Mathematical Sciences: Free Boundary Problems and Inter- facial Dynamics

本文的主要目的是研究一类描述超曲面几何运动的自由边界问题，分析一类描述界面现象的演化方程和系统的渐近行为，将自由边界问题的解与演化方程或系统的渐近行为联系起来，并提出计算自由边界问题解的数值格式。在自由边界问题的研究中，我们将重点研究Hele-Shaw自由边界问题，即寻找一个超曲面的运动，该运动的法向速度等于一个函数的法向导数在超曲面上的跳跃，该函数在超曲面的互补域中是调和的，并且其在超曲面上的边值等于超曲面的平均曲率。在渐近行为的研究中，研究将着眼于模拟二元合金中相分离和相粗化现象的Cahn- Hilliard方程，以及描述固体/清算过程中温度和液固相变化的相场模型。这个项目涉及应用数学中的微分方程。功能分析和几何技术将用于分析这些方程。由于这里研究的问题在数学和物理科学中是非常基础的，因此本项目中开发的方法可以应用于许多其他当前活跃的研究领域，从而推进对界面现象的认识。特别是，本研究可应用于先进材料的研究。＊＊＊

9404773 Chen The primary objective of this proposal is to study certain free boundary problems which describe geometric motions of hypersurfaces, to analyze the asymptotic behavior of certain evolution equations and systems describing interfacial phenomena, to connect the solutions of the free boundary problems with the asymptotic behaviors of evolution equations or systems, and to develop numerical schemes calculating the solutions to the free boundary problems. In the study of free boundary problems, the research will place particular emphasis on a Hele-Shaw free boundary problem , which is to find the motion of a hypersurface such that the normal velocity of the motion is equal to the jump across the hypersurface of the normal derivative of a function that is harmonic in the domain complementary to the hypersurface and whose boundary value on the hypersurface is equal to the mean curvature of the hypersurface. In the study of the asymptotic behavior, the research will aim at the Cahn- Hilliard equation modeling the phase separation and phase coarsening phenomena in a binary alloy, and at the phase- field model describing the temperature and liquid-solid phase changes in a solid/liquidation process. This project deals with differential equations of applied mathematics. Techniques of functional analysis and geometry will be used to analyze these equations. Since the problems studied here are very basic in mathematical and physical sciences, the methods developed in this project can be applied to many other currently active research areas and consequently advance the knowledge of interfacial phenomena. In particular, applications of this research can be made to advanced materials research. ***