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−1x + 𝑏(𝑡)(x/|x|) for any value of 𝛼 = 1̸ or any positive integer 𝑁 = 1̸ . Then, we show that blowup phenomenon occurs when 𝛼=𝑁=1 and 𝑐 2 (0) + 𝑐(0) < 0 ̇ . As a corollary, the blowup properties of solutions with velocity of the form (𝑎(𝑡)/𝑎(𝑡)) ̇ x + 𝑏(𝑡)(x/|x|) are obtained. Our analysis includes both the isentropic case (𝛾 > 1) and the isothermal case (𝛾 = 1).