Blowup Phenomenon of Solutions for the IBVP of the Compressible Euler Equations in Spherical Symmetry

The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of the N-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the form c(t)|x|(α−1) x + b(t)(x/|x|) for any value of α ≠ 1 or any positive integer N ≠ 1. Then, we show that blowup phenomenon occurs when α = N = 1 and [Formula: see text]. As a corollary, the blowup properties of solutions with velocity of the form [Formula: see text] are obtained. Our analysis includes both the isentropic case (γ > 1) and the isothermal case (γ = 1).