Recently, a new singularity formation scenario for the 3D axi-symmetric Euler equation and the 2D inviscid Boussinesq system has been proposed by Hou and Luo (Multiscale Model Simul 12(4):1722–1776, 2014, PNAS 111(36):12968–12973, 2014) based on extensive numerical simulations. As the first step to understand the scenario, models with simplified sign-definite Biot–Savart law and forcing have recently been studied in Choi et al. (Commun Pure Appl Math 70(11):2218–2243, 2017, Commun Math Phys 334:1667–1679, 2015), Do et al. (J Nonlinear Sci, 2016. arXiv:1604.07118), Hoang et al. (J Differ Equ 264:7328–7356, 2018), Hou and Liu (Res Math Sci 2, 2015), Kiselev and Tan (Adv Math 325:34–55, 2018). In this paper, we aim to bring back one of the complications encountered in the original equation—the sign changing kernel in the Biot–Savart law. This makes analysis harder, as there are two competing terms in the fluid velocity integral whose balance determines the regularity properties of the solution. The equation we study here is based on the CKY model introduced in Choi et al. (2015). We prove that finite time blow up persists in a certain range of parameters.

中文翻译：

---

奇异Boussinesq模型的分析

最近，Hou和Luo提出了3D轴对称Euler方程和2D无粘性Boussinesq系统的新奇点形成方案（Multiscale Model Simul 12（4）：1722-1776，2014，PNAS 111（36）：12968 –12973，2014）基于广泛的数值模拟。作为了解这种情况的第一步，最近在Choi等人中研究了具有简化的符号确定的Biot-Savart定律和强迫的模型。（通用公共应用数学70（11）：2218-2243，2017，公共数学物理334：1667-1679，2015），Do等。（J Nonlinear Sci，2016. arXiv：1604.07118），Hoang等。（J Differ Equ 264：7328–7356，2018），Hou和Liu（Res Math Sci 2，2015），Kiselev和Tan（Adv Math 325：34–55，2018）。在本文中，我们旨在带回原始方程式中遇到的复杂问题之一-比奥—萨瓦特定律中的符号变化核。这使分析更加困难，因为在流速积分中有两个竞争项，其平衡决定了解决方案的规则性。我们在这里研究的方程是基于Choi等人介绍的CKY模型。（2015）。我们证明有限的时间爆炸在一定范围的参数中持续存在。