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We are concerned with globally defined entropy solutions to the Eulerequations for compressible fluid flows in transonic nozzles with generalcross-sectional areas. Such nozzles include the de Laval nozzles and other moregeneral nozzles whose cross-sectional area functions are allowed at the nozzleends to be either zero (closed ends) or infinity (unbounded ends). To achievethis, in this paper, we develop a vanishing viscosity method to constructglobally defined approximate solutions and then establish essential uniformestimates in weighted $L^p$ norms for the whole range of physical adiabaticexponents $\gamma\in (1, \infty)$, so that the viscosity approximate solutionssatisfy the general $L^p$ compensated compactness framework. The viscositymethod is designed to incorporate artificial viscosity terms with the naturalDirichlet boundary conditions to ensure the uniform estimates. Then suchestimates lead to both the convergence of the approximate solutions and theexistence theory of globally defined finite-energy entropy solutions to theEuler equations for transonic flows that may have different end-states in theclass of nozzles with general cross-sectional areas for all $\gamma\in (1,\infty)$. The approach and techniques developed here apply to other problemswith similar difficulties. In particular, we successfully apply them toconstruct globally defined spherically symmetric entropy solutions to the Eulerequations for all $\gamma\in (1, \infty)$.

Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows