Blowup Phenomena for the Compressible Euler and Euler-Poisson Equations with Initial Functional Conditions

摘要

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.