Title: 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,
Abstract: We study the motion of isentropic gas in nozzles. This is a major subject in
fluid dynamics. In fact, the nozzle is utilized to increase the thrust of
rocket engines. Moreover, the nozzle flow is closely related to astrophysics.
These phenomena are governed by the compressible Euler equation, which is one
of crucial equations in inhomogeneous conservation laws.
In this paper, we consider its unsteady flow and devote to proving the global
existence and stability of solutions to the Cauchy problem for the general
nozzle. 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 in
the present paper is to remove this restriction, that is, we consider large
data. Although the subject is important in Mathematics, Physics and
engineering, it remained open for a long time. The problem seems to lie in a
bounded estimate of approximate solutions, because we have only method to
investigate the behavior with respect to the time variable. To solve this, we
first introduce a generalized invariant region. Compared with the existing
ones, its upper and lower bounds are extended constants to functions of the
space variable. However, we cannot apply the new invariant region to the
traditional difference method. Therefore, we invent the modified Godunov
scheme. The approximate solutions consist of some functions corresponding to
the upper and lower bounds of the invariant regions. These methods enable us to
investigate the behavior of approximate solutions with respect to the space
variable. The ideas are also applicable to other nonlinear problems involving
similar difficulties.