Mathematical Sciences: Thermistor Problems and Interfacial Dynamics

本课题主要研究两个方面：（1）热敏电阻 建模，其中涉及两个退化的非线性系统 椭圆型偏微分方程;（2）界面 涉及超曲面运动的动力学， 抛物型偏微分方程解的水平集 方程，以及这两个运动之间的关系。 为 第一个问题，研究人员计划使用经典的 椭圆型偏微分方程分析，变分的 不等式、自由边界分析和一些复分析 建立的存在性，唯一性，和规律性的 解决方案，并提供计算的数值方案， 电压的解和基本关系 (or电流）及其总电阻率， 这对于设计和测试热敏电阻器是必不可少的。 为 界面动力学问题，研究人员将使用 奇异摄动方法，粘性方法，泛函 分析等等，建立一些基于物理的标准， 界面的运动，并提供一些数学 现代模式与现代模式之间的一致性 凝固理论及相关领域的经典模型。 热敏电阻器是一种热敏电阻器， 已广泛应用于电流调节、开关、气体 检测、控制和报警等。研究者将使用 数学工具，以找到之间的关系， 晶闸管，从而为实际设计提供指导方针 和试验. 在凝固的情况下，界面是 固体和液体的边界。 界面的运动是 对凝固理论至关重要。 许多好的模型 关于界面的运动已经被提出。 的 调查人员计划严格证明这些关系， 特别是各种模型的共同点，并发现 基于物理的界面运动标准， 目前的模式提供了多种选择。

This project contains two research areas: (1) thermistor modeling, which involves a nonlinear system of two degenerate elliptic partial differential equations, and (2) interfacial dynamics which involves the motion of hypersurfaces, the motion of level sets of solutions of parabolic partial differential equations, and the relations between these two motions. For the first problem, the investigator plans to use the classical analysis of elliptic partial differential equations, variational inequalities, free boundary analysis, and some complex analysis to establish the existence, uniqueness, and regularity of the solutions and to provide numerical schemes for the calculation of the solutions and the fundamental relations between the voltage (or current) applied to the thermistor and its total resistivity, which is essential for designing and testing thermistors. For the interfacial dynamics problem, the investigator will use singular perturbation methods, viscosity methods, functional analysis, etc., to establish some physically based criteria for the motion of the interfaces, and to provide some mathematical justification on the uniformity between the modern models and the classical models in solidification theory and related areas. A thermistor is a heat sensitive electrical resistor which has been widely used in current regulation, switching, gas detection, control and alarms, etc. The investigator will use mathematical tools to find relations among the characteristics of thermistors, thereby providing guidelines for practical design and testing. Interface, in the solidification case, is the boundary between solid and liquid. Motion of the interface is essential to the solidification theory. Many good models concerning the motion of interfaces have been proposed. The investigator plans to justify rigorously these relations, especially the common aspects of various models, and find physically based criteria for the motion of the interfaces where the current models give multiple choices.