Analysis of a singular Boussinesq model
Analysis of a singular Boussinesq model
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DOI：
10.1007/s40687-018-0176-7
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发表时间：
2018-12
影响因子：
1.2
通讯作者：
A. Kiselev;Hang Yang
中科院分区：
文献类型：
--
作者：
A. Kiselev;Hang Yang
最近，Hou和Luo(多尺度模式SIMUL 12(4)：1722-1776,2014，PNAS111(36)：12968-12973,2014)基于广泛的数值模拟，提出了三维轴对称欧拉方程和二维无粘性Boussinesq系统的一种新的奇性形成情景。作为理解这一情景的第一步，Choi等人最近研究了具有简化的符号定Biot-Savart定律和强迫的模型。(Commun Pure Appl Math 70(11)：2218-2243,2017，Commun Math Phys 334：1667-1679,2015)，Do等人。(《非线性科学》，2016。Arxiv：1604.07118)，Hoang et al.(J Different EQU 264：7328-7356,2018)，侯和刘(Res Math Sci 2,2015)，Kiselev和Tan(ADV Math 325：34-55,2018)。在这篇文章中，我们的目的是恢复原始方程中遇到的一个复杂问题--毕奥-萨伐尔定律中的变号核。这增加了分析的难度，因为在流体速度积分中有两个相互竞争的项，它们的平衡决定了解的正则性。本文所研究的方程是基于Choi等人提出的CKY模型。(2015)。我们证明了有限时间爆破在一定的参数范围内持续存在。
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.