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

Abstract: 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 $\gamma\in (1, \infty)$.