Analysis of Some Nonlinear Problems in Electromagnetic Fields

数学科学：电磁场中一些非线性问题的分析[摘要]尹洪明本项目研究电磁场和热传导理论中出现的几个非线性问题。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成。目的是分析是否发生热失控（爆炸），如果在加热过程中发生，如何防止它。更复杂的自由边界问题，模拟加热过程中的相变，也在研究中。第二个问题领域涉及对p-拉普拉斯型演化系统的研究，该系统模拟了超导理论中的经典Bean临界状态模型。这项工作的重点是了解正常和超导状态之间的相变是如何发生的。本项目研究涉及电磁场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。这项工作的结果将有助于更好地理解电磁场的复杂动力学，并将为某些实验现象提供解释。此外，这项研究将产生对麦克斯韦方程组和非线性热传导研究普遍感兴趣的数学结果。

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.