分数阶偏微分方程的正则性问题

Fractional partial differential equations are a class of integro-differential equations. Since they are global equations, the theorems and methods of traditional partial differential equations cannot be simply generalized. .. There are mainly two kinds of methods to study fractional partial differential equations. One way is stochastic, while another way is analytic. The analytic methods mainly consist of two approaches. The first method is employing Fourier analysis, and mainly using the known theory of singular integrals. The second one is the localization technique, and mainly utilizing the known skills of elliptic/parabolic equations... In recent years many important results for fractional partial differential equations have been obtained in Hölder spaces. Nevertheless, numerous problems are still unsolved because of the complexity of fractional partial differential equations. Meanwhile, there are only limited interesting results for fractional partial differential equations in Lebesgue spaces. The main reason is that fractional partial differential equations are global equations. Therefore, it is extremely difficult to estimate the distribution functions of the gradients of solutions... In this project, we will do research on the regularity problems for fractional partial differential equations, with the aim to obtain similar results to second order partial differential equations and to find the special properties of fractional partial differential equations.

分数阶偏微分方程是一类积分-微分方程。由于分数阶偏微分方程是一个全局的方程，传统偏微分方程的理论和研究方法并不能简单推广到分数阶偏微分方程上。..研究分数阶偏微分方程的方法主要有两大类。一类是随机的方法，另一类是分析的方法。而分析的方法目前主要分为两种。一种利用Fourier分析，主要借助奇异积分的成熟理论。另一种办法是局部化方法，主要利用椭圆或抛物型方程的技巧。..近年来分数阶偏微分方程的研究在Hölder空间取得了很多重要的成果。由于分数阶偏微分方程的复杂性，大量的问题仍然没有解决。同时，分数阶偏微分方程在Lebesgue空间的令人感兴趣研究工作甚少。原因是分数阶偏微分方程是一个全局的方程，因而估计方程解的梯度的分布函数极为困难。..本项目将研究分数阶偏微分方程的正则性问题，旨在获得与二阶偏微分方程相似的结果，同时发现分数阶偏微分方程的特有性质。

本项目主要研究内容包括非线性分数阶抛物方程古典解、熵解和重整化解的存在性、唯一性、等价性和正则性等基本问题，拟线性椭圆和抛物方程的弱解、熵解和重整化解的存在性、唯一性、等价性和正则性等核心问题，线性和非线性抛物-抛物方程组的古典解或弱解的存在性、唯一性等基础问题。在项目的资助下，项目组成员积极组织和参加了丰富的学术活动，在相关领域获得一大批高水平的研究成果，解决了一批引人关注的重要问题。这些成果发表在国内外重要影响力的学术杂志上，产生了显著的效果。