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G. Luo, T. Hou (2013)
Potentially singular solutions of the 3D axisymmetric Euler equationsProceedings of the National Academy of Sciences, 111
D. Chae (2006)
Global regularity for the 2D Boussinesq equations with partial viscosity termsAdvances in Mathematics, 203
J. Beale, Tosio Kato, A. Majda (1984)
Remarks on the breakdown of smooth solutions for the 3-D Euler equationsCommunications in Mathematical Physics, 94
Wanrong Yang, Q. Jiu, Jiahong Wu (2018)
The 3D incompressible Boussinesq equations with fractional partial dissipationCommunications in Mathematical Sciences, 16
A. Majda, A. Bertozzi (2001)
Vorticity and incompressible flow
Tam Do, A. Kiselev, Xiaoqian Xu (2016)
Stability of Blowup for a 1D Model of Axisymmetric 3D Euler EquationJournal of Nonlinear Science, 28
T. Hou, Congming Li (2004)
GLOBAL WELL-POSEDNESS OF THE VISCOUS BOUSSINESQ EQUATIONSDiscrete and Continuous Dynamical Systems, 12
Kyudong Choi, A. Kiselev, Yao Yao (2013)
Finite Time Blow Up for a 1D Model of 2D Boussinesq SystemCommunications in Mathematical Physics, 334
A. Gill (1982)
Atmosphere-Ocean Dynamics
C. Doering, Jiahong Wu, Kun Zhao, Xiaoming Zheng (2018)
Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusionPhysica D: Nonlinear Phenomena
A. Kiselev, V. Sverák (2013)
Small scale creation for solutions of the incompressible two dimensional Euler equationarXiv: Analysis of PDEs
Dhanapati Adhikari, C. Cao, Jiahong Wu (2010)
The 2D Boussinesq equations with vertical viscosity and vertical diffusivityJournal of Differential Equations, 249
A Kiselev, V Šverák (2014)
Small scale creation for solutions of the incompressible two dimensional Euler equationAnn. Math. (2), 180
T. Hou, Pengfei Liu (2014)
Self-similar singularity of a 1D model for the 3D axisymmetric Euler equationsResearch in the Mathematical Sciences, 2
(1987)
Geophysical Fluid Dyanmics
Dhanapati Adhikari, C. Cao, Haifeng Shang, Jiahong Wu, Xiaojing Xu, Z. Ye (2011)
Global regularity results for the 2D Boussinesq equations with vertical dissipationJournal of Differential Equations, 260
Bo-Qing Dong, Wenjuan Wang, Jiahong Wu, Hui Zhang (2019)
Global regularity results for the climate model with fractional dissipationDiscrete & Continuous Dynamical Systems - B
A. Kiselev, Changhui Tan (2016)
Finite time blow up in the hyperbolic Boussinesq systemarXiv: Analysis of PDEs
Kyudong Choi, T. Hou, A. Kiselev, G. Luo, V. Sverák, Yao Yao (2017)
On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler EquationsCommunications on Pure and Applied Mathematics, 70
V. Yudovich, Nikolay Tikhonov, V. Yudovich (2003)
Eleven Great Problems of Mathematical HydrodynamicsMoscow Mathematical Journal, 3
Jiahong Wu, Xiaojing Xu, Z. Ye (2018)
The 2D Boussinesq equations with fractional horizontal dissipation and thermal diffusionJournal de Mathématiques Pures et Appliquées
V. Hoang, Betul Orcan-Ekmekci, M. Radosz, Han Yang (2016)
Blowup with vorticity control for a 2D model of the Boussinesq equationsJournal of Differential Equations
E. Saw, D. Kuzzay, D. Faranda, A. Guittonneau, A. Guittonneau, F. Daviaud, C. Wiertel-Gasquet, V. Padilla, B. Dubrulle (2016)
Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flowNature Communications, 7
G. Luo, T. Hou (2014)
Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical InvestigationMultiscale Model. Simul., 12
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.
Research in the Mathematical Sciences – Springer Journals
Published: Dec 19, 2018
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