Blowup for 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 C 2 solutions with a bounded domain, ‖ X ( t ) ‖ ⩽ X 0 , where ‖ ⋅ ‖ denotes the volume and X 0 is a positive constant. In particular, we show that if the cosmological constant Λ < M / X 0 , with the total mass M, then the non-trivial C 2 solutions in R N with the initial condition Ω 0 i j ( x ) = 1 2 [ ∂ i u j ( 0 , x ) − ∂ j u i ( 0 , x ) ] = 0 blow up at a finite time.