Characterization of the Transonic Stationary Solutions of the Hydrodynamic Escape Problem

In this talk we characterize the transonic stationary solutions of the hydrodynamic escape problem (HEP), which is significant on studying the evolution of atmospheres of planets. The characterization is established by the geometric singular perturbation method (GSP) on the wind solution of HEP. We will discuss the existence and non-existence of the adiabatic wind solution with or without heat effect. By analyzing the wind solutions, we can verify the smooth stationary solutions under the effect of conduction. The singularity at the sonic points can be moved to the thermal critical points under the effect of conduction. Through introducing the artificial viscosity, we can study the dynamics at the thermal critical points and sonic points. Such smooth transonic solution is a regular perturbation of the adiabatic wind solution. Discontinuous standing shock solutions and their stationary layer profiles will also be discussed.