Free boundary problems and Viscosity solutions.

提案DMS-0401436 PI：大卫杰里森和克里斯蒂娜金（麻省理工学院）标题：自由边界问题和粘性解摘要该项目的主要重点是在非线性微分方程中的问题，其中边界是未知的，必须确定：所谓的“自由边界”。PI研究具有非局部结构的自由边界问题，其中自由边界的速度依赖于解的整体特征。我们的目标是更好地理解这类问题的以下几个方面：整体时间内解的存在性和唯一性;自由边界在时间到达零时或无穷大时的渐近行为;等待时间现象;以及自由边界的正则性。PI采用粘性解的概念来处理前面提到的自由边界问题。这种方法已成功地用于研究非线性偏微分方程和局部运动规律，如“平均曲率运动”。这种方法的最大优点是对夹断等拓扑转变的处理很好。粘性溶液理论的关键是建立比较原理。这一性质反过来又导致解的存在，并允许进一步研究其性质。利用比较原理的定量形式和调和分析的技巧，进一步研究了解的性质，并以经典的Stefan融冰问题为例进行了讨论。在斯特凡问题中，一个是模拟极地冰盖的演变，而感兴趣的问题是水和冰之间的界面的位置，作为时间的函数。本建议的方法适用的具体问题还包括：Hele-Shaw问题，该问题模拟了两个平行板之间的狭窄单元中的流体运动;火焰前沿;以及油和水之间的界面。这是一个建模和计算远远领先于数学分析的领域。发展一个定义明确的界面一般运动概念的主要障碍是，初始光滑边界在光滑速度下运动可能在有限时间内发展奇异性。火焰传播中界面的碰撞和分裂就是一个例子。很自然地，人们会想知道夹断后解的连续性是否是唯一确定的，或者在单一时刻是否需要额外的本构信息。奇点对自由边界另一部分的影响是另一个有趣的问题。 PI计划从本地和全球角度调查这些问题。

Proposal DMS-0401436PIs: David Jerison and Christina Kim (MIT)Title: Free boundary problems and viscosity solutionsABSTRACTThe main focus of this project is on problems in nonlinear differentialequations in which the boundary is unknown and has to be determined: aso-called 'free boundary'. The PI is interested in the free boundaryproblems with nonlocal structure, where velocities of the free boundariesdepend on global characteristics of the solutions. The goal is to gain abetter understanding of the following aspects of such problems: theexistence and uniqueness of solutions in global time; asymptotic behaviorof the free boundary as time reaches zero or infinity; waiting timephenomena; and the regularity properties of the free boundaries. The PIemploys the notion of viscosity solutions to address the behavior ofaformentioned free boundary problems. This approach has been succesfullyused to study nonlinear PDEs and local laws of motions such as 'motion bymean curvature'. The great advantage of this approach is its seamlesshandling of topological transitions such as pinch-off. The key-step of thetheory of viscosity solutions is to establish a comparison principle. Thisproperty leads, in turn, to the existence of solutions and allows for thefurther study of their properties. Techniques coming from quantitativeversions of the comparison principle and harmonic analysis are employed tostudy further properties of solutions.The classical Stefan problem of melting ice is an example of the freeboundary problems. In the Stefan problem, one is modeling the evolution ofthe polar ice caps, and the question of interest is the location, as afunction of time, of the interface between water and ice. The particularproblems to which the methods of the present proposal apply also include:theHele-Shaw problem which models fluid motion in a narrow cell betweentwo parallel plates; flame fronts; and the interface between oil andwater in a flow. This is an area where modeling and computation are farahead of mathematical analysis. The main obstacle for developing awell-defined notion for general motion of interfaces is that initiallysmooth boundaries moving under smooth velocities may develop singularitiesin finite time. The hitting and splitting of interfaces in flamepropagation is an example. It is natural to wonder whether thecontinuation of the solutions is uniquely determined after pinch-off, orwhether additional constitutive information might be required at thesingular time. The impact of the singularity to the other part of the freeboundary is another interesting question. The PI plans to investigatethese questions in local and global perspectives.