几类具有非局部扩散的发展方程的数学理论

Until this century, the mathematical theory of evolution equations with nonlocal diffusion attracted some scholars’ interests from the PDEs aspects. Currently, many mathematicians, including L Caffarelli、L Silvestre and J L Vazquez etc., pay their special attentions to this kind of equations, but it seems that few Chinese scholars have studied these problems...In this project, we will investigate the mathematical theory of several kinds of evolution equations with nonlocal diffusion. In particular, we will study the global existence, regularity, large time asymptotic behavior, regularity of free boundary and dynamical limits of Cauchy problem for the diffusion equations of Riesz type, the generalized porous medium equations with nonlocal diffusion and the diffusion equations of Landau type. We also consider the homogeneous Dirichlet boundary problem for the generalized porous medium equations with nonlocal diffusion. ..Due to the appearance of nonlinearity and nonlocal diffusion, there are many essential difficulties for this project. We will use the modern theory of parabolic partial differential equations and the techniques from harmonic analysis (for instance, the De Giorgi iteration, singular integrals, …) to overcome these difficulties. We will discover the difference between the nonlocal diffusion and the classical diffusion, which reveals the effects of long-range interaction in diffusion.

具有非局部扩散的发展方程的严格数学理论是本世纪才引起偏微分方程专家注意的课题。当前，包括 L Caffarelli、L Silvestre 和 J L Vazquez 等数学家都十分关注这类方程的研究，相比而言，国内似乎很少有数学家专注于这类问题。..本项目主要研究几类具有非局部扩散的重要发展方程的数学理论，重点研究Riesz型非局部扩散方程、多孔介质型非局部扩散方程、Landau型非局部扩散方程的Cauchy问题解的整体存在性、正则性、大时间渐近行为、自由边界的正则性以及动力学极限等；我们也将考虑多孔介质型非局部扩散方程的齐次Dirichlet 边值问题。..非线性和非局部的同时出现给这类问题的研究带来了极大的困难。我们将充分利用抛物与椭圆型偏微分方程的现代理论和调和分析技巧，特别是De Giorgi类型的迭代、奇异积分估计等，研究非局部扩散与经典扩散的异同，揭示扩散的长程相互作用。

本项目主要利用抛物与椭圆型偏微分方程的现代理论和调和分析技巧研究了具有非局部扩散的偏微分方程的适定性理论、大时间行为等，特别是研究了如下六个方面的内容：..i). 具有信号吸引与信号产生的趋化-流体耦合方程组的整体适定性、一致有界性、小对流极限、快信号扩散极限等；ii). 对数灵敏系数方程组、吸引-排斥方程组的整体存在性与一致有界性等；iii). 不具有磁扩散的不可压缩 MHD 方程、可压缩 Hall-MHD 方程等的整体适定性与自由边值问题的局部适定性等；iv). 有界域上具有弱非齐次初值的 Boltzmann 方程的整体适定性、Boltzmann 方程到带热传导的不可压缩流体力学方程组的逼近、具有弱角奇异性的非截断 Vlasov-Maxwell-Boltzmann 方程组的整体适定性、相对论 Vlasov-Maxwell-Fokker-Planck 方程组的整体适定等；v). 2×2弱耦合 Schrödinger 方程组的系数识别问题等；vi). 修正的 b-family 方程解的爆破和解的持久性等。..项目组全体成员在项目执行期潜心研究，圆满完成了预定的研究目标，特别是基于这些研究成果已在 Math Z、Cal Var、Math Mod Meth Appl Sci、Ann Sc Norm Super Pisa、J Differ Eq、Nonlinearity、Sci China Math、J Functional Anal、Inverse Problem 等高水平数学期刊上发表学术论文 20 余篇，得到了国内外同行的高度评价，已被 SCI 引用 200 余次。