Free Boundary Problems and Reaction-Diffusion Systems

DMS Award AbstractAward #： 0203991 PI： Chen，Xinfu机构： 爱丁堡大学课程： 应用数学项目经理：Catherine Mavriplis职务：自由边界问题和反应扩散问题本项目讨论自然界中常见的界面现象。 这个项目有两个目标。一个是建立一个描述可溶性物质在液体环境中运动的适用模型。另一个是利用现有的反应扩散系统和自由边界模型来研究某些在工业应用中很重要的界面现象。当涉及化学反应时，可使用的现有模型是反应-扩散系统，如活化剂-抑制剂、FitzHugh-Nagumo、Belousov-Zhabotinsky、Gray-Scott和Giererer-Meinhardt。大部分的重点将在这里慢扩散-快反应方程和他们的奇异极限，即自由边界问题。调查的很大一部分将是找到解决方案有关的界面现象，如旅行点，尖峰，目标模式，棋盘图案，旋转螺旋波，和自我复制的行为。对自由边界问题和反应扩散系统中的模式进行深入的研究，将使人们能够预测和控制看似复杂却又自然的界面动力学。 它还可以发展数学理论和思想。作为一个特殊的例子，某些不具有唯一性的演化方程，因此经典上被认为是不适定的，将被仔细研究，并被视为良好的模型，因为它们反映了不可观察的局部微观变化可能影响整个可观察的宏观结果的性质。 在化学、生物学和许多应用科学中，化学物质在液体环境中由于溶解或化学反应而移动是很常见的。通常圆形是稳定的，圆形的点像弹性物体一样移动和相互作用，有自己特殊的“动量定律”。该项目的一个目的是开发，至少对于液体介质中的可溶性固体，一个模型，使人们能够计算一个移动点的速度和反射原理-当一个移动点从另一个点或容器的物理边界反弹时，入射角和出射角之间的关系。为斑点的运动开发的模型预计将在许多领域找到有用的工业应用，如水污染控制，化学测试，化学处理等。这个项目将涉及研究生，并激发他们对具有实际意义的数学问题的兴趣。日期：2002年6月14日

DMS Award AbstractAward #: 0203991PI: Chen, XinfuInstitution: University of PittsburghProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: FREE BOUNDARY PROBLEMS AND REACTION-DIFFUSION SYSTEMSThis project addresses interfacial phenomena that are commonplace in nature. There are two goals in this project. One is to develop an applicable model describing motion of soluble substances in a liquid environment. The other is to use existing reaction--diffusion systems and free boundary models to study certain interfacial phenomena that are important in industrial applications. When chemical reactions are concerned, existing models to be used are reaction--diffusion systems such as the activator--inhibitor, FitzHugh--Nagumo, Belousov-Zhabotinsky, Gray--Scott, and Gierer--Meinhardt. Most of the focus here will be on slow diffusion--fast reaction equations and their singular limits, namely, free boundary problems. A large part of the investigation will be to find solutions related to interfacial phenomena, such as traveling spots, spikes, target patterns, checkerboard patterns, rotating spiral waves, and self--replicating behavior. The underlying investigation for patterns in free boundary problems and reaction--diffusion systems will enable people to predict and to control the seemingly complicated yet natural interfacial dynamics. It can also develop mathematical theories and ideas. As a particular example, certain evolution equations that do not possess uniqueness and thus classically are regarded as ill--posed, will be carefully investigated and regarded as good models since they reflect the nature in which an unobservable local microscopic change may affect the whole observable macroscopic outcome. Such a research will be guided by the current industrial needs.In chemistry, biology, and many applied sciences, it is common that chemical substances move around in a liquid environment, due to dissolution or chemical reaction. Quite often circular shapes are stable and circular spots move and interact each other like elastic objects, with their own special ``momentum laws". One purpose of the project is to develop, at least for soluble solids in a liquid medium, a model enabling one to calculate the speed of a traveling spot and the reflection principle---the relationship between the incoming and outgoing angles when a traveling spot bounces off another spot or the physical boundary of a container. The model developed for the motion of spots is expected to find useful industrial applications in many areas such as water contamination control, chemical testing, chemical processing, etc. This project will involve graduate students and stimulate their interest in mathematical problems of practical importance.Date: June 14, 2002