Free Boundary Problems and nonlinear PDEs

首席研究员过去的研究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极限。目前的项目的目标是获得一个更好的理解的一般性质的自由边界问题使用各种概念的弱解，最大原理型参数，调和分析，测度理论和相互作用的粒子系统的帮助。具体的问题正在调查的是（i）全球时间的存在性和唯一性;（ii）统一的概念弱解;（iii）正则性;（iv）长期行为的解决方案和（v）几何性质的自由边界。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，这些边界以由介质中的不均匀性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，特别是在非变分的情况下。非线性界面运动在物理应用中自然出现，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发展，在一般情况下，有限时间奇点的接口-如水流捏成液滴-在严格的分析中产生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发生以及在什么情况下界面的行为表现出稳定性都是很重要的。

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.