具分数阶耗散的Boussinesq方程的整体适定性研究

Boussinesq方程是描述大气和海洋运动的一类地球物理学模型。源于实际的物理现象，用分数阶扩散算子代替标准的拉普拉斯算子被广泛应用于概率论、金融学、材料科学及量子力学等领域。这使得研究具分数阶耗散的Boussinesq方程有重要的数学理论和应用价值。关于这类方程的整体适定性问题是非线性偏微分方程研究的前沿领域之一。. 本项目拟研究具分数阶耗散的不可压缩Boussinesq方程的整体适定性。二维时，拟利用Littlewood-Paley理论及现代分析中的工具和方法，建立精细的分数阶拉普拉斯算子非线性下界估计和交换子估计，研究次临界、临界、超临界情形解的整体适定性。三维时，研究各向异性的分数阶耗散情形解的整体适定性。. 预期成果将给出解决不可压缩流体方程的整体适定性的一些较系统的方法，丰富和发展非线性偏微分方程的数学理论，为不可压缩流体的实际应用提供数学支持。

The Boussinesq equations model geophysical flows such as atmospheric fronts and oceanic circulation. The fractional Laplacian generalizes the standard Laplacian operator and has many applications in probability, finance, material sciences and quantum mechanics, etc. The global well-posedness problem on this model with fractional dissipation is one of the most intensively investigated topics and is very significant in mathematics and applications. . The projects proposed here are based on our previous investigations and we intend to solve some of the open problems. We consider the global well-posedness of incompressible Boussinesq equations with fractional dissipation. For two-dimensional case, our investigation is divided into three categories: subcritical, critical, and supercritical. To do so, we will establish accurate lower bounds on the fractional Laplacian operator as well as sharp upper bounds for commutators related to the nonlinear terms. We will extensively make use of Littlewood-Paley theory and other tools in model harmonic analysis. For three-dimensional case, we are interested in the structure of the anisotropic fractional Laplacian and establish global regularity. In particular, we would like to develop methods to effectively use the fractional dissipation to balance the nonlinearity.. We aim to provide a systematic approach to the global well-posedness problem on partial differential equations modeling fluids when the fractional dissipation is present. It is our hope that this project will lay a solid foundation for future mathematical research on partial differential equations with fractional dissipation and for applications of these equations in the modeling of many fluid phenomena.