New method for blowup of the Euler-Poisson system

In this paper, we provide a new method for establishing the blowup of C² solutionsfor the pressureless Euler-Poisson system with attractive forces for Rᴺ(N ≥ 2) with ρ(0, x₀) > 0 and Ω0ij(x₀) = 1/2[∂iuj(0, x₀) - ∂jui(0, x₀)]= 0 at somepoint x0 ∈ Rᴺ. By applying the generalized Hubble transformation div u(t, x0(t)) =Na˙(t )/a(t ) to a reduced Riccati differential inequality derived from the system, wesimplify the inequality into the Emden equation ä(t) = - Λ/a(t ) ᴺ⁻¹ , a(0) = 1, a˙(0) =div u(0, x₀)/N . Known results on its blowup set allow us to easily obtain theblowup conditions of the Euler-Poisson system. Copyright © 2016 AIP Publishing.