Questions on Diffusive Phenomena

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

This project considers several fundamental models arising in fluid dynamics and one from chemotaxis. These models are: the Navier-Stokes and Euler equations, which describe hydrodynamics phenomena; polymeric equations, which model noninteracting polymer chains; the two-dimensional quasi-geostrophic (2DQG) equations, which describe meteorological phenomena; and, finally, the damped Boussinesq system, which models the propagation of water waves in shallow water. Part of the project relates to the central questions for nonlinear partial differential equations, namely, regularity, formation of turbulence, and the possible construction of explicit solutions. Specifically, the principal investigator is concerned with the construction of special solutions to the Euler and Platak-Keller-Segel equations (the latter is a well-known model for chemotaxis phenomena) or, in the other direction, with establishing that such solutions cannot exist. It is clear that such constructions are physically meaningful. A central part of the proposal is concerned with the stability of solutions to the Navier-Stokes equations. The work proposed in connection with the other three fluid models: the Polymeric, the 2DQG, and the Boussinesq equations is expected to yield new information on the long-time behavior of the solutions. All of the models considered in this project have the potential for interesting applications. The principal investigator's interest in these models stems from the possibility of working on a more applied side of the subject than she has in the past, in particular, to be able to have an interdisciplinary interaction with other scientists. In what follows the discussion focuses on two of the main models mentioned above: the Navier-Stoke equations and the polymeric equations. The main attraction with respect to the Navier-Stokes equations is that, as a model for viscous flows, it is used to study, among many other things, blood flow. The hope is that a broad theoretical understanding of the equation will lead eventually to the ability to test for the behavior of the real flow, and perhaps predict its long-time behavior. The polymer equations in its original form is obtained by coupling the Navier-Stokes equations with an equation that describes the time evolution of the probability density function of the position of a particle. A polymer is a substance composed of a large molecular mass consisting of repeated structural units (monomers) connected by covalent chemical bonds, which exist due to the sharing of electrons between atoms. The attraction-repulsion stability that is caused by the common electron is what characterizes the covalent bonding. The idea of covalent bonding between long chains of atoms was introduced in a ground-breaking and controversial paper by Hermann Staudinger in 1920 (Nobel Laureate in Chemistry, 1953). The simplest model to account for noninteracting polymer chains is the so-called dumbbell model. A dumbbell consists of two beads connected by an elastic spring. One can imagine that in this model the beads represent the atoms, while the elastic spring plays the role of the covalent bond. This simple model is what the project will seek to understand first. Here again the stress is on the long-time behavior of the motion that the dumbells undergo. All the problems considered in this proposal can be used as basis for work with undergraduate students and for Ph.D. projects.