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 theN-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the formc(t)xα-1x+b(t)(x/x)for any value ofα≠1or any positive integerN≠1. Then, we show that blowup phenomenon occurs whenα=N=1andc2(0)+c˙(0)<0. As a corollary, the blowup properties of solutions with velocity of the form(a˙t/at)x+b(t)(x/x)are obtained. Our analysis includes both the isentropic case(γ>1)and the isothermal case(γ=1).