分数阶Stokes方程和分数阶Navier-Stokes方程的理论和数值方法研究

The study of the fractional differential equations has a very long history, and is attracting increasing attention in last decade. Several research groups have contributed to the theoretical work and numerical investigations. Recently sub-diffusion and sup-diffusion problems of the incompressible flows related to the fractional Stokes equation and the fractional Navier-Stokes equation has been the subject of many research studies in the community of PDE theory. This project aims to propose variational framework of the fractional stokes equation and the fractional Navier-Stokes equation, establish the well-posedness of the weak problem, construct an efficient spectral method, and analysis its convergence and stability.. For the sub-diffusion problem, due to the non-local nature of fractional Laplacian operator, an unexpected numerical boundary layer will be generated near the domain boundary. To overcome this difficulty, a variable order fractional model is proposed and spectral method is used to solve this problem. For the sup-diffusion problem, as far as we know, the study of weak solution theory and the construction of high order numerical schemes have not yet been carried out. A key research content of this project is to construct the weak formulation of the problem, establish the well-posedness result by classical saddle-point theory. Then propose and analyze a stable spectral method, and derive the error estimates for the numerical solution.

分数阶微分方程的研究有很长的历史，并在近十多年得到快速的发展。几个国内外研究团队为其理论工作和数值研究做出了贡献。最近，与分数阶Stokes方程和分数阶Navier-Stokes有关的不可压流的次扩散和超扩散问题成为了偏微分方程理论圈子许多学者的研究课题。本项目旨在提出分数阶Stokes方程和分数阶Navier-Stokes方程的变分框架，导出弱问题的适定性，构造有效的谱方法，并分析其收敛性和稳定性。. 对于次扩散问题，由于分数阶Laplace算子的非局部性质，使得在区域边界附近会产生一个非预期的数值边界层。为了克服这一困难，将引入变分数阶模型，并通过谱方法去数值求解。对于超扩散问题，据目前所知，还没有弱解理论和高精度数值格式构造方面的研究。本项目的一个重点研究内容就是构造该问题的弱形式，通过传统的鞍点理论证明解的适定性。然后提出并分析一种稳定的谱方法，导出数值解的误差估计。

我们将分数阶Stokes和分数阶Navier-stokes问题分解成一些较为简单的问题。对空间分数阶FitzHugh-Nagumo模型，我们时间上采用2阶Crank-Nicolson格式，空间采用Legendre谱方法进行求解；对高维FitzHugh-Nagumo模型，我们时间上给出了一阶和二阶格式，空间上采用傅里叶谱方法进行求解；对带有非局部Lagrange乘子的相场水晶模型，我们基于IEQ方法在时间上提出一种二阶格式，空间采用傅里叶谱方法进行求解。对相场模型Allen-Cahn模型，我们提出一种新的时间上蛙跳格式结合空间上采用傅里叶谱的数值方法进行求解。上述罗列的格式在处理非线性部分，均采用了线性化处理，使得格式易于推广至高维以及其他非线性问题。所有方法都在理论上均证明了收敛性、稳定性和适定性，并都通过相关的数值例子验证了理论结果。