Nonlinear Partial Differential Equations in Applied Mathematics

[9704155] Hunter这个建议是研究一些模拟各种物理和生物系统的非线性偏微分方程。第一个方程是一个自由边值问题，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。第二个方程是一个反应扩散方程系统，它模拟了胰腺中β细胞的胰岛。亨特教授的研究目的是了解β细胞的集体动力学和胰岛空间模式的形成。第三个方程是二维Burgers方程，它模拟了弱冲击波在两个空间维度上的衍射和反射，在这方面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非线性引力波的传播。亨特教授建议研究这个方程，目的是理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。所提出的研究涉及对工程、生物学和基础科学中重要系统的一些数学模型的研究。一个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。第三种模型描述了由跨音速或超音速飞机、涡轮和许多其他高速流体流动产生的冲击波的传播。人们对冲击波的数学理论仍然知之甚少。第四个模型是描述爱因斯坦广义相对论中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结构。理解该理论预测的主要障碍是爱因斯坦方程的非线性。这项工作是为了增加对这种非线性影响的理解。

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.