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

作者：angenent@math.wisc.edu（西古尔德Angenent），注释日期： 3/10/98 4：39 PM优先级：正常收件人：cberenst at nsf 11主题：回复：DMS-9800894 - PI旨在研究非线性椭圆和扩散方程， 研究它们的解可能具有的奇异性。的特定 待研究的方程包括曲线缩短（及其变体） 平均曲率流，调和映射流，里奇流和多孔介质方程。所提出的方法是找到形式渐近 展开的解决方案附近的奇点，然后设计分析 方法，以决定是否与发现的奇异性类型的解决方案可以存在或不存在。 对非线性扩散方程及其解的奇性有一个很好的理解，可以解决微分学中的一些老问题， 几何学，就像它过去所做的那样。除了这个纯粹的数学 有趣是，非线性扩散方程出现在广泛的科学领域， 工程背景。例如，多孔介质方程描述了粘性材料（如油）通过多孔介质（砂）的流动。所以... 被称为“曲线缩短“的技术已经被用于增强和 自动分析视频图像，特别是医学MRI图像。 对非线性扩散方程的深入理解， 科学家和工程师用一套新的数学运算， 可以用于图像处理。 --西古尔德昂让特

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