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 $C^{2}$ solutions with a bounded domain, $\left\Vert X(t)\right\Vert \leq X_{0}$, where $\left\Vert\cdot\right\Vert $ denotes the volume and $X_{0}$ is a positive constant. In particular, we show that if the cosmological constant $\Lambda<M/X_{0}$, with the total mass $M$, then the non-trivial $C^{2}$ solutions in $R^{N}$ with the irrotational initial condition blow up at a finite time.