Nonlinear Fokker-Planck Equations and Hybrid Stochastic Deterministic Systems

该项目是研究变分问题，非线性偏微分方程，和随机微分方程，描述了随机微观系统与宏观确定性领域的许多自由度之间的积极相互作用。 研究具有多个自由度的相互作用粒子的熔体，导出了度量空间和伪度量空间上的Onsager方程。 该项目的第一部分是阐明这种系统的结构并对零温度极限进行分类。 这些系统在流体中的悬浮导致耦合到Navier-Stokes方程的非线性Fokker-Planck方程。 这些系统的全球存在的PDE分析是该项目的第二部分。 除了这种分析，行波的存在，数值记录，将进一步研究。 确定性模型与混合、随机-确定性模型（McKean意义上的非线性）之间的联系是本项目的第三个目标。本项目解决复杂物质模型的理论和计算中出现的基本问题。 复杂物质包括诸如血液的生物流体以及诸如泡沫和凝胶的合成物质。 复杂物质的特征是存在复杂的、相互作用的网络，这些网络由嵌入、影响并反过来影响基质或溶剂的非常小的物体组成。 对这些相互作用的基本理解对于诸如制造具有极端性质的材料或在细胞水平上递送药物等应用的进展至关重要。 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.