Blowup phenomena for the compressible Euler and Euler-Poisson equations with initial functional conditions

We study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations in Rᴺ. For time t ≥ 0, we can define a functional H(t) associated with the solution of the equations and some testing function f. When the pressure function P of the governing equations is of the form P = Kpᵞ, where p is the density function, K is a constant, and y > 1, we can show that the nontrivial C¹ solutions with nonslip boundary condition will blow up in finite time if H(0) satisfies some initial functional conditions defined by the integrals of f. Examples of the testing functions include rᴺ⁻¹ln(r + 1), rᴺ⁻¹ eʳ, rᴺ⁻¹(r³− 3r²+ 3r + Ɛ), rᴺ⁻¹sin((π/2)(r/R)), and rᴺ⁻¹sinh r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given. Copyright © 2014 Sen Wong and Manwai Yuen.