Existence and Blow-Up of Compressible Spherically Symmetric Euler Equations with Vacuum Free Boundary

This paper studies the compressible spherically symmetric Euler equations with a vacuum free boundary, a fundamental model for astrophysical gas dynamics. We rigorously resolve an open problem by proving that nontrivial homogeneous linear velocity solutions exist if, and only if, the adiabatic exponent γ=4/3, the critical value for monatomic gases and radiative stellar atmospheres. Using qualitative analysis of the reduced planar dynamical system, we characterize the flow’s global existence and finite-time blow-up behavior, establish a sharp existence threshold, and derive an explicit upper bound for the blow-up time. Quantitative energy estimates via Bernoulli’s head verify the physical consistency of solutions in both regimes. Our results complete the classification of self-similar solutions in this class, laying a rigorous theoretical foundation for planetary atmospheric gas flows and providing a practical criterion for predicting blow-up.