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- Title
Analysis of a singular Boussinesq model.
- Authors
Kiselev, Alexander; Yang, Hang
- Abstract
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.
- Subjects
BOUSSINESQ equations; MATHEMATICAL singularities; EULER equations; BIOT-Savart law; COMPUTER simulation
- Publication
Research in the Mathematical Sciences, 2018, Vol 6, Issue 1, p1
- ISSN
2522-0144
- Publication type
Article
- DOI
10.1007/s40687-018-0176-7