Profiling singularities of geometric PDE

分析几何偏微分方程的奇点

• 批准号: 1205270
• 负责人: Dan Knopf
• 金额: $ 16.07万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205270&HistoricalAwards=false
• 关键词: Profiling; singularities; geometric; PDE

This project is concerned with nonlinear parabolic partial differential equations (PDE) and systems satisfied by geometric objects evolving by geometrically natural quantities such as curvature. These PDE are used in programs to evolve given geometries toward ones which are in suitable senses "optimal" or "canonical," and which are thus amenable to classification. But because these PDE generically develop singularities, a classification of those singular behaviors is necessary for the successful completion of those programs. Analyzing formation of (finite- or infinite-time) singularities is the unifying goal of this project. A key approach is the use of matched asymptotics, a technique that can provide the most precise description of the set of points on which a solution becomes singular, and of the behavior of the solution in a space-time neighborhood of that singularity. Major objectives of this proposal include: (1) removing symmetry hypotheses in asymptotic singularity analysis for mean curvature flow (MCF) and Ricci flow (RF), thereby proving that certain singularity profiles are "universal" in a rigorous sense; (2) constructing and analyzing (non-generic) Type-II RF singularities, which form more slowly than the natural parabolic rate and thus feature faster curvature blow-up; (3) constructing codimension-2 RF singularities and studying their asymptotics, genericity, and stability; (4) constructing and studying new examples of RF local singularity formation for complex surfaces (and complex manifolds of higher dimension) with applications to the classification of singularity models in those dimensions; (5) studying stability (properly understood) of product structures and related curvature conditions preserved by RF in low dimensions; (6) showing that singular profiles of geometric PDE depend continuously on their initial data, with applications to topology; and (7) studying formation and stability of infinite-time RF singularity models under distinct convergence schemes designed to provide asymptotics at temporal infinity, along with other geometric information.The theory of geometric partial differential equations (PDE) has surprising similarities with nonlinear hyperbolic and dispersive equations. Furthermore, the PDE that arise in curvature flows are remarkably similar to equations that model heat propagation, the movement of oil in shale and thin films, combustion in porous media, and certain effects in plasma physics. In all of these applications, the underlying models are fundamentally nonlinear, a property which causes the associated PDE to develop various critical or singular behaviors. The utility of these models requires a precise mathematical understanding of these behaviors. This project will further develop mathematical techniques, particularly matched asymptotic expansions, that should help the analysis of singularity formation in these varied systems and applications.

该项目与非线性抛物线偏微分方程（PDE）有关，以及通过诸如曲率等几何自然量演变而成的几何对象所满足的系统。这些PDE用于在程序中进化给给定的几何形状，以适当的感觉“最佳”或“规范”，因此可以对分类进行分类。但是，由于这些PDE通常会发展出奇异性，因此对这些程序的成功完成是必要的。分析（有限或无限）奇点的形成是该项目的统一目标。一种关键方法是使用匹配的渐近学，该技术可以提供最精确的描述，以对解决方案变为奇异点，以及在该奇异性的时空社区中解决方案的行为。该提案的主要目标包括：（1）在平均曲率流（MCF）和RICCI流（RF）的渐近奇异性分析中删除对称性假设，从而证明某些奇异性曲线在严格的意义上是“通用的”； （2）构建和分析（非生成）II型RF奇点，其形成比自然抛物线速率慢，从而具有更快的曲率爆炸； （3）构建Codimension-2 RF奇点并研究其渐近学，通用性和稳定性； （4）构建和研究RF局部奇异性形成的新示例，以在这些维度中的奇异模型分类中应用复杂的表面（以及更高维度的复杂歧管）； （5）研究RF在低维度保留的产品结构和相关曲率条件的稳定性（正确理解）； （6）表明几何PDE的奇异曲线不断取决于其初始数据，并应用于拓扑； （7）研究无限时间RF奇异模型在不同的收敛方案下的形成和稳定性，旨在在时间无穷大的情况下提供渐近性的渐近性，以及其他几何信息。几何部分偏微分方程（PDE）的理论令人惊讶地与非线性超高和分散方程式相似。此外，曲率流中产生的PDE与模拟热传播的方程式，页岩和薄膜中的油的运动，多孔介质中的燃烧以及血浆物理学的某些影响非常相似。在所有这些应用中，基本模型从根本上是非线性的，该属性会导致相关PDE发展各种关键或奇异行为。 这些模型的实用性需要对这些行为有精确的数学理解。该项目将进一步开发数学技术，尤其是相匹配的渐近扩展，这将有助于分析这些各种系统和应用中的奇异性形成。