Communications in Mathematical Sciences

Volume 14 (2016)

Number 7

Regularity criteria for the 2D Boussinesq equations with supercritical dissipation

Pages: 1999 – 2022

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n7.a10

Authors

Jingna Li (Department of Mathematics, Jinan University, Guangzhou, China)

Haifeng Shang (School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China)

Jiahong Wu (Department of Mathematics, Oklahoma State University, Stillwater, Ok., U.S.A.)

Xiaojing Xu (School of Mathematical Sciences, Beijing Normal University, Beijing, China; and Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, China)

Zhuan Ye (School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, China)

Abstract

This paper focuses on the 2D incompressible Boussinesq equations with fractional dissipation, given by $\Lambda^{\beta} u$ in the velocity equation and by $\Lambda^{\beta} \theta$ in the temperature equation, where $\Lambda = \sqrt{-\Delta}$ denotes the Zygmund operator. Due to the vortex stretching and the lack of sufficient dissipation, the global regularity problem for the supercritical regime $\alpha + \beta \lt 1$ remains an outstanding problem. This paper presents several regularity criteria for the supercritical Boussinesq equations. These criteria are sharp and reflect the level of difficulty of the supercritical Boussinesq problem. In addition, these criteria are important tools in understanding some crucial properties of Boussinesq solutions such as the eventual regularity.

Keywords

Boussinesq equations, fractional dissipation, global well-posedness

2010 Mathematics Subject Classification

35B65, 35Q35, 76B03

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Published 14 September 2016