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

Probabilistic Approach to Singular Free Boundary Problems and Applications

奇异自由边界问题的概率方法及其应用

基本信息

批准号：
2108680
负责人：
Mykhaylo Shkolnikov
金额：
$ 30.41万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2108680&HistoricalAwards=false
关键词：
Probabilistic Approach Singular Free Boundary

项目摘要

A quantitative understanding of many phenomena in science and engineering, such as the growth of crystals and human tissue, the phase segregation of mixtures, the collective behavior of neurons in the brain and of financial institutions, and the manufacturing of alloys, requires mathematical models of dynamically evolving surfaces. The first models of this kind, so-called free boundary problems, were proposed as early as 1831 and became ubiquitous across science and engineering in the second half of the 20th century. Despite the considerable attention devoted to free boundary problems, the understanding of the solutions they generate is still very limited. In this project, the investigator will use a novel approach on free boundary problems, grounded in modern probability theory to advance the understanding of this important class of mathematical models. A particular focus of the project will be on the desirable irregularities present in these models capturing, for example, the rapid growth of a crystal or neural synchronization, as well as the non-smooth nature of the surface of a tumor or between the components of an alloy. Integral parts of the project are supervision of graduate and undergraduate research, and graduate and undergraduate training in probability theory and stochastic analysis.Free boundary problems provide a universal mathematical framework for many phenomena in the natural and social sciences as well as engineering. While the mathematical formulation of free boundary problems dates back to the 19th century and their analysis has received much attention in the 20th century, the understanding of their solutions is still limited. This is largely due to the singularities oftentimes exhibited by the solutions, either in time, capturing for example the rapid growth of a crystal or neural synchronization, or in space, encapsulating for example the non-smooth nature of the surface of a tumor or between the components of an alloy. This project builds on the analysis for the one phase supercooled Stefan problem in one space dimension, a prototypical example of a singular free boundary problem. The aim is to expand the scope of the novel probabilistic solution concept they introduced to include a large set of singular free boundary problems, such as the one phase supercooled Stefan problem for the nonlinear heat equation and its two-phase analogue in one space dimension, the supercooled Stefan problem in multiple space dimensions, possibly with kinetic undercooling and/or surface tension, and the Hele-Shaw problem in multiple space dimensions. In addition, the investigator intends to identify these solutions with large scale limits of random growth models defined by means of interacting particle systems.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.

要定量理解科学和工程中的许多现象，如晶体和人体组织的生长，混合物的相分离，大脑和金融机构神经元的集体行为，以及合金的制造，需要动态演变表面的数学模型。这种类型的第一个模型，即所谓的自由边界问题，早在1831年就被提出，并在20世纪下半叶在科学和工程中变得无处不在。尽管人们对自由边界问题给予了相当大的关注，但对它们所产生的解的理解仍然非常有限。在这个项目中，研究人员将使用一种基于现代概率论的关于自由边界问题的新方法来促进对这类重要数学模型的理解。该项目的一个特别重点将是这些模型中存在的令人满意的不规则性，例如，捕捉晶体的快速生长或神经同步，以及肿瘤表面或合金成分之间的非光滑性质。该项目的组成部分是指导研究生和本科生的研究，以及研究生和本科生在概率理论和随机分析方面的培训。自由边界问题为自然科学、社会科学和工程学中的许多现象提供了一个通用的数学框架。虽然自由边界问题的数学公式可以追溯到19世纪，其分析在20世纪受到了极大的关注，但对其解的理解仍然有限。这在很大程度上是由于溶液经常表现出的奇点，要么在时间上捕捉到例如晶体或神经同步的快速生长，要么在空间上包裹例如肿瘤表面或合金成分之间的非光滑性质。这个项目建立在对一维空间中一相过冷Stefan问题的分析的基础上，这是一个奇异自由边界问题的典型例子。其目的是扩展他们引入的新的概率解概念的范围，以包括大量的奇异自由边界问题，如非线性热方程的单相过冷Stefan问题及其在一维空间中的两相模拟，多维空间中的过冷Stefan问题(可能具有动力学过冷和/或表面张力)，以及多维空间中的Hele-Shaw问题。此外，研究人员打算通过相互作用的粒子系统定义的随机增长模型的大规模限制来确定这些解决方案。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。