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 R (N). 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 = Kρ (γ), where ρ is the density function, K is a constant, and γ > 1, we can show that the nontrivial C (1) 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 r (N−1)ln(r + 1), r (N−1) e (r), r (N−1)(r (3) − 3r (2) + 3r + ε), r (N−1)sin((π/2)(r/R)), and r (N−1)sinh r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.