Dynamics Of The Two Dimensional Boussinesq System And A Liouville Type Theorem

Two aspects of the two dimensional Boussinesq equations are discussed in this thesis. In the first part, we investigate the evolution along the trajectories of the major quantities of the Boussinesq system. These include velocity, pres-sure, temperature, vorticity, the force on the fluid or their derivatives. Form the Boussinesq equations, we derive ordinary differential equations governing their evolutions and then deduce results from them. In the second part, a Liouville type theorem for the Boussinesq system without viscousity and thermal diffusivi-ty is proved. A condition on the pressure that guarantees the triviality of the flow is given. There are a lot of research on the Boussinesq system. Many of them are on well-posedness and regularity issues and relative few of them discuss the properties of solutions. It is well known that if either the viscousity or thermal diffusivity is non-zero, global smooth solutions of the two dimensional Boussinesq system exist. This thesis is to investigate the properties of these solutions.