Communications in Mathematical Sciences

Volume 14 (2016)

Number 7

Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation

Pages: 1963 – 1997

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n7.a9

Authors

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)

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

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

Abstract

The incompressible Boussinesq equations serve as an important model in geophysics as well as in the study of Rayleigh–Bénard convection. One generalization is to replace the standard Laplacian operator by a fractional Laplacian operator, namely $(-\Delta)^{\alpha/2}$ in the velocity equation and $(-\Delta)^{\beta/2}$ in the temperature equation. This paper is concerned with the two-dimensional (2D) incompressible Boussinesq equations with critical dissipation $(\alpha + \beta = 1)$ or supercritical dissipation $(\alpha + \beta \lt 1)$. We prove two main results. This first one establishes the global-in-time existence of classical solutions to the critical Boussinesq equations with $(\alpha + \beta = 1)$ and $0.7692 \approx \frac{10}{13} \lt \alpha \lt1$. The second one proves the eventual regularity of Leray–Hopf type weak solutions to the Boussinesq equations with supercritical dissipation $(\alpha + \beta \lt 1)$ and $0.7692 \approx \frac{10}{13} \lt \alpha \lt1$.

Keywords

2D Boussinesq equations, generalized supercritical SQG, global regularity, eventual regularity

2010 Mathematics Subject Classification

35B65, 35Q35, 76D03

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