Solutions of euler-poisson equations for gaseous stars

In this paper, we study the Euler-Poisson equations governing gas motion under self-gravitational force. We are interested in the evolution of the gaseous stars, for which the density function has compact support. We establish existence theory for the stationary solutions and describe the behavior of the solutions near the vacuum boundary. The boundary behavior thus obtained agrees with the physical boundary condition proposed and studied in [L, LY] for both Euler equations with damping and the Euler-Poisson equations. Existence, non-existence, uniqueness and instability of the stationary solutions with vacuum are also discussed in terms of the adiabatic exponent and the entropy function. And the phenomena of the blowup, that is, the collapsing of the star to a single point with finite mass, as well as the drifting of part of the star to infinity in space are also studied and shown to agree with the conjecture from the physical considerations.