Free Boundary Problems and Viscosity solutions

自由边界问题和粘性解建议的研究摘要克里斯蒂娜我金PI建议继续她的计划对自由边界问题的解决方案的研究与海勒-肖流和斯特凡问题。PI计划研究（i）时间上的全局存在性和唯一性;（ii）正则性;（iii）解的长时间行为;以及（iv）有界域中自由边界的几何性质。她还建议研究均匀化的自由边界在周期性介质中，移动与振荡正常速度所造成的不均匀性的媒体。一个例子是水滴在不规则表面上扩散的动力学。目前的目标是证明有效自由边界速度的存在性、唯一性和正则性。预期的结果将提供洞察的影响，在媒体上的自由边界的全球行为的不均匀性。自由边界问题出现在流体力学和材料科学中，涉及边界不完全未知的域中的偏微分方程的解。描述边界是问题的一个重要部分。 一个例子是火焰传播，其中重要的是跟踪燃烧区和未燃烧区之间的界面的演变。在这个项目中要研究的问题包括描述固体-流体相变的方程组，在一个狭窄的细胞中的流体流动，和表面上的液滴。这些问题已经被广泛的实验和计算研究，但很少被证明有关theequations模型这些系统.该项目旨在提供有关解决方案的定性和定量信息，以解释实验和数值模拟中观察到的各种现象。

Free Boundary Problems and Viscosity SolutionsAbstract of Proposed Research Christina I KimThe PI proposes to continue her program on the study of solutions of the free boundary problems associated with Hele-Shaw flows and Stefan problems. The PI plans to investigate (i) the global existence in time, and uniqueness; (ii) regularity properties; (iii) the long-time behavior of solutions; and (iv) geometric properties of the free boundaries in bounded domains. She also proposes to study homogenization of the free boundaries in periodic 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 current goal is to prove the existence, uniqueness and regularity properties of the effective free boundary velocity. The anticipated results will provide insight into the effect of inhomogeneities in the media on the global behavior of the free boundary. Free boundary problems arise throughout fluid mechanics and material science and involve the solution of partial differential equations in a domain whose boundary is not entirely unknown. Describing the boundary is an important part of the problem. An example is flame propagation, where it is important to track the evolution of the interface between burnt and un-burnt zones. The problems to be studied in this project include systems of equations describing solid-fluid phase transitions, fluid flow in a narrow cell, and liquid droplets on a surface. These problems have been extensively studied experimentally and computationally but very little has been proved about theequations that model these systems. This project aims to provide qualitative and quantitative information about the solutions that will explain various phenomena observed in experiments and numerical simulations.