end-states in the
class of nozzles with general cross-sectional areas for all $\gamma\in (1,
\infty)$. The approach and techniques developed here apply to other problems
with similar difficulties. In particular, we successfully apply them to
construct globally defined spherically symmetric entropy solutions to the Euler
equations for all $\gamma\in (1, \infty)$.
