权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Dynamic Free Boundary Problems

动态自由边界问题

基本信息

批准号：
1900804
负责人：
Inwon Kim
金额：
$ 29.2万
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：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900804&HistoricalAwards=false
关键词：
Dynamic Free Boundary Problems

项目摘要

Dynamic free boundary problem refers to solving partial differential equations in a domain whose evolution is a priori unknown. One example is the problem of melting ice, where the interface of ice and water is determined dynamically by the distribution of temperature, or solution of the heat equation, in the water region. A particular focus is given on problems where solutions are given with very little regularity. A good example is the problem of drops sliding on tilted surface. In this case the shape of the drop can change drastically when the velocity is increased, and a singularity (corner) develops at the rear of the drop when the velocity exceed a certain critical value. At higher speeds the tail of the drop may break into another component (pearling). Another example is oil drops moving up towards water in a container when initially water, the heavier fluid, is poured on top of oil. The project aims towards a better understanding of the properties of these problems and provide a framework for developing accurate computer-based numerical simulations. Graduate students will be trained by participating in the research.The problems to be discussed arise in a variety of physical phenomena, including the motion of capillary drops, crowd motion, tumor growth, and filtration of fluids in porous media. The presence of lower-dimensional structure is ubiquitous in the physical literature, either as a boundary of a domain or a singular part of an evolution. Many nonlinear PDE problems which are otherwise well-understood face significant challenges when they are coupled with a moving interface, even in seemingly simple settings. Besides the nonlinearity of the problem, the difficulty lies in the non-locality of the problem, in the sense that the behavior of solutions depends on the global geometry of the free boundary. Dynamic problems face an additional difficulty coming from the lack of a priori regularity of the free boundaries. By studying proposed problems we aim to develop general methods to deal with such difficulties. In addition to energy and integral estimates, it will be often necessary to introduce geometric methods to understand pointwise behavior of the moving interface.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

动力学自由边界问题是指在一个区域上求解偏微分方程，而该区域的演化是先验未知的。一个例子是融化冰的问题，其中冰和水的界面由水域中的温度分布或热方程的解动态地确定。一个特别的重点是给的问题，解决方案给出了很少的规律性。一个很好的例子是水滴在倾斜表面上滑动的问题。在这种情况下，当速度增加时，液滴的形状可以急剧变化，并且当速度超过某个临界值时，在液滴的后部形成奇点（拐角）。在较高的速度下，液滴的尾部可能会断裂成另一个组分（珍珠状）。另一个例子是，当最初将水（较重的流体）倒在油的顶部时，油滴向上移动到容器中的水。该项目旨在更好地了解这些问题的性质，并为开发精确的基于计算机的数值模拟提供一个框架。研究生将通过参与研究进行培训。要讨论的问题出现在各种物理现象中，包括毛细滴的运动，群体运动，肿瘤生长和多孔介质中流体的过滤。低维结构的存在在物理文献中是普遍存在的，无论是作为域的边界还是演化的奇异部分。许多非线性偏微分方程的问题，否则很好地理解面临着重大的挑战，当他们与移动界面耦合，即使在看似简单的设置。除了问题的非线性，困难在于问题的非局部性，在这个意义上，解决方案的行为取决于自由边界的整体几何。动力学问题面临着一个额外的困难，来自缺乏先验的规则性的自由边界。通过研究提出的问题，我们的目标是制定一般的方法来处理这些困难。除了能量和积分估算外，通常还需要引入几何方法来理解运动界面的逐点行为。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。