9704155 Hunter The proposal is to study a number of nonlinear partial differential equations which model various physical and biological systems. The first equation is a free-boundary value problem which models the deposition of surfactant on a surface by a liquid droplet. There is a complex interaction between the motion of the droplet on the surface and the deposition of surfactant. The second equation is a system of reaction-diffusion equations which models an islet of beta-cells in the pancreas. The aim of Professor Hunter's research is to understand the collective dynamics of the beta-cells and the formation of spatial patterns in an islet. The third equation is a two dimensional Burgers equation which models the diffraction and reflection of weak shock waves in two space dimensions, where there are longstanding discrepancies between theory and experiment. The fourth equation is a modulation equation which describes the propagation of nonlinear gravitational waves in Einstein's theory of general relativity. Professor Hunter proposes to study this equation with the aim of understanding the formation of space-time singularities in a nonlinear gravitational wave. Mathematical models of physical and biological systems provide a way to understand, predict, and control the behavior of those systems. The proposed research involves the study of a number of mathematical models of systems of importance in engineering, biology, and basic science. One model describes the deposition of surfactant on a surface by droplets. Surface deposition has many applications in materials science, including the printing of very small structures on surfaces for the control of fluids, and as an experimental means of preparing liquid crystal display screens. A second model describes the collective behavior of beta-cells in the pancreas. These cells produce insulin and play a central role in diseases like diabetes. A third model describes the propagation of shock waves, which are generated by transonic or supersonic aircraft, in turbines, and in many other high-speed fluid flows. The mathematical theory of shock waves remains very poorly understood. The fourth model is a set of equations which describe the propagation of nonlinear gravitational waves in Einstein's general theory of relativity. The general theory of relativity is the fundamental physical theory of gravity, and it describes the large scale cosmological structure of the universe. The main obstacle to understanding the predictions of the theory is the nonlinearity of the Einstein equations. This work is directed towards an increased understanding of the effects of this nonlinearity.
