Blowup for C² solutions of the N-dimensional Euler–Poisson equations in Newtonian cosmology

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 the Ω₀ᵢj(x)=½[∂ᵢuʲ(0,x)−∂juⁱ(0,x)]=0 blow up at a finite time.  Copyright © 2014 Elsevier B.V. All rights reserved.