Abstract
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.
Highlights
The compressible isentropic Euler (δ = 0) or Euler-Poisson (δ = ±1) equations for fluids can be written as ρt + ∇ ⋅ = 0, ρ [ut + (u ⋅ ∇) u] + ∇P = ρ∇Φ, (1)ΔΦ (t, x) = δα (N) ρ, where α(N) is a constant related to the unit ball in RN
In the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations in RN
If K > 0, it is a system with pressure
Summary
When δ = 1, the system comprises the Euler-Poisson equations with repulsive forces and can be applied as a semiconductor model [3]. Perthame [8] studied the blowup results for the 3-dimensional pressureless system with repulsive forces (δ = 1). For pressureless fluids (K = 0) or γ > 1, and the nontrivial classical C1 solutions (ρ, V) with radial symmetry and the first boundary condition (6), we have the following results. (a) For the attractive forces (δ = −1), if H0 satisfies the following initial functional condition: 2R. (b) For the nonattractive forces (δ = 0 or 1), if H0 satisfies the following initial functional condition: H0 = ∫ f (r) V0dr > 0,.
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