不可压缩Boussinesq方程和Euler方程的若干问题研究

不可压缩Boussinesq方程及Euler方程描述了流体运动中的对流现象且被广泛应用到航天航空及气象学等领域，其研究一直是数学物理界长期关注的焦点问题之一。特别地，上述方程的适定性问题是近年来极具吸引力的课题之一。我们将在已有的研究基础上，发展新颖的技巧和有效的方法来研究Boussinesq方程及Euler方程的柯西问题。.(1)、分别利用谱分析及对解映射恰当分解，证明Boussinesq方程在Sobolev空间及Euler方程在Besov空间中解对初值的非一致连续依赖性；.(2)、利用Littlewood-Paley理论，建立三维具变粘性及弱耗散的Boussinesq方程的整体适定性；.(3)、利用线性化方法，构造三维Boussinesq方程的某类整体大解。.本项目的研究内容具有鲜明的物理背景及重要的数学意义。预期研究成果将深入理解流体方程的内在机制并为计算机数值模拟提供理论支撑。

The incompressible Boussinesq equations and Euler equations which describe the convection phenomenon in the motion of fluids have been extensively applied to aeronautics and astronautics, meteorology and so on. The research on them has long been a focus of mathematics and physics. Particularly, the well-posedness problem of the above equations in three dimension has been a fascinating object of research in recent years. Based on our previous works, we will develop some new technics and effective methods to study the Cauchy problem of the Boussinesq equations and Euler equations. .Firstly, we will prove the non-uniform dependence on initial data for the Boussinesq equations in Sobolev space and Euler equations in Besov space by utilizing the spectral analysis and decomposing the solution maps in a suitable way respectively. .Secondly, we will establish the global well-posedness of the three dimensional Boussinesq equations with variable viscosity and weak thermal diffusion by using the Littlewood-Paley theory. .Lastly, we will construct some large solutions to the three dimensional Boussinesq equations with the help of the linearized method..The contents of this project have the profound physical background and important mathematical significance. We hope that the research results will offer insight into the underlying mechanisms of the fluid dynamics and provide theoretical support for numerical simulations in the computer science.