Motion By Curvature In Phase Transitions

摘要：多年来，PI一直从事各种自由边界问题的研究，研究了多孔介质中流体的重力流动、多维单相Hele-Shaw问题、一相和两相Mullins-Sekerka模型、带表面张力的拟平稳Stefan问题、表面扩散流等模型经典解的存在性、唯一性、规律性和定性性质。这项研究解决了一些长期存在的开放性问题。在这个项目中，PI将继续研究平均曲率驱动的曲面的几何演化问题。这些模型广泛应用于材料科学、物理和化学中，以模拟相变、畴生长和界面控制晶体生长。数学前沿的进步必然会对材料科学产生影响。Mullins-Sekerka模型是一种非局部几何演化规律，其中传播界面的法向速度取决于一个函数的法向导数在界面上的跳跃，该函数在任何一侧都是调和的，并且等于传播界面上的平均曲率。它被引入到研究零比热材料的凝固和清算中，引起了人们的广泛关注。Alikakos、Bates和Chen的重要贡献是将该模型与Cahn-Hilliard方程的奇异极限联系起来。Cahn-Hilliard方程是一个四阶抛物方程，被广泛用作二元合金熔体中相分离和粗化现象的模型。这个模型也被用来解释相变中的老化或奥斯特瓦尔德成熟。一般来说，一级相变动力学的特征是在第一个阶段中，新相的小液滴从旧相中产生，例如，在过冷液体中形成固体。第一阶段称为成核，产生大量的小颗粒。在下一阶段，细胞核以牺牲旧阶段为代价迅速生长。当相区形成时，新相的质量接近平衡，过冷量较小，但存在较大的表面积。下一阶段，相区域的构型变得粗糙化，相区域的几何形状变得越来越简单，最终趋向于给定体积下最小表面积的区域。这一过程的驱动力来自于降低界面能的需要。在寻找描述奥斯特瓦尔德成熟的理论方面已经付出了相当大的努力，Mullins-Sekerka模型是一个突出的候选者。表面扩散流动和中间表面扩散流动是模拟形态变化的几何演化问题，其中表面扩散和界面动力学是传递机制。这些定律构成了一类动力学问题，其中体积守恒，驱动力是表面能的减少。

PI: Gieri Simonett DMS-9801337 ABSTRACT: Over the last years the PI has worked on various free boundary problems and has studied existence, uniqueness, regularity, and qualitative properties of classical solutions for such models as the gravitational flow of a fluid in a porous medium, the multi-dimensional one-phase Hele-Shaw problem, the one and two-phase Mullins-Sekerka model,the quasi-stationary Stefan problem with surface tension, and the surface diffusion flow.This research led to the solution of some long-standing open problems. In this project the PI will continue to study geometric evolution problems for surfaces driven by mean curvature. These models are widely used in material sciences, physics, and chemistry to model phase changes, domain growth,and interface controlled crystal growth. Progress on the mathematical front will necessarily have an impact in material sciences. The Mullins-Sekerka model is a nonlocal geometric evolution law in which the normal velocity of a propagating interface depends on the jump across the interface of the normal derivative of a function which is harmonic on either side and which equals the mean curvature on the propagating interface. It was introduced to study solidification and liquidation of materials of zero specific heat and has attracted considerable attention since then.Important contributions by Alikakos, Bates and Chen have tied this model to a singular limit for the Cahn-Hilliard equation, a fourth order parabolic equation which is widely used as a model for phase separation and coarsening phenomena in a melted binary alloy. This model has also been proposed to account for aging or Ostwald ripening in phase transitions. In general, the kinetics of a first order phase transition is characterized by a first stage where small droplets of a new phase are created out of the old phase, e.g., solid formation in an undercooled liquid. The first stage, called nucleation,yields a large number of small particles .During the next stage the nuclei grow rapidly at the expense of the old phase.When the phase regions are formed,the mass of the new phase is close to equilibrium and the amount of undercooling is small,but large surface area is present.At the next stage, the configuration of phase regions is coarsened, and the geometric shape of the phase regions become simpler and simpler, eventually tending to regions of minimum surface area with given volume. The driving force of this process comes from the need to decrease the interfacial energy. There have been considerable effortsin finding a theory which describes Ostwald ripening, and the Mullins-Sekerka model is a prominent candidate.The surface diffusion flow and the intermediate surface diffusion flow are geometric evolution problems which model morphological changes where surface diffusion and interface kinetics are the transport mechanisms. These laws constitute a class of dynamic problems where the volume is conserved and the driving force is surface energy reduction.