Dynamics of Partial Differential Equations

Volume 13 (2016)

Number 4

Evolution of the two-dimensional Boussinesq system

Pages: 333 – 350

DOI: https://dx.doi.org/10.4310/DPDE.2016.v13.n4.a3


Tai-Man Tang (School of Mathematics and Computational Sciences, Xiangtan University, Hunan, China)

Ye Xie (School of Mathematics and Computational Sciences, Xiangtan University, Hunan, China)


The smooth evolutions along the trajectories of the main physical quantities of the two dimensional Boussinesq system with viscousity and thermal diffusivity not both non-zero are studied. Specifically, for a spatially $H^m$ solution with $m \gt 4$ (only $m \gt 3$ is needed for some result), quantities including the speed, vorticity, temperature gradient and their stretching rates are shown to evolve smoothly along the trajectories. Conclusions on their evolutions are obtained. Results on some of the stretching rates give information on the evolutions of the relative sizes of some basic quantities. When the viscousity and thermal diffusivity are zero, it is not known if smooth solutions exist globally and we study the dichotomy between finite time singularity and the long time behaviors of the main quantities. If either the viscousity or thermal diffusivity is non-zero, it is known that smooth solutions are global and this investigation provides some information about them by describing the dynamics of the main quantities.


Boussinesq equations, asymptotic behavior, blow up, dynamics, global smooth solution

2010 Mathematics Subject Classification

Primary 35Q86. Secondary 76B99, 76D99.

Published 16 December 2016