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We study the motion of isentropic gas in nozzles. This is a major subject influid dynamics. In fact, the nozzle is utilized to increase the thrust ofrocket engines. Moreover, the nozzle flow is closely related to astrophysics.These phenomena are governed by the compressible Euler equation, which is oneof crucial equations in inhomogeneous conservation laws.  In this paper, we consider its unsteady flow and devote to proving the globalexistence and stability of solutions to the Cauchy problem for the generalnozzle. The theorem has been proved in (Tsuge in Arch. Ration. Mech. Anal.209:365-400 (2013)). However, this result is limited to small data. Our aim inthe present paper is to remove this restriction, that is, we consider largedata. Although the subject is important in Mathematics, Physics andengineering, it remained open for a long time. The problem seems to lie in abounded estimate of approximate solutions, because we have only method toinvestigate the behavior with respect to the time variable. To solve this, wefirst introduce a generalized invariant region. Compared with the existingones, its upper and lower bounds are extended constants to functions of thespace variable. However, we cannot apply the new invariant region to thetraditional difference method. Therefore, we invent the modified Godunovscheme. The approximate solutions consist of some functions corresponding tothe upper and lower bounds of the invariant regions. These methods enable us toinvestigate the behavior of approximate solutions with respect to the spacevariable. The ideas are also applicable to other nonlinear problems involvingsimilar difficulties.

Global entropy solutions to the compressible Euler equations in the  isentropic nozzle flow for large data: Application of the modified Godunov  scheme and the generalized invariant regions