Criterion of singularity formation for radial solutions of the pressureless Euler-Poisson equations in exceptional dimension

Abstract

Spatial dimensions 1 and 4 play an exceptional role for radial solutions of the pressureless Euler-Poisson equations. Namely, for spatial dimensions other than 1 and 4, any nontrivial solution of the Cauchy problem blows up in finite time (except for special cases), whereas for dimensions 1 and 4 there exists a neighborhood of trivial initial data in the $C^1$-norm such that the corresponding solution preserves the initial smoothness globally. This is explained by the fact that only in these dimensions all Lagrangian trajectories are periodic with the same period. For dimension 1, a criterion for the formation of a singularity in terms of initial data was known, however, for the much more difficult case of dimension 4, there was no such result. In this paper, we fill this gap. We also describe a class of problems to which the technique used here can be extended with minor modifications.