New method for blowup of the Euler-poisson system

Abstract

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.