Blowup for $C^2$ 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 $C^2$ solutions with a bounded domain, $\Vert X(t)\Vert \le X_0$, where $\Vert \cdot \Vert$ 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.

10 Pages

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