On the Riccati dynamics of 2D Euler-Poisson equations with attractive forcing

TL;DR

This paper analyzes the Riccati dynamics of 2D Euler-Poisson equations with attractive forcing, demonstrating conditions for global solutions and constructing auxiliary systems to understand blow-up behavior.

Contribution

It introduces a Riccati system for 2D Euler-Poisson equations, establishing criteria for global solutions and analyzing blow-up amplification through an auxiliary 3D system.

Findings

Global solutions exist if blow-up growth is at most exponential.

Vorticity accelerates divergence and influences blow-up.

Numerical examples illustrate theoretical results.

Abstract

The Euler-Poisson (EP) system describes the dynamic behavior of many important physical flows. In this work, a Riccati system that governs two-dimensional EP equations 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 further amplify the blow-up behavior of a flow. The growth of these blow-up amplifying terms are related to the Riesz transform of density, which lacks a uniform bound makes it difficult to study global solutions of the multi-dimensional EP system. We show that the Riccati system can afford to have global solutions, as long as the growth rate of blow-up amplifying terms is not higher than exponential, 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. Some numerical examples are also presented.