Mathematical Sciences: Partial Differential Equations: Free Boundary Problems

9703842弗里德曼，拟议的研究将集中于开发新的分析方法来解决自由边界问题。第一类问题是用于模拟涂层流动的Navier-Stokes方程。第二个主题将是Stefan类型的问题，其中自由边界有非常大的振荡，这涉及均化和边界层；例如，在化学气相沉积中会出现这样的模型。下一个主题将是弹性材料中裂纹的扩展，其中自由边界随时间移动到该区域。其他研究领域将涉及椭圆型变分不等式的非线性系统和粒子在流体中的运动。需要研究的问题出现在工业过程和设计的数学建模中。它们包括照相胶片的涂层、用于改善声音传输的磁头的设计、半导体的加工、浸泡的流变性以及金属裂纹的分析。根据我们过去的经验，预计将开发的分析方法将使人们能够开发出非常有效的数值方案来计算这些模型的感兴趣的量，从而可以在更短的时间内设计出改进的工业产品。

9703842 Friedman The proposed research will concentrate on the development of new analytic methods for solving free boundary problems. The first class of such problems are the Navier-Stokes equations used to model coating flow. The second topic will be Stefan type problems where the free boundary has very large oscillations and this involve homogenization and boundary layers; such models arise, for example, in chemical vapor deposition. Next topic will be the propagation of a crack in elastic material, where the free boundary moves, with time, into the region. Other areas of research will involve nonlinear systems of elliptic variational inequalities and motion of particles in fluid. The problems to be studied arise in mathematical modeling of industrial processes and design. They include the coating of photographic film, the design of magnetic heads for improved sound transmission, processing of semiconductors, rheological properties of immersions, and the analysis of cracks in metals. Based on our past experience, it is anticipated that the analytic methods which will be developed will enable one to develop very efficient numerical schemes for computing quantities of interest for these models, so that improved industrial products can be designed in shorter time.