In this paper, we study the blowup phenomena for the regular solutions of the isentropic relativistic Euler-Poisson equations with a vacuum state in spherical symmetry. Using a general family of testing functions, we obtain new blowup conditions for the relativistic Euler-Poisson equations. We also show that the proposed blowup conditions are valid regardless of the speed requirement, which was one of the key constraints stated in "Y. Geng, Singularity Formation for Relativistic Euler and Euler-Poisson Equations with Repulsive Force, Commun.

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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−1ln(r + 1), r N−1 e r, r N−1(r 3 − 3r 2 + 3r + ε), r N−1sin((π/2)(r/R)), and r N−1sinh r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.

The blowup phenomenon for the initial-boundary value problem of the non-isentropic compressible Euler equations is investigated. More precisely, we consider a functional F(t) associated with the momentum and weighted by a general test function f and show that if F(0) is sufficiently large, then the finite time blowup of the solutions of the non-isentropic compressible Euler equations occurs. As the test function f is a general function with only mild conditions imposed, a class of blowup conditions is established.

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