不可压Boussinesq方程和MHD方程的数学理论研究

Boussinesq方程和MHD方程分别是地球流体力学和磁流体力学中的基本方程。基于分数阶拉普拉斯算子在概率论、金融学、材料科学及量子力学等领域的广泛应用，研究具分数阶耗散Boussinesq方程和MHD方程有重要的数学理论及应用价值，是非线性偏微分方程研究的前沿领域之一。. 本项目拟采用新的尺度变换、谱分析方法和现代分析中的工具，研究具分数阶耗散和部分粘性的不可压Boussinesq方程和MHD方程解的整体正则性、大时间性态等数学理论。通过项目实施给出解决流体方程正则性问题的一些较系统的方法；深入刻画耗散项与传输项之间的平衡关系，为经典N-S方程千禧年问题提供新的视角；丰富和发展对无粘流体方程爆破问题的认知研究，为不可压流体的实际应用提供数学支持。

This project focuses on the Boussinesq equations and the magnetohydrodynamic (MHD) equations with partial or fractional dissipation. The Boussinesq equations play an important role in the study of many geophysical fluids while the MHD equations are the center piece in modeling many electrically conducting fluids such as plasmas. Due to the wide applicability of the fractional Laplace operator in probability, finance, fluid dynamics and quantum mechanics, the study of the Boussinesq and MHD equations with fractional dissipation has recently gained a lot of momentum and has gradually become the forefront of research on nonlinear PDEs.. The goal of this project is to solve several global regularity and large-time behavior problems on the fractionally or partially dissipated Boussinesq and MHD equations. We will employ various advanced approaches such as modern tools from harmonic analysis, spectral analysis and scaling invariant properties. This project will produce a systematic method on solving the global regularity problems for fluid PDEs with fractional dissipation. Especially we will provide a strategy on how to balance the nonlinearity by the fractional dissipation. It is hoped that this project will shed light on the Millennium Prize problem on the 3D Navier-Stokes equations and lay a foundation for practical applications on imcompressible fluids.