Nonlinear Partial Differential Equations

非线性偏微分方程

• 批准号: 9971713
• 负责人: Daniel Phillips
• 金额: $ 14.2万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971713&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; Equations

The principal investigator will study mathematical models for superconductivity and liquid crystals. These materials are described by models posed at a variety of length scales.The intent of the research is to investigate the qualitative features of the solutions to these models, analyze the thresholds at which the models predict change in the materials' basic characteristics (changes of phase) and investigate the connections between the solutions of different models at different length scales. The methods to be employed are analysis of partial differential equations, asymptotic methods, homogenization, and numerical simulation.Superconductors are used in electronic sensing devices, magnets and power transmission.Applications for liquid crystals are visual displays and optical screens.These are examples of materials for which their characteristic features (their phases) can change dramatically if subjected to electro-magnetic fields or variations in temperature.This phenomenon is a great asset (essentially as if one has two materials in the same location) but it is also a difficult feature to control.The goal of the project is to add to the understanding of the behavior of the models and their interrelations so that these materials can be used more effectively. This will be done by means of mathematical analysis of these models in general settings.

首席研究员将研究超导和液晶的数学模型。这些材料由不同长度尺度下的模型描述，本研究的目的是研究这些模型解的定性特征，分析模型预测材料基本特性变化（相变）的阈值，并研究不同长度尺度下不同模型解之间的联系。所采用的方法是偏微分方程分析、渐近方法、均匀化和数值模拟。超导体用于电子传感器件，磁铁和电力传输。液晶的应用是视觉显示器和光学屏幕。这些是材料的例子，它们的特性特征如果受到电磁场或温度变化的影响，（它们的相位）会发生巨大的变化。这种现象是一种巨大的财富（基本上就像一个人在同一位置有两种材料），但它也是一个难以控制的特征。该项目的目标是增加对模型行为及其相互关系的理解，以便更有效地使用这些材料。 这将通过在一般设置中对这些模型进行数学分析来完成。