Communications in Mathematical Sciences

Volume 16 (2018)

Number 3

The 3D incompressible Boussinesq equations with fractional partial dissipation

Pages: 617 – 633



Wanrong Yang (School of Mathematics and Information Sciences, Beifang University of Nationalities, Yinchuan, Ningxia, China)

Quansen Jiu (School of Mathematics, Capital Normal University, Beijing, China)

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


The system of the 3D Boussinesq equations is one of the most important models for geophysical fluids. The fundamental problem of whether or not reasonably smooth solutions to the 3D Boussinesq equations with the standard Laplacian dissipation can blow up in a finite time is an outstanding open problem. The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations, but also are physically relevant and mathematically important. This paper focuses on a system of the 3D Boussinesq equations with fractional partial dissipation and proves that any $H^1$-initial data always leads to a unique and global-in-time solution. The result of this paper is part of our efforts devoted to the global well-posedness problem on the Boussinesq equations with minimal dissipation.


3D Boussinesq equations, fractional partial dissipation, global regularity

2010 Mathematics Subject Classification

35Q35, 76D03

Yang was supported by NNSFC (No. 11601011), by NSF of Ningxia (No. NZ16092), and by the Higher Education Specialized Research Fund of Ningxia (No. NGY2015140). Q. Jiu is supported by NNSFC (No. 11671273 and No. 11231006). J. Wu is supported by NSF grant DMS 1614246, by the AT&T Foundation at Oklahoma State University, and by NNSFC (No. 11471103, a grant awarded to B. Yuan).

Received 16 July 2017

Received revised 22 November 2017

Accepted 22 November 2017

Published 30 August 2018