Existence and uniqueness of generalized stationary waves for viscous gas flow through a nozzle with discontinuous cross section

In this paper we study the existence and uniqueness of the generalized stationary waves for one-dimensional viscous isentropic compressible flows through a nozzle with discontinuous cross section. Following the geometric singular perturbation technique, we establish the existence and uniqueness of inviscid and viscous stationary waves for the regularized systems with mollified cross section. Then, the generalized inviscid stationary waves are classified for discontinuous and expanding or contracting nozzles by the limiting argument. Moreover, we obtain the generalized viscous stationary waves by using Helly's selection principle. However, due to the choices of mollified cross section functions, there may exist multiple transonic standing shocks in the generalized stationary waves. A new entropy condition is imposed to select a unique admissible standing shock in generalized stationary wave. We show that, such admissible solution selected by the entropy condition, admits minimal total variation and has minimal enthalpy loss across the standing shock in the limiting process.