Probabilistic Approach to Singular Free Boundary Problems and Applications

科学和工程中许多现象的定量理解，如晶体和人体组织的生长，混合物的相分离，大脑和金融机构中神经元的集体行为，以及合金的制造，需要动态演变表面的数学模型。第一个这类模型，即所谓的自由边界问题，早在1831年就被提出，并在世纪后半叶在科学和工程领域变得无处不在。尽管自由边界问题得到了相当多的关注，但对它们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新的方法来解决自由边界问题，以现代概率论为基础，以促进对这一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理想的不规则性，例如，晶体或神经同步的快速生长，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研究生和本科生研究的监督，以及研究生和本科生在概率论和随机分析方面的培训。自由边界问题为自然科学和社会科学以及工程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表述可以追溯到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异性，或者在时间上，捕获例如晶体或神经同步的快速生长，或者在空间上，封装例如肿瘤表面的非光滑性质，或者在合金的组分之间。这个项目建立在对一维空间中的单相过冷Stefan问题的分析基础上，这是奇异自由边界问题的一个典型例子。其目的是扩大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，例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类似物，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，和多重空间维度的Hele-Shaw问题。此外，研究人员打算确定这些解决方案与随机增长模型的大规模限制定义的相互作用粒子systems.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.