Mathematical Sciences: Thermistor Problems and Interfacial Dynamics

本项目包括两个研究领域：(1)热敏电阻建模，涉及两个退化椭圆型偏微分方程的非线性系统；(2)界面动力学，涉及超曲面的运动，抛物型偏微分方程解的水平集的运动，以及这两种运动之间的关系。对于第一个问题，研究者计划使用椭圆偏微分方程的经典分析、变分不等式、自由边界分析和一些复分析来建立其存在性、唯一性、给出了热敏电阻的解的计算方法和热敏电阻的电压（或电流）与总电阻率之间的基本关系，这对热敏电阻的设计和测试至关重要。对于界面动力学问题，研究者将使用奇异摄动法、粘度法、泛函分析等，建立一些基于物理的界面运动准则。并对凝固理论及相关领域中现代模型与经典模型的一致性提供了一定的数学依据。热敏电阻是一种热敏电阻器，广泛应用于电流调节、开关、气体检测、控制和报警等领域。研究者将使用数学工具来发现热敏电阻特性之间的关系，从而为实际设计和测试提供指导。界面，在凝固情况下，是固体和液体之间的边界。界面运动是凝固理论的基础。关于界面运动，人们提出了许多很好的模型。研究者计划严格地证明这些关系，特别是各种模型的共同方面，并为当前模型提供多种选择的界面运动找到基于物理的标准。

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.