Nonlinear Partial Differential Equations in Applied Mathematics

9704155 猎人 该建议是研究一些非线性偏微分方程，模拟各种物理和生物系统。 第一个方程是一个自由边界值问题，它模拟了表面活性剂通过液滴在表面上的沉积。 液滴在表面的运动与表面活性剂的沉积之间存在着复杂的相互作用。第二个方程是反应扩散方程组，其模拟胰腺中的胰岛β细胞。亨特教授的研究目的是了解β细胞的集体动力学和胰岛中空间模式的形成。 第三个方程是二维Burgers方程，它模拟了弱激波在二维空间中的衍射和反射，其中理论和实验之间存在长期的差异。 第四个方程是一个调制方程，它描述了爱因斯坦广义相对论中的非线性引力波的传播。亨特教授提出研究这个方程，目的是了解非线性引力波中时空奇点的形成。 物理和生物系统的数学模型提供了一种理解、预测和控制这些系统行为的方法。拟议中的研究涉及一些数学模型的重要性，在工程，生物学和基础科学系统的研究。一种模型描述了表面活性剂通过液滴在表面上的沉积。 表面沉积在材料科学中有许多应用，包括在表面上印刷非常小的结构以控制流体，以及作为制备液晶显示屏的实验手段。第二个模型描述了胰腺中β细胞的集体行为。这些细胞产生胰岛素，并在糖尿病等疾病中发挥核心作用。第三个模型描述了跨音速或超音速飞机、涡轮机和许多其他高速流体流动中产生的冲击波的传播。 冲击波的数学理论仍然知之甚少。第四个模型是爱因斯坦广义相对论中描述非线性引力波传播的一组方程。 广义相对论是引力的基本物理理论，它描述了宇宙的大尺度宇宙学结构。 理解该理论预言的主要障碍是爱因斯坦方程的非线性。 这项工作是针对增加了解这种非线性的影响。

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.