  The hydrodynamic escape problem (HEP), which is characterized by a free
boundary value problem of Euler equation with gravity and heat, is crucial for
investigating the evolution of planetary atmospheres. In this paper, the global
existence of transonic solutions to the HEP is established using the
generalized Glimm method. The new version of Riemann and boundary-Riemann
solvers, are provided as building blocks of the generalized Glimm method by
inventing the contraction matrices for the homogeneous Riemann (or
boundary-Riemann) solutions. The extended Glimm-Goodman wave interaction
estimates are investigated for obtaining a stable scheme and positive gas
velocity, which matches the physical observation. The limit of approximation
solutions serves as an entropy solution of bounded variations. Moreover, the
range of the hydrodynamical region is also obtained.
