Singularities of Nonlinear Heat Equations; and Related Problems in the Calculus of Variations

作者：Angenent@math.wic.edu(Sigurd Angenent)注意日期：3/10/98 4：39 PM优先级：正常至：nsf11 cberenst主题：Re：dms-9800894PI旨在通过研究非线性椭圆型方程和扩散方程的解可能具有的奇异性来研究它们。要研究的特殊方程包括曲线缩短(及其变化)平均曲率流、调和映射流、Ricci流和多孔介质方程。所提出的方法是在奇点附近找到解的形式渐近展开，然后设计分析方法来判断所发现的奇点类型的解是否存在。很好地理解非线性扩散方程及其解的奇异性，可能会导致微分几何中老问题的解决，就像它过去所做的那样。除了这种纯粹的数学兴趣外，非线性扩散方程还出现在广泛的科学和工程背景中。多孔介质方程描述了粘性物质(如油)在多孔介质(沙)中的流动。所谓的“曲线缩短”已经被用于视频图像的增强和自动分析，特别是医学MRI图像。对非线性扩散方程的良好理解为计算机科学家和工程师提供了一套新的数学运算，他们可以在图像处理中使用。--西格德原始人

Author: angenent@math.wisc.edu (Sigurd Angenent) at NOTE Date: 3/10/98 4:39 PM Priority: Normal TO: cberenst at nsf11 Subject: Re: DMS-9800894 ------------------------------- Message Contents ------------------------------- Here is the one page description of my research proposal which you requested. The PI intends to study nonlinear elliptic and diffusion equations by investigating the singularities their solutions may have. The particular equations to be studied include Curve Shortening (and variations thereof) Mean Curvature Flow, the Harmonic Map Flow, the Ricci Flow and the Porous Medium Equation. The proposed method is to find formal asymptotic expansions of solutions near their singularities, and then to devise analytical methods to decide whether or not solutions with the found singularity types can exist or not. A good understanding of nonlinear diffusion equations and the singularities of their solutions may lead to the resolutions of old problems in differential geometry, as it has done in the past. Apart from this purely mathematical interest, nonlinear diffusion equations appear in a wide range of science and engineering contexts. E.g. the Porous Medium Equation describes the flow of a viscous material (like oil) through a porous medium (sand). So- called "Curve Shortening " has been used both for enhancement and for automated analysis of video images, and in particular medical MRI images. A good understanding of nonlinear diffusion equations provides computer scientists and engineers with a new set of mathematical operations which they can use in image processing. --Sigurd Angenent