Qualitative Studies of Some Partial Differential Equations and Systems

一些偏微分方程和系统的定性研究

• 批准号: 0140604
• 负责人: Changfeng Gui
• 金额: $ 7.1万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0140604&HistoricalAwards=false
• 关键词: Qualitative; Studies; Some; Partial; Differential

PI: Changfeng Gui, University of ConnecticutDMS-0140604Abstract:The PI will work on two projects on qualitative properties of partial differential equations. In the first project, he will study phase transitions modeled by the Allen-Cahnequation and its generalized vector equations, which are the gradient flows of the Allen-Cahn energy with either double-well potentials or multiple-well potentials of equal depths. The objectives are to determine the existence, uniqueness and stability of some special configurations, and then to further study the fine structure near the interfaces or nearthe triple junction and the dynamics of these sharp interfaces andtriple junctions. One of the immediate goals is to classify the optimalconfigurations of anti-phases. This is indeed to solve a conjecture ofDe Giorgi which concerns the one dimensional symmetry ofcertain entire solutions of the Euler-Lagrange equations of theAllen-Cahn energy. The De Giorgi conjecture is also related to the studyof minimal surfaces, and is still open in dimensions bigger than three.In the second project the PI will study the Gierier-Meinhardtsystems of equations arising in biological pattern formations and chemicalreactions. In particular, the proposor will try to show mathematicallythe existence of some special concentration patterns and to understandtheir stabilities.Diffusion is a very common phenomenon, and it generates verycomplex structures when several different substances are involved.Phase transitions, pattern formations are well observed in materialsciences, in biological and chemical reactions, and they are closelyrelated to diffusions. Special nonlinear partial differential equationsand systems are used to model these phenomena mathematically. Theproposor hope to develop analytic techniques to obtain goodqualitative properties for solutions of these equations, which will leadto better numerical methods for simulations. The study will help tounderstand the finer structures and long time behaviors of phasetransitions, pattern formations, and other similar phenomena.

主要研究者：Changfeng Gui，University of ConnecticutDMS-0140604摘要：PI将致力于两个关于偏微分方程定性性质的项目。 在第一个项目中，他将研究由艾伦-卡恩方程及其广义矢量方程建模的相变，这些方程是艾伦-卡恩能量的梯度流，具有相同深度的双阱势或多阱势。 其目的是确定某些特殊构型的存在性、唯一性和稳定性，进而进一步研究这些尖锐的界面或三重结附近的精细结构以及这些界面和三重结的动力学性质。 当前的目标之一是对反相的最佳构型进行分类。这实际上是为了解决De Giorgi关于Allen-Cahn能量的Euler-Lagrange方程某些整体解的一维对称性的猜想。 De Giorgi猜想也与极小曲面的研究有关，并且在大于3的维度上仍然是开放的。在第二个项目中，PI将研究生物图案形成和化学反应中产生的Gierier-Meinhardt方程组。特别是，建议者将试图说明一些特殊的浓度模式的存在，并了解他们的稳定性。扩散是一种非常普遍的现象，当涉及到几种不同的物质时，它会产生非常复杂的结构。相变，图案的形成在材料科学，生物和化学反应中很好地观察到，它们与扩散密切相关。 用特殊的非线性偏微分方程和系统对这些现象进行数学建模. 作者希望发展分析技术来获得这些方程的解的良好的定性性质，这将导致更好的数值模拟方法。 该研究将有助于理解相变、斑图形成和其他类似现象的精细结构和长时间行为。