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# Communications in Mathematical Sciences

## Volume 15 (2017)

### Number 5

### On the global regularity of the 2D critical Boussinesq system with $\alpha \gt 2/3$

Pages: 1325 – 1351

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n5.a6

#### Authors

#### Abstract

This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation $(\alpha , \beta) : \alpha + \beta =1$. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data, as long as $\alpha \gt 2/3$. This improves upon some recent works [Q. Jiu, C. Miao, J. Wu and Z. Zhang, *SIAM J. Math. Anal.*, 46:3426–3454, 2014] and [A. Stefanov and J. Wu, *J. Anal. Math.*, 2015].

The main new idea is the introduction of a new, second generation Hmidi–Keraani–Rousset type, change of variables, which further improves the linear derivative in temperature term in the vorticity equation. This approach is then complemented by a new set of commutator estimates (in both negative and positive index Sobolev spaces!), which may be of independent interest.

#### Keywords

Boussinesq equations, fractional dissipation, global regularity

#### 2010 Mathematics Subject Classification

35B65, 35Q35, 76B03

F. Hadadifard has been partially supported by a graduate research fellowship through a grant NSF-DMS #1313107. A. Stefanov’s research has been supported in part by NSF-DMS #1313107 and #1614734.

Received 3 October 2016

Published 26 June 2017