  The Euler-Poisson(EP) system describes the dynamic behavior of many important
physical flows. In this work, a Riccati system that governs the flow's gradient
is studied. The evolution of divergence is governed by the Riccati type
equation with several nonlinear/nonlocal terms. Among these, the vorticity
accelerates divergence while others suppress divergence and enhance the finite
time blow-up of a flow. The growth of the latter terms are related to the Riesz
transform of density and non-locality of these terms make it difficult to study
global solutions of the multi-dimensional EP system. Despite of these, we show
that the Riccati system can afford to have global solutions under a suitable
condition, and admits global smooth solutions for a large set of initial
configurations. To show this, we construct an auxiliary system in 3D space and
find an invariant space of the system, then comparison with the original 2D
system is performed. The present work generalizes several previous so-called
restricted/modified EP models.
