关于一些流体动力学方程的数学理论

In this project we mainly research the well-posedness, regularity and else properties of the solutions to some important fluid dynamics equations with the help of classical functional method and some modern mathematical tools and methods such as Fourier localization method from harmonic analysis, De giorgi iteration method, and so on. The details are as follows..(1) We research the partial regularity for the suitable weak solutions to some fluid dynamics equations closely related to incompressible Navier-Stokes equations, with the help of classical blow-up method studying the partial regularity problem and some modern methods such as harmonic extension, De giorgi iteration method, and so on;.(2) We research the backward uniqueness about the fractal dissipative equation and the regularity problem about some fluid dynamics equations with fractional order dissipation by some tools such as the harmonic extension of the frational order dissipative operator;.(3) We research the losing estimates about the transport-diffusion equation with fractional dissipation by using the Fourier localization method and Besov spaces theory;.(4) We research the box-counting dimension of the possible singularity points set for the suitable weak solutions to some fluid dynamics equations closely related to incompressible Navier-Stokes equations by establishing some more subtle localization estimates.

本项目主要是利用经典的泛函分析方法和Fourier局部化方法等调和分析方法及De Giorgi迭代方法等现代的数学工具和方法来研究流体动力学方程（组）解的适定性、正则性和其他相关性质。具体内容是：.（1）利用blow-up方法等经典的部分正则性问题的研究方法和调和延拓、De Giorgi迭代方法等现代的研究方法研究不可压缩Navier-Stokes方程及相关流体动力学方程的适当弱解的部分正则性问题；.（2）利用分数阶耗散算子的调和延拓等工具研究分数阶耗散方程的倒向唯一性和带有分数阶耗散的流体动力学方程的正则性问题；.（3）利用Fourier局部化方法和Besov空间理论研究带有分数阶耗散的输运扩散方程的导数损失估计问题;.（4）通过建立适当弱解的一些更精细的局部型估计研究不可压缩Navier-Stokes方程及相关流体动力学方程的适当弱解的奇异点集的box-counting维数问题。

本项目主要是利用了经典的泛函分析方法和Fourier局部化方法等调和分析方法及De Giorgi迭代方法等现代的数学工具和方法来研究流体动力学方程（组）解的适定性、正则性和其他相关性质。本项目主要研究了不可压缩Navier-Stokes方程的适当弱解的部分正则性问题，分数阶Navier-Stokes方程、Navier-Stokes-Coriolis方程、Magnetohydrodynamics方程、chemotaxis-Navier-Stokes方程和带有分数阶耗散的不可压缩流体动力学方程的适定性与正则性问题等流体动力学方程的数学理论。本项目在国内外期刊上共发表相关学术论文30篇，部分论文提出了一些新的研究方法，解决了一些较为困难的问题，改进了一些已有的研究结果，本项目的这些研究成果有助于加深对流体动力学方程数学理论的理解。