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Qualitative Studies of Some Partial Differential Equations and Systems

一些偏微分方程和系统的定性研究

基本信息

批准号：
1601885
负责人：
Changfeng Gui
金额：
$ 24万
依托单位：
University of Texas at San Antonio
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601885&HistoricalAwards=false
关键词：
Qualitative Studies Some Partial Differential

项目摘要

The Allen-Cahn equation was first proposed by materials scientists as a model for studying phase separation between different materials as well as phase transitions between different phases of the same materials. An important aspect of the equation is that it enables the display of interfaces separating different physical regions of interest. Such interfaces often share significant features with soap bubbles or, in more precise mathematical terminology, with minimal surfaces. The equation has also found applications in many other area of science and engineering such as astrophysics and image processing. In addition to its significance in advancing knowledge in mathematics and other sciences, the Allen-Cahn equation provides an excellent tool for training students and junior researchers in interdisciplinary research. The principal investigator will engage in both the research and the training activities related to the equation and will involve students at both the undergraduate and the graduate levels, using the theoretical study of the equation to develop better algorithms for medical image analysis. Postdoctoral fellows and junior researchers will also participate and be trained in the project. The principal investigator plans to study saddle solutions and traveling wave solutions of Allen-Cahn-type equations, including the classic scalar equations with double well potentials and vector-valued equations with multiple well potentials. He will focus on the existence of special saddle solutions with prescribed level sets as well as on the level set structure of solutions of finite Morse index, in particular, on the relation between the level sets of solutions and minimal surfaces. He intends to use various identities, as well as Morse index information, to develop new approach for these nonmonotone, nonminimizing solutions. The long-term goal of the project is to understand general entire solutions to both scalar and vector-valued Allen-Cahn equations and to gain insight into the stability and dynamics of triple junctions or quadruple junctions. The nodal sets or singularities of the solutions will receive special attention in the study, for not only do they play an important role in the theoretical analysis of the equation, but they also represent in applications the interfaces or junctions of interfaces of different phases or grain boundaries in materials such as crystalline alloys. The project will broaden the participation of underrepresented minorities, including Hispanic students, in mathematical and interdisciplinary research.

Allen-Cahn方程最早由材料科学家提出，作为研究不同材料之间相分离以及相同材料不同相之间相变的模型。该方程的一个重要方面是，它能够显示分离不同物理感兴趣区域的界面。这种界面通常与肥皂泡或更精确的数学术语中的最小表面共享重要特征。该方程还在许多其他科学和工程领域，如天体物理学和图像处理中得到应用。除了在数学和其他科学知识的进步意义，艾伦-卡恩方程提供了一个很好的工具，培养跨学科研究的学生和初级研究人员。主要研究者将参与与方程相关的研究和培训活动，并将涉及本科生和研究生水平的学生，利用方程的理论研究来开发更好的医学图像分析算法。博士后研究员和初级研究人员也将参与该项目并接受培训。主要研究者计划研究Allen-Cahn型方程的鞍解和行波解，包括经典的双阱势标量方程和多阱势向量值方程。他将专注于存在的特殊鞍解决方案与规定的水平集以及水平集结构的解决方案的有限莫尔斯指数，特别是，对之间的关系水平集的解决方案和最小的表面。他打算使用各种身份，以及莫尔斯指数信息，开发新的方法，这些非单调，非最小化的解决方案。该项目的长期目标是了解标量和矢量值Allen-Cahn方程的一般整体解，并深入了解三重结或四重结的稳定性和动力学。解的节点集或奇点将在研究中受到特别关注，因为它们不仅在方程的理论分析中起着重要作用，而且在应用中还代表了不同相或晶界的界面或界面的接合点，如晶体合金。该项目将扩大包括西班牙裔学生在内的代表性不足的少数民族在数学和跨学科研究中的参与。