具有吸引强迫的二维Euler-泊松方程的Riccati动力学
On the Riccati dynamics of 2D Euler-Poisson equations with attractive forcing
摘要：
Euler-Poisson（EP）系统描述了许多重要物理流的动态行为。在这项工作中，Riccati系统，治理二维EP方程进行了研究。发散的演变由Riccati型方程与几个非线性/非局部项。其中，涡度加速发散，而其他进一步放大流动的爆破行为。这些爆破放大项的增长与密度的Riesz变换有关，而Riesz变换缺乏一个统一的边界，这使得研究多维EP系统的整体解变得困难。我们证明了Riccati系统只要爆破放大项的增长率不高于指数增长率，就可以有全局解，并且对于一个大的初始配置集，可以有全局光滑解.为了证明这一点，我们在三维空间中构造了一个辅助系统，并找到了系统的不变空间，然后与原始的二维系统进行了比较。最后给出了数值算例。
英文摘要：
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.
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