Free Boundary Problems and nonlinear PDEs

这位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介质极限。本项目的目标是利用各种弱解的概念，借助极大值原理、调和分析、度量理论和相互作用的粒子系统，更好地理解自由边界问题的一般性质。所研究的具体问题是：(I)全局时间的存在唯一性；(Ii)弱解的统一概念；(Iii)正则性；(Iv)解的长时间行为；(V)自由边界的几何性质。另一个项目是研究周期性和随机介质中自由边界的均匀化，由介质中的不均匀引起的振荡法向速度运动。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很大的挑战，特别是在非变分环境下。本文的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是自然而然的，在物理学和应用数学文献中得到了广泛的研究。一个经典的例子是冰块或雪晶的融化，其中感兴趣的中心问题是冰/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--例如水流挤压成液滴--给严格的分析带来了相当具有挑战性的困难。在理论和实践中，重要的是要了解发生什么类型的奇点，以及在什么情况下界面的行为表现出稳定性。

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.