Profiling singularities of geometric PDE

该项目涉及非线性抛物型偏微分方程（PDE）和由几何自然量（如曲率）演变的几何对象所满足的系统。这些PDE在程序中用于将给定的几何形状演变为适当意义上的“最优”或“规范”，因此可以进行分类。但是由于这些PDE通常会产生奇异性，因此对这些奇异行为进行分类对于成功完成这些程序是必要的。分析（有限时间或无限时间）奇点的形成是这个项目的统一目标。一个关键的方法是使用匹配渐近，这是一种技术，可以提供最精确的描述，解在其上成为奇异点，以及解在该奇异点的时空邻域中的行为。本文的主要目标包括：(1)消除平均曲率流（MCF）和Ricci流（RF）渐近奇点分析中的对称性假设，从而证明某些奇点剖面在严格意义上是“普遍的”；(2)构造和分析（非一般）ii型RF奇点，其形成速度比自然抛物线速率慢，因此具有更快的曲率爆炸；(3)构造协维-2 RF奇点并研究其渐近性、泛型和稳定性；(4)构建和研究复杂曲面（及高维复杂流形）射频局部奇点形成的新实例，并将其应用于这些维度奇点模型的分类；(5)研究低维RF保存的产品结构稳定性（适当理解）和相关曲率条件；(6)证明几何偏微分方程的奇异轮廓连续依赖于它们的初始数据，并将其应用于拓扑学；(7)研究在不同收敛方案下的无限时RF奇点模型的形成和稳定性，这些方案旨在提供时间无穷远处的渐近性以及其他几何信息。几何偏微分方程的理论与非线性双曲方程和色散方程有着惊人的相似之处。此外，曲率流中产生的偏微分方程与模拟热传播、页岩和薄膜中油的运动、多孔介质中的燃烧以及等离子体物理中的某些效应的方程非常相似。在所有这些应用程序中，底层模型基本上是非线性的，这一特性导致相关的PDE产生各种关键或奇异的行为。这些模型的效用要求对这些行为有精确的数学理解。该项目将进一步发展数学技术，特别是匹配渐近展开式，这将有助于分析这些不同系统和应用中的奇点形成。

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.