Qualitative Studies of Some Partial Differential Equations and Systems

摘要：本项目将开展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allen - cah方程及其广义矢量方程模拟的相变，这些方程是具有等深度的双井势或多井势的Allen-Cahn能量的梯度流动。目的是确定一些特殊构型的存在性、唯一性和稳定性，进而进一步研究界面附近或三结附近的精细结构以及这些尖锐界面和三结的动力学。当前的目标之一是对反相的最佳配置进行分类。这实际上是为了解决de Giorgi的一个猜想，该猜想涉及到allen - cahn能量的欧拉-拉格朗日方程的某些完整解的一维对称性。De Giorgi猜想也与最小曲面的研究有关，并且在大于3的维度上仍然是开放的。在第二个项目中，PI将研究生物模式形成和化学反应中出现的方程的giier - meinhardt系统。特别地，作者将尝试用数学方法证明某些特殊浓度模式的存在，并理解它们的稳定性。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、模式形成在材料科学、生物和化学反应中都有很好的观察，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。作者希望发展解析技术来获得这些方程解的良好定性性质，这将导致更好的数值模拟方法。这项研究将有助于理解相变、模式形成和其他类似现象的精细结构和长时间行为。

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.