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|∝⁻¹x+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 c²(0) + ć (0) < 0. As a corollary, the blowup properties of solutions with velocity of the form (à(t)/a(t))x + b(t)(x|x|) are obtained. Our analysis includes both the isentropic case (ƴ > 1) and the isothermal case (ƴ = 1) . Copyright © 2016 Ka Luen Cheung and Sen Wong.