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

Abstract EN

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)xα-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)+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).