Nonlinear Fokker-Planck Equations and Hybrid Stochastic Deterministic Systems

本课题研究描述有限多自由度随机微观系统与宏观确定场之间主动相互作用的变分问题、非线性偏微分方程组和随机微分方程组。对具有有限多个自由度的相互作用粒子熔体的研究导致了度量空间和伪距离空间上的Onsager方程。该项目的第一部分是阐明这类系统的结构并对其零温极限进行分类。这些体系流体中的悬浮导致了非线性福克-普朗克方程与纳维尔-斯托克斯方程的耦合。对这些系统的全球存在性的偏微分方程分析是该项目的第二部分。除了这一分析外，还将进一步研究行波的存在，并用数值方法加以记录。这个项目的第三个目的是研究确定性模型和混合的随机-确定性模型之间的联系，这些模型在McKean意义上是非线性的。这个项目解决了复杂物质模型理论和计算中出现的基本问题。复杂物质包括血液等生物流体，也包括泡沫和凝胶等合成物质。复杂物质的特征是存在由非常小的物体组成的复杂的、相互作用的网络，这些网络被嵌入、影响并反过来影响基质或溶剂。对这些相互作用的基本了解对于在应用方面取得进展至关重要，例如制造具有极端性能的材料或在细胞水平上输送药物。该项目旨在揭示指导此类系统建模和计算的基本原则。

The project is to study variational problems, nonlinear partial differential equations, and stochastic differential equations that describe the active interactions between stochastic microscopic systems with finitely many degrees of freedom and macroscopic deterministic fields. The study of melts of interacting particles with finitely many degrees of freedom leads to Onsager equations on metric and pseudometric spaces. A first part of the project is to elucidate the structure and classify the zero temperature limits of such systems. The suspensions in fluids of these systems lead to nonlinear Fokker-Planck equations coupled to Navier-Stokes equations. The PDE analysis of global existence for these systems is a second part of the project. In addition to this analysis, the existence of traveling waves, documented numerically, will be further investigated. The connection between deterministic models and hybrid, stochastic-deterministic models, nonlinear in the sense of McKean, is the third aim of this project.This project addresses fundamental problems that arise in the theory and computation of models of complex matter. Complex matter includes biological fluids such as blood and also synthetic matter such as foams and gels. Complex matter is characterized by the presence of complex, interacting networks of very small objects being embedded, influenced, and in turn influencing a matrix or a solvent. The basic understanding of these interactions is crucial for progress in applications such as the manufacturing of materials with extreme properties or the delivery of drugs at the cellular level. The project is aimed at uncovering fundamental principles that guide the modeling and computation of such systems.