Blowup for [formula omitted] solutions of the N-dimensional Euler–Poisson equations in Newtonian cosmology

Abstract

Pressureless Euler–Poisson equations with attractive forces are standard models in Newtonian cosmology. In this article, we further develop the spectral dynamics method and apply a novel spectral-dynamics-integration method to study the blowup conditions for C2 solutions with a bounded domain, ‖X(t)‖⩽X0, where ‖⋅‖ denotes the volume and X0 is a positive constant. In particular, we show that if the cosmological constant Λ<M/X0, with the total mass M, then the non-trivial C2 solutions in RN with the initial condition Ω0ij(x)=12[∂iuj(0,x)−∂jui(0,x)]=0 blow up at a finite time.