Analysis of Some Nonlinear Problems in Electromagnetic Fields

美国国家科学基金会奖摘要- DMS-0102261数学科学：电磁场中的一些非线性问题的分析摘要DMS-0102261尹洪明本项目研究电磁场和热传导理论中的几个非线性问题。 第一个问题领域涉及感应加热过程的研究。 数学模型由非线性麦克斯韦方程组和非线性热方程组组成。 目的是分析是否发生热失控（爆炸），以及如果在加热过程中发生热失控，如何防止热失控。 更复杂的自由边界问题，在加热过程中模拟相变，也正在研究中。 第二个问题领域涉及调查的p-Laplacian类型的演化系统，模型的经典Bean的临界态模型的超导理论。 这项工作的重点是了解正常状态和超导状态之间的相变是如何发生的。本项目研究在涉及电磁场的情况下热流的数学模型。 这些模型对于许多工业应用都很重要。 这项工作的结果将导致更好地理解电磁场的复杂动力学，并将提供某些实验现象的解释。 此外，这项调查将产生的数学结果的普遍利益的研究麦克斯韦方程组和非线性热传导。

NSF Award Abstract - DMS-0102261 Mathematical Sciences: Analysis of Some Nonlinear Problems in Electromagnetic Fields AbstractDMS-0102261 Hong-Ming YinThis project investigates several nonlinear problems arising from the theory of electromagnetic fields and heat conduction. The first problem area concerns investigation of induction heating processes. The mathematical model consists of nonlinear Maxwell's equations coupled with a nonlinear heat equation. The objective is to analyze whether or not thermal runaway (blowup) happens and how to prevent it if it does occur during a heating process. More complicated free boundary problems, modeling phase change during the heating process, are also under study. The second problem area concerns investigation of a p-Laplacian type of evolution system that models the classical Bean's critical-state model in the superconductivity theory. The effort focuses on understanding how the phase change takes place between normal and superconducting states. This project investigates mathematical models of heat flow in situations that involve electromagnetic fields. These models are important for many industrial applications. The results of this work will lead to a better understanding the complicated dynamics of electromagnetic fields and will provide explanations for certain experimental phenomena. In addition, this investigation will produce mathematical results of general interest for the study of Maxwell's equations and nonlinear heat conduction.