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



Fazel Hadadifard (Department of Mathematics, University of Kansas, Lawrence, Ks., U.S.A.)

Atanas Stefanov (Department of Mathematics, University of Kansas, Lawrence, Ks., U.S.A.)


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.


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