分数阶 Boussinesq 方程的适定性研究

In this project, we are devoted to the study of the incompressible Boussinesq equations with fractional Laplacian dissipation and the corresponding inviscid problems. The Boussinesq equations act as an important model in the geophysical dynamics and have comprehensive backgrounds in atmospheric fronts as well as oceanic circulation and so on. The main contents of this project are: the global well-posedness for the two-dimensional incompressible Boussinesq equations with fractional Laplacian dissipation, long time asymptotic behavior of the corresponding global solution, the inviscid limited problems, the eventual regularity of global weak solutions and the Beal-Kato-Majda criterion for the inviscid Boussinesq equations etc., especially, we shall also investigate the relation between the order of fractional Laplacian operator and the global well-posedness of the above problems, furthermore, we shall deeply study this class of operators, such that we can show some properties of solution to the corresponding inviscid problems. We mainly apply Harmonic analysis, microlocal analysis including Littlewood-Paley theory, Bony's paraproduct and pseudodifferential theory, and combine with bilinear estimate、commutator estimate、energy methods and functional spaces theory etc. to consider our problems. We hope that the results to be established in this project can be helpful and referencable to the diffusion equations with the fractional Laplacian operator, inviscid equations and other fluid mechanics systems, meanwhile this results can enrich the theory and approaches of studying the fluid equations.

本项目致力于研究具有分数阶耗散的不可压缩Boussinesq方程以及它所对应的无粘性方程。Boussinesq方程是地球物理动力学中的重要模型，在海洋环流,大气等领域中有广泛的应用背景。本项目所研究的内容主要包括讨论具有分数阶耗散的二维不可压缩Boussinesq方程的整体适定性、长时间渐近性态、无粘性极限、弱解最终正则性以及无粘性Boussinesq方程的Beal-Kato-Majda爆破准则等问题,着重探讨分数阶Laplace算子的耗散指标对整体适定性的影响，并通过对该算子的讨论去探寻相应的无粘性方程解的性质。我们主要利用调和分析、微局部分析特别是Littlewood-Paley分解、Bony的仿微分技术和拟微分算子理论，双线性估计、交换子估计并结合能量方法以及函数空间理论等来研究此类问题。期望本项目的研究能给相应的具有分数阶耗散方程、无粘性方程以及其他的流体动力学方程的研究提供参考，

本项目研究不可压缩Boussinesq方程及相关方程的适定性问题。所研究方程是流体动力学中的基本方程，是非线性偏微分方程研究的中心问题之一，在天气预报、航空航天、海洋生态等领域中有极强的应用背景。本项目主要探讨了如下流体动力学方程的整体适定性问题：具有各向异性分数阶耗散的Boussinesq方程、粘性项依赖温度的Boussinesq方程、各向异性分数阶耗散的准地转方程、分数阶磁流体方程、分数阶微极流体方程、分数阶热带气候模型等。目前对于这些流体动力学方程的耗散指标做到最大范围，并得到了一些重要的不等式以及有意思的理论结果，这对进一步讨论超临界以及完全无粘情形的流体动力学方程有着很重要的意义。项目负责人与项目组成员在本项目资助下已经在本专业著名期刊发表SCI学术论文，其中包括JMPA、JDE、Nonlinearity、JNS、DCDS-A。