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# 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

#### 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

Published 14 September 2016