Free Boundary Problems and nonlinear PDEs

自由边界问题和非线性偏微分方程

• 批准号: 0970072
• 负责人: Inwon Kim
• 金额: $ 14.2万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0970072&HistoricalAwards=false
• 关键词: Free; Boundary; Problems; nonlinear; PDEs

The principal investigator's past research has focused on interface problems arising in a variety of physical phenomena such as phase separation, fluid dynamics, materials science, and continuum limits of nonequilibrium particle systems. The goal of the current project is to gain a better understanding of general properties of free boundary problems using various notions of weak solutions, with the help of maximum-principle-type arguments, harmonic analysis, measure theory, and interacting particle systems. Specific questions under investigation are (i) global-time existence and uniqueness; (ii) uniting notions of weak solutions; (iii) regularity properties; (iv) long-time behavior of solutions and (v) geometric properties of free boundaries. Another project is to study homogenization of the free boundaries in periodic and random media, moving with oscillating normal velocity caused by inhomogeneities in the media. An example is the dynamics of water droplets spreading on an irregular surface. The lower dimensional nature of free boundaries bring major challenges to the homogenization, especially in nonvariational settings. The goal is to prove existence, uniqueness, and stability of the effective free boundary problem in periodic and random media.Nonlinear interface motions arise naturally in physical applications, and they have been extensively studied in the physics and applied mathematics literature. A classical example is the melting of ice pieces or snow crystals, where the central issue of interest is the evolution of the ice/water/air interface. The highly nonlinear structure and the development, in general, of finite-time singularities on the interface -- such as a stream of water pinching into droplets -- give rise to rather challenging difficulties in the rigorous analysis. It is important, both in theory and in practice, to understand what type of singularities occur and under what circumstances the behavior of the interfaces exhibits stability.

首席研究员过去的研究主要集中在各种物理现象中出现的界面问题，如相分离、流体动力学、材料科学和非平衡粒子系统的连续统极限。当前项目的目标是利用各种弱解的概念，在最大原理型论证、谐波分析、测量理论和相互作用粒子系统的帮助下，更好地理解自由边界问题的一般性质。正在调查的具体问题是：(i)全球时间的存在性和唯一性；（ii）统一弱解的概念；（iii）正则性；（iv）解的长时性和(v)自由边界的几何性质。另一个项目是研究周期和随机介质中自由边界的均匀化，它们以振荡的法向速度运动，这是由介质中的不均匀性引起的。一个例子是水滴在不规则表面上扩散的动力学。自由边界的低维性质给均匀化带来了重大挑战，特别是在非变分环境中。目的是证明周期和随机介质中有效自由边界问题的存在性、唯一性和稳定性。非线性界面运动在物理应用中自然出现，在物理和应用数学文献中得到了广泛的研究。一个典型的例子是冰块或雪晶的融化，其核心问题是冰/水/空气界面的演变。高度非线性的结构和界面上有限时间奇点的发展，一般来说，如水流挤压成水滴，给严格的分析带来了相当具有挑战性的困难。无论在理论上还是在实践中，理解什么类型的奇点会发生，以及在什么情况下接口的行为会表现出稳定性是很重要的。