This project will study a large set of nonlinear partial differential equations that describe the dynamics of complex fluids and immersed lower-dimensional geometric objects. These partial differential equations involve nonlinear couplings of transport, phase-field, Fokker-Planck, and geometric evolutionary equations, along with the Navier-Stokes equations. They may be of both parabolic and hyperbolic nature and possess sharp interfaces, singularities, or multiple scales. These are fascinating and challenging problems that are directly motivated by problems in biology, physics, fluid mechanics, and materials science. A thorough understanding of them will require new ideas and methods. The project encompasses a concrete set of problems, builds on some partial results, and proceeds under a detailed plan that describes the efforts to be made and approaches to be taken. One expects exciting new developments in the theory and interesting applications to other fields.
Many biological fluids (blood, for example) fall under the heading of "complex fluids." The study of complex fluids also arises in materials science, medicine, and physics. Understanding the dynamics of these fluids, particularly in the presence of immersed geometric objects, is a challenging problem and is very important for various applications, including the design of medical and high-tech devices. It is, in general, expensive to perform experiments with complex fluids in order to collect a good set of data about them. It is also not easy to model their dynamics nor to do accurate computer simulations of their behavior. Some theoretical analysis of mathematical models of complex fluids (such as that to be undertaken in this project) will not only lead to improved qualitative understanding of them, which is important for advances in our basic knowledge, but also provide insights into the nature of problems involving such fluids in a way that could help to set up reliable and effective numerical schemes for investigating them. The latter would provide data (equivalent to that produced by numerous physical experiments) that could prove useful for applications.
