Global solutions for the two dimensional Euler-Poisson system with attractive forcing

TL;DR

This paper establishes conditions under which the two-dimensional Euler-Poisson system with attractive forcing admits global smooth solutions, by analyzing a Riccati system governing flow gradients and constructing an auxiliary 3D system.

Contribution

It introduces a novel approach using a Riccati system and auxiliary 3D analysis to prove global existence of solutions for the 2D Euler-Poisson system with attractive forcing.

Findings

Global solutions exist under certain conditions.

The Riccati system analysis reveals flow divergence behavior.

Auxiliary 3D system aids in establishing invariance and comparison.

Abstract

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.