Nonlinear Fokker-Planck Equations and Hybrid Stochastic Deterministic Systems

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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.