Nonlinear heat equations applied to geometry and mechanics

应用于几何和力学的非线性热方程

• 批准号: 0101124
• 负责人: Sigurd Angenent
• 金额: $ 19.2万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-15 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0101124&HistoricalAwards=false
• 关键词: Nonlinear; heat; equations; applied; geometry

The PI intends to continue the study of singularities of nonlinear heatequations in differential geometry and mathematical physics, such as TheRicci flow, Harmonic Map flow, Mean curvature flow, and the Porous mediumequation. Possible singularities will be analyzed by means of matchedasymptotic expansions. In addition to this the PI hopes to continue his collaboration withgroups in medical imaging, serving at least as a source of theoreticknowledge for such groups and perhaps collaborating more intensely.Nonlinear diffusion equations occur in many forms in mathematical physicsand engineering, as well as in pure mathematics (differential geometry).Understanding of the solutions to such equations can be gained by studyingtheir singularities. E.g. singularities of a solution to an equationmodeling a chemical reaction in a reactor may correspond to explosion asopposed to steady reaction. The PI intends to study such singularities. In addition, the PI intends to continue his collaboration with aresearch group in Medical Imaging. The practical problems of how toautomatically process 3D fMRI images all belong to a branch of mathematicscalled differential geometry. A detailed knowledge of differential geometryhas proved to be very useful in medical imaging in the past, and promissesto stay important. Conversely, medical imaging problems may serve as asource of new mathematical problems and theories.

PI计划继续研究微分几何和数学物理中非线性热方程的奇异性，如里奇流、谐波图流、平均曲率流和多孔介质。用匹配渐近展开式分析可能的奇异点。除此之外，PI希望继续与医学成像小组合作，至少为这些小组提供理论知识的来源，也许会更密切地合作。非线性扩散方程以多种形式出现在数学、物理、工程以及纯数学（微分几何）中。对这类方程的解的理解可以通过研究它们的奇点来获得。例如，模拟反应堆中化学反应的方程的解的奇点可能对应于爆炸而不是稳定反应。PI打算研究这样的奇点。此外，PI打算继续与医学成像研究小组合作。如何自动处理3D功能磁共振成像图像的实际问题都属于数学的一个分支，称为微分几何。微分几何的详细知识在过去的医学成像中被证明是非常有用的，并且有望保持重要地位。相反，医学成像问题可以作为新的数学问题和理论的来源。